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2008, VOLUME 3
PROGRESS IN PHYSICS
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The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics
PROGRESS IN PHYSICS
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JULY 2008

CONTENTS

VOLUME 3

I. I. Haranas and M. Harney Detection of the Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
R. T. Cahill Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
F. Smarandache and V. Christianto The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
P. Wagener A Classical Model of Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A. A. Ungar On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
P.-M. Robitaille A Critical Analysis of Universality and Kirchhoﬀ’s Law: A Return to Stewart’s Law of Thermal Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
P.-M. Robitaille Blackbody Radiation and the Carbon Particle . . . . . . . . . . . . . . . . . . . . . . . . 36
A. Khazan The Roˆle of the Element Rhodium in the Hyperbolic Law of the Periodic Table of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
V. Chrisatianto and F. Smarandache What Gravity Is. Some Recent Considerations . . . . 63
U. K.W. Neumann Models for Quarks and Elementary Particles — Part III: What is the Nature of the Gravitational Field? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
U. K.W. Neumann Models for Quarks and Elementary Particles — Part IV: How Much Do We Know of This Universe? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A. Sharma The Generalized Conversion Factor in Einstein’s Mass-Energy Equation . . . . . 76
E. A. Isaeva On the Necessity of Aprioristic Thinking in Physics . . . . . . . . . . . . . . . . . . . . . . 84 S. M. Diab and S. A. Eid Potential Energy Surfaces of the Even-Even 230-238U Iso-
topes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
LETTERS
E. Goldfain A Brief Note on “Un-Particle” Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
F. Smarandache International Injustice in Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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July, 2008

PROGRESS IN PHYSICS

Volume 3

Detection of the Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer

Ioannis Iraklis Haranas and Michael Harneyy
Department of Physics and Astronomy, York University, 314A Petrie Building, North York, Ontario, M3J-1P3, Canada E-mail: ioannis@yorku.ca
y841 North 700 West, Pleasant Grove, Utah, 84062, USA
E-mail: michael.harney@signaldisplay.com
General Relativity predicts the existence of relativistic corrections to the static Newtonian potential which can be calculated and veriﬁed experimentally. The idea leading to quantum corrections at large distances is that of the interactions of massless particles which only involve their coupling energies at low energies. In this short paper we attempt to propose the Sagnac intrerferometric technique as a way of detecting the relativistic correction suggested for the Newtonian potential, and thus obtaining an estimate for phase diﬀerence using a satellite orbiting at an altitude of 250 km above the surface of the Earth.

1 Introduction

The potential acting between to masses M and m that sepa-
rated from their centers by a distance r is:

V (r) =

GM r

m

;

(1)

where s the Newton’s constant of gravitation. This potential
is of course only approximately valid [1]. For large masses and or large velocities the theory of General Relativity predicts that there exist relativistic corrections which can be calculated and also veriﬁed experimentally [2]. In the microscopic distance domain, we could expect that quantum mechanics, would predict a modiﬁcation in the gravitational potential in the same way that the radiative corrections of quantum electrodynamics leads to a similar modiﬁcation of the Coulombic interaction [3].
Even though the theory of General Relativity constitutes a very well deﬁned classical theory, it is still not possible to combine it with quantum mechanics in order to create a satisﬁed theory of quantum gravity. One of the basic obstacles that prevent this from happening is that General Relativity does not actually ﬁt the present paradigm for a fundamental theory that of a renormalizable quantum ﬁeld theory. Gravitational ﬁelds can be successfully quantized on smooth-enough space-times [4], but the form of gravitational interactions is such that they induce unwanted divergences which can not be absorbed by the renormalization of the parameters of the minimal General Relativity [5]. Somebody can introduce new coupling constants and absorb the divergences then, one is unfortunately led to an inﬁnite number of free parameters. In spite the diﬃculty above quantum gravity calculations can predict long distance quantum corrections.
The main idea leading to quantum corrections at large distances is due to the interactions of massless particles which

only involve their coupling energies at low energies, something that it is known from the theory of General Relativity, even though at short distances the theory of quantum grav-
ity diﬀers resulting to ﬁnite correction of the order, O cG3r3 .
The existence of a universal long distance quantum correction to the Newtonian potential should be relevant for a wide class of gravity theories. It is well known that the ultraviolet behaviour of Einstein’s pure gravity can be improved, if higher derivative contributions to the action are added, which in four dimensions take the form:

RR + R2;

(2)

where and are dimensionless coupling constants. What

makes the diﬀerence is that the resulting classical and quan-

tum corrections to gravity are expected to signiﬁcantly alter

the gravitational potential qat short distances comparable to
that of Planck length `P = Gc3 = 10 35 m, but it should not
really aﬀect its behaviour at long distances. At long distances

is the structure of the Einstein-Hilbert action that actually de-

termines that. At this point we should mentioned that some

of the calculation to the corrections of the Newtonian gravi-

tational potential result in the absence of a cosmological con-

stant which usually complicates the perturbative treatment

to a signiﬁcant degree due to the need to expand about a non-

ﬂat background.

In one loop amplitude computation one needs to calculate

all ﬁrst order
relativistic O

cGo2crm2re2ctioannsdinthGe ,quwahnitcuhmwmillecinhcalnuidcael

both the
O Gc3

corrections to the classical Newtonian potential [6].

2 The corrections to the potential
Our goal is not to present the details of the one loop treatment that leads to the corrections of the Newtonian gravita-

Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer

3

Volume 3

PROGRESS IN PHYSICS

July, 2008

tional potential but rather sate the result and then use it in our
calculations. Valid in order of G2 we have that the corrected
potential now becomes [6]:

V (r) =

GM r

m


1

G (M + m) 2c2r

122 G 15c3r2


:

(3)

Observing (3) we see that in the correction of the static
Newtonian potential two diﬀeqrent length scales are involved.
First, the Planck length `P = Gc3 =10 35 m and second the Schwarzschild radii of the heavy sources rsch = 2GcM2 n . Fur-
thermore there are two independent dimensionless parame-
ters which appear in the correction term, and involve the ratio
of these two scales wit respect to the distance r. Presumably
for meaningful results the two length scales are much smaller
than r.

3 Perturbations due to oblateness J2

Because the Earth’s gravitational potential is not that of a perfect spherical body, we can approximate its potential as a spherical harmonic expansion of the following form:

V (r; ) =

"

GM m r

1

X 1
n=2

Jn

 Re r

n

Pn

(sin

#
)

=

=

GM m r

[Vo

+

VJ2

+

VJ3

+

:

:

:

]

;

(4)

where:
r = geocentric distance,  = geocentric latitude. Re = means equatorial radius of the Earth, Pn = Legendre polynomial of degree n and order zero, Jn = Jn0 jonal harmonics of order zero, that depend on the latitude  only,
and the ﬁrst term GMm=r now describes the potential of a
homogeneous sphere and thus refers to Keplerian motion, the remaining part represents the Earth’s oblateness via the zonal harmonic coeﬃcients and [7]

V0 = 1

VJ2

=

J2 2

 Re 2 r

3 sin2

VJ3

=

J3 2

 Re 3 r

5 sin3

9

1

>>>>>>=

3 sin  >>>>>>;

(5)

similarly [8]

J2 J3

= 1,082.6 10 6 = 2.53 10 6

)
:

(6)

Therefore equation (4) can be further written:

V (r; ) =

GM m r


1

X 1
n=2


Jn

Re r

n

Pn(sin

)

G

(M + 2rc2

m)


=

(6a)

=

GM m r

[Vo

+

VJ2

+

VJ3

+

:::

VRelativistic] :

Since J2 is 400 larger that any other Jn coeﬃcients, we
can disregard them and write the following expression for the Earth’s potential function including only the relativistic correction and omitting the quantum corrections as being very small we have:

V (r; ) =

GMem r

+

GMemRe2J2 r3

3 2

sin2

+

G2Mem (Me c2r2

+

m)

:

1 2


+

(7)

Since we propose a satellite in orbit that carries the Sagnac instrument it will of a help to express equation (7) for the potential in terms of the orbital elements. We know that
sin  = sin i sin(f +!) where i is the inclination of the orbit, f is the true anomaly and ! is the argument of the perigee. Ignoring long and short periodic terms (those containing ! and f) we write (7) in terms of the inclination as follows:

V (r; ) =

GMem r

+

3GMemRe2J2 2r3

 sin2i 2

+

G2Mem (Me c2r2

+

m)

1 3


+

(8)

therefore the corresponding total acceleration that a mass m

at r > Re has becomes:

gtot =

1 @ m @r

GMem r

+

3GMemRe2J2 2r3

 sin2i 2

1 3

+

G2Mem (M r2c2

+

m)

(9)

so that:

gtot =

GMe r2

+

9 GMeRe2 J2 2r4

 sin2 2

i

1 3


+

+

G2Me (Me c2r3

+

m)

:

(10)

4 Basic Sagnac interferometric theory
The Sagnac interferometer is based on the Sagnac eﬀect, reported by G. Sagnac in 1913 [8]. Two beams are sent in opposite directions around the interferometer until they meet

4

Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer

July, 2008

PROGRESS IN PHYSICS

Volume 3

rs = (c2

h
Rs2 2s) 1

 Rc2s

GrM2 e

+

9

82Rs2N
GM2re4Re2J2


s Rs
sin2i 2

13s++Gr232Mcv2eo2rb

h
1

cos h 2Rcs s 1 + Rsc s

1iii

(15)


q


rs = (c2

Rs2

82Rs2N s Rs

h


2s) 1 + Rc2s GrM2 e

3GM4erR4 e2J2

s

+ 2
Gr23Mc2e2

h (ReG+Mzoerhb)
1 cos 2Rcs

s

1 + Rsc s

1iii

(16)

rs

=h


c2 1+Rs Gc2MRe2e


1

82Rs2N

s


1

+

Rs2c2

2s


Rs

s+

r
2 GRMee 1

 2zRoerb

3G4Mc2eRRe4e2

J2


1

 4zRoerb

GR2e3Mc4e2


1

h
3zRoerb 1

 zRoreb
h
cos 2Rcs s 1+ Rsc s

1iii

(17)

again to create a phase pattern. By rotating the interferometer in the direction of either the clockwise (CW) or counterclockwise (CCW) beam, a phase diﬀerence results between the two beams that its given by:

rs

=

8 2 Rs2ag N (c2 a2

 2)

;

(11)

where is the angular velocity of the interferometer, Rsag is the radius of the interferometer, N is the number of turns of ﬁber around the radius and  is the frequency of light in the
ﬁber. Let us now assume that the Sagnac interferometer and its
light laser beams are in the region of space around the Earth where the gravitational potential is given by equation (3) and let us further assume that the quantum correction to the potential is really negligible. If the Sagnac light loop area has a unit vector that is perpendicular to the acceleration of gravity vector, then the motion of the interferometer will exhibit a red-shift that will be given by:

frs = 1

f
c2V

=1

f
gcco2r z

;

(12)

where V is the diﬀerence in the potential between to diﬀer-

ent points P1 and P2, and gcor is the corrected or total acceleration of gravity and z is the diﬀerence in vertical distance

between the two beams as the interferometer coil rotates. This

distance z that the laser beams see is given by:

8

2

39

<

z

=

Rsag

1
:

cos

4

c

2 1+

Rsag
Rsag c

=
5 :
;

(13)

This Sagnac eﬀect can also be ampliﬁed by an interferometer that is in orbit, where the orbital velocity of the interferometer with respect to the Earth’s surface produces an increased phase shift. Both terms involved in the acceleration of gravity in the ﬁrst one:

rs

=

82Rs2agN  c2 Rs2ag

Rsag 21

+ vor2b
gtco2tz

(14)

 using (14) and taking into account that M m we further
obtain (15), where M is the source of the gravitational ﬁeld = the mass of the Earth in our case Me, and R is the radius of the massive body = Re, and r = Re + zorb it’s orbital height
plus Earth radius for an Earth-based satellite. This Sagnac eﬀect can also be ampliﬁed by an interfer-
ometer that is in orbit, where the orbital velocity of the interferometer with respect to the Earth’s surface produces an increased phase shift. Both terms involved in the acceleration of gravity in the ﬁrst one:

5 Sagnac in circular orbit of known inclination

Let now a Sagnac interferometer be aboard a satellite in a

circular polar orbit of inclination i = 90 degrees. If the incli-

nation is 90 degrees the velocityqat some height

zterambosvien2th2ie

1 3

=

1 6

surface

and the orbital of the Earth is

vorb = (ReG+MZeorb) and (6) takes the form (16) can be ﬁnally

written as (17).

6 Sagnac in elliptical orbit of known inclination

If now a satellite is carrying a Sagnac device is in an elliptical
orbit of eccentricity e and semi-major axis a we have that the
radial orbital vector and the orbital velocity are given by:

v2

=


GMe

2 r

r

(f )

=

a1 1+e

e2 cos f

;

1 a

=

GMe a

2

(1+e cos f) (1 e2)

(18)

1 ; (19)

where f is the true anomaly of the orbit. Substituting now in
(8) we obtain (20).
If we use the fact that GMe = n2a3 where n is the mean
motion of the satellite, equation (20) can be further written as (21).
When the satellite approaches perigee its orbital velocity will increase, so we will expect to see a higher phase diﬀerence than any other point of the orbit, and similarly the eﬀect

Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer

5

Volume 3

PROGRESS IN PHYSICS

July, 2008

82Rs2N

s


1

+

Rs2c2

2s


Rs

s+

2

r GMa e

 1+e2+2e cos 1 e2

f


rs

=h c2 1+Rs

 GMc2ea(21(+1e

cos f)2 e2)2

3GM4ecR2e2aJ42((11+ee2

cos )4

f

)4

G2

Mc4ea23(1(1+eec2o)s3f

)3

h
1

cos h 2Rcs s 1+ Rsc s

1iii

(20)

82Rs2N

s


1

+

Rs2c2

2s


Rs

s+

2na r 1+e21+2ee2cos f

rs

=h c2 1

+

Rs

 n2a(1+e cos f)2 c2(1 e2)2

3n2R4ce22Ja2((11+ee2c)o4s f)4

n4a3(1+e cos c4(1 e2)3

f )3

h
1

cos h 2Rcs s 1 + Rsc s

1iii

(21)

82Rs2N

s


1

+

Rs2c2

2s


Rs

s+

r

2

GMa e

1+e 1e

rs

(perigee)

=

c2

h
1

+

Rs


c2aG2(M1 ee)2

43cG2aM4(e1Re2eJ)24

c4

aG32(1Me2e)3


h
1

h
cos 2Rcs s 1 + Rsc s

1iii

(24)

rs

(perigee)

= c2

h
1

+

Rs


82Rs2N s 1 +
 c2(n12ae)2 4c32na2(R1e2Je2)4

Rs2c2c42sn(14a3e)R3 s

s+
h
1

q

2 cos

nh a2Rcs11+s ee

1

+

Rsc

s

1iii

(25)

82Rs2N

s


1

+

Rs2c2

2s


Rs

s+

2

r GMa e

  1e 1+e

rs

(apogee)

=

c2

h
1

+

Rs


c2 aG2 (M1+e e)2

43cG2aM4(e1R+e2eJ)24

c4

aG32(1M+e2e)3


h
1

cos h 2Rcs s 1 + Rsc s

1iii

(26)

rs

(apogee)

= c2

h
1

+

Rs


82Rs2N s 1 +
 c2(n12+ae)2 4c32na2(R1+e2Je2)4

Rs2c2c42sn(14+a3e)R3 s

s+
h
1

q

 2na
h

1+e 1e

cos 2Rcs s 1

+

Rsc

s

1iii

(27)

will be minimum at the point of apogee because the satellite’s velocity is minimal. The distance at perigee and apogee are given by the equations below:

rpg = a (1 e) )

(22)

rapg = a (1 + e)

also the corresponding velocities are:

vp2g

=

GM a

1 1

+

e e

9 >>>=

va2pg

=

GMe a

1 e 1+e

;
>>>;

(23)

therefore the phase diﬀerence detected by the Sagnac due to the contribution of the Earth’s oblateness plus relativistic correction to the potential at perigee and apogee can be written as (24) or again (25).
Similarly the phase diﬀerence at apogee can be written as (26) or again (27).

For this last case of the elliptical orbit in (25) and (26)

where the Sagnac interferometer is on the satellite and we as-

sume
a=8

R1s06=m1,me,=

=2
0.2,

1014 Hz, N Re = 6.378

=10160m6,etesrs=w4e00arraridv/eseact,

the following values for :

 (perigee) = 3.57 10 16 radians,

 (apogee) = 2.44 10 16 radians.

These values are based on the dominant potential correction in (11) of section 3 which is the ﬁrst term in (11) or the Newtonian correction:

Newtonian correction = 2.17 10 16 radians.

In comparison, the second and third terms in (11) are the oblateness and relativistic corrections respectively and they produce the following values based on the given parameters:

Oblateness correction = 8.52 10 20, Relativistic correction = 7.91 10 26.

So by comparison of the values above, the Newtonian correction is much easier to measure.

6

Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer

July, 2008

PROGRESS IN PHYSICS

Volume 3


q


rs = (c2

a2s


2s) 1 +

as

82a2sN s as s +  2 1+(e2+e 1) cos 2acs s (1+ asc s )
1+e cos 2acs s (1+ asc s ) 1

(ReG+Mzoerb) 1
Gr2Mc2e

3G4Mre4Rc2e2J2

Gr23Mc4e2

(30)

rs

=


c2 1+

as


82a2sN s 1 + 1+(e2+e 1) cos 2acs s (1+ asc s )
1+e cos 2acs s (1+ asc s ) 1

a2sc2

2


as

s+

r
2 GRMee 1

 zRoreb

1


GRe2Mc2e 1

 2zRoerb

3G4MRee4Rc2e2J2


1

 4zRoerb

GR2e3Mc4e2


1

 3zRoerb

(31)

82a2sN

s


1

+

a2sc2

2s


as

s+

2

r GMa e

 1+e2+2e cos f 1 e2


rs=


c2 1+ as

1+(e2+e 1) cos 1+e cos 2acs

2acs s (1+ asc s ) s (1+ asc s ) 1

1


GMc2ea(21(+1e

cos f e2)2

)2

3GM4ecR2e2aJ42((11+ee2)c4os f)4

G2Mc4ea23(1(1+eec2o)s3f)3

(32)

82a2sN

s


1

+

a2sc2

2s


as

s+

r

2

GMa e

1+e 1e

rs (perigee)=


c2 1 +

aa

1+(e2+e 1) cos 1+e cos 2acs

2acs s (1+ asc s ) s (1+ asc s ) 1

1

 c2aG2(M1 ee)2

43cG2aM4(e1Re2eJ)24

c4aG32(1Me2e)3

(33)

82a2sN

s


1

+

a2sc2

2s


as

s+

r

2

GMa e

1e 1+e

rs (apogee)=


c2 1 +

as

1+(e2+e 1) cos 1+e cos 2acs

2acs s (1+ asc s ) s (1+ asc s ) 1

1

 c2 aG2 (M1+e e)2

43cG2aM4(e1R+e2eJ)24

c4aG32(1M+e2e)3

(34)

The  values given above may be more easily measured
using a QPSK-modulator inserted in the CCW or CW beam path to improve phase resolution. Also, the use of higher wavelengths (factor of 10 higher in frequency) will increase resolution.

we can ﬁnally write for (13):

z

=

h
as 1+(e2 +e


1)
n

n
cos

2acs

s

1+

1 + e cos 2acs s 1 + asc s

as1c os

1oi
:

(29)

7 We suggest a Sagnac with an elliptic ﬁber loop

To attempt increasing the resolution of the phase diﬀerence of

the Sagnac interferometer let us now propose a Sagnac loop,

that has the shape of an ellipse that rotates with an angular

velocity . In this case it can be shown that the height dif-

ference between two points on the ellipse can be given by:

"
z=a 1+

e2 + e 1 1 + e cos

#
cos  :

(28)

To check the validity of the formula we derived we can set
e=0 which is the case of a circular Sagnac ﬁber optical path
we can see that the (13) in now retrieved since
Rsag = aloop(sag) = as is the semi major axis of the ellip-
tical ﬁber loop. When the ellipse spins with angular velocity
that would force it to trace out a circle whose radius r, will
be that of the semi-major axis a of the ellipse, and therefore

8 Circular orbit formula for the phase diﬀerence of the Sagnac
Let now as before have a Sagnac interferometer be aboard a
satellite in a circular polar orbit of inclination i = 90 degrees. ItEhfaetrhtoherbiisintcavlloirnvbae(tlcioiorcnci)tyi=sa9q t 0so(dRmeeGge+Mrhzeoeeersibg)thhatenzdtea(rb6mo)vtseainkthe2es2isthuerf13afoc=remo16f(a3tnh0de)
that can be ﬁnally written as (31).
9 Sagnac in elliptical orbit of known inclination
If now a satellite is carrying a Sagnac device is in an elliptical
orbit of eccentricity e and semi-major axis a we have that the
radial orbital vector and the orbital velocity are given by (32). At perigee the equation (32) becomes (33) and also (34).

Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer

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For (33) and (34) above the following values are com-
puted assuming e = 0.2,  = 2 1014 Hz, a = 8 106 meters,

N = 1 (because the orbit is the Sagnac loop), Rsag = Rperigee

or Rapogee as and apogee =

d6et1e0rm4inreadd/bseyc(w22e),ﬁndp,erigee

=

0.001

rad/sec,

 (perigee) = 6.05 1010 radians,

 (apogee) = 2.36 1010 radians.

These values are for measuring the dominant Newtonian contribution as described in Section 6. To detect relativistic contribution which is 3.64 10 10 smaller than the Newtonian contribution the corresponding phase-shifts from (33) and (34) are:
 (perigee) = 22 radians,  (apogee) = 8.59 radians.

Thus, the relativistic contribution in (11) of Section 3 is easily measurable using a Sagnac interferometer where the satellites in orbit are the Sagnac loop. In this scenario, the light path can be implemented by transmitting laser beams from one satellite to the next satellite in orbit ahead of it. Also, by using the maximum spacing possible between satellites in orbit this will allow line of site transmission while reducing the number of satellites required for the Sagnac loop. With the potential to measure such small relativistic corrections, the merit of using satellites to implement a large Sagnac
loop of radius Rs = Rap or Rper is well worth considering.

Submitted on March 27, 2008 Accepted on April 04, 2008

References
1. Donoghue J. F. arXiv: gr-qc/9310024. 2. Bjorken J. D. and Drell S. Relativistic quantum mechanics and
relativistic quantum ﬁeld theory. Mc Graw-Hill, New York 1964. 3. t’ Hooft G. and Veltman M. Ann. Inst. H. Poincare, 1974, v. 20, 69. 4. Capper D. M., Leibrand G. and Medrano R. M. Phys. Rev. D, 1973, v. 8, 4320. 5. Goroﬀ M. and Sagnotti A. Nucl. Phys. B, 1986, v. 22, 709. 6. Hamber H. W. and Liu S. arXiv: hep-th/9505128. 7. Blanco V. M. and Mc Cuskey S. W. Basic physics of the Solar system. Addison-Wesley Publishing Inc., 1961. 8. Sagnac G. R. Acad. Sci., 1913, v. 157, 708. 9. Camacho A. General Relativity and Gravitation, 2004, v. 36, no. 5, 1207–1211. 10. Roy A. E. Orbital motion. Adam Hilger, Third Edition, 1988.

July, 2008

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Volume 3

Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy

Reginald T. Cahill
School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia E-mail: Reg.Cahill@ﬂinders.edu.au
Doppler shift observations of spacecraft, such as Galileo, NEAR, Cassini, Rosetta and MESSENGER in earth ﬂybys, have all revealed unexplained speed “anomalies” — that the Doppler-shift determined speeds are inconsistent with expected speeds. Here it is shown that these speed anomalies are not real and are actually the result of using an incorrect relationship between the observed Doppler shift and the speed of the spacecraft — a relationship based on the assumption that the speed of light is isotropic in all frames, viz invariant. Taking account of the repeatedly measured light-speed anisotropy the anomalies are resolved ab initio. The Pioneer 10/11 anomalies are discussed, but not resolved. The spacecraft observations demonstrate again that the speed of light is not invariant, and is isotropic only with respect to a dynamical 3-space. The existing Doppler shift data also oﬀers a resource to characterise a new form of gravitational waves, the dynamical 3-space turbulence, that has also been detected by other techniques. The Einstein spacetime formalism uses a special deﬁnition of space and time coordinates that mandates light speed invariance for all observers, but which is easily misunderstood and misapplied.

1 Introduction
Planetary probe spacecraft (SC) have their speeds increased, in the heliocentric frame of reference, by a close ﬂyby of the Earth, and other planets. However in the Earth frame of reference there should be no change in the asymptotic speeds after an earth ﬂyby, assuming the validity of Newtonian gravity, at least in these circumstances. However Doppler shift observations of spacecraft, such as Galileo, NEAR, Cassini, Rosetta and MESSENGER in earth ﬂybys, have all revealed unexplained speed “anomalies” — that the Doppler-shift determined speeds are inconsistent with expected speeds [1–6]. Here it is shown that these speed anomalies are not real and are actually the result of using an incorrect relationship between the observed Doppler shift and the speed of the spacecraft — a relationship based on the assumption that the speed of light is isotropic in all frames, viz invariant. Taking account of the repeatedly measured light-speed anisotropy the anomalies are resolved ab initio.
The speed of light anisotropy has been detected in at least 11 experiments [7–17], beginning with the MichelsonMorley 1887 experiment [7]. The interferometer observations and experimental techniques were ﬁrst understood in 2002 when the Special Relativity eﬀects and the presence of gas were used to calibrate the Michelson interferometer in gas-mode; in vacuum mode the Michelson interferometer cannot respond to light speed anisotropy [18, 19], as conﬁrmed in vacuum resonant cavity experiments, a modern version of the vacuum-mode Michelson interferometer [20]. So far three diﬀerent experimental techniques have given consistent results: gas-mode Michelson interferometers [7–11, 16],

coaxial cable RF speed measurements [12–14], and opticalﬁber Michelson interferometers [15, 17]. This light speed anisotropy reveals the existence of a dynamical 3-space, with the speed of light being invariant only with respect to that 3space, and anisotropic according to observers in motion relative to that ontologically real frame of reference — such a motion being conventionally known as “absolute motion”, a notion thought to have been rendered inappropriate by the early experiments, particularly the Michelson- Morley experiment. However that experiment was never null — they reported a speed of at least 8km/s [7] using Newtonian physics for the calibration. A proper calibration of the Michelson-Morley apparatus gives a light speed anisotropy of at least 300km/s. The spacecraft Doppler shift anomalies are shown herein to give another technique that may be used to measure the anisotropy of the speed of light, and give results consistent with previous detections.
The numerous light speed anisotropy experiments have also revealed turbulence in the velocity of the 3-space relative to the Earth. This turbulence amounts to the detection of sub-mHz gravitational waves — which are present in the Michelson and Morley 1887 data, as discussed in [21], and also present in the Miller data [8, 22] also using a gas-mode Michelson interferometer, and by Torr and Kolen [12], DeWitte [13] and Cahill [14] measuring RF speeds in coaxial cables, and by Cahill [15] and Cahill and Stokes [17] using an optical-ﬁber interferometer. The existing Doppler shift data also oﬀers a resource to characterise this new form of gravitational waves.
There has been a long debate over whether the Lorentz 3space and time interpretation or the Einstein spacetime inter-

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W

V

sv

Earth
© V

Earth

c vi E i
'
c + vi

V '
SC

Fig. 1: Spacecraft (SC) earth ﬂyby trajectory, with initial and ﬁnal asymptotic velocity V, diﬀering only by direction. The Doppler shift is determined from Fig. 2 and (1). Assuming, as conventionally done, that the speed of light is invariant in converting measured Doppler shifts to deduced speeds, leads to the so-called ﬂyby anomaly, namely that the incoming and outgoing asymptotic speeds
appear to be diﬀer, by V1. However this eﬀect is yet another
way to observe the 3-space velocity vector, as well as 3-space wave
eﬀects, with the speed of light being c and isotropic only with re-
spect to this structured and dynamical 3-space. The ﬂyby anomalies demonstrate, yet again, that the invariance of the speed of light is merely a deﬁnitional aspect of the Einstein spacetime formalism, and is not based upon observations. A neo-Lorentzian 3-space and time formalism is more physically appropriate.
pretation of observed SR eﬀects is preferable or indeed even experimentally distinguishable. What has been discovered in recent years is that a dynamical structured 3-space exists, so conﬁrming the Lorentz interpretation of SR [22, 24, 25], and with fundamental implications for physics. This dynamical 3-space provides an explanation for the success of the SR Einstein formalism. Indeed there is a mapping from the physical Lorentzian space and time coordinates to the nonphysical spacetime coordinates of the Einstein formalism — but it is a singular map in that it removes the 3-space velocity with respect to an observer. The Einstein formalism transfers dynamical eﬀects, such as length contractions and clock slowing eﬀects, to the metric structure of the spacetime manifold, where these eﬀects then appear to be merely perspective eﬀects for diﬀerent observers. For this reason the Einstein formalism has been very confusing. Developing the Lorentzian interpretation has lead to a new account of gravity, which turns out to be a quantum eﬀect [23], and of cosmology [21, 22, 26, 27], doing away with the need for dark matter and dark energy. So the discovery of the ﬂyby anomaly links this eﬀect to various phenomena in the emerging new physics.
2 Absolute motion and ﬂyby Doppler shifts
The motion of spacecraft relative to the Earth are measured by observing the direction and Doppler shift of the transponded RF transmissions. As shown herein this data gives another technique to determine the speed and direction of the dynamical 3-space, manifested as a light speed anisotropy. Up to now the repeated detection of the anisotropy of the speed of

Fig. 2: Asymptotic ﬂyby conﬁguration in Earth frame-of-reference,
with spacecraft (SC) approaching Earth with velocity V. The de-
parting asymptotic velocity will have a diﬀerent direction but the
same speed, as no force other than conventional Newtonian gravity
is assumed to be acting upon the SC. The Dynamical 3-space ve-
locity is v(r; t), which causes the outward EM beam to have speed c vi, and inward speed c + vi, where vi = v cos(i), with i the
angle between v and V.

light has been ignored in analysing the Doppler shift data,
causing the long-standing anomalies in the analysis [1–6].
In the Earth frame of reference, see Fig. 2, let the trans-
mitted signal from earth have frequency f, then the corresponding outgoing wavelength is 0 = (c vi)=f, where vi = v cos(i). This signal is received by the SC to have period Tc = 0=(c vi +V ) or frequency fc = (c vi +V )=0.
The signal is re-transmitted with the same frequency, and so
has wavelength i = (c + vi V )=fc, and is detected at earth with frequency fi = (c + vi)=i. Then overall we obtain

fi

=

c c+

+ vi

vi

V

c

vi + V c vi

f:

(1)

Ignoring the projected 3-space velocity vi, that is, assum-
ing that the speed of light is invariant as per the usual literal

interpretation of the Einstein 1905 light speed postulate, we

obtain instead

fi

=

c+ c

V V

f

:

(2)

The use of (2) instead of (1) is the origin of the putative

anomalies. The Doppler shift data is usually presented in the

form of speed anomalies. Expanding (2) we obtain

fi f

=

fi f

f

=

2V c

+:::

(3)

From the observed Doppler shift data acquired during a ﬂyby, and then best ﬁtting the trajectory, the asymptotic hy-
perbolic speeds Vi1 and Vf1 are inferred, but incorrectly so,
as in [1]. These inferred asymptotic speeds may be related to an inferred asymptotic Doppler shift:

fi f

=

fi f

f

=

2Vi1 c

+:::

(4)

In practice the analysis is more complex as is the doppler shift technology. The analysis herein is suﬃcient to isolate and quantify the light-speed anisotropy eﬀect.

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Parameter
Date
V1 km/s
i deg i deg f deg f deg
v deg(hrs) v deg
v km/s i deg f deg
(O) V1 mm/s (P) V1 mm/s

GLL-I
Dec 8, 1990 8.949 266.76 12.52 219.97 34.15
108.8(7.25) 76
420 90.5 61.8
3.92 0.3 3.92 0.1

GLL-II
Dec 8, 1992 8.877 219.35 34.26 174.35 4.87
129.0(8.6) 80
420 56.4 78.2
4.6 1.0 4.60 0.6

NEAR
Jan 23, 1998 6.851 261.17 20.76 183.49 71.96
108.8(7.25) 76
450 81.8 19.6
13.46 0.01 13.40 0.1

Cassini
Aug 18, 1999 16.010 334.31 12.92 352.54 20.7
45.0(3.0) 75
420 72.6 76.0
21 0.99 1.0

Rosetta
Mar 4, 2005 3.863 346.12 2.81 246.51 34.29
130.5(8.7) 80
420 95.3 60.5
1.80 0.03 1.77 0.3

M’GER
Aug 2, 2005 4.056 292.61 31.44 227.17 31.92
168.0(11.2) 85
420 124.2 55.6
0.02 0.01 0.025 0.03

Table 1: Earth ﬂyby parameters from [1] for spacecraft Galileo (GLL: ﬂybys I and II), NEAR, Cassini, Rosetta and MESSENGER
(M’GER). V1 is the average osculating hyperbolic asymptotic speed, and are the right ascension and declination of the incoming (i) and outgoing (f) osculating asymptotic velocity vectors, and (O) V1 is the putative “excess speed” anomaly deduced by assuming that the speed of light is isotropic in modeling the doppler shifts, as in (4). The observed (O) V1 values are from [1], and after correcting for atmospheric drag in the case of GLL-II, and thruster burn in the case of Cassini. (P) V1 is the predicted “excess speed”, using (7), taking
account of the known light speed anisotropy and its eﬀect upon the doppler shifts, using v and v as the right ascension and declination of
the 3-space ﬂow velocity, having speed v, which has been taken to be 420 km/s in all cases, except for NEAR, see Fig. 3. The values on (P) V1 indicate changes caused by changing the declination by 5% — a sensitivity indicator. The angles i and f between the 3-space velocity and the asymptotic initial/ﬁnal SV velocity V are also given. The observed doppler eﬀect is in exceptional agreement with the
predictions using (7) and the previously measured 3-space velocity. The ﬂyby doppler shift is thus a new technique to accurately measure
the dynamical 3-space velocity vector, albeit retrospectively from existing data. Note: By ﬁne tuning the v and v values for each ﬂyby a
perfect ﬁt to the observed (O) V1 is possible. But here we have taken, for simplicity, the same values for GLL-I and NEAR.

