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ISSUE 2005
PROGRESS IN PHYSICS
VOLUME 2
ISSN 1555-5534
The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics
PROGRESS IN PHYSICS
A quarterly issue scientific journal, registered with the Library of Congress (DC). This journal is peer reviewed and included in the abstracting and indexing coverage of: Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sweden), Referativnyi Zhurnal VINITI (Russia), etc.
ISSN: 1555-5534 (print) ISSN: 1555-5615 (online) 4 issues per annum
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JULY 2005
CONTENTS
VOLUME 2
Chief Editor Dmitri Rabounski rabounski@yahoo.com
Associate Editors Prof. Florentin Smarandache smarand@unm.edu Dr. Larissa Borissova lborissova@yahoo.com Stephen J. Crothers thenarmis@yahoo.com
Department of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA
Copyright c Progress in Physics, 2005
All rights reserved. Any part of Progress in Physics howsoever used in other publications must include an appropriate citation of this journal. Authors of articles published in Progress in Physics retain their rights to use their own articles in any other publications and in any way they see fit.
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S. J. Crothers On the Geometry of the General Solution for the Vacuum Field of the Point-Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
D. Rabounski A Theory of Gravity Like Electrodynamics. . . . . . . . . . . . . . . . . . . 15
L. Borissova Gravitational Waves and Gravitational Inertial Waves in the General Theory of Relativity: A Theory and Experiments. . . . . . . . . . . . . . . . . . . 30
C. Castro On Geometric Probability, Holography, Shilov Boundaries and the Four Physical Coupling Constants of Nature. . . . . . . . . . . . . . . . . . . . . . . . 63
S. J. Crothers On the Generalisation of Kepler’s 3rd Law for the Vacuum Field of the Point-Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
S. J. Crothers On the Vacuum Field of a Sphere of Incompressible Fluid. . . . . . 76 E. Harokopos Power as the Cause of Motion and a New Foundation of Clas-
sical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
S. Capozziello, S. De Martino, S. Tzenov Hydrodynamic Covariant Symplectic Structure from Bilinear Hamiltonian Functions. . . . . . . . . . . . . . . . . . . . 92
D. Rabounski, L. Borissova, F. Smarandache Quantum Causality Threshold in the General Theory of Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
R. T. Cahill, C. M. Klinger Bootstrap Universe from Self-Referential Noise. . 108
F. Smarandache Verifying Unmatter by Experiments, More Types of Unmatter, and a Quantum Chromodynamics Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A variety of books can be downloaded free from the E-library of Science: http://www.gallup.unm.edu/∼smarandache
ISSN: 1555-5534 (print) ISSN: 1555-5615 (online)
Standard Address Number: 297-5092 Printed in the United States of America
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July, 2005
PROGRESS IN PHYSICS
Volume 2
On the Geometry of the General Solution for the Vacuum Field of the Point-Mass
Stephen J. Crothers Sydney, Australia
E-mail: thenarmis@yahoo.com
The black hole, which arises solely from an incorrect analysis of the Hilbert solution, is based upon a misunderstanding of the significance of the coordinate radius r. This quantity is neither a coordinate nor a radius in the gravitational field and cannot of itself be used directly to determine features of the field from its metric. The appropriate quantities on the metric for the gravitational field are the proper radius and the curvature radius, both of which are functions of r. The variable r is actually a Euclidean parameter which is mapped to non-Euclidean quantities describing the gravitational field, namely, the proper radius and the curvature radius.
1 Introduction
The variable r has given rise to much confusion. In the conventional analysis, based upon the Hilbert metric, which is almost invariably and incorrectly called the “Schwarzschild” solution, r is taken both as a coordinate and a radius in the spacetime manifold of the point-mass. In my previous papers [1, 2] on the general solution for the vacuum field, I proved that r is neither a radius nor a coordinate in the gravitational field (Mg, gg), as Stavroulakis [3, 4, 5] has also noted. In the context of (Mg, gg) r is a Euclidean parameter in the flat spacetime manifold (Ms, gs) of Special Relativity. Insofar as the point-mass is concerned, r specifies positions on the real number line, the radial line in (Ms, gs), not in the spacetime manifold of the gravitational field, (Mg, gg). The gravitational field gives rise to a mapping of the distance D = r − r0 between two points r, r0 ∈ into (Mg, gg). Thus, r becomes a parameter for the spacetime manifold associated with the gravitational field. If Rp ∈ (Mg, gg) is the proper radius, then the gravitational field gives rise to a mapping ψ,
ψ : r − r0 ∈ ( − −) → Rp ∈ (Mg, gg) , (A)
where 0 Rp < ∞ in the gravitational field, on account of Rp being a distance from the point-mass located at the point Rp(r0) ≡ 0.
The mapping ψ must be obtained from the geometrical properties of the metric tensor of the solution to the vacuum field. The r-parameter location of the point-mass does not have to be at r0 = 0. The point-mass can be located at any point r0 ∈ . A test particle can be located at any point r ∈ . The point-mass and the test particle are located at the end points of an interval along the real line through r0 and r. The distance between these points is D = r − r0 . In (Ms, gs), r0 and r may be thought of as describing 2spheres about an origin rc = 0, but only the distance between
these 2-spheres enters into consideration. Therefore, if two test particles are located, one at any point on the 2-sphere r0 = 0 and one at a point on the 2-sphere r = r0 on the radial line through r0 and r, the distance between them is the length of the radial interval between the 2-spheres, D = r − r0 . Consequently, the domain of both r0 and r is the real number line. In this sense, (Ms, gs) may be thought of as a parameter space for (Mg, gg), because ψ maps the Euclidean distance D = r − r0 ∈ (Ms, gs) into the nonEuclidean proper distance Rp ∈ (Mg, gg): the radial line in (Ms, gs) is precisely the real number line. Therefore, the required mapping is appropriately written as,
ψ : r − r0 ∈ (Ms, gs) → Rp ∈ (Mg, gg) . (B)
In the pseudo-Euclidean (Ms, gs) the polar coordinates are r, θ, ϕ, but in the pseudo-Riemannian manifold (Mg, gg) of the point-mass and point-charge, r is not the radial coordinate. Conventionally there is the persistent misconception that what are polar coordinates in Minkowski space must also be polar coordinates in Einstein space. This however, does not follow in any rigorous way. In (Mg, gg) the variable r is nothing more than a real-valued parameter, of no physical significance, for the true radial quantities in (Mg, gg). The parameter r never enters into (Mg, gg) directly. Only in Minkowski space does r have a direct physical meaning, as mapping (B) indicates, where it is a radial coordinate. Henceforth, when I refer to the radial coordinate or r-parameter I always mean r ∈ (Ms, gs).
The solution for the gravitational field of the simple configurations of matter and charge requires the determination of the mapping ψ. The orthodox analysis has completely failed to understand this and has consequently failed to solve the problem.
The conventional analysis simply looks at the Hilbert metric and makes the following unjustified assumptions, tacitly or otherwise;
S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass
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(a) The variable r is a radius and/or coordinate of some kind in the gravitational field.
(b) The regions 0 < r < 2m and 2m < r < ∞ are both valid.
(c) A singularity in the gravitational field must occur only where the Riemann tensor scalar curvature invariant (Kretschmann scalar) f = RαβγδRαβγδ is unbounded.
The orthodox analysis has never proved these assumptions, but nonetheless simply takes them as given, finds for itself a curvature singularity at r = 0 in terms of f , and with legerdemain reaches it by means of an ad hoc extension in the ludicrous Kruskal-Szekeres formulation. However, the standard assumptions are incorrect, which I shall demonstrate with the required mathematical rigour.
Contrary to the usual practise, one cannot talk about extensions into the region 0 < r < 2m or division into R and T regions until it has been rigorously established that the said regions are valid to begin with. One cannot treat the r-parameter as a radius or coordinate of any sort in the gravitational field without first demonstrating that it is such. Similarly, one cannot claim that the scalar curvature must be unbounded at a singularity in the gravitational field until it has been demonstrated that this is truly required by Einstein’s theory. Mere assumption is not permissible.
2 The basic geometry of the simple point-mass
The usual metric gs of the spacetime manifold (Ms, gs) of Special Relativity is,
ds2 = dt2 − dr2 − r2 dθ2 + sin2 θdϕ2 . (1)
The foregoing metric can be statically generalised for the simple (i. e. non-rotating) point-mass as follows,
ds2 = A(r)dt2−B(r)dr2 −C(r) dθ2 + sin2 θdϕ2 , (2a)
A, B, C > 0 ,
where A, B, C are analytic functions. I emphatically remark that the geometric relations between the components of the metric tensor of (2a) are precisely the same as those of (1).
The standard analysis writes (2a) as,
ds2 = A(r)dt2 − B(r)dr2 − r2 dθ2 + sin2 θdϕ2 , (2b)
and claims it the most general, which is incorrect. The form of C(r) cannot be pre-empted, and must in fact be rigorously determined from the general solution to (2a). The physical features of (Mg, gg) must be determined exclusively by means of the resulting gμμ ∈ (Mg, gg), not by foisting upon (Mg, gg) the interpretation of elements of (Ms, gs) in the misguided fashion of the orthodox relativists who, having written (2b), incorrectly treat r in (Mg, gg) precisely as the r in (Ms, gs).
With respect to (2a) I identify the coordinate radius, the r-parameter, the radius of curvature, and the proper radius as follows:
(a) The coordinate radius is D = |r − r0|. (b) The r-parameter is the variable r.
(c) The radius of curvature is Rc = C(r).
(d) The proper radius is Rp = B(r)dr.
The orthodox motivation to equation (2b) is to evidently obtain the circumference χ of a great circle, χ ∈ (Mg, gg) as,
χ = 2πr ,
to satisfy its unproven assumptions about r. But this equation is only formally the same as the equation of a circle in the Euclidean plane, because in (Mg, gg) it describes a nonEuclidean great circle and therefore does not have the same meaning as the equation for the ordinary circle in the Euclidean plane. The orthodox assumptions distort the fact that r is only a real parameter in the gravitational field and therefore that (2b) is not a general, but a particular expression, in which case the form of C(r) has been fixed to C(r) = r2. Thus, the solution to (2b) can only produce a particular solution, not a general solution in terms of C(r), for the gravitational field. Coupled with its invalid assumptions, the orthodox relativists obtain the Hilbert solution, a correct particular form for the metric tensor of the gravitational field, but interpret it incorrectly with such a great thoroughness that it defies rational belief.
Obviously, the spatial component of (1) describes a sphere of radius r, centred at the point r0 = 0. On this metric r r0 is usually assumed. Now in (1) the distance D between two points on a radial line is given by,
D = |r2 − r1| = r2 − r1 .
(3)
Furthermore, owing to the “origin” being usually fixed at r1 = r0 = 0, there is no distinction between D and r. Hence r is both a coordinate and a radius (distance). However, the correct description of points by the spatial part of (1) must still be given in terms of distance. Any point in any direction is specified by its distance from the “origin”. It is this distance which is the important quantity, not the coordinate. It is simply the case that on (1), in the usual sense, the distance and the coordinate are identical. Nonetheless, the distance from the designated “origin” is still the important quantity, not the coordinate. It is therefore clear that the designation of an origin is arbitrary and one can select any r0 ∈ as the origin of coordinates. Thus, (1) is a special case of a general expression in which the origin of coordinates is arbitrary and the distance from the origin to another point does not take the same value as the coordinate designating it. The “origin” r0 = 0 has no intrinsic meaning. The relativists and the mathematicians have evidently failed to understand
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this elementary geometrical fact. Consequently, they have managed to attribute to r0 = 0 miraculous qualities of which it is not worthy, one of which is the formation of the black hole.
Equations (1) and (2a) are not sufficiently general and so their forms suppress their true geometrical characteristics. Consider two points P1 and P2 on a radial line in Euclidean 3-space. With the usual Cartesian coordinates let P1 and P2 have coordinates (x1, y1, z1) and (x2, y2, z2) respectively. The distance between these points is,
D = (x1 − x2)2 + (y1 − y2)2 + (z1 − z2)2 = (4)
= |x1 − x2|2 + |y1 − y2|2 + |z1 − z2|2 0 .
If x1 = y1 = z1 = 0, D is usually called a radius and so written D ≡ r. However, one may take P1 or P2 as an origin for a sphere of radius D as given in (4). Clearly, a general description of 3-space must rightly take this feature into account. Therefore, the most general line-element for the gravitational field in quasi-Cartesian coordinates is,
ds2 = F dt2 − G dx2 + dy2 + dz2 −
2
(5)
− H |x − x0|dx + |y − y0|dy + |z − z0 |dz ,
where F, G, H > 0 are functions of
D = |x − x0|2 + |y − y0|2 + |z − z0 |2 = |r − r0| ,
The spatial part of (7) describes a sphere of radius D =
= r − r0 , centred at the arbitrary point r0 and reaching to some point r ∈ . Indeed, the curvature radius Rc of (7) is,
Rc = r − r0 2 = r − r0 ,
(8)
and the circumference χ of a great circle centred at r0 and reaching to r is,
χ = 2π r − r0 .
(9)
The proper radius (distance) Rp from r0 to r on (7) is,
|r − r0 |
r
Rp =
d r−r0 =
0
r0
r − r0 r − r0
dr = r−r0 . (10)
Thus Rp ≡ Rc ≡ D on (7), owing to its pseudoEuclidean nature.
It is evident by similar calculation that r ≡ Rc ≡ Rp in (1). Indeed, (1) is obtained from (7) when r0 = 0 and r r0 (although the absolute value is suppressed in (1) and (7a)). The geometrical relations between the components of the metric tensor are inviolable. Therefore, in the case of (1), the following obtain,
D = |r| = r ,
√ Rc = |r|2 = r2 = r ,
and P0 (x0, y0, z0 ) is an arbitrary origin of coordinates for a sphere of radius D centred on P0 .
Transforming to spherical-polar coordinates, equation (5) becomes,
ds2 = − H|r − r0|2dr2 + F dt2−
− G dr2 + |r − r0|2dθ2 + |r − r0|2 sin2 θdϕ2 = (6)
= A(D)dt2 − B(D)dr2 − C(D) dθ2 + sin2 θdϕ2 ,
where A, B, C > 0 are functions of D = |r − r0|. Equation (6) is just equation (2a), but equation (2a) has suppressed the significance of distance and the arbitrary origin and is therefore invariably taken with D ≡ r 0, r0 = 0.
In view of (6) the most general expression for (1) for a sphere of radius D = |r − r0|, centred at some r0 ∈ , is therefore,
χ = 2π|r| = 2πr , (11)
|r|
r
Rp = d |r| = dr = r .
0
0
However, equation (1) hides the true arbitrary nature of the origin r0. Therefore, the correct geometrical relations have gone unrecognized by the orthodox analysis. I note, for instance, that G. Szekeres [6], in his well-known paper of 1960, considered the line-element,
ds2 = dr2 + r2dω2 ,
(12)
and proposed the transformation r = r − 2m, to allegedly carry (12) into,
ds2 = dr2 + (r − 2m)2 dω2 .
(13)
ds2 = dt2 − dr2 −
2
r − r0
dθ2 + sin2 θdϕ2
=
(7a)
The transformation to (13) by r = r − 2m is incorrect: by it Szekeres should have obtained,
= dt2−
r−r0
2
dr2−
|r−r0 |2
r−r0
2
dθ2+ sin2 θdϕ2
=
(7b)
= dt2 −
2
2
d|r−r0| − r−r0
dθ2 + sin2 θdϕ2
.
(7c)
ds2 = dr2 + (r + 2m)2 dω2 .
(14)
If one sets r = r − 2m, then (13) obtains from (12). Szekeres then claims on (13),
S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass
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“Here we have an apparent singularity on the sphere r = 2m, due to a spreading out of the origin over a sphere of radius 2m. Since the exterior region r > 2m represents the whole of Euclidean space (except the origin), the interior r < 2m is entirely disconnected from it and represents a distinct manifold.”
In the usual way one obtains the solution to (16) as,
ds2 = Rc − α dt2 − Rc
−
Rc Rc − α
dRc2 − Rc2(dθ2 + sin2 θdϕ2) ,
His claims about (13) are completely false. He has made an incorrect assumption about the origin. His equation (12) describes a sphere of radius r centred at r = 0, being identical to the spatial component of (1). His equation (13) is precisely the spatial component of equation (7) with r0 =2m and r r0, and therefore actually describes a sphere of radius D = r − 2m centred at r0 = 2m. His claim that r = 2m describes a sphere is due to his invalid assumption that r = 0 has some intrinsic meaning. It did not come from his transformation. The claim is false. Consequently there is no interior region at all and no distinct manifold anywhere. All Szekeres did unwittingly was to move the origin for a sphere from the coordinate value r0 = 0 to the coordinate value r0 = 2m. In fact, he effectively repeated the same error committed by Hilbert [8] in 1916, an error, which in one guise or another, has been repeated relentlessly by the orthodox theorists.
It is now plain that r is neither a radius nor a coordinate in the metric (6), but instead gives rise to a parameterization of the relevant radii Rc and Rp on (6).
Consider (7) and introduce a test particle at each of the points r0 and r. Let the particle located at r0 acquire mass. The coordinates r0 and r do not change, however in the gravitational field (Mg, gg) the distance between the pointmass and the test particle, and the radius of curvature of a great circle, centred at r0 and reaching to r in the parameter space (Ms, gs), will no longer be given by (11).
The solution of (6) for the vacuum field of a pointmass will yield a mapping of the Euclidean distance D = |r − r0| into a non-Euclidean proper radius RP (r) in the pseudo-Riemannian manifold (Mg, gg), locally generated by the presence of matter at the r-parameter r0 ∈ (Ms, gs), i. e. at the invariant point Rp(r0) ≡ 0 in (Mg, gg).
Transform (6) by setting,
χ
Rc =
C(D(r)) = , 2π
(15)
D = |r − r0| .
Then (6) becomes,
ds2 = A∗(Rc)dt2 − B∗(Rc)dRc2 −
− Rc2 dθ2 + sin2 θdϕ2 .
(16)
α = 2m ,
which by using (15) becomes,
√
ds2 =
C −α √
dt2 −
C
√ C
−√
C2
r− r0
2
dr2 − C(dθ2 + sin2 θdϕ2) ,
C − α 4C |r− r0|
that is,
√
ds2 =
C −α √
dt2 −
C
√
(17)
C −√
C 2 dr2 − C(dθ2 + sin2 θdϕ2) ,
C − α 4C
which is the line-element derived by Abrams [7] by a different method. Alternatively one could set r = Rc in (6), as Hilbert in his work [8] effectively did, to obtain the familiar Droste/Weyl/(Hilbert) line-element,
ds2 = r − α dt2 − r dr2 −
r
r−α
(18)
− r2(dθ2 + sin2 θdϕ2) ,
and then noting, as did J. Droste [9] and A. Eddington [10], that r2 can be replaced by a general analytic function of r without destroying the spherical symmetry of (18). Let that function be C(D(r)), D = |r − r0|, and so equation (17) is again obtained. Equation (18) taken literally is an incomplete particular solution since the boundary on the r-parameter has not yet been rigorously established, but equation (17) provides a way by which the form of C(D(r)) might be determined to obtain a means by which all particular solutions, in terms of an infinite sequence, may be constructed, according to the general prescription of Eddington. Clearly, the correct form of C(D(r)) must naturally yield the Droste/Weyl/(Hilbert) solution, as well as the true Schwarzschild solution [11], and the Brillouin solution [12], amongst the infinitude of particular solutions that the field equations admit. (Fiziev [13] has also shown that there exists an infinite number of solutions for the point-mass and that the Hilbert black hole is not consistent with general relativity.)
In the gravitational field only the circumference χ of a great circle is a measurable quantity, from which Rc and Rp
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S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass
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are calculated. To obtain the metric for the field in terms of χ, use (15) in (17) to yield,
ds2 =
2πα 1−
χ
dt2 −
2πα −1 dχ2
1− χ
4π2 −
where r0 is the parameter space location of the point-mass. Clearly 0 Rp < ∞ always and the point-mass is invariantly located at Rp(r0) ≡ 0 in (Mg, gg), a manifold with boundary.
From (20),
χ2 − 4π2
dθ2 + sin2 θdϕ2
,
(19)
Rp(r0) ≡ 0 = C(r0) C(r0) − α +
α = 2m . Equation (19) is independent of the r-parameter entirely.
+ α ln
C(r0) + C(r0) − α ,
K
Since only χ is a measurable quantity in the gravitational field, (19) constitutes the correct solution for the gravitational field of the simple point-mass. In this way (19) is truely the only solution to Einstein’s field equations for the simple point-mass.
and so,
√ C(r0) ≡ α, K = α.
Therefore (20) becomes
The only assumptions about r that I make are that the
point-mass is to be located somewhere, and that somewhere
is r0 in parameter space (Ms, gs), the value of which must be obtained rigorously from the geometry of equation (17),
and that a test particle is located at some r = r0 in parameter space, where r, r0 ∈ .
The geometrical relationships between the components
Rp(r) = + α ln
C(|r−r0|) C(|r−r0|) − α +
(21)
C(|r−r0|) + C(|r−r0|) − α
√
,
α
of the metric tensor of (1) must be precisely the same in (6), (17), (18), and (19). Therefore, the circumference χ of a
r, r0 ∈ ,
great circle on (17) is given by,
and consequently for (19),
χ = 2π C(D(r)) ,
2πα < χ < ∞ .
and the proper distance (proper radius) Rp(r) on (6) is,
Rp(r) = B(D(r))dr .
Taking B(D(r)) from (17) gives,
Rp(D) =
√
CC
√
√ dr =
C−α 2 C
= C(D) C(D) − α +
(20)
+ α ln
C(D) + C(D) − α ,
K
D = |r − r0| ,
Equation (21) is the required mapping. One can see that r0 cannot be determined: in other words, r0 is entirely arbitrary. One also notes that (17) is consequently singular only when r = r0 in which case g00 = 0, Cn(r0) ≡ α, and Rp(r0) ≡ 0. There is no value of r that makes g11 = 0. One therefore sees that the condition for singularity in the gravitational field is g00 = 0; indeed g00(r0) ≡ 0.
Clearly, contrary to the orthodox claims, r does not determine the geometry of the gravitational field directly. It is not a radius in the gravitational field. The quantity Rp(r) is the non-Euclidean radial coordinate in the pseudoRiemannian manifold of the gravitational field around the point Rp = 0, which corresponds to the parameter point r0.
Now in addition to the established fact that, in the case of the simple (i .e . non-rotating) point-mass, the lower bound on the radius of curvature C(D(r0)) ≡ α, C(D(r)) must also satisfy the no matter condition so that when α = 0, C(D(r)) must reduce to,
K = const. The relationship between r and Rp is,
as r → r0±, Rp(r) → 0+ ,
C(D(r)) ≡ r − r0 2 = (r − r0)2 ;
(22)
and it must also satisfy the far-field condition (spatially asympotically flat),
or equivalently, as D → 0+, Rp(r) → 0+ ,
C (D(r))
lim
r→±∞
r − r0
2 → 1.
(23)
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When r0 = 0 equation (22) reduces to, C(|r|) ≡ r2 ,
According to (24), Cn(D(r0)) ≡ α is a scalar invariant, being independent of the value of r0. Nevertheless the field is singular at the point-mass. By (21),
and equation (23) reduces to,
C (|r|)
lim
r→±∞
r2
→ 1.
and so,
lim
r → ±∞
|r
Rp2 − r0|2
=
1
,
Furthermore, C(r) must be a strictly monotonically increasing function of r to satisfy (15) and (21), and C (r) = 0 ∀ r = r0 to satisfy (17) from (2a). The only general form for C(D(r)) satisfying all the required conditions (the Metric Conditions of Abrams [7]), from which an infinite sequence of particular solutions can be obtained [1] is,
2
Cn(D(r)) =
r − r0 n + αn
n
,
(24)
n ∈ +, r ∈ , r0 ∈ ,
where n and r0 are arbitrary. Then clearly, when α = 0, equations (7) are recovered from equation (17) with (24),
and when r0 = 0 and α = 0, equation (1) is recovered. According to (24), when r0 = 0 and r r0, and n is
taken in integers, the following infinite sequence of particular
solutions obtains,
C1(r) = (r + α)2 (Brillouin’s solution [12])
C2(r) = r2 + α2
C3(r)
=
(r3
+
α3)
2 3
(Schwarzschild’s solution [11])
C4(r)
=
(r4
+
α4)
1 2
,
etc.
When r0 = α and r ∈ +, and n is taken in integers, the following infinite sequence of particular solutions is
obtained,
C1(r) = r2 (Droste/Weyl/(Hilbert) [9, 14, 8])
C2(r) = (r − α)2 + α2
C3(r)
=
[(r
−
α)3
+
α3]
2 3
C4(r)
=
[(r
−
α)4
+
α4]
1 2
,
etc.
The Schwarzschild forms obtained from (24) satisfy Eddington’s prescription for a general solution.
By (17) and (24) the circumference χ of a great circle in the gravitational field is,
1
χ = 2π Cn(r) = 2π |r − r0|n + αn n ,
(25)
and the proper radius Rp(r) is, from (21),
1
1
Rp(r) = |r−r0|n +αn n |r−r0|n+αn n −α +
1
1
(26)
+ α ln
|r−r0|n +αn
+ 2n √
|r−r0|n +αn
n −α
.
α
lim
Rp2
= lim
Rp2
|r−r0|2 = 1 .
r → ±∞ Cn(D(r))
r → ±∞ Cn(D(r)) |r−r0 |2
Now
the
ratio
χ Rp
>
2π
for
all
finite
Rp,
and
χ lim = 2π , r→±∞ Rp
χ lim = ∞ , r→r0± Rp
so Rp(r0) ≡ 0 is a quasiregular singularity and cannot be extended. The singularity occurs when parameter r = r0, irrespective of the values of n and r0. Thus, there is no sense in the orthodox notion that the region 0 < r < α is
an interior region on the Hilbert metric, since r0 = 0 on that metric. Indeed, by (21) and (24) r0 = α on the Hilbert metric. Equation (26) amplifies the fact that it is the distance
D = |r − r0| that is mapped from parameter space into the proper radius (distance) in the gravitational field, and a dis-
tance must be 0.
Consequently, strictly speaking, r0 is not a singular point in the gravitational field because r is merely a parameter for
the radial quantities in (Mg, gg); r is neither a radius nor a coordinate in the gravitational field. No value of r can
really be a singular point in the gravitational field. However,
r0 is mapped invariantly to Rp = 0, so r = r0 always gives rise to a quasiregular singularity in the gravitational field, at
Rp(r0) ≡ 0, reflecting the fact that r0 is the boundary on the r-parameter. Only in this sense should r0 be considered a singular point.
The Kretschmann scalar f = RαβγδRαβγδ for equation (17) with equation (24) is,
12α2
12α2
f = [Cn(D(r))]3 =
6. |r − r0|n + αn n
(27)
Taking the near-field limit on (27),
12
lim
r → r0±
f
=
α4
,
so
f (r0) ≡
12 α4
is
a scalar
invariant,
irrespective
of
the values
of n and r0, invalidating the orthodox assumption that the
singularity must occur where the curvature is unbounded.
Indeed, no curvature singularity can arise in the gravitational
8
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field. The orthodox analysis claims an unbounded curvature singularity at r0 = 0 in (18) purely and simply by its invalid initial assumptions, not by mathematical imperative. It incorrectly assumes Cn(r) ≡ Rp(r) ≡ r, then with its additional invalid assumption that 0 < r < α is valid on the Hilbert metric, finds from (27),
lim f (r) = ∞ ,
r → 0+
thereby satisfying its third invalid assumption, by ad hoc construction, that a singularity occurs only where the curvature invariant is unbounded.
The Kruskal-Szekeres form has no meaning since the rparameter is not the radial coordinate in the gravitational field at all. Furthermore, the value of r0 being entirely arbitrary, r0 = 0 has no particular significance, in contrast to the mainstream claims on (18).
The value of the r-parameter of a certain spacetime event depends upon the coordinate system chosen. However, the proper radius Rp(D(r)) and the curvature radius Cn(D(r)) of that event are independent of the coordinate system. This is easily seen as follows. Consider a great circle centred at the point-mass and passing through a spacetime event. Its circumference is measured at χ. Dividing χ by 2π gives,
χ
= 2π
Cn(D(r)) .
Putting
χ 2π
=
Cn(D(r)) into (21) gives the proper ra-
dius of the spacetime event,
χχ
Rp(r) =
− α + α ln 2π 2π
χ 2π
+
χ 2π
−α
√
,
α
2πα χ < ∞,
which is independent of the coordinate system chosen. To find the r-parameter in terms of a particular coordinate system set,
χ =
2π
Cn(D(r)) =
1
r − r0 n + αn
n
,
so
|r − r0| =
χ
n
− αn
1
n
.
2π
Thus r for any particular spacetime event depends upon the arbitrary values n and r0, which establish a coordinate system. Then when r = r0, Rp = 0, and the great circumference χ = 2πα, irrespective of the values of n and r0. A truly coordinate independent description of spacetime events
has been attained.
The mainstream insistence, on the Hilbert solution (18),
without proof, that the r-parameter is a radius of sorts in the gravitational field, the insistence that its r can, without
proof, go down to zero, and the insistence, without proof, that a singularity in the field must occur only where the curvature is unbounded, have produced the irrational notion of the black hole. The fact is, the radius always does go down to zero in the gravitational field, but that radius is the proper radius Rp (Rp = 0 corresponding to a coordinate radius D = 0), not the curvature radius Rc, and certainly not the r-parameter.
There is no escaping the fact that r0 = α = 0 in (18). Indeed, if α = 0, (18) must give,
ds2 = dt2 − dr2 − r2 dθ2 + sin2 θdϕ2 ,
the metric of Special Relativity when r0 = 0. One cannot set the lower bound r0 = α = 0 in (18) and simultaneously keep α = 0 in the components of the metric tensor, which is
effectively what the orthodox analysis has done to obtain the
black hole. The result is unmitigated nonsense. The correct
form of the metric (18) is obtained from the associated Schwarzschild form (24): C(r) = r2, r0 = α. Furthermore, the proper radius of (18) is,
r
Rp(r) =
α
r dr ,
r−α
and so Rp(r) =
√√
r+ r−α
r (r − α) + α ln
√
.
α
Then,
r → α+ ⇒ D = |r − α| = (r − α) → 0 ,
and in (Mg, gg), r2 ≡ C(r) → C(α) = α2 ⇒ Rp(r) → Rp(α) = 0 .
Thus, the r-parameter is mapped to the radius of cur-
vature
C (r)
=
χ 2π
by
ψ1,
and
the
radius
of
curvature
is
mapped to the proper radius Rp by ψ2. With the mappings
established the r-parameter can be mapped directly to Rp
by ψ(r) = ψ2 ◦ ψ1(r). In the case of the simple point-mass
the mapping ψ1 is just equation (24), and the mapping ψ2 is
given by (21).
The local acceleration of a test particle approaching the
point-mass along a radial geodesic has been determined by
N. Doughty [15] at,
a
=
√ −grr
(−g rr )
|gtt,r |
.
2gtt
(28)
For (17) the acceleration is,
a=
3
2Cn4
α
1.
1
2
Cn2 − α
S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass
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Then,
lim a = ∞,
r → r0±
since Cn(r0) ≡ α2; thereby confirming that matter is indeed present at the point Rp(r0) ≡ 0.
In the case of (18), where r ∈ +,
α
a= 3
1,
2r 2 (r − α) 2
“. . . it consists of two disjoint regions, 0 < r < 2m, and r > 2m, separated by the singular hypercylinder r = 2m.”
which again betrays the same unproven assumptions. I now draw attention to the following additional problems
with the Kruskal-Szekeres form.
(a) Applying Doughty’s acceleration formula (28) to the Kruskal-Szekeres form, it is easily found that,
and r0 = α by (24), so,
lim a = ∞ .
r → α+
Y. Hagihara [16] has shown that all those geodesics which do not run into the boundary at r = α on (18) are complete. Now (18) with α < r < ∞ is a particular solution by (24), and r0 = α is an arbitrary point at which the point-mass is located in parameter space, therefore all those geodesics in (Mg, gg) not running into the point Rp(r0) ≡ 0 are complete, irrespective of the value of r0.
Modern relativists do not interpret the Hilbert solution over 0 < r < ∞ as Hilbert did, instead making an arbitrary distinction between 0 < r < α and α < r < ∞. The modern relativist maintains that one is entitled to just “choose” a region. However, as I have shown, this claim is inadmissible. J. L. Synge [17] made the same unjustified assumptions on the Hilbert line-element. He remarks,
“This line-element is usually regarded as having a singularity at r = α, and appears to be valid only for r > α. This limitation is not commonly regarded as serious, and certainly is not so if the general theory of relativity is thought of solely as a macroscopic theory to be applied to astronomical problems, for then the singularity r = α is buried inside the body, i. e. outside the domain of the field equations Rmn = 0. But if we accord to these equations an importance comparable to that which we attach to Laplace’s equation, we can hardly remain satisfied by an appeal to the known sizes of astronomical bodies. We have a right to ask whether the general theory of relativity actually denies the existence of a gravitating particle, or whether the form (1.1) may not in fact lead to the field of a particle in spite of the apparent singularity at r = α.”
M. Kruskal [18] remarks on his proposed extension of the Hilbert solution into 0 < r < 2m,
“That this extension is possible was already indicated by the fact that the curvature invariants of the Schwarzschild metric are perfectly finite and well behaved at r = 2m∗.”
which betrays the very same unproven assumptions. G. Szekeres [6] says of the Hilbert line-element,
lim a = ∞.
r → 2m−
But according to Kruskal-Szekeres there is no matter at r = 2m. Contra-hype.
(b) As r → 0, u2 −v2 → −1. These loci are spacelike, and therefore cannot describe any configuration of matter or energy.
Both of these anomalies have also been noted by Abrams in his work [7]. Either of these features alone proves the Kruskal-Szekeres form inadmissible.
The correct geometrical analysis excludes the interior Hilbert region on the grounds that it is not a region at all, and invalidates the assumption that the r-parameter is some kind of radius and/or coordinate in the gravitational field. Consequently, the Kruskal-Szekeres formulation is meaningless, both physically and mathematically. In addition, the so-called “Schwarzschild radius” (not due to Schwarzschild) is also a meaningless concept - it is not a radius in the gravitational field. Hilbert’s r = 2m is indeed a point, i. e. the “Schwarzschild radius” is a point, in both parameter space and the gravitational field: by (21), Rp(2m) = 0.
The form of the Hilbert line-element is given by Karl Schwarzschild in his 1916 paper, where it occurs there in the equation he numbers (14), in terms of his “auxiliary parameter” R. However Schwarzschild also includes there
1
the equation R = r3 + α3 3 , having previously established the range 0 < r < ∞. Consequently, Schwarzschild’s auxiliary parameter R (which is actually a curvature radius) has the lower bound R0 = α = 2m. Schwarzschild’s R2 and Hilbert’s r2 can be replaced with any appropriate analytic function Cn(r) as given by (24), so the range and the boundary on r will depend upon the function chosen. In the case of Schwarzschild’s particular solution the range is 0 < r < ∞
2
(since r0 = 0, C3(r) = r3 + α3 3 ) and in Hilbert’s particular solution the range is 2m < r < ∞ (since r0 = 2m, C1(r) = r2).
The geometry and the invariants are the important properties, but the conventional analysis has shockingly erred in its geometrical analysis and identification of the invariants, as a direct consequence of its initial invalidated assumptions about the r-parameter, and clings irrationally to these assumptions to preserve the now sacrosanct, but nonetheless ridiculous, black hole.
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The only reason that the Hilbert solution conventionally breaks down at r = α is because of the initial arbitrary and incorrect assumptions made about the parameter r. There is no pathology of coordinates at r = α. If there is anything pathological about the Hilbert metric it has nothing to do with coordinates: the etiology of a pathology must therefore be found elewhere.
There is no doubt that the Kruskal-Szekeres form is a solution of the Einstein vacuum field equations, however that does not guarantee that it is a solution to the problem. There exists an infinite number of solutions to the vacuum field equations which do not yield a solution for the gravitational field of the point-mass. Satisfaction of the field equations is a necessary but insufficient condition for a potential solution to the problem. It is evident that the conventional conditions (see [19]) that must be met are inadequate, viz.,
1. be analytic;
2. be Lorentz signature;
3. be a solution to Einstein’s free-space field equations;
geometry resulting from Einstein’s gravitational tensor. The variations of θ and ϕ displace the proper radius vector, Rp(r0) ≡ 0, over the spherical surface of finite area 4πα2, as noted by Brillouin. Einstein’s theory admits nothing more pointlike.
Objections to Einstein’s formulation of the gravitational tensor were raised as long ago as 1917, by T. Levi-Civita [20], on the grounds that, from the mathematical standpoint, it lacks the invariant character actually required of General Relativity, and further, produces an unacceptable consequence concerning gravitational waves (i.e they carry neither energy nor momentum), a solution for which Einstein vaguely appealed ad hoc to quantum theory, a last resort obviated by Levi-Civita’s reformulation of the gravitational tensor (which extinguishes the gravitational wave), of which the conventional analysis is evidently completely ignorant: but it is not pertinent to the issue of whether or not the black hole is consistent with the theory as it currently stands on Einstein’s gravitational tensor.
4. be invariant under time translations;
3 The geometry of the simple point-charge
5. be invariant under spatial rotations; 6. be (spatially) asymptotically flat; 7. be inextendible to a wordline L; 8. be invariant under spatial reflections;
The fundamental geometry developed in section 2 is the same for all the configurations of the point-mass and the pointcharge. The general solution for the simple point-charge [2] is,
9. be invariant under time reflection;
10. have a global time coordinate.
This list must be augmented by a boundary condition at the location of the point-mass, which is, in my formulation of the solution, r → r0± ⇒ Rp(r) → 0. Schwarzschild actually applied a form of this boundary condition in his analysis. Marcel Brillouin [12] also pointed out the necessity of such a boundary condition in 1923, as did Abrams [7] in more recent years, who stated it equivalently as, r → r0 ⇒ C(r) → α2. The condition has been disregarded or gone unrecognised by the mainstream authorities. Oddly, the orthodox analysis violates its own stipulated condition for a global time coordinate, but quietly disregards this inconsistency as well.
Any constants appearing in a valid solution must appear in an invariant derived from the solution. The solution I obtain meets this condition in the invariance, at r = r0, of the circumference of a great circle, of Kepler’s 3rd Law [1, 2], of the Kretschmann scalar, of the radius of curvature C(r0) = α2, of Rp(r0) ≡ 0, and not only in the case of the point-mass, but also in all the relevant configurations, with or without charge.
The fact that the circumference of a great circle approaches the finite value 2πα is no more odd than the conventional oddity of the change in the arrow of time in the “interior” Hilbert region. Indeed, the latter is an even more violent oddity: inconsistent with Einstein’s theory. The finite limit of the said circumference is consistent with the
ds2 =
α 1− √
q2 +
dt2 −
α 1− √
q2 −1
+
×
Cn Cn
Cn Cn
(29)
×
Cn2 dr2 4Cn
−
Cn(dθ2
+
sin2
θdϕ2) ,
2
Cn(r) =
r − r0 n + βn
n
,
β = m + m2 − q2 , q2 < m2 ,
n ∈ +, r, r0 ∈ .
where n and r0 are arbitrary. From (29), the radius of curvature is given by,
1
Rc =
Cn(r) =
r − r0 n + βn
n
,
which gives for the near-field limit,
lim
r → r0±
Cn(r) =
Cn(r0) = β = m +
The expression for the proper radius is,
m2 − q2 .
Rp(r) = C(r) − α C(r) + q2 +
+ m ln
C(r) − m + C(r) − α C(r) + q2 .
m2 − q2
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Then
lim
r → r0±
Rp(r)
=
Rp (r0 )
≡
0
.
The
ratio
χ Rp
> 2π
for
all
finite
Rp,
and
χ
lim
= 2π ,
r → ±∞ Rp(r)
χ
lim
=∞,
r → r0± Rp(r)
so Rp(r0) ≡ 0 is a quasiregular singularity and cannot be extended.
Now, since the circumference χ of a great circle is the only measurable quantity in the gravitational field, the unique solution for the field of the simple point-charge is,
Equation (29) is singular only when g00 = 0; indeed g00(r0) ≡ 0. Hence, 0 g00 1.
Applying Doughty’s acceleration formula (28) to
equation (29) gives,
m Cn(r) − q2
a=
.
Cn(r) Cn(r) − α Cn(r) + q2
Then,
mβ − q2
lim a =
=∞,
r → r0±
β2 β2 − αβ + q2
confirming that matter is indeed present at Rp(r0) ≡ 0.
ds2 =
2πα 4π2q2 1 − χ + χ2
dt2 −
2πα 4π2q2 −1 dχ2
− 1 − χ + χ2
4π2 −
(30)
−
χ2 4π2
(dθ2
+
sin2
θdϕ2)
,
2π m + m2 − q2 < χ < ∞ .
Equation (30) is entirely independent of the rparameter.
In terms of equation (29), the Kretschmann scalar takes the form [21],
2
8 6 m Cn(r) − q2 + q4
f (r) =
Cn4 (r)
, (31)
so
8 6 mβ − q2 2 + q4
r
lim
→ r0±
f
(r)
=
f
(r0
)
=
β8
2
8 6 m2 + m m2 − q2 − q2 + q4
=
,
(m + m2 − q2)8
which is a scalar invariant. Thus, no curvature singularity can arise in the gravitational field of the simple point-charge.
The standard analysis incorrectly takes Cn(r) ≡ Rp(r) ≡ r, then with this assumption, and the additional invalid assumption that 0 < r < ∞ is true on the Reissner-Nordstrom solution, obtains from equation (31) a curvature singularity at r = 0, satisfying, by an ad hoc construction, its third invalid assumption that a singularity can only arise at a point where the curvature invariant is unbounded.
4 The geometry of the rotating point-charge
The usual expression for the Kerr-Newman solution is, in Boyer-Lindquist coordinates,
ds2
=
Δ ρ2
dt − a sin2 θdϕ
2
−
(32)
sin2 θ − ρ2
r2 + a2 dϕ − adt 2 − ρ2 dr2 − ρ2dθ2, Δ
L a= ,
ρ2 = r2 + a2 cos2 θ ,
m
Δ = r2 − rα + a2 + q2, 0 < r < ∞ .
This metric is alleged to have an event horizon rh and a static limit rb, obtained by setting Δ = 0 and g00 = 0 respectively, to yield,
rh = m ± m2 − a2 − q2
rb = m ± m2 − q2 − a2 cos2 θ .
These expressions are conventionally quite arbitrarily taken to be,
rh = m + m2 − a2 − q2
rb = m + m2 − q2 − a2 cos2 θ ,
apparently because no-one has been able to explain away the meaning of the the “inner” horizon and the “inner” static limit. This in itself is rather disquieting, but nonetheless accepted with furtive whispers by the orthodox theorists. It is conventionally alleged that the “region” between rh and rb is an ergosphere, in which spacetime is dragged in the direction of the of rotation of the point-charge.
The conventional taking of the r-parameter for a radius in the gravitational field is manifest. However, as I have shown, the r-parameter is neither a coordinate nor a radius
12
S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass
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in the gravitational field. Consequently, the standard analysis is erroneous.
I have already derived elsewhere [2] the general solution for the rotating point-charge, which I write in most general form as,
ds2
=
Δ ρ2
dt − a sin2 θdϕ
2
−
sin2 θ − ρ2
Cn + a2
dϕ − adt 2 − ρ2 Cn2 dr2 − ρ2dθ2 , Δ 4Cn
2
Cn(r) =
r − r0 n + βn
n
,
n∈ +,
(33)
r, r0 ∈ , β = m + m2 − q2 − a2 cos2 θ ,
a2 + q2 < m2,
L a= ,
m
ρ2 = Cn + a2 cos2 θ ,
Δ = Cn − α Cn + q2 + a2 ,
where n and r0 are arbitrary. Once again, since only the circumference of a great circle
is a measurable quantity in the gravitational field, the unique general solution for all configurations of the point-mass is,
where the proper radius Rp(r0) ≡ 0. The standard analysis incorrectly takes (36) for the “radius” of its static limit.
It is evident from (35) and (36) that the radius of curvature depends upon the direction of radial approach. Therefore, the spacetime is not isotropic. Only when a = 0 is spacetime isotropic. The point-charge is always located at Rp(r0) ≡ 0 in (Mg, gg), irrespective of the value of n, and irrespective of the value of r0. The conventional analysis has failed to realise that its rb is actually a varying radius of curvature, and so incorrectly takes it as a measurable radius in the gravitational field. It has also failed to realise that the location of the point-mass in the gravitational field is not uniquely specified by the r-coordinate at all. The point-mass is always located just where Rp = 0 in (Mg, gg) and its “position” in (Mg, gg) is otherwise meaningless. The test particle has already encountered the source of the gravitational field when the radius of curvature has the value Cn(r0) = β. The so-called ergosphere also arises from the aforesaid misconceptions.
When θ = 0 the limiting radius of curvature is,
Cn(r0) = β = m + m2 − q2 − a2 ,
(37)
ds2
=
Δ ρ2
sin2 θ − ρ2
dt − a sin2 θdϕ
2
−
χ2 4π2
+
a2
dϕ − adt
2
−
ρ2 dχ2 Δ 4π2
− ρ2dθ2 ,
a2 + q2 < m2,
L a= ,
ρ2 = χ2 + a2 cos2 θ , (34)
m
4π2
Δ
=
χ2 4π2
−
αχ 2π
+
q2
+ a2 ,
2π m + m2 − q2 − a2 cos2 θ < χ < ∞ .
and
when
θ
=
π 2
,
the
limiting
radius
of
curvature
is,
Cn(r0) = β = m + m2 − q2 ,
which is the limiting radius of curvature for the simple pointcharge (i. e. no rotation) [2].
The standard analysis incorrectly takes (37) as the “radius” of its event horizon.
If q = 0, then the limiting radius of curvature when θ = 0 is,
Equation (34) is entirely independent of the r-parameter. Equation (34) emphasizes the fact that the concept of a
point in pseudo-Euclidean Minkowski space is not attainable in the pseudo-Riemannian gravitational field. A point-mass (or point-charge) is characterised by a proper radius of zero and a finite, non-zero radius of curvature. Einstein’s universe admits of nothing more pointlike. The relativists have assumed that, insofar as the point-mass is concerned, the Minkowski point can be achieved in Einstein space, which is not correct.
The radius of curvature of (33) is,
Cn(r) =
|r − r0|n + βn
1
n,
(35)
which goes down to the limit,
lim
r → r0±
Cn(r) =
Cn(r0) = β =
(36)
= m + m2 − q2 − a2 cos2 θ ,
Cn(r0) = β = m + m2 − a2 ,
(38)
and
the
limiting
radius
of
curvature
when
θ=
π 2
is,
Cn(r0) = β = 2m = α ,
which is the radius of curvature for the simple point-mass.
The radii of curvature at intermediate azimuth are given
generally by (36). In all cases the near-field limits of the radii
of curvature give Rp(r0) ≡ 0. Clearly, the limiting radius of curvature is minimum at the
poles and maximum at the equator. At the equator the effects
of rotation are not present. A test particle approaching the
rotating point-charge or the rotating point-mass equatorially
experiences the effects only of the non-rotating situation of
each configuration respectively. The effects of the rotation
manifest
only
in
the
values
of
azimuth
other
than
π 2
.
There
is
no rotational drag on spacetime, no ergosphere and no event
horizon, i. e. no black hole.
S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass
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The effects of rotation on the radius of curvature will necessarily manifest in the associated form of Kepler’s 3rd Law, and the Kretschmann scalar [22].
I finally remark that the fact that a singularity arises in the gravitational field of the point-mass is an indication that a material body cannot collapse to a point, and therefore such a model is inadequate. A more realistic model must be sought in terms of a non-singular metric, of which I treat elsewhere [23].
Dedication
I dedicate this paper to the memory of Dr. Leonard S. Abrams: (27 Nov. 1924 — 28 Dec. 2001).
References
1. Crothers S. J. On the general solution to Einstein’s vacuum field and it implications for relativistic degeneracy. Progress in Physics, 2005, v. 1, 68–73.
2. Crothers S. J. On the ramifications of the Schwarzschild spacetime metric. Progress in Physics, 2005, v. 1, 74–80.
3. Stavroulakis N. A statical smooth extension of Schwarzschild’s metric. Lettere al Nuovo Cimento, 1974, v. 11, 8 (see also in www.geocities.com/theometria/Stavroulakis-3.pdf).
4. Stavroulakis N. On the Principles of General Relativity and the SΘ(4)-invariant metrics. Proc. 3rd Panhellenic Congr. Geometry, Athens, 1997, 169 (see also in www.geocities.com/theometria/Stavroulakis-2.pdf).
5. Stavroulakis N. On a paper by J. Smoller and B. Temple. Annales de la Fondation Louis de Broglie, 2002, v. 27, 3 (see also in www.geocities.com/theometria/Stavroulakis-1.pdf).
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10. Eddington A. S. The mathematical theory of relativity. Cambridge University Press, Cambridge, 2nd edition, 1960.
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14
S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass
July, 2005
PROGRESS IN PHYSICS
Volume 2
A Theory of Gravity Like Electrodynamics
Dmitri Rabounski
E-mail: rabounski@yahoo.com
This study looks at the field of inhomogeneities of time coordinates. Equations of motion, expressed through the field tensor, show that particles move along time lines because of rotation of the space itself. Maxwell-like equations of the field display its sources, which are derived from gravitation, rotations, and inhomogeneity of the space. The energy-momentum tensor of the field sets up an inhomogeneous viscous media, which is in the state of an ultrarelativistic gas. Waves of the field are transverse, and the wave pressure is derived from mainly sub-atomic processes — excitation/relaxation of atoms produces the positive/negative wave pressures, which leads to a test of the whole theory.
Contents
1. Inhomogeneity of observable time. Defining the field . . 15 2. The field tensor. Its observable components: gravita-
tional inertial force and the space rotation tensor . . . . . . 17 3. Equations of free motion. Putting the acting force in-
to a form like Lorentz’s force . . . . . . . . . . . . . . . . . . . . . . . 18 4. The field equations like electrodynamics . . . . . . . . . . . . . 19 5. Waves of the field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6. Energy-momentum tensor of the field . . . . . . . . . . . . . . . . 21 7. Physical properties of the field . . . . . . . . . . . . . . . . . . . . . . . 22 8. Action of the field without sources . . . . . . . . . . . . . . . . . . . 23 9. Plane waves of the field under gravitation is neglec-
ted. The wave pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 10. Physical conditions in atoms . . . . . . . . . . . . . . . . . . . . . . . . 26 11. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1 Inhomogeneity of observable time. Defining the field
The meaning of Einstein’s General Theory of Relativity consists of his idea that all properties of the world are derived from the geometrical structure of space-time, from the worldgeometry, in other words. This is a way to geometrize physics. The introduction of his artificial postulates became only of historical concern subsequent to his setting up of the meaning of the theory — all the postulates are naturally contained in the geometry of a four-dimensional pseudoRiemannian space with the sign-alternating signature (−+++) or (+−−−) he assigned to the basic space-time of the theory.
Verification of the theory by experiments has shown that the four-dimensional pseudo-Riemannian space satisfies our observable world in most cases. In general we can say that all that everything we can obtain theoretically in this space geometry must have a physical interpretation.
Here we take a pseudo-Riemannian space with the signature (+−−−), where time is real and spatial coordinates are imaginary, because the observable projection of a fourdimensional impulse on the spatial section of any given observer is positive in this case. We also assign to spacetime Greek indices, while spatial indices are Latin∗.
As it is well-known [1], dS = m0cds is an elementary action to displace a free mass-bearing particle of rest-mass m0 through a four-dimensional interval of length ds. What happens to matter during this action? To answer this question let us substitute the square of the interval ds2 = gαβ dxαdxβ into the action. As a result we see that
dS = m0c ds = m0c gαβ dxαdxβ ,
(1)
so the particle moves in space-time along geodesic lines (free motion), because the field carries the fundamental metric tensor gαβ. At the same time Einstein’s equations link the metric tensor gαβ to the energy-momentum tensor of matter through the four-dimensional curvature of space-time. This implies that the gravitational field is linked to the field of the space-time metric in the frames of the General Theory of Relativity. For this reason one regularly concludes that the action (1) displacing free mass-bearing particles is produced by the gravitational field.