However expanding (1) we obtain, for the same Doppler

shift
Vi1

fi f

c 2

=

fi f

f

c 2

=


1

+

vi2 c2


V

+

:::

(5)

where V is the actual asymptotic speed. Similarly after the

ﬂyby we obtain

Vf1

ff f

c 2

=

ff f f

c 2

=


1

+

vf2 c2


V

+:::

(6)

and we see that the “asymptotic” speeds Vi1 and Vf1 must
diﬀer, as indeed ﬁrst noted in the data by [3]. We then obtain the expression for the so-called ﬂyby anomaly

V1 = Vf1 Vi1 = vf2 c2 vi2 V + : : :

=

v2 c2

cos(f )2

cos(i)2 V1 + : : :

(7)

where here V  V1 to suﬃcient accuracy, where V1 is the
average of Vi1 and Vf1, The existing data on v permits

We ignore terms of order vV =c2 within the parentheses, as in practice they are much smaller than the v2=c2 terms.

ab initio predictions for V1, and as well a separate least-
squares-ﬁt to the individual ﬂybys permits the determination of the average speed and direction of the 3-space velocity, relative to the Earth, during each ﬂyby. These results are all remarkably consistent with the data from the 11 previous laboratory experiments that studied v. Note that whether the
3-space velocity is +v or v is not material to the analysis herein, as the ﬂyby eﬀect is 2nd order in v.
3 Earth ﬂyby data analaysis
Eqn. (7) permits the speed anomaly to be predicted as the
direction and speed v of the dynamical 3-space is known,
as shown in Fig. 3. The ﬁrst determination of its direction was reported by Miller [8] in 1933, and based on extensive observations during 1925/1926 at Mt.Wilson, California, using a large gas-mode Michelson interferometer. These observations conﬁrmed the previous non-null observations by Michelson and Morley [7] in 1887. The general characteris-
tics of v(r; t) are now known following the detailed analysis
of the experiments noted above, namely its average speed, and removing the Earth orbit eﬀect, is some 420 30km/s,

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Fig. 3: Southern celestial sphere with RA and Dec shown. The 4 dark blue points show the consolidated results from the Miller gas-mode Michelson interferometer [8] for four months in 1925/1926, from [22]. The sequence of red points show the running daily average RA and Dec trend line, as determined from the optical ﬁber interferometer data in [17], for every 5 days, beginning September 22, 2007. The lightblue scattered points show the RA and Dec for individual days from the same experiment, and show signiﬁcant turbulence/wave eﬀects.
The curved plots show iso-speed V1 “anomalies”: for example for v = 420 km/s the RA and Dec of v for the Galileo-I ﬂyby must lie
somewhere along the “Galileo-I 420” curve. The available spacecraft data in Table 1, from [1], does not permit a determination of a unique v during that ﬂyby. In the case of “Galileo-I” the curves are also shown for 420 30 km/s, showing the sensitivity to the range of speeds discovered in laboratory experiments. We see that the “Galileo-I” December ﬂyby has possible directions that overlap with the December data from the optical ﬁber interferometer, although that does not exclude other directions, as the wave eﬀects are known to be large. In
the case of NEAR we must have v 440 km/s otherwise no ﬁt to the NEAR V1 is possible. This demonstrates a ﬂuctuation in v of at least +20 km/s on that ﬂyby day. This plot shows the remarkable concordance in speed and direction from the laboratory techniques with the ﬂyby technique in measuring v, and its ﬂuctuation characteristics. The upper-left coloured disk (radius = 8 ) shows concordance for September/August interferometer data and Cassini ﬂyby data ( MESSENGER data is outside this region — but has very small V1 and large uncertainty), and the same, lower disk, for December/January/February/March data (radius = 6 ). The moving concordance eﬀect is
undertsood to be caused by the earth’s orbit about the Sun, while the yearly average of 420 30 km/s is a galaxy related velocity. Directions for each ﬂyby v were selected and used in Table 1.

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from direction right ascension v = 5.5 2hr, declination v = 70 10 S — the center point of the Miller data in Fig. 3,
together with large wave/turbulence eﬀects, as illustrated in Fig. 4. Miller’s original calibration technique for the interferometer turned out to be invalid [22], and his speed of approximately 208 km/s was recomputed to be 420 30 km/s in [19,22], and the value of 420 km/s is used here as shown in Table 1. The direction of v varies throughout the year due to the Earth-orbit eﬀect and low frequency wave eﬀects. A more recent determination of the direction was reported in [17] using an optical-ﬁber version of the Michelson interferometer, and shown also in Fig. 3 by the trend line and data from individual days. Directions appropriate to the date of each ﬂyby were approximately determined from Fig. 3.
The SC data in Table 1 shows the values of V1 and V1
after determining the osculating hyperbolic trajectory, as discussed in [1], as well as the right ascension and declination of
the asymptotic SC velocity vectors Vi1 and Vf1. In computing the predicted speed “anomaly” V1 using (7) it is only necessary to compute the angles i and f between the
dynamical 3-space velocity vector and these SC incoming and outgoing asymptotic velocities, respectively, as we assume
j j here that v = 420 kms, except for NEAR as discussed in
Fig. 3 caption. So these predictions are essentially ab initio in that we are using 3-space velocities that are reasonably well known from laboratory experiments. The observed Doppler eﬀects are in exceptional agreement with the predictions using (7) and the previously measured 3-space velocity. The ﬂyby anomaly is thus a new technique to accurately measure the dynamical 3-space velocity vector, albeit retrospectively from existing data.

Fig. 4: Speeds vP , of the 3-space velocity v projected onto the hori-
zontal plane of the Miller gas-mode Michelson interferometer, plotted against local sidereal time in hours, for a composite day, with data collected over a number of days in September 1925. The data shows considerable ﬂuctuations, from hour to hour, and also day to day, as this is a composite day. The dashed curve shows the nonﬂuctuating best-ﬁt variation over one day, as the Earth rotates, causing the projection onto the plane of the interferometer of the velocity of the average direction of the space ﬂow to change. The maximum projected speed of the curve is 417 km/s, and the min/max occur at approximately 5 hrs and 17 hrs sidereal time (right ascension); see Fig. 3 for September. Analysing Millers’s extensive data set from 1925/26 gives average speed, after removing earth orbit eﬀect, of 420 30 km/s, and the directions for each month shown in Fig. 3.
metric of the induced spacetime, merely a mathematical con-
struct having no ontological signiﬁcance, is related to v(r; t)
according to [21, 22, 27]

4 New gravitational waves

ds2 = dt2

(dr

v(r; t)dt)2 c2

=

g

dx dx :

(8)

Light-speed anisotropy experiments have revealed that a dy-
namical 3-space exists, with the speed of light being c, in vac-
uum, only with respect to to this space: observers in motion
“through” this 3-space detect that the speed of light is in gen-
eral diﬀerent from c, and is diﬀerent in diﬀerent directions.
The dynamical equations for this 3-space are now known and
involve a velocity ﬁeld v(r; t), but where only relative veloc-
ities are observable locally — the coordinates r are relative
to a non-physical mathematical embedding space. These dy-
namical equations involve Newton’s gravitational constant G
and the ﬁne structure constant . The discovery of this dy-
namical 3-space then required a generalisation of the Max-
well, Schro¨dinger and Dirac equations. The wave eﬀects al-
ready detected correspond to ﬂuctuations in the 3-space ve-
locity ﬁeld v(r; t), so they are really 3-space turbulence or
wave eﬀects. However they are better known, if somewhat in-
appropriately, as “gravitational waves” or “ripples” in “space-
time”. Because the 3-space dynamics gives a deeper under-
standing of the spacetime formalism we now know that the

The gravitational acceleration of matter, a quantum eﬀect,

and of the structural patterns characterising the 3-space, are

given by [21, 23]

g

=

@v @t

+

(v

r)v

(9)

and so ﬂuctuations in v(r; t) may or may not manifest as a
gravitational acceleration. The ﬂyby technique assumes that the SC trajectories are not aﬀected — only the light speed anisotropy is signiﬁcant. The magnitude of this turbulence depends on the timing resolution of each particular experiment, and was characterised to be sub-mHz in frequency by Cahill and Stokes [14]. Here we have only used asymptotic osculating hyperbolic trajectory data from [1]. Nevertheless even this data suggests the presence of wave eﬀects. For example the NEAR data requires a speed in excess of 440 km/s, and probably closer to 450 km/s, whereas the other ﬂybys are consistent with the average of 420 km/s from laboratory experiments. So here we see ﬂyby evidence of ﬂuctuations in
the speed v.

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Data exists for each full ﬂyby, and analysis of that data using the new Doppler shift theory will permit the study and characterisation of the 3-space wave turbulence during each ﬂyby: essentially the ﬂybys act as gravitational wave detectors. These gravitational waves are much larger than predicted by general relativity, and have diﬀerent properties.

5 Pioneer 10/11 anomalies

The Pioneer 10//11 spacecraft have been exploring the outer

solar system since 1972/73. The spacecraft have followed

escape hyperbolic orbits near the plane of the ecliptic, after

earlier planet ﬂybys. The Doppler shift data, using (2), have

revealed an unexplained anomaly beyond 10 AU [28]. This

manifests
= (2.92

as an
0.44)

unmodelled 10 18 s/s2,

increasing

blue

shift

d dt

(

f f

)

=

corresponding to a constant in-

ward 10

sun-directed acceleration of a =
8 cm/s2, averaged from Pioneer 10

dV dt

= (8.74

and Pioneer

1.33)
11 data.

However the Doppler-shift data from these spacecraft has

been interpreted using (2), instead of (1), in determining the

speed, which in turn aﬀects the distance data. Essentially this

implies that the spacecraft are attributed with a speed that is
too large by vc22 VD, where VD is the speed determined using
(2). This then implies that the spacecraft are actually closer
to the Sun by the distance vc22 RD, where RD is the distance
determined using (2). This will then result in a computed

spurious inward acceleration, because the gravitational pull

of the Sun is actually larger than modelled, for distance RD.
However this correction to the Doppler-shift analysis appears

not to be large enough to explain the above mention acceler-

ation anomaly. Nevertheless re-analysis of the Pioneer 10/11

data should be undertaken using (1).

6 Conclusions
The spacecraft earth ﬂyby anomalies have been resolved. Rather than actual relative changes in the asymptotic inward and outward speeds, which would have perhaps required the invention of a new force, they are instead direct manifestations of the anisotropy of the speed of light, with the Earth having a speed of some 420 30 km/s relative to a dynamical 3-space, a result consistent with previous determinations using laboratory experiments, and dating back to the Michelson-Morley 1887 experiment, as recently reanalysed [18, 19, 21]. The ﬂyby data also reveals, yet again, that the 3space velocity ﬂuctuates in direction and speed, and with results also consistent with laboratory experiments. Hence we see a remarkable concordance between three diﬀerent laboratory techniques, and the newly recognised ﬂyby technique. The existing ﬂyby data can now be re-analysed to give a detailed charaterisation of these gravitational waves. The detection of the 3-space velocity gives a new astronomical window on the galaxy, as the observed speeds are those relevant

to galactic dynamics. The dynamical 3-space velocity eﬀect also produces very small vorticity eﬀects when passing the Earth, and these are predicted to produce observable eﬀects on the GP-B gyroscope precessions [29].
A special acknowledgement to all the researchers who noted and analysed the spacecraft anomalies, providing the excellent data set used herein. Thanks also to Tom Goodey for encouraging me to examine these anomalies.
Submitted on March 30, 2008 Accepted on April 10, 2008
References
1. Anderson J.D., Campbell J.K., Ekelund J.E., Ellis J. and Jordan J.F. Anomalous orbital-energy changes observed during spaceraft ﬂybys of Earth. Phys. Rev. Lett., 2008, v. 100, 091102.
2. Anderson J.D., Campbell J.K. and Nieto M.M. New Astron. Rev., 2007, v. 12, 383.
3. Antreasian P.G. and Guinn J.R. AIAA Paper No. 984287 presented at the AIAA/AAS Astrodynamics Specialist Conference and Exhibit (Boston, August 10-12, 1998), and available at http://www.issibern.ch/teams/ Pioneer/pa-literature.htm
4. Morley T. and Budnik F. Proc. Intl. Symp. on Space Technology and Science, 2006, v. 25, 593.
5. Guman M.D., Roth D.C., Ionasescu R., Goodson T.D., Taylor A.H. and Jones J.B. Adv. Astron. Sci., 2000, v. 105, 1053.
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7. Michelson A.A. and Morley E.W. Am. J. Sc., 1887, v. 34, 333–345.
8. Miller D.C. Rev. Mod. Phys., 1933, v. 5, 203–242.
9. Illingworth K.K. Phys. Rev., 1927, v. 3, 692–696.
10. Joos G. Ann. d. Physik, 1930, v. 7, 385.
11. Jaseja T.S. et al. Phys. Rev. A, 1964, v. 133, 1221.
12. Torr D.G. and Kolen P. In: Precision Measurements and Fundamental Constants, Taylor B.N. and Phillips W.D. eds. Natl. Bur. Stand. (U.S.), Spec. Pub., 1984, 617, 675.
13. Cahill R.T. The Roland DeWitte 1991 experiment. Progress in Physics, 2006, v. 3, 60–65.
14. Cahill R.T. A new light-speed anisotropy experiment: absolute motion and gravitational waves detected. Progress in Physics, 2006, v. 4, 73–92.
15. Cahill R.T. Optical-ﬁber gravitational wave detector: dynamical 3-space turbulence detected. Progress in Physics, 2007, v. 4, 63–68.

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16. Mune´ra H.A., et al. In: Proceedings of SPIE, 2007, v. 6664, K1-K8, eds. Roychoudhuri C. et al.
17. Cahill R.T. and Stokes F. Correlated detection of submHz gravitational waves by two optical-ﬁber interferometers. Progress in Physics, 2008, v. 2, 103–110.
18. Cahill R.T. and Kitto K. Michelson-Morley experiments revisited. Apeiron, 2003, v. 10(2), 104–117.
19. Cahill R.T. The Michelson and Morley 1887 experiment and the discovery of absolute motion. Progress in Physics, 2005, v. 3, 25–29.
20. Braxmaier C. et al. Phys. Rev. Lett., 2002, 88, 010401; Mu¨ller H. et al. Phys. Rev. D, 2003, 68, 116006-1-17; Mu¨ller H. et al. Phys. Rev. D, 2003, v. 67, 056006; Wolf P. et al. Phys. Rev. D, 2004, v. 70, 051902-1-4; Wolf P. et al. Phys. Rev. Lett., 2003, v. 90, no. 6, 060402; Lipa J.A., et al. Phys. Rev. Lett., 2003, v. 90, 060403.
21. Cahill R.T. Dynamical 3-space: a review. arXiv: 0705.4146.
22. Cahill R.T. Process physics: from information theory to quantum space and matter. Nova Science Pub., New York, 2005.
23. Cahill R.T. Dynamical fractal 3-space and the generalised Schro¨dinger equation: Equivalence Principle and vorticity eﬀects. Progress in Physics, 2006, v. 1, 27–34.
24. Levy J. From Galileo to Lorentz. . . and beyond. Apeiron, Montreal, 2003.
25. Guerra V. and de Abreu R. Relativity: Einstein’s lost frame. Extramuros, 2005.
26. Cahill R.T. Dynamical 3-space: supernovae and the Hubble expansion — the older Universe without dark energy. Progress in Physics, 2007, v. 4, 9–12.
27. Cahill R.T. A quantum cosmology: no dark atter, dark energy nor accelerating Universe. arXiv: 0709.2909.
28. Nieto M.N., Turyshev S.G., Anderson J.D. The Pioneer anomaly: the data, its meaning, and a future test. arXiv: gr-qc/0411077.
29. Cahill R.T. Novel Gravity Probe B frame-dragging eﬀect. Progress in Physics, 2005, v. 3, 30–33.

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The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited

Florentin Smarandache and Vic Christiantoy

Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA

ySciprint.org

—

a

Free

E-mail: smarand@unm.edu Scientiﬁc Electronic Preprint Server,

http://www.sciprint.org

E-mail: admin@sciprint.org

The present article discusses Neutrosophic logic view to Schro¨dinger’s cat paradox. We argue that this paradox involves some degree of indeterminacy (unknown) which Neutrosophic logic can take into consideration, whereas other methods including Fuzzy logic cannot. To make this proposition clear, we revisit our previous paper by oﬀering an illustration using modiﬁed coin tossing problem, known as Parrondo’s game.

1 Introduction

The present article discusses Neutrosophic logic view to

Schro¨dinger’s cat paradox. In this article we argue that this

paradox involves some degree of indeterminacy (unknown)

which Neutrosophic logic can take into consideration,

whereas other methods including Fuzzy logic cannot.

In the preceding article we have discussed how Neutro-

sophic logic view can oﬀer an alternative method to solve the

well-known problem in Quantum Mechanics, i.e. the Schro¨-

dinger’s cat paradox [1, 2], by introducing indeterminacy of

the outcome of the observation.

In other article we also discuss possible re-interpretation

of quantum measurement using Uniﬁcation of Fusion Theo-

ries as generalization of Information Fusion [3, 4, 5], which

results in proposition that one can expect to neglect the prin-

ciple of “excluded middle”; therefore Bell’s theorem can be

considered as merely tautological. [6] This alternative view

of Quantum mechanics as Information Fusion has also been

proposed by G. Chapline [7]. Furthermore this Information

Fusion interpretation is quite consistent with measurement

theory of Quantum Mechanics, where the action of measure-

ment implies information exchange [8].

In the ﬁrst section we will discuss basic propositions of

Neutrosophic probability and Neutrosophic logic. Then we

discuss solution to Schro¨dinger’s cat paradox. In subsequent

section we discuss an illustration using modiﬁed coin tossing

problem, and discuss its plausible link to quantum game.

While it is known that derivation of Schro¨dinger’s equa-

tion is heuristic in the sense that we know the answer to which

the algebra and logic leads, but it is interesting that Schro¨-

dinger’s equation follows logically from de Broglie’s grande

loi de la Nature [9, p.14]. The simplest method to derive

Schro¨dinger’s equation is by using simple wave as [9]:

@2 @x2

exp(ikx)

=

k2 exp(ikx) :

(1)

By deriving twice the wave and deﬁning:

k

=

2mv h

=

mv

=

px

;

(2)

where px, represents momentum at x direction, and ratio-
nalised Planck constants respectively.
By introducing kinetic energy of the moving particle, T ,
and wavefunction, as follows [9]:

T

=

mv2 2

=

p2x 2m

=

2
2m

k2;

(3)

and

(x) = exp(ikx) :

(4)

Then one has the time-independent Schro¨dinger equation

from [1, 3, 4]:

@2 2m @x2

(x) = T

(x) :

(5)

It is interesting to remark here that by convention physicists assert that “the wavefunction is simply the mathematical function that describes the wave” [9]. Therefore, unlike the wave equation in electromagnetic ﬁelds, one should not consider that equation [5] has any physical meaning. Born suggested that the square of wavefunction represents the probability to observe the electron at given location [9, p.56]. Although Heisenberg rejected this interpretation, apparently Born’s interpretation prevails until today.
Nonetheless the founding fathers of Quantum Mechanics (Einstein, De Broglie, Schro¨dinger himself) were dissatisﬁed with the theory until the end of their lives. We can summarize the situation by quoting as follows [9, p.13]:

“The interpretation of Schro¨dinger’s wave function (and of quantum theory generally) remains a matter of continuing concern and controversy among scientists who cling to philosophical belief that the natural world is basically logical and deterministic.”

Furthermore, the “pragmatic” view of Bohr asserts that for a given quantum measurement [9, p.42]:

“A system does not possess objective values of its physical properties until a measurement of one of them is made; the act of measurement is asserted to force the system into an eigenstate of the quantity being measured.”

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In 1935, Einstein-Podolsky-Rosen argued that the axiomatic basis of Quantum Mechanics is incomplete, and subsequently Schro¨dinger was inspired to write his well-known cat paradox. We will discuss solution of his cat paradox in subsequent section.

2 Cat paradox and imposition of boundary conditions

As we know, Schro¨dinger’s deep disagreement with the Born interpretation of Quantum Mechanics is represented by his cat paradox, which essentially questioning the “statistical” interpretation of the wavefunction (and by doing so, denying the physical meaning of the wavefunction). The cat paradox has been written elsewhere [1, 2], but the essence seems quite similar to coin tossing problem:
“Given p = 0.5 for each side of coin to pop up, we
will never know the state of coin before we open our palm from it; unless we know beforehand the “state” of the coin (under our palm) using ESP-like phenomena. Prop. (1).”
The only diﬀerence here is that Schro¨dinger asserts that the state of the cat is half alive and half dead, whereas in the coin problem above, we can only say that we don’t know the state of coin until we open our palm; i.e. the state of coin is indeterminate until we open our palm. We will discuss the solution of this problem in subsequent section, but ﬁrst of all we shall remark here a basic principle in Quantum Mechanics, i.e. [9, p.45]:

“Quantum Concept: The ﬁrst derivative of the wavefunction of Schro¨dinger’s wave equation must be single-valued everywhere. As a consequence, the wavefunction itself must be single-valued everywhere.”

The above assertion corresponds to quantum logic, which can be deﬁned as follows [10, p.30; 11]:

P _Q = P +Q PQ:

(6)

As we will see, it is easier to resolve this cat paradox by releasing the aforementioned constraint of “singlevaluedness” of the wavefunction and its ﬁrst derivative. In fact, nonlinear ﬂuid interpretation of Schro¨dinger’s equation (using the level set function) also indicates that the physical meaning of wavefunction includes the notion of multivaluedness [12]. In other words, one can say that observation of
spin-half electron at location x does not exclude its possibility
to pop up somewhere else. This counter-intuitive proposition will be described in subsequent section.

3 Neutrosophic solution of the Schro¨dinger cat paradox
In the context of physical theory of information [8], Barrett has noted that “there ought to be a set theoretic language which applies directly to all quantum interactions”. This is because the idea of a bit is itself straight out of classical set

theory, the deﬁnitive and unambiguous assignment of an element of the set {0,1}, and so the assignment of an information content of the photon itself is fraught with the same difﬁculties [8]. Similarly, the problem becomes more adverse because the fundamental basis of conventional statistical theories is the same classical set {0,1}.
For example the Schro¨dinger’s cat paradox says that the quantum state of a photon can basically be in more than one place in the same time which, translated to the neutrosophic set, means that an element (quantum state) belongs and does not belong to a set (a place) in the same time; or an element (quantum state) belongs to two diﬀerent sets (two different places) in the same time. It is a question of “alternative worlds” theory very well represented by the neutrosophic set theory. In Schro¨dinger’s equation on the behavior of electromagnetic waves and “matter waves” in quantum theory, the wave function, which describes the superposition of possible states may be simulated by a neutrosophic function, i.e. a function whose values are not unique for each argument from the domain of deﬁnition (the vertical line test fails, intersecting the graph in more points).
Therefore the question can be summarized as follows [1]:
“How to describe a particle  in the inﬁnite micro-
2 2 : universe that belongs to two distinct places P1 and P2
in the same time?  P1 and  P1 is a true con-
tradiction, with respect to Quantum Concept described above.”
Now we will discuss some basic propositions in Neutrosophic logic [1].
3a Non-standard real number and subsets
 Let T,I,F be standard or non-standard real subsets ] 0, 1+[,
with sup T = t sup, inf T= t inf,
sup I = i sup, inf I = i inf,
sup F = f sup, inf F = f inf,
and n sup = t sup + i sup + f sup,
n inf = t inf + i inf + f inf.
Obviously, t sup, i sup, f sup 1+; and t inf, i inf, f inf 0, whereas n sup 3+ and n inf 0. The subsets T, I, F are not
necessarily intervals, but may be any real subsets: discrete or continuous; single element; ﬁnite or inﬁnite; union or intersection of various subsets etc. They may also overlap. These real subsets could represent the relative errors in determining t, i, f (in the case where T, I, F are reduced to points).
For interpretation of this proposition, we can use modal logic [10]. We can use the notion of “world” in modal logic, which is semantic device of what the world might have been like. Then, one says that the neutrosophic truth-value of a
statement A, NLt(A) = 1+ if A is “true in all possible
worlds.” (syntagme ﬁrst used by Leibniz) and all conjunctures, that one may call “absolute truth” (in the modal logic

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it was named necessary truth, as opposed to possible truth),
whereas NLt(A) = 1 if A is true in at least one world at some
conjuncture, we call this “relative truth” because it is related to a “speciﬁc” world and a speciﬁc conjuncture (in the modal logic it was named possible truth). Because each “world” is dynamic, depending on an ensemble of parameters, we introduce the sub-category “conjuncture” within it to reﬂect a particular state of the world.
In a formal way, let’s consider the world W as being generated by the formal system FS. One says that statement A belongs to the world W if A is a well-formed formula (wﬀ ) in W, i.e. a string of symbols from the alphabet of W that conforms to the grammar of the formal language endowing W. The grammar is conceived as a set of functions (formation rules) whose inputs are symbols strings and outputs “yes” or “no”. A formal system comprises a formal language (alphabet and grammar) and a deductive apparatus (axioms and/or rules of inference). In a formal system the rules of inference are syntactically and typographically formal in nature, without reference to the meaning of the strings they manipulate.
Similarly for the Neutrosophic falsehood-value,
NLf (A) = 1+ if the statement A is false in all possible worlds, we call it “absolute falsehood”, whereas NLf (A) = 1
if the statement A is false in at least one world, we call it “relative falsehood”. Also, the Neutrosophic indeterminacy
value NLi(A) = 1 if the statement A is indeterminate in all
possible worlds, we call it “absolute indeterminacy”, whereas
NLi(A) = 1 if the statement A is indeterminate in at least
one world, we call it “relative indeterminacy”.
3b Neutrosophic probability deﬁnition
Neutrosophic probability is deﬁned as: “Is a generalization of the classical probability in which the chance that an event A occurs is t% true — where t varies in the subset T, i% indeterminate — where i varies in the subset I, and f% false
— where f varies in the subset F. One notes that NP(A) =
(T, I, F)”. It is also a generalization of the imprecise probability, which is an interval-valued distribution function.
The universal set, endowed with a Neutrosophic probability deﬁned for each of its subset, forms a Neutrosophic probability space.
3c Solution of the Schro¨dinger’s cat paradox
Let’s consider a neutrosophic set a collection of possible lo-
cations (positions) of particle x. And let A and B be two
neutrosophic sets. One can say, by language abuse, that any
particle x neutrosophically belongs to any set, due to the per-
centages of truth/indeterminacy/falsity involved, which varies
between 0 and 1+. For example: x (0.5, 0.2, 0.3) belongs to A (which means, with a probability of 50% particle x is in a position of A, with a probability of 30% x is not in A, and the rest is undecidable); or y (0, 0, 1) belongs to A (which

normally means y is not for sure in A); or z (0, 1, 0) belongs
to A (which means one does know absolutely nothing about
z’s aﬃliation with A).
[ More general, x ((0.2–0.3), (0.40–0.45) [0.50–0.51],
{0.2, 0.24, 0.28}) belongs to the set A, which means:
— with a probability in between 20-30% particle x is in
a position of A (one cannot ﬁnd an exact approximate because of various sources used);
— with a probability of 20% or 24% or 28% x is not in A;
— the indeterminacy related to the appurtenance of x to
A is in between 40–45% or between 50–51% (limits included).
The subsets representing the appurtenance, indeterminacy,
and falsity may overlap, and n sup = 30% + 51% + 28% >
100% in this case. To summarize our proposition [1, 2], given the Schro¨-
dinger’s cat paradox is deﬁned as a state where the cat can be dead, or can be alive, or it is undecided (i.e. we don’t know if it is dead or alive), then herein the Neutrosophic logic, based on three components, truth component, falsehood component, indeterminacy component (T, I, F), works very well. In Schro¨dinger’s cat problem the Neutrosophic logic oﬀers the possibility of considering the cat neither dead nor alive, but undecided, while the fuzzy logic does not do this. Normally indeterminacy (I) is split into uncertainty (U) and paradox (conﬂicting) (P).
We have described Neutrosophic solution of the Schro¨dinger’s cat paradox. Alternatively, one may hypothesize four-valued logic to describe Schro¨dinger’s cat paradox, see Rauscher et al. [13, 14].
In the subsequent section we will discuss how this Neutrosophic solution involving “possible truth” and “indeterminacy” can be interpreted in terms of coin tossing problem (albeit in modiﬁed form), known as Parrondo’s game. This approach seems quite consistent with new mathematical formulation of game theory [20].

4 An alternative interpretation using coin toss problem

Apart from the aforementioned pure mathematics-logical approach to Schro¨dinger’s cat paradox, one can use a wellknown neat link between Schro¨dinger’s equation and FokkerPlanck equation [18]:

D

@2p @z2

@ @z

p

@p @z

@p @t

=

0

:

(7)

A quite similar link can be found between relativistic classical ﬁeld equation and non-relativistic equation, for it is known that the time-independent Helmholtz equation and Schro¨dinger equation is formally identical [15]. From this reasoning one can argue that it is possible to explain Aharonov eﬀect from pure electromagnetic ﬁeld theory; and therefore it seems also possible to describe quantum mechan-

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ical phenomena without postulating the decisive role of “observer” as Bohr asserted. [16, 17]. In idiomatic form, one can expect that quantum mechanics does not have to mean that “the Moon is not there when nobody looks at”.
With respect to the aforementioned neat link between Schro¨dinger’s equation and Fokker-Planck equation, it is interesting to note here that one can introduce “ﬁnite diﬀerence” approach to Fokker-Planck equation as follows. First, we can deﬁne local coordinates, expanded locally about a
point (z0, t0) we can map points between a real space (z; t) and an integer or discrete space (i; j). Therefore we can sam-
ple the space using linear relationship [19]:

(z; t) = (z0 + i; t0 + j ) ;

(8)

where  is the sampling length and  is the sampling time.
Using a set of ﬁnite diﬀerence approximations for the FokkerPlanck PDE:

@p @z

=

A1

=

p (z0

+ ; t0

 ) p (z0 2

; t0

) ;

(9)

@2p @z2

=

2A2

=

= p (z0

; t0

)

2p (z0; t0 2

 ) +p (z0 +; t0

 ) ; (10)

and

@p @t

=

B1

=

p

(z0;

t0)

p (z0; t0

) :

(11)

We can apply the same procedure to obtain:

@ @z

= A1

=

(z0 +; t0

 ) (z0 2

; t0

 ) : (12)

Equations (9–12) can be substituted into equation (7) to yield the required ﬁnite partial diﬀerential equation [19]:

p (z0; t0) = a 1 p (z0 ; t0  ) a0 p (z0; t0  ) +

+ a+1 p (z0 + ; t0  ) :

(13)

This equation can be written in terms of discrete space by using [8], so we have:

pi;j = a 1 pi 1;j 1 + a0 pi;j 1 + a+1 pi+1;j 1 : (14)

plied external ﬁeld.

With respect to the aforementioned Neutrosophic solu-

tion to Schro¨dinger’s cat paradox, one can introduce a new

“indeterminacy” parameter to represent conditions where the

outcome may be aﬀected by other issues (let say, apparatus

setting of Geiger counter). Therefore equation (14) can be

written as:

pi;j

=

1 2
+

a0


"
pi;j 1

pi 1;j 1 +

+


1 2

+

"


pi+1;j

1;

(17)

 where unlike the bias parameter ( 1/200), the indeterminacy

parameter can be quite large depending on the system in ques-

tion. For instance in the Neutrosophic example given above,

we can write that:

  0.2

0.3

=

k


d t


1

=

k


t d


0.50:

(18)

The only problem here is that in original coin tossing, one

cannot assert an “intermediate” outcome (where the outcome

is neither A nor B). Therefore one shall introduce modal logic

deﬁnition of “possibility” into this model. Fortunately, we

can introduce this possibility of intermediate outcome into

Parrondo’s game, so equation (17) shall be rewritten as:

pi;j

=

1 2

"

+ (2)


 pi 1;j 1 +

pi;j

1

+


1 2

+

"


 pi+1;j 1 ;

(19)

 For instance, by setting  0.25, then one gets the ﬁnite

diﬀerence equation:

pi;j = (0.25 ") pi 1;j 1 + (0.5) pi;j 1 +
+ (0.25 + ") pi+1;j 1 ; (20)
which will yield more or less the same result compared with Neutrosophic method described in the preceding section.
For this reason, we propose to call this equation (19): Neutrosophic-modiﬁed Parrondo’s game. A generalized expression of equation [19] is:

Equation (14) is precisely the form required for Parron-

do’s game. The meaning of Parrondo’s game can be described

in simplest way as follows [19]. Consider a coin tossing prob-

lem with a biased coin:

phead

=

1 2

";

(15)

where " is an external bias that the game has to “overcome”.

This bias is typically a small number, for instance 1/200. Now

we can express equation (15) in ﬁnite diﬀerence equation (14)

as follows:

pi;j

=

1 2


"

pi

1;j

1 +0 pi;j

1


+

1 2

+


"

pi+1;j

1:

(16)

Furthermore, the bias parameter can be related to an ap-

pi;j = (p0 " ) pi 1;j 1 + (z) pi;j 1 +
+ (p0 + " ) pi+1;j 1 ; (21)
where p0, z represents the probable outcome in standard coin
tossing, and a real number, respectively. For the practical
meaning of , one can think (by analogy) of this indetermi-
nacy parameter as a variable that is inversely proportional to
the “thickness ratio” (d=t) of the coin in question. Therefore using equation (18), by assuming k = 0.2, coin thickness = 1.0 mm, and coin diameter d = 50 mm, then we get d=t = 50, or  = 0.2(50) 1 = 0.004, which is negligible. But
if we use a thick coin (for instance by gluing 100 coins alto-
gether), then by assuming k = 0.2, coin thickness = 100 mm,

F. Smarandache and V. Christianto. The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited

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and coin diameter d = 50 mm, we get d=t = 0.5, or  = 0.2(0.5) 1 = 0.4, which indicates that chance to get out-
come neither A nor B is quite large. And so forth. It is worth noting here that in the language of “modal
logic” [10, p.54], the “intermediate” outcome described here
} is given name ‘possible true’, written A, meaning that “it is
not necessarily true that not-A is true”. In other word, given
that the cat cannot be found in location x, does not have to mean that it shall be in y.
Using this result (21), we can say that our proposition in the beginning of this paper (Prop. 1) has suﬃcient reasoning; i.e. it is possible to establish link from Schro¨dinger wave equation to simple coin toss problem, albeit in modiﬁed form. Furthermore, this alternative interpretation, diﬀers appreciably from conventional Copenhagen interpretation.
It is perhaps more interesting to remark here that Heisenberg himself apparently has proposed similar thought on this problem, by introducing “potentia”, which means “a world devoid of single-valued actuality but teeming with unrealized possibility” [4, p.52]. In Heisenberg’s view an atom is certainly real, but its attributes dwell in an existential limbo “halfway between an idea and a fact”, a quivering state of attenuated existence. Interestingly, experiments carried out
by J. Hutchison seem to support this view, that a piece of
metal can come in and out from existence [23]. In this section we discuss a plausible way to represent the
Neutrosophic solution of cat paradox in terms of Parrondo’s game. Further observation and theoretical study is recommended to explore more implications of this plausible link.
5 Concluding remarks
In the present paper we revisit the Neutrosophic logic view of Schro¨dinger’s cat paradox. We also discuss a plausible way to represent the Neutrosophic solution of cat paradox in terms of Parrondo’s game.
It is recommended to conduct further experiments in order to verify and explore various implications of this new proposition, including perhaps for the quantum computation theory.
Acknowledgment
The writers would like to thank to D. Rabounski for early discussion concerning interpretation of this cat paradox.
Submitted on March 31, 2008 / Accepted on April 10, 2008
References
1. Smarandache F. An introduction to the Neutrosophic probability applied in quantum physics. Bull. Pure and Appl. Sci., Physics, 2003, v. 22D, no. 1, 13–25.
2. Smarandache F. and Christianto V. The Neutrosophic logic view to Schro¨dinger’s cat paradox. Progr. in Phys., 2006, no. 2.

3. Smarandache F. and Christianto V. A note on geometric and information fusion interpretation of Bell’s theorem and quantum measurement. Progress in Physics, 2006, no. 4.
4. Smarandache F. and Christianto V. Multivalued logic, neutrosophy and Schro¨dinger equation. Hexis, Phoenix (AZ), 2006, p.52–54.
5. Smarandache F. and Dezert J. Advances and applications of DSmT for information fusion. American Research Press, Rehoboth (NM), 2004.
6. Kracklauer A. La theorie de Bell, est-elle la plus grande meprise de l’histoire de la physique? Ann. Fond. Louis de Broglie, 2000, v. 25, 193.
7. Chapline G. arXiv: adap-org/9906002; quant-ph/9912019; Granik A. and Chapline G. arXiv: quant-ph/0302013.
8. Zurek W. (ed.) Complexity, entropy and the physics of information. Addison-Wesley Publ., 1990, p.378.
9. Hunter G. Quantum chemistry: wave mechanics applied to atoms and molecules. Lecture notes. Chapter 1: Genesis of Quantum Mechanics. 2001, p.14, 26, 42.
10. deVries A. Algebraic hierarchy of logics unifying fuzzy logic and quantum logic. arXiv: math.LO/0707.2161, p.30, 54.
11. Aerts D. Description of many separated physical entities without the paradoxes encountered in Quantum Mechanics. Found. Phys., 1982, v. 12, no. 12, p.1142, 1149–1155.
12. Jin S., Liu H., Osher S., and Tsai R. Computing multivalued physical observables for the semiclassical limits of the Schro¨dinger equation. J. Comp. Phys., 2005, v. 205, 222–241.
13. Rauscher E.A. and Targ R. The speed of thought: investigation of complex spacetime metric to describe psychic phenomena. J. Scientiﬁc Exploration, 2001, v. 15, no. 3, 344–354.
14. Rauscher E.A. and Amoroso R. The physical implications of multidimensional geometries and measurement. Intern. J. Comp. Anticipatory Systems, D. Dubois (ed.), 2006.
15. Lu J., Greenleaf J., and Recami E. Limited diﬀraction solutions to Maxwell and Schro¨dinger equation. arXiv: physics/9610012.
16. Aharonov Y., et al. arXiv: quant-ph/0311155.
17. Goldstein S. Quantum theory without observers — part one. Physics Today, March 1998, 42–46.
18. Ho C.-L. and Sasaki R. Deformed Fokker Planck equations. arXiv: cond-mat/0612318.
19. Allison A., et al. State space visualization and fractal properties of Parrondo’s game. arXiv: cond-mat/0205536; cond-mat/ 0208470.
20. Wu J. A new mathematical representation of game theory. arXiv: quant-ph/0404159.
21. Smarandache F. Uniﬁcation of fusion theories (UFT). Intern. J. Appl. Math. and Stat., 2004, v. 2, 1–14.
22. Smarandache F. An in-depth look at information fusion rules and uniﬁcation of fusion theories. Invited speech at NASA Langley Research Center, Hampton, VA, USA, November 5, 2004.
23. Smarandache F., Christianto V., Khrapko R., Yuhua F., Hutchison J. Unfolding labyrinth: open problems in physics, mathematics, astrophysics, and other areas of science. Hexis, Phoenix (AZ), 2006.

20

F. Smarandache and V. Christianto. The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited

July, 2008

PROGRESS IN PHYSICS

Volume 3

A Classical Model of Gravitation

Pieter Wagener
Department of Physics, NMMU South Campus, Port Elizabeth, South Africa E-mail: Pieter.Wagener@nmmu.ac.za
A classical model of gravitation is proposed with time as an independent coordinate. The dynamics of the model is determined by a proposed Lagrangian. Applying the canonical equations of motion to its associated Hamiltonian gives conservation equa-
tions of energy, total angular momentum and the z component of the angular momen-
tum. These lead to a Keplerian orbit in three dimensions, which gives the observed values of perihelion precession and bending of light by a massive object. An expression for gravitational redshift is derived by accepting the local validity of special relativity at all points in space. Exact expressions for the GEM relations, as well as their associated Lorentz-type force, are derived. An expression for Mach’s Principle is also derived.

1 Introduction
The proposed theory is based on two postulates that respectively establish the dynamics and kinematics of a system of particles subject to a gravitational force. The result is a closed particle model that satisﬁes the basic experimental observations of the force.
The details of applications and all derivations are included in the doctoral thesis of the author [1].

2 Postulates

The model is based on two postulates:

Postulate 1: The dynamics of a system of particles subject

to gravitational forces is determined by the Lagrangian,

L=

m0(c2

+

v2)

exp

R r

;

(1)

where m0 is gravitational rest mass of a test body mov-

ing at velocity v
body of mass M ,

in

=the1=vpici1nity

of a massive, central
v2=c2, R = 2GM=c2

is the Schwarzschild radius of the central body.

Postulate 2: Special Relativity (SR) is valid instantan-eously and locally at all points in the reference system of the central massive body. This gives the kinematics of the system.

3 Conservation equations

Applying the canonical equations of motion to the Hamilto-

nian, derived from the Lagrangian, leads to three conservation

equations:
E=

m0c2

eR=r
2

= total energy = constant ;

(2)

L = eR=r M ;

(3)

= total angular momentum = constant ;

Lz = eR=r m0r2 sin2 _ ;

(4)

= z component of L = constant ;

where M = (r m0v). Equations (2), (3) and (4) give the
quadrature of motion:

d du

=

 e2Ru L2

u2

EeRu  L2

1=2
;

(5)

where u = 1=r, L =jLj and is deﬁned by

jMj

= m0r2

d dt

:

(6)

Expanding the exponential terms to second degree yields

a diﬀerential equation of generalized Keplerian form,

d du

= (au2 + bu + c)

1=2;

(7)

where

u

=

1 r

a

=

R2(4 2L2

E)

b

=

R(2 L2

E)

c

=

1

E L2

9

1

>>>>>>>>>>>>>=

>>>>>>>>>>>>>; ;

(8)

and the convention m0 = c = 1 was used.
Integrating (7) gives the orbit of a test particle as a gener-

alized conic,

u = K(1 + cos k ) ;

(9)

where the angles are measured from = 0, and

k = ( a) 12 ;

(10)

K=

b 2a

;

(11)


=1

4ac b


21

:

(12)

Pieter Wagener. A Classical Model of Gravitation

21

Volume 3

PROGRESS IN PHYSICS

July, 2008


7 Lorentz-type force equation

The corresponding force equation is found from the associated Euler-Lagrange equations:

p_ = Em + m0v H ;

(16)

c

r0


M
c

deﬂection
=2

where

p = m0 r_ = m0v ;

(17)

m

=

m0
2

;

(18)

E=

^r

GM r2

;

(19)

H

=

GM (v c2r3

r) :

(20)


Fig. 1: Deﬂection of light.

The force equation shows the deviation from Newton’s law of gravitation. The above equations are analogous to the gravitoelectromagnetic (GEM) equations derived by Mashhoon [2] as a lowest order approximation to Einstein’s ﬁeld
equations for v  c and r  R.

4 Gravitational redshift

Assuming the validity of d = dt of SR at each point in
space and taking frequencies as the inverses of time, (2) yields

 = 0 e R=2r (0 = constant),

(13)

which, to ﬁrst approximation in exp( R=2r), gives the ob-
served gravitational redshift.

5 Perihelion precession

In the case of an ellipse ( < 1), the presence of the coeﬃcient k causes the ellipse not to be completed after a cycle of  = 2 radians, i.e. the perihelion is shifted through a cer-
tain angle. This shift, or precession, can be calculated as (see

Appendix 9):


=

a

3R (1

2)

;

(14)

where a is the semi-major axis of the ellipse. This expression

gives the observed perihelion precession of Mercury.

8 Mach’s Principle

An ad hoc formulation for Mach’s Principle has been pre-

sented as [3, 4]

G

Lc2 M

;

(21)

where: L = radius of the universe,

M = mass of the universe mass of the distant stars.

This relation can be found by applying the energy relation

of (2) to the system of Fig. 2.

distant stars
M2

M1 L
©

6 Deﬂection of light

We deﬁne a photon as a particle for which v = c. From (2) it follows that E = 0 and the eccentricity of the conic section
is found to be (see Appendix 9)

=

r0 R

;

(15)

where r0 is the impact parameter. Approximating r0 by the radius of the sun, it follows that > 1. From Fig. 1 we see
that the trajectory is a hyperbola with total deﬂection equal to
2R=r0. This is in agreement with observation.

Fig. 2: Mutual gravitational interaction between a central mass M1 and the distant stars of total mass M2.

The potential at M2 due to M1 is and the potential of the shell at M1 is

12==GGMM21==LL==RR21cc22==22LL.

Furthermore, since M1 and M2 are in relative motion, the

value of will be the same for both of them. Applying (2)

to the mutual gravitational interaction between the shell of

distant stars and the central body then gives

E

=

M1c2

exp

R2 L

=

M2c2

exp

R1 L

:

22

Pieter Wagener. A Classical Model of Gravitation

July, 2008

PROGRESS IN PHYSICS

Volume 3

 Since L > R2 R1 we can realistically approximate the  exponential to ﬁrst order in R2=L. After some algebra we get
R2 L, which gives the Mach relation,

2GM2 Lc2


1:

9 Comparison with General Relativity

The equations of motion of General Relativity (GR) are approximations to those of the proposed Lagrangian. This can be seen as follows.
The conservation equations of (2), (3) and (4) can also be derived from a generalized metric,
ds2 = e R=rdt2 eR=r(dr2 + r2d2 + r2 sin2 d2): (22)

Comparing this metric with that of GR,


ds2 = 1

R r


dt2

1

1

R r

dr2

r2d2

r2 sin2 d2; (23)

we note that (23) is a ﬁrst order approximation to the time and radial coeﬃcients, and a zeroth order approximation to the angular coeﬃcients of (22). It implies that all predictions of GR will be accommodated by the Lagrangian of (1) within the orders of approximation.
Comparing (5) with the corresponding quadrature of GR,

d du

=

1

E J2

+

uRE J2

 1=2
u2 + Ru3 ;

(24)

we note that it diﬀers from the Newtonian limit, or the Keple-
rian form of (7), by the presence of the Ru3 term. The form
of this quadrature does not allow the conventional Keplerian
orbit of (9).

where a = (r+ + r )=2 is the semi-major axis of the approx-
imate ellipse. Substituting these values in (8) gives

a=

3R a(1

2)

1:

(27)

Substituting this value in (25) gives (14).

A.2 Deﬂection of light

We ﬁrst have to calculate the eccentricity

this case,


=1

4ac b2

1=2

:

For a photon, setting v = c in (8) gives


2=

1

+

L2 R2


:

of the conic for (28)

At the distance of closest approach, r = r0 = 1=u0, we have d=du = 0; so that from (5):

L2

=

e2Ru0 u20

=

r02 e2R=r0:

(29)

From (28) and (29), and ignoring terms of ﬁrst and higher

order in R=r0, we ﬁnd


r0 R

:

(30)

For a hyperbola cos  = 1= , so that (see Fig. 1):

sin = 1=
)  1= ) 2  2R=r0 = total deﬂection.

Submitted on April 07, 2008 Accepted on April 10, 2008

Appendix A.1 Precession of the perihelion

After one revolution of 2 radians, the perihelion of an el-

lipse given by the conic of (9) shifts through an angle  =

=

2 k

2 or, from (10), as

 = 2 ( a) 1=2 1 ;

(25)

where a is given by (8). The constants of motion E and L

are found from the boundary conditions of the system, i.e.

du=d = 0 at u = 1=r and 1=r+, where r+ and r are the

maximum and minimum radii respectively of the ellipse. We

ﬁnd [1]

E


1

+

R 2a

R2 L2


2R a (1

2)

9 >>=
;
>>;

(26)

References
1. Wagener P. C. Principles of a theory of general interaction. PhD thesis, University of South Africa, 1987. An updated version is to be published as a book during 2008.
2. Mashhoon. Gravitoelectromagnetism: a brief review. In L. Iorio, editor, Measuring Gravitomagnetism: A Challenging Enterprise, pages 29–39. Nova Publishers, 2007. arXiv: gr-qc/ 0311030.
3. Chiu H.-Y. and Hoﬀmann W. P. (eds.). Gravitation and Relativity, Ch. 7., W. A. Benjamin Inc., New York, 1964.
4. Sciama D. W. The physical foundations of General Relativity. Doubleday, 1969.

Pieter Wagener. A Classical Model of Gravitation

23

Volume 3

PROGRESS IN PHYSICS

July, 2008

On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly

Abraham A. Ungar
Dept. of Mathematics, North Dakota State University, Fargo, North Dakota 58105-5075, USA E-mail: Abraham.Ungar@ndsu.edu
Einstein’s special relativity is a theory rich of paradoxes, one of which is the recently discovered Relativistic Invariant Mass Paradox. According to this Paradox, the relativistic invariant mass of a galaxy of moving stars exceeds the sum of the relativistic invariant masses of the constituent stars owing to their motion relative to each other. This excess of mass is the mass of virtual matter that has no physical properties other than positive relativistic invariant mass and, hence, that reveals its presence by no means other than gravity. As such, this virtual matter is the dark matter that cosmologists believe is necessary in order to supply the missing gravity that keeps galaxies stable. Based on the Relativistic Invariant Mass Paradox we oﬀer in this article a model which quantiﬁes the anomalous acceleration of Pioneer 10 and 11 spacecrafts and other deep space missions, and explains the presence of dark matter and dark energy in the universe. It turns out that the origin of dark matter and dark energy in the Universe lies in the Paradox, and that the origin of the Pioneer anomaly results from neglecting the Paradox. In order to appreciate the physical signiﬁcance of the Paradox within the frame of Einstein’s special theory of relativity, following the presentation of the Paradox we demonstrate that the Paradox is responsible for the extension of the kinetic energy theorem and of the additivity of energy and momentum from classical to relativistic mechanics. Clearly, the claim that the acceleration of Pioneer 10 and 11 spacecrafts is anomalous is incomplete, within the frame of Einstein’s special relativity, since those who made the claim did not take into account the presence of the Relativistic Invariant Mass Paradox (which is understandable since the Paradox, published in the author’s 2008 book, was discovered by the author only recently). It remains to test how well the Paradox accords with observations.

1 Introduction

Einstein’s special relativity is a theory rich of paradoxes, one

of which is the Relativistic Invariant Mass Paradox, which

was recently discovered in [1], and which we describe in Sec-

tion 5 of this article. The term mass in special relativity usu-

ally refers to the rest mass of an object, which is the Newto-

nian mass as measured by an observer moving along with the

object. Being observer’s invariant, we refer the Newtonian,

rest mass to as the relativistic invariant mass, as opposed to

the common relativistic mass, which is another name for en-

ergy, and which is observer’s dependent. Lev B. Okun makes

the case that the concept of relativistic mass is no longer even

pedagogically useful [2]. However, T. R. Sandin has argued

otherwise [3].

As we will see in Section 5, the Relativistic Invariant

Mass Paradox asserts that the resultant relativistic invariant

mass m0 of a system S of uniformly moving N particles ex-

ceeds the
1; : : : ; N,

osufmitsocfonthsetitrueelanttivpiasrttiiccliensv,amria0n>t mPasNks=es1

mk, mk,

k=
since

the contribution to m0 comes not only from the masses mk

of the constituent particles of S but also from their speeds

relative to each other. The resulting excess of mass in the

resultant relativistic invariant mass m0 of S is the mass of
virtual matter that has no physical properties other than positive relativistic invariant mass and, hence, that reveals itself by no means other than gravity. It is therefore naturally identiﬁed as the mass of virtual dark matter that the system
S possesses. The presence of dark matter in the universe
in a form of virtual matter that reveals itself only gravitationally is, thus, dictated by the Relativistic Invariant Mass Paradox of Einstein’s special theory of relativity. Accordingly, (i) the fate of the dark matter particle(s) theories as well as (ii) the fate of their competing theories of modiﬁed Newtonian dynamics (MOND [4]) are likely to follow the fate of the eighteenth century phlogiston theory and the nineteenth century luminiferous ether theory, which were initiated as ad hoc postulates and which, subsequently, became obsolete.
Dark matter and dark energy are ad hoc postulates that account for the observed missing gravitation in the universe and the late time cosmic acceleration. The postulates are, thus, a synonym for these observations, as C. La¨mmerzahl, O. Preuss and H. Dittus had to admit in [5] for their chagrin. An exhaustive review of the current array of dark energy theories is presented in [6].

24

Abraham A. Ungar. On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly

July, 2008

PROGRESS IN PHYSICS

Volume 3

The Pioneer anomaly is the anomalous, unmodelled acceleration of the spacecrafts Pioneer 10 and 11, and other spacecrafts, studied by J. D. Anderson et al in [7] and summarized by S. G. Turyshev et al in [8]. In [7], Anderson et al compared the measured trajectory of a spacecraft against its theoretical trajectory computed from known forces acting on the spacecraft. They found the small, but signiﬁcant discrepancy known as the anomalous, or unmodelled, acceleration directed approximately towards the Sun. The inability to explain the Pioneer anomaly with conventional physics has contributed to the growing interest about its origin, as S. G. Turyshev, M. M. Nieto and J. D. Anderson pointed out in [9]. It is believed that no conventional force has been overlooked [5] so that, seemingly, new physics is needed. Indeed, since Anderson et al announced in [7] that the Pioneer 10 and 11 spacecrafts exhibit an unexplained anomalous acceleration, numerous articles appeared with many plausible explanations that involve new physics, as C. Castro pointed out in [10].
However, we ﬁnd in this article that no new physics is needed for the explanation of both the presence of dark matter/energy and the appearance of the Pioneer anomaly. Rather, what is needed is to cultivate the Relativistic Invariant Mass Paradox, which has recently been discovered in [1], and which is described in Section 5 below.
Accordingly, the task we face in this article is to show that the Relativistic Invariant Mass Paradox of Einstein’s special relativity dictates the formation of dark matter and dark energy in the Universe and that, as a result, the origin of the Pioneer anomaly stems from the motions of the constituents of the Solar system relative to each other.

2 Einstein velocity addition vs. Newton velocity addition

The improved way to study Einstein’s special theory of rela-

tivity, oﬀered by the author in his recently published book [1],

enables the origin of the dark matter/energy in the Universe

and the Pioneer anomaly to be determined. The improved

study rests on analogies that Einsteinian mechanics and its

underlying hyperbolic geometry share with Newtonian me-

chanics and its underlying Euclidean geometry. In particu-

lar, it rests on the analogies that Einsteinian velocity addition

shares with Newtonian velocity addition, the latter being just the common vector addition in the Euclidean 3-space R3.

Einstein addition of R3,

is a binary operation in the ball R3c

R3c = fv 2 R3 : kvk < cg

(1)

of all relativistically admissible velocities, where c is the
speed of light in empty space. It is given by the equation

u

v

=

1

1

+

uv
c2

 u

+

1
u

v

+

1 c2

1

u
+

u

(u


v)u

(2)

where u is the gamma factor

v=

r
1

1
kvk2
c2

(3)

k k itnhaRt t3ch,eabnadllwRh3ceirneheraintsdfrom

are the inner its space R3.

product and norm Counterintuitively,

Einstein addition is neither commutative nor associative.

Einstein gyrations gyr[u; v] 2 Aut(R3c; ) are deﬁned by

the equation

gyr[u; v]w = (u v) (u (v w))

(4)

2 for all
of the

u; v; w
Einstein

groRu3cp,oaidnd(Rth3ce;y

turn out to be automorphisms
). We recall that a groupoid is

a non-empty space with a binary operation, and that an au-

2 fttohof(emuRgo3cyrpvroah)ntiist=oomnfisto(sofuefl)fathtghefra(oEtvuirn)pesofstoipedreinca(tlRgsl r3cuto;h;uevp)obiiidnsRa(ra3cRy.3coTo;nopee-e)rtmoaat-priooehnnaae,suitmtzhoeaamtpthoiarsf-t,

phisms of the groupoid, gyrations are also called gyroauto-

morphisms.

Thus, gyr[u; v] of the deﬁnition in (4) is the gyroautomor-
2 puwh;(iviusnmR(3cvoRifn3c ,twotthh)te)hateiEntraieRnklsae3ctts.eivitnhisetigrceraollaluytpivoaidisdmtici(asRslli3cyb;laedv)me, liosgsceiibtnyleeravt(eeuldocivbtyy) The gyrations, which possess their own rich structure,

measure the extent to which Einstein addition deviates from

commutativity and associativity as we see from the following

identities [1, 11, 12]:

u v = gyr[u; v](v u) u (v w) = (u v) gyr[u; v]w (u v) w = u (v gyr[u; v]w) gyr[u; v] = gyr[u v; v] gyr[u; v] = gyr[u; v u]

Gyrocommutative Law Left Gyroassociative Right Gyroassociative Left Loop Property Right Loop Property

Einstein addition is thus regulated by its gyrations so that Einstein addition and its gyrations are inextricably linked. In-
deed, the Einstein groupoid (R3c ; ) forms a group-like math-
ematical object called a gyrocommutative gyrogroup [13], which was discovered by the author in 1988 [14]. Interestingly, Einstein gyrations are just the mathematical abstraction of the relativistic Thomas precession [1, Sec. 10.3].
The rich structure of Einstein addition is not limited to its gyrocommutative gyrogroup structure. Einstein addition admits scalar multiplication, giving rise to the Einstein gyrovector space. The latter, in turn, forms the setting for the Beltrami-Klein ball model of hyperbolic geometry just as vector spaces form the setting for the standard model of Euclidean geometry, as shown in [1].
Guided by the resulting analogies that relativistic mechanics and its underlying hyperbolic geometry share with classical mechanics and its underlying Euclidean geometry, we

Abraham A. Ungar. On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly

25

Volume 3

PROGRESS IN PHYSICS

July, 2008

are able to present analogies that Newtonian systems of particles share with Einsteinian systems of particles in Sections 3 and 4. These analogies, in turn, uncover the Relativistic Invariant Mass Paradox in Section 5, the physical signiﬁcance of which is illustrated in Section 6 in the frame of Einstein’s special theory of relativity. Finally, in Sections 7 and 8 the Paradox reveals the origin of the dark matter/energy in the Universe as well as the origin of the Pioneer anomaly.

3 Newtonian systems of particles

In this section we set the stage for revealing analogies that a
Newtonian system of N particles and an Einsteinian system of N particles share. In this section, accordingly, as opposed Rto3S, eacntdiomn 04,isvtkh,ekN=ew0t;o1n;i:a:n:r;eNsu,latarnetNmeawstsoonfiathnevceoloncsittiiteuseinnt masses mk, k = 1; : : : ; N of a Newtonian particle system S.
Accordingly, let us consider the following well known classical results, (6) – (8) below, which are involved in the
calculation of the Newtonian resultant mass m0 and the clas-
sical center of momentum (CM) of a Newtonian system of particles, and to which we will seek Einsteinian analogs in Section 4. Thus, let

S = S(mk; vk; 0; N) ;

vk 2 R3

(5)

be an isolated Newtonian system of N noninteracting material particles the k-th particle of which has mass mk and
Newtonian uniform velocity vk relative to an inertial frame
0, k = 1; : : : ; N. Furthermore, let m0 be the resultant mass of S, considered as the mass of a virtual particle located at the center of mass of S, and let v0 be the Newtonian velocity relative to 0 of the Newtonian CM frame of S. Then,

1

=

1 m0

X N k=1

mk

(6)

and

v0

=

1 m0

X N
mkvk
k=1

u

+

v0

=

1 m0

X N
mk(u
k=1

+ vk)

9 >>>>>=
;
>>>>>;

(7)

u; vk2R3, mk > 0, k = 0; 1; : : : ; N. Here m0 is the Newto-
nian mass of the Newtonian system S, supposed concentrated at the center of mass of S, and v0 is the Newtonian velocity
relative to 0 of the Newtonian CM frame of the Newtonian
system S in (5). It follows from (6) that m0 in (6) – (7) is given by the
Newtonian resultant mass equation

X N

m0 = mk :

(8)

k=1

The derivation of the second equation in (7) from the ﬁrst equation in (7) is immediate, following (i) the distributive law of scalar-vector multiplication, and (ii) the simple relation-
ship (8) between the Newtonian resultant mass m0 and its constituent masses mk, k = 1; : : : ; N.

4 Einsteinian systems of particles

In this section we present the Einsteinian analogs of the New-

tonian expressions (5) – (8) listed in Section 3. The presented

analogs are obtained in [1] by means of analogies that result

from those presented in Section 2.

k

In this section, accordingly, as opposed
= 0; 1; : : : ; N, are Einsteinian velocities

to in

Section R3c , and

3, vk,
m0 is

the Einsteinian resultant mass, yet to be determined, of the

masses mk, k = 1; : : : ; N, of an Einsteinian particle system S.

In analogy with (5), let

S = S(mk; vk; 0; N);

vk 2 R3c

(9)

be an isolated Einsteinian system of N noninteracting material particles the k-th particle of which has invariant mass mk and Einsteinian uniform velocity vk relative to an inertial frame 0, k = 1; : : : ; N. Furthermore, let m0 be the resultant mass of S, considered as the mass of a virtual particle located at the center of mass of S (calculated in [1, Chap. 11]),

and let v0 be the Einsteinian velocity relative to 0 of the Einsteinian center of momentum (CM) frame of the Einsteinian
system S in (9). Then, as shown in [1, p. 484], the relativistic

analogs of the Newtonian expressions in (6) – (8) are, respec-

tively, the following Einsteinian expressions in (10) – (12),

v0

=

1 m0

X N
mk
k=1

vk

9 >>>>>=

u

v0

=

1 m0

X N
mk
k=1

u

vk

>>>>>;

(10)

and

v0 v0 =

1 m0

X N k=1

mk

vk vk

u

v0 (u

v0)

=

1 m0

X N
mk
k=1

u

vk (u

vk)

9 >>>>>=
;
>>>>>;

(11)

u; vk2R3c, mk > 0, k = 0; 1; : : : ; N. Here m0,

m0

= v u u u u t

X N !2 X N
mk + 2 mjmk(

k=1

j;k=1

j<k

vj vk

1) (12)

is the relativistic invariant mass of the Einsteinian system S, supposed concentrated at the relativistic center of mass of S

26

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(calculated in [1, Chap. 11]), and v0 is the Einsteinian ve- MATLAB) the identities in (10) – (12) in order to gain con-

locity relative to 0 of the Einsteinian CM frame of the Ein- ﬁdence in their validity, before embarking on reading several

steinian system S in (9).

necessary chapters of [1].

5 The relativistic invariant mass paradox of Einstein’s 6 The physical signiﬁcance of the paradox in Einstein’s

special theory of relativity

special theory of relativity

In analogy with the Newtonian resultant mass m0 in (8),
which follows from (6), it follows from (10) that the Ein-
steinian resultant mass m0 in (10) – (11) is given by the
elegant Einsteinian resultant mass equation (12), as shown in [1, Chap. 11].
The Einsteinian resultant mass equation (12) presents a Paradox, called the Relativistic Invariant Mass Paradox, since, in general, this equation implies the inequality

X N

m0 > mk

(13)

k=1

so that, paradoxically, the invariant resultant mass of a system

may exceed the sum of the invariant masses of its constituent

particles.

The paradoxical invariant resultant mass equation (12) for

m0 is the relativistic analog of the non-paradoxical Newtonian resultant mass equation (8) for m0, to which it reduces
in each of the following two special cases:

(i ) The Einsteinian resultant mass m0 in (12) reduces to

! 1 the Newtonian resultant mass m0 in (8) in the limit as

c

; and

(ii ) The Einsteinian resultant mass m0 in (12) reduces to the Newtonian resultant mass m0 in (8) in the special case when the system S is rigid, that is, all the internal motions in S of the constituent particles of S relative to each other vanish. In that case vj vk = 0 so that vj vk = 1 for all j; k = 1; N . This identity, in turn,
generates the reduction of (12) to (8).

The second equation in (11) follows from the ﬁrst equation in (11) in full analogy with the second equation in (7), which follows from the ﬁrst equation in (7) by the distributivity of scalar multiplication and by the simplicity of (8). However, while the proof of the latter is simple and well known, the proof of the former, presented in [1, Chap. 11], is lengthy owing to the lack of a distributive law for the Einsteinian scalar multiplication (see [1, Chap. 6]) and the lack of
a simple relation for m0 like (8), which is replaced by (12).
Indeed, the proof of the former, that the second equation in (11) follows from the ﬁrst equation in (11), is lengthy, but accessible to undergraduates who are familiar with the vector space approach to Euclidean geometry. However, in order to follow the proof one must familiarize himself with a large part of the author’s book [1] and with its “gyrolanguage”, as indicated in Section 2.
It is therefore suggested that interested readers may corroborate numerically (using a computer software like

In this section we present two classically physical signiﬁcant results that remain valid relativistically owing to the Relativistic Invariant Mass Paradox, according to which the rel-
ativistic analog of the classical resultant mass m0 in (8) is, paradoxically, the relativistic resultant mass m0 in (12).
To gain conﬁdence in the physical signiﬁcance that results from the analogy between
(i ) the Newtonian resultant mass m0 in (8) of the Newtonian system S in (5) and
(ii) the Einsteinian invariant resultant mass m0 in (12) of the Einsteinian system S in (9)

we present below two physically signiﬁcant resulting analogies. These are:

(1) The Kinetic Energy Theorem [1, p. 487]: According to

this theorem,

where

K = K0 + K1 ;

(14)

(i) K0 is the relativistic kinetic energy, relative to a
given observer, of a virtual particle located at the
relativistic center of mass of the system S in (9), with the Einsteinian resultant mass m0 in (12);
and

(ii) K1 is the relativistic kinetic energy of the constituent particles of S relative to its CM; and
(iii) K is the relativistic kinetic energy of S relative to
the observer.

The Newtonian counterpart of (14) is well known; see, for instance, [15, Eq. (1.55)]. The Einsteinian analog in (14) was, however, unknown in the literature since the
Einsteinian resultant mass m0 in (12) was unknown in
the literature as well till its ﬁrst appearance in [1]. Accordingly, Oliver D. Johns had to admit for his chagrin that “The reader (of his book; see [15, p. 392]) will be disappointed to learn that relativistic mechanics does not have a theory of collective motion that is as elegant and complete as the one presented in Chapter 1 for Newtonian mechanics.”
The proof that m0 of (12) is compatible with the va-
lidity of (14) in Einstein’s special theory of relativity is presented in [1, Theorem 11.8, p. 487].

(2) Additivity of Energy and Momentum: Classically, en-
ergy and momentum are additive, that is, the total en-
ergy and the total momentum of a system S of particles
is, respectively, the sum of the energy and the sum of
momenta of its constituent particles. Consequently,

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also the resultant mass m0 of S is additive, as shown
in (8). Relativistically, energy and momentum remain
additive but, consequently, the resultant mass m0 of S
is no longer additive. Rather, it is given by (12), which is the relativistic analog of (8).
The proof that m0 of (12) is compatible with the ad-
ditivity of energy and momentum in Einstein’s special theory of relativity is presented in [1, pp. 488–491].

Thus, the Einsteinian resultant mass m0 in (12) of the Einsteinian system S in (9) is the relativistic analog of the

Newtonian resultant mass m0 in (8) of the Newtonian system S in (5). As such, it is the Einsteinian resultant mass m0 in
(12) that is responsible for the extension of the validity of (14)

and of the additivity of energy and momentum from classical

to relativistic mechanics.

However, classically, mass is additive. Indeed, the New-

ttohneicaonnrsetistuuletannt tpamrtaiscslems, 0me0qu=alsPtNkh=e 1smumk,

of as

the we

masses of see in (8).