Let us find which field will manifest by the action (1) as a source of free motion, if the space-time interval ds therein is written with quantities which would be observable by a real observer located in the four-dimensional pseudo-Riemannian space.
A formal basis here is the mathematical apparatus of physically observable quantities (the theory of chronometric invariants), developed by Zelmanov in the 1940’s [2, 3]. Its essence is that if an observer accompanies his reference body,
∗Alternatively, Landau and Lifshitz in their The Classical Theory of Fields [1] use the space signature (−+++), which gives an advantage in certain cases. They also use other notations for tensor indices: in their book space-time indices are Latin, while spatial indices are Greek.
D. Rabounski. A Theory of Gravity Like Electrodynamics
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July, 2005
his observable quantities are projections of four-dimensional
quantities on his time line and the spatial section — chrono-
metrically invariant quantities, made by projecting operators
bα
=
dxα ds
and
hαβ = −gαβ + bαbβ
which
fully
define
his
real reference space (here bα is his velocity with respect
to his real references). Thus, the chr.inv.-projections of a
world-vector
Qα
are
bαQα
=
√Q0 g00
and
hiαQα = Qi,
while
chr.inv.-projections of a world-tensor of the 2nd rank Qαβ
are
bαbβ Qαβ
=
Q00 g00
,
hiαbβ Qαβ
=
√Qi0 , g00
hiαhkβ Qαβ
= Qik.
Physically observable properties of the space are derived
from
the
fact
that
chr.inv.-differential
operators
∗∂ ∂t
=
√1 g00
∂ ∂t
and
∗∂ ∂xi
=
∂ ∂xi
+
1 c2
vi
∗∂ ∂t
are
non-commutative
∗∂2 ∂xi∂t
−
−
∗∂2 ∂t ∂xi
=
1 c2
Fi
∗∂ ∂t
and
∗∂2 ∂xi∂xk
−
∂
∗∂2 xk∂xi
=
2 c2
Aik
∗∂ ∂t
,
and
also from the fact that the chr.inv.-metric tensor hik may not
be stationary. The observable characteristics are the chr.inv.-
vector of gravitational inertial force Fi, the chr.inv.-tensor of
angular velocities of the space rotation Aik, and the chr.inv.-
tensor of rates of the space deformations Dik, namely
1
Fi
=
√ g00
∂w ∂xi
−
∂vi ∂t
,
√
w
g00 = 1 − c2
(2)
1 Aik = 2
∂vk ∂xi
−
∂vi ∂xk
1 + 2c2 (Fivk − Fkvi) ,
(3)
√
Dik =
1 2
∗∂hik ∂t
,
Dik
1 =−
∗∂hik
,
2 ∂t
Dkk
=
∗∂
ln ∂t
h ,
(4)
where
w
is
gravitational
potential,
vi
=
−c
√g0i g00
is
the
linear
velocity
of
the
space
rotation,
hik = −gik +
1 c2
vivk
is
the
chr.inv.-metric tensor, and also h = det hik , h g00 = −g,
g = det gαβ . Observable inhomogeneity of the space is
set up by the chr.inv.-Christoffel symbols Δijk = himΔjk,m, which are built just like Christoffel’s usual symbols Γαμν = = gασΓμν,σ using hik instead of gαβ.
A four-dimensional generalization of the main chr.inv.-
quantities Fi, Aik, and Dik (by Zelmanov, the 1960’s [4])
is: Fα = −2c2bβ aβα, Aαβ = chμαhνβ aμν , Dαβ = chμαhνβ dμν ,
where
aαβ
=
1 2
(∇α
bβ
−
∇β
bα),
dαβ
=
1 2
(∇α
bβ
+
∇β
bα).
In this way, for any equations obtained using general
covariant methods, we can calculate their physically observ-
able projections on the time line and the spatial section of
any particular reference body and formulate the projections
in terms of their real physically observable properties, from
which we obtain equations containing only quantities mea-
surable in practice.
Expressing ds2 = gαβ dxαdxβ through the observable time interval
dτ
=
1 c
bαdxα
=
w 1 − c2
dt
−
1 c2
vidxi
(5)
and also the observable spatial interval dσ2 = hαβ dxαdxβ = = hik dxidxk (note, bi = 0 for an observer who accompanies
his reference body), we come to the formula
ds2 = c2dτ 2 − dσ2.
(6)
Using this formula, we can write down the action (1) to displace a free mass-bearing particle in the form
dS = m0c bαbβ dxαdxβ − hαβ dxαdxβ .
(7)
If the particle is at rest with respect to the observer’s
reference body, then its observable displacement along his
spatial section is dxi = 0, so its observable chr.inv.-velocity
vector
equals
zero;
vi
=
dxi dτ
=
0.
Such
a
particle
moves
only
along time lines. In this case, in the accompanying reference
frame, we have hαβ dxαdxβ = hik dxidxk = 0 hence the
action is
dS = m0c bαdxα,
(8)
so the mass-bearing particle moves freely along time lines because it is carried solely by the vector field bα.
What is the physical meaning of this field? The vector bα is the operator of projection on time lines (non-uniform,
in general case) of a real observer, who accompanies his reference body. This implies that the vector field bα defines
the geometrical structure of the real space-time along time
lines. Projecting an interval of four-dimensional coordinates dxα onto the time line of a real observer in his accompanying
reference frame, we obtain the interval of real physical time
dτ
=
1 c
bαdxα
=
1−
w c2
dt
−
1 c2
vidxi
he
observes.
For
his
measurements in the same spatial point, in other words, along
the same time line, dτ =
1−
w c2
dt. This formula and the
previous one lead us to the conclusion that the components
of the observer’s vector bα define a “density” of physically
observable time in his accompanying reference frame. As it
is easy to see, the observable time density depends on the
gravitational potential and, in the general case, on the rotation of the space. Hence, the vector field bα in the accompanying
reference frame is the field of inhomogeneity of observable
time references. For this reason we will call it the field of
density of observable time.
In the same way, a field of the tensor hαβ = −gαβ + bαbβ projecting four-dimensional quantities on the observer’s spa-
tial section is the field of density of the spatial section.
From the geometric viewpoint, we can illustrate the conclusions in this way. The vector field bα and the tensor field
hαβ of the accompanying reference frame of an observer, located in a four-dimensional pseudo-Riemannian space,
“split” the space into time lines and a spatial section, proper-
ties of which (such as inhomogeneity, anisotropy, curvature,
etc.) depend on the physical properties of the observer’s
reference body. Owing to this “splitting” process, the field
16
D. Rabounski. A Theory of Gravity Like Electrodynamics
July, 2005
PROGRESS IN PHYSICS
Volume 2
of the fundamental metric tensor gαβ, containing the geometrical structure of this space, “splits” as well (7). Its “transverse component” is the time density field, a fourdimensional potential of which is the monad vector bα. The “longitudinal component” of this splitting is the field of density of the spatial section.
In accordance with the above, two particular cases of time density fields are possible. These are:
1. If a time density field has Hik = 0 and Ei = 0, then the field is strictly of the “electric” kind. This particular case corresponds to a holonomic (non-rotating) space filled with gravitational force fields;
2 The field tensor. Its observable components: gravitational inertial force and the space rotation tensor
Chr.inv.-projections of the four-dimensional vector potential bα of a time density field are, respectively
ϕ = √b0 = 1 ,
qi = bi = 0 .
(9)
g00
Emulating the way that Maxwell’s electromagnetic field
tensor is introduced, we introduce the tensor of a time density
field as the rotor of its four-dimensional vector potential
Fαβ
=
∇α
bβ
−
∇β
bα
=
∂bβ ∂xα
−
∂bα ∂xβ
.
(10)
Taking into account that F00 = F 00 = 0, as for any antisymmetric tensor of the 2nd rank, after some algebra we obtain the other components of the field tensor Fαβ
1√ F0i = c2 g00 Fi ,
1 Fik = c
∂vi ∂xk
−
∂vk ∂xi
,
(11)
F0∙0∙
=
1 − c3
vkF k,
F0∙i∙
=
1√ − c2 g00
F i,
(12)
Fk∙0∙
=
1 −√
g00
1 c2
Fk
+
2 c2
vmAmk
−
1 c4
vkvmF
m
,
(13)
Fk∙i∙
=
1 c3
vkF i +
2 c
A∙ki∙
,
F
ik
=
2 −
Aik,
c
(14)
F 0k
=
1 −√
g00
1 c2
Fk
+
2 c2
vmAmk
.
(15)
We denote chr.inv.-projections of the field tensor just like the chr.inv.-projections of the Maxwell tensor [5], to display their physical sense. We will refer to the time projection
2. A time density field is of the “magnetic” kind, if therein Ei = 0 and Hik = 0. This is a non-holonomic space, where fields of gravitational inertial forces are homogeneous or absent. This case is possible also if, according to the chr.inv.-definition of the force
1
Fi
=
√ g00
∂w ∂xi
−
∂vi ∂t
,
√
w
g00 = 1− c2 , (18)
where the first term — a force of gravity would be reduced by the second term — is a centrifugal force of inertia.
In addition to the field tensor Fαβ, we introduce the field pseudotensor F ∗αβ dual and in the usual way [1]
F ∗αβ
=
1 2
Eαβμν Fμν
,
F∗αβ
=
1 2
Eαβμν F μν ,
(19)
where the four-dimensional completely antisymmetric dis-
criminant tensors
E αβ μν
=
e√αβμν −g
√ and Eαβμν = eαβμν −g,
transforming regular tensors into pseudotensors in inhomoge-
neous anisotropic pseudo-Riemannian spaces, are not phys-
ically observable quantities. The completely antisymmetric unit tensor eαβμν , being defined in a Galilean reference frame
in Minkowski space [1], does not have this quality either.
Therefore we employ Zelmanov’s chr.inv.-discriminant tensors εαβγ = bσEσαβγ and εαβγ = bσEσαβγ [2], which in the
accompanying reference frame are
εikm
=
eikm √
,
√ εikm = eikm h .
(20)
h
Using components of the field tensor Fαβ, we obtain chr.inv.-projections of the field pseudotensor, which are
H ∗i
=
√F0∗∙∙i g00
1 =−
c
εikmAkm
2 =−
c
Ω∗i,
(21)
Ei
=
√F0∙i∙ g00
=
1 − c2
F i,
Ei
=
hik E k
=
1 − c2
Fi
(16)
of the field tensor Fαβ as “electric”. The spatial projection
Hik = F ik =− 2 Aik, c
Hik
=
himhknF
mn
=−
2 c
Aik
(17)
of the field tensor will be referred to as “magnetic”. So, we arrive at physical definitions of the components:
The “electric” observable component of a time density field manifests as the gravitational inertial force Fi. The “magnetic” observable component of a time density field manifests as the angular velocity Aik of the space rotation.
E ∗ik
=
F ∗ik
=
1 c2
εikmFm
,
(22)
where
Ω∗i
=
1 2
εikmAkm
is
the
chr.inv.-pseudovector
of
an-
gular velocities of the space rotation. Their relations to the
field tensor chr.inv.-projections express themselves just like
any chr.inv.-pseudotensors [5, 6], by the formulae
H ∗i
=
1 2
εimnHmn ,
H∗i
=
1 2
εimn H mn ,
(23)
εipq H∗i
=
1 2
εipq εimn H mn
=
H pq ,
(24)
εikpH∗p = Eik ,
E∗ik = −εikmEm ,
(25)
D. Rabounski. A Theory of Gravity Like Electrodynamics
17
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July, 2005
where εipqεimn = δmp δnq − δmq δnp , see [1, 5, 6] for details. We introduce the invariants J1 = Fαβ F αβ and J2 =
= Fαβ F ∗αβ for a time density field. Their formulae are
J1
=
Fαβ F αβ
=
4 c2
Aik Aik
−
2 c4
FiF
i,
(26)
J2
=
Fαβ F ∗αβ
=
8 − c3
Fi Ω∗i,
(27)
so the time density field can be spatially isotropic (one of the invariants becomes zero) under the conditions:
• the invariant AikAik of the space rotation field and the
invariant Fi F i of the gravitational inertial force field
are
proportional
one
to
another
Aik Aik
=
1 2c2
Fi
F
i;
• Fi Ω∗i = 0, so the acting gravitational inertial force Fi
is orthogonal to the space rotation pseudovector Ω∗i;
• both of the conditions are realized together.
3 Equations of free motion. Putting the acting force into a form like Lorentz’s force
Time lines are geodesics by definition. In accordance with the
least action principle, an action replacing a particle along a
geodesic line is minimum. Actually, the least action principle
implies that geodesic lines are also lines of the least action.
This is the physical viewpoint.
We are going to consider first a free mass-bearing particle,
which is at rest with respect to an observer and his reference
body. Such a particle moves only along a time line, so it
moves solely because of the action of the inhomogeneity of
time coordinates along the time line — a time density field.
The action that a time density field expends in displacing
a free mass-bearing particle of rest-mass m0 at dxα has the value dS = m0c bαdxα (8). Because of the least action,
variation of the action integral along geodesic lines equals
zero
b
δ dS = 0 ,
(28)
a
which, after substituting dS = m0c bαdxα, becomes δ abdS =
= m0c δ
abbαdxα= m0c
b a
δ
bαdxα
+
m0c
abbαdδxα
where
abbαdδxα = bαδxα
b
a−
abdbαδxα =−
abdbαδxα. Because
δbα
=
∂bα ∂xβ
δxβ
and
dbα
=
∂bα ∂xβ
dxβ ,
b
b
δ dS = m0c δ
a
a
∂bβ ∂xα
−
∂bα ∂xβ
dxβ δ xα .
(29)
This variation is zero, so along time lines we have
m0c
∂bβ ∂xα
−
∂bα ∂xβ
dxβ = 0 .
(30)
This condition, being divided by the interval ds, gives general covariant equations of motion of the particle
m0c Fαβ U β = 0 ,
(31)
wherein Fαβ is the time density field tensor and U β is the particle’s four-dimensional velocity∗.
Taking chr.inv.-projections of (31) multiplied by c2, we
obtain chr.inv.-equations of motion of the particle
m0
c3
F√0σU σ g00
= 0,
m0c2F∙iσ∙ U σ = 0 ,
(32)
where the scalar equation gives the work to displace the
particle, and the vector equations its observable acceleration.
It is interesting to note that the left side of the equations,
which is the acting force, both in the general covariant form
and its chr.inv.-projections we have obtained, has the same
form as Lorentz’s force, which displaces charged particles in
electromagnetic fields [5]. From the mathematical viewpoint
this fact implies that the time density field acts on mass-
bearing particles as the electromagnetic field moves electric
charge.
Taking ds2 = c2dτ 2 − dσ2 = c2dτ 2
1
−
v2 c2
, that is for-
mula (4) into account, we obtain
U α = dxα =
1
dxα ,
(33)
ds
c
1
−
v2 c2
dτ
U0
=
1 c2
vkvk
+
1
√ g00
1
−
v2 c2
,
Ui =
1
vi. (34)
c
1
−
v2 c2
Using the components obtained for the field tensor Fαβ
(11–15) and taking into account that the observable velocity of the particle we are considering is vi = 0, we transform
the chr.inv.-equations of motion (32) into the final form. The
scalar equation becomes zero, while the vector equations
become
m0F i
=
0
or,
substituting
Ei
=
−
1 c2
Fi
(16),
m0c2Ei = 0 ,
(35)
leading us to the following conclusions:
1. The “electric” and the “magnetic” components of a
time density field do not produce work to displace
a free mass-bearing particle along time lines. Such a
particle falls freely along its own time line under the
time density field;
2. In this case Ei = 0, so the particle falls freely along its
own time line, being carryied solely by the “magnetic”
component
Hik
=
−
2 c
Aik
=
0
of
the
time
density
field;
3. Inhomogeneity of the spatial section (the chr.inv.-
Christoffel symbols Δijk) or its deformations (the chr.
inv.-deformation rate tensor Dik) do not have an effect
on free motion along time lines.
∗Do
not
confound
this
vector
Uα
=
dxα ds
with the
vector
bα
=
dxα ds
:
they are built on different dxα. The vector bα contains displacement of the
observer with respect to his reference body, while the vector U α contains
displacement of the particle.
18
D. Rabounski. A Theory of Gravity Like Electrodynamics
July, 2005
PROGRESS IN PHYSICS
Volume 2
In
other
words,
the
“magnetic”
component
Hik
=
−
2 c
Aik
of a time density field “screws” particles into time lines (a
very rough analogy). There are no other sources which could
cause particles to move along time lines, because observable
particles with the whole spatial section move from past into
future, hence Hik = 0 everywhere in our real world. So, our
real space is strictly non-holonomic, Aik = 0.
This purely mathematical result brings us to the very
important conclusion that under any conditions a real space
is non-holonomic at the “start”, that is, a “primordial non-
orthogonality” of the real spatial section to time lines. Cond-
itions such as three-dimensional rotations of the reference
body, are only additions, intensifying or reducing this startrotation of the space, depending on their relative directions∗.
We are now going to consider the second case of free
motion — the general case, where a free mass-bearing particle
moves freely not only along time lines, but also along the
spatial section with respect to the observer and his reference
body. Chr.inv.-equations of motion in this general case had
been deduced by Zelmanov [2]. They have the form
not produce any work to displace free mass-bearing particles, only the space deformation field produces the work.
In particular, a mass-bearing particle can be moved freely along the spatial section, solely because of the field of time density. As it easy to see from equations (37), this is possible under the following conditions
Dikvivk = 0 ,
Dki
=
1 −
2
Δink vn ,
(38)
so it is possible in the following particular cases:
• if the spatial section has no deformations, Dik = 0;
• if, besides the absence of the deformations (Dik = 0), the spatial section is homogeneous, Δink = 0.†
The scalar equations of motion (37) also show that, under the particular conditions (38), the energy dE to displace the particle at dxi equals the work
dE = −mc2Eidxi
(39)
dE dτ
− mFivi
+ mDikvivk
=
0,
E = mc2
d (mvi) −mF i+2m dτ
Dki +A∙ki∙
vk+mΔinkvnvk=0.
the “electric” field component Ei expends for this displacement. The vector equations of motion in this particular case
(36) show that the “electric” and the “magnetic” components of the acting field of time density accelerate the particle just like external forces‡
Let us express the equations through the “electric” and
the “magnetic” observable components of the acting field of
time
density.
Substituting
E
i
=
−
1 c2
F
i
and
Hik
=
−
2 c
Aik
into the Zelmanov equations (36), we obtain
dE dτ
+ mc2Eivi
+ mDikvivk
=
0,
d (mvi) + mc2 dτ
Ei
+
1 c
H ik vk
+
(37)
+ 2mDki vk + mΔinkvnvk = 0.
dpi dτ
= −mc2
Ei
+
1 c
H ik vk
.
(40)
Looking at the right sides of equations (39, 40), we see that they have a form identical to the right sides of the chr.inv.-equations of motion of a charged particle in the electromagnetic field [5]. This implies also that the field of time density acts on mass-bearing particles as an electromagnetic field moves electric charge.
4 The field equations like electrodynamics
From this we see that a free mass-bearing particle moves freely along the spatial section because of the factors:
1. The particle is carried with a time density field by its “electric” Ei = 0 and “magnetic” Hik = 0 components;
2. The particle is also moved by forces which manifest as an effect of inhomogeneity Δink and deformations Dik of the spatial section. As we can see from the scalar equation, the field of the space inhomogeneities does
∗A similar conclusion had also been given by the astronomer Kozyrev [7], from his studies of the interior of stars. In particular, besides the “start” self-rotation of the space, he had come to the conclusion that additional rotations will produce an inhomogeneity of observable time around rotating bulky bodies like stars or planets. The consequences should be more pronounced in the interaction of the components of bulky double stars [8]. He was the first to used the term “time density field”. It is interesting that his arguments, derived from a purely phenomenological analysis of astronomical observations, did not link to Riemannian geometry and the mathematical apparatus of the General Theory of Relativity.
As is well-known, the theory of the electromagnetic field, in a pseudo-Riemannian space, characterizes the field by a system of equations known also as the field equations:
• Lorentz’s condition stipulates that the four-dimensional vector potential Aα of the field remains unchanged just like any four-dimensional vector in a four-dimensional pseudo-Riemannian space
∇σ Aσ = 0 ;
(41)
• the charge conservation law (the continuity equation) shows that the field-inducing charge cannot be destroyed, but merely re-distributed in the space
∇σ jσ = 0 ,
(42)
†However the first condition Dik = 0 would be sufficient. ‡Here the chr.inv.-vector pi = mvi is the particle’s observable impulse.
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where jα is the four-dimensional current vector; its
observable projections are the chr.inv.-charge density
scalar
ρ
=
c
√1 g00
j0
and
the
chr.inv.-current
density
vector ji, which are sources inducing the field;
• Maxwell’s equations show properties of the field, ex-
pressed by components of the field tensor Fαβ and its dual pseudotensor F ∗αβ. The first group of the Maxwell equations contains the field sources ρ and ji,
the second group does not contain the sources
∇σ
F ασ
=
4π c
jα,
∇σ F ∗ασ = 0 .
(43)
We can put all the equations into chr.inv.-form, employ-
ing Zelmanov’s formula [2] for the divergence of a vector Qα, where he expressed the divergence through chr.inv.projections ϕ = √Q0 and qi = Qi of this vector
g00
∇σ
Qσ
=
1 c
∗∂ϕ + ϕD
∂t
+
∗∇i
qi
−
1 c2
Fi qi,
(44)
where we use his notation for chr.inv.-divergence
√
∗∇i qi
=
∗∂qi ∂xi
+
qi
∗∂ ln ∂xi
h
=
∗∂qi ∂xi
+
qiΔjji .
(45)
In particular, the chr.inv.-Maxwell equations, which are
chr.inv.-projections of the Maxwell general covariant equa-
tions (43), had first been obtained for an arbitrary field
potential by del Prado and Pavlov [9], Zelmanov’s students,
at Zelmanov’s request. The equations are
∗∇i
Ei
−
1 c
H ikAik
=
4πρ

∗
∇k
H
ik
−
1 c2
Fk
H
ik
−
1 c
∗∂Ei +EiD
4π =
ji
I, 
∂t
c
(46)
∗∇i
H∗i −
1 c
E ∗ikAik
=
0

∗
∇k
E
∗ik −
1 c2
Fk
E ∗ik
−
1 c
∗∂H∗i + H∗iD
II, =0 
∂t
(47)
where D = hikDik, being the spur of the deformation rate
tensor, is the rate of expansion of an elementary volume.
Actually, the obtained Lorentz condition (48) implies that
the value of an elementary volume filled with a time density
field remains unchanged under its deformations.
We now collect chr.inv.-projections of the tensor of a
time density field Fαβ and of the field pseudotensor F ∗αβ
together:
Ei
=−
1 c2
Fi,
H
ik
=−
2 c
Aik,
H ∗i
=−
2 c
Ω∗i,
E∗ik =
=
1 c2
εikmFm.
We
also
take
Zelmanov’s
identities
for
the
chr.inv.-discriminant tensors [2] into account
∗∂εimn ∂t
= εimn D ,
∗∂εimn = −εimnD ,
(49)
∂t
∗∇k εimn = 0 ,
∗∇k εimn = 0 .
(50)
Substituting the chr.inv.-projections into (46, 47) along
with the obtained Lorentz condition D = 0 (48), we arrive at
Maxwell-like chr.inv.-equations for a time density field
1 c2
∗∇i
Fi
−
2 c2
Aik Aik
=
− 4πρ
2 c
∗∇k
Aik
−
2 c3
Fk Aik
−
1 c3
∗∂F i ∂t
= − 4π ji c
 
I, 
(51)
∗∇i
Ω∗i
+
1 c2
Fi
Ω∗i
=
0
∗∇k (εikmFm) −
1 c2
εikmFkFm + 2
∗∂Ω∗i ∂t
=0
 

II,
(52)
so that the field-inducing sources ρ and ji are
1 ρ = − 4πc2
∗∇i F i − 2AikAik
,
(53)
ji
=
1 −
2π
∗∇k Aik
−
1 2πc2
Fk
Aik
−
1 4πc2
∗∂F i ∂t
.
(54)
The “charge” conservation law ∇σ jσ = 0 (the continuity equation), after substituting chr.inv.-projections ϕ = cρ and qi = ji of the “current” vector jα, takes the chr.inv.-form
From the mathematical viewpoint, equations of the field
are a system of 10 equations in 10 unknowns (the Lorentz
condition, the charge conservation law, and two groups of
the Maxwell equations), which define the given vector field Aα and its inducing sources in a pseudo-Riemannian space. Actually, equations like these should exist for any four-
dimensional vector field, a time density field included. The
only difference should be that the equations should be chang-
ed according to a formula for the specific vector potential. We are going to deduce such equations for the field bα
we are considering — equations of a time density field. Because ϕ = 1 and qi = 0 are chr.inv.-projections of the
potential bα of a time density field, the Lorentz condition ∇σ bσ = 0 for a time density field bα becomes the equality
D = 0,
(48)
1 ∗∂ c2 ∂t
Aik Aik
1 ∗∂Aik ∗∂2Aik + c2 Fi ∂xk − ∂xi∂xk +
+
1 c2
FiΔjj
k
+
∗∂Δjjk ∂xi
+
ΔjjiΔllk
Aik −
(55)
−
1 2c2
F i ∗∂Δjji ∂t
−
1 c4
Fi
Fk
Aik
=
0,
The Lorentz condition (48), the Maxwell-like equations (51, 52), and the continuity equation (55) we have obtained are chr.inv.-equations of a time density field.
5 Waves of the field
Let us turn now to d’Alembert’s equations. We are going to obtain the equations for a time density field.
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d’Alembert’s operator = gαβ ∇α∇β, being applied to a field, may or may not be zero. The second case is known as the d’Alembert equations with field-inducing sources, while the first case is known as the d’Alembert equations without sources. If the field has no sources, then the field is free. This is a wave. So, the d’Alembert equations without sources are equations of propagation of waves of the field.
From this reason, the d’Alembert equations for the vector potential bα of a time density field without the sources
bα = 0
(56)
are the equations of propagation of waves of the time density field. Chr.inv.-projections of the equations are
where U σ is the four-dimensional velocity of the particle. The
left side of the equations has the dimensions [ gramme/sec ] as
well as a four-dimensional force. Because of motion along
only time lines, such particle moves solely under the action
of a time density field whose tensor is Fαβ. If this free-moving particle is not a point-mass, then it
can be represented by a current jα of the time density field.
On the other hand, such currents are defined by the 1st group
∇σF ασ
=
4π c
j
α
of
the
Maxwell-like
equations
of
the
field.
In this case equations of motion (59), drawing an analogy
with an electromagnetic field current, take the form
μF∙ασ∙jσ = 0 .
(60)
bσ bσ = 0 ,
hiσ bσ = 0 .
(57)
We substitute chr.inv.-projections ϕ = 1 and qi = 0 of the field potential bα into this. Then, taking into account that the Lorentz condition for the field bα is D = 0 (48), after some
algebra we obtain the chr.inv.-d’Alembert equations for the
time density field without sources
1 c2
FiF i
−
Dik Dik
=
0
,
1 c2
∗∂F i ∂t
+ hkm
∗∂Dmi ∂xk
+
∗∂A∙mi ∙ ∂xk
+
(58)
+ Δikn Dmn − A∙mn∙ − Δnkm Dni − A∙ni∙ = 0 .
Unfortunately,
a
term
like
1 a2
∂2qi ∂t2
containing
the
linear
speed a of the waves is not present, because of qi = 0. For this
reason we have no possibility of saying anything about the
speed of waves traveling in time density fields. At the same
time the obtained equations (58) display numerous specific
peculiarities of a space filled with the waves:
1. The rate of deformations of a surface element in waves of a time density field is powered by the value of the acting gravitational inertial force Fi. If Fi = 0, the observable spatial metric hik is stationary;
2. If a space, filled with waves of a time density field, is homogeneous Δikn = 0 and also the acting force field is stationary Fi = const, the spatial structure of the space deformations is the same as that of the space rotation field.
6 Energy-momentum tensor of the field
Proceeding from the general covariant equations of motion along only time lines, we are going to deduce the energymomentum tensor for time density fields. It is possible to do this in the following way.
The aforementioned equations m0c Fαβ U β = 0 (31), being taken in contravariant (upper-index) form, are
m0cF∙ασ∙U σ = 0 ,
(59)
The numerical coefficient μ here is a new fundamental constant. This new constant having the dimension[gramme/sec] gives the dimensions [ gramme/cm2×sec2 ] to the left side of the equations, making the left side a current of the acting four-dimensional force (59) through 1 cm2 per 1 second. The numerical value of this constant μ can be found from measurements of the wave pressure of a time density field, see formula (101) below. However it does not exclude that future studies of the problem will yield an analytic formula for μ, linking it to other fundamental constants.
Chr.inv.-projections of the equations (60)
μ√F0σjσ = 0 , g00
μF∙iσ∙ jσ = 0 ,
(61)
after substituting the Fαβ components (11–15) take the form
μEk jk = 0 ,
μc
ρEi
−
1 c
H∙ik∙
jk
= 0,
(62)
where Ei is the “electric” observable component and Hik is the “magnetic” observable component of the time density field. Sources ρ and ji inducing the field are defined by the
1st group of the Maxwell-like chr.inv.-equations (51). Actually, the term∗
f α = μF∙ασ∙jσ
(63)
on the left side of the general covariant equations of motion
(60) can be transformed with the 1st Maxwell-like group
∇β F σβ
=
4π c
jσ
to
the
form
fα
=
μc 4π
Fασ
∇β
F
σβ
which
is
μc fα = 4π
∇β FασF σβ
− F σβ ∇β Fασ
,
(64)
where we express the second term in the form F σβ ∇β Fασ =
=
1 2
F
σβ
(∇β Fασ
+
∇σ Fβα )
=
−
1 2
F
σβ
(∇βFσα +
∇σ Fαβ)
=
=
−
1 2
F
σβ ∇σFαβ
=
1 2
F
σβ
∇αFσβ .
Using
this
formula,
we
transform the current f α (63) to the form
μc fα = 4π ∇β
−FασF βσ
+
1 4
δαβ FpqF pq
,
(65)
∗From the physical viewpoint, this term is a current of the acting fourdimensional force, produced by the time density field.
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so we write the current f α in the form
f α = ∇β T αβ
(66)
just as electrodynamics does to deduce the energy-momentum tensor T αβ of electromagnetic fields. In this way, we obtain
the energy-momentum tensor of a time density field, which is
T αβ = μc 4π
−F∙ασ∙F βσ
+
1 4
gαβ FpqF pq
,
(67)
the form of which is the same as the energy-momentum tensor of electromagnetic fields [1, 5] to within the coefficient of its dimension. It is easy to see that the tensor is symmetric, so its spur is zero, Tσσ = gαβ T αβ = 0.
So forth we deduce the chr.inv.-projections of the energymomentum tensor of a time density field
q = T00 , g00
J i = √c T0i , g00
U ik = c2 T ik. (68)
After substituting the required components of the field tensor Fαβ (11–15), we obtain
μ q=
4πc
Aik Aik
+
1 2c2
FkF
k
,
(69)
where
αik = βik +
1 3
αhik
is
the
viscous
strength
tensor
of
the field. Zelmanov called αik the viscosity of the 2nd kind
(here α = hikαik = αnn is its spur). Its anisotropic part βik,
called the viscosity of the 1st kind, manifests as anisotropic
deformations of the space. The quantity p0 is that pressure inside the medium, which equalizes its density in the absence of viscosity, p is the true pressure of the medium∗. It is easy to
see that the viscous strength tensors αik and βik are chr.inv.quantities by their definitions.
By extracting the viscous strength tensors αik and βik from the formula of the strength tensor Uik of a time density field, we are going to deduce the equation of state of the
field. Transforming U ik (71) into covariant form and also keep-
ing the formula for q (69) in the mind, we write
Uik = −q c2hik −
μc −
4π
4Aim Am ∙k ∙
+
1 c2 FiFk
−
1 c2
Fm
F
m
hik
,
(73)
which, after equating to Uik = p0hik − αik (72), gives the equilibrium pressure in the field
p0 = −qc2,
(74)
Ji
=
−
μ 2πc
Fk
Aik,
(70)
U ik
=
μc −
4π
4Ai∙m∙ Amk
+
1 c2
F
i
F
k
+
+ ApqApqhik
−
1 2c2
Fp
F
phik
.
(71)
In accordance with dimensions, the chr.inv.-projections have the following physical meanings:
• the time observable projection q [ gramme/cm×sec2 ] is the energy [ gm×cm2/sec2 ] this time density field contains in 1 cm3. Actually, the chr.inv.-scalar q is the observable density of the field;
• the mixed observable projection Ji [ gramme/sec3 ] is the energy the time density field transfers through 1 cm2 per second, in other words, this is the observable density of the field momentum;
• the spatial observable projection U ik [ gm×cm/sec4 ] is the tensor of the field momentum flux observable density, in other words, the field strength tensor.
7 Physical properties of the field
It has been proven by Zelmanov [10], that the chr.inv.-field strength tensor U ik, can be written in covariant (lower index)
form as follows
Uik = p0hik − αik = phik − βik ,
(72)
while the viscous strength tensor of the field is
μc αik = 4π
4Aim Am ∙k ∙
+
1 c2
FiFk
−
1 c2
FmF m hik
. (75)
Because the spur of this tensor αik, as it is easy to see, is
not
zero,
α
=
hik
αik
=−
μc π
Aik Aik
+
1 2c2
FkF
k
= 0,
the
tensor
αik
=
βik
+
1 3
αhik
has
the
non-zero
anisotropic
part
μc βik = 4π
4Aim Am ∙k ∙
+
1 c2 FiFk
−
(76)
−
1 3c2
FmF m hik
+
4 3
Amn
Amnhik
,
so viscous strengths of time density fields are anisotropic.
It is also easy to see that this anisotropy increases with the value ApqApq of the space rotation.
Because the viscous strengths αik are anisotropic, the equilibrium pressure p0 = −qc2 and the true pressure p inside the medium are different. The true pressure is
μc p=
12π
Aik Aik
+
1 2c2
Fk
F
k
,
(77)
∗The equation of state of a medium is the relation between the pressure
p inside the medium and its density q. In a non-viscous medium or where
the viscous strengths are isotropic, the true pressure p is the same as the
equilibrium pressure p0. The equation of state of a dust medium has the
form
p = 0.
Ultra-relativistic
gases
have
the
equation
of
state
p=
1 3
qc2
.
The equation of state of matter inside atomic nuclei is p = qc2. Vacuum and
μ-vacuum have the equation of state p = −qc2, see [5].
22
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which gives the equation of state for time density fields
p = 1 qc2.
(78)
3
Finally, we write the strength tensor Uik = phik − βik of a time density field in the form
Uik
=
1 3
q c2hik
−
βik
.
(79)
So, we can conclude for the physical properties of time density fields:
1. In general, a time density field is a non-stationary dis-
tributed medium, because its density may be q = const.
The field becomes stationary q = const under stationary
space rotation Aik = const, and stationary gravitational inertial force Fi = const;
2. A time density field bears momentum, because Ji =
=−
μ 2πc
Fk Aik
= 0.
So,
the
field
can
transfer
impulse.
The field does not transfer impulse Ji = 0, if the space
does not rotate Aik = 0. The absence of gravitation does not affect the field’s transfer of impulse, because
the “inertial” part of the force Fi remains unchanged even in the absence of gravitational fields;
3. A time density field is an emitting medium Ji = 0 in a
non-holonomic (rotating) space. In a holonomic (non-
rotating) space the field does not produce radiations;
4. A time density field is a viscous medium. The viscosity αik (75), derived from non-zero rotation of the space or from gravitational inertial force, is anisotropic. The anisotropy βik increases with the space rotation speed. The field is viscous anisotropic anyhow, because its viscous strengths would be αik = 0 and βik = 0 only if both Aik = 0 and Fi = 0. But in this case the field density would be q = 0, so the field itself is not there;
5. Therefore the equilibrium pressure p0 does not possess
a physical sense for time density fields; only the true
pressure
is
real
p
=
p0
−
1 3
α;
6.
The
equation
of
state
for
time
density
fields
is
p
=
1 3
q c2
(78) indicating that such fields are in the state of an
ultrarelativistic gas — at positive density of the med-
ium its inner pressure becomes positive, the medium
is compressed.
dS = dSm + dSmf + dSf =
=
m0
c
ds
+
e c
Aαdxα
+
a
Fαβ
F
αβ
dV
dt
,
(80)
where Aα is the four-dimensional electromagnetic field po-
tential, Fαβ = ∇α Aβ −∇β Aα is the electromagnetic field
tensor, dV = dxdydz is an elementary thee-dimensional vol-
ume filled with this field.
The first term Sm is “that part of the action which depends
only on the properties of the particles, that is, just the action
for free particles. . . . The quantity Smf is that part of the action which depends on the interaction between the particles
and the field. . . . Finally Sf is that part of the action which
depends only on the properties of the field itself, that is, Sf is the action for a field in the absence of charges”.
Because the action Sf must depend only on the field
properties, the action must be taken over the space volume,
filled with the field. The action must be scalar: only the 1st
field invariant J1 = Fαβ F αβ has this property. The 2nd field invariant J2 = Fαβ F ∗αβ is pseudoscalar, not scalar, leading
to the detailed discussion in Landau and Lifshitz.
“The numerical value of a depends on the choice of
units for measurement of the field. . . . From now on we
shall use the Gaussian system of units; in this system a is a
dimensionless
quantity
equal
to
1 16π
”.
According to §27 of The Classical Theory of Fields
we
have
dSf
=
a
Fαβ
F αβdV
dt
=
1 16πc
Fαβ
F αβdΩ,
where
dΩ = c dtdV = cdtdxdydz is an elementary space (four-
dimensional) volume. So the action (80) takes the final form
dS
=
m0c ds
+
e c
Aαdxα
+
1 16πc
Fαβ
F αβdΩ
.
(81)
According to this consideration, we write an elementary action for the whole system consisting of a time density field and a single mass-bearing particle, which falls freely along time lines in a pseudo-Riemannian space, as follows
dS = dSm + dSmt = m0cds + amt Fαβ F αβdΩ = (82)
= m0c bαdxα + amt Fαβ F αβdΩ ,
where Fαβ is the time density field tensor, amt is a constant consisting of other fundamental constants.
The first term Sm is that part of the action for the interaction between the particle and the time density field carrying it into motion along time lines. The second term
8 Action of the field without sources
According to §27 of The Classical Theory of Fields [1], an elementary action for a whole system consisting of an electromagnetic field and a single charged particle, which are located in a pseudo-Riemannian space, contains three parts∗
∗In accordance with the least action principle, this action must have a minimum, so the integral of the action between a pair of world-points
and the action itself must be positive. A negative action could give rise to a quantity with arbitrarily ”large” negative values, which cannot have a minimum. Because in The Classical Theory of Fields Landau and Lifshitz take a pseudo-Riemannian space with the signature (−+++), they write in §3 that “. . . the clock at rest always indicates a greater time interval than the moving one”. Therefore they put “minus” before the action. To the contrary, we stick to a pseudo-Riemannian space with Zelmanov’s signature (+−−−), because in this case three-dimensional observable impulse is positive. In a space with such a signature, a regular observer takes his own flow of observable time positive always, dτ > 0. Any particle, moving from past into future, has also a positive count of its own time coordinate dt > 0 with respect to the observer’s clock. Therefore we put “plus” before the action.
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July, 2005
Smt, depending only on the field properties, is the action
for the field in the absence of its sources. In the absence
of time density fields the second term Smt is zero, so only
Sm = m0cds remains here. A time density field is absent if
the space is free of rotation Aik = 0 and gravitational inertial forces Fi = 0, therefore if the conditions g0i = 0 and g00 = 1 are true. This situation is possible in a pseudo-Riemannian
space with a unit diagonal metric, which is the Minkowski
space of the Special Theory of Relativity, where there is
no gravitational field and no rotation. But in considering real
space, we are forced to take a time density field into account.
So we need to consider the terms Sm and Smt together.
The constant amt, according to its dimension, is the same
as the constant μ in the energy-momentum tensor of time
density
fields,
taken
with
the
numerical
coefficient
a
=
1 16π
,
in the Gaussian system of units.
As a result, we obtain the action (82) in the final form
dS
= dSm + dSmt
=
m0c
bαdxα
+
μ 16π
Fαβ
F
αβ dΩ
.
(83)
Because an action for a system is expressed through La-
grange’s function L of the system as dS = Ldt, we take the
action
dSmt
in
the
form
dSmt =
μc 16π
Fαβ
F
αβ
dV
dt
=
Ldt,
for the Lagrangian of an elementary volume dV = dxdydz
of the field. We therefore obtain the Lagrangian density in
time density fields
Λ
=
μc 16π
Fαβ
F
αβ
=
μ 4πc
Aik Aik
−
1 2c2
FiF i
.
(84)
The term AikAik here, being expressed through the space rotation angular velocity pseudovector Ω∗i, is
the chr.inv.-vector of its momentum density Ji (70) can be
written as follows
Ji
=
μ −
2πc
Fk Aik
=
μc −
4π
Ek H ik .
(86)
We are going to consider a particular case, where the
field depends on only one coordinate. Waves of such a field
traveling in one direction are known as plane waves.
We assume the field depends only on the axis x1 = x, so
only
the
component
J1
=−
μ 2πc
Fk A1k
of
the
field’s
chr.inv.-
momentum density vector is non-zero. Then a plane wave of
the field travels along the axis x1 = x. Assuming the space
rotating in xy plane (only the components A12 = −A21 are
non-zeroes) and replacing the tensor Aik with the space
rotation angular velocity pseudovector Ω∗m in the form
εmik Ω∗m
=
1 2
εmik εmpq
Apq
=
1 2
δpi δqk −δpk δqi Apq = Aik, we
obtain
J1
=
−
μ 2πc
F2A12
=
−
μ 2πc
F2
ε123 Ω∗3
.
(87)
It is easy to see that while a plane wave of the field travels along the axis x1 = x, the field’s “electric” and “magnetic” strengths are directed along the axes x2 = y and x3 = z, i. e. orthogonal to the direction the wave travels. Therefore waves travelling in time density fields are transverse waves.
Following the arguments of Landau and Lifshitz in §47 of The Classical Theory of Fields [1], we define the wave pressure of a field as the total flux of the field energymomentum, passing through a unit area of a wall. So the pressure Fi is the sum
Fi = Tik nk + Tik nk
(88)
AkmAkm = εkmn Ω∗nAkm = 2Ω∗n Ω∗n,
(85)
because
εnkmΩ∗n
=
1 2
εnpq εnkm Apq
=
1 2
δkpδmq −δkqδmp
Apq =
= Akm
and
Ω∗n =
1 2
εnkmAkm.
So
the
space
rotation
plays
the first violin, defining the Lagrangian density in time den-
sity fields. Rotation velocities in macro-processes are incom-
mensurably small in comparison with rotations of atoms and
particles. For instance, in the 1st Bohr orbit in an atom of
hydrogen, measuring the value of Λ in the units of the energymomentum constant μ, we have Λ 9.1×1021μ. On the Earth’s surface near the equator the value is Λ 2.8×10−20μ,
so it is in order of 1042 less than in atoms. Therefore, because
the Lagrangian of a system is the difference between its
kinetic and potential energies, we conclude that time density
fields produce their main energy flux in atoms and sub-atomic
interactions, while the energy flux produced by the fields of
macro-processes is negligible.
9 Plane waves of the field under gravitation is neglected. The wave pressure
In general, because the electric and the magnetic strengths
of
a
time
density
field
are
Ei
=−
1 c2
Fi
and
H ik
=−
1 c
Aik,
of the spatial components of the energy-momentum tensor
Tαβ in a wave, falling on the wall, and of the energymomentum tensor Tαβ in the reflected wave, projected onto the unit spatial vector n(k) orthogonal to the wall surface.
Because the chr.inv.-strength tensor of a field is Uik = = c2 hiαhkβ T αβ = c2 Tik [2], we obtain
1 Fi = c2
Uik nk + Uik nk
,
(89)
where Uik = c2 Tik and Uik = c2 Tik are the chr.inv.-strength tensors in the falling wave and in the reflected wave. So the three-dimensional wave pressure vector Fi has the property of chronometric invariance.
Using our formulae for the density q (68) and the strength tensor Uik (71) obtained for time density fields, we are going to find the pressure a wave of such field exerts on a wall.
We consider the problem in a weak gravitational field, assuming its potential w and the attracting force of gravity negligible. We can do this because formulae (68) and (71) contain gravitation in only higher order terms. So the space rotation plays the first violin in the wave pressure Fi in time density fields, gravitational inertial forces act there only because of their inertial part.
24
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Volume 2
A plane wave travels along a single spatial direction: we
assume axis x1= x. In this case the chr.inv.-field strength
tensor Uik has the sole non-zero component U11. All the
other components of the strength tensor Uik are zero, which
simplifies this consideration.
We assume the space rotating around the axis x3 = z (the
rotation is in the xy-plane) at a constant angular velocity
Ω. In this case A12 =−A21 =−Ω, A13 = 0, A23 = 0, so the components of the rotation linear velocity vi = Aikxk are
v1 =−Ωy, v2 = Ωx, v3 = 0. Then the components of the act-
ing
gravitational
inertial
force
will
be
F1
=−
∂v1 ∂t
=Ω
∂y ∂t
=
= Ωv2 = Ω2x,
F2
=−
∂v2 ∂t
=
Ω2 y,
F3 = 0.
Because
in
this
case AikAik = 2A12A12 = 2Ω2 and A1mAm ∙1 ∙ = A1mAmnh1n
= A12A21h11=−Ω2h11, we obtain
μ q=
4πc
2Ω2 +
1 2c2
Ω4
x2 + y2
,
(90)
μc U11 = 4π
2Ω2
h11
−
1 c2
Ω4
x2
+
1 2c2
Ω4
x2+ y2
h11
.
(91)
We assume a coefficient of the reflection as the ratio between the density of the field energy q in the reflected wave to the energy density q in the falling wave. Actually, because q = q, the reflection coefficient is the energy loss of the field after the reflection.
We assume x = x0 = 0 at the reflection point on the surface of the wall. Then we have U11 = qc2h11, which, after substituting into (89), gives the pressure
F1 = (1 + ) q h11 n1
(92)
that a plane wave of a time density field exerts on the wall. To bring this formula into final form in a Riemannian
space becomes a problem, because the coordinate axes are curved there, and inhomogeneous. For this reason we cannot define the angles between directions in a Riemannian space itself, the angle of incidence and the angle of reflection of a wave for instance. At the same time, to consider this problem in the Minkowski space of the Special Theory of Relativity, as done by Landau and Lifshitz for the pressure of plane electromagnetic waves [1], would be senseless — because in Minkowski space we have g00 = 1 and g0i = 0, then Fi = 0 and Aik = 0, which implies no time density fields there.
To solve this problem correctly for a Riemannian space, let us introduce a locally geodesic reference frame, following Zelmanov. We therefore introduce a locally geodesic reference frame at the point of reflection of a wave on the surface of a wall. Within infinitesimal vicinities of any point of such a reference frame the fundamental metric tensor is
1 g˜αβ = gαβ + 2
∂2g˜αβ ∂x˜μ∂x˜ν
(x˜μ −xμ)(x˜ν −xν ) + . . . ,
(93)
the higher order terms, values of which can be neglected. Therefore, at any point of a locally geodesic reference frame the fundamental metric tensor can be considered constant, while the first derivatives of the metric (the Christoffel symbols) are zero.
As a matter of fact, within infinitesimal vicinities of any point located in a Riemannian space, a locally geodesic reference frame can be set up. At the same time, at any point of this locally geodesic reference frame a tangential flat Euclidean space can be set up so that this reference frame, being locally geodesic for the Riemannian space, is the global geodesic for that tangential flat space.
The fundamental metric tensor of a flat Euclidean space is constant, so the values of g˜μν , taken in the vicinities of a point of the Riemannian space, converge to the values of the tensor gμν in the flat space tangential at this point. Actually, this means that we can build a system of basis vectors e(α), located in this flat space, tangential to curved coordinate lines of the Riemannian space.
In general, coordinate lines in Riemannian spaces are curved, inhomogeneous, and are not orthogonal to each other (if the space is non-holonomic). So the lengths of the basis vectors may sometimes be very different from unity.
We denote a four-dimensional vector of infinitesimal displacement by dr =(dx0, dx1, dx2, dx3), so that dr = e(α)dxα, where components of the basis vectors e(α) tangential to the coordinate lines are e(0)= {e0(0), 0, 0, 0}, e(1)= {0, e1(1), 0, 0}, e(2)= {0, 0, e2(2), 0}, e(3)= {0, 0, 0, e2(3)}. The scalar product of the vector dr with itself is drdr = ds2. On the other hand, the same quantity is ds2 = gαβ dxαdxβ. As a result we have gαβ = e(α)e(β)= e(α)e(β) cos (xα; xβ). So we obtain
g00 = e2(0) ,
g0i = e(0)e(i) cos (x0; xi) , (94)
gik = e(i)e(k) cos (xi; xk) .
(95)
The gravitational potential is w = c2(1−√g00). So the
time basis having the
vector length
ee((00))=t√angg0e0n=tia1l−tocw2t,heis
time line x0 smaller than
= ct, unity
the greater is the gravitational potential w.
The space rotation linear velocity vi and, according to it,
the chr.inv.-metric tensor hik are
vi = −c e(i) cos (x0; xi) ,
(96)
hik = e(i)e(k) cos(x0; xi)cos(x0; xk)−cos(xi; xk) . (97)
Harking back to the formula for the pressure F1 (92), that a plane wave of a time density field traveling along the axis x1 = x exerts on a wall, we have
F1 = (1+ ) q cos2(x0; x1)+1 n(1)e2(1) cos(x1; n1), (98)
i. e. its components at a point, located in the vicinities, are because according to the signature (+−−−), the spatial codifferent from those at the point of reflection to within only ordinate axes in the pseudo-Riemannian space are directed
D. Rabounski. A Theory of Gravity Like Electrodynamics
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July, 2005
opposite to the same axes xi in the tangential flat Euclidean
space.
We denote cos (x1; n1) = cos θ, where θ is the angle of
reflection. Assuming e(1) = 1, n(1) = 1, v(1) = v we obtain
the
field
density
q
=
μ 2πc
Ω2
1+
v2 4c2
, so that the wave pres-
sure FN = F1 cos θ normal to the wall surface is
FN = (1 +
v2 ) 1 + c2
q cos2 θ ,
(99)
which, for low rotational velocities gives∗
FN = (1 + ) q cos2 θ ,
q = μ Ω2. 2πc
(100)
Most of rotations we observe are slow. The maximum of
the known velocities is that for an electron in the 1st Bohr
orbit
(vb = 2.18×108 cm/sec).
Therefore
the
ratio
v2 c2
,
taking
reaches a maximum numerical value of only 5.3×10−5.
The presence of wave pressure in time density fields
provides a way of measuring the numerical value of the
energy-momentum constant μ, specific for such fields. For instance, a gyroscope, rotating around the axis x3 = z, will
be a source of circular waves of the field of time density
propagating in the xy-plane. In this case the chr.inv.-field
strength tensor Uik has the non-zero components U11, U12, U21. It is easy to calculate that the normal wave pressure of a circular wave will be different from the pressure of a plane
wave (99) in only higher order terms. The same situation applies for spherical waves†. Therefore the normal pressure
exerted by the waves on a wall orthogonal to the direction x1 = x, shall be
μ FN = 2πc (1 +
) Ω2
(101)
to within the higher order terms withheld. Rotations at 6×103
rpm (Ω = 100 rps) are achievable in modern gyroscopes, rotations in atoms are much greater, taking their maximum angular velocity to 4.1×1016 rps in the 1st Bohr orbit. A
torsion balance registers forces, values of which are about 10−5 dynes. Then in accordance with the formula (101), if the wave pressure in an experiment is FN ≈ 10−5 din/cm2, derived from atomic transformations, the constant’s numerical value will be in the order of μ ≈ 10−28 gramme/sec.