Relativistically, in contrast, mass is not additive. Indeed, the

mEiansssteesinoiafnthreescuolntasntittumenatsspamrt0iclmesa,yme0xceedPtNkh=e1smumk,

of as

the we

see from (12). Accordingly, from the relativistic viewpoint,

the resultant mass m0 in (12) of a galaxy that consists of
stars that move relative to each other exceeds the sum of

the masses of its constituent stars. This excess of mass re-

veals its presence only gravitationally and, hence, we iden-

tify it as the mass of dark matter. Dark matter is thus vir-

tual matter with positive mass, which reveals its presence

only gravitationally. In particular, the dark mass mdark of the Einsteinian system S in (9), given by (16) below, is the

mass of virtual matter called the dark matter of S. To con-

trast the real matter of S with its virtual, dark matter, we

call the former bright (or, luminous, or, baryonic) matter.

The total mass m0 of S, which can be detected gravitationally, is the composition of the bright mass mbright of the real, bright matter of S, and the dark mass mdark of the virtual, dark matter of S. This mass composition, presented in

(15) – (17) in Section 7 below, quantiﬁes the eﬀects of dark

matter.

7 The origin of the dark matter

the equation

q

m0 = m2bright + m2dark

(17)

The mass mbright in (15) is the Newtonian resultant mass of the particles of the Einsteinian system S in (9). These par-

ticles reveal their presence gravitationally, as well as by radi-

ation that they may emit and by occasional collisions.

In contrast, the mass mdark in (16) is the mass of virtual matter in the Einsteinian system S in (9), which reveals its

presence only gravitationally. In particular, it does not emit

radiation and it does not collide. As such, it is identiﬁed with

the dark matter of the Universe.

In our expanding universe, with accelerated expansion

[16], relative velocities between some astronomical objects

are signiﬁcantly close to the speed of light c. Accordingly,

2 1 sloincciteiegsavmmaRf3cacatpoprrsoavchatphperosapcehed

when their relative veof light, it follows from

(16) that dark matter contributes an increasingly signiﬁcant

part of the mass of the universe.

8 The origin of the dark energy
Under diﬀerent circumstances dark matter may appear or disappear resulting in gravitational attraction or repulsion. Dark matter increases the gravitational attraction of the region of each stellar explosion, a supernova, since any stellar explosion creates relative speeds between objects that were at rest relative to each other prior to the explosion. The resulting generated relative speeds increase the dark mass of the region, thus increasing its gravitational attraction. Similarly, relative speeds of objects that converge into a star vanish in the process of star formation, resulting in the decrease of the dark mass of a star formation region. This, in turn, decreases the gravitational attraction or, equivalently, increases the gravitational repulsion of any star formation inﬂated region. The increased gravitational repulsion associated with star formation results in the accelerated expansion of the universe, ﬁrst observed in 1998; see [6, p. 1764], [17] and [18, 19]. Thus, according to the present special relativistic dark matter/energy model, the universe accelerated expansion is a late time cosmic acceleration that began at the time of star formation.

Let

X N

mbright = mk

(15)

k=1

and

mdark

=

v u u u u t2

X N j;k=1

mj

mk(

vj vk

1)

(16)

j<k

so that the Einsteinian resultant mass m0 in (12) turns out to be a composition of an ordinary, bright mass mbright of real matter and a dark mass mdark of virtual matter according to

9 The origin of the Pioneer anomaly
The Einsteinian resultant mass m0 of our Solar system is given by the composition (17) of the bright mass mbright and the dark mass mdark of the Solar system. The bright mass mbright of the Solar system equals the sum of the Newtonian
masses of the constituents of the Solar system. Clearly, it is
time independent. In contrast, the dark mass mdark of the
Solar system stems from the speeds of the constituents of the Solar system relative to each other and, as such, it is time dependent.

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The Pioneer 10 and 11 spacecrafts and other deep space missions have revealed an anomalous acceleration known as the Pioneer anomaly [7, 8]. The Pioneer anomaly, described in the introductory section, results from an unmodelled acceleration, which is a small constant acceleration on top of which there is a smaller time dependent acceleration. A brief summary of the Pioneer anomaly is presented by K. Tangen, who asks in the title of [20]: “Could the Pioneer anomaly have a gravitational origin?”
Our answer to Tangen’s question is aﬃrmative. Our dark matter/energy model, governed by the Einsteinian resultant
mass m0 in (15) – (17), oﬀers a simple, elegant model that
explains the Pioneer anomaly. The motion of any spacecraft in deep space beyond the Solar system is determined by the Newtonian law of gravity where the mass of the Solar sys-
tem is modelled by the Einsteinian resultant mass m0 in (17) rather than by the Newtonian resultant mass m0 in (8). It is the contribution of the dark mass mdark to the Einsteinian resultant mass m0 in (15) – (17) that generates the Pioneer
anomaly. Ultimately, our dark matter/energy model, as dictated by
the paradoxical Einsteinian resultant mass m0 in (12), will
be judged by how well the model accords with astrophysical and astronomical observations. Since our model is special relativistic, only uniform velocities are allowed. Hence, the model can be applied to the solar system, for instance, under the assumption that, momentarily, the solar system can be viewed as a system the constituents of which move uniformly.
Submitted on April 13, 2008 Accepted on April 15, 2008
References

9. Turyshev S. G., Nieto M. M., and Anderson J. D. Lessons learned from the pioneers 10/11 for a mission to test the pioneer anomaly. arXiv: gr-qc/0409117.
10. Castro C. (Anti) de Sitter relativity, modiﬁed Newtonian dynamics, noncommutative phase spaces and the Pioneer anomaly. Adv. Stud. Theor. Phys., 2008, v. 2(7), 309–332.
11. Ungar A. A. Beyond the Einstein addition law and its gyroscopic Thomas precession: The theory of gyrogroups and gyrovector spaces. Vol. 117 of Fundamental Theories of Physics, Dordrecht, Kluwer Academic Publishers Group, 2001.
12. Ungar A. A. Analytic hyperbolic geometry: Mathematical foundations and applications. World Scientiﬁc Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.
13. Ungar A. A. Thomas precession and its associated grouplike structure. Amer. J. Phys., 1991, v. 59, no. 9, 824–834.
14. Ungar A. A. Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett., 1988, v. 1, no. 1, 57–89.
15. Johns O. D. Analytical mechanics for relativity and quantum mechanics. New York, Oxford University Press, 2005.
16. Brax P. The cosmological constant problem. Contemp. Physics, 2004, v. 45, no. 3, 227–236.
17. Kirshner R. P. The extravagant universe: exploding stars, dark energy, and the accelerating cosmos. Princeton (NJ), Princeton University Press, 2000.
18. Ries A. G. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J., 1998, v. 116, 1009–1038.
19. Perlmutter S. Measurements of ! and  from 42 high-redshift
supernovae. Astrophys. J., 1999, v. 517, 565–586.
20. Tangen K. Could the Pioneer anomaly have a gravitational origin? arXiv: gr-qc/0602089.

1. Ungar A. A. Analytic hyperbolic geometry and Albert Einstein’s special theory of relativity. World Scientiﬁc Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
2. Okun L. B. The concept of mass. Phys. Today, 1989, v. 46(6), 31–36.
3. Sandin T. R. In defense of relativistic mass. Amer. J. Phys., 1991, v. 59(11), 1032–1036.
4. Bekenstein J. D. The modiﬁed Newtonian dynamics — MOND and its implications for new physics. Contemp. Physics, 2007, v. 47(6), 387–403.
5. La¨mmerzahl C., Preuss O., and Dittus H. Is the physics within the Solar system really understood? arXiv: gr-qc/0604052.
6. Copeland E. J., Sami M., and Tsujikawa S. Dynamics of dark energy. Intern. J. Modern Phys. D, 2006, v. 15, no. 11, 1753– 1935.
7. Anderson J. D., Laing P. A., Lau E. L., Liu A. S., Nieto M. M., and Turyshev S. G. Study of the anomalous acceleration of pioneer 10 and 11. Phys. Rev. D, 2002, v. 65, no. 8, 082004, 1–50.
8. Turyshev S. G., Toth V. T., Kellogg L. R., Lau E. L., and Lee K. J. The study of the Pioneer anomaly: new data and objectives for new investigation. arXiv: gr-qc/0512121.

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A Critical Analysis of Universality and Kirchhoﬀ’s Law: A Return to Stewart’s Law of Thermal Emission
Pierre-Marie Robitaille
Dept. of Radiology, The Ohio State University, 130 Means Hall, 1654 Upham Drive, Columbus, Ohio 43210, USA E-mail: robitaille.1@osu.edu
It has been advanced, on experimental (P.-M. Robitaille, IEEE Trans. Plasma Sci., 2003, v. 31(6), 1263–1267) and theoretical (P.-M. Robitaille, Progr. Phys., 2006, v. 2, 22–23) grounds, that blackbody radiation is not universal and remains closely linked to the emission of graphite and soot. In order to strengthen such claims, a conceptual analysis of the proofs for universality is presented. This treatment reveals that Gustav Robert Kirchhoﬀ has not properly considered the combined eﬀects of absorption, reﬂection, and the directional nature of emission in real materials. In one instance, this leads to an unintended movement away from thermal equilibrium within cavities. Using equilibrium arguments, it is demonstrated that the radiation within perfectly reﬂecting or arbitrary cavities does not necessarily correspond to that emitted by a blackbody.

1 Introduction
Formulated in 1858, Stewart’s Law [1] states that when an object is studied in thermal equilibrium, its absorption is equal to its emission [1]. Stewart’s formulation leads to the realization that the emissive power of any object depends on its temperature, its nature, and on the frequency of observation. Conversely, Gustav Kirchhoﬀ [2–4] reaches the conclusion that the emissive power of a body is equal to a universal function, dependent only on its temperature and the frequency of interest, and independent of its nature and that of the enclosure. He writes: “When a space is surrounded by bodies of the same temperature, and no rays can penetrate through these bodies, every pencil in the interior of the space is so constituted, with respect to its quality and intensity, as if it proceeded from a perfectly black body of the same temperature, and is therefore independent of the nature and form of the bodies, and only determined by the temperature” (see [4], p. 96–97).
At the same time, Max Planck, in his Theory of Heat Radiation, reminds us that: “. . . in a vacuum bounded by totally reﬂecting walls any state of radiation may persist” (see [5], §51). Planck is aware that a perfect reﬂector does not necessarily produce blackbody radiation in the absence of a perfect absorber [6]. It is not simply a matter of waiting a sufﬁcient amount of time, but rather the radiation will “persist” in a non-blackbody, or arbitrary, state. Planck re-emphasizes this aspect when he writes: “Every state of radiation brought about by such a process is perfectly stationary and can continue inﬁnitely long, subject, however, to the condition that no trace of an emitting or absorbing substance exists in the radiation space. For otherwise, according to Sec. 51, the distribution of energy would, in the course of time, change through the releasing action of the substance irreversibly, i.e., with an increase of the total entropy, into the stable distribu-

tion corresponding to black radiation” (see [5], §91). Planck suggests that if an absorbing substance is present, blackbody radiation is produced. Such a statement is not supported scientiﬁcally. In fact, a perfect absorber, such as graphite or soot, is required [6–8].
Recently, I have stated [6–8] that cavity radiation was not universal and could only assume the normal distribution (i.e. that of the blackbody) when either the walls of the cavity, or the objects it contains, were perfectly absorbing. These ideas are contrary to the expressed beliefs of Kirchhoﬀ and Planck. Therefore, they deserve further exposition by revisiting Kirchhoﬀ’s basis for universality. In combination with a historical review of blackbody radiation [8], such an analysis demonstrates that claims of universality were never justiﬁed [6–8].
2.1 Kirchhoﬀ’s ﬁrst treatment of his law
Kirchhoﬀ’s ﬁrst presentation of his law [2] involved two
plates, C and c, placed before one another (see Fig. 1). Nei-
ther plate was perfectly absorbing, or black. Behind each
plate, there were mirrors, R and r; which ensured that all the
radiation remained between the plates. Kirchhoﬀ assumed
that one of the plates, c, was made of a special material which
absorbed only one wavelength and transmitted all others. This assumption appears to have formed the grounds for the most strenuous objections relative to Kirchhoﬀ’s ﬁrst derivation [9–11]. Kirchhoﬀ moved to insist (see [9] for a treatment in English) that, under these conditions, at a certain temperature and wavelength, all bodies had the same ratio of emissive and absorptive powers.
The fallacy with Kirchhoﬀ’s argument lays not only in the
need for a special material in the second plate, c, as so many
have hinted [9–11]. The most serious error was that he did not consider the reﬂection from the plates themselves. He treated

30 Pierre-Marie Robitaille. A Critical Analysis of Universality and Kirchhoﬀ’s Law: A Return to Stewart’s Law of Thermal Emission

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Fig. 1: Schematic representation of Kirchhoﬀ’s ﬁrst proof [2]. C and c represented objects of a speciﬁed nature (see text). R and r corresponded to perfectly reﬂecting mirrors. Note that Kirchhoﬀ Rha0dannedgrle0.cted the reﬂection from the surfaces of C and c denoted as

the reﬂection as coming only from the mirrors placed behind

the plates. But this dealt with the problem of transmission,

not reﬂection. As a result, Kirchhoﬀ ignored the reﬂection

produced by the surfaces of the plates.

The total radiation leaving from the surface of each plate,

given thermal equilibrium, is obtained, not only by its emis-

sion, E e), and

r(eoﬂreec)ti,obnu,tRra0 t(hoerr

rb0y).

the It

sum of is only

its emission, E
when the plates

(or are

black that surface reﬂection can be neglected. Consequently,

aifddKrierscshehdoﬀ(Ri0n=sisrts0 =th0a)t,

surface reﬂection itself he simply proves that

need not the ratio

be of

emission to absorption is the same for all blackbodies, not

for all bodies. The entire argument, therefore, is ﬂawed be-

cause Kirchhoﬀ ignored the surface reﬂection of each plate,

and is considering all reﬂection as originating from the per-

fectly reﬂecting mirrors behind the plates. A proper treat-

ment from

would not lead
plate C was E

to
+

Run0 ivneortsasilmityp,lysinEc,e

wthheetroetaRl 0raddeinaotitoens

the tal

reﬂection from surface C radiation from plate c was

(es+eerF0,ign.o1t).simSipmlyilaer,lyw, htheeretor-0

denotes the reﬂection from surface c. The mirrors, R and r,

are actually dealing only with transmission through plates C

and c. The conceptual diﬃculty when reviewing this work is

that Kirchhoﬀ apparently treats reﬂection, since mirrors are

present. In fact, he dismisses the issue. The mirrors cannot

treat the reﬂection oﬀ the surfaces of C and c. They deal with

transmission. Kirchhoﬀ’s incorrect visualization of the eﬀect

of reﬂection is also a factor in his second proof.

2.2 Kirchhoﬀ’s second treatment of his law
Kirchhoﬀ’s second treatment of his law [3, 4] is much more interesting conceptually and any error will consequently be more diﬃcult to locate. The proof is complex, a reality rec-

Fig. 2: Schematic representation of Kirchhoﬀ’s second proof [3, 4]. The cavity contained three openings, labeled 1, 2, and 3. There was
also a plate, P , which was perfectly transmitting for the frequency
and polarization of interest, and perfectly reﬂecting for all others. While the existence of such a plate can be the source of objections relative to Kirchhoﬀ’s proof [10], the discussion in this work does not center on the nature of the plate. Idealized objects can be assumed as valid as they represent (more or less) mathematical exten-
sions of physical observations (see text). A black screen, S, was
used to prevent radiation from traveling directly between openings 1 and 3. An object, which was either perfectly absorbing or arbitrary, was placed in the enclosure located behind opening 1. The key to Kirchhoﬀ’s proof relied on rapidly changing the covering of opening 3, from a perfect concave mirror to a perfectly absorbing surface. In Kirchhoﬀ’s initial presentation, the entire cavity was perfectly absorbing [3, 4]. However, Kirchhoﬀ extended his result to be independent of the nature of the walls, making it acceptable to consider the entire cavity as perfectly reﬂecting (see text).
ognized by Stewart in his Reply: “I may remark, however, that the proof of the Heidelburg Professor is so very elaborate that I fear it has found few readers either in his own country or in this” [12].
Kirchhoﬀ began by imagining a cavity whose walls were perfectly absorbing (see Fig. 2). In the rear of the cavity was an enclosure wherein the objects of interest were placed. There were three openings in the cavity, labeled 1, 2, and 3. He conceived that openings 2 and 3 could each be sealed with a perfectly absorbing surface. As a result, when Kirchhoﬀ did this, he placed his object in a perfectly absorbing cavity [6]. He eventually stipulated that the experiment was independent of the nature of the walls, in which case the cavity could be viewed as perfectly reﬂecting [6]. Yet, as has been previously highlighted [6], the scenario with the perfectly reﬂecting cavity required, according to Planck, the introduction of a minute particle of carbon [5, 8]. Hence, I have argued that Kirchhoﬀ’s analysis was invalid on this basis alone [6]. By carefully considering Kirchhoﬀ’s theoretical constructs, the arguments against blackbody radiation, within a perfect reﬂecting enclosure, can now be made from a slightly diﬀerent perspective.
Kirchhoﬀ’s analysis of his cavity (see Fig. 2) was ingenious. He set strict conditions for the positions of the walls

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which linked the openings 1 and 2, and which contained opening 3. The key was in the manner wherein opening 3 was handled. Kirchhoﬀ permitted opening 3 to be covered either with a perfect absorber or with a perfect concave mirror. He then assumed that equilibrium existed in the cavity and that he could instantaneously change the covering at opening 3. Since equilibrium was always preserved, Kirchhoﬀ could then treat the rays within the cavity under these two diﬀerent conditions and, hence, infer the nature of the radiation within the cavity at equilibrium.
Kirchhoﬀ initially demonstrated that, if the enclosed object and the cavity were perfectly absorbing, the radiation was denoted by the universal function of blackbody radiation. He then replaced the object with an arbitrary one, and concluded, once again, that the radiation was black. Kirchhoﬀ’s presentation was elegant, at least when the cavity was perfectly absorbing. The Heidelburg Professor extended his ﬁndings to make them independent of the nature of the walls of the enclosure, stating that the derivation was valid, even if the walls were perfectly reﬂecting. He argued that the radiation within the cavity remained blackbody radiation. Let us revisit what Kirchhoﬀ had done.
Since the walls can be perfectly reﬂecting, this state is adopted for our analysis. Opening 3 can once again be covered, either by a concave mirror or by a perfectly absorbing surface. An arbitrary object, which is not a blackbody, is placed in the cavity. The experiment is initiated with the perfect concave mirror covering opening 3. As shown in Section 3.1.2, under these conditions, the cavity contains radiation whose nature depends not on the cavity, but on the object. This radiation, in fact, is not black. This can be seen, if the object was taken as perfectly reﬂecting. The arbitrary radiation is weaker at all frequencies. Thus, when an arbitrary object is placed in the enclosure, the intensity of the radiation within the cavity, at any given frequency, does not correspond to that predicted by the Planckian function (see Section 3.1.2). However, when opening 3 is covered by a perfectly absorbing substance, the radiation in the cavity becomes black (see Sections 3.1.2 and 3.2). The emission from the object is that which the object emits and which it reﬂects. The latter originates from the surface of opening 3 (see Section 3.2). When the perfect absorber is placed over opening 3, the entire cavity appears to hold blackbody radiation. Therefore, by extending his treatment to the perfect reﬂector, Kirchhoﬀ is inadvertently jumping from one form of cavity radiation (case 1: the concave mirror, object radiation) to another (case 2: the perfect absorber, blackbody radiation) when the covering on opening 3 is changed. At that moment, the cavity moves out of equilibrium.
Thus, Kirchhoﬀ’s proof is invalid. This is provided, of course, that the test began with the perfect concave mirror covering opening 3. Only under these circumstances would Kirchhoﬀ’s proof fail. Nonetheless, the experimental proof cannot be subject to the order in which manipulations are ex-

ecuted. This is because the validity of equilibrium arguments is being tested. Consequently, nothing is independent of the nature of the walls. This is the lesson provided to us by Balfour Stewart in his treatise when he analyzes radiation in a cavity temporarily brought into contact with another cavity [8]. Dynamic changes, not equilibrium, can be produced in cavities, if reﬂectors are used. This is the central error relative to Kirchhoﬀ’s second attempt at universality [3, 4].
There are additional minor problems in Kirchhoﬀ’s presentation [3, 4]. In §13 of his proof [3, 4], Kirchhoﬀ is examining an arbitrary object within a perfectly absorbing cavity. It is true that the resultant cavity radiation will correspond to a blackbody, precisely because the walls are perfectly absorbing (see Section 3.1.1). However, Kirchhoﬀ states: “the law §3 is proved under the assumption that, of the pencil which falls from surface 2 through opening 1 upon the body C, no ﬁnite part is reﬂected by this back to the surface 2; further, that the law holds without limitation, if we consider that when the condition is not fulﬁlled, it is only necessary to turn the body C inﬁnitely little in order to satisfy it, and that by such a rotation the quantities E and A undergo only and [sic] inﬁnitely small change” (see [4], p. 92). Of course, real bodies can have diﬀuse reﬂection. In addition, rotation does not ensure that reﬂection back to surface 2 will not take place. Real bodies also have directional spectral emission, such that the eﬀect of rotation on E and A is not necessarily negligible. These complications are of little signiﬁcance within a perfectly absorbing cavity. The radiation within such enclosures is always black (see Section 3.1.1). Conversely, the problems cannot be dismissed in the perfect reﬂector and the entire proof for universality, once again, is invalid.
For much of the 19th century, the understanding of blackbody radiation changed little, even to the time of Planck [11]. No laboratory proof of Kirchhoﬀ’s Law was ever produced, precisely because universality could not hold. Only theoretical arguments prevailed [10]. Yet, such ﬁndings cannot form the basis for a law of physics. Laws stem from experiments and are fortiﬁed by theory. They are not born de novo, using mathematics without further validation. It is not possible to ensure that black radiation exists, within a perfectly reﬂecting cavity, without recourse at least to a carbon particle [6, 8]. In fact, this is the route which Planck utilized in treating Kirchhoﬀ’s Law [5, 8].
3 Thermal equilibrium in cavities
A simple mathematical treatment of radiation, under conditions of thermal equilibrium, begins by examining the fate of the total incoming radiation, , which strikes the surface of an object. The various portions of this radiation are either absorbed (A), reﬂected (R), or transmitted (T) by the object. If normalized, the sum of the absorbed, reﬂected, or trans-
mitted radiation is equal to +  +  = 1. Here, absorptivity,
, corresponds to the absorbed part of the incoming radia-

32 Pierre-Marie Robitaille. A Critical Analysis of Universality and Kirchhoﬀ’s Law: A Return to Stewart’s Law of Thermal Emission

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tion/total incoming radiation. Similarly, the reﬂectivity, , is

the reﬂected part of the incoming radiation/total incoming ra-

diation. Finally, the transmissivity,  , involves the transmitted

part of the incoming radiation/total incoming radiation. If all

objects under consideration are fully opaque, then 1 = + .

Stewart’s Law [1] states that, under conditions of thermal

equilibrium, the ability of an object to absorb light, , is ex-

actly equal to its ability to emit light, ". Nonetheless, for this

presentation, Stewart’s Law is not assumed to be valid [1].

The question arises only in the ﬁnal Section 4.2, when two

objects are placed within a perfectly reﬂecting cavity. Emis-

sivity, ", is standardized relative to lamp-black [8] and, for

such a blackbody, it is equal to 1. For a perfect reﬂector, the

emissivity, ", is 0. All other objects hold values of emissiv-

ity between these two extremes. If thermal equilibrium is not

established, then " and are not necessarily equal [8]. If a cubical cavity is considered with walls P 1, P 2, P 3,
P 4, P 5 (top surface), and P 6 (bottom surface), the following can be concluded at thermal equilibrium: since P 1 and P 3

are equal in area and opposite one another, then the total radi-

ation from these walls must be balanced, p1 p3 = 0. Sim-

ilarly, p2 p4 = 0 and p5 p6 = 0. As such, p1 = p3

and p2 = p4. If one considers pairs of adjacent walls, then

( p1 + p2) ( p3 + p4) = 0. It is possible to conclude that

p1 = p2 = p3 = p4 and, using symmetry, it can ﬁnally be

concluded that
quently, with
+ p4 + p5 +

npp6o1)r=.mFaolpirz2aa=ntioonpp,a3q=uecc=pa4v=16it(y,

p5 = p6. Consep1 + p2 + p3 +
the total radiation

coming from the cavity, T , is given by T = "c c + c c =

= "c c + (1 c) c. This states that the total emission from

the cavity must be represented by the sum of its internal emis-

sion and reﬂection. If the cavity is constructed from perfectly

absorbing walls, c = 1, c = 0, yielding T = "c c. The cavity is black and "c must now equal 1, by necessity. Stew-

art’s Law [1] has now been proved for blackbodies. If the cav-

ity is made from perfectly reﬂecting walls, at thermal equi-

librium, "c c + (1 c) c = 0. There is also no source of radiation inside the cavity ("c = 0) and (1 c) c = 0, leading explicitly to c = 0. Because c = 0, the total radiation monitored T = "c c + c c = 0.

These conclusions can be extended to perfectly absorb-

ing and reﬂecting cavities of rectangular (or arbitrary) shapes.

The central point is that a perfectly reﬂecting cavity can sus-

tain no radiation, a ﬁrst hint that universality cannot be valid.

Planck only obtains blackbody radiation, in such cavities, by

invoking the action of a carbon particle [6, 8]. This special

case will be treated in Sections 3.1.1 and 3.2.

3.1 An object in a perfect cavity

At thermal equilibrium, the total emission from the surface

of the object, so, is equal to that from the surface of the

cavity, sc. therefore be

When normalizing, the total emission,

as follows:

T

=

1 2

so

+

1 2

sc. The

T , will total ra-

diation from the surface of the object is equal to that which

it emits plus that which it reﬂects, so = ["o o + o c], and

similarly for the surface of the cavity, sc = ["c c + c o].

Therefore, at equilibrium, ["o o + o c] = ["c c + c o] or

o["o c] = c["c o]. Solving for either o or c, we

obtain that

o=

c

["c ["o

o] c]

and

c=

o

["o ["c

c o

] ]

.

3.1.1 An arbitrary object in a perfectly absorbing cavity

InTsu=ch12ac"aosec"[[c""co=

1,
o] c]

c = 0. + o

Since

c

+

1 2

T
"c

=
c

1 2
+

so
c

+

1 2

c

["c ["o

sc, o] c]

then . It

is readily shown that T = c. Note that no use of Stewart’s

Law [1] was made in this derivation. In any case, when an

object is placed within a cavity, which is perfectly absorbing,

the emitted spectrum is independent of the object and depends

only on the nature of the cavity. A blackbody spectrum is pro-

duced. This was the condition which prevailed over much of

the 19th century when cavities were often lined with soot [8].

If the radiation was independent of the nature of the walls, or

of the object, it was because the walls were coated with this

material [8].

3.1.2 An arbitrary object in a perfectly reﬂecting cavity

In such acase

T

=

1 2

"o

"c = 0, o + o

c =

o

["o ["c

1.ocS]] inc+e

T =

1 2

"c

1 2 o

so

+

1 2

sc,

["o ["c

c] o]

+

c

then o.

It is readily shown that T = o. Note, once again, that no

use of Stewart’s Law [1] was made in this derivation. When

an object is placed within a cavity which is perfectly reﬂect-

ing, the emitted spectrum is determined only by the object

and is independent of the nature of the cavity. If the object

is perfectly absorbing, like a carbon particle [6, 8], a black-

body spectrum will be obtained. Furthermore, if an arbitrary

object is placed within a cavity, which is perfectly reﬂecting,

the emitted spectrum is dependent only on the nature of the

object. One observes object radiation, not blackbody radi-

ation, because the object was never black a priori. This is

the condition which Kirchhoﬀ has failed to realize when he

extended his treatment to be independent of the nature of the

walls in his 1860 proof [3, 4], as seen in Section 2.

3.1.3 An arbitrary object in an arbitrary cavity

Co+rTo12nas=lti"edcren21racstu+i"vcoehlycao,g+cwe[[n""ecoeorhaaloocvc]][[""eacos.eI.TnocS]]t=ihnic+s12eca21"sTeo=,"tcch21[[e""ocoe[[s""xoocpoc+r]]eo+cs12]]si+oosncs,cctchaoen+n-

not

+

1 2

be further simpliﬁed and sc, can be maintained.

the initial form,

T

=

1 2

so +

Therefore, the total radiation

emitted from such a cavity is a mixture depending on both

the characteristics of the object and the walls of the cavity.

Pierre-Marie Robitaille. A Critical Analysis of Universality and Kirchhoﬀ’s Law: A Return to Stewart’s Law of Thermal Emission 33

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PROGRESS IN PHYSICS

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This highlights that cavities do not always contain black radiation and that universality is invalid [6–8].

3.2 An arbitrary object and a carbon particle in a perfectly reﬂecting cavity

If thermal equilibrium exists between an opaque object, o, a

carbon particle, p, and a cavity, c, then ["o o+o p+o c]

["p p + p o + p c] + ["c c + c o c p] = 0. Since

the cavity is perfectly reﬂecting, c = 0, "c = 0, and c = 1,

yielding, "o o + o P "p p p o + o p = 0, and

with rearrangement, ("o +o 1) p "p p +(1 p) o = 0.

If we take Stewart’s Law ("p = p; "o = o) as valid [1], we

can see that "o + o = 1, and then (1 p) o = "p p,

leading directly to o = p. Alternatively, we may no-

tice that, by deﬁnition, o = 1 o and p = 1 p, then,

o = ("p

"o+
p

o)

p. If we take the particle to be black, we

can simplify to o = (1 "o + o) p. Therefore, if we then

observe the radiation in the cavity and ﬁnd it to be black, since

the particle is also black, Stewart’s law is veriﬁed. This is be-

cause o will be black and equal to p only when "o = o.
The problem can be examined from a slightly diﬀerent

angle in order to yield a little more insight, but the same con-

clusions hold. Because the objects are in a perfect reﬂector,

then the radiation coming oﬀ their surfaces can be expressed

as so = "o o + o p and sp = "p p + p o. Given
thermal equilibrium, the production of radiation from each

object must be equal, so = sp, and thus "o o + o p =

= "p p +
tion 3.1).

p
If

o. the

Consequently, o particle is black,

= "p

["p
=["1o

o] p] and

p (see Sec-
p = 0, and

o=
must

(1b"eoop) ropd.uAcisnagresaulttooftatlheremmailsesqiounilibwrihuimch, theapopbejeacrst

black in nature. o must equal p. All solutions involve
o + "o = 1, which as stated above, is a proof of Stewart’s Law ("o = o). The object takes the appearance of a
blackbody through the sum of its emission and reﬂection. The

presence of completely black radiation within a cavity ﬁlled

in this manner constitutes an explicit veriﬁcation of Stewart’s

Law [1], as mentioned above. Since such cavities are known

to be black, Stewart’s Law has been proven. In fact, we have

returned to the ﬁrst portion of Section 3.1.2. The eﬀect is the

same as if the walls of the cavity were perfectly absorbing.

This is the point Planck failed to realize when he placed the

carbon particle within the perfectly reﬂecting cavity and gave

it a catalytic function [5, 6, 8].

4 Conclusions
Nearly 150 years have now passed since Gustav Robert Kirchhoﬀ ﬁrst advanced his Law of Thermal Radiation. Kirchhoﬀ’s Law [2–4] was far reaching. Its universal nature had a profound eﬀect on the scientists of the period. At the time, many of these men were trying to discover the most

general laws of nature. Hence, the concept of universality had great appeal and became ingrained in the physics literature. As a result, Kirchhoﬀ’s Law has endured, despite controversy [10], until this day. Recently, I have questioned universality [6, 7]. It is doubtful that Kirchhoﬀ’s Law can long survive the careful discernment of those physicists who wish to further pursue this issue.
At the same time, Kirchhoﬀ’s Law seems inseparably tied to Max Planck’s equation [13]. As such, could a reevaluation of Kirchhoﬀ’s ideas compromise those of Max Planck [13]? In the end, it is clear that this cannot be the case [8]. Planck’s solution to the blackbody problem remains valid for cavities which are perfectly absorbing. Thus, physics loses nothing
of the Planck and Boltzmann constants, h and k, which were
born from the study of heat radiation [1, 8]. That blackbody radiation loses universal signiﬁcance also changes nothing, in fact, relative to the mathematical foundations of quantum theory. However, the same cannot be said relative to experimental ﬁndings [8]. In the end, the physics community may well be led to reconsider some of these positions [8].
Balfour Stewart [1] preceded Kirchhoﬀ [2–4] by nearly two years in demonstrating, under equilibrium, the equality between absorptivity and emissivity. Stewart’s treatment, unlike Kirchhoﬀ’s, does not lead to universality [1, 8, 9, 14] but, rather, shows that the emissive power of an object is dependent on its nature, its temperature, and the frequency of observation. This is true even within cavities, provided that they do not contain a perfect absorber. It is only in this special circumstance that the nature of the object is eliminated from the problem. Yet, this is only because the nature of the carbon itself controls the situation. Stewart also properly treats emission and reﬂection in his Treatise [14]. Despite popular belief to the contrary [9], Stewart’s interpretation is the correct solution. Conversely, Kirchhoﬀ’s formulation, not only introduced error, but provided justiﬁcation for setting temperatures inappropriately. I have repeatedly expressed concern in this area [6–8]. It can be argued that Stewart’s analysis lacked mathematical sophistication [9]. Stewart himself [12] counters the point [8]. Nonetheless, it is doubtful that the important consequences of Stewart’s work can continue to be ignored. Justice and the proper treatment of experimental data demand otherwise.
Acknowledgements
The author would like to thank Luc Robitaille for assistance in ﬁgure preparation.
Dedication
This work is dedicated to the memory of my beloved mother, Jacqueline Alice Roy (May 12, 1935 – December 02, 1996).
Submitted on May 06, 2008 / Accepted on May 09, 2008 First published online on May 12, 2008 / Corrected on June 25, 2008

34 Pierre-Marie Robitaille. A Critical Analysis of Universality and Kirchhoﬀ’s Law: A Return to Stewart’s Law of Thermal Emission

July, 2008

PROGRESS IN PHYSICS

References
1. Stewart B. An account of some experiments on radiant heat, involving an extension of Pre´vost’s theory of exchanges. Trans. Royal Soc. Edinburgh, 1858, v. 22(1), 1–20. (Also found in Harper’s Scientiﬁc Memoirs, edited by J. S. Ames: The Laws of Radiation and Absorption: Memoirs of Pre´vost, Stewart, Kirchoﬀ, and Kirchoﬀ and Bunsen, translated and edited by D. B. Brace, American Book Company, New York, 1901, 21–50).
2. Kirchhoﬀ G. U¨ ber den Zusammenhang zwischen Emission und Absorption von Licht und. Wa¨rme. Monatsberichte der Akademie der Wissenschaften zu Berlin, sessions of Dec. 1859, 1860, 783–787.
3. Kirchhoﬀ G. U¨ ber das Verha¨ltnis zwischen dem Emissionsvermo¨gen und dem Absorptionsvermo¨gen. der Ko¨rper fu¨r Wa¨rme und Licht. Poggendorfs Annalen der Physik und Chemie, 1860, v. 109, 275–301. (English translation by F. Guthrie: Kirchhoﬀ G. On the relation between the radiating and the absorbing powers of diﬀerent bodies for light and heat. Phil. Mag., 1860, ser. 4, v. 20, 1–21).
4. Kirchhoﬀ G. On the relation between the emissive and the absorptive power of bodies for light and heat. (Reprinted from Investigations of the Solar Spectrum and the Spectra of the Chemical Elements, 2nd Edition, Berlin, Ferd. Dummler’s Publishing House, 1866, Gesammelte Abhandlungen, 571–598, Leipzig, 1882 as found in Harper’s Scientiﬁc Memoirs, edited by J. S. Ames: The Laws of Radiation and Absorption: Memoirs of Pre´vost, Stewart, Kirchoﬀ, and Kirchoﬀ and Bunsen, translated and edited by D. B. Brace, American Book Company, New York, 1901, 73–97).
5. Planck M. The theory of heat radiation. P. Blakiston’s Son & Co., Philadelphia, PA, 1914.
6. Robitaille P. M. L. On the validity of Kirchhoﬀ’s law of thermal emission. IEEE Trans. Plasma Sci., 2003, v. 31(6), 1263–1267.
7. Robitaille P. M. L. An analysis of universality in blackbody radiation. Prog. in Phys., 2006, v. 2, 22–23.
8. Robitaille P. M. L. Blackbody radiation and the carbon particle. Prog. in Phys., 2008, v. 3, 36.
9. Siegel D. M. Balfour Stewart and Gustav Robert Kirchhoﬀ: two independent approaches to Kirchhoﬀ’s law. Isis, 1976, v. 67(4), 565–600.
10. Schirrmacher A. Experimenting theory: the proofs of Kirchhoﬀ’s radiation law before and after Planck. Munich Center for the History of Science and Technology, 2001. http://mzwtg. mwn.de/arbeitspapiere/Schirrmacher 2001 1.pdf
11. Cotton A. The present status of Kirchhoﬀ’s law. Astrophys. J., 1899, v. 9, 237–268.
12. Stewart B. Reply to some remarks by G. Kirchhoﬀ in his paper “On the History of Spectrum Analysis”. Phil. Mag., 1863, ser. 4, v. 25, 354–360.
13. Planck M. U¨ ber das Gesetz der Energieverteilung im Normalspektrum. Annalen der Physik, 1901, v. 4, 553–563.
14. Stewart B. An elementary treatise on heat. Clarendon Press, Oxford, U.K., 1888 (online: http://books.google.com).

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Blackbody Radiation and the Carbon Particle
Pierre-Marie Robitaille
Dept. of Radiology, The Ohio State University, 130 Means Hall, 1654 Upham Drive, Columbus, Ohio 43210, USA
E-mail: robitaille.1@osu.edu
Since the days of Kirchhoﬀ, blackbody radiation has been considered to be a universal process, independent of the nature and shape of the emitter. Nonetheless, in promoting this concept, Kirchhoﬀ did require, at the minimum, thermal equilibrium with an enclosure. Recently, the author stated (P.-M. Robitaille, IEEE Trans. Plasma Sci., 2003, v. 31(6), 1263–1267; P.-M. Robitaille, Progr. in Phys., 2006, v. 2, 22–23), that blackbody radiation is not universal and has called for a return to Stewart’s law (P.-M. Robitaille, Progr. in Phys., 2008, v. 3, 30–35). In this work, a historical analysis of thermal radiation is presented. It is demonstrated that soot, or lampblack, was the standard for blackbody experiments throughout the 1800s. Furthermore, graphite and carbon black continue to play a central role in the construction of blackbody cavities. The advent of universality is reviewed through the writings of Pierre Pre´vost, Pierre Louis Dulong, Alexis The´re`se Petit, Jean Baptiste Joseph Fourier, Sime´on Denis Poisson, Fre´de´rick Herve´ de la Provostaye, Paul Quentin Desain, Balfour Stewart, Gustav Robert Kirchhoﬀ, and Max Karl Ernst Ludwig Planck. These writings illustrate that blackbody radiation, as experimentally produced in cavities and as discussed theoretically, has remained dependent on thermal equilibrium with at least the smallest carbon particle. Finally, Planck’s treatment of Kirchhoﬀ’s law is examined in detail and the shortcomings of his derivation are outlined. It is shown once again, that universality does not exist. Only Stewart’s law of thermal emission, not Kirchhoﬀ’s, is fully valid.

1 Introduction
If real knowledge is to be derived from an equation, it is often necessary to reassess the experiments that gave it life. A thorough evaluation of these developments, relative to Planck’s equation [1, 2], can be found in Hans Kangro’s Early History of Planck’s Radiation Law [3]. Kangro reminds us of the need to study important milestones relative to physical ideas: “Only concern with details appearing in sources reveals — often unexpectedly — what has really happened historically, and allowed something to be divined from that history as to ‘how it really happened’” [3; p. 3]. He then sets forth a fascinating account of the history of the law [1, 2] which gave birth to modern physics. Kangro’s work [3] is unique for its balance relative to experimental methods and theoretical foundations. It covers, in considerable detail, the period from Kirchhoﬀ to Planck [3]. Hoﬀmann’s work [4] is also valuable since it is short, well written, and reviews the experiments from which Planck formulated his equation [1, 2]. Kuhn’s text [5] centers on the theoretical basis of Planck’s law. It has been the subject of substantial justiﬁed criticism, primarily for advancing that Planck was not the ﬁrst to introduce quantized processes [6–8]. It is by using such works, and the collection of the scientiﬁc literature, that we may revisit the days of Planck [9–16] and judge, with perhaps greater insight than our forefathers, the soundness of the claims on which universality in blackbody radiation rests.

At the onset, it should be emphasized that the validity of Planck’s equation [1, 2], as a mathematical solution to the blackbody problem, is not being disputed in any way. The accuracy and merit of Planck’s equation [1, 2] has been established beyond question. Nonetheless, two aspects of Planck’s formulation are being brought to the forefront. First, that Planck [1, 2, 9–16], Einstein [17, 18], and all of physics have yet to ascribe a direct physical process for the production of blackbody radiation [19]. That is to say, blackbody radiation remains unlinked to a speciﬁc and identiﬁable physical entity (such as the nucleus, the electron, etc). Second, that blackbody radiation is not universal, contrary to what Kirchhoﬀ has concluded [20–22] and Planck believed [1, 2, 9].
I have previously stated that Kirchhoﬀ’s law [20–22], and, as a necessary result, Planck’s law [1, 2] and blackbody radiation, are not universal in nature [23–25]. Kirchhoﬀ’s conclusions hold only for objects in thermal equilibrium with a perfectly absorbing enclosure [23]. Under these conditions, Kirchhoﬀ’s cavities act, in essence, as transformers of light [23]. Any object placed within them will give a total emission which is the sum of its own emission and the reﬂection of the emission from the cavity wall. Consequently, the entire cavity appears black [23, 25]. Outside the restrictions imposed by such a cavity, universality does not exist [23–25]. As for Kirchhoﬀ’s law, it holds only under very limited experimental conditions: the walls of these cavities, or the objects they contain, must be perfectly absorbing (see [25] for a proof).

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Otherwise, Kirchhoﬀ’s law in its widest sense (i.e. universality) does not hold [23]. However, that section of Kirchhoﬀ’s law speciﬁcally addressing the equality between emissivity and absorptivity at equilibrium is valid. This is Stewart’s law [26], not Kirchhoﬀ’s [20–22], as will be seen below.
In Planck’s words (see [9; §44]), Kirchhoﬀ’s law of thermal emission holds that: “With these assumptions, according to equations (46), (45), and (43), Kirchhoﬀ’s law holds,
E=A = I = d cos  d K d, i.e., the ratio of the emissive
power to the absorbing power of any body is independent of the nature of the body”. The implications of Kirchhoﬀ’s law are best summarized in the words of its originator: “When a space is surrounded by bodies of the same temperature, and no rays can penetrate through these bodies, every pencil in the interior of the space is so constituted, with respect to its quality and intensity, as if it proceeded from a perfectly black body of the same temperature, and is therefore independent of the nature and form of the bodies, and only determined by the temperature. . . In the interior of an opaque glowing hollow body of given temperature there is, consequently, always the same brightness whatever its nature may be in other respects” [22; §17]. Kirchhoﬀ’s law states that, for all bodies, the ratio of emissive to absorbing power is a function of only wavelength and temperature, given thermal equilibrium with an enclosure. All that Kirchhoﬀ knew about his universal function, in 1859, was that its value was zero in the visible range at low temperatures, non-zero at high temperatures, and nonzero at the longer wavelengths at all temperatures [3; p. 7]. Planck [1, 2], in 1900, eventually deﬁned the function on the right side of Kirchhoﬀ’s law [20–22].
Given thermal equilibrium within an enclosure, Kirchhoﬀ’s law [20–22] states that the ability of an object to emit a photon is equal to its ability to absorb one. This aspect of Kirchhoﬀ’s work [20–22], properly called Stewart’s law [25, 26], is not being questioned. If equilibrium holds, the equality between emissivity and absorptivity has been experimentally demonstrated (see [25] for a complete discussion). It is only when objects are permitted to radiate freely, that equality may fail. Discussions on this issue have been published [27–29]. It has been argued that the equality between absorptivity and emissivity may, in fact, still be applicable for freely radiating bodies, provided that “the distribution over material states is the equilibrium condition” [27]. At the same time, it should be realized that, under all non-equilibrium conditions, these laws collapse [20–22, 25, 26].
The vast experimental knowledge relative to thermal emission reveals that virtually all materials fall far short of exhibiting blackbody behavior. Yet, Max Thiesen, a pupil of Kirchhoﬀ, in 1900 stated that: “we have become accustomed to treat radiation independently of the emitting body” and therefore, this radiation should “be designated simply as black radiation” [3; p. 184]. Experimental reality illustrates that nothing in nature behaves like a blackbody. Kirchhoﬀ’s statement that: “In the interior of an opaque glowing hol-

low body of given temperature there is, consequently, always the same brightness whatever its nature may be in other respects” [22; Brace, p. 97] is incorrect without much further consideration. Even graphite and soot produce the desired result only over a limited range of conditions. It remains true that “diﬀerent bodies . . . radiate diﬀerent kinds of heat” as published in the ﬁrst issue of Nature in 1869 [30]. An examination of thermal emissivity plots is suﬃcient to conﬁrm these statements [31]. Not a single object in nature is a blackbody. Hence, it is reasonable to wonder why this concept has so captivated physics. In studying blackbody radiation, it will be demonstrated that radiation within an enclosed body is not necessarily black [25], as Kirchhoﬀ’s law erroneously dictates [20–22].
If this subject matter remains important after all these years [1, 2, 20–22], it is because so much of physics, and more speciﬁcally astrophysics, is tied to the concept of universality in blackbody radiation. Agassi highlights the importance of Kirchhoﬀ’s law for astrophysics: “Browsing through the literature, one may ﬁnd an occasional use of Kirchhoﬀ’s law in some experimental physics, but the only place where it is treated at all seriously today is in the astrophysical literature” [32]. As a result, in astrophysics, if a thermal spectrum is observed which displays, or even approximates, a Planckian (or normal) distribution, temperatures are immediately inferred. For this reason, the fall of universality heralds, in the most profound and far-reaching manner, a new dawn in this sub-discipline. Should universality be reconsidered, there are signiﬁcant consequences for our models of the Sun and relative to the temperatures of the stars [33–35]. The validity
 of the 3 K microwave background temperature would be
questioned [36–41] and with it, perhaps, the entire framework of cosmology [33, 42]. Kirchhoﬀ’s law of thermal emission [20–22] may well be the simplest law in physics, but it is clear that, upon its validity, rests the very foundation of modern astrophysics.
Given these facts, it is unusual that Planck has advanced an equation [1, 2] which remains unlinked to any real physical process or object. Sadly, it is somewhat as a result of Kirchhoﬀ’s law that Planck remained unable to link his equation to a physical cause. The problem was an extremely serious one for Planck, and the fact that his hands were tied by universality is no more evident than in the helplessness he displays in the following quotation: “On the contrary, it may just as correctly be said that in all nature there is no process more complicated than the vibrations of black radiation. In particular, these vibrations do not depend in any characteristic manner on the special processes that take place in the centers of emission of the rays, say on the period or the damping of emitting particles; for the normal spectrum is distinguished from all other spectra by the very fact that all individual diﬀerences caused by the special nature of the emitting substances are perfectly equalized and eﬀaced. Therefore to attempt to draw conclusions concerning the special properties of the parti-

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cles emitting the rays from the elementary vibrations in the rays of the normal spectrum would be a hopeless undertaking” [9; §111].
Yet, it is primarily universality that makes this task a “hopeless undertaking”. Planck, in fact, realized that vibrating atoms, electrons, or particles of some sort, must be responsible for the process of thermal emission. He speciﬁcally believed that the answer might be found by studying the electron and devoted much of his life to this topic [5; pp. 133– 134, 198–199, 245]. But, unfortunately, Planck never makes the link to a real physical species, and the electron itself is not the proper lone candidate. Planck’s belief that the answer lay in electron theory is explicitly contained in his letter to Paul Ehrenfest on July 6, 1905 in which he states: “But perhaps it is not out of the question to make progress in the following way. If one assumes that resonator oscillations are produced by the motion of electrons. . . ” [5; p. 132]. Lorenz had already been successful, in deriving the radiation equation for long wavelengths (the Rayleigh-Jeans solution), using the analysis of electrons [5; p. 190].
Surprisingly, the real solution to the blackbody radiation problem has never been discovered [19]. Even Albert Einstein, in 1909, expressed frustration in this regard in a letter to H. A. Lorentz: “I cherish the hope that you can ﬁnd the right way, if indeed you ﬁnd the reasons given in the paper for the untenability of the current foundations to be at all valid. But if you should deem those reasons to be invalid, then your counterarguments could perhaps furnish the key to the real solution of the radiation problem” [18; p. 105]. The problem was never solved. As late as 1911, Einstein continues to express his frustration to Lorentz: “I am working on the case of damped resonators; it involves quite a lot of calculation. The case of the electrons in the magnetic ﬁeld, which I already mentioned in Brussels, is interesting, but not as much as I had thought in Brussels. Electrons in a spatially variable magnetic ﬁeld are oscillators with variable frequency. If one neglects the radiation, then statistical mechanics yields the distribution law at every location if it is known at one location. If that location is ﬁeld-free, then Maxwell’s distribution holds there; from this one concludes it must hold everywhere. This leads of course to Jean’s formula. Nevertheless, to me the thing seems to show that mechanics does not hold even in the case of the electron moving in the magnetic ﬁeld. I am telling you this as an argument against the view that mechanics ceases to hold at the point where more than two things interact with each other. Anyway, the h-disease looks ever more hopeless” [18; p. 228]. Blackbody radiation was never linked to a direct physical process. Yet, according to Kuhn, Einstein pointed out that “not only the vibrations of electrons but also those of charged ions must, contribute to the blackbody problem” [5; p. 210]. Nonetheless, Kuhn goes on to write that by the early 1910s “while the nature of Planck’s oscillators and of the corresponding emission process remained a mystery, the black-body problem could provide no further clues to

physics” [5; p. 209]. In 1910, Peter Debye, derives Planck’s law by quantizing the vibration modes of the electromagnetic ﬁeld without recourse to oscillators [5; p. 210]. Albert Einstein would soon obtain it using his coeﬃcients [17]. But the nature of the emitter was not identiﬁed [19]. In fact, in both cases, physics moved increasingly outside the realm of physical reality and causality.
Astrophysics believes that nothing of known physical origin is needed to obtain a blackbody spectrum. All that is required is a mathematical construct involving photons in thermal equilibrium and this, well outside the conﬁnes of a solid enclosure, as demanded by the experimental constraints surrounding blackbody radiation. Astrophysics has no need of the physical lattice, of some physical species vibrating within the conﬁnes of a structural physical assembly. But, if a thermal spectrum is to be produced, it is precisely this kind of physical restriction which must exist [19, 23]. However, as long as the idea that blackbody radiation is independent of the nature of the walls prevails, there can be no correction of this situation. It is the very formation of Kirchhoﬀ’s law [20–22] which must be brought into question, if any progress is to be made toward linking Planck’s equation [1, 2] to the physical world and if astrophysics is to reform the manner in which it treats data. For these reasons, we now embark on the review of the ﬁndings which led to the concept of universality. Overwhelming evidence will emerge (see also [23–25]) that this concept is erroneous and should be reconsidered.
2 Experimental production of black radiation
2.1 The 19th century and the lampblack standard
Wedgwood published his delightful analysis on the production of light from heated substances in 1792 [43]. The works are noteworthy and pleasant to read because 1) they deﬁne the “state of the art” just prior to the 19th century, 2) they examine a plethora of substances, and 3) they possess wonderful historical descriptions of antecedent works. The experiments contained therein are nothing short of elegant for the period. Even at this time, the emission within a cylinder, either polished or blackened (presumably covered with lampblack), had already gained the attention of science [43]. Wedgwood realized that it did not matter, if heat entered the substance of interest through light, or through friction [43]. Much was already known about thermal radiation, but confusion remained.
The experimental aspects of the science of thermal radiation really began with the release of Leslie’s An Experimental Inquiry into the Nature and Propagation of Heat [44]. In this classic work, Leslie describes how all objects emit light, but also that they have very diﬀerent emissive powers, even at the same temperature [44; pp. 81, 90, 110]. This was well understood throughout the 19th century [45, 46]. Leslie opens his work as follows: “The object I chieﬂy proposed, was to discover the nature, and ascertain the properties of

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what is termed Radiant Heat. No part of physical science appeared so dark, so dubious and neglected” [44; p. X]. Ironically, Leslie’s last sentence rings somewhat true, even 200 years later.
Using reﬂectors made of tin, Leslie analyzed radiation emitted from the sides of a cube made of “block tin”. At least one side was kept polished, one side was often coated with lampblack, and the other two were used to place miscellaneous substances, like tin foil, colored papers, or pigments [44; p. 8]. In order to maintain a constant temperature, the cubes were ﬁlled with water. The key to Leslie’s experiments was a diﬀerential thermometer. By positioning various faces of the cube towards the reﬂector and placing his thermometer at the focal point, he soon discovered that polished metals give much less radiant heat than soot. He also realizes that the power to absorb or emit heat is somehow conjoined [44; p. 24]. It is interesting that, in his very ﬁrst experiment, Leslie examines lampblack. It would become, for the rest of the 1800s, the means by which radiation would be calibrated.
Lampblack, the oxidation product of oil lamps, was not only a suitable material for coating surfaces and generating blackbodies over the course of the 1800s, it rapidly became the standard of radiation. By 1833, the Reverend Baden Powell, whose son was to form the Scouting movement, already writes that: “all experimenters have usually blackened their thermometer” [47; p. 276]. In 1848, G. Bird notes how lampblack has become a reference standard in the study of emission [48; p. 516]. Stewart refers repeatedly to lampblack invoking that soot had become the standard by which all radiation was to be measured: “The reason why lampblack was chosen as the standard is obvious; for, it is known from Leslie’s observations, that the radiating power of a surface is proportional to its absorbing power. Lampblack, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power” [26; §4]. He directly refers to lampblack heat [49; p. 191]. His experiments with lampblack are covered below in the context of the theoretical formulation of the law of radiation. Silliman’s work is particularly valuable in that it was completed in 1861 [50]. It not only gives a well written and thorough account of the current state of knowledge in heat radiation, but it restates the central role of lampblack: “Lampblack is the only substance which absorbs all the thermal rays, whatever be the source of heat” [50; p. 442].
Langley re-emphasizes the extensive use of lampblack in his paper on solar and lunar spectra: “I may reply that we have lately found an admirable check upon the eﬃciency of our optical devices in the behavior of that familiar substance lampblack, which all physicists use either on thermometers, thermopiles, or bolometers” [51]. In 1893, Clerke writes of the “lampblack standard” in her tremendous work on the history of Astronomy [52; p. 271]. Tillman, in the 4th edition of his Elementary Lessons in Heat, summarizes well the be-

lief that prevailed throughout the 1800s: “Lampblack is the most perfect absorber and radiator, it being devoid of both reﬂecting and diﬀusive power. Its absorbing power is also most nearly independent of the source of heat. It absorbs all rays nearly alike, the luminous as well as the dark ones. Lampblack is accordingly taken as the standard surface of absorption, absorbing in the greatest degree every variety of ray which fall upon it. It is consequently, also , when hot, the typical radiator, giving out the maximum amount of heat which any substance at the same temperature could possibly give out; moreover, it gives out the maximum amount of each kind of heat that can be given out by any body at that temperature” [46; p. 92]. Tillman does recall Langley’s discovery that, in the infrared, lampblack was nearly transparent [51]. In any event, the role of lampblack in thermal radiation was well established by the end of the 19th century.
In his textbook on physics, published for the 7th time in 1920, Watson provides an elaborate description of the use of lampblack in coating both thermometers and surfaces for the study of comparative emission between objects [53]. He describes the lampblack standard as follows: “Lampblack, although it does not absorb quite the whole of the incident radiation, yet possesses the property of absorbing very nearly, if not quite, the same proportion of the incident radiation whatever the wave-length, and so this substance is taken as a standard” [53; p. 301].
A review of the blackbody literature for the 19th century reveals that blackbodies were produced either from graphite itself or from objects covered with lampblack (soot) or paints, which contained soot or bone black [54]. That is not to say that other substances were not used. Kangro [3] outlines an array of studies where experimentalists, over a small region of the spectrum, used diﬀerent materials (platinum black, copper oxide, iron oxide, thorium oxide, etc). Nonetheless, graphite and soot take precedence over all other materials, precisely because their absorbance extends over such a wide range of wavelengths. Conversely, all other materials exhibit disadvantages, either because of their suboptimal emissivity, or due to their limited frequency ranges [31]. There are problems in visualizing the infrared, even with platinum black. Kangro explains: “They (Lummer and Kurlbaum) changed to a platinum box as being more easily heated electrically and better suited to exact temperature measurement, then they used a platinum roll and ﬁnally a platinum cylinder the interior of which was blackened with iron oxide, and also divided by diaphragms the whole enclosed in a large asbestos cylinder” [3; p. 159]. They also report “the defective absorption of long wavelengths by Platinum black with which their bolometers were coated” as a possible source of error [3; p. 159]. Lummer and Kurlbaum made their 1898 cavity from platinum blackened with a mixture of chromium, nickel, and cobalt oxide [4]. Nonetheless, in order to properly visualize the longest wavelengths, the method of residual rays, developed by Rubens, was utilized [4]. These were critical experiments

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for Planck. Yet, since platinum black could not reach elevated temperatures, in 1903, Lummer and Pringsheim would design a new blackbody with graphite walls [4]. This design has endured, essentially unchanged, until the present day [4].
2.2 The 19th century and the general state of knowledge
In 1833, Powell gave his excellent report on radiant heat [47]. By this time, the amount of radiation was known to be inversely related to conductive power [47; p. 266]. The more an object conducted thermal radiation, the better it acted as a reﬂector and the worst it was as an emitter/absorber. Based on the experiments of William Ritchie [55], it was also known that the absorptive power of a substance was directly related to its emissive power [47, p. 265]. Pre´vost’s theory on thermal equilibrium, the famous Theory of Exchanges [56–58] was understood [47; p. 261]. Herschel’s studies with infrared radiation were complete and the blocking action of glass was established [47; pp. 269–272]. While Herschel had discovered infrared radiation in 1800 [59], it was not until Langley, that infrared radiation could be accurately monitored [51]. At the time, Langley observed that lampblack was very nearly transparent to infrared radiation. Using prisms, it was also known that, on opposite sides of the spectrum, there existed “isothermal points” [47; p. 296]. Prisms played an important role in the early classiﬁcation of the quality of light and heat by separation into colors [47; pp. 291–296]. Interestingly, Powell takes a sidestep relative to liquids and writes in his conclusion: “In liquids, it has been disputed whether there can be radiation; and they are worse conductors than solids” [47; p. 300]. Silliman notes that, even at the time of Kirchhoﬀ, there remained some debate as to the relation between absorptive and emissive powers [50; p. 441], with de la Provostaye, Desains, and Melloni highlighting that these were not always equivalent. Given this general state of knowledge during the 19th century, we now move to the most important areas of experimentation, Pre´vost’s Theory of Exchanges [56–58] and cavity radiation at thermal equilibrium.
2.3 The 19th century and cavity radiation
Pierre Pre´vost advanced his powerful Theory of Exchanges just as the 19th century came to life [56–58]. In formulating his law, Pre´vost invokes the enclosure: “. . . I will suppose the two portions to be enclosed in an empty space, terminated on all sides by impenetrable walls” [56; in Brace, p. 5]. He then moves to develop his Theory of Exchanges [56–58]. This theory was critical to Kirchhoﬀ’s thinking when the concept of universality was formulated [20–22]. As such, it is important to understand how Pre´vost’s theory was viewed, not simply at the time of its formulation, but in the days of Kirchhoﬀ. This knowledge can be gained by examining Balfour Stewart’s summary of Pre´vost’s theory. Stewart recounts the central ideas of equilibrium with an enclosure in his Treatise [49]. He summarizes Pre´vost’s ﬁndings as follows: “1. If an

enclosure be kept at a uniform temperature, any substance surrounded by it on all sides will ultimately attain that temperature. 2. All bodies are constantly giving out radiant heat, at a rate depending upon their substance and temperature, but independent of the substance or temperature of the bodies that surround them. 3. Consequently when a body is kept at uniform temperature it receives back just as much heat as it gives out” [49; p. 215].
With Pre´vost, nearly 70 years before Kirchhoﬀ, the real study of cavity radiation began. At the same time, the understanding of cavity radiation really grew near the 1820s. This was when the experimental work of Dulong and Petit [60] with cavities took place. Simultaneously, theoretical studies of heat were being forged by Fourier [61–67] and Poisson [68, 69]. Fourier’s works are particularly important in that they represent the most far-reaching theoretical analysis of heat and cavities in this time frame.
The paper by Dulong and Petit [60] is a major milestone in experimental science and it is diﬃcult to do it justice in a brief treatment. Thus, let us concentrate not on the ﬁrst section dealing with the measurements of temperatures, the dilatation of solids, and the speciﬁc heats of materials, but rather on the second section. This section addresses the laws of cooling derived within an enclosure. Of course, Kirchhoﬀ’s law of thermal emission [20–22] deals with radiation under equilibrium conditions. Conversely, the results of Dulong and Petit examine a dynamic process [60]. While they do not directly apply, the studies by Dulong and Petit form the experimental basis for the works that follow and are crucial to understanding cavity radiation. Dulong and Petit recognized the importance of distinguishing the eﬀects of gas particles and radiative emission in cooling [60]. By examining the cooling of water and liquids in enclosures of varying shapes, they conclude that the rate of cooling is independent of the shape of the walls of the enclosure, on its size, and on the nature of the liquid [60; p. 245]. Note how this conclusion is reminiscent of Kirchhoﬀ’s law [20–22]. Importantly, they observe that the rate of cooling is dependent on the state of the surface of the enclosure [60; p. 245].
Dulong and Petit continue their inquiry into the laws of cooling by building a copper enclosure, the inner surface of which they cover with lampblack [60; p. 247]. They place a thermometer at the center of the enclosure. The outer surface of the thermometer is either silvered or left in its glassy state [60; p. 250]. Using a pump, a balloon (containing various gases of interest), and a barometer attached to the enclosure, they deduce the law of cooling. Dulong and Petit accomplished their goal by varying the gas pressure within the enclosure while monitoring the drop in temperature of the previously heated thermometer. Initially, ignoring the eﬀect of gases and working near vacuum, they quickly realize that the rate of cooling depends on the nature of the thermometer surface, and this even within the blackened cavity [60; p. 260]. The rates of cooling of the two thermometers were

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proportional to one another, not equal [60; p. 260]. They arrive at a simple general law of cooling that applies to all bodies [60; p. 263]. Finally, by repeating the same experiments with gases at diﬀerent pressures, they derive a law of cooling with two terms depending on radiation and the eﬀect of the gas. They infer that the ﬁrst term depends on the nature, the size, and the absolute temperature of the enclosure, while the second term depends only on the characteristics of the gas [60; p. 288]. Dulong and Petit’s work is not revisited in a substantial manner until de la Provostaye and Desain publish their Me´moires [70–75].
De la Provostaye and Desain published their second Memoir on the Radiation of Heat in 1848, more than 10 years before the formulation of Kirchhoﬀ’s law of thermal emission [71]. The authors open their work by stating (all translations from French were made by the author): “We must know how the quantity of heat emitted by a surface of a determined size depends on its temperature, its proper nature, its state, on the direction of the emission” [71; p. 358]. They then highlight: “but that we (scientists) have not, up to this day, introduced into the solution questions of equilibrium and of movement of the heat” [71; p. 358].
The authors revisit Dulong and Petit’s experiments with gases using a half liter cylinder, blackened interiorly with lampblack (noir de fume´e), in which they can introduce gases. They were never able to conﬁrm the exact relation of Dulong and Petit and, therefore, present a more elaborate equation to describe the law of cooling [71; p. 369]. The paper contains a relevant caveat in that the authors report that it is not always easy to obtain a black surface, even with lampblack paste. They resort to the ﬂame of a lamp to resurface the object of interest in order that its emission becomes truly independent of angle of observation [71; p. 398]. However, the bulk of our concern is relative to their work on the approach towards thermal equilibrium within an enclosure [71; pp. 406–431].
They recall that Fourier has proved: “1) that within a blackened enclosure without reﬂective power, equilibrium is established from element to element, 2) that the equilibrium is maintained in the same manner if we restore to one of the elements a reﬂective power, as long as we admit, in the ﬁrst instance, that the absorbing and reﬂecting powers are complementary; and in the second place, that the emissive power is equal to the absorptive power, 3) that the same will hold, if we restore a certain reﬂective power to all the elements” [71; p. 406].
De la Provostaye and Desain highlight that the enclosure must be blackened for Fourier’s conclusions to hold, but the latter does not always speciﬁcally state if his cavity is blackened interiorly. Nonetheless, Fourier’s derivations make the assumption that the wall of the enclosure follows Lambert’s law [66]. As such, the objects can be viewed as placed within a perfectly absorbing cavity. De la Provostaye and Desain make the point as follows: “The demonstration supposes, what the author (Fourier) seems in fact to have

recognized for himself (Annales de Chimie et de Physiques,

tome XXVII, page 247 (see [66]) in his last Memoires, that

the radiating body is stripped of all reﬂective power. It would

therefore be not at all general. . . ” [71; p. 408].

De la Provostaye and Desain begin their studies by plac-

ing a hypothetical thermometer in a spherical cavity and make

no assumptions other than stating that diﬀuse reﬂection does

not occur. They permit, therefore, that both the cavity and

the thermometer can sustain normal reﬂection and emission.

Assuming that reﬂective power does not depend on the an-

gle of incidence, they permit the rays to travel throughout the

cavity and follow the progression of the rays over time, until

equilibrium is reached. The authors conclude that the radia-

tion inside such a cavity will not follow Lambert’s law [71;

p. 414]. The result is important because it directly contradicts

Kirchhoﬀ’s assertion that the radiation inside all cavities must

be black [20–22]. They then restrict their treatment to the

consideration of angles below 60˚ or 70˚, in order to reach a

simpliﬁed form for the laws of cooling.

Like Dulong and Petit [60], de la Provostaye and Desain

[70–75] are not concerned exclusively with thermal equilib-

rium, but rather, they are examining the velocity of cooling,

the path to equilibrium. They provide important insight into

the problem, as the following excerpt reveals: “When in an

blackened enclosure with an invariable temperature t, we in-

troduce a thermometer at the same temperature and a body

either warmer or colder, but maintained always at the same

degree T , the thermometer will warm or cool, and, following

the reciprocal exchanges of heat, it will attain a ﬁnal temper-

aemtuirsesiv,ewphoowseervEal0ueo,f

function of T and t, depends also on the its surface, of that E of the source, and

of their forms, sizes and reciprocal distances” [71; p. 424].

Siegel [76] highlights appropriately that de la Provostaye

and Desains deﬁned the emissive power E of a body as a

fraction of the radiant emission of the blackbody where f(t)

is the emission of the blackbody, and the emission of the body

is Ef(t) [74; p. 431]. In contrast, Kirchhoﬀ deﬁnes emission

simply as E, which, in fact, corresponds to de la Provostaye

and Desain’s Ef(t) [76]. Consequently, the universal func-

tion f(t) is incorporated into Kirchhoﬀ’s law, even when it

does not seem to be the case [76].