Of course this is a crude supposition, based on the pre-
cision limits of measurement. Anyhow, the exact numerical
value of the energy-momentum constant μ will be ascertained from special measurements with a torsion balance.
∗Formula (100) is the same as FN = (1+ ) q cos2 θ — the normal
pressure exerted by a plane electromagnetic wave in Minkowski space, (see
§47 in The Classical Theory of Fields [1]). So the wave pressure of a time
density field depends on the reflection coefficient 0
1 in the same
way as the pressure of electromagnetic waves.
†In a real experiment such a gyroscope, being an arbitrarily thin disc,
will be a source of spherical waves of a time density field which propagates
in all spatial directions. The waves will merely have a maximum amplitude
in the gyroscope’s rotation plane xy.
10 Physical conditions in atoms
So we have obtained formulae for chr.inv.-projections of the energy-momentum tensor of time density fields, which are physically observable characteristics of such fields — the energy density q (69), the momentum density Ji (70), and the strength tensor Uik (71).
The formulae must be valid everywhere, the inside of atoms included. At the same time, physical conditions in atoms are subject to Bohr’s quantum postulates. For an external observer, an atom can be represented as a tiny gyroscope, the rotations of which are ruled by the quantum laws. The quantised rotations of electrons are sources of a time density field, which shall be perceptible, because of the super-rapid angular velocities up to the maximum value in the 1st Bohr orbit Ωb = 4.1×1016 rps. This is a way of formulating the physical conditions under which a time density field exists in atoms.
Taking the above into account, we formulate the physical conditions with postulates, which result from the application of Bohr’s postulates to a time density field in atoms.
POSTULATE I A time density field in an atom remains unchanged in the absence of external influences. The atom radiates or absorbs waves of the time density field only in transitions of the electrons between their stationary orbits.
Naturally, when an atom is in a stable state, all its electrons are located in their orbits. Such a stable atom, having a set of quantum orbital angular velocities, must possess numerous quantum states of the time density field. The quantum states are set up with the second postulate‡.
POSTULATE II A time density field is quantised in atoms. Its energy density and the momentum density take quantum numerical values which, in accordance with the quantization of electron orbits, in n-th stationary orbit are
μ qn = 2πc
1
+
vn2 4c2
vn2 Rn2
,
(102)
‡To introduce the second postulate we assume a reference frame in
an atom, where an electron rotates around the nucleus at the angular
velocity Ω in the xy-plane. Then A12 = −A21 = −Ω, A13 = 0,
A23 = 0. So out of all components of Ω∗i only Ω∗3 is non-zero: Ω∗3 =
=
1 2
ε3mn
Amn
=
1 2
ε312A12 + ε321A21
=
ε312 √
A12
=
e√312
√h
A12
=−
√Ω h
and
Ω∗3
=
1 2
ε3mn Amn
=
ε312 A12
=
e312
h A12 =−
h Ω. In cal-
culating h = det hik , it should be noted that the components of
the space rotation linear velocity vi = Aik xk in this reference frame
are
v1 = −Ωy ,
v2 = Ωx,
v3 = 0.
We
obtain
h11
=
1+
1 c2
Ω2
y2,
h22
=
1+
1 c2
Ω2
x2,
h12
=−
1 c2
Ω2
xy
,
h33 = 1.
Then
h = det
hik
=
=
h11h22−(h12)2= 1 +
1 c2
Ω2(x2
+ y2).
In the 1st Bohr orbit we have
1 c2
Ω2 (x2
+
y2) =
1 c2
Ω2 R2
= 5.3×10−5,
so
we
can
set
h≈1
to
within
the higher order terms withheld. Harking back to the formulae for Ω∗3 and
Ω∗3, we see that the space rotates in atoms at a constant angular velocity
Ω∗3 = −Ω, Ω∗3 = −Ω, then in the assumed reference frame we have
AikAik = 2A12A12 = 2Ω∗3 Ω∗3 = 2Ω2,
and
also
F1
=
−
∂v1 ∂t
= Ω2 x,
F2
=−
∂v2 ∂t
= Ω2 y ,
F3
= 0,
which
is
taken
into
account
in
Postulate
II.
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D. Rabounski. A Theory of Gravity Like Electrodynamics
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PROGRESS IN PHYSICS
Volume 2
Jn =
(JiJ i)n
=
μ 2πc
Ω3n Rn
=
μ 2πc
vn3 Rn2
.
(103)
Calculating the field density in neighbouring levels n and
n+1, we take into account that the n-th orbital radius relates to the 1st Bohr radius as Rn = n2Rb. As a result we obtain
q = qn− qn+1 =
=
μ 2πc
Ω2b
11 n6 − (n+1)6
+
vb2 4c2
11 n8 − (n+1)8
,
(104)
so the difference between the field density in the neighbour levels is inversely proportional to n7, and n 1 gives
q
=
qn
− qn+1
≈
1 n7
3μ πc
Ω2b ,
(105)
and q → 0 for quantum numbers n → ∞.
Theoretically, the non-zero field density, q = 0, must
result in a flux of the field momentum (this flux is set up by
the
field
strength
tensor
Uik
=
1 3
q c2hik−βik).
So
an
electron,
moving in its orbit, should be radiating a momentum flux of
the time density field (waves of the field). Because of the
momentum loss in the radiation, the electron’s own angular
velocity would decrease, contradicting the experimental facts
on the stability of atoms in the absence of external influences.
To obviate this contradiction the third postulate is,
POSTULATE III An atom radiates a quantum portion of momentum flux of a time density field, when an electron transits from the n-th quantum level to the (n+1)-th level in the atom. When an electron transits from the (n+1)-th level to the n-th level, the atom absorbs the same portion of the momentum flux, which is
U 11= U1n1−U1n1+1=
=
μc 2π
Ω2b
11 n6 − (n+1)6
−
vb2 4c2
11 n8 − (n+1)8
.
(106)
We assume in this formula that the atom radiates/absorbs
a plane wave of a time density field, which travels along the x1 = x axis. Taking this formula with n 1, we have
U 11
=
U1n1
− U1n1+1
≈
1 n7
3μc π
Ω2b ,
(107)
which, for quantum numbers n → ∞, gives U 11 → 0. So for quantum numbers n 1 we have the ratio
U 11 = qc2.
(108)
In accordance with the correspondence principle, any result of quantum theory at high quantum numbers must coincide with the relevant classical results; any difference being imperceptible. We therefore take into consideration the formulae for q (69) and Uik (71) in atoms, obtained by the methods of the classical theory of fields, under the condition
h
≈
1.
As
a
result
we
get
the
formulae
q
=
μ 2πc
Ω2
1
+
v2 4c2
≈
μ 2πc
Ω2
and
Uik =
μc 2π
Ω2
h11 −
v2 2c2
+
v2 4c2
h11
≈
μc 2π
Ω2,
leading to the same relationship U11 = qc2 that quantum
theory has given (108). So the correspondence principle is
valid for time density fields in atoms.
Postulate III has two consequences:
CONSEQUENCE I An atom undergoing excitation radiates the momentum flux of a time density field, producing a positive wave pressure in the field.
Calculating this positive pressure, orthogonal to the surface of a wall (here θ is the angle of reflection, is the reflection coefficient) for quantum numbers n 1, we obtain
FN = (1 + ) q cos2 θ .
(109)
CONSEQUENCE II An atom undergoing relaxation absorbs the momentum flux of a time density field. In this case the wave pressure in a time density field near the atom becomes negative.
As a matter of fact, this negative pressure around a relaxing atom should be
FN = − (1 + ) q cos2 θ .
(110)
That is, in accordance with this theory, excitation of atoms causes radiation of waves of the time density field. An effect derived from the radiation should be the positive pressure of the waves. On the other hand, relaxing atoms, absorbing waves of the time density field, should be sources of negative wave pressure.
It is interesting that this effect is opposite to that which atoms produce in an electromagnetic field — it is well-known that relaxing atoms radiate electromagnetic waves, so they produce a positive wave pressure in an electromagnetic field.
The predicted repulsion/attraction produced by atomic processes, being outside the actions of electromagnetic or gravitational fields, are peculiarities of only the theory of the time density field herein. So the given conclusions open up wide possibilities for checking the whole theory in practice.
In particular, for instance, if a torsion balance registered the repulsing/attracting wave pressure FN derived from subatomic excitation/relaxation processes, we will have obtained the numerical value of the energy-momentum constant μ for time density fields. After substituting q (105) into the wave pressure FN, assuming cos θ = 1, we arrive at a formula for experimental calculations
μ
=
πc n7 3 Ω2b
FN (1 +
. )
(111)
A torsion balance registered such forces at ∼10−5 dynes
in prior experiments. The torsion balance had a 2-in long
nylon thread, 15 μm in diameter, and a 3-in long wooden
D. Rabounski. A Theory of Gravity Like Electrodynamics
27
Volume 2
PROGRESS IN PHYSICS
July, 2005
balance suspended in the ratio 8:1 of the length. The balance had a reflecting shield at the end of the long arm and a lead load on the short arm. The torsion balance was located inside a box isolated from air convection and light radiation. Chemical reactions of the opposite directions, processes of crystallization and dissolution were sources of a time density field acting on the torsion balance. Prof. Kyril Stanyukovich and Dr. Larissa Borissova assisted me in the experiments that were repeated a number of time during a period of 2 years in Moscow (Russia). The balance underwent deviations of up to 90◦ in directions predicted by this theory.
Even heating up bodies and cooling down bodies gave the same thermal influence, moving the balance in opposite directions, according to the theory, so the discovered phenomenon is outside thermal influences on torsion balance.
The techniques and measurements are very simply, and could therefore be reproduced in any physical laboratory. Anyway the experiments should be continued, with the aim of determining the exact numerical value of the energymomentum constant μ for time density fields through formula (111).
11 Conclusions
Let us collect the main results of this analysis.
By projecting an interval of four-dimensional coordinates
dxα onto the time line of an observer, who accompanies his
references (bi = 0), we obtain an interval of physical time
dτ
=
1 c
bαdxα
he observes. √
Observations
at
the
same
spatial
point give dτ = g00 dt, so the operator of projection on time
lines bα defines observable inhomogeneity of time references
in the accompanying reference frame.
So, observable inhomogeneities of time references can
be represented as a field of “density” of observable time τ . The projecting operator bα is the field “potential”, chr.inv.projections of which are ϕ = 1 and qi = 0.
The field tensor Fαβ = ∇α bβ −∇β bα for time density
fields was introduced as well as Maxwell’s electromagnetic
f=ie−ld2ctAeniskora.reItsdecrhirv.iendv.-fproromjecthtieongsraEviit=ati−onc1a2lFiinearntidal
Hik= force
and rotation of the space. We referred to the Ei and Hik
as the “electric” and “magnetic” observable components of
the time density field, respectively. We also introduced the field pseudotensor F ∗αβ, dual of the Fαβ, and also the field
invariants.
Equations of motion of a free mass-bearing particle, being
expressed through the Ei and Hik, group them into an acting
force of a form similar to the Lorentz force. In particular if the
particle moves only along time lines, it moves solely because
of the “magnetic” component Hik = 0 of a time density field. In other words, the space rotation Aik effectively “screws”
particles into the time lines. Because observable particles
with the whole spatial section move from past into future,
a “starting” non-holonomity, Aik = 0, will exist in our real
space that is a “primordial non-orthogonality” of the real
spatial section to the time lines. Other physical conditions
(gravitation, rotation, etc.) are only augmentations that in-
tensify or reduce this starting-rotation of the space.
A system of equations of a time density field consists of
Lorentz’s condition ∇σ bσ = 0, two groups of Maxwell-like
equations,
∇σF ασ =
4π c
jα
and
∇σF ∗ασ = 0,
and
the
con-
tinuity equation ∇σ jσ = 0, which define the main properties
of the field and its-inducing sources. All the equations have
been deduced here in chr.inv.-form.
The energy-momentum tensor T αβ we have deduced for
time density fields has the following observable projections: chr.inv.-scalar q of the field density; chr.inv.-vector Ji of the field momentum density, and chr.inv.-tensor U ik of the field
strengths. Their specific formulas define physical properties
of such fields:
1. A time density field is non-stationary distributed medium q = const, it becomes stationary, q = const, under stationary rotation, Aik = const, of the space and stationary gravitational inertial force Fi = const;
2. The field bears momentum (Ji = 0 in the general case), so it can transfer impulse;
3. In a rotating space, Aik = 0, the field is an emitting medium;
4. The field is viscous. The viscosity αik is anisotropic. The anisotropy increases with the space rotation speed;
5.
The
equation
of
state
of
the
field
is
p
=
1 3
q
c2,
so
the
field is like an ultrarelativistic gas: at positive density
the pressure is positive — the medium compresses.
For a plane wave of the field considered, we have concluded that waves of the time density fields are transverse. The wave pressure in the fields is derived from atomic and sub-atomic transformations mainly, because of huge rotational velocities. Exciting atoms produces a positive wave pressure in the time density field, while the wave pressure resulting from relaxing atoms is negative. This effect is opposite to that of the electromagnetic field — relaxing atoms radiate γ-quanta, producing a positive pressure of light waves.
Experimental tests have a basis in the predicted repulsion/ attraction, produced by sub-atomic processes, being outside of known effects of electromagnetic or gravitational fields, which are peculiarities only of this theory. A torsion balance registered such forces at ∼10−5 dynes in prior experiments. The registered repulsion/attraction is outside thermal effects on the torsion balance.
The results we have obtained in this study imply that even if inhomogeneity of time references is a tiny correction to ideal time, a field of the inhomogeneities that is a time density field, manifest as gravitational and inertial forces, has a more fundamental effect on observable phenomena, than those previouly supposed.
28
D. Rabounski. A Theory of Gravity Like Electrodynamics
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PROGRESS IN PHYSICS
Volume 2
Acknowledgements
References
I began this research in 1984 when I, a young scientist in those years, started my scientific studies under the direction of Prof. Kyril Stanyukovich, who in the 1940’s was already a well-known expert in the General Theory of Relativity. At that time we found an article of 1971, wherein Prof. Nikolai Kozyrev (1908–1983) reported on his experiments with a torsion balance [7].
His high precision torsion balance, having unequal arms, registered weak forces of attraction/repulsion at 10−5– 10−6 dynes; the forces derived from creative/destructive processes in his laboratory. Proceeding from his considerations, redistribution of energy should produce a non-uniformity of time that generates a force field of attraction or repulsion depending on the creative/destructive direction of the redistributing energy process.
Kozyrev did not put this idea into any mathematical form. So his propositions, having a purely phenomenological a basis, remained without a theory. Kozyrev was the famous experimental physicist and astronomer of the 20th century who discovered volcanic activity in the Moon (1958), the atmosphere of Mercury (1963), and many other phenomena. His authority in physical experiment was beyond any doubt.
In those years Stanyukovich was head of the Department of Fundamental Theoretical Metrology, Surface and Vacuum Scientific Centre, State Committee for Standards (Moscow, Russia). Dr. Larissa Borissova worked in his Department. We were all close friends, despite our age difference. We had a good time working with Stanyukovich, who was friendly in his conversations on different scientific problems.
I was interested by Kozyrev’s experiments with the torsion balance, therefore Stanyukovich proposed me that I investigate them, meaning that it would be helpful for me to build the required theory. During 1984–1985 I twice visited the laboratory of late Kozyrev in Pulkovo Astronomical Observatory, near Leningrad. Then, in Moscow, we made a copy of his torsion balance, and modified it to make it more sensitive.
Stanyukovich was right — the experiments were approbated with strictly positive result. We repeated the experiments for some of his colleagues, in particular for Dr. Vitaly Schelest.
More than 15 years were required for the development of this theory. I finished the whole theory only in 2004. Stanyukovich had died; only Larissa Borissova and I remain. Our years of friendly conversations with Stanyukovich, and his patient personal instructions reached us by all his experience in theoretical physics. Actually, we are beholden to him. Today I would like to do only one thing — satisfy the hopes of my teacher.
Finally I am grateful to my colleagues: Stephen J. Crothers for some editing this paper, and Larissa Borissova who checked all my calculations that here.
1. Landau L. D. and Lifshitz E. M. The classical theory of fields. GITTL, Moscow, 1939 (ref. with the 4th final exp. edition, Butterworth–Heinemann, 1980, 428 pages).
2. Zelmanov A. L. Chronometric invariants. Dissertation, 1944. First published: CERN, EXT-2004-117, 236 pages.
3. Zelmanov A. L. Chronometric invariants and co-moving coordinates in the general relativity theory. Doklady Acad. Nauk USSR, 1956, v. 107 (6), 815–818.
4. Zelmanov A. L. Orthometric form of monad formalism and its relations to chronometric and kinemetric invariants. Doklady Acad. Nauk USSR, 1976, v. 227 (1), 78–81.
5. Borissova L. B. and Rabounski D. D. Fields, vacuum, and the mirror Universe. Editorial URSS, Moscow, 2001, 272 pages (the 2nd revised ed.: CERN, EXT-2003-025).
6. Rabounski D. D. The new aspects of General Relativity. CERN, EXT-2004-025, 117 pages.
7. Kozyrev N. A. On the possibility of experimental investigation of the properties of time. Time in Science and Philosophy, Academia, Prague, 1971, 111–132.
8. Kozyrev N. A. Physical peculiarities of the components of double stars. Colloque “On the Evolution of Double Stars”, Comptes rendus, Communications du Observatoire Royal de Belgique, ser. B, no. 17, Bruxelles, 1967, 197–202.
9. Del Prado J. and Pavlov N. V. Private communications to A. L. Zelmanov, Moscow, 1968–1969.
10. Zelmanov A. L. To relativistic theory of anisotropic inhomogeneous Universe. Proceedings of the 6th Soviet Conference on Cosmogony, Nauka, Moscow, 1959, 144–174.
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Gravitational Waves and Gravitational Inertial Waves in the General Theory of Relativity: A Theory and Experiments
Larissa Borissova
E-mail: lborissova@yahoo.com
This research shows that gravitational waves and gravitational inertial waves are linked to a special structure of the Riemann-Christoffel curvature tensor. Proceeding from this a classification of the waves is given, according to Petrov’s classification of Einstein spaces and gravitational fields located therein. The world-lines deviation equation for two free particles (the Synge equation) is deduced and that for two forceinteracting particles (the Synge-Weber equation) in the terms of chronometric invariants — physical observable quantities in the General Theory of Relativity. The main result drawn from the deduced equations is that in the field of a falling gravitational wave there are not only spatial deviations between the particles but also deviations in the time flow. Therefore an effect from a falling gravitational wave can manifest only if the particles located on the neighbouring world-lines (both geodesics and nongeodesics) are in motion at the initial moment of time: gravitational waves can act only on moving neighbouring particles. This effect is purely parametric, not of a resonance kind. Neither free-mass detectors nor solid-body detectors (the Weber pigs) used in current experiments can register gravitational waves, because the experimental statement (freezing the pigs etc.) forces the particles of which they consist to be at rest. In aiming to detect gravitational waves other devices should be employed, where neighbouring particles are in relative motion at high speeds. Such a device could, for instance, consist of two parallel laser beams.
Contents
1. Introduction and advanced results . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2. Theoretical bases for the possibility of registering gravi-
tational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3. Invariant criteria for gravitational waves and their link to
Petrov’s classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4. Basics of the theory of physical observable quantities . . . . . . . . 40 5. Gravitational inertial waves and their link to the chrono-
metrically invariant representation of Petrov’s classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6. Wave properties of Einstein’s equations . . . . . . . . . . . . . . . . . . . . . 47 7. Expressing the Synge-Weber equation (the world-lines deviation equation) in the terms of physical observable quantities and its exact solutions . . . . . . . . . . . . . . . . . . . . . . . 50 8. Criticism of Weber’s conclusions on the possibility of detecting gravitational waves by solid-body detectors of the resonance kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 9. Criticism of Weber’s theory for detecting gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1 Introduction and advanced results
The fact that gravitational waves have not yet been discovered has attracted the attention of experimental physicists over the last decade. Initial interest in gravitational waves arose in 1968–1971 when Joseph Weber, professor at Maryland University (USA), carried out his first experiments with
gravitational antennae. He registered weak signals, in common with all his independent antennae, which were separated by up to 1000 km [1]. He supposed that some processes at the centre of the Galaxy were the origin of the registered signals. However such an interpretation had a significant drawback: the frequency of the observed signals (more than 5 per month) meant that the energy spent by the signal’s source, located at the centre of the Galaxy, should be more than M c2×103 per annum (M is the mass of the Sun, c is the velocity of light). This energy expenditure is a fantastic value, if we accept today’s bounds on the age of the Galaxy [2, 3, 4].
In 1972 the experiments were approbated by the a common group of researchers working at Moscow University and the Institute of Space Research (Moscow, Russia). Their antennae were similar to Weber’s antennae, but they were separated by 20 km. The registering system in their antennae was better than that for the Weber detectors, making the whole system more sensitive. But. . . 20 days of observations gave no signals that would be more than noise [5].
The experiments were continued in 1973–1974 at laboratories in Rochester University, Bell Company, and IBM in USA [6, 7], Frascati, Mu¨nich, Meudon (Italy, Germany, France) [8], Glasgow University (Scotland) [9] and other laboratories around the world. The experimental systems used in these attempts were more sensitive than those of the Weber detectors, but none registered the Weber effect.
Because theoretical considerations showed that huge
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gravitational waves should be accompanied by other radiations, the researchers conducted a search for radio outbreaks [10] and neutron outbreaks [11]. The result was negative. At the same time it was found that Weber’s registered effects were related to solar and geomagnetic activities, and also to outbreaks of space beams [12, 13].
The search for gravitational waves has continued. Higher precision and more sensitive modifications of the Weber antennae (solid detectors of the resonance kind) are used in this search. But even the second generation of Weber detectors have not led scientists to the expected results. Besides gravitational antennae of the Weber kind, there are antennae based on free masses. Such detectors consist of two freely suspended masses located far from one another, within the visibility of a laser range-finder. Supposed deviations of the masses, derived from a gravitational wave, should be registered by the laser beam.
So gravitational waves have not been discovered in experiments. Nonetheless it is accepted by most physicists that the discovery of gravitational waves should be one of the main verifications of the General Theory of Relativity. The main arguments in support of this thesis are:
1. Gravitational fields bear an energy described by the energy-momentum pseudotensor [14, 15];
2. A linearized form of the equations of Einstein’s equations permits a solution describing weak plane gravitational waves, which are transverse;
3. An energy flux, radiated by gravitational waves, can be calculated through the energy-momentum pseudotensor of the field [14, 15];
4. Such waves, because of their physical nature, are derived from instability of components of the fundamental metric tensor (this tensor plays the part of the four-dimensional gravitational potential).
These theoretical considerations were placed into the foreground of the theory for detecting gravitational waves, the main part in the theory being played by the theoretical works of Joseph Weber, the pioneer and famous expert in the detection of gravitational waves [16]. His main theoretical claim was that he deduced equations of deviation of worldlines — equations that describe relative oscillations of two non-free particles in a gravitational field, particles which are connected by a force of non-gravitational nature. Equations of deviation of geodesic lines, describing relative oscillations of two free particles, was obtained earlier by Synge [17]. In general, relative oscillations of test-particles, both free particles and linked (interacting) particles, are derived from the space curvature∗, given by the Riemann-Christoffel fourdimensional tensor. Equality to zero of all its components in an area is the necessary and sufficient condition for the four-
∗As it is well-known, the space curvature is linked to the gravitational field by the Einstein equations.
dimensional space (space-time) to be flat in the area under consideration, so no gravitational fields exist in the area.
Thus the Synge-Weber equation provides a means for the calculation of the relative oscillations of test-particles, derived from the presence of the space curvature (gravitational fields). Weber proposed a gravitational wave detector consisting of two particles connected by a spring that imitates a non-gravitational interaction between them. In his analysis he made the substantial supposition that the under action of gravitational waves the model will behave like a harmonic oscillator where the forcing power is in the RiemannChristoffel curvature tensor. Weber made calculations and theoretical propositions for the behaviour of this model. This model is known as the quadrupole mass-detector [17].
The Weber calculations served as theoretical grounds for creating a whole industry, the main task of which has been the building of resonance type detectors, known as the Weber detectors (the Weber pigs). It is supposed that the body of a Weber detector, having cylindrical form, should be deformed under the action of a gravitational wave. This deformation should lead to a piezoelectric effect. Thus, oscillations of atoms in the cylindrical pig, resulting from a gravitational wave, could be registered. To amplify the effect in measurements, the level of noise was lowered by cooling the cylinder pigs down to temperature close to 0 K.
But the fact that gravitational waves have not yet been discovered does not imply that the waves do not exist in Nature. The corner-stone of this problem is that the Weber theory of detection is linked to a search for waves of only a specific kind — weak transverse waves of the space deformation (weak deformation transverse waves). However, besides the Weber theory, there is the theory of strong gravitational waves, which is independent of the Weber theory. Studies of the theory of strong gravitational waves reached its peak in the 1950’s.
Generally speaking, all theoretical studies of gravitational waves can be split into three main groups:
1. The first group consists of studies whose task is to give an invariant definition for gravitational waves. These are studies made by Pirani [18, 19], Lichnerowicz [20, 21], Bel [22, 23, 24], Debever [25, 26, 27], He´ly [28], Trautman [29], Bondi [19, 30], and others.
2. The second group joins studies around a search for such solutions to the Einstein equations for gravitational fields, which, proceeding from physical considerations, could describe gravitational radiations. These are studies made by Bondi [31], Einstein and Rosen [32, 33], Peres [34], Takeno [35, 36], Petrov [37], Kompaneetz [38], Robinson and Trautman [39], and others.
3. The task of works related to the third group is to study gravitational inertial waves, covariant with respect of transformations of spatial coordinates and also invariant with respect of transformations of time [40, 41].
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The studies are based on the theory of physically observable quantities — Zelmanov’s theory of chronometric invariants [42, 43].
Most criteria for gravitational waves are linked to the structure of the Riemann-Christoffel curvature tensor, hence one assumes space curvature the source of such waves.
Besides these three main considerations, the theory of gravitational waves is directly linked to the algebraical classification of spaces given by Petrov [37] (Petrov classification), according to which three kinds for spaces (gravitational fields) exist. They are dependent on the structure of the Riemann-Christoffel curvature tensor:
1. Fields of gravitation of the 1st kind are derived from island distributions of masses. An instance of such a field is the that of a spherical distribution of matter (a spherical mass island) described by the Schwarzschild metric [44]. Spaces containing such fields approach a flat space at an infinite distance from the gravitating island.
2–3. Spaces containing gravitational fields of the 2nd and 3rd kinds cannot asymptotically approach a flat space even, if they are empty. Such spaces can be curved themselves, independently of the presence of gravitating matter. Such fields satisfy most of the invariant definitions given to gravitational waves [40, 45, 46, 47].
It should be noted that the well-known solution that gives weak plane gravitational waves [14, 15] is related to fields of the sub-kind N of the 2nd kind by Petrov’s classification (see p. 38). Hence the theory of weak plane gravitational waves is a particular case of the theory of strong gravitational waves. But, bseides this well-studied particular case, the theory of strong gravitational waves contains many other approaches to the problem and give other methods for the detection of gravitational waves, different to the Weber detectors in principle (see [48], for instance).
We need to look at the gravitational wave problem from another viewpoint, by studying other cases of the theory of strong gravitational waves not considered before. Exploring such new approaches to the theory of gravitational waves is the main task of this research.
At the present time there are many solutions of the gravitational wave problem, but none of them are satisfactory. The principal objective of this research is to extract that which is common to every one of the theoretical approaches.
We will see further that this analysis shows, according to most definitions given for gravitational waves, that a gravitational field is assumed a wave field if the space where it is located has the specific curvature described by numerous particular cases of the Riemann-Christoffel curvature tensor.
Note that we mean the Riemannian (four-dimensional) curvature, whose formula contains accelerations, rotations, and deformations of the observer’s reference space. Analysis of most wave solutions to the gravitational field equations
(Einstein’s equations) shows that such gravitational waves have a deformation nature — they are waves of the space deformations. The true nature of gravitational waves can be found by employing the mathematical methods of chronometric invariants (the theory of physically observable quantities in the General Theory of Relativity), which show that the space deformation (non-stationarity of the spatial observable metric) consists of two factors:
1. Changes of the observer’s scale of distance with time (deformations of the 1st kind);
2. Possible vortical properties of the acting gravitational inertial force field (deformations of the 2nd kind).
Waves of the space deformations (of the 1st or 2nd kind) underlie the detection attempts of the experimental physicists.
Because such gravitational waves are expected to be weak, one usually uses the metric for weak plane gravitational waves of the 1st kind (which are derived from changes of the distance scale with time).
The basis for all the experiments is the Synge-Weber equation (the world-lines deviation equation), which sets up a relation between relative oscillations of test-particles and the Riemann-Christoffel curvature tensor. Unfortunately Joseph Weber himself gave only a rough analysis of his equation, aiming to describe the behaviour of a quadrupole mass-detector in the field of weak plane gravitational waves. In his analysis he assumed (without substantial reasons) that space deformation waves of the 1st kind must produce a resonance effect in a quadrupole mass-detector.
However, it would be more logical way, making no assumptions or propositions, to solve the Synge-Weber equation aiming exactly. Weber did not do this, limiting himself instead to only rough bounds on possible solutions.
In this research we obtain exact solutions to the SyngeWeber equation in the fields of weak plane gravitational waves. As a result we conclude that the expected relative oscillations of test-particles, which originate in the space deformation waves of the 1st kind, are not of the resonance kind as Weber alleged from his analysis, but are instead parametric oscillations.
This deviation between our conclusion and Weber’s false conclusion is very important, because oscillations of a parametric kind appear only if test-particles are moving∗, whilst in Weber’s statement of the experiment the particles are at rest in the observer’s laboratory reference frame. All activities in search of gravitational waves using the Weber pigs are concentrated around attempts to isolate the bulk pigs from external affects — experimental physicists place them in mines in the depths of mountains and cool them to 2 K,
∗In other words, if their velocities are different from zero. Parametric oscillations merely add their effect to the relative motion of the moving particles. Parametric oscillations cannot be excited in a system of particles which are at rest with respect to each other and the observer.
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so particles of matter in the pigs can be assumed at rest with respect to one another and to the observer. At present dozens of Weber pigs are used in such experiments all around the world. Experimental physicists spend billions and billions of dollars yearly on their experiments with the Weber pigs.
Parametric oscillations do not appear in resting particles, so the space deformation waves of the 1st kind can not excite parametric oscillations in the Weber pigs. Therefore the gravitational waves expected by scientists cannot be registered by solid-body detectors of the resonance kind (the Weber pigs).
Even so, everything said so far does not mean rejection of the experimental search for gravitational waves. We merely need to look at the problem from another viewpoint. We need to remember the fact that our world is not a three-dimensional space, but a four-dimensional space-time. For this reason we need to turn our attention to the fact that relative deviations of particles in the field of gravitational waves have both spatial components and a time component. Therefore it would be reasonable to propose an experiment by which, having a detector under the influence of gravitational waves, we could register both relative displacements of particles in the detector and also corrections to time flow in the detector due to the waves (the second task is much easier from the technical viewpoint).
Here are two aspects for consideration. First, in solving the Synge-Weber equations we must take its time component into account; we must not neglect the time component. Second, we should turn our attention to possible experimental effects derived from gravitational waves of the 2nd (deformation) kind, which appear if the acting gravitational inertial force field is vortical, as it will be shown further that in this case there is a field of the space rotation (stationary or non-stationary)∗. Such experiments, aiming to register gravitational waves of the 2nd kind are progressive because they are much simpler and cheaper than the search for waves of the 1st kind.
2 Theoretical bases for the possibility of registering gravitational waves
Gravitational waves were already predicted by Einstein [37], but what space objects could be sources of the waves is not a trivial problem. Some link the possibility of gravitational radiations to clusters of black holes. Others await powerful gravitational radiations from super-dense compact stars of radii close to their gravitational radii† r ∼ rg. Although the
∗There are well-known Hafele-Keating experiments concerned with displacing standard clocks around the terrestrial globe, where rotation of the Earth space sensibly changes the measured time flow [49, 50, 51, 52].
†According to today’s mainstream concepts, the gravitational radius rg of an object is that minimal distance from its centre to its surface, starting from which this object is in a special state — collapse. One means that any object going into collapse becomes a “black hole”. From the purely mathematical viewpoint, under collapse, the potential w of the gravitational field of the object merely reaches its upper ultimate numerical value w = c2.
“black hole solution”, being under substantial criticism from the purely mathematical viewpoint [53, 54, 55], makes objects like black holes very doubtful, the existence of superdense neutron stars is outside of doubt between astronomers. Gravitational waves at frequencies of 102–104 Hz should also be radiated in super-nova explosions by explosion of their super-dense remains [56].
The search for gravitational waves, beginning with Weber’s observations of 1968–1971, is realized by using gravitational antennae, the most promising of which are:
1. Solid-body detectors (the Weber cylinder pigs);
2. Antennae built on free masses.
A solid-body detector of the Weber kind is a massive cylindrical pig of 1–3 metres in length, made with high precision. This experiment supposes that gravitational waves are waves of the space deformation. For this reason the waves cause a piezoelectic effect in the pig, one consequence of which is mechanical oscillations at low frequencies that can be registered in the experiment. It is supposed that such oscillations have a resonance nature. An immediate problem is that such resonance in massive pigs can be caused by very different external processes, not only waves of the space deformation. To remove other effects, experimental physicists locate the pigs in deep tunnels in mountains and cool the pigs down to temperature close to 0 K.
An antenna of the second kind consists of two masses, separated by Δl ∼ 103–104 metres, and a laser range-finder which should register small changes of Δl. Both masses are freely suspended. This experiment supposes that waves of the space deformation should change the distance between the free masses, and should be registered by the laser rangefinder. It is possible to use two satellites located in the same orbit near the Earth, having a range-finder in each of the satellites. Such satellites, being in free fall along the orbit, should be an ideal system for measurements, if it were not for effects due to the terrestrial globe. In practice it would be very difficult to divorce the effect derived from waves of the space deformation (supposed gravitational waves) and many other factors derived from the inhomogeneity of the Earth’s gravitational field (purely geophysical factors).
The mathematical model for such an antenna consists of two free test-particles moving on neighbouring geodesic lines located infinitely close to one another. The mathematical model for a solid-body detector (a Weber pig) consists of two test-masses connected by a spring that gives a model for elastic interactions inside a real cylindrical pig, in which changes reveal the presence of a wave of the space deformation.
From the theoretical perspective, we see that the possibi-
lity of registering waves of the space deformation (supposed gravitational waves) is based on the supposition that particles which encounter such a wave should be set into relative oscillations, the origin of which is the space curvature. The
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strong solution for this problem had been given by Synge for
free particles [17]. He considered a two-parameter family of geodesic lines xα = xα(s, v), where s is a parameter
along the geodesics, v is a parameter along the direction orthogonal to the geodesics (it is taken in the plane normal
to the geodesics). Along each geodesic line v = const. He introduced two vectors
U α = ∂xα , ∂s
V α = ∂xα , ∂v
(2.1)
where α = 0, 1, 2, 3 denotes four-dimensional (space-time) indexes. The vectors satisfy the condition
DU α DV α
=
,
∂v
∂s
(2.2)
(where D is the absolute derivative operator) that can be easy verified by checking the calculation. The parameter v is different for neighbouring geodesics; the difference is dv. Therefore, studying relative displacements of two geodesics Γ(v) and Γ(v + dv), we shall study the vector of their infinitesimal relative displacement
ηα = ∂xα dv = V αdv . ∂v
(2.3)
The deviation of the geodesic line Γ(v + dv) from the geodesic line Γ(v) can be found by solving the equation [17]
D2V α D DV α D DU α
ds2
= ds ds
= ds dv
=
=
D dv
DU α ds
+
Rα∙ β∙γ∙ ∙δ U β U δ V γ ,
(2.4)
where Rα∙ β∙γ∙ ∙δ is the Riemann-Christoffel curvature tensor. This equality has been obtained using the relation [17]
D2V α dsdv
−
D2V α dvds
= Rα∙ β∙γ∙ ∙δ U β U δ V γ .
(2.5)
For two neighbour geodesic lines, the following relation is obviously true
DU α ds
=
dU α ds
+ Γαμν U μU ν
=
0,
(2.6)
where Γαβγ are Christoffel’s symbols of the 2nd kind. Then (2.4) takes the form
The quantity UαU α = gαβU αU β takes the numerical
value +1 for non-isotropic geodesics (substantial particles)
or 0 for isotropic geodesics (massless light-like particles).
Therefore
UαV α = const .
(2.10)
In the particular case where the vectors Uα and V α are orthogonal to each to other at a point, where UαV α is true, the orthogonality remains true everywhere along the Γ(v).
Thus relative accelerations of free test-particles are caused by the presence the space curvature (Rα∙ β∙γ∙ ∙δ = 0), and linear velocities of the particles are determined by the
geodesic equations (2.6).
Relative accelerations of test-particles, connected by a force Φα of non-gravitational nature, are determined by the Synge-Weber equation [16]. The Synge-Weber equation is
the generalization of equation (2.8) for that case where the
particles, each having the rest-mass m0, are moved along non-geodesic world-lines, determined by the equation
DU α ds
=
dU α ds
+ Γαμν U μU ν
=
Φα m0c2
.
(2.11)
In this case the world-lines deviation equation takes the
form
D2ηα ds2
+
Rα∙ β∙γ∙ ∙δ U β U δ ηγ
=
1 m0c2
DΦα dv
dv ,
(2.12)
which describes relative accelerations of the interacting masses. In this case
∂ ∂s
(Uα
ηα
)
=
1 m0c2
Φαηα,
(2.13)
so the angle between the vectors U α and ηα does not remain constant for the interacting particles.
Equations (2.8) and (2.12) describe relative accelerations of free particles and interacting particles, respectively. Then, to obtain formulae for the velocity U α it is necessarily to solve the geodesic equations for free particles (2.6) and the world-line equations for interacting particles (2.11). We consider the equations (2.8) and (2.12) as a mathematical base, with which we aim to calculate gravitational wave detectors: (1) antennae built on free particles, and (2) solidbody detectors of the resonance kind (the Weber detectors).
D2V α ds2
+ Rα∙ β∙γ∙ ∙δ U β U δ V γ
=
0,
or equivalently,
D2ηα ds2
+ Rα∙ β∙γ∙ ∙δ U β U δ ηγ
=
0.
(2.7) (2.8)
It can be shown [17] that,
∂ ∂s
(UαV
α)
=
Uα
DV α ds
=
Uα
DU α dv
=
1 2
∂ ∂v
(UαU α)
.
(2.9)
3 Invariant criteria for gravitational waves and their link to Petrov’s classification
From the discussion in the previous paragraphs, one concludes that a physical factor enforcing relative displacements of test-particles (both free particles and interacting particles) is the space curvature — a gravitational field wherein the particles are located.
Here the next question arises. How well justified is the statement of the gravitational wave problem?
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Generally speaking, in the General Theory of Relativity,
there is a problem in describing gravitational waves in a
mathematically correct way. This is a purely mathematical
problem, not solved until now, because of numerous diffic-
ulties. In particular, the General Theory of Relativity does
not contain a satisfactory general covariant definition for
the energy of gravitational fields. This difficulty gives no
possibility of describing gravitational waves as traveling
energy of gravitational fields.
The next difficulty is that when one attempts to solve
the gravitational wave problem using the classical theory
of differential equations, he sees that the gravitational field
equations (the Einstein equations) are a system of 10 non-
linear equations of the 2nd order written with partial de-
rivatives. No universal boundary conditions exist for such
equations.
The gravitational field equations (the Einstein equations)
are
1
Rαβ − 2 gαβ R = −κ Tαβ + λ gαβ ,
(3.1)
where Rαβ = R∙σα∙σ∙ ∙β is Ricci’s tensor, R = gαβRαβ is the
scalar
curvature,
κ
=
8πG c2
is
Einstein’s
constant
for
gravi-
tational fields, G is Gauss’ constant of gravitation, λ is the
cosmological constant (λ-term).
When studying gravitational waves, one assumes λ = 0.
Sometimes one uses a particular case of the Einstein equa-
tions (3.1)
Rαβ = κ gαβ ,
(3.2)
in which case the space, where the gravitational field is located, is called an Einstein space. If κ = 0, we have an empty space (without gravitating matter). But even in empty spaces (κ = 0) gravitational fields can exist, if the spaces are of the 2nd and 3rd kinds by Petrov’s classification.
In accordance with the classical theory of differential equations, those gravitational fields that describe gravitational waves are determined by solutions of the Einstein equations with initial conditions located in a characteristic surface. A wave is a Hadamard break in the initial characteristic surface; such a surface is known as the wave front. The wave front is determined as the characteristic isotropic surface S {Φ(xα) = 0} for the Einstein equations. Here the scalar function Φ satisfies the eikonal equation [20, 21]
gαβ ∇α∇β = 0 ,
(3.3)
where ∇α denotes covariant differentiation with respect to Riemannian coherence with the metric gαβ. The trajectories along which gravitational waves travel (gravitational rays) are bicharacteristics of the field equations, having the form
dxα dτ
=
gασ ∇σΦ ,
(3.4)
where τ is a parameter along lines of the geodesic family.
But the general solution of the Einstein equations with initial conditions in the hypersurface is unknown. For this reason the next problem arises: it is necessary to formulate an effective criterion which could determine solutions to the Einstein equations with initial conditions in the characteristic hypersurface.
There is another difficulty: there is no general covariant d’Alembertian which, being in its clear form, could be included into the Einstein equations.
Therefore, solving the gravitational wave problem reduces to the problem of formulating an invariant criterion which could determine this family of the field equations as wave equations.
Following this approach, analogous the classical theory of differential equations, we encounter an essential problem. Are functions gαβ(xσ) smooth when we set up the Cauchy problem for the Einstein equation? A gravitational wave is interpreted as Hadamard break for the curvature tensor field in the initial characteristic hypersurface. The curvature tensor field permits a Hadamard break only if the functions gαβ(xσ) permit breaks in their first derivatives. In accordance with Hadamard himself [20], the second derivatives of gαβ can have a break in a surface S {Φ(xα) = 0}
[∂ρσ gαβ] = aαβ lρ lσ , (lα ≡ ∂αΦ) (3.5)
only if a Hadamard break in the curvature tensor field [Rαβγδ] satisfies the equations [21]
lλ[Rμαβν ] + lα[Rμβλν ] + lβ [Rμλαν ] = 0 .
(3.6)
Proceeding from such an interpretation of the characteristic hypersurface for the Einstein equations, and also supposing that a break [Rαβγδ] in the curvature tensor Rαβγδ located in the front of a gravitational wave is proportional to the tensor itself, Lichnerowitz [20, 21] formulated this criterion for gravitational waves:
Lichnerowitz’ criterion The space curvature Rαβγδ = 0 determines the state of “full gravitational radiations”, only if there is a vector lα = 0 satisfying the equations
lμRμαβν = 0 , lλRμαβν + lαRμβλν + lβ Rμλαν = 0 ,
(3.7)
and thus the vector lα is isotropic (lαlα = 0). If Rαβ = 0 (the space is free of masses, so it is empty), the equations (3.7) determine the state of “clear gravitational radiations”.
There is also Zelmanov’s invariant criterion for gravitational waves [40]∗, it is linked to the Lichnerowitz criterion.
∗This criterion is named for Abraham Zelmanov, although it had been published by Zakharov [40]. This happened because Zelmanov gave many of his unpublished results, his unpublished criterion included, to Zakharov, who completed his dissertation under Zelmanov’s leadership at that time.
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Zelmanov proceeded from the general covariant generalization given for the d’Alembert wave operator
σ σ
≡
gρσ ∇ρ∇σ
.
(3.8)
Zelmanov’s criterion The space determines the state of gravitational radiations, only if the curvature tensor: (a) is not a covariant constant quantity (∇σRμαβγ = 0); (b) satisfies the general covariant condition
σ σ
Rμαβν
=
0.
(3.9)
Thus, as it was shown in [40], any empty space that satisfies the Zelmanov criterion also satisfies the Lichnerowitz criterion. On the other hand, any empty space that satisfies the Lichnerowitz criterion (excluding that trivial case where ∇σRμαβγ = 0) also satisfies the Zelmanov criterion.
There are also other criteria for gravitational waves, introduced by Bel, Pirani, Debever, Maldybaeva and others [58]. Each of the criteria has its own advantages and drawbacks, therefore none of the criteria can be considered as the final solution of this problem. Consequently, it would be a good idea to consider those characteristics of gravitational wave fields which are common to most of the criteria. Such an integrating factor is Petrov’s classification — the algebraic classification of Einstein spaces given by Petrov [37], in the frame of which those gravitational fields that satisfy the condition (3.2) are classified by their relation to the algebraic structure of the Riemann-Christoffel curvature tensor.
As is well known, the components of the RiemannChristoffel tensor satisfy the identities
gαβ, can be sign-alternating, having a signature dependent
on the signature of the gαβ. So, for Minkowski’s signature
(+−−−), the signature of the gab is (+++−−−).
Mapping the curvature tensor Rαβγδ onto the metric
bivector space RN , we obtain the symmetric tensor Rab
(a, b = 1, 2, . . . N ) which can be associated with a lambda-
matrix
(Rab − Λ gab) .
(3.12)
Solving the classic problem of linear algebra (reducing the lambda-matrix to its canonical form along a real distance), we can find a classificaton for Vn under a given n. Here the specific kind of an Einstein space we are considering is set up by a characteristic of the lambda-matrix. This kind remains unchanged in that area where this characteristic remains unchanged.
Bases of elementary divisors of the lambda-matrix for any Vn have an ordinary geometric meaning as stationary curvatures. Naturally, the Riemannian curvature Vn in a twodimensional direction is determined by an ordinary (singlesheet) bivector V αβ = V(α1)V(β2), of the form
K
=
Rαβγδ V αβ V γδ gαβγδ V αβ V γδ
.
(3.13)
If V αβ is not ordinary, the invariant K is known as the bivector curvature in the given vector’s direction. Mapping K onto the bivector space, we obtain
K = RabV aV b , gabV aV b
a, b = 1, 2, . . . N.
(3.14)
Rαβγδ =−Rβαγδ =−Rαβδγ = Rγδαβ , Rα[βγδ] = 0 . (3.10)
Because of (3.10), the curvature tensor is related to tensors of a special family, known as bitensors. They satisfy two conditions:
1. Their covariant and contravariant valencies are even;
Ultimate numerical values of the K are known as stationary curvatures taken at a given point, and the vectors V a corresponding to the ultimate values are known as stationary
not simple bivectors. In this case
V αβ = V(α1)V(β2) ,
(3.15)
2. Both covariant and contravariant indices of the tensors
are split into pairs and inside each pair the tensor
Rαβγδ is antisymmetric.
A set of tensor fields located in an n-dimensional Rie-
mannian space is known as a bivector set, and its represent-
ation at a point is known as a local bivector set. Every anti-
symmetric pair of indices αβ is denoted by a common index
a,
and
the
number
of
the
common
indices
is
N
=
n
(n − 2
1)
.
It is evident that if n = 4 we have N = 6. Hence a bitensor
Rαβγδ → Rab, located in a four-dimensional space, maps itself into a six-dimensional bivector space. It can be metrised
by introducing the specific metric tensor
gab → gαβγδ ≡ gαγ gβδ − gαδ gβγ .
(3.11)
The tensor gab (a, b = 1, 2, . . . N ) is symmetric and nondegenerate. The metric gab, given for the sign-alternating
so the stationary curvature coincides with the Riemannian
curvature Vn in the given two-dimension direction. The problem of finding the ultimate values of K is the
same as finding those vectors V a where the K takes the ultimate values, that is, the same as finding undoubtedly
stationary directions. The necessary and sufficient condition of stationary state of the V a is
∂ ∂V a K = 0 .
(3.16)
The problem of finding the stationary curvatures for Einstein spaces had been solved by Petrov [40]. If the space signature is sign-alternating, generally speaking, the stationary curvatures are complex as well as the stationary bivectors relating to them in the Vn.
For four-dimensional Einstein spaces with Minkowski signature, we have the following theorem [40]:
36
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THEOREM Given an ortho-frame gαβ = {+1, −1, −1, −1}, there is a symmetric paired matrix (Rab)
MN Rab = N −M ,
(3.17)
where M and N are two symmetric square matrices of the 3rd order, whose components satisfy the relationships
m11 +m22 +m33 = −κ , n11 +n22 +n33 = 0 . (3.18)
After transformations, the lambda-matrix (Rab − Λ gab) where gαβ = {+1, +1, +1, −1, −1, −1} takes the form
α1
M = 0
0 
β1 N = 0
0
0
0
α2 +1 0  ,
0 α2 −1
 00
β2 1  ,
1 β2
(3.21)
where α1 + 2α2 = −κ , β1 + 2β2 = 0 (so in this case there are 2 independent parameters determining the space structure by an invariant form),
The 3rd Kind
(Rab − Λ gab) =
M + iN + Λ ε
0
=
≡
0
M − iN + Λ ε
(3.19)
Q(Λ) 0
≡
0 Qˉ(Λ) ,
where Q(Λ) and Qˉ(Λ) are three-dimensional matrices, the elements of which are complex conjugates, ε is the threedimensional unit matrix. The matrix Q(Λ) can have only one of the following types of characteristics:
(1) [111]; (2) [21]; (3) [3]. It is evident that the initial lambda-matrix can have only one characteristic drawn from:
(1) [111, 111]; (2) [21, 21]; (3) [3, 3].
The bar in the second half of a characteristic implies that elementary divisors in both matrices are complex conjugates. There is no bar in the third kind because the elementary divisors there are always real.
Taking a lambda-matrix of each of the three possible kinds, Petrov deduced the canonical form of the matrix (Rab) in a non-holonomic ortho-frame [40]
The 1st Kind
MN
(Rab) =
, N −M
α1 0 0
M =  0 α2 0  ,
0 0 α3
β1 0 0
N =  0 β2 0  ,
0 0 β3
(3.20)
where
3 s=1
αs
=
−κ
,
3 s=1
βs
=
0
(so in this case there
are 4 independent parameters, determining the space struct-
ure by an invariant form),
The 2nd Kind
MN
(Rab) =
, N −M
MN (Rab) = N −M ,
−
κ 3
1
 0
M = 1
−
κ 3
0 ,
0
0
−
κ 3
00 0
N =  0 0 −1  ,
0 −1 0
(3.22)
so no independent parameters determining the space structure by an invariant form exist in this case.
Thus Petrov had solved the problem of reducing a lambdamatrix to its canonical form along a real path in a space of the sign-alternating metric. Although this solution is obtained only at given point, the classification obtained is invariant because the results are applicable to any point in the space.
Real curvatures take the form
Λs = αs + iβs ,
(3.23)
in gravitational fields (spaces) of the 3rd kind, where the
quantities
Λs
are
real:
Λ1
=
Λ2
=
Λ3
=
−
κ 3
.
Values of some stationary curvatures in gravitational
fields (spaces) of the 1st and 2nd kinds can be coincident. If
they coincide, we have sub-kinds of the fields (spaces). The
1st kind has 3 sub-kinds: I (Λ1 = Λ2 = Λ3); D (Λ2 = Λ3);
O (Λ1 = Λ2 = Λ3). If the space is empty (κ = 0) the kind
O means the flat space. The 2nd kind has 2 sub-kinds: II
(Λ1 = Λ2); N (Λ1 = Λ2). Kinds I and II are called basic kinds. In empty spaces (empty gravitational fields) the stationary
curvatures become the unit value Λ = 0, so the spaces (fields)
are called degenerate.
Studying the algebraic structure of the curvature tensor
for known solutions to the Einstein equations, it was shown
that the most of the solutions are of the 1st kind by Petrov’s
classification. The curvature decreases with distance from
a gravitating mass. In the extreme case where the distance
becomes infinite the space approaches the Minkowski flat
space. The well-known Schwarzschid solution, describing
a spherically symmetric gravitational field derived from a
spherically symmetric island of mass located in an empty
space, is classified as the sub-kind D of the 1st kind [44].