3 Cavity radiation
3.1 The Stewart-Kirchhoﬀ dispute
Balfour Stewart [26] preceded Kirchhoﬀ [20–22] by at least 2 years in the treatment of radiation at thermal equilibrium. Both Kirchhoﬀ and Stewart built on the idea, initially advanced by Pre´vost [56–58], and expanded upon by Fourier [61–67], Poisson [68, 69], Dulong [60], Petit [60], de la Provostaye [70–75], Desains [70–75], and surely others, that thermal equilibrium existed between objects at the same temperature in the presence of conﬁnement [49; p. 196]. The Stewart-Kirchhoﬀ conﬂict is one of the darkest moments in

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Gustav Robert Kirchhoﬀ (12 March 1824 – 17 October 1887)

Balfour Stewart (1 November 1828 – 19 December 1887)

the history of science and it has been the subject of an excellent review [76]. This public quarrel is worth revisiting, not only because it is a powerful example of how science must not be performed, but also because it is very likely that the dispute between these men, and the international involvement of their collaborators [76] was directly responsible for the persistence of universality. If Stewart and Kirchhoﬀ had better communicated, Kirchhoﬀ might have yielded and the erroneous concept of universality, might have been retracted.
However, nationalistic passions were inﬂamed to such a measure that reason and scientiﬁc truth were moved to secondary positions. The animosity between Germany and British scientists would eventually reach the boiling point when, in 1914, Planck and 92 other learned men signed the Appeal to the Cultured Peoples of the World [16; pp. 70–ﬀ]. Planck apparently signed the Appeal without examining its contents. Wien, for his part, insisted that British scientists “appropriated discoveries made in Germany, confused truth and falsehood, argued in bad faith, and . . . that England was the worst enemy of the Reich” [16; p. 72]. He urged that German scientists avoid, as much as possible, publication in British journals [16; p. 72]. Planck, for his part, refused to sign Wien’s manifesto [16; pp. 70–ﬀ]. While the StewartKirchhoﬀ aﬀair cannot bear all the responsibility for these tragic developments, and while other scientiﬁc battles also raged [76], it is relatively certain that the situation played an early role in the building of such misconceptions.
The papers from Stewart and Kirchhoﬀ which caused this conﬂict were all published in The London, Edindurgh, and

Dublin Philosophical Magazine and Journal of Science. Kirchhoﬀ was able to have access to the English literature, primarily through the assistance of F. Guthrie and Henry E. Roscoe. The latter translated many of Kirchhoﬀ’s works into English for Philosophical Magazine. Roscoe had studied and published with Bunsen who, in turn, eventually became Kirchhoﬀ’s key collaborator.
Stewart opens the discourse by publishing, in 1858, “An account of some experiments on radiant heat, involving an extension of Pre´vost’s Theory of Exchanges” [26]. It will be discovered below that, in fact, it is Stewart’s work which reached the proper conclusion, not Kirchhoﬀ’s [25]. Yet, Stewart’s Account [26] has been forgotten, in large part, because, unlike Kirchhoﬀ’s papers [20–22], it did not arrive at universality as Seigel emphasizes [76].
The battle really begins when F. Guthrie translates Kirchhoﬀ’s paper and places it in Philosophical Magazine [21], the journal where Stewart’s work had appeared just two years earlier. Kirchhoﬀ is rapidly criticized for failure to cite prior work, not only relative to Stewart, but relative to other seminal discoveries [76]. With the aid of Roscoe [77–80], he publishes in 1863, “Contributions towards the history of spectrum analysis and of the analysis of the solar atmosphere” [81] in which he seems to dismiss the importance of Stewart’s contributions. Kirchhoﬀ writes: “This proof cannot be a strict one, because experiments which have only taught us concerning more and less, cannot strictly teach us concerning equality” [81]. Kirchhoﬀ highlights that Stewart is not treating an enclosure in his experiments, but extends his con-

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clusions to these objects [81]. In the end, Kirchhoﬀ’s Contributions [81] is not impolite. . . but it is tough.
For Stewart, Kirchhoﬀ’s Contributions [81] is viewed as an attack which must be immediately countered [82]. Stewart opens his rebuttal by stating: “In the course of his remarks the learned author has reviewed in a somewhat disparaging manner some researches of mine on radiant heat, in consequence of which I am forced to reply, although very unwillingly, and desiring much to avoid a scientiﬁc controversy, especially with Professor Kirchhoﬀ as an opponent” [82]. In his own defense, Stewart then adds: “nor did I omit to obtain the best possible experimental veriﬁcation of my views, or to present this to men of science as the chief feature, grounding theory upon the experiments, rather than deducing the experiments from the theory” [82]. This powerful charge by Stewart, in the end, forms the entire argument against Kirchhoﬀ’s proof [82]. Kirchhoﬀ’s results can never be validated by experiments, and Stewart, as an expert in heat radiation, must have recognized this to be the case [82].
Stewart closes his defense as follows: “Although I preceded Kirchhoﬀ nearly two years in my demonstration, I did not hesitate to acknowledge that his solution had been independently obtained; but, as a general principle, I cannot consent to admit that when a man of science has proved a new law and is followed by another who from the same premises deduces the same conclusions, the latter is justiﬁed in depreciating the labours of the former because he conceives that his solution is more complete. Will Kirchhoﬀ himself willingly forego his own claims in favour of any one who shall in the future ages devise (if this be possible) a simpler and more convincing demonstration than that which has been given us by the Hiedelberg Professor? I feel, Sir, that, as an historian of science, you will acknowledge the justice of these remarks, and join me in regretting that one who has so eminently distinguished himself in original investigation should have chosen to superadd to his functions as a discoverer those of a severe and hostile critic upon the labours of those men who have worked at the same subject with himself, and by all of whom he has been treated with the utmost possible consideration” [82].
The Stewart-Kirchhoﬀ dispute reached such a magnitude that Kirchhoﬀ, it seems, never again publishes in Philosophical Magazine, even though Bunsen, for his part, continues to utilize the journal. Stewart remained at a profound disadvantage, as he did not beneﬁt from a relationship similar to that between Kirchhoﬀ, Bunsen, and Roscoe. Roscoe would reprint Kirchhoﬀ’s infamous Contributions [81] in his Spectrum Analysis [80; pp. 115–122]. However, in this version [80; pp. 115–122], all text referring to Stewart has been removed without comment. It is impossible to understand Roscoe’s motivation for the attenuated version. Roscoe may have suﬀered for having translated the letter. Alternatively, Kirchhoﬀ’s Contributions [81] might not ﬁt in its entirety within the context of the other lectures. In any event,

Roscoe‘s Spectrum Analysis is a strange ode to Kirchhoﬀ, which lacks broad scientiﬁc review. Regrettably, it seems that Roscoe made no attempt to reconcile the Kirchhoﬀ-Stewart matter through proper and continuing scientiﬁc discourse.
In the end, Kirchhoﬀ and Stewart each fell short of the mark. However, Kirchhoﬀ’s error was more serious [20–22], since it has theoretical consequences to this day. As for Balfour Stewart, had he presented a better theoretical case [26], the course of physics may have followed a diﬀerent path. Kirchhoﬀ, for example, correctly highlighted that Stewart’s proof should not use the index of refraction, but rather, the square of the index [81]. Stewart conceded the point [76, 82]. For Kirchhoﬀ, Stewart’s proof was possibly true, not necessarily true [81]. Siegel elegantly clariﬁed Kirchhoﬀ’s concerns [76]. These shortcomings in Stewart’s derivation hinder the search for truth. Finally, had nationalistic sentiments not been aroused [76], it might have been easier to resolve the conﬂict.
3.2 Balfour Stewart
In examining Stewart’s writings [26, 49, 82–85], we discover, as Brace highlights, “the comprehensiveness of his mind and the originality of his genius” [83; p. 72]. Many of Stewart’s [26, 82–85] ideas are contained in his Elementary Treatise on Heat [49] and the later reﬂects his positions at the end of his life. As such, our discussion will begin ﬁrst with the examination of this work and close with the review of his 1858 and 1859 papers [26, 83].
By the time Stewart writes his Treatise, he clearly recognizes that all substances display at least selective absorption of light [49; p. 191]. He comments on the probable identity of heat and light and writes: “The facts detailed in this chapter all tend to shew that radiant light and heat are only varieties of the same physical agent, and also that when once the spectrum of a luminous object has been obtained, the separation of the diﬀerent rays from one another is physically complete; so that if we take any region of the visible spectrum, its illuminating and heating eﬀect are caused by precisely the same rays” [49; p. 195]. He continues: “Furthermore, we have reason to suppose that the physical distinction between diﬀerent parts of the spectrum is one of wave length, and that rays of great wave length are in general less refracted than those of small wave length” [49; p. 196].
Stewart’s thoughts with respect to radiation within a cavity are important, not only because they provide us insight into the proper analysis of the enclosures, but also because they clearly outline what was known just prior to Planck. Stewart’s comments relative to these experiments are summarized once again in his Treatise: “. . . let us for our present purpose imagine to ourselves a chamber of the following kind. Let the walls which surround this chamber be kept at a constant temperature, say 100˚C, and let them be covered with lampblack — a substance which reﬂects no heat, or at least

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very little; — also let there be a thermometer in the enclosure. It is well known that this thermometer will ultimately indicate the temperature of the surrounding walls. . . Suppose that the outside of the bulb of the thermometer of last article is covered with tinfoil, so that its reﬂecting power is considerable. Now according to the Theory of Exchanges this thermometer is constantly radiating heat towards the lampblack, but it is receiving just as much heat as it radiates. Let us call radiation of lampblack 100, and suppose that 80 of these 100 rays which strike the thermometer are reﬂected back from its tinfoil surface, while the remaining 20 are absorbed. Since therefore the thermometer is absorbing 20 rays, and since nevertheless its temperature is not rising, it is clear that it must be also radiating 20 rays, that is to say, under such circumstances its absorption and radiation must be equal to one another. If we now suppose the outside of the bulb to be blackened instead of being covered with tinfoil, the thermometer will absorb nearly all the 100 rays that fall upon it, and just as in the previous case, since its temperature is not rising, it must be radiating 100 rays. Thus we see that when covered with tinfoil it only radiated 20 rays, but when blackened it radiates 100. The radiation from a reﬂecting metallic surface ought therefore, if our theory be true, to be much less than from a blackened one. This has been proved experimentally by Leslie, who shewed that good reﬂectors of heat are bad radiators. Again, we have seen that in the case of the bulb covered with tinfoil 80 of the 100 rays which fell upon it were reﬂected back, and we have also seen that 20 were radiated by the bulb. Hence the heat reﬂected plus the heat radiated by this thermometer in the imaginary enclosure (author underscoring text) will be equal to 100, that is to say, it will be equal to the lampblack radiation from the walls of the enclosure. We may generalize this statement by saying that in an enclosure of constant temperature the heat reﬂected plus the heat radiated by any substance will be equal to the total lampblack radiation of that temperature, and this will be the case whether the reﬂecting substance be placed inside the enclosure or whether it form a part of the walls of the enclosure” [49; pp. 199–201].
Stewart reaches this conclusion for an enclosure whose walls have been covered with lampblack [49]. In that case, the heat inside the enclosure will correspond to that from lampblack, as I have shown [25]. In the pages which follow [49], Stewart goes on to explain that his law holds, in a manner which is independent of the nature of the walls, provided that both radiation and reﬂection are included. He also illustrates independence relative to wall shape. Importantly, he invokes the work of de la Provostaye and Desains with silver and lampblack to demonstrate that the total radiation inside an enclosure containing a silver surface will also be equal to 100, where 2.2 parts arise from the emission of silver itself and 97 parts from the reﬂection of lampblack. Stewart realizes that the value of 100 is only achieved in the presence of lampblack. The nature of the wall was immaterial simply because lampblack was always present. In fact, it appears that

Stewart was actually contemplating enclosures which contain both reﬂective surfaces and absorbing ones, as seen in his section 227: “It has already been stated (Art. 204) that the stream of radiant heat continually proceeding through an enclosure of which the walls are kept at a constant temperature depends only on the temperature of the walls, and not on the nature of the various substances of which they are composed; the only diﬀerence being that for metals this stream is composed partly of radiated and partly also of reﬂected heat, while for lampblack it is composed wholly of radiated heat. This may be expressed by saying that this stream depends upon or is a function of the temperature, and of it alone; but there is the following very important diﬀerence between a reﬂecting and lampblack surface, as representing this stream of radiant heat. It is only when a reﬂecting surface forms part of a complete enclosure of the same temperature as itself, that the radiated and reﬂected heat from this surface together represent the whole stream of heat; for if we bring it for a moment into another enclosure of lower temperature, the reﬂected heat is altered, and although the radiation will for a short time continue nearly constant, yet this radiation will not represent the whole stream of heat due to the temperature of the surface. On the other hand, if a lampblack surface be placed in the above position, since the stream of heat which ﬂows from it is entirely independent of the reﬂexion due to neighboring bodies, the heat which it radiates when brought for a moment into an enclosure of lower temperature than itself will truly represent the stream of radiant heat due to the temperature of the lampblack” [49; pp. 221–222]. One can see that reﬂecting materials provide very diﬀerent conditions than lampblack within enclosures. That is, within an enclosure under dynamic conditions, objects which are partially or fully reﬂecting cannot indeﬁnitely support black radiation. They simply emit their own radiation and reﬂect the heat incident upon their surface. Through this discussion, Stewart demonstrates that thermal equilibrium would be disturbed when a perfect absorber is replaced with a reﬂector, bringing about dynamic rather than equilibrium conditions. This was an important insight relative to the analysis which I recently provided [25] of Kirchhoﬀ’s second proof [21, 22].
In order to examine the velocity of temperature change, Stewart invokes a thin copper globe lined with lampblack: “Having now considered the law of cooling as representing with much accuracy the quantity of heat given out by a black substance at diﬀerent temperatures, we come next to the relation between the temperature and the quality or nature of the heat given out. And here we may remark that the laws which connect the radiation of a black body with its temperature, both as regards to the quantity and the quality of the heat given out, hold approximately for bodies of indeﬁnite thickness which are not black, — thus, for instance, they would hold for a metallic surface, which would represent very nearly a lampblack surface, with the radiation diminished a certain number of times. These laws would not, however, hold

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exactly for a white surface, such as chalk; for this substance behaves like lampblack with respect to rays of low temperature, while it is white for rays of high temperature, and the consequence of this will be that its radiation will increase less rapidly than that of a lampblack surface. In like manner, these laws will not hold exactly for coloured surfaces” [49; p. 230]. Note how these statements are directly contradictory to what Kirchhoﬀ requires. For Stewart, there is no universality and this is a major distinction between his work and that of his adversary [25].
With regards speciﬁcally to a black surface, Stewart writes (see page 231): “1. The spectrum of the radiant heat and light given out by a lampblack surface is continuous, embracing rays of all refrangibilities between certain limits on either side. . . 2. We have reason to think that as the temperature rises, the spectrum of a black substance is extended in the direction of greatest refrangibility, so as to embrace more and more of the violet and photographic rays” [49; p.231]. Stewart goes on to discuss thin plates of glass and explains how they cannot be compared to lampblack, as their radiation with increasing temperature will be substantially diﬀerent [49; p. 232].
It is clear that if scientists of the period coated the walls of their enclosure with lampblack, that emission would be independent of the nature of the walls themselves, precisely because lampblack was coating these walls. After all, Stewart fully realizes that silver, for instance, has a total emission much below lampblack [49; pp. 201–206]. Stewart used an enclosure coated with lampblack to arrive at the following laws: “1. The stream of radiant heat is the same throughout, both in quantity and quality; and while it depends on the temperature it is entirely independent of the materials or shape of the enclosure. 2. This stream is unpolarized. 3. The absorption of a surface in such an enclosure is equal to its radiation and this holds for every kind of heat” [49; p. 206]. That is how the concept of independence of the nature of the walls entered the literature. Nothing, in fact, was independent. The walls were simply coated with lampblack [49; pp. 201–206]. This was such an obvious part of these experiments, during the 19th century, that it is likely that most scientists, unlike Balfour Stewart, simply neglected to report their common practice. As a result, future generations who followed the theoretical avenues of Kirchhoﬀ, actually came to believe that the nature of the walls was unimportant and the vital role of the soot coating was forgotten.
Stewart’s law stated that absorption was equal to radiation for every kind of heat [26, 49, 76, 82]. This was true under equilibrium conditions. However, Kirchhoﬀ objected [81] to this formulation by Stewart [26], since he believed that Stewart had inappropriately extended the results of his experimental ﬁnding to include equality whereas proportionality was all that had been proven [76, 81]. In any event, the fact remains that Stewart’s conclusion [26, 49, 82], not Kirchhoﬀ’s [20– 22], was correct. It alone was supported by the experimental

ﬁndings and, unlike Kirchhoﬀ’s law [20–22], made no claims of universality [76].
The central portion of Stewart’s proof considers a continuous plate of rock salt positioned between two plates covered with lampblack [26; §12]. The idea is both simple and powerful. Stewart immediately reaches the result that “the absorption of a plate equals, its radiation, and that for every description of heat” [26; §19]. Then, Stewart considers radiation internal to a substance: “Let AB, and BC be two contiguous, equal, and similar plates in the interior of a substance of indeﬁnite extent, kept at a uniform temperature” [26; §20]. Stewart is invoking the same restriction found for thermal equilibrium with an enclosure. However, he moves to the interior of a body, apparently in order to avoid dealing with surface reﬂection [82]. Seigel [76] highlights this point. Kirchhoﬀ believes that Stewart has not properly treated the enclosure [81]. The point is weak as Stewart’s entire treatment is based on the ideas of Pre´vost [55–57].
Stewart is clearly working within the conﬁnes of Pre´vost’s Theory of Exchanges [26, 56–58]. Considering the equilibrium between lampblack and an arbitrary surface at thermal equilibrium, he writes “. . . hence the total quantity of heat radiated and reﬂected which leaves the surface. . . (is) the same as if the substance had been lampblack, the only diﬀerence being, that, in the case of lampblack, all this heat is radiated, whereas in other substances only part is radiated, the remainder being reﬂected heat” [26; §31]. He continues: “Although we have considered only one particular case, yet this is quite suﬃcient to make the general principle plain. Let us suppose we have an enclosure whose walls are of any shape, or any variety of substances (all at a uniform temperature), the normal or statical condition will be, that the heat radiated and reﬂected together, which leaves any portion of the surface, shall be equal to the radiated heat which would have left that same portion of the surface, if it had been composed of lampblack. . . Let us suppose, for instance, that the walls of this enclosure were of polished metal, then only a very small quantity of heat would be radiated; but this heat would be bandied backwards and forwards between the surfaces, until the total amount of radiated and reﬂected heat together became equal to the radiation of lampblack” [26; §32]. These passages are quite similar to Kirchhoﬀ‘s with the distinction that universality is never invoked. Stewart realizes that the lampblack surface within the enclosure is essential.
Stewart’s manner of addressing the problem is lacking, as Siegel highlights [76], especially for Kirchhoﬀ [81]. A review of this work [76] provides a suﬃcient discussion. Stewart advances an initial attempt at the correct solution to the radiation puzzle, but the presentation was not suﬃcient, at least for his adversary. Surprisingly, in his Reply to Kirchhoﬀ in 1863, Stewart seems embarrassed [76] relative to reﬂection writing: “I shall only add that it was attempted, as far as possible, to disengage the proof, theoretical and experimental, from the embarrassment of considering surface reﬂexion”

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[82]. If reﬂection is neglected, however, almost by deﬁnition, the radiation must be black [25]. Consequently, all attempts to address the issue devoid of surface reﬂection can never yield the proper conclusion relative to the existence of universality. Stewart reaches the proper answer because he does include reﬂection in his papers [26, 83] and within his Treatise [49]. Within an enclosure containing a lampblack surface and another object, he reminds us that “the reﬂection plus the radiation of the body at any temperature equals the lampblack radiation at that temperature” [83; §44]. The proper consideration of reﬂection is key [25] and though Stewart may have had weaknesses in his presentation, he did ascertain the truth.
3.3 Gustav Kirchhoﬀ and his law
It can be said that Kirchhoﬀ’s law of thermal emission [20– 22], through its claims of the universal nature of radiation within enclosures, represents one of the most profound dismissals of experimental science in the history of physics. The great mass of experimental evidence speaks against universality of radiation within cavities. Cavity radiation only assumes the normal distribution (i.e. that of the blackbody) when either the walls of the cavity, or at least one of the objects it contains, are perfectly absorbing [23, 25]. In fact, the proof that Kirchhoﬀ’s law does not hold, in its universal form, does not require extensive mathematical or experimental arguments, only simple ones [23–25].
Schirrmacher [86] emphasizes that, at the time Planck formulated his law, a solid proof of Kirchhoﬀ’s remained absent. Furthermore, he highlights that, as late as 1912, Hilbert was arguing that Kirchhoﬀ’s law still lacked proof [86]. Hilbert makes this statement in spite of Planck’s attempt to prove the law in his Theory of Heat Radiation [9]. Schirrmacher also outlines that nearly all attempts to advance universality were met with a refutation [86; p. 16]. Sadly, these corrections never prevailed.
De la Provostaye was one of the ﬁrst to oﬀer an analysis of cavity radiation following Kirchhoﬀ, in 1863 [87]. In his work, de la Provostaye deduces that the radiation within a perfectly absorbing cavity must be black [87]. He also infers that a cavity, a portion of whose walls are perfectly absorbing, and which contains an object of arbitrary emittance and reﬂectance, must also contain normal (or blackbody) radiation [87]. Like Kirchhoﬀ, he attempts to extend his ﬁndings to a perfectly reﬂecting cavity. At ﬁrst, he concedes that a fully reﬂecting cavity must be devoid of radiation. At this point, de la Provostaye should have ceased as the question was resolved; but strangely . . . he continues. Prompted perhaps by the quest for Kirchhoﬀ’s universality [20–22], he permits radiation to enter the perfectly reﬂecting cavity and immediately moves to show that such radiation must be black [87]. As a result, de la Provostaye stumbles in a manner quite similar to Kirchhoﬀ and his paper does not, in fact, form a refutation of Kirchhoﬀ’s law [87]. De la Provostaye simply objected that Kirch-

hoﬀ, by introducing perfect reﬂectors, essentially dictated the result which he sought [86].
De la Provostaye’s analysis of cavity radiation is particularly important, because he was an expert in the subject. He had dealt with enclosures on an experimental basis and must have known from the work of his own hands, that Kirchhoﬀ’s law could not hold, in its universal form. This is why he presents the second case discussed above where at least a portion of the cavity walls remained perfectly absorbing. De la Provostaye did overreach in his conclusions [87] in a manner not dissimilar from Kirchhoﬀ [20–22].
In any event, de la Provostaye’s theoretical objections relative to the absence of a perfectly reﬂecting mirror was not the central problem for Kirchhoﬀ [25]. While many followed de la Provostaye’s initial objection, refutations always seemed to be based on arguments such as perfectly reﬂecting mirrors do not exist, neither do perfectly diathermanous (or transparent) bodies, or bodies which can only absorb one wavelength. Such idealized substances are utilized in various proofs of Kirchhoﬀ’s law [86]. Unfortunately, since Kirchhoﬀ’s law is based on a theoretical extension of experimental reality, the fact that idealized objects do not exist is not suﬃcient to overturn Kirchhoﬀ’s position [25]. Hence, the law has prevailed, even though experimental reality is well established against its claims as de la Provostaye and Stewart must have realized.
The only way to refute Kirchhoﬀ’s law is to show that some section of its treatment either fails to consider an essential aspect of physical reality or that, through its derivation, Kirchhoﬀ himself violates the thermal equilibrium, which he required as a precondition [25]. Both of these complications have been brought to the forefront [25]. Kirchhoﬀ’s law is not valid for two reasons: ﬁrst, the importance of reﬂection is not properly included and second, Kirchhoﬀ’s model gives rise, under certain conditions, to a violation of thermal equilibrium [25].
Physics is in a diﬃcult position relative to Kirchhoﬀ’s law, since the modern relationship between radiation and absorption, under equilibrium conditions, is based upon this work. At the same time, Kirchhoﬀ’s claims of universality given enclosure are strictly invalid [25]. A perfect absorber must be present. The only means of rectifying this situation is to ﬁnally acknowledge the merit of Stewart’s contributions [26, 49, 83].
3.4 Max Planck and cavity radiation
3.4.1 Whence the carbon particle
In the ﬁrst preface of his book The Theory of Heat Radiation Planck mentions that he has “deviated frequently from the customary methods of treatment, wherever the matter presented or considerations regarding the form of presentation seems to call for it, especially in deriving Kirchhoﬀ’s laws. . . ” [9; p. xi]. Yet, when one reads Planck’s text, the precise nature of the deviations cannot be ascertained and the

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origin of the carbon particle remains a mystery. Since the exposition deals with Kirchhoﬀ, one could be led to assume that the idea came from Kirchhoﬀ [23]. Planck, after all, was a strict theoretician. He relied on experimentalists to give him insight in the particle used for the generation of blackbody radiation. Still, we are never told speciﬁcally that Kirchhoﬀ invoked the carbon particle [23]. It is certain that, at the time of Kirchhoﬀ, virtually all blackbodies were covered with lampblack. Hence, radiation in a cavity whose inner walls were coated with lampblack would have been observed to be independent of the nature of the walls. This simple observation may well have prompted Kirchhoﬀ and Planck to reach for physically profound statements relative to universality while minimizing the role of soot.
The origin of the carbon particle is surely of historical interest. However, with regards to physics, its existence causes concern, not its historical origin. How a particle of carbon entered the perfectly reﬂecting cavity and involved the actions of Kirchhoﬀ, Planck, or another scientist, alters nothing relative to the consequences for universality [23]. What remain critical are Kirchhoﬀ’s claims that blackbody radiation was independent of the nature of the walls of the cavity, whether these were absorbing, transparent or reﬂecting to radiation, provided that thermal equilibrium was maintained [21, 22]. Planck’s invocation of the carbon particle [9] shatters all these arguments [23, 25] and, as such, it is important to repeat the many words of Planck relative to the need for a tiny piece of carbon.
We begin by recalling how Planck himself was well aware that real blackbodies are formed using lampblack. Nothing here is independent of the nature of the walls: “Now, since smooth non-reﬂecting surfaces do not exist . . . it follows that all approximately black surfaces which may be realized in practice (lampblack, platinum black). . . ” [9; §11]. Relative to the carbon particle itself, the ﬁrst key passages come at the end of Part I: “Thus far all the laws derived in the preceding sections for diathermanous media hold for a definite frequency, and it is to be kept in mind that a substance may be diathermanous for one color and adiathermanous for another. Hence the radiation of a medium completely enclosed by absolutely reﬂecting walls is, when thermodynamic equilibrium has been established for all colors for which the medium has a ﬁnite coeﬃcient of absorption, always the stable radiation corresponding to the temperature of the medium such as is represented by the emission of a black body. Hence this is brieﬂy called “black” radiation. On the other hand, the intensity of colors for which the medium is diathermanous is not necessarily the stable black radiation, unless the medium is in a state of stationary exchange of radiation with an absorbing substance” [9; §50]. Planck recognizes that the presence of a perfectly absorbing substance is required within the perfect reﬂector. If this condition is not fulﬁlled, Planck reminds us immediately that: “. . . in a vacuum bounded by totally reﬂecting walls any state of radiation may persist”

[9; §51]. As such, Planck is fully aware that the perfect reﬂector can never produce blackbody radiation in the absence of a perfect absorber. It is not simply a matter of waiting a sufﬁcient amount of time, but rather, the radiation will persist in a non-blackbody or arbitrary state. He re-emphasizes this aspect clearly “Every state of radiation brought about by such a process is perfectly stationary and can continue inﬁnitely long, subject, however, to the condition that no trace of an emitting or absorbing substance exists in the radiation space. For otherwise, according to Sec. 51, the distribution of energy would, in the course of time, change through the releasing action of the substance irreversibly, i.e., with an increase of the total entropy, into the stable distribution corresponding to black radiation” [9; §91].
Planck soon brings the carbon particle front and center: “But as soon as an arbitrarily small quantity of matter is introduced into the vacuum, a stationary state of radiation is gradually established. In this the radiation of every color which is appreciably absorbed by the substance has intensity
K corresponding to the temperature of the substance and determined by the universal function (42) for q = c, the inten-
sity of radiation of the other colors remaining intermediate. If the substance introduced is not diathermanous for any color, e.g., a piece of carbon however small, there exists at the stationary state of radiation in the whole vacuum for all colors
the intensity K of black radiation corresponding to the temperature of the substance. The magnitude of K regarded as a function of  gives the spectral distribution of black radi-
ation in a vacuum, or the so-called normal energy spectrum, which depends on nothing but the temperature. In the normal spectrum, since it is the spectrum of emission of a black body, the intensity of radiation of every color is the largest which a body can emit at that temperature at all” [9; §51].
“It is therefore possible to change a perfectly arbitrary radiation, which exists at the start in the evacuated cavity with perfectly reﬂecting walls under consideration, into black radiation by the introduction of a minute particle of carbon. The characteristic feature of this process is that the heat of the carbon particle may be just as small as we please, compared with the energy of radiation contained in the cavity of arbitrary magnitude. Hence, according to the principle of conservation of energy, the total energy of radiation remains essentially constant during the change that takes place, because the changes in the heat of the carbon particle may be entire neglected, even if its changes in temperature would be ﬁnite. Herein the carbon particle exerts only a releasing (auslo¨send) action. Thereafter the intensities of the pencils of diﬀerent frequencies originally present and having diﬀerent frequencies, directions, and diﬀerent states of polarization change at the expense of one another, corresponding to the passage of the system from a less to a more stable state of radiation or from a state of smaller to a state of larger entropy. From a thermodynamic point of view this process is perfectly analogous, since the time necessary for the process is not essential,

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to the change produced by a minute spark in a quantity of oxy-hydrogen gas or by a small drop of liquid in a quantity of supersaturated vapor. In all these cases the magnitude of the disturbance is exceedingly small and cannot be compared with the magnitude of the energies undergoing the resultant changes, so that in applying the two principles of thermodynamics the cause of the disturbance of equilibrium, viz., the carbon particle, the spark, or the drop, need not be considered. It is always a case of a system passing from a more or less unstable into a more stable state, wherein, according to the ﬁrst principle of thermodynamics, the energy of the system remains constant, and, according to the second principle, the entropy of the system increases” [9; §52]. Planck views the carbon particle simply as a catalyst. He does not recognize that it has a vital function as a perfect absorber. This is a critical oversight, as demonstrated in my review of thermal equilibrium within a perfectly reﬂecting cavity containing a carbon particle [25].
Planck invokes the carbon particle repeatedly throughout his text. This issue is so central to the discussion at hand that all these sections must be brought forth. He writes: “For the following we imagine a perfectly evacuated hollow cylinder with an absolutely tight-ﬁtting piston free to move in a vertical direction with no friction. A part of the walls of the cylinder, say the rigid bottom, should consist of a black body, which temperature T may be regulated arbitrarily from the outside. The rest of the walls including the inner surface of the piston may be assumed to be totally reﬂecting. Then, if the
piston remains stationary and the temperature, T , constant,
the radiation in the vacuum will, after a certain time, assume the character of black radiation (Sec. 50) uniform in all di-
rections. The speciﬁc intensity, K, and the volume density, u, depend only on the temperature, T , and are independent of the volume, V , of the vacuum and hence the position of the
piston” [9; §61]. “Let us also consider a reversible adiabatic process. For
this it is necessary not merely that the piston and the mantle but also that the bottom of the cylinder be assumed as completely reﬂecting, e.g., as white. Then the heat furnished on
compression or expansion of the volume of radiation is Q = 0
and the energy of radiation changes only by the value pdV of the external work. To insure, however, that in a ﬁnite adiabatic process the radiation shall be perfectly stable at every instant, i.e., shall have the character of black radiation, we may assume that inside the evacuated cavity there is a carbon particle of minute size. This particle, which may be assumed to possess an absorbing power diﬀering from zero for all kinds of rays, serves merely to produce stable equilibrium of the radiation in the cavity (Sec. 51 et seq.) and thereby to ensure the reversibility of the process, while its heat contents may be taken as so small compared with the energy of radi-
ation, U , that the addition of heat required for an apprecia-
ble temperature change of the particle is perfectly negligible. Then at every instant in the process there exists absolutely

stable equilibrium of radiation and the radiation has the temperature of the particle in the cavity. The volume, energy, and entropy of the particle may be entirely neglected” [9; §68].
“Let us ﬁnally, as a further example, consider a simple case of an irreversible process. Let the cavity of volume V, which is elsewhere enclosed by absolutely reﬂecting walls, be uniformly ﬁlled with black radiation. Now let us make a small hole through any part of the walls, e.g., by opening of a stopcock, so that the radiation may escape into another completely evacuated space, which may also be surrounded by rigid, absolutely reﬂecting walls. The radiation will at ﬁrst be of a very irregular character; after some time, however, it will assume a stationary condition and will ﬁll both com-
municating spaces uniformly, its total volume being, say, V 0.
The presence of a carbon particle will cause all conditions of black radiation to be satisﬁed in the new state” [9; §69].
“If the process of irreversible adiabatic expansion of the
radiation from the volume V to the volume V 0 takes place as
just described with the single diﬀerence that there is no carbon particle present in the vacuum, after the stationary state of radiation is established, as will be the case after a certain time on account of the diﬀuse reﬂection from the walls of the
cavity, the radiation in the new volume V 0 will not any longer
have the character of black radiation, and hence no deﬁnite temperature . . . If a carbon particle is afterwards introduced into the vacuum, absolutely stable equilibrium is established by a second irreversible process, and, the total energy as well as the total volume remaining constant, the radiation assumes the normal energy distribution of black radiation and the en-
tropy increases to the maximum value S0. . . ” [9; §70].
“Hence, on subsequent introduction of a carbon particle into the cavity, a ﬁnite change of the distribution of energy is obtained, and simultaneously the entropy increases further to
the value S0 calculated in (82)” [9; §103].
Throughout The Theory of Heat Radiation, Planck invokes the carbon particle as a vital determinant of blackbody radiation. Only in the section of the derivation of Wien’s law does he try to minimize the importance of his catalyst. However, in this case, the derivation starts with the presence of a blackbody spectrum a priori. One could argue that Planck goes through great pains to explain that he does not need the particle when, in fact, he has already invoked it to produce the radiation he requires as a starting point. The discussion is well worth reading precisely for the number of times that the carbon particle is utilized: “The starting point of Wien’s displacement law is the following theorem. If the black radiation contained in a perfectly evacuated cavity with absolutely reﬂecting walls is compressed or expanded adiabatically and inﬁnitely slowly, as described above in Sec. 68, the radiation always retains the character of black radiation, even without the presence of a carbon particle. Hence the process takes place in an absolute vacuum just as was calculated in Sec. 68 and the introduction, as a precaution, of a carbon particle is shown to be superﬂuous. But this is true only in this special

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case, not at all in the case described in Sec. 70. . . ” [9; §71]. “Let the completely evacuated hollow cylinder, which is at
the start ﬁlled with black radiation, be compressed adiabatically and inﬁnitely slowly to a ﬁnite fraction of the original volume. If, now, the compression being completed, the radiation were no longer black, there would be no stable thermodynamic equilibrium of the radiation (Sec. 51). It would then be possible to produce a ﬁnite change at constant volume and constant total energy of radiation, namely, the change to the absolutely stable state of radiation, which would cause a ﬁnite increase of entropy. This change could be brought about by the introduction of a carbon particle, containing a negligible amount of heat as compared with the energy of radiation. This change, of course, refers only to the spectral
density of the radiation uv, whereas the total density of the
energy u remains constant. After this has been accomplished, we could, leaving the carbon particle in the space, allow the cylinder to return adiabatically and inﬁnitely slowly to its original volume and then remove the carbon particle. The system will then have passed through a cycle without any external changes remaining. For heat has been neither added nor removed, and the mechanical work done on compression has been regained on expansion, because the latter, like the radiation pressure, depends only on the total density u of the energy of radiation, not on its spectral distribution. Therefore, according to the ﬁrst principle of thermodynamics, the total energy of radiation is at the end just the same as at the beginning, and hence also the temperature of the black radiation is again the same. The carbon particle and its changes do not enter into the calculation, for its energy and entropy are vanishingly small compared with the corresponding quantities of the system. The process has therefore been reversed in all details; it may be repeated any number of times without any permanent change occurring in nature. This contradicts the assumption, made above, that a ﬁnite increase in entropy occurs; for such a ﬁnite increase, once having taken place, cannot in any way be completely reversed. Therefore no ﬁnite increase in entropy can have been produced by the introduction of the carbon particle in the space of radiation, but the radiation was, before the introduction and always, in the state of stable equilibrium” [9; §71].
In reading these sections, it is almost as if Planck has entered into a duel with the carbon particle. He tries to minimize its role, even though it is strictly necessary to his success. In any event, as I have shown [25], when Planck (or Kirchhoﬀ) places the carbon particle inside the perfectly reﬂecting cavity, it is as if the entire cavity had been lined with soot [23]. Thermal equilibrium arguments are powerful, and one of their interesting aspects is that equilibrium does not depend on the extent of the interacting surfaces. This aﬀects only the amount of time required to reach equilibrium, not the nature of the radiation present under equilibrium conditions. Planck’s catalyst is a perfect absorber, and therefore, given equilibrium, it controls the entire situation. The carbon parti-

cle does not simply lead to a distribution of radiation which would have occurred even in its absence.