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Invariant criteria for gravitational waves are linked to the algebraic structure of the curvature tensor, which should be associated with a given criterion from the aforementioned types. The most well-known solutions, which are interpreted as gravitational waves, are attributed to the sub-kind N (of the 1st kind). Other solutions are attributed to the 2nd kind and the 3rd kind. It should be noted that spaces of the 2nd and 3rd kinds cannot be flat anywhere, because components of the curvature tensor matrix Rab contain +1 and −1. This makes asymptotical approach to a curvature of zero impossible, i .e. excludes asymptotical approach to Minkowski space. Therefore, because of the structure of such fields, gravitational fields in a space of the 2nd kind (the sub-kind N) or the 3rd kind, are gravitational waves of the curvature traveling everywhere in the space. Pirani [18] holds that gravitational waves are solutions to gravitational fields in spaces of the 2nd kind (the sub-kind N) or the 3rd kind by Petrov’s classification. The following solutions are classified as sub-kind N: Peres’ solution [34] where he describes flat gravitational waves
satisfies the Zelmanov criterion.
With these theorems we obtain the general relation between the Zelmanov criterion for gravitational wave fields located in empty spaces and the Lichnerowicz criterion for “pure gravitational radiations”:
An empty V4, satisfying the Zelmanov criterion for gravitational wave fields, also satisfies the Lichnerowicz criterion for “pure gravitational radiations”. Conversely, any empty Vn, satisfying the Lichnerowicz criterion (excluding the sole trivial Vn described by the metric 3.27), satisfies the Zelmanov criterion. The relation between the criteria in the general case is still an open problem.
In [40] it was shown that all known solutions to the Einstein equations in vacuum, which satisfy the Zelmanov and Lichnerowicz criteria, can be obtained as particular cases of the more generalized metric whose space permits a covariant constant vector field lα
∇σlα = 0 .
(3.28)
ds2 = (dx0)2 − 2α(dx0 + dx3)2 − − (dx1)2 − (dx2)2 − (dx3)2;
(3.24)
Takeno’s solution [35]
ds2 = (γ + ρ)(dx0)2 − 2ρdx0dx3 − α(dx1)2 − (3.25)
− 2δdx1dx2 − β(dx2)2 + (ρ − γ)(dx3)2,
where α = α(x1 − x0), and γ, ρ, β, δ are functions of (x3 = x0); Petrov’s solution [37], studied also by Bondi,
Pirani and Robertson in another coordinate system [19]
ds2 = (dx0)2 − (dx1)2 + α(dx2)2 + + 2βdx2dx3 + γ(dx3)2,
(3.26)
where α, β, γ are functions of (x1 + x0). A detailed study of relations between the invariant criteria
for gravitational waves and Petrov’s classification had been undertaken by Zakharov [40]. He proved:
THEOREM In order that a given space satisfies the state of “pure gravitational radiations” (in the Lichnerowicz sense), it is a necessary and sufficient condition that the space should be of the sub-kind N by Petrov’s algebraical classification, characterized by equality to zero of the values of the curvature tensor matrix Rab in the bivector space. THEOREM An Einstein space that satisfies Zelmanov’s criterion can only be an empty space (κ = 0) of the sub-kind N. And conversely, any empty space V4 of the sub-kind N (excluding the sole symmetric space∗ of this kind), that is described by the metric
ds2 = 2dx0dx1 − sh2dx0(dx2)2 − sin2 dx0(dx3)2, (3.27)
It is evident that condition (3.10) leads automatically to
the first condition (3.7), hence this empty V4 is classified as sub-kind N by Petrov’s classification and, also, there the vector lα, playing a part of the gravitational field wave vector, is isotropic lαlα = 0 and unique. According to Eisenhart’s theorem [60], the space V4 containing the unique isotropic covariant constant vector lα (the absolute parallel vector field lα, in other words), has the metric
ds2 = ε(dx0)2 + 2dx0dx1 + 2ϕdx0dx2 + (3.29)
+ 2ψdx0dx3 + α(dx2)2 + 2γdx2dx3 + β(dx3)2,
where ε, ϕ, ψ, α, β, γ are functions of x0, x2, x3, and lα = δ1α. The metric (3.29), satisfying equations (3.2), is the exact solution to the Einstein equations for vacuum, and satisfies the Zelmanov and Lichnerowicz gravitational wave criteria. This solution generalizes well-known solutions deduced by Takeno, Peres, Bondi, Petrov and others, that satisfy the aforementioned criteria [40].
The metric (3.29), taken under some additional conditions [30], satisfies the Einstein equations in their general form (3.1) in the case where λ = 0 and the energy-momentum tensor Tαβ describes an isotropic electromagnetic field where Maxwell’s tensor Fμν satisfies the conditions
Fμν F μν = 0 ,
Fμν F ∗μν = 0 ,
(3.30)
F ∗μν
=
1 2
ημνρσ Fρσ
is
the
pseudotensor
dual
of
the
Maxwell
tensor, ημνρσ is the discriminant tensor. Direct substitution
shows that this metric satisfies the following requirements:
the Riner-Wheeler condition [61]
∗A space is called symmetric, if its curvature tensor is a covariant constant, i. e. if it satisfies the condition ∇σRαβγδ = 0.
R = 0,
RαρRρβ
=
1 4
δαβ (RρσRρσ)
= 0,
(3.31)
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and also the Nordtvedt-Pagels condition [62]
ημεγσ Rδγ,σRετ − Rδε,σRγτ ,
(3.32)
where Rδγ,σ = gσμ∇μRδγ , δβα = gβα. From the physical viewpoint we have an interest in isotro-
pic electromagnetic fields because an observer who accompanies it should be moving at the velocity of light [18, 21]. Hence, isotropic electromagnetic fields can be interpreted as fields of electromagnetic radiation without sources. On the other hand, according to Eisenhart theorem [60], a space V4 with the metric (3.29) permits an absolute parallel vector field lα = δ1α. Taking this fact and also the Einstein equations into account, we conclude that the vector lα considered in this case satisfies the Lichnerowicz criterion for “full gravitational radiations”.
Thus the metric (3.29), satisfying the conditions
1 Rαβ − 2 gαβR = −κTαβ ,
Tαβ
=
1 4
Fρσ
F
ρσ
gαβ
− FασFβ∙ σ∙ ,
Fαβ F αβ = 0 ,
Fαβ F ∗αβ = 0
(3.33)
and under the additional condition [30]
R2323 = R0232 = R0323 = 0 ,
(3.34)
is the exact solution to the Einstein equations which describes co-existence of both gravitational waves and electromagnetic waves. This solution does not satisfy the Zelmanov criterion in the general case, but the solution satisfies it in some particular cases where Tαβ = 0, and also under Rαβ = 0.
Wave properties of recursion curvature spaces were studied in [63]. A recursion curvature space is a Riemannian space having a curvature which satisfies the relationship
For the metric (3.37) there is only one component of the
Ricci
tensor
that
is
not
zero,
R00
=
−1 2
∂2ψ ∂x2 2
,
in
the
metric
(3.38) only
R11
=
−
1 2
∂2ψ ∂x2 2
is not
zero. Einstein spaces with
such metrics can only be empty (κ = 0) and flat (Rαβγδ = 0).
This can be proven by checking that both metrics satisfy
conditions (3.31) and (3.32), which describe isotropic elec-
tromagnetic fields.
Both metrics are interesting from the physical viewpoint:
in these cases the origin of the space curvature is an isotropic
electromagnetic field. Moreover, if we remove this field from
the space, the space becomes flat. Besides these there are few
metrics which are exact solutions to the Einstein-Maxwell
equations, related to the class of isotropic electromagnetic
fields. Neither of the said metrics satisfy the Zelmanov and
Lichnerowicz criteria.
Minkowski’s signature permits only two metrics for non-
simple recursion curvature spaces. They are the metric
ds2 = ψ(x0, x2, x3)(dx0)2 + 2dx0dx1 + + K22(dx2)2 + 2K23dx2dx3 + K33(dx3)2,
(3.39)
K22 < 0 ,
K22K33 − K223 < 0 ,
wherein ψ = χ1(x0)(a22(x2)2 + 2a23x2x3 + a33(x3)2) + + χ2(x0)x2 + χ3(x0)x3, and the metric
ds2 = 2dx0dx1 + ψ(x1, x2, x3)(dx1)2 + + K22(dx2)2 + 2K23dx2dx3 + K33(dx3)2,
(3.40)
wherein ψ = χ1(x1)(a22(x2)2 + 2a23x2x3 + a33(x3)2) + + χ2(x1)x2 + χ3(x1)x3. Here aij, Kij (i, j = 2, 3) are constants.
Both metrics satisfy the conditions Rαβ = κgαβ only if κ = 0, reducing to the single relationship
∇σRαβγδ = lσRαβγδ .
(3.35)
Because of Bianchi’s identity, such spaces satisfy
lσRαβγδ + lαRβσγδ + lβ Rσαγδ = 0 .
(3.36)
Total classification for recursion curvature spaces had been given by Walker [64]. His results [64] were applied to the basic space-time of the General Theory of Relativity, see [65] for the results. For the class of prime recursion spaces∗, we are particularly interested in the two metrics
ds2 =ψ(x0, x2)(dx0)2+2dx0dx1−(dx2)2−(dx3)2, (3.37)
ds2 = 2dx0dx1 + ψ(x1, x2)(dx1)2 − (3.38)
− (dx2)2 − (dx3)2, ψ > 0 .
∗A recursion curvature space is known as prime or simple, if it contains n − 2 parallel vector fields, which could be isotropic or non-isotropic. Here n is the dimension of the space.
K33a22 + K22a33 − 2K23a23 = 0 .
(3.41)
In this case both metrics are of the sub-kind N by Petrov’s classification. It is interesting to note that the metric (3.40) is stationary and, at the same time, describes “pure gravitational radiation” by Lichnerowicz. Such a solution was also obtained in [65].
In the general case (Rαβ = κgαβ) the metrics (3.39) and (3.40) satisfy conditions (3.32) and (3.33), so the metrics are solutions to the Einstein-Maxwell equations that describe co-existing gravitational waves and electromagnetic waves without sources. In this general case both metrics satisfy the Zelmanov and Lichnerowicz invariant criteria. The solution (3.40) is stationary.
All that has been detailed above applies to gravitational waves as waves of the space curvature, which exist in any reference frame.
Additionally it would be interesting to study another approach to the gravitational radiation problem, where the
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main issue is gravitational inertial waves, connected to the given reference frame of an observer. This new approach is linked directly to the mathematical apparatus of physically observable quantities (the theory of chronometric invariants), introduced by Zelmanov in 1944 [42, 43]. In order to understand the true results given by gravitational wave experiments it is necessary to master this mathematical apparatus, which is described concisely in the in the next section.
4 Basics of the theory of physical observable quantities
In brief, the essence of the mathematical apparatus of physic-
ally observable quantities (the theory of chronometric invari-
ants), developed by Zelmanov in 1940’s [42, 43] is that, if
an observer accompanies his reference body, his observable
quantities are projections of four-dimensional quantities on
his time line and the spatial section — chronometrically invar-
iant
quantities,
made
by
the
projecting
operators
bα
=
dxα ds
and hαβ = −gαβ + bαbβ which fully define his real reference
space (here bα is his velocity with respect to his real refer-
ences). The chr.inv.-projections of a world-vector Qα are
bαQα =
√Q0 g00
and hiαQα = Qi, while
chr.inv.-projections of
a
world-tensor
of
the
2nd
rank
Qαβ
are
bαbβ Qαβ
=
Q00 g00
,
hiαbβ Qαβ
=
√Qi0 g00
,
hiαhkβ Qαβ
=
Qik.
Physically
observable
properties of the space are derived from the fact that the chr.
inv.-differential
operators
∗∂ ∂t
=
√1 g00
∂ ∂t
and
∗∂ ∂xi
=
∂ ∂xi
+
+
1 c2
vi
∗∂ ∂t
are
non-commutative,
so
that
∗∂2 ∂xi∂t
−
∗∂2 ∂t ∂xi
=
=
1 c2
Fi
∗∂ ∂t
and
∗∂2 ∂xi∂xk
−
∗∂2 ∂xk∂xi
=
2 c2
Aik
∗∂ ∂t
,
and
also
from the fact that the chr.inv.-metric tensor hik may not
be stationary. The observable characteristics are the chr.inv.-
vector of gravitational inertial force Fi, the chr.inv.-tensor of angular velocities of the space rotation Aik, and the chr.inv.tensor of rates of the space deformations Dik, namely
1
Fi
=
√ g00
∂w ∂xi
−
∂vi ∂t
,
√
w
g00 = 1 − c2
(4.1)
1 Aik = 2
∂vk ∂xi
−
∂vi ∂xk
1 + 2c2 (Fivk − Fkvi) ,
(4.2)
√
Dik =
1 2
∗∂hik ∂t
,
Dik = − 1 ∗∂hik , 2 ∂t
Dkk
=
∗∂
ln ∂t
h ,
(4.3)
where
w
is
the
gravitational
potential,
vi
=
−c
√g0i g00
is
the
linear
velocity
of
the
space
rotation,
hik = −gik +
1 c2
vivk
is the metric chr.inv.-tensor, and h = det hik , h g00 = −g,
g = det gαβ . Observable inhomogeneity of the space is set up by the chr.inv.-Christoffel symbols Δijk = himΔjk,m,
which are built just like Christoffel’s regular symbols Γαμν = = gασΓμν,σ, but using hik instead of gαβ.
In this way, any equations obtained using general covar-
iant methods we can calculate their physically observable
projections on the time line and the spatial section of any
particular reference body and formulate the projections with
its real physically observable properties. From this we arrive
at equations containing only quantities measurable in practice. Expressing ds2 = gαβ dxαdxβ through the observable time interval
dτ
=
1 c
bαdxα
=
w 1 − c2
dt
−
1 c2
vidxi
(4.4)
and also the observable spatial interval dσ2 = hαβ dxαdxβ = = hik dxidxk (note that bi = 0 for an observer who accom-
panies his reference body). We arrive at the formula
ds2 = c2dτ 2 − dσ2.
(4.5)
From an“external” viewpoint, an observer’s threedimensional space is the spatial section x0 = ct = const. At any point of the space-time a local spatial section (a local space) can be placed orthogonal to the time line. If there exists a space-time enveloping curve for such local spaces, then it is a spatial section everywhere orthogonal to the time lines. Such a space is called holonomic. If no enveloping curve exists for such local spaces, so there only exist spatial sections locally orthogonal to the time lines, such a space is called non-holonomic. A spatial section, placed in a holonomic space, is everywhere orthogonal to the time lines, i. e. g0i = 0 is true there. In the presence of g0i = 0 we have vi = 0, hence Aik = 0. This implies that non-holonomity of the space and its three-dimensional rotation are the same. In a non-holonomic space g0i = 0 and Aik = 0. Hence Aik = 0 is the necessary and sufficient condition of holonomity of the space. So Aik is the tensor of the space non-holonomity.
Zelmanov had also found that the chr.inv.-quantities Fi and Aik are linked to one another by two identities
∗∂Aik + 1 ∂t 2
∗∂Fk ∂xi
−
∗∂Fi ∂xk
= 0,
(4.6)
∗∂Akm ∂xi
+
∗∂Ami ∂xk
+
∗∂Aik ∂xm
+
1 + 2 (FiAkm + FkAmi + FmAik) = 0 ,
(4.7)
which are known as Zelmanov’s identities. Components of the usual Christoffel symbols
Γαμν
=
1 2
gασ
∂gμσ ∂xν
+
∂gνσ ∂xμ
−
∂gμν ∂xσ
.
(4.8)
are linked to the chr.inv.-Christoffel symbols
Δijk
=
1 2
him
∗∂hjm ∂xk
+
∗∂hkm ∂xj
−
∗∂hjk ∂xm
,
(4.9)
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and other chr.inv.-chractersitics of the accompanying reference space of the given observer by the relations
Dki
+ A∙ki∙
=
c √
g00
Γi0k
−
g0k Γi00 g00
,
(4.10)
Fk
=
c2 −
Γk00
,
g00
giαgkβ Γm αβ = hiqhksΔm qs. (4.11)
Here is the four-dimensional generalization of the chr.inv.-
quantities Fi, Aik, and Dik (by Zelmanov, the 1960’s [57]):
Fα = −2c2bβ aβα, Aαβ = chμαhνβ aμν , Dαβ = chμαhνβ dμν ,
where
aαβ
=
1 2
(∇α
bβ
−
∇β
bα),
dαβ
=
1 2
(∇α
bβ
+
∇β
bα).
Zelmanov also deduced formulae for chr.inv.-projections
of the Riemann-Christoffel tensor [42]. He followed the
same procedure by which the Riemann-Christoffel tensor
was built, proceeding from the non-commutativity of the
second derivatives of an arbitrary vector taken in the given
space. Taking the second chr.inv.-derivatives of an arbitrary
vector
∗∇i ∗∇k Ql − ∗∇k ∗∇i Ql =
2Aik ∗∂Ql c2 ∂t
+ Hl.k..ij∙Qj ,
(4.12)
he obtained the chr.inv.-tensor
Hl.k..ij∙
=
∗∂Δjil ∂xk
−
∗∂Δjkl ∂xi
+ Δm il Δjkm
− Δm klΔjim ,
(4.13)
which is like Schouten’s tensor from the theory of nonholonomic manifolds [59]. The tensor Hl.k..ij differs algebraically from the Riemann-Christoffel tensor because of the
presence of rotation of the space Aik in the formula (4). Nevertheless its generalization gives the chr.inv.-tensor
1 Clkij = 4 (Hlkij − Hjkil + Hklji − Hiljk) ,
(4.14)
which possesses all the algebraic properties of the Riemann-
Christoffel tensor in this three-dimensional space. Therefore
Zelmanov called Ciklj the chr.inv.-curvature tensor, which actually is the tensor of the observable curvature of the
observer’s spatial section. This tensor, describing the observ-
able curvature of the three-dimensional space of an observer,
possesses all the properties of the Riemann-Christoffel curva-
ture tensor in the three-dimensional space and, at the same
time, the property of chronometric invariance. Its contraction
Ckj = Ck∙∙i∙ji∙ = himCkimj ,
C = Cjj = hlj Clj (4.15)
gives the chr.inv.-scalar C whose sense is the observable
three-dimensional curvature of this space.
Substituting the necessary components of the Riemann-
Christoffel tensor into the formulae for its chr.inv.-projections
X ik =−c2
R0∙i∙∙0k∙ g00
,
Y
ijk=−c
R√0∙i∙j∙k∙ g00
,
Zijkl= c2Rijkl,
and
by
lo-
wering indices Zelmanov obtained the formulae
Xij
=
∗∂Dij ∂t
−
Dil + A∙il∙ (Djl + Ajl) +
+
1 2
(∗∇iFj
+
∗∇j Fi)
−
1 c2
FiFj
,
(4.16)
Yijk = ∗∇i (Djk + Ajk) − ∗∇j
Dik + Aik + 2
+ c2 Aij Fk ,
(4.17)
Ziklj = DikDlj − DilDkj + AikAlj − AilAkj + (4.18)
+ 2Aij Akl − c2Ciklj ,
where we have Y(ijk) = Yijk + Yjki + Ykij = 0 just like the Riemann-Christoffel tensor. Contraction of the spatial observable projection Ziklj step-by-step gives
Zil = DikDlk − DilD + AikA∙lk∙ + 2AikAk∙l∙ − c2Cil , (4.19)
Z = hilZil = DikDik − D2 − AikAik − c2C . (4.20)
Besides these considerations, taken in an observer’s accompanying reference frame, Zelmanov considered a locally geodesic reference frame that can be introduced at any point of the pseudo-Riemannian space. Within infinitesimal vicinities of any point of such a reference frame the fundamental metric tensor is
1 g˜αβ = gαβ + 2
∂2g˜αβ ∂x˜μ∂x˜ν
(x˜μ−xμ)(x˜ν −xν ) + . . . , (4.21)
i. e. its components at a point, located in the vicinities, are different to those at the point of reflection to within only the higher order terms, values of which can be neglected. Therefore, at any point of a locally geodesic reference frame the fundamental metric tensor can betaken as constant, while the first derivatives of the metric (the Christoffel symbols) are zero.
As a matter of fact, within infinitesimal vicinities of any point located in a Riemannian space, a locally geodesic reference frame can be defined. At the same time, at any point of this locally geodesic reference frame, a tangential flat Euclidean space can be defined so that this reference frame, being locally geodesic for the Riemannian space, is the global geodesic for that tangential flat space.
The fundamental metric tensor of a flat Euclidean space is constant, so values of g˜μν, taken in the vicinities of a point of the Riemannian space converge to values of the tensor gμν in the flat space tangential at this point. Actually, this means that we can build a system of basis vectors e(α), located in this flat space, tangential to curved coordinate lines of the Riemannian space.
In general, coordinate lines in Riemannian spaces are curved, inhomogeneous, and are not orthogonal to each other (if the space is non-holonomic). So the lengths of the basis vectors may be sometimes very different from unity.
We denote a four-dimensional vector of infinitesimal displacement by dr = (dx0, dx1, dx2, dx3). Then dr = e(α)dxα, where components of the basis vectors e(α) tangential to the coordinate lines are e(0)= {e0(0), 0, 0, 0}, e(1)= {0, e1(1), 0, 0}, e(2)= {0, 0, e2(2), 0}, e(3)= {0, 0, 0, e2(3)}. The scalar product
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of the vector dr with itself is drdr = ds2. On the other hand, the same quantity is ds2 = gαβ dxαdxβ. As a result we have gαβ = e(α)e(β)= e(α)e(β) cos (xα; xβ). So we obtain
g00 = e2(0) ,
g0i = e(0)e(i) cos (x0; xi) ,
(4.22)
gik = e(i)e(k) cos (xi; xk) .
(4.23)
The
gravitational
potential
is
w
=
c2
√ (1− g00).
So,
the
time basis having the
vector length
ee((00))=t√angg0e0n=tia1l −tocw2thies
time line x0 smaller than
= ct, unity
the greater is the gravitational potential w.
The space rotation linear velocity vi and, according to it,
the chr.inv.-metric tensor hik are
vi = −c e(i) cos (x0; xi) ,
(4.24)
hik = e(i)e(k) cos(x0; xi)cos(x0; xk)−cos(xi; xk) . (4.25)
This representation enablkes us to see the geometric sense of physical quantities measurable in experiments, because we represent them through pure geometric characteristics of the observer’s space — the angles between coordinate axes etc.
This completes the basics of Zelmanov’s mathematical apparatus of chronometric invariants (physically observable quantities) that will be employed below with the aim of studying the gravitational wave problem.
5 Gravitational inertial waves and their link to the chronometrically invariant representation of Petrov’s classification
Of all the experimental statements on the General Theory of Relativity, including the search for gravitational wave experiments, the most important case is that where the observer is at rest with respect to his laboratory reference frame and all physical standards located in it. Quantities measured by the observer in an accompanying reference frame are chronometrically invariant quantities (see the previous paragraph for the details). Keeping this fact in mind, Zelmanov formulated his chronometrically invariant criterion for gravitational waves. This criterion is invariant only for transformations of coordinates of that reference system which is at rest with respect to the laboratory references (the body of reference). Such an approach, in contrast to the invariant approach, permits us to interpret the results of measurement in terms of physically observable quantities, providing thereby a means of comparing results given by the theory of gravitational waves to results obtained from real physical experiments.
In order to solve the problem of interpretation of experimental data on gravitational waves it is appropriate to consider a more general case — fields of gravitational inertial waves. Such fields are more general because they are applicable to both gravitational fields and the inertial field of
the observer’s reference frame. The mathematical method
that we propose to apply to this problem joins both fields
into a common field. The method itself does not differ for
each field: to set an invariant difference between gravitational
fields and the observer’s inertial field would be possible only
by introducing an additional invariant criterion.
Gravitational waves are determined independently of
both spatial coordinate frames and space-time reference fra-
mes. In contrast to gravitational waves, gravitational inertial
waves are determined only in the reference frame of an
observer, who observes them. They are determined with pre-
cision to within so-called “inner” transformations of coordi-
nates
(a) x˜0 = x˜0(x0, x1, x2, x3)
 
(b) x˜i = x˜i(x1, x2, x3) ,
∂x˜i ∂x0 = 0
(5.1)
which does not change the space-time reference frame itself.
Invariance with respect to (5.1) splits into invariance
with respect to (5.1a), so-called chronometric invariance,
and also invariance with respect to (5.1b), so-called spatial
invariance. Therefore a definition given for gravitational
inertial waves should be:
(1) chronometrically invariant;
(2) spatially covariant.
We then have a basis by which we introduce the chro-
nometrically invariant spatially covariant d’Alembert oper-
ator [40]∗
∗
=
hik ∗∇i ∗∇k
−
1 a2
∗∂2 ∂t2
,
(5.2)
where hik = −gik is the chr.inv.-metric tensor (the phys-
ically observable metric tensor) in its contravariant (upper-
index) form, ∗∇i is the symbol for the chr.inv.-derivative (the chr.inv.-analogue to the covariant derivative symbol ∇σ),
a is the linear velocity at which attraction of gravity spreads,
∗∂ ∂t
is
the
symbol
for
the
chr.inv.-derivative
with
respect
to
time.
A chronometrically invariant criterion for gravitational
inertial waves, formulated according to Zelmanov’s idea, is:
Zelmanov’s chr.inv.-criterion Chr.inv.-quantities f , char-
acterising the observer’s reference space, such as the
gravitational inertial force vector Fi, the space non-
holonomity (self-rotation) tensor Aik, the space defor-
mation rate tensor Dik, the spatial curvature tensor Ciklj, and also scalar quantities, built on them, and also
the Riemann-Christoffel curvature tensor’s chr.inv.components Xij, Y ijk, Ziklj must satisfy equations
of the form
∗ f = A,
(5.3)
∗This approach to the gravitational inertial wave problem was developed by Zelmanov, although it had first been published by Zakharov because the latter prepared his dissertation under Zelmanov’s leadership: see footnote on page 35.
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where A is an arbitrary function of four-dimensional world-coordinates, which has no more than first order derivatives of the f .
The Zelmanov chr.inv.-criterion (5.3) was applied in analyzing well-known solutions to the Einstein equations in emptiness [40]. This criterion is true for the metrics (3.25) in that case where the gravitational inertial force vector F i is the wave function. But, at the same time, most of the invariant criteria for gravitational waves are related to some conditions and limitations imposed on the curvature tensor. Therefore it would be most interesting to study relations between gravitational wave criteria and gravitational inertial wave criteria in that case where the Riemann-Christoffel curvature tensor’s chr.inv.-components Xij, Y ijk, Ziklj are the wave functions.
What is the relation between the Zelmanov invariant criterion (3.9) and his chr.inv.-criterion (5.3)? This problem was solved by Zakharov [40, 58]. His method was to express equation (3.9) in chr.inv.-form. In chr.inv..-form (in the terms of physically observable quantities) equation (3.9) takes the form
∗ Xij = Ai(j1) , ∗ Y ijk = Ai(j2k) , ∗ Ziklj = Ai(k3)lj , (5.4)
where Ai(j1), Ai(j2k) , Ai(k3)lj are chronometrically invariant and spatially invariant tensors, which have no more than first order derivatives of the wave functions Xij, Y ijk, Ziklj.
Thus those gravitational fields that satisfy the Zelmanov
invariant criterion also satisfy the Zelmanov chr.inv.-criterion
(5.3), where the Riemann-Christoffel curvature tensor’s physically observable components Xij, Y ijk, Ziklj play the
part of wave functions.
The necessary condition for gravitational inertial waves
is the fact that the chr.inv.-d’Alembert operator (5.2) is non-
trivial, mathematically expressed as follows:
1.
Chr.inv.-quantities f
are non-stationary, i. e.
∗∂f ∂t
= 0;
2. The quantities f are inhomogeneous, i. e. ∗∇ifk = 0.
The wave functions Xij (4.16), Yijk (4.17) and Ziklj (4.18) satisfy these requirements only if the mechanical chr.inv.-characteristics of the observer’s reference space (the chr.inv.-quantities Fi, Aik, Dik) and the geometric chr.inv.characteristic of the space (the chr.inv.-quantity Ciklj) also satisfy these requirements. Zelmanov himself in [42] formulated conditions of inhomogeneity inside a finite region located in the observer’s space
∗∇i Fk = 0 , ∗∇j Aik = 0 , ∗∇j Dik = 0 , ∗∇j Cik = 0 .
(5.5)
It is evident that under these conditions the wave functions Xij, Y ijk, Ziklj shall be inhomogeneous.
The origin of non-stationary states of the gravitational inertial force vector Fi (4.1) is the non-stationarity of the
gravitational potential w or the linear velocity of the space
rotation vi, consisting the force. Identities (4.6) and (4.7), linking quantities Fi and Aik, lead us to conclude that the
source of non-stationary states of vi is the vortical nature of the vector Fi, i. e. ∗∇kFi −∗∇iFk = 0. The origin of non-
stationary states of the space deformation rate Dik (4.3)
and the space observable curvature Ciklj (4.14) is non-
stationarity of the physical observable metric tensor hik,
see [42],
hik
= −gik +
g0ig0k g00
= −gik +
1 c2
vi vk .
(5.6)
Thus, the origin of non-stationary states of the wave
functions Xij, Y ijk, Ziklj is the non-stationarity of com-
ponents of the fundamental metric tensor gαβ, namely:
(1)
g00 =
1
−
w c2
2
;
(2)
g0i
=
−
1 c
vi
1
−
w c2
;
(3)
gik
=
−hik
+
1 c2
vi vk
.
We consider each of the cases here,mindful of the need to
find theoretical grounds for gravitational wave experiments:
1. Non-stationary states of g00 manifest as a result of time changes of the gravitational potential w. In experiments this non-stationarity is derived from very different geophysical sources, which, in a particular case, are due to changes in solar activity;
2. Non-stationary states of mixed components g0i are derived from the non-stationarity of the space rotation
linear velocity vi and the gravitational potential w. The qinutaenrvtiatlieosfg0o0bsaenrdvagb0ilearteimineclduτde=d√ingt0h0edfto+rm√uglga00if0ordaxni [42, 43]. Thus under non-stationary states of g00 and g0i in the observer’s laboratory (his reference frame) a standard clock located there should have some cor-
rections (which change with time) with respect to a
standard clock located in an region where the quanti-
ties g00 and g0i are stationary.
3. Non-stationary states of gik are usually considered as deformations of the three-dimensional space. But the
theory of physically observable quantities introduces
substantial corrections to this thesis. The approach of
Classical Mechanic looks at the spatial deformations as
1 2
∂gik ∂t
,
but
the
theory
of
physically
observable
quant-
ities, taking properties of the observer into account,
gives rise to a corrected formula for the spatial deform-
ations
which
is
Dik =
√1 2 g00
∂ ∂t
−gik
+
1 c2
vi
vk
.∗
∗The presence of the minus sign here is a consequence of the fact that
we use the signature (+−−−), where plus is related to the time coordinate
while minus is attributed to spatial coordinates. The minus sign has been
chosen for the gik in the hik formula, because in this case the observable spatial interval dσ = hik dxidxk is positive, which is an important fact in the theory of physically observable quantities [42, 43].
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The formulae coincide in that particular case where
g00 = 1 (w = 0) and g0i = 0 (vi = 0). If Fi = 0, according (to 4.6) the space rotation is stationary. If vi = 0,
Aik = 0. Thus the necessary and sufficient condition
to make w and vi simultaneously zero is Fi = 0 and Aik = 0 [42, 43]. In this case the observer’s reference frame falls freely and is free of rotations. Such refer-
ence frames are known as synchronous [15], because
there all clocks can be synchronized. Moreover, in
this case time can be integrated: in calculations of
the time interval dτ = dt between any two events, the
integral of dτ is independent of the way we take this
integral between the events (the path of integration).
If Fi = 0 but Aik = 0, it is impossible to synchronize
all the clocks simultaneously, but the synchronization
itself can be realized because of the proportionality √
dτ = g00 dt there. If Aik = 0, the synchronization is
impossible in principle, because the integral of dτ =
√ = g00 dt
+
√g0i g00
dxi
depends
on
the
path
of
integ-
ration [42, 43].
Synchronous reference frames, because of their simplicity and associated simple calculations, are of broad utility in the General Theory of Relativity. In particular, they are used in relativistic cosmology and the gravitational wave problem. For instance, the well-known metric of weak plane gravitational waves takes the form [14, 15]
ds2 = c2dt2 − (dx1)2 − (1 − a)(dx2)2 + + 2 b dx2dx3 − (1 + a)(dx3)2,
(5.7)
where a = a(ct ± x1), b = b(ct ± x1). So in this metric there is no gravitational potential (w = 0) as soon as there is no space rotation (vi = 0). The condition w = 0 prohibits the ultimate transit to Newton’s theory of gravity. For this reason we arrive at an important conclusion:
Weak plane gravitational waves are derived from sources other than gravitational fields of masses∗.
An analogous situation arises in relativistic cosmology, where, until now, the main part is played by the theory of a homogeneous isotropic universe. Foundations of this theory are built on the metric of a homogeneous isotropic space [42]
ds2 = c2dt2 −
− R2
(dx1)2 + (dx2)2 + (dx3)2
1
+
k 4
(dx1)2 + (dx2)2 + (dx3)2
2,
(5.8)
R = R(t) , k = 0, ±1 .
When one substitutes this metric into the Einstein equations taken with a specific value of the cosmological constant
∗See §7 and §8 below for detailed calculations for the effect due to weak plane gravitational waves in solid-body detectors of the Weber kind (the Weber pigs) and also in antennae built on free masses.
(λ = 0, λ < 0, λ > 0), he obtains a spectra of solutions, which are known as Friedmann’s cosmological models [42].
Taking our previous conclusion on the origin of weak plane gravitational waves into account, we come to a new and important conclusion:
No gravitational fields derived from masses exist in any Friedmann universe. Moreover, any Friedmann universe is free of space rotations.
Currently there is no indubitable observational data supporting the absolute rotation of the Universe. This problem has been under considerable discussion between astronomers and physicists over last decade, and remains open. Rotations of bulk space bodies like planets, stars, and galaxies are beyond any doubt, but these rotations do not imply the absolute rotation of the whole Universe, including the absolute rotation of its gravitational field if one will describe it by the Friedmann models.
Looking back at the question of whether or not gravitational inertial waves exist, or whether or not non-stationary states of the wave functions Xij, Y ijk, Ziklj exist, we conclude that non-stationary states of the quantities are derived from:
1. A vortical nature of the field of the acting gravitational inertial force Fi ;
2. Non-stationary states of the spatial components gik of the fundamental metric tensor gαβ .
In the first case, the effect of gravitational inertial waves manifests as non-stationary corrections to the observer’s time flow.
In the second case, the observer’s time flow remains unchanged, but gravitational waves are waves of only the space deformation. Such pure deformation waves will deform a detector itself, so one simply waits for a gravitational wave to cause a resonance effect in a solid-body detector of the Weber kind [16]. Whether this conclusion is true or false will be considered in §7 and §8. Here we consider only the general theory of gravitational inertial waves and its relation to the invariant theory of gravitational waves.
As we showed above, those gravitational fields that satisfy the Zelmanov invariant criterion (3.9) also satisfy the Zelmanov chr.inv.-criterion (5.3), where the wave functions f are the Riemann-Christoffel tensor’s observable components Xij, Y ijk, Ziklj. As it was shown in the previous paragraph, “empty gravitational fields” (we mean gravitational fields permeating empty spaces, where no mass islands of matter exist) that satisfy the Zelmanov invariant criterion (3.9) are related to the 2nd kind (the sub-kind N) by Petrov’s classification. Therefore it is appropriate to specify the algebraical kinds of the Riemann-Christoffel tensor in terms of physically observable quantities (chronometric invariants).
The whole problem of representing Petrov’s classification in chronometrically invariant form has been solved in [66]. This solution, obtained Petrov in general covariant form [37],
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was obtained for an ortho-frame, taken at an arbitrary fixed point of the space.
Chr.inv.-components of the Riemann-Christoffel curvature tensor have the properties
Xij = Xji , Y[ijk] = 0 ,
Xkk = −κ , Yijk = −Yikj .
(5.9)
Equations (4.16), (4.17), (4.18) in an ortho-frame are
Xij = −c2R0i0j ,
Yijk = −cR0ijk , Ziklj = c2Riklj .
(5.10)
where
x11 0 0 x = 0 x22 0 ,
0 0 x33
y11 0 0 y = 0 y22 0 ,
0 0 y33
(5.17)
x11 + x22 + x33 = −κ , y11 + y22 + y33 = 0 . (5.18)
Using (5.11) we also express values of the stationary curvatures Λi (i = 1, 2, 3) through the Riemann-Christoffel tensor’s physically observable components
When we write equations Rαβ = κgαβ in the orth-frame, we take the relationships (5.10) into account. Then, introducing three-dimensional matrices x and y such that
1 x ≡ xik = − c2 Xik ,
y≡
yik
1 =−
2c
εimnY
∙ mn k∙∙
,
(5.11)
where εimn is the three-dimensional discriminant tensor, we represent the six-dimensional matrix Rab as follows
xy Rab = y −x ,
a, b = 1, 2, . . . 6 , (5.12)
satisfying the relations
x11 + x22 + x33 = −κ , y11 + y22 + y33 = 0 . (5.13)
Now, let us compose a lambda-matrix
x + Λε y
Rab − Λgab =
, y −x − Λε
(5.14)
where ε is the three-dimensional unit matrix. Then, after transformations, we reduce this lambda-matrix to the form
x+iy +Λε
0
Qˉ(Λ) 0
0
− x − iy −Λε = 0 Qˉ(Λ) . (5.15)
The initial lambda-matrix can have one of the following characteristics:
1
i
Λ1 = − c2 X11 + c Y123 ,
1
i
Λ2 = − c2 X22 + c Y231 ,
1
i
Λ3 = − c2 X33 + c Y312 .
(5.19)
Thus, the components Xik are included in the real parts of the stationary curvatures Λi, and components Yijk are included in the imaginary parts. In spaces of the sub-kind D
(Λ2 = Λ3) we have: X22 = X33, Y231 = Y312. In spaces of the
sub-kind
O
(Λ1
=
Λ2
=
Λ3)
we
have:
X11
=
X22
=
X33
=
−
κ 3
,
Y123 = Y231 = Y312. Hence Einstein spaces of the sub-kind O
have only real curvatures, while being empty they are flat.
For the 2nd kind we have
The 2nd Kind
xy Rab = y −x ,
x11
0
0
x = 0 x22 +1 0 ,
0
0 x33 −1
y11 0 0 y = 0 y22 1 ,
0 1 y22
(5.20)
where
(1) [111, 111]; (2) [21, 21]; (3) [3, 3]. (5.16)
Then, using Petrov’s had obtained the canonical form of the matrix Rab in the non-holonomic ortho-frame for each of the three kinds of the curvature tensor [37], we express the matrix Rab through components of the chr.inv.-tensors Xij and Yijk [66]. We obtain
The 1st Kind
xy
Rab =
, y −x
x11 + x22 + x33 = −κ , x22 − x33 = 2 , y11 + 2y22 = 0 .
The stationary curvatures are
1
i
Λ1 = − c2 X11 + c Y123 ,
1
i
Λ2 = − c2 X22 − 1 + c Y231 ,
1
i
Λ3 = − c2 X33 + 1 + c Y312 .
(5.21) (5.22)
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From this we conclude that values of the stationary curvatures Λ2 and Λ3 can never become zero, so Einstein spaces (gravitational fields) of the 2nd kind are curved in any case — they cannot approach Minkowski flat space.
In spaces of the sub-kind N (Λ1 = Λ2) in an ortho-frame the relations are true
X11 = X22 − c2 = X33 + c2, Y123 = Y231 = Y312 = 0 ,
(5.23)
so the stationary curvatures are real. In an empty space the matrices x and y become degenerate (its determinant becomes zero). For this reason spaces of the sub-kind N are degenerate, and, respectively, gravitational fields in spaces of the sub-kind N are known as gravitational fields of the 2nd degenerate kind by Petrov’s classification. In emptiness (κ = 0) some elements of the matrices x and y take the numerical values +1 and −1 thereby making an ultimate transition to the Minkowski flat space impossible.
For the 3rd kind we have
The 3rd Kind
xy Rab = y −x ,
010 x= 1 0 0 ,
000
(5.24)
00 0 y = 0 0 −1 .
0 −1 0
Here the stationary curvatures are zero and both of the matrices x and y are degenerate. Einstein spaces of the 3rd kind can only be empty (κ = 0), but, at the same time, they can never be flat.
From the equations deduced for the canonical form of the matrix Rab , we conclude: Yijk = 0 can be true only in gravitational fields of the 1st kind, which are derived from island masses of matter in emptiness or vacuum. Therefore we conclude that those gravitational fields where Yijk = 0 is true in the observer’s accompanying reference frame can only be of the 1st kind, having stationary curvatures which are real.
Furthermore, in accordance with most of the criteria, the presence of gravitational waves is linked to spaces of the 2nd (N) kind and the 3rd kind, where the matrix yik has components equal to +1 or −1. Moreover, in fields of the 2nd (N) and 3rd kinds the values +1 or −1 are attributed also to components of the matrix x. This implies that:
Those spaces which contain gravitational fields, satisfying the invariant criteria for gravitational waves, are curved independently of whether or not they are
empty (Tαβ = 0) or filled with matter (in such spaces Tαβ = gαβ). In any case, gravitational radiations are derived from interaction between two observable components Xij, Yijk of the Rimeann-Christoffel curvature tensor.
The classification of gravitational fields built here applies only to Einstein spaces, because solving this problem for spaces of general kind, where Tαβ = κgαβ, would be very difficult, for mathematical reasons. Considering the details of these difficulties, we see that, having an arbitrary distribution of matter in a space, the matrix Rab , taken in a nonholonomic ortho-frame, is not symmetrically doubled; on the contrary, the matrix takes the form
xy Rab = y z ,
(5.25)
where the three-dimensional matrices x, y, z are built on the following elements, respectively∗
1 xik = − c2 Xik ,
zik
=
1 c2
εimn εkpq Z mnpq ,
yik
=
1 2c
εimn
Y
∙mn k∙∙
,
(5.26)
and y implies transposition. It is evident that reduction of this matrix to its canonical form is a very difficult problem.
Nevertheless Petrov’s classification permits us to con-
clude:
The physically observable components Xij and Y ijk of the Riemann-Christoffel curvature tensor are different in their physical origin†. Metrics can exist where Y ijk = 0 but Xij = 0 and Ziklj = 0. Such spaces are of the 1st kind by Petrov’s classification; they have real stationary curvatures. Such spaces do not satisfy the invariant criteria for gravitational waves. Thus no wave fields of gravity exist in spaces where Y ijk = 0 but Xij = 0 and Ziklj = 0.
And further:
In solutions of the Einstein equations there are no metrics where Y ijk = 0 but Xij = 0 and Ziklj = 0. Thus in wave fields of gravity Y ijk = 0 and Xij = 0 (and as well Ziklj = 0: see the footnote) everywhere
and always.
∗In ortho-frames there is no difference between upper and lower
indices
(see
[37]).
For
this
reason
we
can
write
zik
=
1 c2
εimnεkpq Zmnpq
and
yik
=
1 2c
εimn Ykmn
instead
of
zik =
1 c2
εimnεkpq Zmnpq
and
yik
=
1 2c
εimn
Y
∙mn k∙∙
in
formula
(3.26).
This
note
relates
to
all
formulae
written in an ortho-frame. We met a similar case in formula (5.11),
where we can also write y ≡
yik
=
−
1 2c
εimn Ykmn
instead of
y≡
yik
=
−
1 2c
εimn
Y
∙mn k∙∙
.
†We do not mention the third observable component Ziklj , because in
an ortho-frame the matrices x and z are connected by the equation x = −z.
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We will show that in Einstein spaces filled with gravi-
tational fields where the Riemann-Christoffel tensor’s observable components Xij, Y ijk, Ziklj play a part of the wave functions, the quantity Xij is analogous to the electric component of an electromagnetic field, while Y ijk is anal-
ogous to its magnetic component. All this will be discussed
in §7.
6 Wave properties of Einstein’s equations
In §2 we have showed that the gravitational field equations (the Einstein equations) do not contain a general covariant d’Alembert operator derived from the fundamental metric tensor gαβ (where gαβ is considered as a “four-dimensional gravitational potential”). Nevertheless this problem has been solved in linear approximation in the case where gravitational fields are occupy an empty space (Rαβ = 0, “empty gravitational fields”) [14, 15]. In this case a gravitational field is considered as a tiny addition to a flat space background described by the Minkowski metric. Thus
gαβ = δαβ + γαβ ,
(6.1)
where δαβ are components of the fundamental metric tensor in a Galilean reference frame δαβ = {+1, −1, −1, −1}, and γαβ describes weak corrections for the gravitational fields. The contravariant fundamental metric tensor gαβ to within
the first order approximation of the γαβ is
gαβ = δαβ − γαβ ,
(6.2)
so the determinant of the tensor gαβ is
g = − (1 + γ) , γ = det γαβ . (6.3)
The requirement that components of the “additional” metric γαβ must be infinitesimal fixes a prime reference frame. If this requirement is true in a reference frame, it will also be true after transformations
where
≡
gαβ
∂
∂2 xα∂xβ
=
1 c2
∂2 ∂t2
−Δ,
∂2
∂2
∂2
Δ = ∂x1 2 + ∂x2 2 + ∂x3 2 .
(6.8)
Here is the d’Alembert operator, Δ is the Laplace
operator. The calibrating requirements (6.6) are true in any
metric γαβ only if the quantities ξα are solutions of the
equation
ξα = 0 .
(6.9)
In [15] the requirement
γαβ = 0
(6.10)
was imposed on the quantities γαβ, which is interpreted as the equation of weak gravitational waves in emptiness — this formula (6.10) is a standard wave equation that describes a wave of the tensor field γαβ, traveling at the velocity c in emptiness.
One usually considers the equation (6.10) as the basis for the claim that the General Theory of Relativity predicts gravitational waves, which travel at the speed of light.
If we have a weak plane gravitational wave, so the field has changes along a single spatial direction (the x1 axis, for instance), the formula (6.10) takes the form
1 ∂2
∂2
c2 ∂t2 − ∂x1 2 γαβ = 0 ,
(6.11)
and solutions of it can be any function of ct ± x1. After numerous transformations of the function γαβ [14, 15] it obtains that in the field of a weak plane gravitational wave only the following components are non-zero: γ22 = −γ33 ≡ a, γ23 ≡ b. Thus, those weak plane gravitational waves that satisfy the Einstein equations in emptiness are transverse.
Thus if some additional requirements are imposed upon the Einstein equations in emptiness, the equations describe weak plane waves of the space deformation, the space metric of which is [15]
x˜α = xα + ξα,
(6.4)
where ξα are infinitesimal quantities ξα 1. Then we have
γ˜αβ
=
γαβ
−
∂ξα ∂xβ
−
∂ξβ ∂xα
.
(6.5)
Because of (6.1), we impose an additional requirement on the tensor γαβ; this requirement is [15]
∂ψα ∂xβ = 0 ,
ψβα
=
γβα
−
1 2
δβα γ
.
(6.6)
Taking (6.6) into account, the Ricci tensor takes the form
1 Rαβ = 2 γαβ ,
(6.7)
ds2 = c2dt2 − (dx1)2 − (1 + a)(dx2)2 + + 2bdx2dx3 − (1 − a)(dx3)2,
(6.12)
where a and b are functions of ct ± x1. The field of gravitation, described by the metric (6.12), is of the sub-kind N by Petrov’s classification, so it satisfies most of invariant criteria for gravitational waves.
The metric (6.12) has been written in a synchronous reference frame, so its space deforms, falls freely, and, at the same time, has no rotations. Hence, under the given assumptions, weak plane gravitational waves are waves of “pure” deformation of the space. This conclusion is the main reason why experimental physicists, and Weber in particular [16], expect that gravitational waves will cause a “pure” deformation effect in detectors.
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Calculations for the interaction between a Weber solidbody detector and a weak plane gravitational wave field will be given in §7. Here we continue our argument for the wave nature of the Einstein equations in strong gravitational fields in the case where matter is arbitrarily distributed in the space. This research will be given in the terms of physically observable quantities for the reason that we will consider situations derived from different factors, generating gravitational wave fields, not only the space deformation.
The Einstein equations in the case where matter is arbitrarily distributed are [42]
∗∂D ∂t
+
DjlDjl
+
AjlAlj
+
∗∇j
−
1 c2
Fj
Fj =
κ =−
ρc2 + U
+ λc2,
2
(6.13)
The quantity Hik, by definition, is
Hik
=Hi∙j∙∙kj∙=
∗∂Δjij ∂xk
−
∗∂Δjik ∂xj
+Δm ij Δjkm−Δm ikΔjjm
,
(6.19)
√
where Δm jm
=
∗∂ ln ∂xj
h.
Taking into account (6.17), (6.19), and also Zelmanov’s
identities (4.6), (4.7) that link Fi and Aik, we reduce (6.18)
to the form
√
∗
∗∂2 ln h 2 hik = 2 ∂xi∂xk − c2
∗∇iFk
+
∗∂Aik ∂t
−
4 − c2
Aij A∙kj∙ − Dij Dkj
2D − c2
Dik + Aik
+
+ 2 hpqΔm pqΔik,m + Δm ij Δjkm −
(6.20)
∗∇j
hij D − Dij − Aij
+
2 c2
Fj Aij
=
κJ i,
(6.14)
∗∂Dik ∂t
− (Dij
+
Aij )
Dkj + A∙kj∙
+ DDik +
+ 3Aij A∙kj∙
+
1 2
(∗∇iFk
+
∗∇kFi)
−
1 c2
FiFk
−
− c2Cik
=
κ 2
ρc2hik + 2Uik − U hik
+λc2hik .
(6.15)
Here ∗∇j denotes the chr.inv.-derivative, while the quan-
tities
ρ
=
T00 g00
,
Ji
=
√cT0i g00
,
U ik = c2T ik
(from
which
we
have U = hikUik) are the chr.inv.-components of the energy-
momentum tensor Tαβ of matter: the physically observable density ρ, the physically observable impulse density vector Ji, and the physically observable stress-tensor U ik.
Zelmanov had deduced [42] that the chr.inv.-spatial cur-
vature tensor Ciklj is linked to a chr.inv.-tensor Hiklj,
which is like Schouten’s tensor [67], by the equation
1 Hlkij = Clkij + c2 2Akj Djl + Aij Akl +
+ AjkDil + AklDij + AliDjk
(6.16)
and
contracted
tensors
Hlk
=
Hl∙i∙k∙
i ∙
and
Clk
=
Cl∙i∙k∙
i ∙
are
re-
lated as follows
1 Hlk = Clk + c2
Akj Dlj + Alj Dkj + AklD
.
(6.17)
Taking
the
definition
Dik
=
1 2
∗∂hik ∂t
into
account,
and
Clk from (6.17), we reduce (6.15) to the form
1 2
∗∂2hik ∂t2
− Dij Dkj
+D
Dik − Aik
+ 2Aij A∙kj∙ +
1 +
2
∗∇iFk − ∗∇kFi
−
1 c2
FiFk
−
c2Hik
=
= κUik + λc2hik .
(6.18)
−
hpm
∗∂ ∂xp
∗∂him ∂xk
+
∗∂hkm ∂xi
+
2
U
+ κ ρhik + c2 Uik − c2 hik + 2λhik ,
where ∗ is the chr.inv.-d’Alembert operator, applied here to the chr.inv.-metric tensor hik (the observable metric tensor of the observer’s three-dimensional space)∗.
If we equate the right part of (6.20) in zero, the whole equation becomes a wave equation with respect to hik, namely
∗
hik
=
1 c2
∗∂ 2 hik ∂t2
−
hjm
∗∂ 2 hik ∂ xj ∂ xm
.
(6.21)
In this case the spatial components of the Einstein equa-
tions describe gravitational inertial waves of the spatial met-
ric hik, which travel at the velocity u = c
1
−
w c2
which
depends on the value of the gravitational potential w. This
coincides with the results recently obtained by Rabounski
[48]. If w = 0, the waves travel at the velocity of light. The
greater is w the smaller is u. The wave’s velocity u becomes zero in the extreme case where w = c2 which occurs under
collapse, hence under collapse gravitational waves stop —
they become standing gravitational waves.
It is evident from the mathematical viewpoint, that redu-
cing the right side of (6.20) to zero is a very difficult task,
because the whole equation is a system of 6 nonlinear equa-
tions of the 2nd order, in which numerous variables are
linked by relationships (6.13) and (6.14). Systems such as
this cannot be solved analytically in general, but we can
obtain solutions for various specific metrics.
Because experimental physicists. in their search for gravi-
tational waves, propound experimental statements for detect-
ing weak wave fields of gravitation, we are going to study a
linearized form of the equation (6.20).
∗Components of the chr.inv.-metric tensor hik satisfy the requirements
∗∇j hik = ∗∇j hki = ∗∇j hik = 0. For this reason we can apply the chr.inv.-
d’Alembert operator ∗
=
1 c2
∗∂2 ∂t2
− hik ∗∇i∗∇k
to
it.
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For (6.20) in emptiness, the linear approximation is∗
√
∗
∗∂2 ln h 2 hik = 2 ∂xi∂xk − c2
∗∇iFk
+
∗∂Aik ∂t
.
(6.22)
As a matter of fact, equation (6.22) describes weak plane
gravitational inertial waves without sources, if the wave field
satisfies the obvious chr.inv.-condition
√
∗∂2 ln h 1 ∂xi∂xk = c2
∗∇iFk
+
∗∂Aik ∂t
.
(6.23)
In other words, the field of the observable metric tensor hik is a wave field if there are some relations between the inhomogeneity of the gravitational inertial force field, the non-stationary rotation of the space, and the volume transformations of the space element, taken in the field†. The condition (6.23) is true for the well-known metric of weak plane gravitational waves (√6.12),√because in the metric (6.12) we have Fi = 0, Aik = 0, h = 1−a2−b2 ≈ 1. Thus:
Weak plane gravitational waves in emptiness are also weak plane gravitational inertial waves of the spatial observable metric hik.
As shown in [41], the metric (6.12) satisfies the Zelmanov chr.inv.-criterion for gravitational waves, where the wave functions are the Riemann-Christoffel tensor’s physically observable components Xij, Y ijk, Ziklj. Hence weak plane gravitational inertial waves (waves of the space curvature) can exist in emptiness, because of the Einstein equations. We have shown above that such wave gravitational fields can also exist in spaces of the sub-kind N by Petrov’s classification (such spaces are curved themselves, and matter contributes only an additional component to the initial curvature). Hence such fields satisfy most of the known invariant criteria for gravitational waves.
As we showed above, on page 46, that fields of gravitational radiations cannot exist in spaces of the 1st kind by Petrov’s classification. In spaces of the 1st kind Y ijk = 0. Therefore it would be logical to express the Einstein equations in the physically observable components Xij, Y ijk, Ziklj of the Riemann-Christoffel curvature tensor, aiming to find relations between the ch.inv.-quantities Xij, Y ijk, Ziklj and the physically observable components of the energy-momentum tensor Tαβ of distributed matter (ρ, Ji, Uik, see page 48).
In chr.inv.-components the Einstein equations become
Zm∙ ∙km∙k∙ = κ ρc2 + U − 2λc2,
Y
im ∙ ∙∙m
=
κJ i ,
Xik − Xhik + Z∙mim∙∙k∙ =
=
κ 2
ρc2hik + 2Uik − U hik
+ λc2hik ,
(6.24)
∗In obtaining this formula, in the initial equation (6.20), we neglect
products of the chr.inv√.-quantities and of their derivatives. †The integral of h dx1dx2dx3 is the volume of an element of the
space. Here the differentials dxi themselves and an interval, where values of the xi change where we take the integral, do not depend on x0 [42].
if matter is distributed arbitrarily. Here X = hikXik is the trace (spur) of the tensor Xik.
From here we see that the physical observable components of the Riemann-Christoffel tensor have different physical origins:
1. Quantities Xij (and as well Ziklj) are linked to the mass density ρ and the stress-tensor Uik;
2. Quantities Y ijk are linked to the impulse density Ji of matter.
As we showed above, on page 46, in all the widely known
metrics which satisfy both the invariant criteria and the chr.inv.-criterion for gravitational waves, we have Y ijk = 0, although Xij (and as well Ziklj) can be zero. This fact leads
us to a very important conclusion:
Gravitational waves and gravitational inertial waves are mainly waves of the field of the Y ijk physically
observable component of the Riemann-Christoffel curvature tensor‡.
But this conclusion does not mean that only waves of the field Y ijk can be discovered. As we will see in §7, relative accelerations of test-particles are derived from wave fields of all three observable components Xij, Y ijk, Ziklj of the Riemann-Christoffel tensor. Our conclusion means:
If in a space, filled with a gravitational field, Y ijk = 0 is true, the structure of the space itself prohibits the gravitational field from being a wave.
Contracting (6.26) and taking (6.24) into account, we
obtain
κ X=
U − ρc2
− 2λc2.
2
(6.25)
In an empty space where there are no λ-fields, the trace
of
X ij
and
the
contracted
quantity
Z
∙ ∙ mk mk ∙ ∙
are
zero,
as
well
as
the
contracted
quantity
Y
im ∙ ∙∙m
.
Thus
the
chr.inv.-Einstein
equations (6.24) in emptiness take the form§
Zm∙ ∙km∙k∙ = 0 ,
X = 0,
Y
im ∙ ∙∙m
=
0,
Xik + Z∙mim∙∙k∙ = 0 ,
(6.26)
so, while the quantities Xik and Ziklj are connected to one another, the quantity Y ijk (which, being non-zero, Y ijk = 0, permits gravitational fields to be a wave) is the independent
observable component of the Riemann-Christoffel tensor.
‡Quadrupole mass-detectors, in particular, solid-body detectors (the
Weber pigs) can only register waves of the Xij component, not waves of Y ijk if its particles are at rest in the initial moment of time (see §7 and
§8 for details). Thus, the Weber experimental statement is false at its base.
§As a matter of fact, equality to zero of inflected forms of a tensor does
not imply that the tensor quantity itself is zero. Thus, equalities X = 0,
Y
im ∙ ∙∙m
=
0,
Zm∙ ∙km∙k∙ = 0
do
not
imply
that
the
quantities
X ik ,
Y
ij k ,
Z iklj
themselves are zero. Therefore the chr.inv.-Einstein equations in emptiness
(6.24) permit gravitational waves if, of course, Y ijk = 0.
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7 Expressing Synge-Weber equation (the world-lines deviation equation) in the terms of physical observable quantities, and its exact solutions
In the previous paragraphs we focused our attention on general criteria, which differentiate gravitational wave fields from other gravitational fields in the General Theory of Relativity. As a result, we have found the main properties of gravitational wave fields.
We are now going to introduce a substantial criticism of the contemporary theoretical foundations of current attempts to detect gravitational waves by solid-body detectors of the resonance kind (the Weber pigs) and quadrupole mass detectors in general.
As we showed in the previous paragraphs, only gravitational fields located in spaces where the Riemann-Christoffel curvature tensor has a specific structure, permit the presence of gravitational waves. Therefore it would be reasonable to design experiments by which a physical detector could register wave changes of the four-dimensional (space-time) curvature∗ — the waves of the Riemann-Christoffel curvature tensor field.
Such a physical detector could be a system of two testparticles: their relative world-trajectories will necessarily undergo changes through the action of a wave of the space curvature. These systems are described by the world-lines deviation equation — the Synge equation of geodesic deviation (2.8) if these are two free particles, and the Synge-Weber equation (2.12) if the particles are connected by a force of non-gravitational nature.
We propose gravitational wave detectors of two possible kinds. The system of two free particles is known as a detector built on free masses. In practice such a detector consists of two freely suspended massive bodies, separated by a suitable distance. The system of two particles connected by a spring is known as a quadrupole mass-detector— this is a detector of the resonance kind, a typical instance of which is the Weber cylindrical pig.
To understand how a graviational wave would affect the different types of detectors we need to make specific calculations for their behaviour in gravitational wave fields. But before making the calculations, it is required to describe the behaviour of two test-particles in regular gravitational fields (of non-wave nature) in the terms of physically observable quantities (chronometric invariants). This analysis will show how different kinds of gravitational inertial waves cause relative deviation (both spatial and time displacements) of two test-particles.
We will solve this problem first for a system two free
∗It is important to note that the expected gravitational waves are waves of the space-time curvature, not merely of the spatial curvature of the threedimensional space. Consequently, waves of the four-dimensional curvature must produce changes not only in the distance between test-particles in a detector, but also in the time flow for the particles.
particles as described by the Synge equation (2.8) where the
right side is zero. The problem for spring-connected particles,
described by the Synge-Weber equation (2.12), will be solved
in the same way except that there will be a non-gravitational
force acting, so that the right side of the equation will be
non-zero.
Relative
accelerations
of
free
test-particles
D2ηα ds2
as
a
whole and the quantity R∙αβ∙γ∙∙δ are derived from components
of the Riemann-Christoffel world-tensor, contracted with
components of the particles’ four-dimensional velocity vector U β and their relative deviation vector ηγ, namely — from the quantity R∙αβ∙γ∙∙δU βU δηγ. To determine what effect is introduced by each observable component of the Riemann-
Christoffel tensor into the spatial and time relative displace-
ments, described by the relative displacement world-vector
ηα, we consider the geodesic deviation equation (2.8), keep-
ing
the
term
D2ηα ds2
as
a
whole
and
the
quantity
R∙αβ∙γ∙∙δ
without
expressing it in terms of the Christoffel symbols and their
derivatives.
As well as any general covariant equation, the geodesic
deviation equation (2.8) can be projected onto the observer’s
time line and spatial section (his three-dimensional space) as
given in [42, 43] or on page 40 herein. Denoting
Mα
≡
D2ηα ds2
+
R∙αβ∙γ∙∙δ U β U δ ηγ
=
0,
(7.1)
let us find equations which are its projection on the time line
√M0 g00
=
√g0α M α g00
=
√ g00
M
0
−
1 c
viM
i
=
0,
and its projection on the spatial section
Mi = 0.
(7.2) (7.3)
To find the equations in expanded form we need first to find the chr.inv.-projections of them, consisting of the quantities ηα and U α. Projections of the ηα onto the time line and spatial section are, respectively
ϕ ≡ √η0 , g00
ni ≡ ηi,
(7.4)
other components of the ηα are expressed through its physically observable components ϕ and ni as follows
η0 = ϕ +√1c vk nk , g00
ϕ ηi = − c vi − ni .
(7.5)
The time and spatial components of the particles’ worldvelocity vector U α are derived from the chr.inv.-definitions
given by the theory of chronometric invariants for the spacetime interval ds and the observable chr.inv.-velocity vector vi
ds = cdτ
v2 1 − c2 ,
vi = dxi , dτ
v2 = hikvivk, (7.6)
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so the required quantities U 0 and U i are
U0 =
1 dt
,
1
−
v2 c2
dτ
Ui = c
vi
.
1
−
v2 c2
(7.7)
A formula for the time function dt/dτ is obtained from∗
gαβ U αU β
= gαβ
dxα dxβ ds ds
= 1,
which can be reduced to the quadratic equation
(7.8)
dt dτ
2
−
c2
2 vivi
1
−
w c2
dt +
dτ
1
+
1
−
w c2
2
1 c2
vivkvivk
−
1
= 0,
(7.9)
which has two solutions
dt
=
1 c2
vivi+1
,
dτ 1
1
−
w c2
dt
=
1 c2
vivi−1 .
dτ 2
1
−
w c2
(7.10)
The first solution is related to a space where time flows
from past into future (a regular observer’s space), the second
solution is related to a space where time flows from future
into past with respect to a regular observer’s time flow (the
mirror Universe [70, 71]). Taking only the first root, U 0 takes
the form
U0 =
1 c2
vivi
+
1
.
1
−
v2 c2
1
−
w c2
(7.11)
Substituting
formulae
(7.5),
(7.7),
(7.11)
into
D2ηα ds2
+
+ R∙αβ∙γ∙∙δU βU δηγ = 0 (7.1), and expressing the components
of the Riemann-Christoffel tensor R∙αβ∙γ∙∙δ in terms of its phys-
ically observable components Xij, Y ijk, Ziklj, we obtain a
formula for the relative spatial oscillations of two free test-
particles
D2ηi 1 ds2 = c2−v2
Y
∙∙i mk ∙
vk−Xmi −
1 c2
Zm∙ ∙ki∙∙nvk
vn
ηm. (7.12)
From this formula we see that:
The relative spatial deviations of two free particles can be caused by all three observable components of the Riemann-Christoffel curvature tensor. Moreover, each of the components acts on the particles in a different way: (1) the field of Xik acts the particles only if they are at rest with respect to the observer’s space references, so the field of Xik can move particles only if they are at rest at the initial moment of time; (2) the fields of Y ijk and Ziklj can displace the particles with respect of each to other only if they are in motion (vi = 0) — the effect of Ziklj is perceptible if the particles move at speeds close to the velocity of light.
∗That is the evident equality.
Thus, with measurement taken by any observer, the physically observable components of the Riemann-Christoffel curvature tensor are of 3 different kinds:
1. The component Xik —of “electric kind”, because it can displace even resting particles;
2. The component Y ijk — of “magnetic kind”, because it can displace only moving particles;
3. The component Ziklj of “magnetic relativistic kind”, because it causes an effect only in particles moving at relativistic speeds.
Besides the observable spatial component ηi of the relative deviation vector ηα there is also its observable time component ϕ, which indicates the difference between time flows measured by clocks located at each of the particles.
We then obtain the relative time deviation equation for two free test-particles
√ D2η0 1 D2ηi g00 ds2 − c vi ds2 =
=
−√g00R∙0β∙γ∙∙δ U β U δ ηγ
+
1 c
viR∙iβ∙∙γ∙δ U β U δ ηγ .
(7.13)
Taking (7.10) into account and substituting the formulae for U 0, η0, U i, ηi into (7.11), then, expressing R∙0β∙γ∙∙δ in terms of physically observable quantities, we reduce formula
(7.13) to its final form
√ D2η0 1 D2ηi g00 ds2 − c vi ds2 =
1 = c2−v2
1 c Xik
ni− ϕ vk c
vk+
1 c
Yimk
vivk
η
m
.
(7.14)
Looking at this formula we note one simple thing about the effect of gravitational waves on the system of two free particles:
The time observable component of the relative deviation vector for two free particles undergoes oscillations due only to the Xik and Y ijk observable components of the Riemann-Christoffel curvature tensor, not its Ziklj component. Moreover, the fields of both the components Xik and Y ijk act on the particles only if they are in motion with respect to the space references. If the particles are at rest with respect to each other and the observer (vi = 0), the fact that the space has a Riemannian curvature makes no difference to the time flow measured in the particles.
It should be added that if the particles are in motion with respect to the space references and the observer, the effect of Xik is both linearly and quadratically dependent on the speed, whilst the effect of Y ijk is only quadratically dependent on the speed.
Thus, there is no complete analogy between the physically observable components of the Riemann-Christoffel curvature tensor and Maxwell’s electromagnetic field tensor.
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The components Xik can be interpreted “electric” only in
relative spatial displacements of two particles. In relative
time deviations between the particles (the difference between the time flow measured in the them both) both Xik and Y ijk
act on them depending on the particles’ velocity with respect
to the space references and the observer, so in this case both Xik and Y ijk are of the “magnetic” kind. Therefore the terms
“electric” and “magnetic” are only applicable relative to
observable components of the Riemann-Chrstoffel curvature
tensor. This terminology is strictly true in that case where
the particles have only relative spatial deviations, while the
time flow is the same on the both world lines.
A formula for the observable relative time deviation
ϕ
=
√η0 g00
between
two
free
particles
can
be
obtained
from
the requirement that the scalar product Uαηα remains un-
changed along geodesic trajectories, so Uαηα = const must
be true along trajectories of free particles. For this reason, if
the vectors U α and ηα are orthogonal, they are orthogonal on
the entire world-trajectory [17]. Formulating the orthogonality condition Uαηα = const in terms of physically observ-
able quantities, we introduce some corrections to the results
obtained in [17].
In terms of physically observable quantities the ortho-
gonality condition Uαηα = const, because it is actually the same as U0η0 + Ukηk = const, reduces to
ϕ
−
1 c
nivi
=
const
×
v2 1 − c2 .
(7.15)
effects will be presented in future article. Here we focus our
attention on particles of non-zero rest-mass m0 = 0 (so-called mass-bearing particles), which are at rest with respect to the
space references and the observer or, alternatively, moving
at sub-light speeds.
In equations (7.10) and (7.12), we kept the second absol-
ute
derivative
D2ηα ds2
of
the
relative
deviation
vector
ηα
as
a
whole, because we were concerned only with the effects in-
troduced by the Riemannian curvatureto the relative spatial
acceleration
D2ηi ds2
and
relative
time
acceleration
D2η0 ds2
of
two free test-particles.
But if we wish to obtain solutions to the world-lines
deviation
equation,
we
need
to
express
the
quantity
D2ηα ds2
and also R∙αβ∙γ∙∙δ in terms of the Christoffel symbols and their
derivatives.
We are now going to obtain solutions to the deviation
equation for geodesic lines (the Synge equation).
Taking the definition
Dηα ds
=
dηα ds
+ Γαμν ημU ν
(7.16)
into account, we obtain
D2ηα ds2
=
d2ηα ds2
+
dΓαμν ds
ημU ν
+
2Γαμν
dημ ds
Uν +
+
Γαμν
ημ
dU ν ds
+ ΓαρσΓρμν ημU ν U σ
=
0.
(7.17)
From this we see that the vectors U α and ηα are orthogonal only if v2 = c2, i. e. U α is isotropic: gαβ U αU β = 0. So if U α and ηα are orthogonal, we have the deviation equation for two isotropic geodesics — world-lines of lightlike particles moving at the velocity of light. We defer this case for the moment and consider only the case of two neighbour non-isotropic geodesics. In the particular case when two particles are moving on neighbouring geodesics, and are at rest with respect to the observer and his references (only the time flow is different in the particles), formula (7.15) leads to ϕ = const.
This formula verifies our previous conclusion that the particles have a time deviation only if they are in motion. The greater their velocity with respect to the space reference and the observer, the greater the deviation between the time flow on both world-lines. Thus measurement of time deviations between two particles in gravitational waves and gravitational inertial waves would be easier in experiments where the particles move at high speeds. In practice such an experimental statement could be realized using light-like particles (in particular, photons). A time deviation of two photons in gravitational wave fields can manifest as changes in the frequencies of two parallel light rays (laser beams, for instance), while a spatial deviation of the photons can manifest as changes in the phases of the light rays. Calculations of these
We write R∙αβ∙γ∙∙δ as
R∙αβ∙γ∙∙δ
=
∂Γαβδ ∂xγ
−
∂Γαβγ ∂xδ
+ Γσβδ Γαγσ
− Γσβγ Γασδ ,
express
dU α ds
via the
geodesic equations
dU α ds
=
−Γαμν U μU ν ,
(7.18) (7.19)
and use the definition
dΓαμν ds
=
∂Γαμν ∂xσ
Uσ.
(7.20)
Using the auxiliary formulae we obtain from (7.17) the Synge equation (the geodesic lines deviation equation) in its final form
d2ηα ds2
+
2Γαμν
dημ ds
Uν
+
∂Γαβδ ∂xγ
U β U δ ηγ
=
0.
(7.21)
This is a differential equation of the 2nd order with respect to the quantity ηα: the equation is a system of 4 differential equations with respect to the quantities η0 and ηi (i = 1, 2, 3). The variable coefficients Γαμν and their derivatives must be taken for that gravitational field, whose
waves act on two free test-particles in our experiment. The
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world-quantities U ν (ν = 0, 1, 2, 3) can be found as solutions to the geodesic equations
dU ν ds
+ ΓνμρU μU ρ
=
0
(7.22)
only if the particles move with respect to the space references and the observer. If the particles are at rest with respect to the observer and his references, the components of their worldvelocity vector U ν are
U0
=
1 √
,
g00
Ui = 0,
(7.23)
and, according to (7.13–7.15), their relative time deviation is zero, ϕ = 0 (the time flow measured on both geodesic lines is the same).
Current detectors used in the search for gravitational wave radiations are of such a construction that the particles therein, which detect the waves, are almost at rest with respect of each other and the observer. Experimental physicists, following Joseph Weber and his methods, think that gravitational waves can cause the rest-particles to undergo a relative displacement. With the current theory of the gravitational wave experiment, the experimental physicists limit themselves to the expected amplitude and energy of waves arriving from a proposed source of a gravitational wave field.
However, to set up the gravitational wave experiment correctly, we need to eliminate all extraneous assumptions and traditions. We merely need to obtain exact solutions to the world-lines deviation equation, applied to detectors of that kind which this experiment uses.
Detectors described by the geodesic lines deviation equation (the Synge equation), which we consider in this section, are known as “antennae built on free masses”. We shall consider such detectors first.
The detectors consist of two freely suspended masses which are at rest with respect of each other and the observer, and separated by an appreciable distance. These could be two mirrors, located in a near-to-Earth orbit, for instance. Each of the mirrors is fitted with a laser range-finder, so we can measure the distance between them with high precision.
In order to interpret the possible results of such an experiment, we need to solve the Synge equation (7.21), expressing its solutions in the terms of physically observable quantities (chronometric invariants). Following “tradition”, we solve the Synge equation for particles which are at rest with respect to each other and the observer’s space references. So we consider that case where the particles’ observable velocities are zero (vi = 0).
At first, because we are going to obtain solutions to the Synge equation in chr.inv.-from, we need to know the physically observable characteristics of the observer’s reference space through which we express the solutions. We find the chr.inv.-characteristics from the geodesic equations taken
in the main chr.inv.-from [42]
dm dτ
−
m c2
Fivi
+
m c2
Dik vi vk
=
0,
(7.24)
d (mvi) + 2m dτ
Dki + A∙ki∙
vk − mF i +
+ mΔinkvnvk = 0 ,
(7.25)
for each of the particles (because both particles are at rest
with respect to one another, their geodesic equations are
the same). Here m is the particle’s relativistic mass, which, because in the case we are considering vi = 0, reduces to
the rest-mass m = m0. Then the geodesic equations take the very simple form
dm = 0,
dτ
(7.26)
mF i = 0 ,
(7.27)
so in this case the chr.inv.-vector of gravitational inertial
force is F i = 0: the particles are in free fall. In this case
we
can
transform
coordinates
so
that
g00 = 0
and
∂g0i ∂t
=0
[42]. This implies that the Synge initial equation (7.19) can
be solved correctly only for gravitational fields where the
potential is weak w = 0 (i. e. g00 = 1) and where the space
rotation
is
stationary
∂Aik ∂t
= 0.
It
should
be
noted
that
the
metric of weak plane gravitational waves, the only metric
used in the theory of gravitational wave experiments, satisfies
these requirements.
Because
ϕ
=
1 c
nivi
(7.15),
in
the
case
we
are
consider-
ing the time observable component ϕ of the relative deviation
vector ηα is zero ϕ = 0. For this reason we consider only the
observable spatial component of the Synge equation (7.21).
In the accompanying reference frame (where the observer
accompanies his references), according to the theory of chro-
nometric invariants [42, 43], in the absence of gravitational
fields w = 0 and also gravitational inertial forces Fi = 0, we
have:
d ds
=
1 c
d dτ
,
U
0
=
√1 g00
=
1
,
U
i
=
1 c
vi,
η0 = −g0iηi,
Γi00
=
−
1 c2
1
−
w c2
2
Fi
= 0,
Γi0k
=
1 c
1
−
w c2
Dki + A∙ki∙+
+
1 c2
vkF
k
=
1 c
Dki + A∙ki∙ .
Employing now the formulae
for the Synge equation (7.21) under vi = 0, we obtain the
Synge equation in chr.inv.-form∗
d2ηi +2
dτ 2
Dki + A∙ki∙
dηk dτ
= 0.
(7.28)
The
quantity
d dτ
=
∗∂ ∂t
+
vi
∗∂ ∂xi
[42,
43]
here
is
d∂ =,
dτ ∂t
(7.29)
∗As we mentioned, if the particles are at rest vi = 0, the chr.inv.-time component of the Synge equation becomes zero.
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so the chr.inv.Synge -equation (7.28) takes its final form
∂2ηi ∂t2 + 2
Dki + A∙ki∙
∂ηk ∂t
= 0.
(7.30)
where D1 and D2 are integration constants. Assuming x1 = 0 at the initial moment of time t = 0, we easily express the integration constants C1, C2, D1, D2 through the initial conditions. Finally, we obtain solutions
We find the exact solution to the Synge chr.inv.-equation (7.30) in the field of weak plane gravitational waves∗. In the case we are considering (vi = 0) we have
Fi = 0 , Aik = 0 , 1 ∂a
D22 = −D33 = 2 ∂t ,
1 ∂b D23 = 2 ∂t .
(7.31)
Substituting the requirements into the initial equation (7.30) we obtain a system of three equations
∂2η1 ∂t2 = 0 ,
(7.32)
∂2η2 ∂a ∂η2 ∂b ∂η3
∂t2
+ ∂t
∂t
− ∂t
∂t
= 0,
(7.33)
∂2η3 ∂a ∂η3 ∂b ∂η2
∂t2
− ∂t
∂t
− ∂t
∂t
= 0.
(7.34)
The solution of (7.32) is
η1 = η(10) + η˙(10)t ,
(7.35)
where η(10) is the particle’s initial deviation, η˙(10) is its initial velocity.
This system can be easy solved in two particular cases of a linear polarized wave: (1) b = 0, and (2) a = 0.
In the first case (b = 0) we obtain
η2
= η˙(20)
Aω t + cos
ωc
ct ± x1
+
η(20)
−
A ω
η˙(20)
,
(7.41)
η3
= η˙(30)
Aω t − cos
ωc
ct ± x1
+
η(30)
−
A ω
η˙(30)
,
(7.42)
where η(20), η(30) and η˙(20), η˙(30) are the initial numerical values of the relative deviation η and relative velocity η˙ of the particles along the x2 and x3 axes, respectively.
We have now obtained the exact solutions to the Synge
equation (the geodesic lines deviation equation). From the
solutions we see,
If at the initial moment of time t = 0, two free particles are at rest with respect to each other and the observer η˙(20) = η˙(30) = 0, weak plane gravitational waves of the deformation kind (waves of the Riemannian curvature) cannot force the particles to go into relative motion. If at the initial moment of time the particles are in motion, the waves augment the particles’ initial motion, accelerating them.
Thus our purely mathematical analysis of detectors built on free masses leads to the final conclusion:
Weak plane gravitational waves of the deformation kind (the Riemannian curvature’s waves) cannot be detected by any antenna composed on free masses, if the masses are at rest with respect to each other and the observer.
∂η2 ∂t
=
C1 e−a,
∂η3 ∂t
=
C2 e+a,
(7.36)
where C1 and C2 are integration constants. Because values of a are weak, we can decompose e−a into series. Then,
assuming higher order terms infinitesimal, we obtain
∂η2 ∂t = C1 (1 − a) ,
∂η3 ∂t = C2 (1 + a) .
(7.37)
Assuming also that a falling gravitational wave is monochrome, bearing a constant amplitude A and a frequency ω,
ω a = A sin
ct ± x1
,
c
(7.38)
we integrate the system (7.37). As a result we obtain
η2
= C1
t+
Aω cos
ωc
ct ± x1
η3
= C2
t−
Aω cos
ωc
ct ± x1
+ D1 , + D2 ,
(7.39) (7.40)
∗Where the metric (5.7) is ds2 = c2dt2 − (dx1)2 − (1 − a)(dx2)2 + + 2bdx2dx3 − (1 + a)(dx3)2.
8 Criticism of Weber’s conclusions on the possibility of detecting gravitational waves by solid-body detectors of the resonance kind
Historically, the first gravitational wave detector was the quadrupole mass-detector built in 1964 by Prof. Joseph Weber with his students David Zipoy and Robert Forward at Maryland University [70]. It was an aluminium cylindrical pig weighing 1.5 tons, suspended by a steel “thread” in a vacuum camera. At the point of connection between the pig and the thread, the pig was covered by a piezoelectric quartz film linked to a highly sensitive voltmeter. Weber expected that a falling gravitational wave should make relative displacements of the butt-ends of the cylindrical pig — extension or compression of the pig. In other words, they expected that falling gravitational waves will deform the pig, necessarily causing a piezoelectric effect in it. Modified by Sinsky [71], the first detector gave a possibility of registering a 10−16cm relative displacement of its butt-ends.
Later, Weber built a system of two pigs. That system worked through the principle of coincident frequencies of
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the signals registered in both pigs. The pigs had a relaxation time about 30 sec, were tuned for the frequency 104 rps, and were separated by 2 km. In 1967 Weber and his team registered coincident signals (to a precision within 0.2 sec) which appeared about once a month [1]. The registered relative displacements of the butt-ends in the pigs were ∼3×10−10cm. Weber supposed that the origin of the observed signals were gravitational wave radiations.
Weber subsequently even used 6 pigs, one of which was located at Argonne National Laboratory (Illinois), the other 5 pigs located in his laboratory at Maryland University. The distance between the laboratories wss about 1000 km. The detectors were tuned for 1660 Hz — the frequency of supposed gravitational radiations excited from collapsing supernovae. During several months of observations, numerous coincident signals were registered [72]. A second cycle of the observations gave the same positive result [73]. Weber interpreted the registered signals as proof that strong gravitational radiations exist in the Galaxy. A peculiarity of those experiments was that the pigs located both in Illinois and Maryland were isolated as much as possible from external electromagnetic and seismic influences.
After Weber’s pioneering experiments, experimental physicists built many similar detectors, much more sensitive than those of Weber. However, in contrast to those of Weber, not one of them registered any signals.
Therefore, using the world-lines deviation theory developed here in the terms of physically observable quantities, we are going to:
(1) investigate what in principle can be registered by a solid-body detector (a Weber pig) and
(2) compare our conclusion with that explanation given by Weber himself for his observed signals.
From the theoretical viewpoint we can conceive of a
solid-body cylindrical detector as consisting of two test-
particles, connected by a spring [16]. It is supposed that
the system falls freely. It is also supposed that at the initial
moment of time, when we start our measurements, the par-
ticles are at rest with respect to us (the observers) and each
other. This is the standard problem statement, not only of
Weber [16] or ourselves, but also of any other theoretical
physicist.
The behaviour of two neighbouring particles in their
motion along their neighbouring world-lines is described
by the world-lines deviation equation. If the particles are not free, but connected by a non-gravitational force Φα
(a spring, for instance), the Synge-Weber equation (2.12)
applies,
namely
D2ηα ds2
+ Rα∙ β∙γ∙ ∙δ U β U δ ηγ
=
1 m0c2
DΦα dv
dv.
This is an inhomogeneous differential equation of the 2nd
order with respect to the relative deviation vector ηα of the
particles. In order to solve the world-lines deviation equation
we
need
to
write
D2ηα ds2
and
DΦα dv
dv
in
expanded
form.
Because both terms contain the Christoffel symbols Γαμν , it would be reasonable to express the components of the
Riemann-Christoffel tensor Rα∙ β∙γ∙ ∙δ in terms of the Γαμν and their derivatives: in collecting similar terms some of them
will cancel out (the same situation arose when we made the
same calculations for the geodesic lines deviation equation).
Using
formulae
(7.17)
and
(7.18),
and
the
quantity
dU α ds
from the world-line equation of a particle moved by a non-
gravitational force Φα (2.11), we obtain
dU α ds
=
−ΓαρσU ρU σ
+
Φα m0c2
.
Expanding
the
formula
for
DΦα dv
dv
DΦα dv
dv
=
∂Φα ∂v
dv
+
Γαμν
Φμ
∂xv ∂v
dv
=
=
∂Φα ∂xσ
ησ
+
Γαμν Φμην
(8.1) (8.2)
and substituting this into the world-lines deviation equation in its initial form (2.12), takeing into account that (8.1) and (8.2), we obtain
d2ηα ds2
+
2Γαμν
dημ ds
Uν
+
∂Γαβδ ∂xγ
U β U δ ηγ
=
=
1 m0c2
∂Φα ∂xσ
ησ.
(8.3)
This is the final form of the world-lines deviation equation for
two test-particles connected by a spring. The quantities ηα
and
Uα
are
connected
by
(2.13):
∂ ∂s
(Uαηα)
=
1 m0c2
Φαηα.
In a gravitational wave detector like Weber’s, the cy-
lindrical pig is isolated as much as possible from external
influences of thermal, electromagnetic, seismic and another
origins. To minimise external influences, experimental physi-
cists place the detectors in mines located deep inside mount-
ains or otherwise deep beneath the terrestrial surface, and
cool the pigs to 2 K. Therefore particles of matter in the
butt-ends and the pig in general, can be assumed at rest with
respect to one another and to the observer.
Following Weber, experimental physicists expect that a
falling gravitational wave will deform the pig, displacing its
butt-ends with respect to each other. Relative displacements
of the butt-ends of a pig are supposed to result in a piezo-
electric effect which can be registered by a piezo-detector. In
other words, experimental physicists expect that oscillations
of the acting gravitational wave field give rise to a force in the
world-lines deviation equation (the Synge-Weber equation),
thereby displaceing the test-particles which were at rest at the
initial moment of time. Oscillations of the acting gravitational
wave field force the butt-ends of the pig to oscillate. As
soon as the frequency of the pig’s oscillations coincides
with the falling wave’s frequency, the amplitude of the pig’s
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oscillations will increase significantly because of resonance,
so the amplitude becomes measurable. Therefore the Weber
detectors are said to be of the resonance kind.
Before ratifying the aforesaid conclusions it would be
reasonable to study the world-lines deviation equation for
two interacting test-particles that model a Weber pig, because
this equation is the theoretical basis of all experimental
attempts to register gravitational waves made by Weber and
his followers during more than 30 years.
We will study this equation, proceeding from its form
(8.3), because formula (8.3) gives a possibility of obtaining exact solutions to the relative deviation vector ηα; not the
initial equation (2.12). Following analysis of the solutions
we will come to a conclusion as to what effect a falling gravitational wave has on the detector∗.
When we need to give a theoretical interpretation of
experimental results, it is very important to analyse the results
in the terms of physically observable quantities because such
quantities can be registered in practice. For this reason we
will study the behaviour of the Weber model (the system
of two particles, connected by a non-gravitational force) in
the terms of physically observable quantities (chronometric
invariants) as we did in §7 when we solved a similar problem
for the system of two free particles.
In detail, our task here is to consider commonly the
world-lines equation (2.11) and the world-lines deviation
equation (8.3), both written in chr.inv.-form. Note that the
relationship
(2.13),
that
is
∂ ∂s
(Uαηα)
=
1 m0c2
Φαηα,
gives
the exact solution for the quantity ϕ. The ϕ is the chr.inv.-
time component of the relative deviation world-vector ηα
with respect to which the world-lines deviation equation (8.3)
is written. For this reason there is no need here to solve the
chr.inv.-time projection of the world-lines deviation equation
(8.3). We solve instead the relationship (2.13).
The world-lines equation (2.11) in chr.inv.-form is
dm dτ
−
m c2
Fivi
+
m c2
Dik vi vk
=
σ c
,
(8.4)
d (mvi)+2m dτ
Dki +A∙ki∙
vk−mF i+Δiknvkvn=f i,
(8.5)
where
σ≡
√Φ0 g00
and
f i ≡ Φi
are
chr.inv.-components
of the
prevailing non-gravitational force Φα. In the case of the
Weber model where the particles are at rest with respect to
the observer (vi = 0), the chr.inv.-equations (8.4) and (8.5)
become
σ = 0,
(8.6)
m0F i = −f i.
(8.7)
The condition σ = 0 comes from the fact that, when a particle is at rest its relativistic mass becomes the restmass m = m0. Thus resting particles are under the action
∗It is evident that equation (8.3) can be solved also for other forcing fields, which can be of a non-wave origin.
of only the spatial observable components f i of the nongravitational force Φα, so that the f i are of the same value as the acting gravitational inertial force F i, but acts in the
opposite direction. Looking at definition (4.1), given by the
theory of physical observable quantities for the gravitational inertial force F i, we see that in this case the non-gravitational force f i acts on a resting particle only if at least one of the
following factor holds:
1.
Inhomogeneity
of the
gravitational potential
∂w ∂xi
= 0;
2. Non-stationarity of the vector of the space rotation
linear
velocity
∂vi ∂t
= 0.
If neither factor holds, F i = 0 and hence f i = 0, in which
case both interacting particles, which are at rest with respect
to each other and the observer, behaviour like free particles: their connecting force Φα does not manifest. Looking at the
well-known metric (5.7) that describes weak plane gravitational waves, we see there that F i = 0, Aik = 0 and hence vi = 0. Therefore:
Weak plane gravitational waves described by the met-
ric (5.7) cannot be registered by a solid-body detect-
or of the resonance kind (a Weber detector).
Writing the relationship (2.13) in chr.inv.-form, we obtain
d
ϕ
−
1 c
nivi 
=
σϕ −
fi ni
,
dτ
1
−
v2 c2
mc
(8.8)
where
again,
ϕ≡
√η0 g00
and
ni ≡ ηi
are
chr.inv.-components
of the relative deviation world-vector ηα.
From this we see that the angle between the vectors Uα and ηα is a variable depending on many factors, including
the velocity vi of the particles. At speeds close to that of
light c, the angle increases. At v = c formula (8.8) becomes
senseless: the denominator on the left side becomes zero.
If both particles are at rest, formula (8.8) becomes
dϕ = − fi ni = Fi ni ,
dτ
m0c
c
(8.9)
so that in the case of interacting rest-particles, in contrast to free ones, there is the time observable component ϕ of the relative deviation vector ηα. This implies that there are not only relative spatial displacements of the particles, but also a deviation between measurements of time made by the clocks of both particles. The “time deviation” ϕ can be found by integrating (8.9). We obtain
1 ϕ=
c
Fi ni + const ,
(8.10)
so the value of the “time deviation” ϕ increases with time.
It
Note
that
ϕ τ
=
0
only
if
the
vector
Fi
(and
hence,
in
this case, also f i) is orthogonal to the vector ni, so that
Fi
ni
=
−
1 m0
fi
ni
=
0.
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The integral (8.10) is the solution to the chr.inv.-time component of the world-lines deviation equation (8.3). This solution for ϕ itself, being a chronometric invariant, is a physically observable quantity.
We are now going to obtain solutions to the remaining three chr.inv.-equations with respect to ηα:
1 c2
d2ηi dτ 2
+
2Γi00
1 c
dη0 dτ
U0
+
2Γik0
1 c
dηk dτ
U0
+
(8.11)
+
∂Γi00 ∂x0
U
0U
0η0
+
∂Γi00 ∂xk
U 0U 0ηk
=
1 m0c2
∂Φi ∂xσ
ησ,
— the chr.inv.-spatial components of the world-lines deviation
equation (8.3), in the case of two rest-particles ds = cdτ . In the left side of (8.11) we substitute the formulae for
the quantities Γi00, Γi00, U 0, η0, given on page 53, and also derivatives of the quantities. Then we transform the right part
of (8.11) as follows
∂Φi ∂xσ
ησ
=
∂fi ∂x0
η0
+
∂fi ∂xk
ηk
=
ϕ ∗∂fi c ∂t
+
∗∂f i ∂xk
nk,
(8.12)
where we use the definitions of the chr.inv.-derivative oper-
ators
(see
page
40):
∗∂ ∂t
=
√1 g00
∂ ∂t
and
∗∂ ∂xi
=
∂ ∂xi
+
1 c2
vi
∗∂ ∂t
.
The initial equation (8.11) becomes
d2ni dτ 2 +2
Dki +A∙ki∙
dnk dτ
−
2 c
dϕ dτ
F
i
+
2 c2
Fk
nk
F
i
−
(8.13)
−
ϕ c
∗∂F ∂t
i
−
∗∂F i ∂xk
nk
=
1 m0
ϕ c
∗∂f ∂t
i
+
∗∂F i ∂xk
nk
.
Owing to the particular conditions (8.7) and (8.9), derived from the requirement that the particles are at rest (vi = 0),
formula (8.13) becomes much more simple
d2ni dτ 2 + 2
Dki +A∙ki∙
dnk dτ
= 0,
(8.14)
which is the final form for the chr.inv.-spatial deviation eq-
uation for two rest-particles, connected by a non-gravitational
force.
Equation (8.14) is like the chr.inv.-spatial deviation equation for two free rest-particles (7.30) — the chr.inv.-spatial
part of the Synge general covariant equation. The difference
is
that
(8.14)
contains
derivatives
d dτ
=
√1 g00
∂ ∂t
,
while
(7.30)
contains ∂ . This difference is derived from the fact that ∂t
(7.30) is applicable to gravitational fields where Fi = 0, the
potential
w
is
neglected
and
hence
∂vi ∂t
=
0,
while
(8.14)
describes the relative deviation of two particles located in
gravitational fields where Fi = 0. The required condition Fi = 0 implies:
1. In this region the gravitational potential is w=0, hence, because the interval of physical observable time is
dτ =
1
−
w c2
dt
−
1 c2
vidxi,
the
time
flow
differences
at different points inside the region. In particular, if
vi = 0, synchronization of clocks located at different points cannot be conserved. In the more general case
where vi = 0, clocks located at different points cannot be synchronized [42, 43];∗
2. If the gravitational inertial force field F i is vortical,
the
space
rotation
is
non-stationary
∂vi ∂t
= 0.
Let us get back to the chr.inv.-spatial equation for two
particles connected by a non-gravitational force (8.14). There
are quantities Dik and Aik, so relative accelerations of the particles can be derived from both the space deformations
and rotation. In this problem statement, w implies that the
gravitational potential of a distant source of gravitational
radiations. So in a gravitational wave experiment we should
specify
the
acting
gravitational
field
as
weak
w c2
≈ 0,
hence
in
the experiment the chr.inv.-gravitational inertial force vector
Fi
(4.1)
becomes
Fi ≈
∂w ∂xi
−
∂vi ∂t
.
There
are
as
well
d dτ
=
∂ ∂t
.
We solve equation (8.14) in two cases, aiming to find
what kind of gravitational field fluctuations were registered
by Weber and his colleagues.
First case: Aik = 0, Dik = 0.
In
this
case
equation
(8.14),
with
w c2
≈ 0,
is
the
same
as
the
chr.inv.-world-lines deviation equation for two free particles
(7.30). As it was shown in §7, with solutions of equation
(7.30) considered, a gravitational wave can affect the system
of two free particles only if the particles are in motion at the
initial moment of time. In that case a gravitational wave can
only augment the initial motion of the particles. If they are at
rest gravitational waves can have no effect on the particles.
∗To realize the condition w = 0 it is not necessary to have a wave gravitational field. In particular, w = 0 is true even in stationary gravitational fields derived from island masses (like Schwarzschild’s metric). Moreover, the phenomenon of different time flow in the Earth gravitational field is well-known from experimental tests of the General Theory of Relativity: a standard clock, located on the terrestrial surface, shows time which is ∼10−9sec different from time measured by the same standard clock, located in a balloon a few kilometers above the terrestrial surface (the difference increases with the duration of the experiment). But such corrections of time are not linked to the presence of gravitational waves.
There time corrections can also be registered, the origin of which are wave changes of the gravitational potential w. They can be interpreted as waves of the gravitational inertial force field F i. In this case corrections to standard clocks, located at different points, should bear a relation to wave changes of w.
The presence of the space rotation vi = 0 changes the time flow as well. Experiments, where a standard clock was moved by a jet plane around the world [49, 50, 51, 52], showed differential time flow with respect to the same standard clock located at rest at the air force base. Such difference of measured time, called the desynchronization correction, depends on the flight direction — with or opposite to Earth’s rotation. Although such corrections are derived from the Earth rotation (the reference space rotation), in the “background” of such corrections there could also be registered additional tiny corrections derived from the rapid stationary rotation field of a massive space body, located far from the Earth.
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When Aik = 0 the chr.inv.-world-lines deviation equation (8.14), describing a Weber detector, coincides with equation (7.30), and we conclude:
We differentiate the first equation with respect to t
x¨ = 2 ωy˙
(8.23)
A Weber detector (a solid-body detector of the resonance kind) will have no response to a falling gravitational wave of the pure deformation kind, if the particles of which the detector is composed are at rest at the initial moment of measurements (the situation assumed in the Weber experiment).
Second case: Dik = 0, Aik = 0.
We assume that the space rotation has a constant angular velocity ω around the x3 axis, while the linear velocity of this rotation is vi = ωk∙i∙xk c. For the background metric, following the classical approach [14, 15], we use the Min-
kowski line element, where the gravitational waves are superimposed as tiny corrections to it. Then the components of vi
are v1 = −ωx2, v2 = ωx1, v3 = 0 ,
(8.15)
and substitute y˙ = x¨/2 ω into the second one. We obtain a harmonic oscillation equation
x¨ + 4ω2x = 0 ,
(8.24)
with
respect
to
the
relative
velocity
x=
∂η1 ∂t
of
the
particles.
The solution to (8.24) is
∂η1 x = ∂t = C1 cos 2ωt + C2 sin 2ωt ,
(8.25)
where C1 and C2 are integration constants, which can be obtained from the initial conditions. Thus we obtain
∂η1
∂η1
1 ∂2η1
= ∂t
cos 2ωt +
∂t (0)
2ω
∂t2
sin 2ωt , (8.26)
(0)
and the space metric in its expanded form is
ds2 = c2dt2 + 2 ω (x2dx1 − x1dx2) dt − − (dx1)2 − (dx2)2 − (dx3)2.
(8.16)
This metric describes the four-dimensional space of a uniformly rotating reference frame, whose rotational linear velocity is negligible with respect to c.
Components of the tensor Aik are
A∙21∙ = ω2∙1∙ = −ω , A∙12∙ = ω1∙2∙ = ω , A∙13∙ = 0 . (8.17)
Substituting (8.17) into the chr.inv.-world-lines deviation equation (8.14) we obtain a system of deviation equations
∂2η1
∂η2
− 2ω = 0,
∂t2
∂t
(8.18)
where terms marked with zero are the initial values of the relative velocity and acceleration of the particles. Integrating (8.26), we obtain
η1
=
η˙(10) 2ω
sin 2ωt
−
η¨(10) 4ω2
cos 2ωt
+
B1
,
(8.27)
where B1 is an integration constant. Obtaining B1 from the initial conditions, we obtain the final formula for η1
η1
=
η˙(10) 2ω
sin 2ωt −
η¨(10) 4ω2
cos 2ωt +
η(10)
+
η¨(10) 4ω2
.
(8.28)
In the same fashion we obtain a formula for η2
η2
=
η˙(20) 2ω
sin 2ωt −
η¨(20) 4ω2
cos 2ωt +
η(20)
+
η¨(20) 4ω2
.
(8.29)
∂2η2
∂η1
∂t2
+ 2ω ∂t
= 0,
(8.19)
∂2η3 ∂t2 = 0 ,
(8.20)
which commonly describe behaviour of two neighboring restparticles in a uniformly rotating reference frame.
Equation (8.20) can be integrated immediately
η3 = η(30) + η˙(30)t ,
(8.21)
where η(30) and η˙(30) are the initial values of the relative displacement and velocity of the particles along the x3 axis.
In integrating equations (8.19) and (8.20), we introduce
the
notation
∂η1 ∂t
≡x
and
∂η2 ∂t
≡ y.
Then
we
have
x˙ − 2ωy = 0 .
y˙ + 2 ωx = 0
(8.22)
By the exact solutions (8.21), (8.28), (8.29), obtained for the world-lines deviation equation taken in chr.inv.-form (8.14), it follows that:
Stationary rotations of the space cannot force two neighbouring particles to initiate relative motion, if they are at rest at the initial moment of time.
In common with the result obtained in §7, where we discussed gravitational wave detectors built on free masses, we arrive at a final conclusion for the possibilities of gravitational wave detectors:
Behaviour of both a gravitational wave detector built on free masses and a sold-body detector (a Weber pig) are similar. The only difference is that a solidbody detector can register both the time observable component and spatial observable components of the relative deviation vector, while a free-mass detector can register only spatial observable deviations. Deformations and stationary rotation of the space do not affect detectors of either kind.
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Thus neither deformations nor stationary rotation of the space can not induce relative motion in the butt-ends of a Weber detector, if they are at rest. However Weber and his teamregistered signals. The question therefore arises:
What signals did Weber register, and why, during the past 30 years, have his signals remained undetected by other researchers using superior detectors of the Weber kind?
We assume that the signals registered by Weber and his team, were much more than noise, and beyond doubt. Therefore, according to our theoretical analysis of the behaviour of solid-body detectors in weak gravitational waves, we make the following suppositions:
1. Weber registered signals which were an effect made in the pig by a vortex of the gravitational inertial force field. In other words, the origin of the signals could be rapid non-stationary rotation of a distant object in the depths of space;
2. The particles of the aluminium cylindrical pig, used by Weber, had substantial thermal motions. In this case parametric oscillations could appear as an effect of a falling gravitational wave. But in order to get such a real effect, the “background thermal oscillations” should be substantial;
3. The signals were registered only by Weber and his team. Not one signal has been registered by other experimental physicist during the subsequent 30 years, using superior detectors of the Weber kind. Either Weber registered gravitational waves derived from a non-stationary rotating object in the Universe, which occurred as a unique and short-lived phenomenon, or his original detector had a substantial peculiarity that made it differ in principle from the detectors used by other scientists.
We consider Weber’s theory, aiming to ascertain what he registered with his solid-body detector.
9 Criticism of Weber’s theory of detecting gravitational waves
In his book in 1960 [16], Weber propounded his theoretical arguments for the detection of gravitational waves by means of a solid-body detector of the resonance kind. He built his theory on the world-lines deviation equation for two particles, connected by a non-gravitational force (a spring, for instance). This is equation (2.12), being a modification of the well-known deviation equation for two free particles deduced by Synge (2.8), is known as the Synge-Weber equation. We considered both equations in detail above.
There is no doubt that the Synge-Weber equation is valid. Our main claim here is that Weber himself, in his analysis of the equation in order to build the theory for detecting gravitational waves, introduced a substantial assumption:
Weber’s assumption 1 A falling gravitational wave should produce relative displacements of the butt-ends of a cylindrical pig.
So he obtained the same principle that he introduced, precluding himself from any possibility of obtaining anything else.
This line of reasoning constitutes a vicious circle. It would be been more reasonable and honest to have solved the world-lines deviation equation. Then he would have obtained exact solutions to the equation as was done in the previous sections herein.
In detail Weber’s assumption 1 leads to the fact that, having a system of two test-particles connected by a spring, the resulting distance vector between them should be [16]
ηα = rα + ξα, rα ξα,
(9.1)
where the initial distance vector rα is the such that
D rα = 0.
ds
(9.2)
He supposed as well that ηα → rα in the ultimate case
where the friction rises infinitely or the Riemann-Christoffel curvature tensor becomes zero R∙αβ∙γ∙∙δ = 0 [16].
Taking the main supposition (9.1) into account, Weber
transforms the Synge-Weber equation (2.12) into
D2ξα ds2
+ Rα∙ β∙γ∙ ∙δ U β U δ (rγ
+ ξγ)
=
1 m0c2
fα,
(9.3)
where f α is the difference between non-gravitational forces
of the particles’ interaction. Weber assumes f α the sum of
the elasticity force f1α = −Kσα ξσ that restores the particles,
and
the
oscillation
relaxing
force
f2α
=
−
c
Dσα
Dξσ ds
,
where
Kσα and Dσα are the elasticity and friction coefficients, re-
spectively. Then (9.3) takes the form
D2ξα ds2
+
1 m0
c
Dσα
Dξσ ds
+
1 m0c2
Kσα
ξσ
=
= −Rα∙ β∙γ∙ ∙δ U β U δ (rγ + ξγ ) .
(9.4)
Weber introduced additional substantial assumptions:
Weber’s assumption 2 The whole detector is in the state of free falling;
Weber’s assumption 3 The reference frame in his laboratory is such that the Christoffel symbols can be assumed zero.