3.4.2 Planck’s derivation of Kirchhoﬀ’s law

Planck’s derivation of Kirchhoﬀ’s law, as presented in

The Theory of Heat Radiation [9; pp. 1–45], brings the reader

to universality, precisely because reﬂection is not fully con-

sidered. Planck’s exposition is elegant and involves two dis-

tinct parts. The ﬁrst deals with radiation within an object [9;

§4–26] and is eerily similar to Stewart’s formulation [26, 82].

The second examines radiation between “two diﬀerent ho-

mogeneous isotropic substances contiguous to each other . . .

and enclosed in a rigid cover impermeable to heat” [9; §35–

39]. By combining these two parts, Planck arrives at a rela-

tionship which is independent of the nature of the materials

in a manner consistent with his belief in universality.

A cursory examination of this derivation [9; pp. 1–45],

suggests that universality must be valid. Planck seems to

properly include reﬂection, at least when discussing the inter-

face between two separate materials [9; §35–39]. He arrives
with ease at Kirchhoﬀ’s law, q2("= ) = q2K, [9; Eq. 42],
involving the square of the velocity of propagation, q, the co-

eﬃcient of emission, ", the coeﬃcient of absorption, , and the universal function, K. This relationship simpliﬁes to the familiar form "=  = K. The Theory of Heat Ra-
diation focuses, later, on the deﬁnition of the universal func-

tion, which of course, is the right side of Planck’s famous

equation [1, 2]:

"


=

2h3 c2

1 eh=kT

1:

Unfortunately, there is a diﬃculty at the very beginning
of the Planck’s elucidation of Kirchhoﬀ’s law.
In order to arrive at universality [20–22], Planck ﬁrst ex-
amines the equilibrium of radiation within an object. He be-
gins by considering only the emission from a single element
d internal to the object and in so doing, is deliberately ig-
noring reﬂection. Planck writes, in deriving Eq. (1), that the
“total energy in a range of frequency from  to  + d emitted in the time dt in the direction of the conical element d by a volume element d ” [9; §6] is equal to dtd d d2". This
will lead directly to Kirchhoﬀ’s law. If Planck had properly
weighed that the total radiation coming from the element d
was equal to the sum of its emission and reﬂection, he would
have started with dtd d d 2(" + ), which would not
lead to universality.
Planck moves on to examine absorption, by imagining
two elements d and d0 which are exchanging radiation
within the same substance [9; §20]. Finally, he views the
total “space density of radiation” in a sphere at the center
of which is a volume element, , receiving radiation from a small surface element, d [9; §22]. In the end, by combining
his results for emission and absorption, Planck demonstrates

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that within an individual substance, K = "= . He writes
the powerful conclusion that “in the interior of a medium in

a state of thermodynamic equilibrium the speciﬁc intensity

of radiation of a certain frequency is equal to the coeﬃcient

of emission divided by the coeﬃcient of absorption of the

medium for this frequency” [9; §26]. This was the ﬂaw in his

presentation. Had Planck fully included reﬂection, he would

have obtained K = (" + )=(  + ).

Yet, this is only the ﬁrst portion of Planck’s walk to uni-

versality. In order to extend his deduction to all substances,

he must ﬁrst bring two diﬀering materials in contact with one

another. He accomplishes this correctly in §35–38. Properly

treating reﬂection in this case, he is led, as was seen above,
to q2("= ) = q2K [9; §38], a statement of universality.
The equation becomes completely independent of the nature

of the substance. But if Planck had properly executed the ﬁrst

portion of his proof [9; §1–26], he would have been led, for

every substance, once again to K = (" + )=(  + ).

In hindsight, there are many problems with Planck’s

derivation. In the ﬁrst section of his proof, he moves to the

inside of an object. He advances that thermal equilibrium

is achieved internally, not through conduction and the vibra-

tion of atoms, but rather through radiation. While it is true,

as Planck believes, that in a state of thermal equilibrium there

can be no net conduction, it cannot be said that there can be no

conduction. In fact, modern condensed matter physics would

surely argue that thermal equilibrium within objects is sus-

tained through conduction, not radiation. Planck like Stewart

before him [26, 76, 82] invokes internal radiation as a central

component of his proof. He does so precisely to avoid deal-

ing
d

awnidthdreﬂ0 eccatniosnu.sHtaeinasosnulmy eesmthisastiothne,

volume elements d ,
not reﬂection. In so

doing, he predetermines the outcome he seeks, beginning as

we have seen with his equation (1) [9; §6].

3.5 Graphite, carbon-black, and the modern age
Graphite and soot, whose commercial forms include carbon black [88] and black carbon [89], continue to be at the center of nearly all blackbody experiments conducted by the National Bureau of Standards and other laboratories. Nonetheless, certain metal blacks [88], namely platinum black and gold black [90–92], have a narrow range of uses as absorbers, especially at long wavelengths. Platinum black is usually prepared by electroplating the surface with platinum. Gold black is particularly interesting as a material. It is produced, by vaporizing the metal onto a substrate until thin gold ﬁlms are generated. In this sense, the conductivity of gold is being structurally limited and the resulting material is black. In the end, the metal blacks are used primarily in the infrared, and their applications, while important, even in the days following Planck, are somewhat limited.
It remains the overwhelming case that the walls of many cavities are still made from graphite [93–97]. However, if

they are made of alternate materials (i.e. brass [98], copper [99], clay [93]), they are either blackened, or smoked with soot [98], or they are covered with black paint [93, 96, 98– 104]. Some of these paints have proprietary contents. Nonetheless, it is relatively certain that they all contain the carbon black pigment [105, 106]. For instance, the author has been able to verify that Aeroglaze Z306 and Z302 both contain carbon black (private communication, Robert Hetzell, Lord Corporation, Erie, PA). The same can be ascertained relative to Nextel Velvet coating P/N101-C10 black. It is true that carbon black, with its extremely high carbon content remains the premium black pigment [105]. Graphite and soot (carbon black, black carbon) continue to absolutely dominate all work with experimental blackbodies.
Even ﬁxed point blackbodies [95] which operate at the freezing points of elements such as gold [95], aluminum, zinc, and tin [100] rely either on graphite [107] walls or cavities coated with black paints. In these ﬁxed point blackbodies, the metal freezing/melting point ensures that the entire surface of the emitter can be temporarily maintained at a unique temperature. Interestingly, the metals themselves appear to be relatively innocuous or transparent to emission by the graphitic, or carbon lined, surfaces of the cavity.
There are restrictions on the quality of freezing point blackbody cavities, and these have been outlined by Geist [108]: “How well the actual radiance approaches the ideal radiance in a given blackbody is often referred to in a qualitative manner as the quality of the blackbody...The principle restriction on the concept of quality...is that it can only be deﬁned for radiation from blackbodies with wall materials whose thermal radiative parameters are independent of wavelength. One important class of freezing point blackbody for which this is not a serious restriction is the class whose cavity walls are constructed from graphite.” A mathematical treatment of laboratory blackbodies reveals that the production of a cavity whose performance will yield a high quality blackbody is not a trivial task [109].
In any event, it remains clear that whether a blackbody is designed to operate at the freezing point of an element or not, graphite [31, 107], or soot (carbon black [105, 106], or black carbon [89]) continue to dominate this ﬁeld.
4 Conclusion
Through the exposition of Kirchhoﬀ’s law, we have been able to highlight that universality does not hold in cavity radiation. The great bulk of experimental evidence leads to this conclusion. Indeed, if blackbody radiation was universal, there would be no need for the National Bureau of Standards to utilize graphite or soot in order to study such processes. The absence of cavities made of arbitrary walls (without any trace of a perfect absorber) is the best physical proof that universality does not hold. Our laboratories require carbon. Nothing further is needed to shatter Kirchhoﬀ’s belief. Nonetheless, even

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the simplest of mathematical considerations suﬃces to illustrate the point [25]. Perfectly reﬂecting cavities, containing no objects, emit no radiation [25]. Perfectly reﬂecting cavities which contain objects emit radiation which is characteristic of these objects [25]. Thus, if a carbon particle is placed within a perfectly reﬂecting cavity, the cavity will be black, irrespective of the size of the particle. This is a testament to the power of thermal equilibrium; but if the particle is small, it may take some time to reach this equilibrium. Perfectly absorbing cavities emit normal, or blackbody radiation [23, 25]. In such a cavity, the proper description of the radiation
from an arbitrary object is (" + )=(  + ) = f(T; )
[25]. This equation echoes Stewart [26, 49, 82]. Conversely,
Kirchhoﬀ incorrectly advanced "=  = f(T; ), leading to
universality [20–22]. Consequently, when examining blackbody radiation, we
are not dealing with a phenomenon of universal signiﬁcance. Rather, we are dealing with a physical process which is extremely limited in its applications. Blackbodies are made of solids, and speciﬁcally relative to practical blackbodies, they are made of graphite. Nature knows no equivalent as is well demonstrated by the review of thermal emissivity tables [31]. Yet, even in the case of radiation from graphite, the physical cause of the process remains remarkably unknown to modern science. The physical species producing blackbody emission has not been concretely identiﬁed [19, 23].
If Planck’s law [1, 2] has not been linked to a physical species, it is in part certain that the formulation of Kirchhoﬀ’s law [20–22], in its creation of universality, hindered the process. At the same time, there is a fundamental diﬃculty in providing a complete physical picture relative to thermal emission. This is because the nature of the oscillators, at the heart of thermal radiation, can change depending on the physical nature of the material being examined. The thermal emission proﬁles of metals are highly aﬀected by their conduction electrons, at least in the sense that their presence acts to prevent emission and favor reﬂection. For each opaque material, a unique emission proﬁle exists [31] and the answer to these problems will most likely involve the use of computational tools, not simple algebraic solutions. It may well be that entire lattices will have to be represented and processed in digital forms, in order to yield meaningful results. Yet, some thermal emission proﬁles, which provide Planck-like behavior, such as graphite, the microwave background (only apparent Planckian behavior), and the emission of the photosphere (only apparent Planckian behavior), may be capable of being solved analytically. A solution for one of these is likely to have broad implications for the others. At the same time, only graphite will remain truly Planckian in nature, as it is the only one restricted to a solid. The microwave background and the photosphere produce only apparent Planckian spectra. Since their physical sources are not solids, their relevant internal bonds (if any) are weak, and they support convection processes which alter the validity of

the temperatures they report [33].

For graphite or soot

"


2h3 c2

1 eh=kT

1

as Planck derived [1, 2]. Conversely, for the Sun and the

microwave background, we can write that

"


2h3 c2

1 eh=kT app

1;

where Tapp is constant. Tapp = T =, where T is the real tem-

perature of the source and  is a variable, with temperature
 dependence, whose value is 1,000 for the photosphere and

100 for the microwave background [33]. Thus, the real tem perature of the photosphere is 1,000 times higher than the

currently accepted temperature [34, 35]. Similarly, the tem-
 perature for the source of the microwave background is 100

times higher than the measured value [33, 39, 40]. These

complications arise because we are dealing with non-solids

outside the conﬁnes of enclosure [23, 33].

If a Planckian approach is used to analyze graphite, the

carbon nucleus can be viewed as the mass and the carbon-

carbon bond as the spring in an oscillator scenario [1, 2].

If the microwave background is conﬁrmed to be from an

oceanic source [33, 36–42], then the oscillators might be en-

tire water molecules, linked through weak hydrogen bond-

ing, vibrating within a ﬂeeting lattice. In this regard, it re-

mains interesting that water can become completely black.

This occurs, for instance, when shock waves from nuclear

explosions propagate in the sea. For the photosphere, if a

hydrogen-based condensed Sun is contemplated [34, 35], the

vibration of protons within a ﬂeeting lattice ﬁeld will have to

be considered. In this case, the electrons might simply oc-

cupy conduction bands. Nonetheless, the nuclei should be

viewed as being conﬁned to a distinct condensed structure

which, though ﬂeeting, is being maintained, perhaps only by

the need to sustain the quantum mechanical requirements to

produce the conduction bands. Physicists versed in the prop-

erties of condensed liquid metallic hydrogen might consider

these questions. Only the future can reveal how mankind

moves forward on linking a given physical species to a center

of emission.

With the loss of the universal function, the proper treat-

ment of materials will involve the long recognized fact that
the ratio of the emission, e, of an object to its absorption, a, is equal to a complex function dependent on its temperature, T , its nature, N , (its shape, the roughness of its sur-

face, its speciﬁc heat, etc.), and the wavelengths of inter-

est, namely e=a = f(T; N; ). Also, e and a, individually,

are functions of these parameters, otherwise, as Agassi high-

lights [30], spectroscopy would be impossible. The afore-

mentioned equation can be simpliﬁed to Kirchhoﬀ’s formu-
lation e=a = f(T; ) only within a perfectly absorbing enclo-

sure or within an enclosure where a perfect absorber is also

present. In all these cases, the object never truly becomes a

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blackbody. Along with its own emission, it simply reﬂects radiation in the cavity and appears to hold blackbody properties. It is diﬃcult to envision how this scenario is of any use in modern physics.
The physics community has persisted in upholding Kirchhoﬀ’s law of thermal emission even though it has been refuted both recently [23–25] and in the past (see [86] for a discussion of the controversy surrounding Kirchhoﬀ’s law). This has occurred despite the fact that graphite and soot are uniquely positioned in all blackbody work with cavities. Nonetheless, some of this hesitance may be due to a certain respect, even reverence, for Kirchhoﬀ and his work. In part, there is also the proximity to Planck himself. Such concerns are unjustiﬁed, in that even if Kirchhoﬀ’s law loses its universal status, nothing changes relative to Planck’s derivation. Planck’s law [1, 2] simply becomes devoid of universal signiﬁcance. It maintains its value relative to the treatment of radiation within perfectly absorbing enclosures and within perfectly reﬂecting enclosures which contain a perfect absorber. Of course, Planck’s equation will no longer extend to simple perfectly reﬂecting enclosures.
At the same time, the merit of k and h, at the heart of
Planck’s law, is not altered. The great changes simply involve the interdict of extending the laws of thermal emission [1, 2, 110, 111], without modiﬁcation, to objects which are not solids [33–42] or enclosed within perfectly absorbing cavities [23–25].
Despite these facts, it may well be that physics remains unwilling to pronounce itself relative to the invalidity of Kirchhoﬀ’s treatment until the consequences of the error become so great that society demands retraction. The reassignment of the microwave background to the Earth [33, 36–42] should eventually provide suﬃcient motivation to act. On that day, a new age in astrophysics will spring forth [34, 35] and we may ﬁnally begin to write the long-awaited ode to Balfour Stewart.
Acknowledgements
The author would like to thank Bernadette Carstensen for the illustrations of Gustav Robert Kirchhoﬀ and Balfour Stewart. The staﬀ of The Ohio State University rare books collection and book depository, especially Mrs. Rebecca Jewett, is also recognized.
Dedication
This work is dedicated to my sister He´le`ne, her husband Gervais Be´dard, their children (Sonia, Karl, and Genevie`ve) and
their grandchildren (Megan Gagne´, Raphae¨l Turcottey, and
The´oge`nes Turcotte) on this, their 30th wedding anniversary (May 22, 1978).
Submitted on May 06, 2008 / Accepted on May 12, 2008 First published online on May 22, 2008 / Corrected on June 25, 2008

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56. Pre´vost P. Me´moire sur l’equilibre du feu. Journal de Physique, 1791, v. 38, 314–322 (translated in Harper’s Scientiﬁc Memoirs (J.S. Ames, Ed.) — The Laws of Radiation and Absorption: Memoirs of Pre´vost, Stewart, Kirchhoﬀ, and Kirchhoﬀ and Bunsen (translated and edited by D.B. Brace), American Book Company, New York, 1901, 1–13).
57. Pre´vost P., Du calorique rayonnant. JJ Paschoud, Geneva & Paris 1809 (Sections are translated in Harper’s Scientiﬁc Memoirs (J.S. Ames, Ed.) — The Laws of Radiation and Absorption: Memoirs of Pre´vost, Stewart, Kirchhoﬀ, and Kirchhoﬀ and Bunsen (translated and edited by D.B. Brace), American book company, New York, 1901, 15–20).
58. Pre´vost P., Recherches physico-me´caniques sur la chaleur. Barde, Manget, Geneve: Me´rigot, Paris, 1792.
59. Herschel W. Investigation of the powers of the prismatic colors to heat and illuminate objects; with remarks, that prove the diﬀerent refrangibility of radiant heat. To which is added, an inquiry into the method of viewing the Sun advantageously, with telescopes of large apertures and high magnifying powers. Phil. Trans., 1800, v. 90, 255–283.
60. Dulong P.L., Petit A.T. Recherches sur la mesure des tempe´ratures et sur les lois de la communication de la chaleur. Journal de L’E´cole Polytechnique, 1820, v. 11(18), 189–294.

61. Fourier J. Me´moire sur la Propagation de la Chaleur dans les Corps Solides. Nouveau Bulletin des Sciences par la Socie´te´ Philomathique de Paris, 1808, v. 1, 112–116.
62. Fourier J. The´orie de la chaleur. Annales de Chimie et de Physique, 1816, v. 3, 350–375.
63. Fourier J. Questions sur la the´orie-physique de la chaleur rayonnante. Annales de Chimie et de Physique, 1817, v. 6, 259–303.
64. Fourier J. The´orie du movement de la chaleur dans les corps solides. Me´moires de l’Acade´mie Royale des Sciences, 1819– 1820, v. 4, 185–555 (section VII deals with the solution for the spherical enclosure).
65. Fourier J. The´orie du movement de la chaleur dans les corps solides (suite). Me´moires de l’Acade´mie Royale des Sciences, 1821–1822, v. 5, 153–246.
66. Fourier J. Re´sume´ the´orique des proprie´te´s de la chaleur rayonnante. Annales de Chimie et de Physique 1824, v. 27, 236–287.
67. Fourier J. Me´moire sur la the´orie analytique de la chaleur. Me´moires de l’Acade´mie Royale des Sciences, 1825, v. 8, 581–622.
68. Poisson S.D. Me´moire sur la distribution de la chaleur dans les corps solides. Journal de l’E´cole Polytechnique. Paris, 1823, v. 12, 249–403.
69. Poisson S.D. Sur la chaleur rayonnante. Annales de Chimie et de Physique, 1824, v. 26, 225–245.
70. de la Provostaye F., Desains P. Me´moire sur le rayonnement de la chaleur. Annales de Chimie et de Physique, 1846, v. 16, 337–425.
71. de la Provostaye F., Desains P. Me´moire sur le rayonnement de la chaleur. Annales de Chimie et de Physique, 1848, v. 18, 358–431.
72. de la Provostaye F., Desains P. Me´moire sur la de´termination du pouvoir absorbant des corps pour la chaleur rayonnante, Annales de Chimie et de Physique, 1850, v. 30, 431–443.
73. de la Provostaye F., Desains P. Me´moire sur la reﬂexion de la chaleur. Annales de Chimie et de Physique, 1850, v. 30, 276–286.
74. de la Provostaye F., Desains P. Me´moire sur la polarisation de la chaleur par re´fraction simple. Annales de Chimie et de Physique, 1850, v. 30, 159–178.
75. de la Provostaye F., Desains P. Me´moire sur la diﬀusion de la chaleur, Annales de Chimie et de Physique, 1852, v. 34, 192–225.
76. Siegel D.M. Balfour Stewart and Gustav Robert Kirchhoﬀ: two independent approaches to Kirchhoﬀ’s Law. Isis, 1976, v. 67(4), 565–600.
77. Roscoe H.E. Letter from Prof. Kirchhoﬀ on the chemical analysis of the solar atmosphere. Phil. Mag., 1861, v. 21, 185–188.
78. Roscoe H.E. On the solar spectrum, and the spectra of the chemical elements (an extract of Kirchhoﬀ). Phil. Mag., 1862, ser. 4., v. 23, 63–64.
79. Roscoe H.E., Bunsen R. Photochemical researches: the optical and chemical extinction of the chemical rays. Phil. Mag., 1858, ser. 4, v. 15, 230–233.

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80. Roscoe H.E. Spectrum analysis. D. Appleton and Co., New York, 1869.
81. Kirchhoﬀ G. Contributions towards the history of spectrum analysis and of the analysis of the solar atmosphere. Phil. Mag., 1863, ser. 4, v. 25, 250–262 (also found in part within Roscoe’s Spectrum Analysis, see [80], 115–122).
82. Stewart B. Reply to some remarks by G. Kirchhoﬀ in his paper “On the history of spectrum analysis”. Phil. Mag., 1863, ser. 4, v. 25, 354–360.
83. Stewart B. Researches on radiant heat. Trans. Royal Soc. Edinburgh., 1859, v. 22(I), 59–73 (also found in Harper’s Scientiﬁc Memoirs (J.S. Ames, Ed.) — The laws of radiation and absorption: memoirs of Pre´vost, Stewart, Kirchhoﬀ, and Kirchhoﬀ and Bunsen (translated and edited by D.B. Brace), American Book Company, New York, 1901, 51–73).
84. Stewart B. On the radiative powers of bodies with regard to the dark or heat-producing rays of the spectrum. Phil. Mag., 1860, ser. 4, v. 20, 169–173.
85. Stewart B. On the light radiated by heated bodies (with supplements added). Phil. Mag., 1860, ser. 4, v. 20, 534–540.
86. Schirrmacher A. Experimenting theory: the proofs of Kirchhoﬀ’s radiation law before and after Planck. Munich Center for the History of Science and Technology, 2001 (available online: http://www.mzwtg.mwn.de/arbeitspapiere/Schirrmacher 2001 1.pdf).
87. de la Provostaye F. Conside´rations the´oriques sur la chaleur rayonnante. Annales de Chimie et de Physique, 1863, v. 67, 5–51.
88. Harris L. The optical properties of metal blacks and carbon blacks. MIT and The Eppley Foundation for Research, Monograph Ser., 1, New Port, R.I., Dec. 1967.
89. Smith D.M., Chughtai A.R. The surface structure and reactivity of black carbon. Colloids and Surfaces A, 1995, v. 105, 47–77.
90. Nelms N., Dowson J. Goldblack coating for thermal infrared detectors. Sensors and Actuators, 2005, v. A120, 403–407.
91. O’Neill P., Ignatiev A, Doland C. The dependence of optical properties on the structural composition of solar absorbers: Gold black. Solar Energy, 1978, v. 21, 465–478.
92. Harris L., McGuiness R.T., Siegel B.M. The preparation and optical properties of gold black. J. Opt. Soc. Am., 1948, v. 38, 582.
93. Murphy A.V., Tsai B.K., Saunders R.D. Comparative calibration of heat ﬂux sensors in two blackbody facilities. J. Res. Nat. Inst. Stand. Technol., 1999, v. 104, 487–494.
94. Murphy A.V., Tsai B.K., Saunders R.D. Transfer calibration validation tests on a heat ﬂux sensor in the 51 mm hightemperature blackbody. J. Res. Nat. Inst. Stand. Technol., 2001, v. 106, 823–831.
95. Navarro N., Bruce S.S., Carol Johnson B., Murphy A.V, Saunders R.D.. Vacuum processing techniques for development of primary standard blackbodies. J. Res. Nat. Inst. Stand. Technol., 1999, v. 104, 253–259.

96. Chahine K., Ballico M., Reizes J., Madadnia J. Optimization of a graphite tube blackbody heater for a thermogage furnace. Int. J. Thermophys., 2008, v. 29, 386–394.
97. Hanssen L.M., Mekhontsev S.N., Zeng J., Prokhorov A.V. Evaluation of blackbody cavity emissivity in the infrared using total integrated scatter measurements. Int. J. Thermophys., 2008, v. 29, 352–369.
98. Birkebak R.C. A technique for measuring spectral emittance. Rev. Scient. Instr., 1972, v. 43 (7), 1027–1030.
99. Fowler J.B. A third generation water bath based blackbody source. J. Res. Nat. Inst. Stand. Technol., 1995, v. 100, 591–599.
100. Schalles M., Bernhard F. Triple-ﬁxed-point blackbody for the calibration of radiation thermometers. Int. J. Thermophys., 2007, v. 28, 2049–2058.
101. Bauch A., Schroder R. Experimental veriﬁcation of the shift of the cesium hyperﬁne transition frequency due to black radiation. Phys. Rev. Let., 1997, v. 78(4), 622–625.
102. Best F.A., Revercomb H.E., Knueson R.O., Tobin D.C., Ellington S.D., Werner M.W., Adler D.P., Garcia R.K., Taylor J.K., Ciganovich N.N., Smith W.L., Bingham G.E., Elwell J.D., Scott D.K. The geosynchronous imaging Fourier transform spectrometer (GIFTS) on-board blackbody calibration system. Proc. Soc. Photo-Optical Instrum. Eng., Fourth Int. Asia-Paciﬁc Environmental Remote Sensing Symposium, 8 November, 2004 (available online: http://www.ssec.wisc.edu/gifts/blackbody/bibliography/ SPIE 2004 5655-10 Best.pdf)
103. Fowler J.B., Carol Johnson B., Rice J.P., Lorenz S.R. The new cryogenic vacuum chamber and black-body source for infrared calibrations at the NIST’s FARCAL facility. Metrologia, 1998, v. 35, 323–327.
104. Fowler J.B. An oil-bath-based 293 K to 473 K blackbody source. J. Res. Natl. Inst. Stand. Technol., 1996, v. 101, 629–637.
105. Buxbaum C. Industrial inorganic pigments. 2nd Edition, Wiley-Interscience, New York, 1998
106. International Carbon Black Association. Carbon black user’s Guide (online: http://carbon-black.org/user guide.html).
107. Burchell T.D. Thermal properties and nuclear energy applications. In: Graphite and Precursors, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 2001, 87–109.
108. Geist J. Note on the quality of freezing point blackbodies. Applied Optics, 1971, v. 10(9), 2188–2190.
109. Geist J. Theoretical analysis of laboratory blackbodies 1: a generalized integral equation. Applied Optics, 1973, v. 12(6), 1325–1330.
110. Wien W. U¨ ber die Energieverteilung in Emissionspektrum eines schwarzen Ko¨rpers. Ann. Phys., 1896, v. 58, 662–669.
111. Stefan J. U¨ ber die Beziehung zwischen der Warmestrahlung und der Temperature. Sitzungsberichte der mathematischnaturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften, Wien 1879, v. 79, 391–428.

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The Roˆle of the Element Rhodium in the Hyperbolic Law of the Periodic Table of Elements
A. Khazan
2265 Ocean Parkway, Apt. 5N, Brooklyn, NY 11223, USA E-mail: albkhazan@list.ru
The roˆle of the element rhodium as an independent aﬃrmation of calculations by the Hyperbolic Law and validity of all its relations is shown herein. The deviation in calculation by this method of the atomic mass of heaviest element is 0.0024%, and its coeﬃcient of scaling 0.001–0.005%.

1 Introduction
The method of rectangular hyperbolas assumes that their peaks (i.e. vertices) should be determine with high accuracy. For this purpose the theorem of Lagrange and the coeﬃcient of scaling calculated by the Author for transition from the system of coordinates of the image of a hyperbola, standard practice of the mathematician, and used in chemistry, are utilized. Such an approach provides a means for calculating the parameters of the heaviest element in the Periodic Table of D. I. Mendeleyev [1].
In the ﬁrst eﬀect of the Hyperbolic Law it is shown that to each direct hyperbola corresponds an adjacent hyberbola:
they intersect on the line Y = 0.5 at a point the abscissa of
which is twice the atomic mass of an element [2]. This fact is clearly illustrated for Be, Ca, Cd in Fig. 1.
Upon close examination of the ﬁgure deeper relationships become apparent:
• From the centre of adjacent hyperbolas (X = 0, Y = 1)
the secants have some points of crossing, the principal
of which lie on the line Y = 0.5 and on the virtual axes
(peaks);
• The secants intersect a direct hyperbola in two points, with gradual reduction of a segment with the increase in molecular mass;
• Behind the virtual axis of adjacent hyperbolas the secants cut a direct hyperbola in only one point;
• In conformity therewith, the magnitude of the abscissa, between a secant and a point of intersection of hyper-
bolas on the line Y = 0.5, also changes;
• For the element rhodium the secant becomes a tangent and also becomes the virtual axis of adjacent hyperbolas.
2 Mathematical motivation
On the basis of the presented facts, we have been led to calculations for 35 elements to establish the laws for the behavior of secants. The results are presented in the table for the following parameters:

• Atomic numbers of elements and their masses;
• Calculated coordinates of peaks of elements (the square root of the atomic mass and coeﬃcient of scaling 20.2895 are used);
• Abscissas of secants on the line Y = 0.5 are deduced
from the equation of a straight lines by two points

(X (X2

X1) X1)

=

(Y (Y2

Y1) Y1)

(column 6);

• Points of intersection of direct and adjacent hyperbolas (column 7);
• Diﬀerence between the abscissas in columns 6 and 7 (column 8);
• Tangent of an inclination of a secant from calculations for column 6.
According to columns 6 and 7 in Fig. 2, dependences which essentially diﬀer from each other are obtained. Abscissas of secants form a curve of complex form which can describe with high reliability (size of reliability of approxi-
mation R2 = 1) only a polynomial of the ﬁfth degree. The second dependency has a strictly linear nature (Y = 2X), and
its straight line is a tangent to a curve at the point (102.9055, 205.811). For clarity the representation of a curve has been broken into two parts: increases in molecular mass (Fig. 3) and in return — up to hydrogen, inclusive (Fig. 4). The strongly pronounced maximum for elements B, C, N, O, F, Ne is observed.
At the end of this curve there is a very important point at which the ordinate is equal to zero, where (the line of rhodium in the table) the data of columns 6 and 7 coincide.
Thus it is unequivocally established that for rhodium the secant, tangent and the virtual axis for an adjacent hyperbola are represented by just one line, providing for the ﬁrst time a means to the necessary geometrical constructions on the basis of only its atomic mass (the only one in the Periodic Table), for the proof of the Hyperbolic Law.
Graphical representation of all reasoning is reﬂected in Fig. 5 from which it is plain that the point with coordinates (205.811, 0.5) is the peak of both hyperbolas, and the peaks

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of Ca and Ta are on both sides of it. Below are the calculations for the basic lines of rhodium on these data:

1. A secant: —

(X 0) (205:811

0)

=

(Y (0:5

1) 1)

;

whence
Y = 0:0024294134X + 1 :

At Y = 0, X = 411.622; in this case coordinates of peak will be: X = 205.811, Y = 0.5.

2. A tangent:— the equation of a direct hyperbola,

Y

=

102:9055 X

;

its derivative at X = 205.811, so

Y0 =

102:9055 205:8112

=

0:0024294134 ;

Y 0:5 = 0:0024294134X + 0:5 :

Finally,
Y = 0:0024294134X + 1 ;

at Y = 0, X = 411:622.

3. A normal: — (the virtual axis),

Y = 0:0024294134X ;

at Y = 1, X = 411:622.