Because of these assumptions, and the condition | r| |ξ|, Weber writes equation (9.4) as follows
d2ξα dt2
+
1 m0
Dσα
dξσ dt
+
1 m0c2
Kσα
ξσ
= −c2Rα∙ 0∙σ∙0∙ rσ.
(9.5)
Looking at the right side of Weber’s equation (9.5) we see his fourth hidden assumption:
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Weber’s assumption 4 Particles located on two neighbouring world-lines in the Weber experimental statement (the butt-ends of his cylindrical pig) are at rest at the initial moment of time, so U i = 0.
In §7, where we considered chr.inv.-equations of motion for two particles connected by a non-gravitational force (8.4) and (8.5), we came to the conclusion: a reference frame where interacting particles (Φα = 0) are at rest (vi = 0) cannot be in astate of free fall. Really, the free fall condition is F i = 0. Equation m0F i = −f i (8.7), which is the chr.inv.form of spatial equations of motion of the interacting particles, implies that when F i = 0, f i = 0. Therefore:
It should be noted that the main structure of motion is
determined by the left (geometrical) side of equations of
motion, while the right side introduces only an additional
effect into the motion.
In my previous articles [74, 75, 76] common exact sol-
utions to the geodesic equations and the deviation equation
had been obtained in the field of weak plane gravitational
waves, described by the metric (6.12). The exact solutions
had been obtained in general covariant form.
The solutions to the equations of motion for a free partic-
le, equations (2.6), in a linear polarized harmonic wave
a
=
A
sin
ω c
(ct
±
x1),
b
=
0
are
as
follows
The Weber assumption 2 is inapplicable to his experimental statement.
U 0 + U 1 = ε = const ,
(9.6)
Moreover, a reference frame where the Christoffel symbols are zero can be applicable only at a point, it is unapplicable to a finite region. At the same time, in the Weber experimental statement, the detector itself is a system of two particles located at the distance η from each other. In a Riemannian space the Riemann-Christoffel curvature tensor is different from zero, so the Riemannian coherence objects (the Christoffel symbols Γαβγ) cannot be reduced to zero by coordinate transformations. We can merely choose a reference system where, at a given point P , the coherence objects are zero (Γαβγ)P = 0. Such a reference frame is known as a geodesic reference frame [37]. Therefore:
The Weber assumption 3 is inapplicable to his experimental statement.
Thus if we retain the rest-condition U i = 0 and the free fall condition in the Weber equation (9.4), there must still be the non-gravitational force Φα = 0. So the Weber equation becomes the free particles deviation equation, which in chr.inv.-form is (7.30).
If we reject free fall in the Weber equation (9.4), but retain U i = 0, it takes the same form as (8.14), which is not a free oscillation equation, in which case weak plane gravitational waves can act on the particles only if they are in motion at the initial moment of time.
Collecting these results we conclude that:
The Weber equation (9.4) is incorrect, because the free fall condition in common with the rest-condition for two neighbouring particles, connected by a nongravitational force, lead to the requirement that this force should be zero, thus contradicting the initial conditions of the Weber experimental statement.
It is evident that in aiming to determine the sort of effects a falling gravitational wave has on a free-mass detector or a Weber detector, it would be reasonable to consider a case where the particles are in motion U i = 0. In this case, before solving the deviation equation (2.8) for two free particles or (2.12) for two interacting particles (depending on the type of detector used), we should solve the equations of motion for free particles (2.6) or forced particles (2.11), respectively.
U1 = − 1 4ε
U(20)
2 e2A sin
+ 2ω
c
(ct±x1
)
+
U(30)
2 e−2A sin
ω c
(ct±x1
)
+ U(10) ,
a
(9.7)
U2
=
U(20)
eA sin
ω c
(ct±x1
)
,
(9.8)
U3
=
U(30)
eA sin
ω c
(ct±x1
)
,
(9.9)
where U(10), U(20), U(30) are the initial values of the particle’s velocity along each of the spatial axes.
From the solutions two important conclusions follow:
1. A weak plane gravitational wave, falling in the x1 direction, acts on free particles only if they have nonzero velocities in directions x2 and x3 orthogonal to the wave motion.
2. The presence of transverse oscillations in the plane (x2, x3) leads also to longitudinal oscillations in the direction x1.
The solutions to the free-particles deviation equation (2.8) in the field of a weak plane gravitational wave are
A η1 =
U(30)
2
−
U(20) 2
2 ε2
η(10) + η˙(10)t ×
× sin ω (ct ± x1) + AL cos ω (ct ± x1) +
c
2εω c
A + η˙(10) −
U(30)
2
−
U(20) 2
2 ε2
ωη(10)
t+
+
η(10)
−
AL 2ε
,
η2 = η˙(20)
A t+
cos
ω
(ct±x1)
ωc
+
η(20)−
A ω
η˙(20)
−
− AU(20) ε
η˙(10)
t − 1 cos ω (ct ± x1) ωc
−
−
η(10)+η˙(10) t
cos
ω c
(ct ±
x1)
+
η˙(10) ω
+η(10)
,
(9.10) (9.11)
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η3 = η˙(30)
t − A cos ω (ct±x1) ωc
+
η(30)+
A ω
η˙(30)
−
+ AU(30) ε
η˙(10)
t−
1
cos
ω (ct
± x1)
ωc
−
−
η(10)+η˙(10) t
cos
ω c
(ct
±
x1)
−
η˙(10) ω
−η(10)
,
Acknowledgements
(9.12)
I am very grateful to Dmitri Rabounski for his help in this research, over many years, and his substantial additions and corrections to it. Many thanks to Stephen J. Crothers for his corrections and help in editing the manuscript.
where
L
=
U(20)η˙(20)
−
U(30)η˙(30)
=
η(10) ε
U(30)
2
−
U(20) 2
.
(9.13)
The solutions η1, η2, η3 are the relative deviations of two
free particles in directions orthogonal to the direction of the
wave’s motion. The deviations are actually generalizations
of the solutions (7.41) and (7.42), where the particles were
at rest. The only difference is that here (9.10–9.12) there are
additional parts, where the particles’ initial velocities U(20)
and U(30) are added.
Here we see that, besides regular harmonic oscillations,
the
term
t
cos
ω c
(ct
±
x1)
describes
oscillations
with
an
am-
plitude that increases without bound with time. Another
substantial difference is that, in contrast to solutions (7.35),
(7.42), (7.43) given for rest-particles, the solutions (9.10),
(9.11), (9.12) contain longitudinal oscillations — they are
described by solution (9.10). Both harmonic oscillations
and unbounded-rising oscillations exist there only if, at the initial moment of time, the particles are in motion along x2 and x3 (orthogonal to the x1 direction of the wave’s motion).
So, we come to our final conclusions on both free-mass
detectors and solid-body detectors of gravitational waves:
The greater the velocities of particles (atoms and molecules) in a gravitational wave detector (built on either free masses or of the Weber kind), the more sensitive is the detector to a falling weak plane gravitational wave. In current experiments researchers cool the Weber pigs to super low temperatures, about 2 K, aiming to minimize the inherent oscillations of the particles of which they consist. This is a counterproductive procedure by which experimental physicists actually reduce the sensitivity of the Weber detectors to practically zero. We see the same vicious drawback in current experiments with free-mass detectors, where such a detector consists of two satellites located in the same orbit near the Earth. Because the observer (a laser range-finder) is located in one of the satellites, both satellites are at rest with respect to each other and the observer. All the current experiments cannot register gravitational waves in principle. In a valid experiment for discovering gravitational waves, the particles of which the detector consists must be in as rapid motion as possible. It would be better to design a detector using two laser beams directed parallel to each other, because of the light velocity of the moving particles (photons). The indicative quantities to be observed are the light frequency and phase.
References
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PROGRESS IN PHYSICS
Volume 2
On Geometric Probability, Holography, Shilov Boundaries and the Four Physical Coupling Constants of Nature
Carlos Castro Center for Theor. Studies of Physical Systems, Clark Atlanta University, Atlanta, Georgia, 30314, USA
E-mail: czarlosromanov@yahoo.com; castro@ctsps.cau.edu
By recurring to Geometric Probability methods, it is shown that the coupling constants,
αEM ; αW ; αC associated with Electromagnetism, Weak and the Strong (color) force are given by the ratios of the ratios of the measures of the Shilov boundaries Q2 = S1 × RP 1; Q3 = S2 × RP 1; S5, respectively, with respect to the ratios of the measures μ[Q5]/μN [Q5] associated with the 5D conformally compactified real Minkowski spacetime Mˉ5 that has the same topology as the Shilov boundary Q5 of the 5 complex-dimensional poly-disc D5. The homogeneous symmetric complex domain D5 = SO(5, 2)/SO(5) × SO(2) corresponds to the conformal relativistic curved 10 real-dimensional phase space H10 associated with a particle moving in the 5D Anti de Sitter space AdS5. The geometric coupling constant associated to the gravitational force can also be obtained from the ratios of the measures involving Shilov boundaries.
We also review our derivation of the observed vacuum energy density based on the
geometry of de Sitter (Anti de Sitter) spaces.
1 The fine structure constant and Geometric Probability
Geometric Probability [21] is the study of the probabilities involved in geometric problems, e. g., the distributions of length, area, volume, etc. for geometric objects under stated conditions. One of the most famous problem is the Buffon’s Needle Problem of finding the probability that a needle of length l will land on a line, given a floor with equally spaced parallel lines a distance d apart. The problem was first posed by the French naturalist Buffon in 1733. For l < d the probability is
1 2π l| cos(θ)| 4l π/2
P=
dθ
=
cos(θ) =
2π 0
d
2πd 0
(1)
2l 2ld = πd = πd2 .
Hence, the Geometric Probability is essentially the ratio of the areas of a rectangle of length 2d, and width l and the area of a circle of radius d. For l > d, the solution is slightly more complicated [21]. The Buffon needle problem provides with a numerical experiment that determines the value of π empirically. Geometric Probability is a vast field with profound connections to Stochastic Geometry.
Feynman long ago speculated that the fine structure constant may be related to π. This is the case as Wyler found long ago [1]. We will based our derivation of the fine structure constant based on Feynman’s physical interpretation of the electron’s charge as the probability amplitude that an electron emits (or absorbs) a photon. The clue to evaluate this probability within the context of Geometric Probability theory is provided by the electron self-energy diagram. Using Feynman’s rules, the self-energy Σ (p) as a function of the el-
ectron’s incoming (outgoing) energy-momentum pμ is given by the integral involving the photon and electron propagator along the internal lines
− iΣ(p) = (−ie)2 ×
×
d4k (2π)4
γμ
i γρ(pρ − kρ) − m
−igμν k2
γν.
(2)
The integral is taken with respect to the values of the photon’s energy-momentum kμ. By inspection one can see that the electron self-energy is proportional to the fine structure constant αEM = e2, the square of the probability amplitude (in natural units of = c = 1) and physically represents the electron’s emission of a virtual photon (off-shell, k2 = 0) of energy-momentum kρ at a given moment, followed by an absorption of this virtual photon at a later moment.
Based on this physical picture of the electron self-energy graph, we will evaluate the Geometric Probability that an electron emits a photon at t = −∞ (infinite past) and reabsorbs it at a much later time t = +∞ (infinite future). The off-shell (virtual) photon associated with the electron selfenergy diagram asymptotically behaves on-shell at the very moment of emission (t = −∞) and absorption (t = +∞). However, the photon can remain off-shell in the intermediate region between the moments of emission and absorption by the electron.
The topology of the boundaries (at conformal infinity) of the past and future light-cones are spheres S2 (the celestial sphere). This explains why the (Shilov) boundaries are essential mathematical features to understand the geometric derivation of all the coupling constants. In order to describe the physics at infinity we will recur to Penrose’s ideas [10]
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of conformal compactifications of Minkowski spacetime by
attaching the light-cones at conformal infinity. Not unlike the
one-point compactification of the complex plane by adding
the points at infinity leading to the Gauss-Riemann sphere.
The conformal group leaves the light-cone fixed and it does
not alter the causal properties of spacetime despite the rescal-
ings of the metric. The topology of the conformal compactification of real Minkowski spacetime Mˉ4 = S3 × S1/Z2 = = S3 × RP 1 is precisely the same as the topology of the Shilov boundary Q4 of the 4 complex-dimensional polydisc D4. The action of the discrete group Z2 amounts to an antipodal identification of the future null infinity I+ with the past null infinity I−; and the antipodal identification of the past timelike infinity i− with the future timelike infinity, i+, where the electron emits, and absorbs the photon, respectively.
Shilov boundaries of homogeneous (symmetric spaces)
complex domains, G/K [7, 8, 9] are not the same as the ordinary topological boundaries (except in some special cases).
The reason being that the action of the isotropy group K of the origin is not necesarily transitive on the ordinary topolo-
gical boundary. Shilov boundaries are the minimal subspaces
of the ordinary topological boundaries which implement the
Maldacena-’T Hooft-Susskind holographic principle [13] in
the sense that the holomorphic data in the interior (bulk)
of the domain is fully determined by the holomorphic data
on the Shilov boundary. The latter has the property that the
maximum modulus of any holomorphic function defined on
a domain is attained at the Shilov boundary.
For example, the poly-disc D4 of 4 complex dimensions is an 8 real-dim Hyperboloid of constant negative scalar curvature that can be identified with the conformal relativistic
curved phase space associated with the electron (a particle)
moving in a 4D Anti de Sitter space AdS4. The polydisc is a Hermitian symmetric homogeneous coset space
associated with the 4D conformal group SO(4, 2) since D4 = SO(4, 2)/SO(4)×SO(2). Its Shilov boundaryShilov (D4) = Q4 has precisely the same topology as the 4D conformally compactified real Minkowski spacetime Q4 = Mˉ4 = = S3 × S1/Z2 = S3 × RP 1. For more details about Shilov boundaries, the conformal group, future tubes and holo-
graphy we refer to the article by Gibbons [12] and [7, 16].
In order to define the Geometric Probability associated with this process of the electron’s emission of a photon at i− (t=−∞), followed by an absorption at i+ (t=+∞), we must take into account the important fact that the photon is on-shell k2 = 0 asymptotically (at t =±∞), but it can move off-shell k2 = 0 in the intermediate region which is represented by the interior of the conformally compactified real Minkowski spacetime Q4 = Mˉ4 = S3 × S1/Z2 = S3 × RP 1.
Denoting by μˆ[Q4] the measure-density (the measurecurrent) whose flux through the future and past celestial spheres S2 (associated with the future/past light-cones) at timelike infinity i+, i−, respectively, is V (S2)μˆ[Q4]. The net
flux through the two celestial spheres S2 at timelike infinity i± requires an overall factor of 2 giving then the value of 2V (S2)μˆ[Q4]. The Geometric Probability is defined by the
ratio of the measures associated with the celestial spheres S2 at i+, i− timelike infinity, where the photon moves on-
shell, relative to the measure of the full interior region of Q4 = Mˉ4 = S3 × S1/Z2 = S3 × RP 1, where the photon can move off-shell, as it propagates from i− to i+:
α = 2V (S2) μˆ[Q4] .
(3)
μ[Q4]
The ratio (μˆ[Q4]/μ[Q4] ) can be re-written in terms of the ratios of the normalized measures of
Mˉ5 = Q5 = Shilov [D5] = S4 ×S1/Z2 = S4 ×RP 1, (4)
namely, in terms of the normalized measures of the conformally compactified 5D Minkowski spacetime. This is achieved as follows [4]
μˆ[Q4] μ[Q4]
=
1 V (S4)
μN [Q5] , μ[Q5]
(5)
resulting from the embeddings (inmersions ) of D4 → D5. The origin of the factor V (S4) in the r. h. s of (5), as
one goes from the ratio of measures in Q4 to the ratio of the measures in Q5, is due to the reduction from the action of the isotropy group of the origin SO(5) × SO(2) on Q5, to the action of the isotropy group of the origin SO(4)×SO(2) on Q4, furnishing an overall reduction factor of V [SO(5)/SO(4)] = V (S4). The 5 complex-dimensional poly-disc D5 = SO(5, 2)/SO(5)×SO(2) is the 10 real-dim Hyperboloid H10 corresponding to the conformal relativistic curved phase space of a particle moving in 5D Anti de Sitter Space AdS5. This picture is also consistent with the KaluzaKlein compactification procedure of obtaining 4D EM from pure Gravity in 5D. The H10 can be embedded in the 11-dim pseudo-Euclidean R9,2 space, with two-time like directions. This is where 11-dim lurks into our construction.
Next we turn to the Hermitian metric on D5 constructed by Hua [8] which is SO(5, 2)-invariant and is based on the Bergmann kernel [15] involving a crucial normalization
factor of 1/V (D5). However, the standard normalized measure μN [Q5] based on the Poisson kernel and involving a normalization factor of 1/V (Q5) is not invariant under the full group SO(5, 2). It is only invariant under the isotropy group of the origin SO(5) × SO(2). In order to construct an invariant measure on Q5 under the full group SO(5, 2) one requires to introduce a crucial factor related to the Jacobian
measure involving the action of the conformalgroupSO(5, 2) on the full bulk domain D5. As explained by [4] one has:
μN [Q5] μ[Q5]
=
1 V (Q5)
||JC−1||
=
1 =
V (Q5)
||JC−1 (JC∗ )−1 ||
=
1 V (Q5)
||JR−1|| =
64
C. Castro. On Geometric Probability, Holography, Shilov Boundaries and the Four Physical Coupling Constants of Nature
July, 2005
PROGRESS IN PHYSICS
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1 =
| det g|−1 =
1
[|
det(g
)|]−
1 4
=
V (Q5)
V (Q5)
(6)
1
1
=
V
[V (Q5)
(D5)] 4 ,
the z dependence of the complex Jacobian is no longer
explicit because the determinant of the SO(5, 2) matrices
is unity.
This explains very clearly the origins of the factor
1
[V (D5)] 4 in Wyler’s formula for the fine structure constant
[1]. This reduction factor of V (Q5) is in this case given by
1
V (D5) 4 .
As
we
shall
see
below,
the
power
of
1 4
is
related
to the inverse of the dim(S4) = 4. This summarizes, briefly,
the role of Bergmann kernel [15] in the construction by Hua
[8], and adopted by Wyler [1], of the Hermitian metric of a
bounded homogenous (symmetric) complex domain. To sum
up, we must perform the reduction from V (Q5) → V (Q5)/
1
V (D5) 4 in the construction of the normalized measure
μN [Q5] . This approach is very different than the interpreta-
tion given by Smith [3] and later adopted by Smilga [5].
Hence, the Geometric Probability ratio becomes
μˆ[Q4] μ[Q4]
=
1 V (S4)
μN [Q5] μ[Q5]
=
(7a)
11
1
1
=
V (S4) V (Q5)
[
V (D5)
]4
≡
. αG
This last ratio, for reasons to be explained below, is nothing but the inverse of the geometric coupling strength of gravity, 1/αG. The relationship to the gravitational constant is based on the definition of the coupling appearing in the Einstein-Hilbert Lagrangian (R/16πG), as follows
(16πG)(m2P lanck) ≡ αEM αG = 8π ⇒
1 8π
1
G = 16π m2P lanck = 2m2P lanck ⇒
Gm2proton
=
1 2
mproton
2
∼ 5.9×10−39,
mP lanck
(7b)
and in natural units = c = 1 yields the physical force strength of Gravity at the Planck Energy scale 1.22×1019 GeV. The Planck mass is obtained by equating the Schwarz-
schild radius 2 G mP lanck to the Compton wavel√ength 1/mP lanck associated with the mass; where mP lanck 2 = = 1.22×1019 GeV and the proton mass is 0.938 GeV. Some a√uthors define the Planck mass by absorbing the factor of
2 inside the definition of mP lanck = 1.22×1019 GeV. The role of the conformal group in Gravity in these ex-
pressions (besides the holographic bulk/boundaryAdS/CF T duality correspondence [13]) stems from the MacDowell-
Mansouri-Chamseddine-West formulation of Gravity based
on the conformal group SO(3, 2) which has the same number of 10 generators as the 4D Poincare group. The 4D vielbein
eaμ which gauges the spacetime translations is identified with the SO(3, 2) generator A[μa5], up to a crucial scale factor R, given by the size of the Anti de Sitter space (de Sitter space) throat. It is known that the Poincare group is the WignerInonu group contraction of the de Sitter Group SO(4, 1) after taking the throat size R = ∞. The spin-connection ωμab that gauges the Lorentz transformations is identified with the SO(3, 2) generator A[μab]. In this fashion, the eaμ, ωμab are encoded into the A[μmn] SO(3, 2) gauge fields, where m, n run over the group indices 1, 2, 3, 4, 5. A word of caution, Gravity is a gauge theory of the full diffeomorphisms group which is infinite-dimensional and which includes the translations. Therefore, strictly speaking gravity is not a gauge theory of the Poincare group. The Ogiovetsky theorem shows that the diffeomorphisms algebra in 4D can be generated by an infinity of nested commutators involving the GL(4, R) and the 4D Conformal Group SO(4, 2) generators.
In [17] we have shown why the MacDowell-MansouriChamseddine-West formulation of Gravity, with a cosmological constant and a topological Gauss-Bonnet invariant term, can be obtained from an action inspired from a BF-ChernSimons-Higgs theory based on the conformal SO(3, 2) group. The AdS4 space is a natural vacuum of the theory. The vacuum energy density was derived to be the geometricmean between the UV Planck scale and the IR throat size of de Sitter (Anti de Sitter) space. Setting the throat size to coincide with the future horizon scale (of an accelerated de Sitter Universe) given by the Hubble scale (today) RH, the geometric mean relationship yields the observed value of the vacuum energy density ρ ∼ (LP )−2(RH )−2 = (LP )−4× × (L2P /RH2 ) ∼ 10−122MP4 lanck. Nottale [23] gave a different argument to explain the small value of ρ based on Scale Relativistic arguments. It was also shown in [17] why the Euclideanized AdS2n spaces are SO(2n − 1, 2) instantons solutions of a non-linear sigma model obeying a double self duality condition.
Therefore, the Geometric Probability αEM for an electron to emit a photon at t = −∞ and to absorb it at t = +∞ agrees with the Wyler’s celebrated expression for the fine structure constant
αEM
=
2V (S2)μˆ[Q4] μ[Q4]
=
(8π) V
1 (S4)
V
1 (Q5)
×
1
(8)
1
9
π5 4
1
× [V (D5)] 4 = 8π4 24×5!
=
,
137.03608
after one inserts the values of the volumes:
π5 V (D5) = 24×5! ,
8π3 V (Q5) = 3 ,
V (S4) = 8π2 . (9) 3
In general
πn V (Dn) = 2n−1n! ,
V (Sn−1) = 2πn/2 , Γ(n/2)
(10a)
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V (Qn)=V (Sn−1×RP 1)=V (Sn−1)×V (RP 1) =
2πn/2
2π(n+2)/2
=
×π =
.
Γ(n/2)
Γ(n/2)
(10b)
Objections were raised to Wyler’s original expression by Robertson [2]. One of them was that the hyperboloids (discs) are not compact and whose volumes diverge since the Lobachevsky metric diverges on the boundaries of the poly-discs. Gilmore explained [2] why one requires to use the Euclideanized regularized volumes as Wyler did. Furthermore, in order to resolve the scaling problems of Wyler’s expression, Gilmore showed why it is essential to use dimensionless volumes by setting the throat sizes of the Anti de Sitter hyperboloids to r = 1, because this is the only choice for r where all elements in the bounded domains are also coset representatives, and therefore, amount to honest group operations. Hence the so-called scaling objections against Wyler raised by Robertson were satisfactory solved by Gilmore [2].
The question as to why the value of αEM obtained in Wyler’s formula is precisely the value of αEM observed at the scale of the Bohr radius aB, has not been solved, to my knowledge. The Bohr radius is associated with the ground ( most stable ) state of the Hydrogen atom [3]. The spectrum generating group of the Hydrogen atom is well known to be the conformal group SO(4, 2) due to the fact that there are two conserved vectors, the angular momentum and the Runge-Lentz vector. After quantization, one has two commuting SU (2) copies SO(4) = SU (2) × SU (2). Thus, it makes physical sense why the Bohr-scale should appear in this construction. Bars [14] has studied the many physical applications and relationships of many seemingly distinct models of particles, strings, branes and twistors, based on the (super) conformal groups in diverse dimensions. In particular, the relevance of two-time physics in the formulation of M, F, S theory has been advanced by Bars for some time. The Bohr radius corresponds to an energy of 137.036×2×13.6 eV ∼ 3.72×103 eV. It is well known that the Rydberg scale, the Bohr radius, the Compton wavelength of electron, and the classical electron radius are all related to each other by a successive scaling in products of αEM .
2 The fiber bundle interpretation of the Wyler formula
Shilov boundary Q5 is written by E|Q5 and locally is the product of Q5 × S4, but this is not true globally. For this
reason one has that the volume V (E|Q5 ) = V (Q5 × S4) = = V (Q5) × V (S4). But instead, V (E|Q5 ) = V (S4) × (V (Q5)/V (D5)1/4).
This is the reasoning behind the construction of the
quantity μˆ[Q4]/μ[Q4] that has the units of a density. Its
inverse μ[Q4]/μˆ[Q4] is the volume associated with the re-
striction of the Fiber Bundle E to the Shilov boundary Q5: V (E|Q5 ) = V (S4) × (V (Q5)/V (D5)1/4).
The reason why one embeds D4 → D5 and Q4 → Q5 is because the space Q4 = S3 × RP 1 is not large enough
to implement the action of the SO(5) group, the compact
version of the Anti de Sitter Group SO(3, 2) that is required
in the MacDowell-Mansouri-Chamseddine-West formulation
of Gravity. However, the space Q5 = S4 × RP 1 is large enough to implement the action of SO(5) via the internal symmetry space S4 = SO(5)/SO(4). This justifies the em-
bedding procedure of D4 → D5. This Fiber Bundle interpretation is not very different from Smith’s interpretation
[3]. Following the Fiber Bundle interpretation of the volume V (E|Q5 ) = V (S4) × (V (Q5)/V (D5)1/4), we will now prove why
2 V (S2)
=
μ(S1) μˆ(S1)
=
8π .
(11)
The space S1 is associated with the U (1) group action and naturally encodes the U (1) gauge invariance linked to Electromagnetism ( EM ). The result of eq-(11) is what will
allow us to define αEM as the ratio of the ratios of suitable measures in S1 and Q4, respectively,
αEM
=
2V (S2) μˆ[Q4] μ[Q4]
=
(μ(S1)/μˆ(S1)) (μ[Q4]/μˆ[Q4])
.
(12)
We may notice that S1 ≡ Q1 (very special case) since the
circle is both the Shilov and ordinary topological boundary of the disc D1. However, Q2 ≡ S1 × S1/Z2 = S1 × RP 1. Once again, we will write the ratio of the measures in Q1 = S1 in terms of the ratio of the normalized measures in Q2 via the reduction from S1 × S1/Z2 to S1. This requires the
embedding (inmersion) of D1 → D2 in order to construct the
measures on D1, Q1 as induced from the measures in D2, Q2
resulting from the embedding (inmersion):
Having found Wyler’s expression from Geometric Probabili-
ty, we shall present a Fiber Bundle interpretation of the Wyler
expression by starting with a Fiber bundle E over the base curved-space D5 = SO(5, 2)/SO(5) × SO(2). The subgroup H=SO(5) of the isotropy group K=SO(5)×SO(2) acts on the Fibers F = S4 (the internal symmetry space). Locally, and only locally, the Fiber bundle E is the product D5 × S4. However, this is not true globally. On the Shilov boundary Q5, the restriction of the Fiber bundle E to the
μˆ(S1) μ(S1)
=
μˆ(Q1) μ(Q1)
=
1 V (S1/Z2)
μN [Q2] μ[Q2]
=
(13)
1
1
=
V (S1/Z2)
. (V (Q2)/V (D2)
Notice that μˆ(S1) as explained before is a measuredensity on S1. Likewise, μˆ(Q4) was a measure-density on Q4. We should not confuse these measure-densities with the normalized measures in one-higher dimension.
66
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July, 2005
PROGRESS IN PHYSICS
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By inserting the values of the measures and using
In general, for other homogeneous complex domains, this
V (S1/Z2) = V (RP 1) = π , V (Q2) = 2π2 = 2π2,
Γ(1)
π2 V (D2) = 2×2! ,
power is given by the inverse of the dimension of the internal symmetry space.
(14) 3 The weak and strong coupling constants from Geometric Probability
it yields then
μ(S1) μˆ(S1)
=
(2π2)
(π)
1 (π2/2×2!)
=
8π
=
2
V (S2)
(15)
as claimed. Therefore, 2V (S2) = μ(S1)/μˆ(S1) = 8π is the crucial factor appearing in Wyler’s formula which admits a
natural Geometric probability explanation which is very dif-
ferent from the different interpretations provided in [3, 4, 5].
The Fiber Bundle interpretation associated with the
U (1) ∼ SO(2) group is the following. The Fiber bundle E is defined over the curved space D2 = SO(2, 2)/SO(2) × SO(2). The subgroup H = SO(2) ∼ U (1) of the isotropy group K = SO(2) × SO(2) acts on the fibers identified with the symmetry space S1 (where the U (1) group acts). The Fiber bundle E locally can be written as D2 × S1 but not globally. The restriction of the Fiber bundle E to the Shilov boundary Q2 = S1 × S1/Z2 = S1 × RP 1 is E|Q2 and locally can be written as Q2 × S1, but not globally. This is why the volume V (E|Q2 ) = V (Q2) × V (S1) but instead it equals (V (Q2)/V (D2)) × V (S1/Z2) = 2V (S2) = 8π.
Concluding, the Geometric Probability that an electron
emits a photon at t =−∞ and absorbs it at t =+∞ is given by the ratio of the ratios of measures, and it agrees with
Wheeler’s ideas that one must normalize the couplings with
respect to the geometric coupling strength of Gravity:
αEM
=
2V
(S2)μˆ[Q4] μ[Q4]
=
(μ(S1)/μˆ(S1)) (μ[Q4]/μˆ[Q4])
=
(16)
11
1
1
= (8π) V (S4) V (Q5) [V (D5)] 4 = 137.03608 .
The second important conclusion that can be derived
from Geometric Probability theory is the general numerical
values of the exponents sn appearing in the factors V (Dn)sn . The normalization factor V (Q5)/V (D5)1/4 in the construc-
tion of the ratio of measures μN [Q5]/μ[Q5] involves in
this
case
powers
of
the
type
V (D5)1/4.
The
power
of
1 4
is related to the inverse of the dim(S4) = 4 (the internal
symmetry space SO(5)/SO(4)). From eq-(13) we learnt that the reduction factor of V (Q2)/V (D2) was V (D2); i. e.
the exponent is unity. The power of unity is related to the inverse of the dim(S1/Z2) = 1. Thus, the arguments based
on Geometric Probability leads to normalized measures by factors of V (Qn)/V (Dn)sn and whose exponents sn are
given by the inverse of the dimensions of the internal symmetry spaces sn = (dim(Sn−1))−1. There is a different interpretation of these factors V (Dn)sn given by Smith [3].
We turn now to the derivation of the other coupling constants. The Fiber Bundle picture of the previous section is essential in our construction. The Weak and the Strong geometric coupling constant strength, defined as the probability for a particle to emit and later absorb a SU (2), SU (3) gauge boson, respectively, can both be obtained by using the main formula derived from Geometric Probability after one identifies the suitable homogeneous domains and their Shilov boundaries to work with. We will show why the weak and strong couplings are given by
αWeak
=
(μ[Q2]/μˆ[Q2]) (μ[Q4]/μˆ[Q4])
=
(μ[Q2]/μˆ[Q2]) αG
=
(17)
= (μ[Q2]/μˆ[Q2]) ,
(8π/αEM )
and
(μ[S4]/μˆ[S4]) (μ[S4]/μˆ[S4])
αColor = (μ[Q4]/μˆ[Q4] =
αG
=
(μ[S4]/μˆ[S4])
(18)
=
.
(8π/αEM )
At this point we must emphasize that we define αweak, αcolor as gw2 , gc2 instead of the conventional (gw2 /4π), (gc2/4π) definitions used in the Renormalization Group program. The Shilov boundary of (D2) is Q2 = S1 × RP 1 but is not large enough to accommodate the action of the
isospin group SU (2). One needs a Fiber Bundle over D3 = = SO(3, 2)/SO(3) × SO(2) whose subgroup H = SO(3) of
the isotropy group K = SO(3) × SO(2) acts on the internal symmetry space S2 (the fibers). Since the coset space SU (2)/U (1) is a double-cover of the S2 as one goes from
the SO(3) action to the SU (2) action one must take into
account an extra factor of 2. This is the reason why one jumps to one-dimension higher from Q2 to Q3 = S2 × RP 1, because the coset SU (2)/U (1) is a double-cover of the sphere S2 = SO(3)/SO(2) and can accommodate the action
of the SU (2) group.
By following the same procedure as above, i. e. by re-
writing the ratio of the measures (μˆ[Q2]/μ[Q2]) in terms of the ratio of the measures (μN [Q3]/μ[Q3]) via the embeddings of D2 → D3, one has
(μˆ[Q2]/μ[Q2]) =
1 V (SU (2)/U (1))
μN [Q3] . μ[Q3]
(19)
Notice that because SU (2) is a 2 − 1 covering of the SO(3), this implies that the measure
V (SU (2)/U (1))= 2V (SO(3)/U (1))= 2V (S2)= 8π . (20)
C. Castro. On Geometric Probability, Holography, Shilov Boundaries and the Four Physical Coupling Constants of Nature
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PROGRESS IN PHYSICS
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As indicated above, because the dimension of the internal
symmetry space is dim(S2) = 2, the construction of the
normalized measure μN [Q3] will require a reduction of
V (Q3) by a factor of V (D3) raised to the power of
(dim(S2))−1
=
1 2
:
μN [Q3] μ[Q3]
=
V
1 (Q3)/V (D3)1/2
=
1 V (Q3)
V (D3)1/2.
(21)
Therefore, the ratio of the measures is
μˆ[Q2] μ[Q2]
=
1 2V (S2)
1 V (Q3)
V (D3)1/2,
(22)
whose Fiber Bundle interpretation is that the volume of the
Fiber Bundle over D3, but restricted to the Shilov boundary Q3, and whose structure group is SU (2) (the double cover of SO(3)), is V (E|Q3 ) = 2V (S2) × (V (Q3)/V (D3)1/2). Thus, that the Geometric probability expression is
αWeak
=
(μ[Q2]/μˆ[Q2]) (μ[Q4]/μˆ[Q4])
=
(μ[Q2]/μˆ[Q2]) (8π/αEM )
=
(23)
=
2V
(S2)V
(Q3)
V
1 (D3)1/2
αEM 8π
= 0.2536,
that corresponds to the weak geometric coupling constant αW at an energy of the order of
E = M = 146 GeV ∼ MW2 + + MW2 − + MZ2 , (24)
special case the Shilov and ordinary topological boundaries of B6 coincide with S5 [3]. Hence, following the same procedures as above, the ratio of the measures in S4 (bound-
ary of B5) can be re-written in terms of the ratio of the measures in S5 (boundary of B6) via the embeddings of
B5 → B6 as follows:
μˆ[S4] μ[S4]
=
V
1 (S
4
)
μN [S5] μ[S5]
=
V
1 (S4)
V
1 (S5)/V
(B6
)1/4
=
(26)
=
V
1 (S4)
V
1 (S5)
V
(B6)1/4,
since the exponent of the reduction factor V (B6)1/4 is given
by
(dim(S
4
))−1=
1 4
.
Notice,
again,
that
μˆ[S4]
is
the
measure-
density in S4 and must not be confused with the normalized
measures.
Therefore, one arrives at
αColor
=
V (S4)
V (S5)
1 V (B6)1/4
αEM 8π
= 0.6286,
(27)
that corresponds to the strong coupling constant at an energy related to the pion masses [3]:
E = 241 MeV ∼ m2π+ + m2π− + m2π0
(28)
and where we have used the expressions:
V (S4) = 8π2 , 3
V (S5) = 4π3 ,
π3 V (B6) = 6 .
(29)
after we have inserted the expressions
V (S2) = 4π ,
V (Q3) = 4π2 ,
π3 V (D3) = 24 ,
(25a)
into the formula (23). The relationship to the Fermi coupling GF ermi goes as follows (after indentifying the energy scale E = M = 146 GeV):
GF
≡
αW M2
⇒ GF
m2proton
=
αW M2
m2proton =
= 0.2536 ×
mproton
2
∼ 1.04×10−5
146 GeV
(25b)
in very good agreement with experimental observations.
Once more, it is unknown why the value of αWeak obtained from Geometric Probability corresponds to the energy
scale related to the W+, W−, Z0 boson mass, after spontaneous symmetry breaking.
Finally, we shall derive the value of αColor from eq(18). Since S4 is not large enough to accommodate the action of the color group SU (3) one needs to work with onedimension higher S5 , that can be interpreted as the boundary of the 6D Ball B6 = SU (4)/U (3) = SU (4)/SU (3) × U (1). Thus, the SU (3) group is part of the isotropy group K = = SU (3) × U (1) that defines the coset space B6. In this
The pions are the known lightest quark/antiquark pairs
that feel the strong interaction [3]. For a detailed analysis of
volumes of compact manifolds (coset spaces) see [24].
Once again, it is unknown why the value of αColor obtained from Geometric Probability (28) corresponds to
the energy scale related to the masses of the three pions
[3]. Masses of the fundamental particles were derived in [3]
based on the definitions that mass is the probability amplitude
for a particle to change direction.
To conclude, by defining the geometric coupling constants α = g2 as the Geometric Probability to emit (and later absorb) a gauge boson, all the three geometric coupling
constants, αEM ; αWeak; αColor are given by the ratios of the ratios of the measures of the Shilov boundaries Q2 = S1× RP 1; Q3 = S2 × RP 1; S5, respectively, with respect to the ratios of the measures μ[Q5]/μN [Q5] associated with the 5D conformally compactified real Minkowski spacetime Mˉ5 that has the same topology as the Shilov boundary Q5 of the 5 complex-dimensional poly-disc D5. The latter corresponds to the conformal relativistic curved 10 real-dimensional phase space H10 associated with a particle moving in the 5D Anti de Sitter space AdS5. The ratios of particle masses, like the proton to electron mass ratio mp/me ∼ 6π5 has also been calculated using the volumes of homogeneous bounded
domains [3, 4].
68
C. Castro. On Geometric Probability, Holography, Shilov Boundaries and the Four Physical Coupling Constants of Nature
July, 2005
PROGRESS IN PHYSICS
Volume 2
It is not known whether this procedure would work for Grand Unified Theories based on the groups
SU (5), SO(10), E6, E7, E8 .
(30)
Beck [6] has obtained all the Standard Model parameters
by studying the numerical minima (and zeros) of certain po-
tentials associated with the Kaneko coupled two-dim lattices
based on Stochastic Quantization methods. The results above
and by Smith [3] are analytical rather than being numerical
[6] and it is not clear if there is any relationship between these
two approaches. Noyes has proposed an iterated numerical
hierarchy based on Mersenne primes Mp = 2p −1 for certain values of p = primes [18] and obtained many numerical
values for the physical parameters. Pitkanen has developed
methods to calculate the physical masses recurring to a p-adic
hierarchy of scales based on Mersenne primes [19].
An important connection between anomaly cancellation
in string theory and perfect even numbers was found in
[22]. These are numbers which can be written in terms of
sums of its divisors, including unity, like 6 = 1 + 2 + 3,
and
are
of
the
form
P
(p)
=
1 2
2p(2p
−
1)
if,
and
only
if,
2p − 1 is a Mersenne prime. Not all values of p = prime
yields primes. The number 211 − 1 is not a Mersenne prime,
for example. The number of generators of the anomaly free
groups SO(32), E8 × E8 of the 10-dim superstring is 496 which is an even perfect number. Another important group
related to the unique tadpole-free bosonic string theory is the SO(213) = SO(8192) group related to the bosonic string
compactified on the E8 × SO(16) lattice. The number of generators of SO(8192) is an even perfect number since 213 − 1 is a Mersenne prime. For an introduction to p-adic
numbers in Physics and String theory see [20]. A lot more
work needs to be done to be able to answer the question: Is
all this just a mere numerical coincidence or is it design?
4 Acknowledgements
We are indebted to Frank (Tony) Smith and C. Beck for numerous discussions about their work. To M. Bowers and C. Handy (CTSPS) for their kind support.
References
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3. Smith Jr F. D., Int. J. Theor. Phys., 1985, v. 24, 155; Int. J. Theor. Phys., 1985, v. 25, 355. From sets to quarks. arXiv: hep-ph/9708379, CERN CDS EXT-2003-087.
4. Gonzalez-Martin G. Physical Geometry. Univ. of Simon Bolivar Publ., Caracas, 2000. The proton/electron geometric mass ratio. arXiv: physics/0009052. The fine structure constant from relativistic groups. arXiv: physics/ 0009051.
5. Smilga W. Higher order terms in the contraction of S0(3, 2). arXiv: hep-th/0304137.
6. Beck C. Spatio-temporal vacuum fluctuations of quantized fields. World Scientific, Singapore 2002.
7. Coquereaux R., Jadcyk A. Reviews in Mathematical Physics, 1990, No. 1, v. 2, 1–44.
8. Hua L. K. Harmonic analysis of functions of several complex variables in the classical domains. Birkhauser, Boston-BaselBerlin, 2000.
9. Faraut J., Kaneyuki S., Koranyi A., Qi-Keng Lu and Roos G. Analysis and geometry on complex homogeneous domains. Progress in Math., v. 185, Birkhauser, Boston-Basel-Berlin.
10. Penrose R., Rindler W. Spinors and space-time. Cambridge University Press, 1986.
11. Hugget S.A., Todd K. P. An introduction to twistor theory. London Math. Soc. Stud. Texts, v. 4, Cambridge Univ. Press, 1985.
12. Gibbons G. Holography and the future tube. arXiv: hep-th/ 9911027.
13. Maldacena J. The large N limit of superconformal theories and supergravity. arXiv: hep-th/9711200.
14. Bars I. Twistors and two-time physics. arXiv: hep-th/0502065.
15. Bergman S. The kernel function and conformal mapping. Math. Surveys, 1970, v. 5, AMS, Providence.
16. Odzijewicz A. Int. Jour. of Theor. Phys., 1986, v. 107, 561–575.
17. Castro C. Mod. Phys. Letts., 2002, v. A17, No. 32, 2095–2103. Class. Quan. Grav., 2003, v. 20, 3577–3592.
18. Noyes P. Bit-strings physics: a discrete and finite approach to natural philosophy. Series in Knots in Physics, v. 27, Singapore, World Scientific, 2001.
19. Pitkanen M. Topological Geometrodynamics I, II. Chaos, Solitons and Fractals, 2002, v. 13, No. 6, 1205, 1217.
20. Vladimorov V., Volovich I. and Zelenov I. p-Adic numbers in mathematical physics. Singapore, World Scientific, 1994. Brekke L., Freund P. Physics Reports, 1993, v. 1, 231.
21. Ambartzumian R. V. (Ed.) Stochastic and integral geometry. Dordrecht (Netherlands), Reidel, 1987. Kendall M. G. and Moran P. A. P. Geometric probability. New York, Hafner, 1963. Kendall W.S, Barndorff-Nielson O. and van Lieshout M. C. Current trends in stochastic geometry: likelihood and computation. Boca Raton (FL), CRC Press, 1998. Klain D. A. and Rota G.-C. Introduction to geometric probability. New York, Cambridge Univ. Press, 1997. Santalo L. A. Integral geometry and geometric probability. Reading (MA) Addison-Wesley, 1976. Solomon H. Geometric probability. Philadelphia (PA), SIAM, 1978. Stoyan D., Kendall W. S. and Mecke J. Stochastic geometry and its applications. New York, Wiley, 1987. http://mathworld.wolfram.com/GeometricProbability.html
22. Frampton P., Kephart T. Phys. Rev., 1999, v. D60, 08790.
23. Nottale L. Fractal spacetime and microphysics, towards the theory of scale relativity. World Scientific, Singapore 1992. Chaos, Solitons and Fractals, 2003, v. 16, 539.
24. Boya L., Sudarshan E. C. G. and Tilma T. Volumes of compact manifolds. arXiv: math-ph/0210033.
C. Castro. On Geometric Probability, Holography, Shilov Boundaries and the Four Physical Coupling Constants of Nature
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On the Generalisation of Kepler’s 3rd Law for the Vacuum Field of the Point-Mass
Stephen J. Crothers Sydney, Australia
E-mail: thenarmis@yahoo.com
I derive herein a general form of Kepler’s 3rd Law for the general solution to Einstein’s vacuum field. I also obtain stable orbits for photons in all the configurations of the point-mass. Contrary to the accepted theory, Kepler’s 3rd Law is modified by General Relativity and leads to a finite angular velocity as the proper radius of the orbit goes down to zero, without the formation of a black hole. Finally, I generalise the expression for the potential function of the general solution for the point-mass in the weak field.
1 Introduction
In previous papers [1, 2] I derived the general solution for Einstein’s vacuum field and showed that black holes do not exist in Einstein’s universe. In those papers I also obtained expressions for Kepler’s 3rd Law for the simple (i. e. nonrotating) point-mass and the simple point-charge. In this paper I obtain expressions for Kepler’s 3rd Law for the rotating point-mass and the rotating point-charge. Owing to the rotation of the source of the field, Kepler’s 3rd Law for the polar orbit is not the same as that for the equatorial orbit, so that stable photon orbits are also different in the polar and equatorial orbits, showing that in the rotating configurations spacetime is no longer isotropic.
The expressions I obtain readily reduce to those I have previously derived for the non-rotating configurations.
ds2
=
Δ ρ2
dt − a sin2 θdϕ
2
−
sin2 θ − ρ2
R2 + a2 dϕ − adt 2 − ρ2 dR2 − ρ2dθ2. Δ
This can be written as,
ds2 = Δ − a2 sin2 θ dt2 − ξ dR2 −
ξ
Δ
− ξdθ2 + a2Δ sin4 θ − R2 + a2 2 sin2 θ dϕ2− (1) ξ
2aΔ sin2 θ − 2a R2 + a2 sin2 θ
−
dtdϕ ,
ξ
where I have previously shown [2, 3] in the case of the rotating point-charge,
2 Definitions
I have already shown [3] that the most general static metric for the point-mass is, ds2 = A(D)dt2 − B(D)dD2 − C(D) dθ2 + sin2 θdϕ2 ,
D = |r − r0| , A, B, C > 0 , where r0 is an arbitrary real number. With respect to this metric I identify the coordinate radius, the r-parameter, the radius of curvature, and the proper radius thus: 1. The coordinate radius is D = |r − r0| . 1. The r-parameter is the variable r . 2. The radius of curvature is R = C(D) . 3. The proper radius is Rp = B(D) dD .
3 The equatorial orbit
2
R2 = Cn(r) =
r − r0 n + βn
n
,
r0 ∈ , r ∈ ,
β = m + m2 − a2 cos2 θ − q2 ,
a2 + q2 < m2 , n ∈ +, ξ = ρ2 = R2 + a2 cos2 θ ,
L a= ,
Δ = R2 − αR + a2 + q2 ,
m
0 < |r − r0| < ∞ ,
where L is the angular momentum, and n and r0 are arbitrary. I have also shown previously that Kepler’s 3rd Law for
the simple point-mass is,
ω2
=
α 2R3
,
(2)
where
lim
r → r0±
Cn(r) = R0 = α = 2m ∀ r0 ,
is a scalar invariant; and for the simple point-charge is,
The general Kerr-Newman form in Boyer-Lindquist coordinates is,
ω2
=
α 2R3
−
q2 R4
,
(3)
70
S. J. Crothers. On the Generalisation of Kepler’s 3rd Law for the Vacuum Field of the Point-Mass
July, 2005
PROGRESS IN PHYSICS
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where, ∀ r0,
lim
r → r0±
Cn(r) = R0 = β = m +
m2 − q2, q2 < m2 ,
is a scalar invariant.
In
the
case
of
the
equatorial
orbit,
θ
=
π 2
and
θ˙ = 0,
so
(1) becomes,
ds2 = Δ − a2 dt2 − ξ dR2 +
ξ
Δ
+ a2Δ − R2 + a2 2 dϕ2 − ξ
(4)
2aΔ − 2a R2 + a2
−
dtdϕ .
ξ
where
ω
=
ϕ˙ t˙
.
The
solutions
for
ω
are,
aαR − 2aq2 ± R2 2αR − 4q2
ω=
a2αR − 2a2q2 − 2R4
.
In order for this to reduce to the non-rotating configurations, the plus sign must be taken so,
aαR − 2aq2 + R2 2αR − 4q2
ω=
a2αR − 2a2q2 − 2R4
,
(8)
2
R2 = Cn(r) =
r − r0 n + βn
n
,
β = m + m2 − q2 , q2 < m2 ,
2
R2 = Cn(r) =
r − r0 n + βn
n
,
β = m + m2 − q2 , q2 < m2 , ξ = R2 , Δ = R2 − αR + a2 + q2 ,
0 < |r − r0| < ∞ . Consider the associated Lagrangian, where the dot indicates ∂/∂τ ,
1 L=
Δ − a2 t˙2 − ξ R˙ 2
+
2ξ
Δ
1 +
a2Δ − R2 + a2 2
ϕ˙ 2 −
2
ξ
(5)
1 2aΔ − 2a R2 + a2 −
t˙ ϕ˙ .
2
ξ
α = 2m ,
0 < |r − r0| < ∞ .
Equation (8) is Kepler’s 3rd Law for the equatorial plane of the rotating point-charge. I remark that the radius of curvature in the equatorial orbit is precisely that for the simple point-charge. The expression for Kepler’s 3rd Law for the equatorial plane of the rotating point-mass is obtained from (8) by setting q = 0,
√ aαR + R2 2αR ω = a2αR − 2R4 ,
2
R2 = Cn(r) =
r − r0 n + αn
n
,
α = 2m ,
Then,
∂L ∂ −
∂R ∂τ
∂L ∂R˙
ξΔ − ξ Δ − a2
=0⇒
2ξ2
t˙2 +
ξ a2Δ − 4R R2 + a2
+
2ξ2
ϕ˙ 2 −
ξ a2Δ − R2 + a2 2
−
2ξ2
ϕ˙ 2 −
(6)
ξ (2aΔ − 4aR) − ξ 2aΔ − 2a R2 + a2
−
2ξ2
t˙ϕ˙ +
Δξ − ξΔ + 2Δ2
R˙2
+
ξ R¨ = 0 . Δ
Taking R = const. reduces (6) to,
ξ a2Δ − 4R R2 + a2 −
− ξ a2Δ − R2 + a2 2 ω2 − − ξ (2aΔ −4aR) −ξ 2aΔ−2a R2 +a2
(7) ω+
+ ξΔ − ξ Δ − a2 = 0 ,
0 < |r − r0| < ∞ ,
in which case the radius of curvature in the equatorial orbit is precisely that for the simple point-mass.
Taking the near-field limit on (8) gives,
aαβ − 2aq2 + β2 2αβ − 4q2
lim ω =
, (9)
r → r0±
a2αβ − 2a2q2 − 2β4
which is a scalar invariant. When a = 0 and q = 0, equation (8) reduces to,
ω2
=
α 2R3
−
q2 R4
,
which recovers Kepler’s 3rd Law (3) for the simple pointcharge. If a = q = 0, equation (8) reduces to,
ω2
=
α 2R3
,
β = α = 2m ,
which recovers Kepler’s 3rd Law (2) for the simple pointmass.
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When a = 0 and q = 0, (9) reduces in the near-field
limit, to
lim
ω2 =
α
q2 −,
r → r0±
2β3 β4
β = m + m2 − q2 ,
which recovers the stable radius of curvature for the photon around the simple point-mass [1].
When n = 1 and r0 = α, equation (13) gives,
Rph−e = Cn(rph−e) = rph−e = 3m ,
the scalar invariant of Kepler’s 3rd Law for the simple point-
charge; and when a = q = 0, (9) reduces to the near-field
limit,
lim
r → r0±
ω2
=
1 2α2
,
α = 2m ,
the scalar invariant for Kepler’s 3rd Law for the simple pointmass, as originally obtained by Karl Schwarzschild [4] for his particular solution.
4 Photons in equatorial orbit
Setting
θ=
π 2
in
(1)
and
setting
(1)
equal
to
zero
gives,
a2Δ − R2 + a2 2 ω2 − (10)
− 2aΔ − 2a R2 + a2 ω + Δ − a2 = 0 ,
This radius is taken incorrectly by the orthodox relativists as
a measurable proper radius in the gravitational field of the
simple point-mass. The actual proper radius associated with
(13) is,
√
√
α3
1+ 3
Rp = 2 + α ln
√ 2
,
which is a scalar invariant for the photon orbit about the point-mass.
The expression for the radius of curvature of the stable photon equatorial orbit for the rotating point-mass is obtained from (12) by setting q = 0, thus
√
√
aαR + R2 2αR aαR − R2 R2 − αR + a2
a2αR − 2R4 = αa2R + a2R2 + R4 ,
2
R2 = Cn(r) =
r − r0 n + αn
n
,
α = 2m ,
from which it follows,
r0 ∈ , n ∈ + .
ϕ˙ a q2 − αR + R2 R2 − αR + a2 + q2
ω = t˙ =
a2q2 − αa2R − a2R2 − R4
. (11) 5 The polar orbit
Equating (8) to (11) gives,
aαR − 2aq2 + R2 2αR − 4q2
a2αR − 2a2q2 − 2R4
=
(12)
a q2 − αR + R2 R2 − αR + a2 + q2
=
a2q2 − αa2R − a2R2 − R4
,
α = 2m ,
2
R2 = Cn(r) =
r − r0 n + βn
n
,
β = m + m2 − q2, q2 < m2 , r0 ∈ , n ∈ + ,
for the radius of curvature Rph−e = R = Cn(rph−e) of the equatorial orbit of a photon for the rotating point-charge. When a = 0 equation (12) reduces to,
3α + 9α2 − 32q2
Rph−e = Cn(rph−e) =
4
,
According to (1), if R = Cn(r) is a function of t,
1
R = R(t, θ) = = Cn(r(t)) = |r(t) − r0|n + βn n ,
β = m + m2 − q2 − a2 cos2 θ ,
so if r˙ = 0, R˙ = 0. In the polar orbit there is no loss of generality in taking
ϕ = const., ϕ˙ = 0. Then (1) becomes,
ds2 = Δ − a2 sin2 θ dt2 − ξ dR2 − ξdθ2 ,
(14)
ξ
Δ
2
R2 = Cn(r) =
r − r0 n + βn
n
,
r0 ∈ , r ∈ ,
β = m + m2 − a2 cos2 θ − q2 ,
a2 + q2 < m2 , n ∈ +, ξ = ρ2 = R2 + a2 cos2 θ ,
L a= ,
Δ = R2 − αR + a2 + q2 ,
m
recovering the stable radius of curvature for the photon orbit about the simple point-charge [2]. When a = q = 0, equation (12) reduces to,
3α
Rph−e = Cn(rph−e) = 2 = 3m ,
(13)
0 < |r − r0| < ∞ . Consider the associated Lagrangian,
1 L=
Δ − a2 sin2 θ t˙2 − ξ R˙ 2 − ξθ˙2
.
2
ξ
Δ
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Then,
When q = 0, a = 0, equation (16) reduces to,
∂L ∂ −
∂R ∂τ
∂L ∂R˙
1 =
2
ξΔ
−
ξ
(Δ −a2 ξ2
sin2
θ)
t˙2
+
(15)
1 −
2
(Δξ
− ξΔ Δ2
) R˙ 2
+
ξ
θ˙2
+ ξ R¨ = 0 . Δ
ω2 =
αR2 − αa2 cos2 θ 2R (R2 + a2 cos2 θ)2
=
(18)
=
αCn − αa2 cos2 θ
√ 2 Cn
(Cn
+
a2
cos2
θ)2
,
If R˙ = 0, then (15) yields,
ω2 =
θ˙2 t˙2
ξΔ =
−ξ
(Δ − a2 sin2 θ)
ξ ξ2
=
β = m + m2 − a2 cos2 θ , a2 < m2 , n ∈ + r0 ∈ , 0 < |r − r0| < ∞ .
αR2 − αa2 cos2 θ − 2q2R
= 2R (R2 + a2 cos2 θ)2 =
(16)
√
=
αCn − αa2 cos2 θ − 2q2 Cn
√ 2 Cn
(Cn
+
a2
cos2
θ)2
,
β=m+
m2 − a2 cos2 θ − q2 , a2 + q2 < m2 , n ∈ + r0 ∈ , 0 < |r − r0| < ∞ .
Equation (16) is Kepler’s 3rd Law for the polar orbit of the rotating point-charge. I remark that the angular velocity depends upon azimuth.
Let a = 0, q = 0, then (16) reduces to,
ω2
=
α 2R3
−
q2 R4
,
2
R2 = Cn(r) =
r − r0 n + βn
n
, β=m+
m2 − q2,
q2 < m2, r0 ∈ , n ∈ + ,
0 < |r − r0| < ∞,
which recovers Kepler’s 3rd Law (3) for the simple pointcharge. Setting a = q = 0 reduces (16) to,
ω2
=
α 2R3
,
2
R2 = Cn(r) =
r − r0 n + αn
n
,
n ∈ +, r(0) ∈ ,
0 < |r − r0| < ∞,
which recovers Kepler’s 3rd Law (2) for the simple pointmass.
Taking the near-field limit on (16),
This is Kepler’s 3rd Law for the polar orbit of the rotating point-mass.
Taking the near-field limit on (18),
lim
r → r0±
ω2
=
αβ2 − αa2 cos2 θ 2β (β2 + a2 cos2 θ)2
,
(19)
which is a scalar invariant, subject to azimuth, for the polar orbit of the rotating point-mass.
Thus, ω varies with azimuth as does R = Cn(r). At the poles of the rotating point-charge,
2
R2 = Cn(r) = |r − r0|n + βn n ,
β = m + m2 − a2 − q2 ,
(20)
ω2
=
αR2 − αa2 − 2q2R 2R (R2 + a2)2
,
and at the equator,
2
R2 = Cn(r) = |r − r0|n + βn n ,
β = m + m2 − q2 ,
(21)
ω2
=
α 2R3
−
q2 R4
.
It is noted that at the momentary equator in a polar orbit, the radius of curvature and Kepler’s 3rd Law are precisely those for the simple point-charge.
In the case of the rotating point-mass, at the poles,
2
R2 = Cn(r) = |r − r0|n + βn n ,
β = m + m2 − a2 ,
(22)
ω2
=
αR2 − αa2 2R (R2 + a2)2
,
and at the equator,
lim
r → r0±
ω2
=
αβ2 − αa2 cos2 θ − 2q2β 2β (β2 + a2 cos2 θ)2
,
(17)
which is a scalar invariant, subject to azimuth, for the polar orbit of the rotating point-charge.
2
R2 = Cn(r) = |r − r0|n + βn n ,
β = 2m = α ,
(23)
ω2
=
α 2R3
.
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At the momentary equator in a polar orbit the radius of curvature and Kepler’s 3rd Law are precisely those for the simple point-mass.
6 Photons in the polar orbit
which is the stable radius of curvature for the photon about the simple point-mass, but which is misinterpreted by the orthodox relativists as a measurable proper radius.
To obtain the stable photon radius of curvature of the polar orbit for the rotating point-mass, set q = 0 in (25),
Setting (14) equal to zero, with R˙ = 0, gives
2Rp3h−p − 3αRp2h−p + 2a2 cos2 θRph−p +
ω2 =
Δ
−
a2 sin2 ξ2
θ
=
R2
− αR + a2 cos2 θ + (R2 + a2 cos2 θ)2
q2
.
(24)
Denote the stable photon radius of curvature for a photon in polar orbit by Rph−p = Cn(rph−p). Then equating (24) to (16) gives,
2Rp3h−p − 3αRp2h−p +
+ αa2 cos2 θ = 0 ,
2
Rp2h−p = Cn(rph−p) = |rph − r0|n + βn n , β = m + m2 − a2 cos2 θ, a2 < m2 , r0 ∈ , n ∈ + .
(28)
+ 2a2 cos2 θ + 4q2 Rph−p + αa2 cos2 θ = 0 ,
2
Rp2h−p = Cn(rph−p) = |rph − r0|n + βn n ,
7 Potential functions in the weak field (25)
In the case of the rotating point-charge,
β = m + m2 − a2 cos2 θ − q2, a2 + q2 < m2 ,
r0 ∈ , n ∈ + .
Equation (25) gives the stable photon radius of curvature in the polar orbit. The orbit has a variable radius of curvature with azimuth.
When a = 0, q = 0, equation (25) reduces to
Δ − a2 sin2 θ
g00 =
ρ2
,
(29)
Δ = Cn(r) − α Cn(r) + a2 + q2 ,
ρ2 = Cn(r) + a2 cos2 θ . The potential Φ for a general metric is given by,
3α + 9α2 − 32q2
Rph−p = Cn(rph−p) =
4
,
2
Cn(rph−p) =
rph−p − r0 n + βn
n
,
(26)
g00 = (1 − Φ)2 = 1 − 2Φ + Φ2 . In the weak field,
g00 ≈ 1 − 2Φ .
β = m + m2 − q2, q2 < m2 ,
Now (29) gives,
r0 ∈ , n ∈ + ,
which recovers the radius of curvature for the stable orbit of a photon about the simple point-charge. When a = q = 0, (25) reduces to,
g00
=
Cn(r)
−
α Cn(r) + a2 cos2 Cn(r) + a2 cos2 θ
θ
+
q2
=
=
1−
α Cn(r) − q2 Cn(r) + a2 cos2 θ
,
3α Rph−p = Cn(rph−p) = 2 ,
(27) so the potential is,
2
Cn(rph−p) =
rph−p − r0 n + αn
n
,
Φ
=
2
α Cn(r) − q2 (Cn(r) + a2 cos2
θ)
,
(30)
α = 2m, r0 ∈ , n ∈ + ,
which recovers the curvature radius for the stable orbit of a photon about the simple point-mass. When n = 1 and r0 = α, equation (27) gives,
2
Cn(r) =
r − r0 n + βn
n
,
r0 ∈ ,
β = m + m2 − a2 cos2 θ − q2 , a2 + q2 < m2 , n ∈ + ,
Rph−p = Cn(rph−p) = rph−p = 3m ,
0 < |r − r0| < ∞ .
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The potential therefore depends upon azimuth. The potential for the rotating point-mass is obtained from
(30) by setting q = 0,
Φ
=
2
α (Cn(r)
Cn(r) + a2 cos2
θ)
,
(31)
2
Cn(r) =
r − r0 n + βn
n
,
r0 ∈ ,
β = m + m2 − a2 cos2 θ , a2 < m2 , n ∈ + ,
0 < |r − r0| < ∞ .
References
1. Crothers S. J. On the general solution to Einstein’s vacuum field and it implications for relativistic degeneracy. Progress in Physics, 2005, v. 1, 68–73.
2. Crothers S. J. On the ramifications of the Schwarzschild spacetime metric. Progress in Physics, 2005, v. 1, 74–80.
3. Crothers S. J. On the geometry of the general solution for the vacuum field of the point-mass. Progress in Physics, 2005, v. 2, 3–14.
4. Schwarzschild K. On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 1916, 189 (arXiv: physics/9905030).
If a = 0 the potential for the simple point-charge is re-
covered from (30),
α
q2
Φ=
−
,
(32)
2 Cn(r) 2Cn(r)
2
Cn(r) =
r − r0 n + βn
n
,
r0 ∈ ,
β = m + m2 − q2 , q2 < m2 , n ∈ + ,
0 < |r − r0| < ∞ ,
and if a = q = 0 the potential for the simple point-mass is
recovered,
α
Φ=
,
(33)
2 Cn(r)
2
Cn(r) =
r − r0 n + αn
n
,
r0 ∈
n∈ +,
0 < |r − r0| < ∞ .
According to (30), orbit in the equatorial gives equations (32) for the simple point-charge. According to (31), orbit in the equatorial gives equations (33) for the simple pointmass. For orbits in the polar, equations (32) and (33) are momentarily realised at the equator for a test particle orbiting the rotating point-charge and the rotating point-mass respectively. Thus, the effects of rotation of the source of the field do not manifest for a test particle in an equatorial orbit.
Taking the near-field limit on (30) gives,
αβ − q2
lim Φ =
,
(34)
r → r0±
2 (β2 + a2 cos2 θ)
β = m + m2 − a2 cos2 θ − q2 , a2 + q2 < m2 .
The potential approaches a finite limit with azimuth. The limiting values for the simpler configurations are easily obtained from (34) in the obvious way.
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On the Vacuum Field of a Sphere of Incompressible Fluid
Stephen J. Crothers Sydney, Australia
E-mail: thenarmis@yahoo.com
The vacuum field of the point-mass is an unrealistic idealization which does not occur in Nature — Nature does not make material points. A more realistic model must therefore encompass the extended nature of a real object. This problem has also been solved for a particular case by K. Schwarzschild in his neglected paper on the gravitational field of a sphere of incompressible fluid. I revive Schwarzschild’s solution and generalise it. The black hole is necessarily precluded. A body cannot undergo gravitational collapse to a material point.
1 Introduction
In my previous papers [1, 2] concerning the general solution for the point-mass I showed that the black hole is not consistent with General Relativity and owes its existence to a faulty analysis of the Hilbert [3] solution. In this paper I shall show that, along with the black hole, gravitational collapse to a point-mass is also untenable. This was evident to Karl Schwarzschild who, immediately following his derivation of his exact solution for the mass-point [4], derived a particular solution for an extended body in the form of a sphere of incompressible, homogeneous fluid [5]. This is also an idealization, and so too has its shortcomings, but represents a somewhat more plausable end result of gravitational collapse.
The notion that Nature makes material points, i. e. masses without extension, I view as an oxymoron. It is evident that there has been a confounding of a mathematical point with a material object which just cannot be rationally sustained. Einstein [6, 7] objected to the introduction of singularities in the field but could offer no viable alternative, even though Schwarzschild’s extended body solution was readily at his hand.
The point-mass and the singularity are equivalent. Abrams [8] has remarked that singularities associated with a spacetime manifold are not uniquely determined until a boundary is correctly attached to it. In the case of the pointmass the source of the gravitational field is identified with a singularity in the manifold. The fact that the vacuum field for the point-mass is singular at a boundary on the manifold indicates that the point-mass does not occur in Nature. Oddly, the conventional view is that it embodies the material point. However, there exists no observational or experimental data supporting the idea of a point-mass or point-charge. I can see no way an electron, for instance, could be compressed into a material point-charge, which must occur if the pointmass is to be admitted. The idea of electron compression is meaningless, and therefore so is the point-mass. Eddington [9] has remarked in similar fashion concerning the electron,
and relativistic degeneracy in general. I regard the point-mass as a mathematical artifice and
consider it in the fashion of a centre-of-mass, and therefore not as a physical object. In Newton’s theory of gravitation, r = 0 is singular, and equivalently in Einstein’s theory, the proper radius Rp(r0) ≡ 0 is singular, as I have previously shown. Both theories therefore share the non-physical nature of the idealized case of the point-mass.
To obtain a model for a star and for the gravitational collapse thereof, it follows that the solution to Einstein’s field equations must be built upon some manifold without boundary. In more recent years Stavroulakis [10, 11, 12] has argued the inappropriateness of the solutions on a manifold with boundary on both physical and mathematical grounds, and has derived a stationary solution from which he has concluded that gravitational collapse to a material point is impossible.
Utilizing Schwarzschild’s particular solution I shall extend his result to a general solution for a sphere of incompressible fluid.
2 The general solution for Schwarzschild’s incompressible sphere of fluid
At the surface of the sphere the required solution must maintain a smooth transition from the field outside the sphere to the field inside the sphere. Therefore, the metric for the interior and the metric for the exterior must attain the same value for the radius of curvature at the surface of the sphere. Furthermore, owing to the extended nature of the sphere, the exterior metric must take the form of the metric for the point-mass, but with a modified invariant containing the factors giving rise to the field, reflecting the non-pointlike nature of the source, thereby treating the source as a mass concentrated at the centre-of-mass of the sphere, just as in Newton’s theory. Schwarzschild has achieved this in his particular case. He obtained the following metric for the field
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inside his sphere,
ds2 =
3 cos χa − cos χ
2
dt2 −
2
(1)
− 3 dχ2 − 3 sin2 χ dθ2 + sin2 θdϕ2 ,
κρ0
κρ0
sin χ =
κρ0
1
η3,
η = r3 + ρ ,
3
ρ=
κρ0 3
−3 2
3 2
sin3
χa
−
9 4
cos χa
1 χa − 2 sin 2χa
,
κ = 8πk2 ,
π 0 χ χa < 2 , where ρ0 is the constant density of the fluid, k2 Gauss’ gravitational constant, and the subscript a denotes values at the surface of the sphere. Metric (1) is non-singular. Schwarzschild’s particular metric outside the sphere is,
ds2 =
α 1−
dt2 −
α 1−
−1
dR2 −
R
R
(2)
− R2 dθ2 + sin2 θdϕ2 ,
account for the extended configuration of the gravitating mass. Consequently, equation (1) becomes,
2
ds2 = 3 cos χa − χ0 − cos χ − χ0 dt2 − 2
(4)
− 3 dχ2 − 3 sin2 χ − χ0 dθ2 + sin2 θdϕ2 ,
κρ0
κρ0
sin χ − χ0 =
κρ0
η
1 3
,
3
3
η = r − r0 + ρ ,
−3
ρ = κρ0 3
2
3 sin3 2
χa − χ0
−
9
1
−
cos 4
χa − χ0
χa − χ0
− sin 2 2
χa − χ0
,
κ = 8πk2 , r0 ∈ , r ∈ , χa ∈ , χ0 ∈ , π
0 |χ − χ0| |χa − χ0| < 2 , and outside the sphere, equation (2) becomes,
ds2 =
α 1−
dt2 −
α 1−
−1
dR2 −
R
R
(5)
− R2 dθ2 + sin2 θdϕ2 ,
R3 = r3 + ρ ,
α=
3 κρ0
sin3
χa
,
π 0 χa < 2 ,
ra r < ∞ .
Metric (2) is non-singular for an extended body. In the case of the simple point-mass (i. e. non-rotating, no
charge) I have shown elsewhere [13] that the general solution is,
√
√
ds2 = ( √Cn−α) dt2− √ Cn
Cn2 dr2−
Cn
( Cn−α) 4Cn
(3)
− Cn(dθ2 + sin2 θdϕ2) ,
2
Cn(r) =
r − r0 n + αn
n
, α = 2m ,
n ∈ + , r ∈ , r0 ∈ ,
0 < |r − r0| < ∞ ,
where n and r0 are arbitrary. Now Schwarzschild fixed his solution for r0 = 0. I note
that his equations, rendered herein as equations (1) and (2),
can be easily generalised to an arbitrary r0 ∈ and arbitrary χ0 ∈ by replacing his r and χ by |r − r0| and |χ − χ0| respectively. Furthermore, equation (3) must be modified to
R3 =
3
r − r0 + ρ,
α=
3 sin3 κρ0
χa − χ0
,
n ∈ + , r0 ∈ , r ∈ , χ0 ∈ , χa ∈ , π
0 |χa − χ0| < 2 ,
|ra − r0| |r − r0| < ∞ ,
and outside the sphere, equation (3) becomes,
√
√
ds2 = ( √Cn−α) dt2− √ Cn
Cn2 dr2−
Cn
( Cn−α) 4Cn
(6)
− Cn(dθ2 + sin2 θdϕ2) ,
2
Cn(r) =
n
r − r0 +
n
n
,
α=
3 κρ0
sin3
χa − χ0
,
=
3 κρ0
3 sin3 2
χa − χ0
−
1
9
1
3
−
cos 4
χa − χ0
χa − χ0
− sin 2 2
χa − χ0
,
r0 ∈ , r ∈ , n ∈ + , χ0 ∈ , χa ∈ ,
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|ra − r0| |r − r0| < ∞ .
The general solution for the interior of the incompressible Schwarzschild sphere is given by (4), and (6) gives the general solution on the exterior of the sphere.
Consider the general form for a static metric for the gravitational field [13],
ds2 = A(D)dt2 − B(D)dD2 − C(D) dθ2 + sin2 θdϕ2 ,
Substituting into (8) for K gives for the proper radius from outside the sphere,
Rp(r) = Rpa + Cn(r) Cn(r) − α −
−
Cn(ra) Cn(ra) − α +
(9)
D = |r − r0| , A, B, C > 0 ∀ r = r0 .
+ α ln
Cn(r) + Cn(ra) +
Cn(r) − α .
Cn(ra) − α
With respect to this metric I identify the real r-parameter, the radius of curvature, and the proper radius thus:
1. The real r-parameter is the variable r.
2. The radius of curvature is Rc = C(D).
3. The proper radius is Rp = B(D) dD. According to the foregoing, the proper radius of the sphere of incompressible fluid determined from inside the sphere is, from (4),
χa
Rp =
3 (χ − χ0) dχ = κρ0 |χ − χ0|
3 κρ0 χa − χ0 .
(7)
χ0
The proper radius of the sphere cannot be determined from outside the sphere. According to (6) the proper radius to a spacetime event outside the sphere is,
Rp =
√ √ Cn √Cn dr =
Cn − α 2 Cn
= K + Cn(r) Cn(r) − α +
(8)
Then by (9), for |r − r0| |ra − r0|
|r − r0| → |ra − r0| ⇒ Rp → Rp+a ,
but Rpa cannot be determined. According to (4) the radius of curvature Rc = Pa at the
surface of the sphere is,
3
Pa = κρ0 sin χa − χ0 .
(10)
Furthermore, inside the sphere,
G 2π ,
Rp and
G lim = 2π , χ → χ± 0 Rp
where G = 2πRc is the circumference of a great circle. But outside the sphere,
G 2π ,
Rp
+ α ln
Cn(r) + Cn(r) − α ,
K = const.
At the surface of the sphere the proper radius from outside has some value Rpa , for some value ra of the parameter r. Therefore, at the surface of the sphere,
with the equality only when Rp → ∞. The radius of curvature of (6) at the surface of the sphere
is Cn(ra) so,
1
Cn(ra) =
n
ra − r0 +
n
n
.
(11a)
At the surface of the sphere the measured circumference Ga of a great circle is,
Rpa = K + Cn(ra) Cn(ra) − α +
Ga = 2πPa = 2π Cn(ra) .
+ α ln
Cn(ra) +
Solving for K,
Cn(ra) − α .
K = Rpa − Cn(ra) Cn(ra) − α −
− α ln
Cn(ra) + Cn(ra) − α .
Therefore, at the surface of the sphere equations (10) and (11a) are equal,
1
n
ra − r0 +
n
n
=
3 κρ0 sin χa − χ0 ,
(11b)
and so, |ra − r0| =
n
1 n
3 κρ0
2
sinn χa − χ0 − n
.
(11c)
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The variable r is just a parameter for the radial quantities Rp and Rc associated with (6). Similarly, χ is also a parameter for the radial quantities Rp and Rc associated with (4). I remark that r0 and χ0 are both arbitrary, and independent of one another, and that |r − r0| and |χ − χ0| do not of themselves denote radii in any direct way. The arbitrary values of the parameter “origins”, r0 and χ0, are simply boundary points on r and χ respectively. Indeed, by (7), Rp(χ0) ≡ 0, and by (9), Rp(ra) ≡ Rpa , irrespective of the values of r0, ra, and χ0. The centre-of-mass of the sphere of fluid is always located precisely at Rp(χ0) ≡ 0. Furthermore, Rp(r) for |r − r0| < |ra − r0| has no meaning since inside the sphere (4) describes the geometry, not (6).
According to (11b), equation (9) can be written as,
Rp(r) = Rpa + Cn(r) Cn(r) − α −
3
−
κρ0 sin χa − χ0
3 κρ0 sin χa − χ0 − α +
(12)
Schwarzschild [5] has also shown that the velocity of light inside his sphere of incompressible fluid is given by,
2 vc = 3 cos χa − cos χ , which generalises to,
2
vc = 3 cos χa − χ0 − cos χ − χ0 .
(14)
At the centre χ = χ0, so vc reaches a maximum value
there of,
2
vc =
,
3 cos χa − χ0 − 1
Equation (14) is singular when cos χa − χ0
=
1 3
,
which
means that there is a lower bound on the possible radii of
curvature for spheres of incompressible, homogeneous fluid,
which is, by (13) and (6),
9
8
Pa (min) = 8 α =
, 3κρ0
(15a)
+ α ln
√
Cn(r) +
3 κρ
sin|χa −χ0 |
+
0
√
Cn (r)−α
,
3 κρ
sin|χa −χ0 |−α
0
α=
3 sin3 κρ0
χa − χ0
.
and consequently, by equation (11a),
|ra − r0|(min) = =
9α
n
−n
1 n
=
8
8 3κρ0
1
n 2
n
−n ,
(15b)
Note
that
in
(4),
|χ − χ0|
cannot
grow
up
to
π 2
,
so
that Schwarzschild’s sphere does not constitute the whole
spherical space, which has a radius of curvature of 3 .
κρ0
From (4) and (6),
α =
Pa
sin2
χa − χ0
,
α
=
κρ0 3
Pa3
.
(13)
The volume of the sphere is,
from which it is clear that a body cannot collapse to a material
point.
From
(13),
a
sphere
of
given
gravitational
mass
α k2
,
cannot have a radius of curvature, determined from outside,
smaller than,
Pa (min) = α ,
so
|ra − r0|(min) = [αn −
n
]
1 n
,
3 V=
κρ0
π
3 χa
2
sin2
|χ
−
χ0|
χ |χ
− −
χ0 χ0|
dχ
×
χ0
2π
α=
3 κρ0
sin3
χa − χ0
.
× sin θdθ dϕ =
3 Kepler’s 3rd Law for the sphere of incompressible fluid
0
0
3
32
1
= 2π κρ0
|χa − χ0| − 2 sin 2|χa − χ0| ,
There is no loss of generality in considering only the equator-
ial
plane,
θ
=
π 2
.
Equation
(6)
then
leads
to
the
Lagrangian,
√
√
so the mass of the sphere is,
1 L=
C −α √
t˙2 −
√
C
2
C
C −α
33
1
M = ρ0V = 4k2 κρ0 |χa − χ0| − 2 sin 2|χa − χ0| . where the dot indicates ∂/∂τ .
√˙C
2
− Cϕ˙ 2
,
S. J. Crothers. On the Vacuum Field of a Sphere of Incompressible Fluid
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Let R = Cn(r). Then,
4 Passive and active mass
∂ ∂τ
∂L ∂R˙
−
∂L ∂R
=
R R−α
R¨ +
α 2R2
t˙2
−
− α R˙ 2 − R ϕ˙ 2 = 0 . 2 (R − α)
Now let R = const. Then,
α 2R2
t˙2 = R
ϕ˙ 2 ,
so
ω2 =
α 2R3
=
α
3
2C 2
=
2
α
3.
n
r − r0 +
n
n
(16)
Equation (16) is Kepler’s 3rd Law for the sphere of incompressible fluid.
Taking the near-field limit gives,
ωa2
=
lim |r−r0| → |ra−r0|+
ω2
=
2
α
3.
n
ra − r0 +
n
n
According to (11b) and (10) this becomes,
ωa2 = 2
3 κρ0
α
α
3
2 sin3
χa − χ0
=. 2Pa3
Finally, using (13),
ωa = sin3 |χ√a − χ0| ,
(17)
α2
α=
3 sin3 κρ0
χa − χ0
.
The relationship between passive and active mass manifests,
owing to the difference established by Schwazschild, be-
tween what he called “substantial mass” (passive mass) and
the gravitational (i .e. active) mass. He showed that the for-
mer is larger than the latter.
Schwarzschild has shown that the substantial mass M is
given by,
3
32
1
M = 2πρ0 κρ0
χa − 2 sin 2 χa ,
π 0 χa < 2 ,
and the gravitational mass is,
αc2 1 m= =
2G 2
3 κρ0
sin3
χa =
κρ0 6
Pa3
=
4π 3
Pa3ρ0
,
Pa = 0
3 κρ0 sin χa ,
π χa < 2 .
I have generalised Schwarzschild’s result to,
3
32
1
M = 2πρ0 κρ0
χa − χ0
− sin 2 2
χa − χ0
,
αc2 1 m= =
2G 2
3 sin3 κρ0
χa − χρ0
=
(19)
=
κρ0 6
Pa3
=
4π 3
Pa3ρ0
,
In contrast, the limiting value of ω for the simple point-
mass [4] is,
1
ω0
=
√ α2
,
Pa = 0
3 κρ0 sin χa − χ0 ,
π
χa − χ0
<, 2
α = 2m .
When Pa is minimum, (17) becomes,
ωa2
=
16 27α
,
where G is Newton’s gravitational constant. Equation (19) is only formally the same as that for the Euclidean sphere, because the radius of curvature Pa is not a Euclidean quantity, and cannot be measured in the gravitational field. (18) The ratio between the gravitational mass and the substantial mass is,
16 6
α=
.
27 κρ0
Clearly, equation (17) is an invariant,
ωa =
κρ0 . 6
m =
M3
2 sin3 χa − χ0
χa − χ0
−
1 2
sin
2
χa − χ0
.
Schwarzschild has shown that the naturally measured fall velocity of a test particle, falling from rest at infinity down to the surface of the sphere of incompressible fluid is,
va = sin χa ,
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S. J. Crothers. On the Vacuum Field of a Sphere of Incompressible Fluid
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which I generalise to,
va = sin χa − χ0 .
The quantity va is the escape velocity.
Therefore,
as
the
escape
velocity
increases,
the
ratio
m M
decreases, owing to the increase in the mass concentration.
In the case of the fictitious point-mass,
m
lim
=1.
|χa−χ0| → 0 M
However, according to equation (14), for an incompressible sphere of fluid,
10. Stavroulakis N. A statical smooth extension of Schwarzschild’s metric. Lettere al Nuovo Cimento, 1974, v. 11, 8 (www.geocities.com/theometria/Stavroulakis-3.pdf).
11. Stavroulakis N. On the Principles of General Relativity and the SΘ(4)-invariant metrics. Proc. 3rd Panhellenic Congr. Geometry, Athens, 1997, 169 (www.geocities.com/theometria/Stavroulakis-2.pdf).
12. Stavroulakis N. On a paper by J. Smoller and B. Temple. Annales de la Fondation Louis de Broglie, 2002, v. 27, 3 (www.geocities.com/theometria/Stavroulakis-1.pdf).
13. Crothers S. J. On the geometry of the general solution for the vacuum field of the point-mass. Progress in Physics, 2005, v. 2, 3–14.
so Finally,
1
cos
χa − χ0
min
=, 3
m ≈ 0.609 .
M max
πm 4
as
χa − χ0
→, → . 2 M 3π
Dedication
I dedicate this paper to the memory of Dr. Leonard S. Abrams: (27 Nov. 1924 — 28 Dec. 2001).
References
1. Crothers S. J. On the general solution to Einstein’s vacuum field and it implications for relativistic degeneracy. Progress in Physics, 2005, v. 1, 68–73.
2. Crothers S. J. On the ramifications of the Schwarzschild spacetime metric. Progress in Physics, 2005, v. 1, 74–80.
3. Hilbert D. Nachr. Ges. Wiss. Gottingen, Math. Phys. Kl., 1917, v. 53, (see also in arXiv: physics/0310104).
4. Schwarzschild K. On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 1916, 189 (see also in arXiv: physics/ 9905030).
5. Schwarzschild K. On the gravitational field of a sphere of incompressible fluid according to Einstein’s theory. Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 1916, 424 (see also in arXiv: physics/9912033).
6. Einstein A., Rosen N. The particle problem in the general theory of relativity. Phys. Rev., 1935, v. 48, 73.
7. Einstein A. The Meaning of Relativity. 5th Edition, Science Paperbacks and Methuen and Co. Ltd., London, 1967.
8. Abrams L. S. Black holes: the legacy of Hilbert’s error. Can. J. Phys., v. 67, 1989, 919 (see also in arXiv: gr-qc/0102055).
9. Eddington A. S. The mathematical theory of relativity, Cambridge University Press, Cambridge, 2nd edition, 1960.
S. J. Crothers. On the Vacuum Field of a Sphere of Incompressible Fluid
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Power as the Cause of Motion and a New Foundation of Classical Mechanics
Efthimios Harokopos
P.O. Box 77034, Paleo Faliro, 17510 Athens, Greece E-mail: makischarokopos@yahoo.com
Laws of motion are derived based on power rather than on force. I show how power extends the law of inertia to include curvilinear motion and I also show that the law of action-reaction can be expressed in terms of the mutual time rate of change of kinetic energies instead of mutual forces. I then compare the laws of motion based on power to Newton’s Laws of Motion and I investigate the relation of power to Leibniz’s notion of vis viva. I also discuss briefly how the metaphysics of power as the cause of motion can be grounded in a modern version of occasionalism for the purpose of establishing an alternative foundation of mechanics. The laws of motion derived in this paper along with the metaphysical foundation proposed come in defense of the hypotheses that time emerges as an ordered progression of now and that gravitation is the effect of energy transfer between an unobservable substance and all matter in the Universe.
1 Introduction
This paper’s central aim is the derivation of laws of motion based on the notion of power rather than on the classical notion of force. Although the derivation of laws of motion is traditionally a subject of mechanics, several references are made herein to the history and philosophy of science. This is necessary because this paper deals primarily with the foundations of mechanics. Specifically, the hypothesis that power is the cause of motion, as contrasted to the Newtonian hypothesis according to which force is the cause of motion, leads to a major revision of the foundations of Classical Mechanics.
Most contemporary philosophers of science focus on the foundational problems of General Relativity and Quantum Mechanics and, unlike their seventeenth-century counterparts, think of Classical Mechanics as unproblematic. Butterfield mentions two errors found in this view that correspond to what he calls the matter-in-motion picture and the particlein-motion picture [1]. According to the matter-in-motion picture, for example, bodies are collections of particles separated by voids, can move in vacuum and interact with each other, whilst their motion is completely determined by Newton’s Second Law. This view has become a part of an “educated layperson’s” common sense nowadays but according to Butterfield it is problematic: it does not offer, amongst other things, any explanation of the mechanism(s) of the assumed interactions but resorts to concepts such as forces acting across an intervening void (“action-at-adistance”).
The failure of modern theories to provide solutions to the foundational problems of Classical Mechanics is partly due to the fact that alternative rigid foundations have not been proposed but issues seem to have been further perplexed.
Quantum uncertainty and the four-dimensional space-time of relativity have taken the place of the determinism and of the unobservable absolute space and universal time of Classical Mechanics. Mysterious action-at-a-distance still prevails in the quantum world and attempts to quantize gravity and unite Quantum Mechanics with General Relativity have failed to this date. In presenting an alternative system of laws of motion based on power, I aim primarily in the investigation of a new foundation, which offers an alternative approach for solutions to some of the unsolved problems of Classical Mechanics.
In a similar way to the matter-in-motion picture, the notion of force has also become part of an “educated layperson’s” common sense, thanks to the empirical support the laws of mechanics have enjoyed over the past 300 years. It is well known, however, that Newton was heavily criticized for his use of the notion of force in an effort to ground his physics on his metaphysics and there is still considerable interest in the metaphysics of his Principia. In Science and Hypothesis, Poincare´ writes [2]:
When are two forces equal? We are told that it is when they give the same acceleration to the same mass, or when acting in opposite directions they are in equilibrium. This definition is a sham.
In Principles of Dynamics, Donald T. Greenwood offers an introduction to the issues raised by Newton’s concept of force [3]:
The concept of force as a fundamental quantity in the study of mechanics has been criticized by various scientists and philosophers of science from shortly after Newton’s enunciation of the laws of motion until the present time. Briefly, the idea of a force, and a field force in particular, was considered to be an
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intellectual construction, which has no real existence. It is merely another name for the product of mass and acceleration, which occurs in the mathematics of solving a problem. Furthermore, the idea of force as a cause of motion should be discarded since the assumed cause and effect relationship cannot be proven. (Italics added)
The questions raised from Newton’s definition of force and postulation of absolute space are well known to the philosophers of science and will be further discussed in sections 4, 5 and 6. In the following two sections, 2 and 3, I will show that using the notion of power as a priori principle, laws of motion can be derived with remarkably different definitions of inertia and action-reaction. I will then argue in section 4, where I discuss the relation of this alternative system of laws to Newton’s, that the existence of a more general principle of motion is even acknowledged by Newton, in his own writings. In section 5, the relation of the notion of power to Leibniz’s notion of vis viva is examined. Then, in section 6, I discuss how the metaphysics of power can be grounded in a modern version of occasionalism for the purpose of establishing an alternative foundation of Classical Mechanics. I argue that the alternative foundation proposed, along with an appropriate space-time structure, support a new hypothesis about time and about the nature of gravitation.
2 The axiom of motion
I begin the derivation of the laws of motion by stating the axiom of motion, an expression relating the velocity and the time rate of change of momentum of a particle, to a scalar quantity called the time rate of change of kinetic energy, also known as (instantaneous) power. The status of this axiom is assumed here to be that of a priori truth as opposed to a self-evident or empirical principle.
Axiom of Motion: The time rate of change of the kinetic energy of a particle is equal to the scalar product of its velocity and time rate of change of its momentum.
Denoting the kinetic energy by Ek and the momentum by p, the axiom of motion can be expressed as follows:
dEk = d p ∙ d r ,
(1)
dt dt dt
where r is the position vector of the particle. The momentum
p is defined as
dr
p=m .
(2)
dt
If the mass m of the particle is independent of time t
and position r, then by combining equations (1) and (2), the
time rate of change of the kinetic energy Ek can be written
as follows:
dEk dt
=
d2r m dt2
∙
dr dt
.
(3)
Corollary I: The kinetic energy of a particle with a constant mass m is given by
1
Ek = 2 mv ∙ v ,
(4)
where v is defined as
dr
v= .
(5)
dt
Proof: From equation (3) we obtain
dEk dt
=
d2r d r m dt2 ∙ dt
=
dr d m∙
dt dt
dr dt
d =m
dt
1 dr dr ∙
2 dt dt
,
which yields
1 dr dr 1
Ek
=
m 2
dt
∙
dt
=
mv ∙ v. 2
(6)
The axiom of motion is the only principle required for deriving the laws of motion, as it will be shown in the next section.
3 The laws of motion
Law of Inertia: If the time rate of change of the kinetic energy of a particle is zero, the particle will continue in its state of motion.
Proof: If the time rate of change of the kinetic energy of a particle is zero, then from equation (3) we obtain
d2r d r
m dt2 ∙ dt = 0 .
(7)
Assuming m remains constant, the following satisfy eq-
uation (7)
dr
dt = v0 ,
(8)
dr
= 0,
(9)
dt
d2r
dt2 ∙ v = 0 ,
(10)
where v0 is a constant. Thus, solutions to equation (7) include motion with a constant velocity v0, given by equation (8), or a state of rest, given by equation (9) and in both these cases the time rate of change of kinetic energy is zero. These are trivial solutions to equation (7) arising when either the velocity or the acceleration of the particle, are null vectors. Yet, these two trivial solutions result in the simplest kinematic states possible and the only two states allowed when there are no forces acting on a particle according to Newton’s First Law. However, if power is postulated as the cause of motion there is another trivial solution, that of uniform circular motion, as it will be shown below.
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General solutions to equation (10) include all curvilinear paths with a constant kinetic energy Ek. The requirement of a constant kinetic energy could have been included in the statement of the law of inertia but this is obviously redundant since, if the time rate of change of the kinetic energy is zero then kinetic energy is constant. Clearly, the states of motion resulting from (8) and (9) are trivial solutions to (10) with zero velocity and zero acceleration, respectively. From equations (5), (6) and (10) we obtain:
d2r dt2
∙
dr dt
=
0
⇔
dr dt
∙
dr dt
=
v∙v
=
2Ek m
= k,
(11)
where k is a constant equal to twice the kinetic energy per unit mass. Thus, all motion paths that satisfy equation (10) also satisfy the following equation
dr dr
∙ =k,
(12)
dt dt
which is equivalent to the statement that the magnitude of velocity, or the speed, must be constant. In the case of motion in a plane, v can be expressed in polar coordinates as follows:
v=
dr ˆr
+
r dθ θˆ.
(13)
dt
dt
From equations (12) and (13) we obtain:
dr 2 +
dθ r
2
= k2.
(14)
dt
dt
A trivial solution to equation (14) is uniform circular
motion given by
r (t) = rˆr(t) ,
(15)
where r is a constant radius and the unit radial vector ˆr rotates at a constant rate dθ/dt. In the context of this law of inertia, if a particle is in uniform circular motion and the time rate of change of its kinetic energy remains zero, the state of uniform circular motion will be maintained. Notice that no claim of any sort is made herein that zero power is the cause of uniform circular motion. Obviously, a zero of something cannot be the real cause of anything. The only claim made is that if a particle is in uniform circular motion -or in any other curvilinear path that satisfies equation (12) — and power, the postulated cause of motion, remains zero then the particle will continue in its state of motion. I would like to stretch this point because, as it will be discussed further in chapter 4, the laws of motion presented in this paper can be considered as an alternative to Newton’s Laws of Motion. Thus, one should refrain from evaluating these laws in the context of Newtonian mechanics, since the two systems of laws are grounded in different metaphysics. The question then of how a particle is set on a uniform circular motion in the first place is a metaphysical one and it will be placed in its proper context in chapter 6.
Non-trivial solutions to equation (14) include motion in a plane where the magnitude of the velocity v remains constant up to sign changes. Such motion possibilities are virtually unlimited, including for instance motion in eightshaped figures and cycloid paths. However, some of these paths may represent physical possibilities and others may not. Uniform circular motion is a physical possibility in both micro and macro scales and this has been confirmed empirically. The choice of specific curvilinear motions over others as an effect of inertia, if power is postulated to be the cause of motion, is the subject of metaphysics discussed in section 6. The law of inertia presented in this section is a statement that the state of such motions is maintained in the absence of a cause, if power is postulated to be the cause of motion. However, the law does not provide a justification for the existence or preference of certain states of motions over others in the absence of a cause of motion.
General solutions to equation (12) in three-dimensional Euclidean space include motion along any curve. It is known from differential geometry that if a curve is regular, then there exists a reparametrization such that the curve has unit speed [4]. Thus, a particle can be made to move with constant speed along any curve in space using proper arc-length reparametrization resulting in constant kinetic energy and as a consequence, zero power.
The law of inertia is a statement about the tendency of particles to maintain their state of motion when the time rate of change in their kinetic energy is zero and this tendency is called inertia. Again, the law of inertia was derived based on the metaphysical hypothesis that power is the cause of motion. A consequence from such hypothesis is that the set of “cause-free” paths now includes all paths where the kinetic energy remains constant, instead of just uniform rectilinear motion and the state of rest defined in Newtonian mechanics. As it will be discussed in section 4.1, from an empirical viewpoint it is irrelevant whether one considers just rectilinear or curvilinear motion as an effect of inertia, since no experiment can be devised to prove that in the case of a freely moving particle. This is because, there is always a cause present affecting the motion of all particles. In the case of Newtonian mechanics, this cause is a gravity force and in the case of the laws of motion discussed in this paper there is always a power cause acting and giving rise to gravitational effects as it will be discussed in chapter 6.
Corollary II: If the time rate of change of the kinetic energy of a particle is zero, linear momentum is conserved.
Proof: As a direct consequence of the law of inertia, if the time rate of change of kinetic energy is zero and the velocity is denoted by v, then from equations (1) and (5) we obtain
dp
∙v = 0.
(16)
dt
By using equation (2) and since v is not the null vector
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in general, we obtain from equation (16) the result:
d(mv)
dt = 0 ⇒ (mv)2 − (mv)1 = 0 ⇒
(17)
⇒ (mv)2 = (mv)1 = mv = const .
Equation (17) is the mathematical statement of the theorem of the conservation of linear momentum [5].
Law of Interaction: To every action there is an equal and opposite reaction; that is, in an isolated system of two particles acting upon each other, the mutual time rate of change of kinetic energies are equal in magnitude and opposite in sign.
Proof: We denote the two interacting particles as m1 and m2.
Furthermore, we denote m1 as the agent causing the action in
the system. The total kinetic energy of the interacting system
of particles is the sum of the kinetic energies of the two
particles:
Ek = Ek1 + Ek2 .
(18)
From equations (1), (5) and (18) we obtain
dEk dt
=
d p1 dt
∙ v1
+
d p2 dt
∙ v2 ,
(19)
where v1 and v2 are the velocities of the two particles with momentum p1 and p2, respectively.
Next, we consider the mutual time rate of change of
kinetic energy imposed by the particles upon each other. The
time rate of change of kinetic energy of particle m2, denoted as Ek2 , is equal to the action imposed on it by particle m1, denoted as Ek12 and given by
dEk2 dt
=
d p2 dt
∙ v2
=
dEk12 dt
.
(20)
The time rate of change of the kinetic energy of particle m1 is equal to the sum of the time rate of change of the kinetic energy of the system due to its action as an agent and that imposed on it by particle m2 in the form of a reaction and denoted as Ek21
horse is reacted by the cart, from equation (21) it may be seen that dEk/dt is equal to zero and the state of motion does not change. Then, in this special case, action is equal to reaction by definition. This can serve the purpose of clearing any confusion that may arise when the action by the horse on the cart is thought to be equal to the total action produced by the horse, a statement that is not true in the most general case.
The philosophical issues arising from the law of interaction will be discussed in more detail in section 4.
Corollary III: In an isolated system of two particles acting upon each other and both having velocity v, the mutual time rate of change of momentum vectors are equal in magnitude and opposite in direction.
Proof: By denoting the mutual momentum vectors by p12 and p21, from equations (1), (5) and (22) we obtain
d p12 ∙ v = − d p21 ∙ v ⇔ d p12 + d p21 ∙ v = 0 . (23)
dt
dt
dt
dt
Since v is not in general a null vector, we obtain the
result:
d p12 = − d p21 .
(24)
dt
dt
In the case where v is orthogonal to the sum of the mutual time rate of change of the momentum vectors of the two particles, then equation (23) will still hold. However, in this case, the mutual time rate of change of momentum vectors will not in general be equal in magnitude and opposite in direction.
The axiom of motion of section 2, together with the law of inertia and the law of interaction, combined further with the axiom of conservation of energy of isolated systems, provide a framework for deriving the differential equations of motion of particles and by extension of rigid bodies in dynamical motion. Next, I will examine the relation of the laws of motion presented in this section to Newton’s Laws of Motion.
4 Power versus force
dEk1 dt
=
dEk dt
+
dEk21 dt
=
d p1 dt
∙ v1 .
(21)
By combining equations (19), (20) and (21), we obtain
the result:
dEk12 = − dEk21 .
(22)
dt
dt
Equation (22) is the mathematical statement of the law of interaction. According to the law, the reaction on a horse pulling on a cart, — to use Newton’s example in the Principia — is equal to the action applied by the horse on the cart. In general, part of the action produced by the horse is used to change its own state of motion and the remaining to change that of the cart. In the case where the total action of the
Newton stated his laws of motion in Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1686 [6]. The Principia was revised by Newton in 1713 and 1726. Using modern terminology, the laws can be stated as follows [3]:
First Law: Every body continuous in its state of rest, or of uniform motion in a straight line, unless compelled to change that state by forces acting upon it.
Second Law: The time rate of change of linear momentum of a body is proportional to the force acting upon it and occurs in the direction in which the force acts.
Third Law: To every action there is an equal and opposite reaction and thus, the mutual forces of two bodies acting
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upon each other are equal in magnitude and opposite in direction.
4.1 Newton’s First Law: A priori truth or an experimental fact?
Newton’s First Law can be deduced from the law of inertia stated in section 3 and specifically from equations (8) and (9), or from corollary II. According to the law of inertia, when the time rate of change of the momentum of a particle is zero, then that particle will either remain at rest or move in a straight line with constant velocity v0.
It is interesting to recall Newton’s comments in Principia that follow the First Law [6]:
Projectiles continue in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are continuously drawn aside from rectilinear motion, do not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, preserve their motions both progressive and circular for a much longer time.
The first part of Newton’s comments regarding the projectile motion is problematic from an empirical perspective. No experiment can be devised where a projectile will move in the absence of gravity. Thus, there can be no cause free motion experiments in the context of Newtonian mechanics in order to observe what the resulting motion would be if the cause were to be removed. Therefore, it seems that Newton was referring to a thought experiment than to a wellestablished empirical fact. Furthermore, in the remaining part of Newton’s comment regarding the First Law, things become even more interesting as he attempts to draw conclusions regarding the validity of the First Law from the motion of rotating bodies, such as spinning disks and planets. This is obviously a peculiar attempt for a connection between the rectilinear motion the First Law deals exclusively with, and rotational motion in the absence of a resisting medium. It appears that Newton’s attempt to provide conclusive empirical support of the First Law is fraught with difficulties simply because no experiments can be devised from which the First Law can be inferred from the phenomena and rendered general by induction. This fact turns out to conflict with Newton’s statement in the general scholium in book III of the Principia [6]:
In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive forces of bodies, and the laws of motion and gravitation, were discovered.
The First Law and specifically the statement that bodies remain at rest or move uniformly in a straight line unless a force acts upon them, does not comply with the rules of
the (experimental) philosophy Newton claims to abide with. The First Law does not deal with circular orbits, even if such orbits were employed by Newton as an example in his attempt to justify it. The First Law is actually an axiom, which must be accepted without proof, and not a statement derived via the use of inductive methodology. This is again due to the fact that no experiment can be devised on our planet for the purpose of observing what the motion of a projectile would be when there is no force acting upon it. According to Newton’s Law of Universal Gravitation, gravity forces act upon a body unless it is set in motion in a region of space sufficiently far away from the influence of other bodies. Is then Newton alluding to the possibility of the existence of a more general First Law similar to the law of inertia of section 3? Let us recall what Poincare´ said [2]:
The Principle of Inertia. — A body under the action of no force can only move uniformly in a straight line. Is this a truth imposed on the mind a` priori? If this be so, how is it that the Greeks have ignored it? How could they have believed that motion ceases with the cause of motion? Or, again, that every body, if there is nothing to prevent it, will move in a circle, the noblest of all forms of motion? If it be said that the velocity of a body cannot change, or there is no reason for it to change, may we not just as legitimately maintain that the position of a body cannot change, or that the curvature of its path cannot change, without the agency of an external cause? Is, then, the principle of inertia, which is not an a` priori truth, an experimental fact? Have there ever been experiments on bodies acted on by no forces? And, if so, how did we know that no forces were acting?
Poincare´ continues with his discussion of the principle of inertia by stating that
Newton’s First Law could be the consequence of a more general principle, of which the principle of inertia is only a particular case.
In turn, I argue that the axiom of motion, equation (1), can serve the role of this more general principle and Newton’s First Law is indeed a special case of a more general law of inertia, such as the one derived in section 3.
Thus, I essentially argue that Newton’s First Law makes reference to phenomena that are just two possibilities within a broader range of possibilities mandated by a more general principle of inertia, such as the law of inertia of section 3. As I will demonstrate in the proceedings, the same holds true with Newton’s Third Law. There, matters are even clearer regarding my argument that Newton’s laws are just a special case of the laws presented in section 3.
4.2 Newton’s Second Law: The metaphysical cause of motion
The mathematical expression of Newton’s Second Law, after a suitable choice of units is made is the following [3]:
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dp d
F = = (mv) .
(25)
dt dt
With the Second Law, Newton defines force as the cause
of motion and equates it to the time rate of change of
momentum. The laws of motion presented in section 3,
based on the axiom of motion, challenge the notion that the
Newtonian force is the cause of motion and the metaphysical
foundation of mechanics. However, in these laws of motion,
the metaphysics of force are replaced by those of the time
rate of change of kinetic energy, also known as power. In
a way analogous to Newton’s Second Law, the axiom of
motion stated in section 2 can be expressed as follows
P = d(Ek) ,
(26)
dt
where P is the (instantaneous) power and Ek the kinetic energy of a particle.
When we say force is the cause of motion, we are talking metaphysics. . .
writes Poincare´ in Science and Hypothesis [2]. This statement made by Poincare´ also applies when the time rate of change of kinetic energy, or power, is defined as the cause of motion. Whether using force or power, the physics of the associated laws of motion must be grounded in some metaphysics and this is done in section 6. It is important to understand that the particular choice of a quantity to assume the role of the cause of motion becomes the link between the empirical world of physics and the metaphysics of what exists and is real. Thus, although one can choose either force or power as the basis of stating laws of motion, the metaphysical foundations of such laws will turn out to be profoundly different. Newton used his notion of force to ground his physics in the metaphysics of absolute space and time. In section 6, I will discuss how the notion of power grounds the physics of the laws of motion of section 3 in the metaphysics of a modern version of Cartesian occasionalism and a dual space-time account. It turns out that the view of the world implied by such metaphysics is very different from the Newtonian or Leibnizian ones.
Besides the difference in metaphysics, the alternative to Newton’s second law given by equation (26) offers an advantage in resolving some philosophical issues regarding the foundations of Classical Mechanics and in particular the need to consider fictitious forces when applying Newton’s Second Law in non-inertial reference frames. In the case of observers at rest in accelerated reference frames in either rectilinear or uniform circular motion, the time rate of change of kinetic energy is zero and thus no additional fictitious power cause is needed to explain the state of motion. Again, this is only true if power is defined as the cause of motion. If force is defined as the cause of motion then in both noninertial reference frames mentioned fictitious causes must be considered. Specifically, in the case of rectilinear motion, observers at rest in an accelerated frame must assume inertial
fictitious forces acting and in the case of observers at rest in a uniformly rotating reference frame, centrifugal forces acting must be assumed.
The same conclusion holds in the case of fictitious Coriolis forces acting on freely moving particles in rotating reference frames. Since such fictitious forces are always orthogonal to the velocity of a particle in motion, for rotating observers it turns out that the time rate of change of kinetic energy of the particle is equal to zero, as obtained by equation (1). The same result is true for observers at rest since in that case the time rate of change of momentum of a freely moving particle is zero. Fictitious forces need to be considered regardless of whether force or power is defined as the cause of motion when a force analysis is carried out. However, when power is defined as the cause of motion, there are no philosophical issues arising from the need to consider fictitious causes of motion in non-inertial reference frames and this is the point just made. Thus, the transition from force to power as the cause of motion leads to a compatibility with the epistemological principle which states that every phenomenon is to receive the same interpretation from any given moving coordinate system. This epistemological principle also plays an important role in the axiomatic foundation of the theory of relativity [7].
4.3 Newton’s Third Law: a special case of a more general action-reaction law?
Newton’s Third Law may be deduced from the law of interaction of section 3 and in particular from equation (24) of corollary III. In the scholium following the Laws of Motion, Newton attempts to provide additional support for the Third Law through a host of observations related to various modes of mechanical interaction between bodies. From the closing comments in the scholium, some interesting conclusions can be drawn [6]:
. . .But to treat of mechanics is not my present business. I was aiming to show by those examples the greater extent and certainty of the third Law of Motion. For if we estimate the action of the agent from the product of its force and velocity and likewise the reaction of the impediment from the product of the velocities of its several parts, and the forces of resistance arising from friction, cohesion, weight, and acceleration of those parts, the action and reaction in the use of all sorts of machines will be found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impressed upon the resisting body, the ultimate action will be always contrary to the reaction. (Italics added)
It is clear that Newton was well aware of the product of velocity and force being a measure of action and of reaction, as defined in the law of interaction of section 3. Newton actually made use of the law of interaction in his scholium above to justify some particular situations where his Third
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Law of action-reaction does not apply directly. But why is it the case that Newton stated his Third Law in terms of forces and not in terms of the product of force and velocity he mentions in his scholium quoted above? Why does it appear that a more general law was used to justify some particular situations Newton’s Third Law does not directly apply to, but the latter was stated as a law of mechanics? The answer can be found in the attempt to model gravity in Newtonian mechanics as the effect of mutual attraction caused by central forces acting at a distance. The Third Law had to be stated in terms of the mutual action-reaction forces being equal in magnitude and opposite in direction to justify the particular form of Newton’s Law of Universal Gravitation. But again, the Third Law fails the requirement set forth by the rules of the experimental philosophy of Newton, for it being deduced from the phenomena; it is just another axiom that must be accepted without proof. Forces acting on different bodies, and especially celestial ones, cannot be experimentally determined to be equal. Only forces acting on the same body can be determined to be equal by experiment.
I have shown that even Newton himself made both indirect and direct use of the notion of power in an attempt to provide a general justification of his Third Law. Can we simply assume that Newton was unaware that there is a single principle that could serve as the basis of a system of laws of mechanics that are in a certain way more general than his laws? I suspect that he was aware of it. But the consequences from stating laws based on this principle of motion would be devastating on the metaphysics of force. If force were to be just an intellectual construction and not the cause of motion, then Newton’s whole system of the world was at stake. Motion then would have to be explained based on some other metaphysics, such as Cartesian occasionalism for example and the notion that all causes are due to God, or Spinoza’s doctrine that everything is a mode of God [8], or even Leibniz’s notion of a living force.
5 Power versus vis viva
Leibniz rejected the doctrine of Cartesian occasionalism and Newtonian substantivalism but his efforts to ground his relationism on the metaphysics of a living force were also met with difficulties. Leibniz realized that for motion to be real, it must be grounded on something that is not mere relation, something absolute and unobservable that serves as its cause [8]. Leibniz stated his laws of motion in his unpublished during his lifetime work Dynamica de Potentia et Legibus Naturae Corporeae in which he attempted to explain the world in terms of the conservation laws of vis viva and momentum of colliding bodies.
The laws of inertia and interaction of section 3 were derived from the axiom of motion of section 2. The latter is related to the living force, or vis viva, defined by Leibniz as being a real metaphysical property of a substance. Leibniz
measure of vis viva is the quantity mv2, in contrast to the Cartesian definition of the quantity of motion being equal to size multiplied by speed, and later redefined by Newton as being equal to the product of mass and velocity. In turn, the axiom of motion stated in section 2 is related to the time rate of change of vis viva, the quantity Leibniz argued is conserved and a real metaphysical property of a substance, in an effort to support his relational account of space-time.
Leibniz’s definition of vis viva as a real metaphysical property of a substance is fraught with difficulties. Roberts has argued that, in his later communications with Samuel Clarke, who was a defender of Newton’s substantivalism, Leibniz seems to commit to a richer space-time structure that can support absolute velocities [9]. Roberts’ work has cast light into a little known, or maybe misinterpreted, aspect of Leibniz’s metaphysics. Specifically, into Leibniz’s efforts to come up with laws of motion based on vis viva being a measure of force, while at the same time his relationism implies a space-time structure that is a well-founded phenomenon. This might be an indication of Leibniz’s later realization that relationism fails unless absolute velocities are supported by a richer space-time structure than what is commonly referred to as Leibnizian space-time. In section 6, I define an account of space-time that can support relationism and absolute velocities in an attempt to ground the physics of the axiom and laws of motion in the metaphysics of power.
Along these lines, in a similar way to the link between the Newtonian force and momentum, the former being the time rate of change of the latter, I argue that vis viva is actually a quantity of motion and power, its time rate of change, is the cause of motion. In this way the similarities between the laws of conservation of momentum and vis viva become evident, because they are both defined as quantities of motion. In essence, I argue, the time rate of change of vis viva is the real metaphysical cause of motion. Of course, such a switch in the definitions is not compatible with Leibniz’s metaphysics. This is because the time rate of change of the kinetic energy of a body moving with constant linear velocity, or even in uniform circular motion, is zero. A zero of something cannot assume the role of a real metaphysical property of a substance and the cause of motion in a Leibnizian world. Despite these metaphysical difficulties I will deal with in more detail in the next section, on the physics side it is clear that the laws of motion of section 3 were derived from a quantity that is proportional to the time derivative of vis viva. Thus, they have a direct link to Leibniz’s Laws of Motion [8]. Specifically, Leibniz’s laws of conservation of vis viva and momentum can be derived from the laws of inertia and interaction of section 3, respectively, but the details are left out.
6 The metaphysics of power
Before I discuss the metaphysics of power and specifically the notion that power is the cause of motion, I will briefly
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review the philosophical debate about the ontology of spacetime. I argue that the space-time debate and the debate about the cause of motion are closely related in the sense that an answer to the former provides an answer to the latter. Thus, I essentially argue that the space-time debate is not a mere philosophical one and its resolution will have a decisive impact on which laws of motion and gravitation are assigned the status of “laws of nature” as opposed to that of mere heuristics.
6.1 The space-time debate
The publication of Newton’s Principia in 1686 was the cause of the start of one of the most interesting debates in the history of the philosophy of science, dealing mainly with the ontology of space-time. Leibniz ignited the debate by arguing that Newton’s substantival space-time, the notion that space and time exist independently of material things and their spatiotemporal relations, was not a well-founded phenomenon. Leibniz confronted Newtonian substantivalists with his relationism, based on which space is defined as the set of (possible) relations among material things and the only well-defined quantities of motion are relative ones [10]. Newton just grounded his physics in the metaphysics of force and absolute space and time. For Newton, the only well-defined quantities of motion are the absolute ones, like absolute position, velocity and acceleration. Substantivalism and relationism then appear in modern literature as two completely different accounts of space-time.
The key issue regarding the space-time debate, which is still alive by the way, is whether it does really make sense to speak of either a substantival or a relational account of space-time. Since diametrically opposite views of this kind have only led to sharp conflict and irreconcilable differences, maybe it would make sense to investigate whether both a substantival and relational space-time is a possibility. This two-level approach seems not to have been considered seriously because it implies a superfluous world. However, both Newtonian substantivalism and Leibnizian relationism are fraught with difficulties. On one hand, the metaphysics of Newtonian force require the postulation of unobservables, like absolute space. On the other hand, in Leibniz’s relationism, for motion to be real, it must be grounded in something that is not mere relation, something absolute and unobservable that serves as its cause, what Leibniz called a vis viva [9]. The differences seem to reconcile when a two-level, or if I may call it a dual, space-time account is postulated and I will throw in here the term substantival relationism.
6.2 From cause-free motion to gravitation
The hypothesis about the duality of space-time just put forward is next examined in the context of gravitation and its observable effects, i. e. the motion of celestial bodies
and free-falling particles. This step is of great importance since any laws of motion must account for all observable phenomena including those that are attributed to gravitation. Newton accomplished the step of grounding the physics of the Laws of Motion to his metaphysics of substantival space and universal time, by assuming that the cause of gravitation was also some type of force. Next, in what was a remarkable achievement in the history of science, he derived the famous Law of Universal Gravitation (LUG). In a similar way, I argue that power is the cause of gravitation in order to maintain a compatibility with the axiom and laws of motion of sections 2 and 3, respectively. Thus, the time rate of change of a potential energy function Ep(r) is the cause of gravitation and equation (1), the axiom of motion, becomes
dEk = d p ∙ d r = − dEp .
(27)
dt dt dt
dt
The law of conservation of mechanical energy can be derived from equation (27) as follows:
dEk dt
= − dEp dt
⇔
d dt (Ek + Ep) = 0
⇔
(28)
⇔ Ek + Ep = const .
The Law of Universal Gravitation may be restated as follows:
Law of Universal Gravitation: All particles move in such a way as for the time rate of change of their kinetic energy to be equal the time rate of change of their potential energy.
In fact, I argue that Newton’s Law of Universal Gravitation is a statement about the form of the potential function Ep(r) in equation (27) and thus it can assume a variety of interpretations regarding mechanisms giving rise to it. If we postulate that energy transfer affects all particles in motion, in accordance with equation (27), this can support the hypothesis that gravitation is the result of energy transfer between all bodies in motion with some substance. Substantival space-time can serve the role of this substance and can facilitate the energy transfer to and from all bodies in motion and in such a way that all spatiotemporal quantities evolve according to certain rules giving rise to the wellknown potential function Ep(r) first discovered by Newton.
Since the above metaphysics are compatible with the concept of a mechanical universe, one could then postulate the existence of some type of mechanism that facilitates the transfer of energy between all bodies in motion and substantival space-time. This mechanism must be part of the substance level, whereas at the phenomenal level its effect is the observed motions. According to this dual scheme, at the phenomenal level the only well-founded quantities of motion are relative ones and space-time is relational, whereas, at the substance level, the only well-defined quantities of motion are the absolute ones and the space-time is substantival.
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6.3 A new foundation of mechanics
The hypothesis just made, attributing gravitation to energy transfer between all bodies in motion and substantival spacetime requires that at every instance something must accomplish this task and bring about the perceived effects. I will relate this to occasionalism in the following way: according to Nicolas Malebranche and other seventeenthcentury Cartesian occasionalists, what we actually call causes are really no more than occasions on which, in accordance with his own laws, God acts to bring about the effect [11]. If one were to replace the notion of God by the notion of a mechanism, then a modern (or mechanical) occasionalist could assert that what we actually call causes are no more than occasions on which a mechanism acts to bring about the effect. In this sense we immediately resolve two more issues: first, time emerges as an ordered progression of instances, or nows, on which the mechanism acts to bring about the effect. Then, the matter-in-motion picture [1] is better illuminated by asserting that all motion and interactions of material bodies are facilitated by a mechanism that operates based on its own rules rather than taking place due to forces or based on rules inherent in the bodies themselves.
The concept of time as a collection of nows is in fact similar to that found in Barbour [12]. The main difference with the view I express here is that time emerges due to the actions of a mechanism hidden in substantival space-time in an orderly fashion and has a direction, i. e. there is an arrow of time. More importantly, the universal clock of Newton is now part of the mechanism that resides in substantival spacetime but at the phenomenal level time and motion cannot be separated because there is no motion without time and no time without motion, i. e. time and motion are inextricably related.
What I argue essentially is that gravitation has an external cause to the phenomenal level and space-time is a substance of some kind that facilitates the energy transfer required for the manifestation of gravitational effects. These ideas may not be completely new. What is new here is the derivation of a system of laws of motion based on the notion of power. Power allows grounding the physics that all phenomena are caused by energy transfer, including those attributed to gravitation, to the metaphysics of substantival space-time being a giant mechanism and a substance. Since the times of the Greeks, Anaximander of Miletus (c. 650 BCE) expressed the view that
The apeiron, from which the elements are formed, is something that is different (from the elements).
Then, Newton argued that all motion must be referenced to an absolute, unobservable space. Even in general relativity space-time retains its substantival account and it exists independently of the events occurring in it [10]. Baker has argued that the space-time of general relativity must be a substance and attempts to support this claim of his based
on the observed expansion of the Universe [13]. Baker’s argument about the requirement of a carrier of gravitational energy from its source to a detector, if it is to be compelling, must apply to all forms of energy transfer traditionally assumed to take place in vacuum. But such generalization can be further coupled with the hypothesis that some causes are external to the world of observable phenomena. In Wiithrich there are references made to the hypothesis that gravity forces have an external cause in an attempt to explain the failure in quantizing the field equations of general relativity [14]. Thus, arguments have already been made in favor of the hypothesis that space-time is some kind of a substance and that any causal connections attributed to gravitation are apparent. Usually, arguments leading to such provocative hypotheses are treated at the level of epistemological skepticism but as McCabe argues the hypothesis, for instance, that our universe is part of a computer simulation implementation generates empirical predictions and it is therefore a falsifiable hypothesis [15]. One question that arises from this discussion is the following: does the existence of external causes imply that our world is some type of virtual reality? My own answer to this important question is both yes and no. Yes, because according to the hypothesis there are external causes to the world of perceived phenomena and thus part of another world. No, because a cause being external and unobservable does not preclude it being part of an all-encompassing entity, which we can call Universe. Therefore, the answer to the question seems to depend on how one defines Universe. But the presence of external causes to the world of observable phenomena must not be rejected a priori on the basis that it leads to the provocative virtual reality hypothesis and experimental physics must pursue seriously its falsification or corroboration. Although such task is highly challenging, the state-of-the-art in precision instrumentation has reached levels that allow the initiation of a program of this nature.
7 Summary
The axiom and laws of motion presented in sections 2 and 3, respectively, are:
Axiom of Motion: The time rate of change of the kinetic energy of a particle is the scalar product of its velocity and time rate of change of its momentum.
Law of Inertia: If the time rate of change of the kinetic energy of a particle is zero, the particle will continue in its state of motion.
Law of Interaction: To every action there is an equal and opposite reaction; that is, in an isolated system of two particles acting upon each other, the mutual time rate of change of kinetic energies are equal in magnitude and opposite in sign.
A restatement of the Law of Universal Gravitation was presented in section 6 as follows:
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Law of Universal Gravitation: All particles move in such a way as for the time rate of change of their kinetic energy to be equal to the time rate of change of their potential energy.
In section 4, I argued that the above laws of motion are, in a certain sense, more general than Newton’s, and that this claim is even supported by Newton’s own writings, especially in the case of the Third Law. Furthermore, in section 5, I discussed the relation of the axiom and laws of motion to Leibniz’s laws of the conservation of vis viva and momentum. I argued that kinetic energy can be defined as a quantity of motion and its time derivative as the cause of motion, in a similar way to the Newtonian force being the time derivative of momentum and a postulated cause of motion.
In section 6, I discussed how the axiom and laws of motion of sections 2 and 3, combined further with a modified version of Cartesian occasionalism and a dual space-time account form an alternative foundation of classical mechanics in the context of a mechanical Universe. Specifically, I proposed a substantival-relational account of space-time and a mechanism residing in the substance level whose actions coordinate all motion and interactions. I argued that the proposed foundation supports the hypothesis about gravitation being the effect of energy transfer between all bodies in motion and substantival space-time and I stated a version of the Law of Universal Gravitation which is compatible with the hypothesis that power is the cause of motion. These metaphysics also provide solutions to some foundational problems of Classical Mechanics, such as the matter-inmotion picture and the emergence and direction of time. Finally, I briefly referred to the ramifications on the nature of our physical reality when the cause of gravitation is considered part of an unobservable substance. I argued that the soundness of the virtual reality or computer simulation hypothesis depends on how Universe is defined. The fact that such hypothesis about the nature of our reality is provocative should not be an excuse for rejecting a priori external causes of motion and gravitation. Theoretical physicists ought to seriously investigate new models incorporating such assumptions about the nature of our physical reality and experimental physicists should pursue their falsification.
5. Meirovitch S. L. Methods of analytical dynamics. McGrawHill, New York, 1970.
6. Newton I. Mathematical principles of natural philosophy. Trans. by Andrew Motte and rev. by Florian Cajori, R. M. Hutchins, ed. in Great Books of the Western World: 34. Newton Huygens. Encyclopaedia Britannica, Chicago, 1952, 1–372.
7. Reichenbach H. The philosophy of space and time. Dover, New York, 1958.
8. Garber S. D. Leibniz: physics and philosophy. N. Jolley, ed. in The Cambridge Companion to Leibniz, Cambridge University Press, Cambridge, 1995, 270–352.
9. Roberts J. T. Leibniz on force and absolute motion. J. Phil. Sci., 2003, v. 70, 553–571.
10. Sklar L. Space, time and space-time. University of California Press, Berkeley, 1997.
11. Brown S. The seventeenth-century intellectual background. N. Jolley, ed. in The Cambridge Companion to Leibniz, Cambridge University Press, Cambridge, 1995, 43–66.
12. Barbour J. The end of time. Oxford University Press, New York, 1999.
13. Baker D. Spacetime substantivalism and Einstein’s cosmological constant. Proc. Phil. Sci. Assoc., 19th Biennial Meeting, PSA2004, 2004.
14. Wiithrich, C. To quantize or not to quantize: fact and folklore in quantum gravity. Proc. Phil. of Sci. Assoc., 19th Biennial Meeting, PSA2004 2004.
15. McCabe G. Universe creation on a computer, 2004. e-print, Pittsburgh Archives/00001891.
References
1. Butterfield J. Between laws and models: some philosophical morals of Lagrangian mechanics, 2004. e-print, Pittsburgh Archives/00001937.
2. Henri P. Science and hypothesis. Dover, New York, 1952. 3. Greenwood D. T. Principles of dynamics. Prentice-Hall, New
Jersey, 1965. 4. O’Neill B. Elementary differential geometry. Academic Press,
San Diego, 1997.
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Hydrodynamic Covariant Symplectic Structure from Bilinear Hamiltonian Functions
Salvatore Capozziello∗, Salvatore De Martino†, Stephan Tzenov†
∗Dipartimento di Scienze Fisiche and INFN sez. di Napoli, Universita´ “Federico II” di Napoli, Complesso Universitario di Monte S. Angelo Edificio N, via Cinthia 45, I-80126 Napoli, Italy
E-mail: capozziello@na.infn.it †Dipartimento di Fisica “E.R. Caianiello” and INFN sez. di Napoli, Universita´ di Salerno,
Via S. Allende, I-84081 Baronissi (SA), Italy E-mail: demartino@sa.infn.it, tzenov@sa.infn.it
Starting from generic bilinear Hamiltonians, constructed by covariant vector, bivector or tensor fields, it is possible to derive a general symplectic structure which leads to holonomic and anholonomic formulations of Hamilton equations of motion directly related to a hydrodynamic picture. This feature is gauge free and it seems a deep link common to all interactions, electromagnetism and gravity included. This scheme could lead toward a full canonical quantization.
1 Introduction
It is well known that a self-consistent quantum field theory of space-time (quantum gravity) has not been achieved, up to now, using standard quantization approaches. Specifically, the request of general coordinate invariance (one of the main features of General Relativity) gives rise to unescapable troubles in understanding the dynamics of gravitational field. In fact, for a physical (non-gravitational) field, one has to assign initially the field amplitudes and their first time derivatives, in order to determine the time development of such a field considered as a dynamical entity. In General Relativity, these quantities are not useful for dynamical determination since the metric field gαβ can evolve at any time simply by a general coordinate transformation. No change of physical observables is the consequence of such an operation since it is nothing else but a relabelling under which the theory is invariant. This apparent “shortcoming” (from the quantum field theory point of view) means that it is necessary a separation of metric degrees of freedom into a part related to the true dynamical information and a part related only to the coordinate system. From this viewpoint, General Relativity is similar to classical Electromagnetism: the coordinate invariance plays a role analogous to the electromagnetic gauge invariance and in both cases (Lorentz and gauge invariance) introduces redundant variables in order to insure the maintenance of transformation properties. However, difficulties come out as soon as one try to disentangle dynamical from gauge variables. This operation is extremely clear in Electromagnetism while it is not in General Relativity due to its intrinsic non-linearity. A determination of independent dynamical modes of gravitational field can be achieved when the theory is cast into a canonical form involving the minimal
number of degrees of freedom which specify the state of the system. The canonical formalism is essential in quantization program since it leads directly to Poisson bracket relations among conjugate variables. In order to realize it in any fundamental theory, one needs first order field equations in time derivatives (Hamilton-like equations) and a (3 + 1)form of dynamics where time has been unambiguously singled out. In General Relativity, the program has been pursued using the first order Palatini approach [6], where metric gαβ is taken into account independently of affinity connections Γγαβ (this fact gives rise to first order field equations) and the so called ADM formalism [7] where (3 + 1)-dimensional notation has led to the definition of gravitational Hamiltonian and time as a conjugate pair of variables. However, the genuine fundament of General Relativity, the covariance of all coordinates without the distinction among space and time, is impaired and, despite of innumerable efforts, the full quantization of gravity has not been achieved up to now. The main problems are related to the lack of a well-definite Hilbert space and a quantum concept of measure for gαβ. An extreme consequence of this lack of full quantization for gravity could be related to the dynamical variables: very likely, the true variables could not be directly related to metric but to something else as, for example, the connection Γγαβ. Despite of this lack, a covariant symplectic structure can be identified also in the framework of General Relativity and then also this theory could be equipped with the same features of other fundamental theories. This statement does not still mean that the identification of a symplectic structure immediately leads to a full quantization but it could be a useful hint toward it.
The aim of this paper is to show that a prominent role in the identification of a covariant symplectic structure is
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played by bilinear Hamiltonians which have to be conserved. In fact, taking into account generic Hamiltonian invariants, constructed by covariant vectors, bivectors or tensors, it is possible to show that a symplectic structure can be achieved in any case. By specifying the nature of such vector fields (or, in general, tensor invariants), it gives rise to intrinsically symplectic structure which is always related to Hamilton-like equations (and a Hamilton-Jacobi-like approach is always found). This works for curvature invariants, Maxwell theory and so on. In any case, the only basic assumption is that conservation laws (in Hamiltonian sense) have to be identified in the framework of the theory.
The layout of the paper is the following. In Sec.II, we give the generalities on the symplectic structure and the canonical description of mechanics. Sec.III is devoted to the discussion of symplectic structures which are also generally covariant. We show that a covariant analogue of Hamilton equations can be derived from covariant vector (or tensor) fields in holonomic and anholonomic coordinates. In Sec. IV, the covariant symplectic structure is casted into the hydrodynamic picture leading to the recovery of the covariant Hamilton equations. Sec.V is devoted to applications, discussion and conclusions.
If {ei} is a vector basis in E2n, the antiscalar product is completely singled out by the matrix elements
wij = [ei, ej ] ,
(8)
where w is an antisymmetric matrix with determinant different from zero. Every antiscalar product between two vectors can be expressed as
[x, y] = wij xiyj ,
(9)
where xi and yj are the vector components in the given basis.
The form of the matrix w and the relation (9) become considerably simpler if a canonical basis is taken into account for w. Since w is an antisymmetric non-degenerate tensor, it is always possible to represent it through the matrix
J=
0I −I 0
,
(10)
where I is a (n × n) unit matrix. Every basis where w can be represented through the form (10) is a symplectic basis. In other words, the symplectic bases are the canonical bases for any antisymmetric non-degenerate tensor w and can be characterized by the following conditions:
2 Generalities on the Symplectic Structure and the Canonical description
In order to build every fundamental theory of physics, it is worth selecting the symplectic structure of the manifold on which such a theory is formulated. This goal is achieved if suitable symplectic conjugate variables and even-dimensional vector spaces are chosen. Furthermore, we need an antisymmetric, covariant tensor which is non-degenerate.
We are dealing with a symplectic structure if the couple
{E2n, w} ,
(1)
[ei, ej ] = 0 , [en+i, en+j ] = 0 , [ei, en+j ] = δij , (11)
which have to be verified for every pair of values i and j ranging from 1 to n.
Finally, the expression of the antiscalar product between two vectors, in a symplectic basis, is
n
[x, y] =
xn+iyi − xiyn+i ,
(12)
i=1
and a symplectic transformation in E2n leaves invariant the antiscalar product
is defined, where E2n is a vector space and the tensor w on E2n associates scalar functions to pairs of vectors, that is
[x, y] = w(x, y),
(2)
which is the antiscalar product. Such an operation satisfies the following properties
S[x, y] = [S(x), S(y)] = [x, y].
(13)
It is easy to see that standard Quantum Mechanics satisfies such properties and so it is endowed with a symplectic structure.
On the other hand a standard canonical description can be sketched as follows. For example, the relativistic Lagrangian of a charged particle interacting with a vector field A(q; s) is
[x, y] = −[y, x]
∀x, y ∈ E2n
(3)
[x, y + z] = [x, y] + [xz] ∀x, y, z ∈ E2n, (4)
mu2
L(q, u; s) =
− eu ∙ A(q; s),
(14)
2
where the scalar product is defined as
a[x, y] = [ax, y]
∀a ∈ R, x, y ∈ E2n (5)
z ∙ w = zμwμ = ημν zμwν ,
(15)
[x, y] = 0
∀y ∈ E2n ⇒ x = 0
(6)
and the signature of the Minkowski spacetime is the usual
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 .
(7) one with
The last one is the Jacobi cyclic identity.
zμ = ημν zν ,
η = diag(1, −1, −1, −1). (16)
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Furthermore, the contravariant vector uμ with components u = u0, u1, u2, u3 is the four-velocity
uμ = dqμ .
(17)
ds
The canonical conjugate momentum πμ is defined as
πμ
=
ημν
∂L ∂uν
= muμ − eAμ ,
(18)
so that the relativistic Hamiltonian can be written in the form
H(q, π; s) = π ∙ u − L(q, u; s).
(19)
Suppose now that we wish to use any other coordinate system xα as Cartesian, curvilinear, accelerated or rotating one. Then the coordinates qμ are functions of the xα, which
can be written explicitly as
and using the well-known identity for connections Γαμν
∂gμν ∂xλ
= Γαλμgαν + Γαλν gαμ ,
(29)
we obtain
Dvλ Ds
=
dvλ ds
− Γμλν vν vμ
= 0.
(30)
Here Dvλ/Ds denotes the covariant derivative of the covariant velocity vλ along the curve xν(s). Using Eqs. (28) and (29) and the fact that the affine connection Γλμν
is symmetric in the indices μ and ν, we obtain the equation of motion for the contravariant vector vλ
Dvλ Ds
=
dvλ ds
+ Γλμν vμvν
=
0.
(31)
Before we pass over to the Hamiltonian description, let us note that the generalized momentum pμ is defined as
qμ = qμ(xα).
(20)
The four-vector of particle velocity uμ is transformed according to the expression
uμ
=
∂qμ ∂xα
dxα ds
=
∂qμ ∂xα
vα,
(21)
where
vμ = dxμ .
(22)
ds
is the transformed four-velocity expressed in terms of the
new coordinates. The vector field Aμ is also transformed as
a vector
Aμ
=
∂xμ ∂qα
Aα.
(23)
In the new coordinate system xα the Lagrangian (14) becomes
L(x, v; s) = gμν
m vμvν − evμAν (x; s) , 2
(24)
where
∂qμ ∂qν
gαβ = ημν ∂xα ∂xβ .
(25)
The Lagrange equations can be written in the usual form
pμ
=
∂L ∂vμ
=
mgμν vν ,
(32)
while, from Lagrange equations of motion, we obtain
dpμ ds
=
∂L ∂xμ .
(33)
The transformation from (xμ, vμ; s) to (xμ, pμ; s) can be
accomplished by means of a Legendre transformation, and
instead of the Lagrangian (24), we consider the Hamilton
function
H(x, p; s) = pμvμ − L(x, v; s).
(34)
The differential of the Hamiltonian in terms of x, p and s is given by
dH
=
∂H ∂xμ
dxμ
+
∂H ∂pμ dpμ
+
∂H ds.
∂s
(35)
On the other hand, from Eq.(34), we have
dH
=
vμdpμ
+pμdvμ
−
∂L ∂vμ
dvμ
−
∂L ∂xμ
dxμ
−
∂L ∂s
ds.
(36)
Taking into account the defining Eq.(32), the second and the third term on the right-hand-side of Eq.(36) cancel out. Eq.(33) can be further used to cast Eq.(36) into the form
d ∂L ∂L
ds ∂vλ − ∂xλ = 0 .
(26)
In the case of a free particle (no interaction with an external vector field), we have
d ds
(gλμ
vμ)
−
1 2
∂gμν ∂xλ
vμvν
=
0.
(27)
Specifying the covariant velocity vλ as
vλ = gλμvμ,
(28)
dH
=
vμdpμ
−
dpμ ds
dxμ
−
∂L ds ,
∂s
(37)
Comparison between Eqs.(35) and (37) yields the Hamilton equations of motion
dxμ ∂H =,
ds ∂pμ
dpμ ds
=
∂H − ∂xμ
,
(38)
where the Hamiltonian is given by
H(x, p; s)
=
gμν 2m pμpν
+
e m
pμAμ.
(39)
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In the case of a free particle, the Hamilton equations can be written explicitly as
dxμ gμν ds = m pν ,
dpλ ds
=
1 ∂gμν
− 2m
∂xλ
pμpν
.
(40)
To obtain the equations of motion we need the expression
∂gμν ∂xλ
= −Γμλαgαν − Γνλαgαμ,
(41)
which can be derived from the obvious identity
∂ ∂xλ
(gμαgαν
)
=
0
,
(42)
and Eq.(29). From the second of Eqs. (40), we obtain
Dpλ Ds
=
dpλ ds
−
Γμλν vν pμ
=
0,
(43)
similar to equation (30). Differentiating the first of the Hamilton equations (40) with respect to s and taking into account equations (41) and (43), we again arrive to the equation for the geodesics (31).
Let us now show that on a generic curved (torsion-free) manifolds the Poisson brackets are conserved. To achieve this result, we need the following identities
gμν
=
gνμ
=
ηαβ
∂xμ ∂qα
∂xν ∂qβ
,
(44)
∂2xλ ∂qα∂qβ
=
−Γλμν
∂xμ ∂qα
∂xν ∂qβ
,
(45)
To prove (45), we differentiate the obvious identity
∂xλ ∂qρ ∂qρ ∂xν
= δνλ .
(46)
As a result, we find
Γλμν
=
∂xλ ∂qρ
∂2qρ ∂xμ∂xν
∂qρ = − ∂xν
∂qσ ∂2xλ ∂xμ ∂qρ∂qσ .
(47)
The next step is to calculate the fundamental Poisson
brackets in terms of the variables (xμ, pν), initially defined using the canonical variables (qμ, πν) according to the rela-
tion
∂U ∂V ∂V ∂U
[U, V ] = ∂qμ ∂πμ − ∂qμ ∂πμ ,
(48)
where U (qμ, πν ) and V (qμ, πν ) are arbitrary functions. Making use of Eqs.(18) and (21), we know that the variables
qμ
⇔
πμ
=
muμ
=
mημν uν
=
mημν
∂qν ∂xα
vα,
(49)
form a canonical conjugate pair. Using Eq.(32), we would like to check whether the variables
xμ
⇔
pμ
=
mgμν vν
=
gμν
ηαλπλ
∂ ∂
xν qα
,
(50)
form a canonical conjugate pair. We have
[U, V ] =
∂U ∂xα
∂xα ∂qμ
+
∂U ∂pσ
ηβλ
πλ
∂ ∂qμ
∂xν gσν ∂qβ
×
×
∂V ∂pα
gαχ
ηρμ
∂xχ ∂qρ
−
−
∂V ∂xα
∂xα ∂qμ
+
∂V ∂pσ
ηβλ
πλ
∂ ∂qμ
∂xν gσν ∂qβ
×
×
∂U ∂pα
gαχ
ηρμ
∂xχ ∂qρ
.
(51)
The first and the third term on the right-hand-side of Eq.(51) can be similarly manipulated as follows
I-st
term
=
∂U ∂xα
∂V ∂pβ
gβχ
ηρμ
∂xχ ∂qρ
∂xα ∂qμ
=
=
gβχgχα
∂U ∂xα
∂V ∂pβ
=
∂U ∂xα
∂V ∂pα
,
(52)
∂V ∂U
III-rd term = − ∂xα ∂pα .
(53)
Next, we manipulate the second term on the right-hand-
side of Eq.(51). We obtain
II-nd
term
=
∂U ∂pσ
∂V ∂pα
gαχ
η
ρμ
∂xχ ∂qρ
ηβλπλ
×
×
∂2xν gσν ∂qμ∂qβ
+
∂xν ∂qβ
∂gσν ∂xγ
∂xγ ∂qμ
=
=
∂U ∂pσ
∂V ∂pα
gαχ
η
ρμ
∂xχ ∂qρ
ηβλπλ
×
×
−gσν Γνγδ
∂xγ ∂qμ
∂xδ ∂qβ
∂xδ + ∂qβ
∂xγ ∂qμ
Γνγσgνδ +Γνγδ gνσ
=
=
∂U ∂pσ
∂V ∂pα
gαχ
g
χγ
η
βλπλ
∂ ∂
xδ qβ
gνδ
Γνγσ
=
=
∂U ∂pσ
∂V ∂pβ
gμν
ηαλπλ
∂ ∂
xν qα
Γμβσ
=
Γλμν
pλ
∂U ∂pν
∂V .
∂pμ
(54)
The fourth term is similar to the second one but with U and V interchanged
IV-th
term
=
−Γλμν
pλ
∂U ∂pμ
∂V ∂pν
.
(55)
In the absence of torsion, the affine connection Γλμν is symmetric with respect to the lower indices, so that the second and the fourth term on the right-hand-side of Eq.(51) cancel each other. Therefore,
∂U ∂V ∂V ∂U
[U, V ] = ∂xμ ∂pμ − ∂xμ ∂pμ ,
(56)
which means that the fundamental Poisson brackets are con-
served. On the other hand, this implies that the variables {xμ, pν} are a canonical conjugate pair.
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As a final remark, we have to say that considering a generic metric gαβ and a connection Γαμν is related to the fact that we are passing from a Minkowski-flat spacetime (local inertial reference frame) to an accelerated reference frame (curved spacetime). In what follows, we want to show that a generic bilinear Hamiltonian invariant, which is conformally conserved, gives always rise to a canonical symplectic structure. The specific theory is assigned by the vector (or tensor) fields which define the Hamiltonian invariant.
3 A symplectic structure compatible with general covariance
The above considerations can be linked together leading to a more general scheme where a covariant symplectic structure is achieved. Summarizing, the main points which we need are: (i) an even-dimensional vector space E2n equipped with an antiscalar product satisfying the algebra (3)-(7); (ii) generic vector fields defined on such a space which have to satisfy the Poisson brackets; (iii) first-order equations of motion which can be read as Hamilton-like equations; (iv) general covariance which has to be preserved.
Such a program can be pursued by taking into account covariant and contravariant vector fields. In fact, it is possible to construct the Hamiltonian invariant
H = V αVα ,
(57)
which is a scalar quantity satisfying the relation
δH = δ(V αVα) = 0 ,
(58)
and then
δV α = −ΓασβV σdxβ.
(62)
Analogously, for the covariant derivative applied to the covariant vector,
DVα = dVα − δVα = ∂βVαdxβ − δVα ,
(63)
and then
DVα = ∂βVαdxβ − ΓσαβVσdxβ,
(64)
and
∇β Vα = ∂β Vα − Γσαβ Vσ .
(65)
The spurious variation is now
δVα = Γσαβ Vσdxβ.
(66)
Developing the variation (58), we have
δH = VαδV α + V αδVα ,
(67)
and
δH dxβ
δV α = Vα dxβ
+
V
α
δVα dxβ
,
(68)
which becomes
δH dxβ
=
δV α dxβ
∂H ∂V α
+
δVα dxβ
∂H ,
∂Vα
(69)
being
∂H ∂V α = Vα,
∂H = V α. ∂Vα
(70)
From Eqs.(62) and (66), it is
being δ a spurious variation due to the transport. It is worth stressing that the vectors V α and Vα are not specified and the following considerations are completely general. Eq.(57) is a so called “already parameterized” invariant which can constitute the “density” of a parameterized action principle where the time coordinate is not distinguished a priori from the other coordinates [8, 9].
Let us now take into account the intrinsic variation of V α. On a generic curved manifold, we have
DV α = dV α − δV α = ∂βV αdxβ − δV α, (59)
δV α dxβ
= −Γασβ V σ
= −Γασβ
∂H ∂Vσ
,
(71)
δVα dxβ
= Γσαβ Vσ
= Γσαβ
∂H ∂V σ
,
(72)
and substituting into Eq.(69), we have
δH dxβ
= −Γασβ
∂H ∂Vσ
∂H ∂V α
+ Γσαβ
∂H ∂Vα
∂H ∂V σ , (73)
and then, since α and σ are mute indexes, the expression
where D is the intrinsic variation, d the total variation and δ the spurious variation due to the transport on the curved manifold. The spurious variation has a very important meaning since, in General Relativity, if such a variation for a given quantity is equal to zero, this means that the quantity is conserved. From the definition of covariant derivative, applied to the contravariant vector, we have
δH dxβ =
Γασβ − Γασβ
∂H ∂Vσ
∂H ∂V α ≡ 0 ,
(74)
is identically equal to zero. In other words, H is absolutely conserved, and this is very important since the analogy with a canonical Hamiltonian structure is straightforward. In fact, if, as above,
DV α = ∂βV αdxβ + ΓασβV σdxβ,
(60)
H = H(p, q)
(75)
and
is a classical generic Hamiltonian function, expressed in the
∇β V α = ∂β V α + Γασβ V σ,
(61) canonical phase-space variables {p, q}, the total variation (in
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a vector space E2n whose dimensions are generically given by pi and qj with i, j = 1, ..., n) is
∂H ∂H
dH = dq + dp ,
(76)
∂q
∂p
and
dH ∂H ∂H = q˙ + p˙ =
dt ∂q ∂p
∂H ∂H ∂H ∂H
=
−
≡ 0,
(77)
∂q ∂p ∂p ∂q
thanks to the Hamilton canonical equations
∂H
∂H
q˙ = , p˙ = − .
(78)
∂p
∂q
Such a canonical approach holds also in our covariant case if we operate the substitutions
V α ←→ p
Vα ←→ q
(79)
and the canonical equations are
δV α dxβ
= −Γασβ
∂H ∂Vσ
dp ∂H
←→ = − , (80)
dt
∂q
δVα dxβ
= Γσαβ
∂H ∂V σ
dq ∂H ←→ = .
dt ∂p
(81)
Finally, conservation laws are given by
δH dxβ =
Γασβ − Γασβ
∂H ∂Vσ
∂H ∂Wα ≡ 0 .
(87)
In our picture, this means that the canonical symplectic structure is assigned in the way in which covariant and contravariant vector fields are related. However, if the Hamiltonian invariant is constructed by bivectors and tensors, equations (86) and (87) have to be generalized but the structure is the same. It is worth noticing that we never used the metric field but only connections in our derivations.
These considerations can be made independent of the reference frame if we define a suitable system of unitary vectors by which we can pass from holonomic to anholonomic description and viceversa. We can define the reference frame on the event manifold M as vector fields e(k) in event space and dual forms e(k) such that vector fields e(k) define an orthogonal frame at each point and
e(k) e(l) = δ((lk)) .
(88)
If these vectors are unitary, in a Riemannian 4-spacetime are the standard vierbiens [5].
If we do not limit this definition of reference frame by orthogonality, we can introduce a coordinate reference frame (∂α, dsα) based on vector fields tangent to line xα = const. Both reference frames are linked by the relations
In other words, starting from the (Hamiltonian) invariant (57), we have recovered a covariant canonical symplectic structure. The variation (67) may be seen as the generating function G of canonical transformations where the generators of q−, p− and t−changes are dealt under the same standard.
At this point, some important remarks have to be done. The covariant and contravariant vector fields can be also of different nature so that the above fundamental Hamiltonian invariant can be generalized as
H = W αVα ,
(82)
or, considering scalar smooth and regular functions, as
H = f (W αVα),
(83)
or, in general
e(k) = eα(k)∂α;
e(k) = e(αk)dxα.
(89)
From now on, Greek indices will indicate holonomic coordinates while Latin indices between brackets, the anholonomic coordinates (vierbien indices in 4-spacetimes). We can prove the existence of a reference frame using the orthogonalization procedure at every point of spacetime. From the same procedure, we get that coordinates of frame smoothly depend on the point. The statement about the existence of a global reference frame follows from this. A smooth field on time-like vectors of each frame defines congruence of lines that are tangent to this field. We say that each line is a world line of an observer or a local reference frame. Therefore a reference frame is a set of local reference frames. The Lorentz transformation can be defined as a transformation of a reference frame
H = f W αVα, BαβCαβ, BαβVαVβ, . . . , (84)
x α = f x0, x1, x2, x3, . . . , xn ,
(90)
where the invariant can be constructed by covariant vectors, bivectors and tensors. Clearly, as above, the identifications
W α ←→ p
Vα ←→ q
(85)
hold and the canonical equations are
δW α dxβ
= −Γασβ
∂H ∂Vσ
,
δVα dxβ
= Γσαβ
∂H ∂W σ
.
(86)
e
α (k)
=
Aαβ B((kl))eβ(l)
,
(91)
where
Aαβ
=
∂x ∂x
α
β,
δ(i)(l)B((ji))B((kl)) = δ(j)(k) .
(92)
We call the transformation Aαβ the holonomic part and transformation B((kl)) the anholonomic part.
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A vector field V has two types of coordinates: holonomic coordinates V α relative to a coordinate reference frame and anholonomic coordinates V (k) relative to a reference frame.
For these two kinds of coordinates, the relation
V (k) = e(αk)V α ,
(93)
holds. We can study parallel transport of vector fields using any form of coordinates. Because equations (90) and (91) are linear transformations, we expect that parallel transport in anholonomic coordinates has the same form as in holonomic coordinates. Hence we write
DV α = dV α + Γαβγ V βdxγ ,
(94)
DV (k) = dV (k) + Γ((kl))(p)V (l)dx(p).
(95)
Because DV α is also a tensor, we get
Γ((kl))(p)
=
eα(l)eβ(p)e(γk)Γγαβ
+
eα(l)eβ(p)
∂e(αk) ∂xβ
.
(96)
where we introduced the anholonomic object ω((ki))(l). Therefore each reference frame has n vector fields
∂(k)
=
∂ ∂x(k)
= eα(k)∂α ,
(103)
which have the commutators
∂(i), ∂(j) = eα(i)∂αeβ(j) − eα(j)∂αeβ(i) e(βm)∂(m) = = eα(i)eβ(j) −∂αe(βm) + ∂β e(βm) ∂(m) = ω((im)()j)∂(m). (104)
For the same reason, we introduce the forms
dx(k) = e(k) = e(βk)dxβ ,
(105)
and a differential of this form is
d2x(k) =d e(αk)dxα = ∂β e(αk)−∂αe(βk) dxα∧dxβ = = −ω((km)()l)dx(k) ∧ dx(l). (106)
Eq.(96) shows the similarity between holonomic and anholonomic coordinates. Let us introduce the symbol ∂(k) for the derivative along the vector field e(k)
∂(k) = eα(k)∂α .
(97)
Then Eq.(96) takes the form
Γ((kl))(p) = eα(l)eβ(p)e(γk)Γγαβ + eα(l)∂(p)e(αk) .
(98)
Therefore, when we move from holonomic coordinates
to anholonomic ones, the connection also transforms the way
similarly to when we move from one coordinate system to an-
other. This leads us to the model of anholonomic coordinates.
The vector field e(k) generates lines defined by the differ-
ential equations
eα(l)
∂τ ∂xα
= δ((lk)) ,
(99)
or the symbolic system
Therefore when ω((ki))(l) = 0, the differential dx(k) is not an exact differential and the system (101), in general, cannot be integrated. However, we can consider meaningful objects which model the solution. We can study how the functions x(i) changes along different lines. The functions x(i) is a natural parameter along a flow line of vector fields e(i). It is defined along any line.
All the above results can be immediately achieved in holonomic and anholonomic formalism considering the equation
H = W αVα = W (k)V(k) ,
(107)
and the analogous ones. This means that the results are independent of the reference frame and the symplectic covariant structure always holds.
4 The hydrodynamic picture
∂τ ∂x(l)
= δ((lk)) .
(100)
Keeping in mind the symbolic system (100), we denote the functional τ as x(k) and call it the anholonomic coordinate. We call the regular coordinate holonomic. Then we can find derivatives and get
∂x(k) ∂xα
= δα(k) .
(101)
The necessary and sufficient conditions to complete the integrability of system (101) are
ω((ki))(l) = eα(k)eβ(l)
∂e(αi) ∂xβ
−
∂e(βi) ∂xα
= 0,
(102)
In order to further check the validity of the above approach, we can prove that it is always consistent with the hydrodynamic picture (see also [10] for details on hydrodynamic covariant formalism).
Let us define a phase space density f (x, p; s) which evolves according to the Liouville equation
∂f ∂s
+
1 m
∂ ∂xμ
(g
μνpν
f
)−
1 2m
∂ ∂pλ
∂gμν ∂xλ pμpν f
=0 .
(108)
Next we define the density (x; s), the covariant current velocity vμ(x; s) and the covariant stress tensor Pμν (x; s) according to the relations
(x; s) = mn d4pf (x, p; s) ,
(109)
98
S. Capozziello, S. De Martino, S. Tzenov. Hydrodynamic Covariant Symplectic Structure from Bilinear Hamiltonian Functions
July, 2005
PROGRESS IN PHYSICS
Volume 2
(x; s)vμ(x; s) = n d4ppμf (x, p; s) ,
(110)
n Pμν (x; s) = m
d4ppμpν f (x, p; s) .
(111)
It can be verified, by direct substitution, that a solution to the Liouville Eq.(108) of the form
1 f (x, p; s) =
mn
(x; s)δ4[pμ − mvμ(x; s)] ,
leads to the equation of continuity
(112)
∂ ∂s
+
∂ ∂xμ
(gμν
vν
) = 0,
(113)
and to the equation for balance of momentum
∂
∂
( ∂s
vμ) +
∂xλ
gλαPαμ
1 ∂gαβ + 2 ∂xμ Pαβ = 0 .
(114)
Taking into account the fact that for the particular solution
(112), the stress tensor, as defined by Eq.(111), is given by
the expression
Pμν (x; s) = vμvν ,
(115)
we obtain the final form of the hydrodynamic equations
∂ ∂s
+
∂ ∂xμ (
vμ) = 0 ,
(116)
∂vμ + vλ ∂s
∂vμ ∂xλ
−
Γνμλvν
=
∂vμ ∂s
+
vλ∇λvμ
=
0
.
(117)
Clearly, the Riemann tensor results from the commutation of covariant derivatives and it can be expressed as the sum of two commutators
Rσρμν = ∂[μ, Γρν]σ + Γρλ[μ, Γλν]σ .
(119)
Furthermore, (anti) commutation relations and cyclic identities (in particular Bianchi’s identities) hold for the Riemann tensor [5].
All these straightforward considerations suggest the presence of a symplectic structure whose elements are covariant and contravariant vector fields, V α and Vα, satisfying the properties (3)-(7). In this case, the dimensions of vector space E2n are assigned by V α and Vα. It is important to notice that such properties imply the connections (Christoffel symbols) and not the metric tensor.
As we said, the invariant (57) is a generic conserved quantity specified by the choice of V α and Vα. If
V α = dxα , ds
(120)
is a 4-velocity, with α =0, 1, 2, 3, immediately, from Eq.(80), we obtain the equation of geodesics of General Relativity,
d2xα ds2
+
Γαμν
dxμ ds
dxν ds
= 0.
(121)
On the other hand, being
δV α = Rβαμν V β dxμ1 dxν2 ,
(122)
It is straightforward to see that, through the substitution vμ → Vμ, Eq.(72) is immediately recovered along a geodesic, that is our covariant symplectic structure is consistent with a hydrodynamic picture. It is worth noting that if ∂vμ in
∂s Eq.(117), the motion is not geodesic. The meaning of this term different from zero is that an extra force is acting on the system.
the result of the transport along a closed path, it is easy to
recover the geodesic deviation considering the geodesic (121) and the infinitesimal variation ξα with respect to it, i. e.
d
2
(xα+ ds2
ξα
)
+Γαμν
(x
+
ξ)
d(xμ+ ds
ξ
μ)
d(xν + ds
ξν
)
=
0,
(123)
which gives, through Eq.(119),
5 Applications, Discussion and Conclusions
d2ξα ds2
=
Rμαλν
dxμ ds
dxν ξλ. ds
(124)
Many applications of the previous results can be achieved specifying the nature of vector (or tensor) fields which define the Hamiltonian conserved invariant H. Considerations in General Relativity and Electromagnetism are particularly interesting at this point. Let us take into account the Riemann tensor Rσρμν . It comes out when a given vector V ρ is transported along a closed path on a generic curved manifold. It is
[∇μ, ∇ν ]V ρ = Rσρμν V σ,
(118)
where ∇μ is the covariant derivative. We are assuming a Riemannian Vn manifold as standard in General Relativity. If connection is not symmetric, an additive torsion field comes out from the parallel transport.
Clearly the symplectic structure is due to the fact that
the Riemann tensor is derived from covariant derivatives
either as
[∇μ, ∇ν ]V ρ = Rσρμν V σ,
(125)
or
[∇μ, ∇ν ]Vρ = RμσνρVσ .
(126)
In other words, fundamental equations of General Relativity are recovered from our covariant symplectic formalism.
Another interesting choice allows to recover the standard Electromagnetism. If V α = Aα, where Aα is the vector potential and the Hamiltonian invariant is
H = AαAα ,
(127)
S. Capozziello, S. De Martino, S. Tzenov. Hydrodynamic Covariant Symplectic Structure from Bilinear Hamiltonian Functions
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