Here are the same calculations for the tabulated data presented:

1. A secant: —

X 205:82145

=

(Y 1) (0:4999746

1) ;

whence

Y = 0:0024294134X + 1 ; Y = 1 ; X = 411:622 :

2. A tangent: —

Y

=

102:9055 X

;

the ﬂuxion at X = 205:821454,

Y0 =

102:9055 205:821452

=

0:0024291667 ;

so

Y 0:4999746 = 0:0024291667(X 205:82145) ;

whence

Y = 0:0024291667X + 0:99994928 ;

Y = 0 ; X = 411:6429 :

3. A normal: —

Y = 0:0024291667X ;

Y = 1 ; X = 411:6638 :

3 Comparative analysis calculations

For a secant the results are identical with the ﬁrst set of calculations above, whereas for a tangent and normal there are some deviations, close to last element calculated.
By the ﬁrst set of calculations above its atomic mass is
411.622; hence the deviation is 411.663243 411.622 = = 0.041243 (0.01%). By the second set the size of a tan-
gent and a normal are close to one another (an average of 411.65335) and have a smaller deviation: 411.663243
411.65335 = 0.009893 (0.0024%). This is due to the tan-
gent of inclination of the virtual axis of a direct hyperbola in the ﬁrst set is a little high.
Using rhodium (Fig. 5) we can check the propriety of a choice of coeﬃcient of scaling. It is necessary to make the following calculations for this purpose:
• Take the square root of atomic mass of rhodium (X = = Y = 10.1442348);
• Divide X0 by X of the peak (205.811/10.1442348 = = 20.2885);
• Divide Y = 10.1442348 by Y0 of the peak (0.5): also
gives 20.2885;
• The diﬀerence by X and Y with the coeﬃcient obtain-
ed, 20.2895, yielding the same size at 0.001 or 0.005%.
Formulae for transition from one system of coordinates to another have been given in the ﬁrst paper of this series.
Using data for peaks, from the table, we get the following results:
Coordinates of peak

X0 = 205:8215; Y0 = 0:49997;

X = Y = 10:1442348;

then

X0 X

=

20:2895;

Y Y0

=

20:2897;

i. e. absolute concurrence (maximum diﬀerence of 0.0009%).

4 The roˆle of the element Rhodium
However, all these insigniﬁcant divergences do not belittle the most important conclusion: that the validity of the Hyperbolic Law is estabished because the data calculated above completely coincide with calculations for rhodium is proved, based only on its atomic mass.
All the calculations for the table were necessary in order to ﬁnd a zero point for rhodium, for which it is possible to do so without calculating the secant, but using only its atomic mass, thereby verifying the Hyperbolic Law.
How to get the correct choice of abscissa of a secant is depicted in Fig. 6 (using beryllium as an example) where instead of its tabulated value, 35.7434, the value equal to twice the point of intersection (36.0488) has been used. Here we

Albert Khazan. The Roˆle of the Element Rhodium in the Hyperbolic Law of the Periodic Table of Elements

57

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Fig. 1

tried to make a start from any ﬁxed point not calculated (similar to the case for rhodium). It has proved to be impossible and has led to a mistake in the deﬁnition of the peak. In Fig. 7 the geometrical constructions for beryllium on the basis of correct settlement of data are given.

the valid axis. The divergence with tabular data is in the order of 0.004%–0.009%.
Thus rhodium provides an independent veriﬁcation of the method of rectangular hyperbolas for the Periodic Table of elements of D. I. Mendeleyev.

5 Conclusions

Submitted on May 08, 2008 Accepted on June 19, 2008

Previously we marked complexity of a choice of peak of a hyperbola of an element in the coordinates, satisfying the condi-
tions Y 1, K X, as on an axis of ordinates the maximum
value being a unit whilst the abscissa can take values in the hundreds of units. The problem has been solved by means of the theorem of Lagrange and the coeﬃcient of scaling deduced. On the basis thereof our further conclusions depended, so it was very important to ﬁnd a method not dependent on our calculations and at the same time allowing unequivocally to estimate the results. Owing to properties of the virtual axis of an rectangular hyperbola on which peaks of all elements lie, it is enough to have one authentic point.
Analyzing the arrangement of the virtual axes of direct and adjacent hyperbolas, we have paid attention to their point of intersection (205.83, 0.5), the abscissa of which is exactly half of atomic mass of the last element. As secants from the
centre X = 0, Y = 1 cut direct hyperbolas any way (Fig. 1),
we have been led to necessary calculations and have obtained a zero point at which the secant coincides with a tangent and

References
1. Khazan A. Upper limit in the Periodic Table of Elements. Progress in Physics, 2007, v. 1, 38.
2. Khazan A. Eﬀect from hyperbolic law in Periodic Table of Elements. Progress in Physics, 2007, v. 2, 83.

58

Albert Khazan. The Roˆle of the Element Rhodium in the Hyperbolic Law of the Periodic Table of Elements

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PROGRESS IN PHYSICS

Volume 3

12

3

4

5

6

7

8

9

El. No. At. mass X0 peak Y0 peak Abs. secant Cross. hyperb. ∆ = 6 7 tan a, secant

H

1 1.0079 20.3695 0.04948

10.715

2.0158

8.6992

0.046664

He

2 4.0026 40.5992 0.0986

22.5163

8.0052

14.5111

0.0222

Li

3 6.941 53.4543 0.12985

30.7155

13.882

16.8335

0.01628

Be

4 9.0122 60.9097 0.14976

35.7434

18.0244

17.719

0.014

B

5 10.811 66.712 0.162055 39.80692 21.622

18.18492 0.01256

C

6 12.0107 70.3162 0.1708

42.4

24.0214

18.3786

0.0117923

N

7 14.0067 75.9345 0.184458 46.5546

28.0134

18.5412

0.01074

O

8 15.9994 81.1565 0.197143 50.5423

31.9988

18.5435

0.009893

F

9 18.9984 88.4362 0.21483

56.3163

37.9968

18.3195

0.008878

Ne 10 20.1797 91.1441 0.2214

58.5311

40.3594

18.1717

0.0085425

Mg 12 24.305 100.0274 0.242983 66.0669

48.61

17.4569

0.007568

S

16 32.065 114.89125 0.27909

79.6849

64.13

15.5549

0.006273

Ca 20 40.078 128.4471 0.31202

93.3508

80.156

13.1948

0.005356

Cr 24 51.9961 146.3042 0.3554

113.484

103.9922

9.4918

0.004406

Zn 30 65.409 164.093 0.3986

136.428

130.818

5.61

0.003665

Br 35 79.904 181.366 0.44057 162.0982 159.808

2.29

0.003085

Zr 40 91.224 193.7876 0.47074 183.075

182.448

0.627

0.002731

Mo 42 95.94 198.7336 0.482757 192.1085 191.88

0.2285

0.002603

Rh 45 102.906 205.82145 0.4999746 205.811

205.811

0

0.00242941

Cd 48 112.411 215.1175 0.52256 225.26

224.822

0.458

0.00221946

Ba 56 137.327 237.7658 0.577573 281.428

274.654

6.774

0.001777

Nd 60 144.242 243.6785 0.591936 298.5785 288.484

10.09455 0.0016746

Sm 62 150.36 248.7926 0.60436 314.417

300.72

13.7

0.00159

Dy 66 162.5

258.6414 0.628283 347.9

325

22.9

0.001437

Yb 70 173.04 266.8976 0.64834 379.48

346.08

33.4

0.0013176

Hf 72 178.49 271.068 0.65847 396.843

356.98

39.863

0.00126

Ta 73 180.948 272.928 0.663

404.923

361.896

43.027

0.0012348

Re 75 186.207 276.8658 0.67255 422.7646 372.414

50.35

0.0011827

Ir

77 192.217 281.2984 0.68332 444.1376

384.434

59.704

0.0011258

Hg 80 200.59 287.3598 0.698

475.8318 401.18

74.6518

0.00105

At 85 210

294.0228 0.71423 514.44

420

94.44

0.000972

Fr 87 223

302.9868 0.736

573.85

446

127.85

0.00087

Th 90 232.038 309.0658 0.75077 620.0472 464.07612

155.971

0.000806

Am 95 243

316.282 0.7683

682.53

486

196.53

0.0007326

Es 99 252

322.0858 0.7824

740.0874 504

236.0874

0.0006756

a) columns 4 and 5 contain coordinates of peaks of rectangular hyperbolas of elements;
b) in a column 6 are presented abscissas the secants which are starting with the peak center (0,1) up to crossings
with line Y = 0:5; at prolongation they cross the valid axis in points peaks;
c) in a column 7 are resulted abscissa points of crossing of a direct and adjacent hyperbola each element presented here;
d) the column 8 contains a diﬀerence between sizes of 6 and 7 columns;
e) in a column 9 tangents of a corner of an inclination of secants are resulted; at an element “rhodium” this line
crosses an axis X in a point with abscissa, equal 411.622, and its position coincides with tangent in peak; 411:66 411:62 = 0:04 or nearly so 0.01% from atomic mass.

Table 1: Results of calculations for some elements of the Periodic Table

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Fig. 2

Fig. 3

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Fig. 4

Fig. 5

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Fig. 6

Fig. 7

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Volume 3

What Gravity Is. Some Recent Considerations

Vic Christianto and Florentin Smarandachey

Sciprint.org — a Free Scientiﬁc Electronic Preprint Server, http://www.sciprint.org

yDepartment

of

E-mail: admin@sciprint.org Mathematics, University of New Mexico,

Gallup,

NM

87301,

USA

E-mail: smarand@unm.edu

It is well-known, that when it comes to discussions among physicists concerning the meaning and nature of gravitation, the room temperature can be so hot. Therefore, for the sake of clarity, it seems worth that all choices were put on a table, and we consider each choice’s features and problems. The present article describes a nonexhaustive list of such gravitation theories for the purpose of inviting further and more clear discussions.

1 Introduction
The present article summarizes a non-exhaustive list of gravitation theories for the purpose of inviting further and more clear discussions. It is well-known, that when it comes to discussions among physicists concerning the meaning and nature of gravitation, the room temperature can be so hot. Therefore, for the sake of clarity, it seems worth that all choices were put on a table, and we consider each choice’s features and problems. Of course, our purpose here is not to say the last word on this interesting issue.
2 Newtonian and non-relativistic approaches
Since the days after Newton physicists argued what is the meaning of “action at a distance” (Newton term) or “spooky action” (Einstein term). Is it really possible to imagine how an apple can move down to Earth without a medium whatsoever?
Because of this diﬃculty, from the viewpoint of natural philosophy, some physicists maintained (for instance Euler with his impulsion gravity), that there should be “pervasive medium” which can make the attraction force possible. They call this medium “ether” though some would prefer this medium more like “ﬂuid” instead of “solid”. Euler himself seems to suggest that gravitation is some kind of “external force” acting on a body, instead of intrinsic force:
“gravity of weight: It is a power by which all bodies are forced towards the centre of the Earth” [3].
But the Michelson-Morley experiment [37] opened the way for Einstein to postulate that ether hypothesis is not required at all in order to explain Lorentz’s theorem, which was the beginning of Special Relativity. But of course, one can ask whether the Michelson-Morley experiment really excludes the so-called ether hypothesis. Some experiments after Michelson seem to indicate that “ether” is not excluded in the experiment setup, which means that there is Earth absolute motion [4, 5].

To accept that gravitation is external force instead of intrinsic force implies that there is distinction between gravitation and inertial forces, which also seem to indicate that inertial force can be modiﬁed externally via electromagnetic ﬁeld [6].
The latter notion brings us to long-time discussions in various physics journals concerning the electromagnetic nature of gravitation, i.e. whether gravitation pulling force have the same properties just as electromagnetic ﬁeld is described by Maxwell equations. Proponents of this view include Tajmar and de Matos [7, 8], Sweetser [9]. And recently Rabounski [10] also suggests similar approach.
Another version of Euler’s hypothesis has emerged in modern way in the form of recognition that gravitation was carried by a boson ﬁeld, and therefore gravitation is somehow related to low-temperature physics (superﬂuid as boson gas, superconductivity etc.). The obvious advantage of superﬂuidity is of course that it remains frictionless and invisible; these are main features required for true ether medium — i.e. no resistance will be felt by objects surrounded by the ether, just like the passenger will not feel anything inside the falling elevator. No wonder it is diﬃcult to measure or detect the ether, as shown in Michelson-Morley experiment. The superﬂuid Bose gas view of gravitation has been discussed in a series of paper by Consoli et al. [11], and also Volovik [12].
Similarly, gravitation can also be associated to superconductivity, as shown by de Matos and Beck [29], and also in Podkletnov’s rotating disc experiment. A few words on Podkletnov’s experiment. Descartes conjectured that there is no gravitation without rotation motion [30]. And since rotation can be viewed as solution of Maxwell equations, one can say that there is no gravitation separated from electromagnetic ﬁeld. But if we consider that equations describing superconductivity can be viewed as mere generalization of Maxwell equations (London ﬁeld), then it seems we can ﬁnd a modern version of Descartes’ conjecture, i.e. there is no gravitation without superconductivity rotation. This seems to suggest the signiﬁcance of Podkletnov’s experiments [31, 32].

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3 Relativistic gravitation theories
Now we will consider some alternative theories which agree with both Newton theory and Special Relativity, but diﬀer either slightly or strongly to General Relativity. First of all, Einstein’s own attempt to describe gravitation despite earlier gravitation theories (such as by Nordstrom [1]) has been inspired by his thought-experiment, called the “falling elevator” experiment. Subsequently he came up with conjecture that there is proper metric such that a passenger inside the elevator will not feel any pulling gravitation force. Therefore gravitation can be replaced by certain speciﬁc-chosen metric.
Now the questions are twofold: (a) whether the propermetric to replace gravitation shall have non-zero curvature or it can be ﬂat-Minkowskian; (b) whether the formulation of General relativity is consistent enough with Mach principle from where GTR was inspired. These questions inspired heated debates for several decades, and Einstein himself (with colleagues) worked on to generalize his own gravitation theories, which implies that he did ﬁnd that his theory is not complete. His work with Strauss, Bergmann, Pauli, etc. (Princeton School) aimed toward such a uniﬁed theory of gravitation and electromagnetism.
There are of course other proposals for relativistic gravitation theories, such as by Weyl, Whitehead etc. [1]. Meanwhile, R. Feynman and some of his disciples seem to be more ﬂexible on whether gravitation shall be presented in the General-Relativity “language” or not.
Recently, there is also discussion in online forum over the question: (a) above, i.e. whether curvature of the metric surface is identical to the gravitation. While most physicists seem to agree with this proposition, there is other argument suggesting that it is also possible to conceive General Relativity even with zero curvature [13, 14].
Of course, discussion concerning relativistic gravitation theories will not be complete without mentioning the PVgravitation theory (Puthoﬀ et al. [15]) and also Yilmaz theory [16], though Misner has discussed weaknesses of Yilmaz theory [17], and Yilmaz et al. have replied back [18]. Perhaps it would be worth to note here that General Relativity itself is also not without limitations, for instance it shall be modiﬁed to include galaxies’ rotation curve, and also it is actually theory for one-body problem only [2], therefore it may be diﬃcult to describe interaction between bodies in GTR.
Other possible approaches on relativistic gravitation theories are using the fact that the “falling-elevator” seems to suggest that it is possible to replace gravitation force with certain-chosen metric. And if we consider that one can ﬁnd simpliﬁed representation of Maxwell equations with Special Relativity (Minkowski metric), then the next logical step of this “metrical” (some physicists prefer to call it “geometrodynamics”) approach is to represent gravitation with yet another special relativistic but with extra-dimension(s). This was ﬁrst conjectured in Kaluza-Klein theory [19]. Einstein

himself considered this theory extensively with Strauss etc. [20]. There are also higher-dimensional gravitation theories with 6D, 8D and so forth.
In the same direction, recently these authors put forth a new proposition using Carmeli metric [21], which is essentially a “phase-space” relativity theory in 5-dimensions.
Another method to describe gravitation is using “torsion”, which is essentially to introduce torsion into Einstein ﬁeld equations. See also torsional theory developed by Hehl, Kiehn, Rapoport etc. cited in [21].
It seems worth to remark here, that relativistic gravitation does not necessarily exclude the possibility of “aether” hypothesis. B. Riemann extended this hypothesis by assuming (in 1853) that the gravitational aether is an incompressible ﬂuid and normal matter represents “sinks” in this aether [34], while Einstein discussed this aether in his Leiden lecture Ether and Relativity.
A summary of contemporary developments in gravitation theories will not be complete without mentioning Quantum Gravity and Superstring theories. Both are still major topics of research in theoretical physics and consist of a wealth of exotic ideas, some or most of which are considered controversial or objectionable. The lack of experimental evidence in support of these proposals continues to stir a great deal of debate among physicists and makes it diﬃcult to draw deﬁnite conclusions regarding their validity [38]. It is generally alleged that signals of quantum gravity and superstring theories may occur at energies ranging from the mid or far TeV scale all the way up to the Planck scale.
Loop Quantum Gravity (LQG) is the leading candidate for a quantum theory of gravitation. Its goal is to combine the principles of General Relativity and Quantum Field Theory in a consistent non-perturbative framework [39]. The features that distinguish LQG from other quantum gravity theories are: (a) background independence and (b) minimality of structures. Background independence means that the theory is free from having to choose an apriori background metric. In LQG one does not perturb around any given classical background geometry, rather arbitrary ﬂuctuations are allowed, thus enabling the quantum “replica” of Einstein’s viewpoint that gravity is geometry. Minimality means that the general covariance of General Relativity and the principles of canonical quantization are brought together without new concepts such as extra dimensions or extra symmetries. It is believed that LQG can unify all presently known interactions by implementing their common symmetry group, the four-dimensional diﬀeomorphism group, which is almost completely broken in perturbative approaches.
The fundamental building blocks of String Theory (ST) are one-dimensional extended objects called strings [40, 41]. Unlike the “point particles” of Quantum Field Theories, strings interact in a way that is almost uniquely speciﬁed by mathematical self-consistency, forming an allegedly valid quantum theory of gravity. Since its launch as a dual res-

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onance model (describing strongly interacting hadrons), ST has changed over the years to include a group of related superstring theories (SST) and a unifying picture known as the M-theory. SST is an attempt to bring all the particles and their fundamental interactions under one umbrella by modeling them as vibrations of super-symmetric strings.
In the early 1990s, it was shown that the various superstring theories were related by dualities, allowing physicists to map the description of an object in one superstring theory to the description of a diﬀerent object in another superstring theory. These relationships imply that each of SST represents a diﬀerent aspect of a single underlying theory, proposed by E. Witten and named M-theory. In a nut-shell, M-theory combines the ﬁve consistent ten-dimensional superstring theories with eleven-dimensional supergravity. A shared property of all these theories is the holographic principle, that is, the idea that a quantum theory of gravity has to be able to describe physics occurring within a volume by degrees of freedom that exist on the surface of that volume. Like any other quantum theory of gravity, the prevalent belief is that true testing of SST may be prohibitively expensive, requiring unprecedented engineering eﬀorts on a large-system scale. Although SST is falsiﬁable in principle, many critics argue that it is un-testable for the foreseeable future, and so it should not be called science [38].
One needs to draw a distinction in terminology between string theories (ST) and alternative models that use the word “string”. For example, Volovik talks about “cosmic strings” from the standpoint of condensed matter physics (topological defects, superﬂuidity, superconductivity, quantum ﬂuids). Beck refers to “random strings” from the standpoint of statistical ﬁeld theory and associated analytic methods (spacetime ﬂuctuations, stochastic quantization, coupled map lattices). These are not quite the same as ST, which are based on “brane” structures that live on higher dimensional spacetime.
There are other contemporary methods to treat gravity, i.e. by using some advanced concepts such as group(s), topology and symmetries. The basic idea is that Nature seems to prefer symmetry, which lead to higher-dimensional gravitation theories, Yang-Mills gravity etc.
Furthermore, for the sake of clarity we have omitted here more advanced issues (sometimes they are called “fringe research”), such as faster-than-light (FTL) travel possibility, warpdrive, wormhole, cloaking theory (Greenleaf et al. [35]), antigravity (see for instance Naudin’s experiment) etc. [36].
4 Wave mechanical method and diﬀraction hypothesis
The idea of linking gravitation with wave mechanics of Quantum Mechanics reminds us to the formal connection between Helmholtz equation and Schro¨dinger equation [22].
The use of (modiﬁed) Schro¨dinger equation has become so extensive since 1970s, started by Wheeler-DeWitt (despite

the fact that the WDW equation lacks observation support). And recently Nottale uses his scale relativistic approach based on stochastic mechanics theory in order to generalize Schro¨dinger equation to describe wave mechanics of celestial bodies [23]. His scale-relativity method ﬁnds support from observations both in Solar system and also in exo-planets.
Interestingly, one can also ﬁnd vortex solution of Schro¨dinger equation, and therefore it is worth to argue that the use of wave mechanics to describe celestial systems implies that there are vortex structure in the Solar system and beyond. This conjecture has also been explored by these authors in the preceding paper. [24] Furthermore, considering formal connection between Helmholtz equation and Schro¨dinger equation, then it seems also possible to ﬁnd out vortex solutions of Maxwell equations [25, 26, 27]. Interestingly, experiments on plasmoid by Bostick et al. seem to vindicate the existence of these vortex structures [28].
What’s more interesting in this method, perhaps, is that one can expect to to consider gravitation and wave mechanics (i.e. Quantum Mechanics) in equal footing. In other words, the quantum concepts such as ground state, excitation, and zero-point energy now can also ﬁnd their relevance in gravitation too. This “classical” implications of Wave Mechanics has been considered by Ehrenfest and also Schro¨dinger himself.
In this regards, there is a recent theory proposed by Gulko [33], suggesting that matter absorbs from the background small amounts of energy and thus creates a zone of reduced energy, and in such way it attracts objects from zones of higher energy.
Another one, by Glenn E. Perry, says that gravity is diffraction (due to the changing energy density gradient) of matter or light as it travels through the aether [33].
We can remark here that Perry’s Diffraction hypothesis reminds us to possible production of energy from physical vacuum via a small ﬂuctuation in it due to a quantum indeterminancy (such a small oscillation of the background can be suggested in any case because the indeterminancy principle). On the average the background vacuum does not radiate — its energy is constant. On the other hand, it experiences small oscillation. If an engine built on particles or ﬁeld interacts with the small oscillation of the vacuum, or at least ”senses the oscillation, there is a chance to get energy from them. Because the physical vacuum is eternal capacity of energy, it is easy to imagine some possible techniques to be discovered in the future to extract this energy.
Nonetheless, diﬀraction of gravity is not a “new hot topic” at all. Such ideas were already proposed in the 1920’s by the founders of relativity. They however left those ideas, even unpublished but only mentioned in memoirs and letters. The main reason was that (perhaps) almost inﬁnitely small energy which can be extracted from such background per second. (In the mean time, there are other vaious proposals suggesting that it is possible to ’extract’ energy from gravitation ﬁeld).

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About Glenn Perry and his theory. There is a drawback that that matter he called “aether” was not properly determined by him. In such a way like that, everything can be “proven”. To produce any calculation for practical purpose, we should have exact data on the subject of this calculation, and compare it with actual experiments.
On the other hand, such an idea could be put into another ﬁeld — the ﬁeld of Quantum Mechanics. That is, to study diﬀraction not gravitational radiation (gravitational waves which is so weak that not discovered yet), but waves of the ﬁeld of the gravitational force — in particular those can be seismic-like waves travelling in the cork of the Earth (we mean not the earthquakes) but in the gravitational ﬁeld of the planet. These seismic-like oscillations (waves) of the gravitational force are known to science, and they aren’t weak: everyone who experienced an earthquake knows this fact.
Other hint from wave aspect of this planet is known in the form of Schumann resonance, that the Earth produces vibration at very-low frequency, which seems to support the idea that planetary mass vibrates too, just as hypothesized in Wave Mechanics (de Broglie’s hypothesis). Nonetheless, there are plenty of things to study on the large-scale implications of the Wave Mechanics.
5 Concluding remarks
The present article summarizes a non-exhaustive list of gravitation theories for the purpose of inviting further and more clear discussions. Of course, our purpose here is not to say the last word on this interesting issue. For the sake of clarity, some advanced subjects have been omitted, such as fasterthan-light (FTL) travel possibility, warpdrive, wormhole, cloaking theory (Greenleaf et al.), antigravity etc. As to the gravitation research in the near future, it seems that there are multiple directions which one can pursue, with which we’re not so sure. The only thing that we can be sure is that everything changes (Heraclitus of Ephesus), including how we deﬁne “what the question is” (Wheeler’s phrase), and also what we mean with “metric”, “time”, and “space”. Einstein himself once remarked that ’distance’ itself is merely an illusion.
Acknowledgment
The ﬁrst writer wishes to thank to Prof. T. Van Flandern for sending his Meta Research Bulletin around 8 years ago. Both writers wish to thank to Dr. J. Sarfatti and P. Zielinski and others for stimulating discussions concerning the meaning of curvature in General Relativity. Special thanks to D. Rabounski for his remarks on Glenn Perry diﬀraction theory; and to E. Goldfain for summary of Loop Quantum gravity and Superstring. Previous discussions with Profs. M. Pitkanen, D. Rapoport, A. Povolotsky, Yoshio Kishi, and others are ac-

knowledged. At the time of writing this paper, P. LaViolette has just released a new book discussing antigravity research.
Submitted on May 02, 2008 / Accepted on June 19, 2008
References
1. http://en.wikipedia.org/wiki/Classical theories of gravitation
2. Millis M. New ways of thinking inertia and gravity. http://www.spinningdisk.org/article0.htm
3. Lamprecht J. Euler’s impulsion gravity. Meta Research Bulletin, 1998, v. 7, no. 3.
4. Consoli M. and Constanzo E. Flat space picture of gravitation vs General Relativity. arXiv: 0710.5613.
5. Consoli M. Modern Michelson-Morley experiment and gravi-
tation induced anisotropy of c. arXiv: gr-qc/0306105.
6. Haisch B. and Rueda A. Geometrodynamics, inertia, and quantum vacuum. arXiv: gr-qc/0106075.
7. De Matos C. J. and Tajmar M. Gravitational Poynting vector. arXiv: gr-qc/0107014.
8. De Matos C. J. and Tajmar M. Coupling of gravitation and electromagnetism in the weak ﬁeld approximation. arXiv: gr-qc/ 0003011.
9. Sweetser D. Classical uniﬁed ﬁeld theory. http://world.std.com/
sweetser/arch/uniﬁed ﬁeld.2001.09.01/uniﬁed ﬁeld.html
10. Rabounski D. A theory of gravity like electrodynamics. Progress in Physics, 2005, v. 2.
11. Consoli M. Gravitational forces from Bose-Einstein condensate. arXiv: hep-ph/0002098.
12. Volovik G. Superﬂuid analogies of cosmological phenomena. arXiv: gr-qc/0005091.
13. Brown P. M. Einstein gravitational ﬁeld. arXiv: physics/ 0204044.
 14. Luminet J.-P. http://luth2.obspm.fr/ luminet/etopo.html
15. Puthoﬀ H. PV representation of general relativity. arXiv: gr-qc/ 9909037.
16. http://en.wikipedia.org/wiki/Yilmaz theory of gravitation
17. Misner C. W. Yilmaz cancels Newton. arXiv: gr-qc/9504050.
18. Alley C. O., et al. Refutation of Misner’s claims. arXiv: gr-qc/ 9506082.
19. Kol B. and Smolkin M. Non-relativistic gravitation: from Newton to Einstein and back. arXiv: 0712.4116; [19a] http://en.wikipedia.org/wiki/Scalar theories of gravitation
20. Huang X. B. Uniﬁcation of gravitation, gauge ﬁeld, and dark energy. arXiv: hep-th/0410266
21. Christianto V. and Smarandache F. Kaluza-Klein-Carmeli metric from Quaternion-Cliﬀord Space, Lorentz force, and some observables. Progress in Physics, 2008, v. 4.
22. Lu L. J., Greeleaf J. F., and Recami E. arXiv: physics/9610012.
23. Celerier M. N. and Nottale L. Generalized macro Schro¨dinger equation from scale relativity. arXiv: gr-qc/0505012.

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24. Smarandache F. and Christianto V. Schro¨dinger equation and the quantization of celestial systems. Progress in Physics, 2006, v. 2.
25. Sowa A. Nonlinear Maxwell theory and electrons in two dimensions. arXiv: cond-mat/9904204.
26. Bialinycki-Birula I. and Bialinycki-Birula Z. arXiv: physics/ 0305012.
27. Bialinycki-Birula I. arXiv: physics/0309112.
28. Bostick W. Experimental study of plasmoids. Phys. Rev., 1957, v. 106, 404–412; http://en.wikipedia.org/wiki/Plasmoid
29. de Matos G. J. and Beck C. arXiv: 0707.1797.
30. Snelders H. A. M. Huygens and Newton’s theory of gravitation. Notes and Records of the Royal Society of London, 1989, v. 43, no. 2.
31. Solomon B. An introduction to gravity modiﬁcation: a guide to using Laithwaite’s and Podkletnov’s experiments and the physics of forces for empirical results. http://www. libreriauniversitaria.it/an-introduction-to-gravity-modiﬁcation/ book/9781599429922
32. http://en.wikipedia.org/wiki/Eugene Podkletnov
33. Perry G. E. Gravity as diﬀraction — proof and consequences. Inﬁnite Energy, 2008, v. 14, no. 79, 25–38.
34. http://en.wikipedia.org/wiki/Mechanical explanations of gravitation
35. http://plus.maths.org/latestnews/jan-apr07/cloaking/index.html
36. Sato R. Physicist use levitation to solve a sticky problem. http://www.dailygalaxy.com/my weblog/2007/08/physicistsuse-.html; [36a] http://www.zamandayolculuk.com/cetinbal/ wwormholew.htm
 37. Pavlovic M. http://users.net.yu/ mrp/chapter5.html
38. http://discovermagazine.com/2006/feb/dialogue-woit
39. See e.g.: Smolin L. Three roads to quantum gravity. Basic Books, New York, 2001.
40. See e.g.: Green M. Superstring theory in new physics for the twenty-ﬁrst century. Cambridge University Press, 2006.
41. ’t Hooft G. Introduction to string theory. http://www.phys.uu.nl/
thooft/lectures/stringnotes.pdf

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Models for Quarks and Elementary Particles — Part III: What is the Nature of the Gravitational Field?

Ulrich K. W. Neumann
Tschidererstr. 3, D-86609 Donauwo¨rth, Germany E-mails: Marianne-Dru.Neumann@t-online.de; elgravi@universum-un.de
The ﬁrst two parts of this article series dealt with the questions: What is a quark? and What is mass? While the present models lead to a physical idea of the mass, the geometrical theory of the general relativity only shows the eﬀect of mass. From the physical idea of mass, from the idea of the resultant vector (EV) as electric ﬂux
and from the ideas relating to the magnetic monopole (MMP) it follows that the gravitational ﬁeld is an electrical ﬁeld. The share of the electrical gravitational ﬂux
on the entire electrical ﬂux of a quark is determined from Newton’s empirical
gravitational constant G. The superposition of the -ﬂuxes of two quark collectives
produces the gravitational force eﬀect between two quark collectives. Gravitational ﬁelds reach inﬁnitely far according to our current ideas. Connected with the quark oscillations hinted in the Parts I and II this results in the idea of the - -ﬂux spreading with inﬁnite speed, having enormous consequences.

1 Introduction
In Parts I and II separate reference is made to the most productive assumptions or ideas relating to the development of the models. In Part I the formal assumptions/ideas are shown, which include the vectors in the constellation of the outer product of a vector with certain angular movements. At the end of Part II it transpires that the locus loop created by the EV is a physical central-symmetrical sinus oscillation in the mass-aﬀected three-quark particle. Other productive ideas are the orthogonal, hyperbolic space with two real axes and an imaginary axis as well as the identiﬁcation of the formal EV with a physical meaning. The EV identiﬁed as electrical ﬂux
with the dimension [Vm] results in the idea of the MMP.
The absolute number of amounts to = 1.8095 10 8[Vm]
according to the network of constants, see [1, page 143]. The massless MMP is an important idea to recognise on the one hand what mass is and on the other hand to develop the quark structure of the massless photon(-likes) from the quark composition of the electron.
2 The meaning of the “fountain”
In Part II the model idea for the composition of the MMP with the surrounding electrical ﬁeld is shown with Fig. 1. Thus, the decisive physical components of a quark are introduced with Part II, not considering the dynamics of these components in mass-aﬀected and massless particles. Relating to Fig. 1 it was not explained what the -ﬁeld is. This is done now.
During the course of the development of the models attempts were made to look behind the facade of Newton’s gravitational equation wherein obviously there was no shortage of incorrect estimates, one way streets and wrong tracks.

Newton’s gravitational constant G included in the equation is one of the many independent quantities of the standard model of physics to be determined empirically.
In Part II it is shown what mass is. The route there com-
mences with the equations of E = m c2 and E = h , re-
sulting in equation 1 of Part II, which can also be described as
edGquuccaeatdnioibnneN(d8ee–wtIeItr)omnoi’fns[eg1dr]a:wvmiittha=ttiho2eneacloeerrlqtCeucatctdi2oi.mnIefKnthsii=osnfGo[rmm5/miVsaAiln2stm5r]ob:-,

G

=

4e 2 elt

c4CaCb n1 n2

:

(1)

In it e are the ﬁne structure constant, C the Compton wavelengths of the elementary particles involved and elt see

below. On the route to clarifying the gravitational equation

the aim is to ﬁnd what the quantities n1 and n2 are and how large they are. If equation (1) is solved for n1 and n2 and the

Compton wavelengths of the nucleons (as mass-richest ele-

p mmeernitcaarlyvpaalurteiclesn)1arens2ub=stintui t=ed3.f9o3r9C1,01th8eiseombptairiniceadl. nu-
The n thus are gigantic numbers. What do these gigantic

numbers stand for?

At this point it is highly productive to use Fig. 1 of Part II. Visible is the MMP that occurs with highest frequencies, which is enclosed by the electrical source ﬂux . Here, by far the predominant part of this source ﬂux is closely connected with the magnetic ﬂux (Maxwell). Only the minute share of the total ﬂux leads to the outside. This share is expressed in the simple relationship:

=

1 n

:

(2)

If the gigantic numbers n are substituted in the equa-

tion (2) (see [1, page 172, equation (8–XIII)]), it follows:

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=

1
3.939

1018

=

= 2.539 10 19 1.8095 10 8 = 4.594 10 27 [Vm]:

This is the minute share of the -ﬁeld ( -ﬁeld or gravitational ﬁeld), leaving the quarks of a three-quark particle (3QT). is shown as a symbolic line in Fig. 1, Part II.
In addition to Newton’s gravitational equation there are further important equations of physics with a similar structure, such as the equations of Coulomb (elec. charges), Rydberg (spectral series) and Schro¨dinger (waves). These equations are diﬀerent forms of the universal equation from [1, page 157]:

elt elt n1 n2 = a b elta eltb : (3)
In it the universal constant elt has the dimension [VAsm],
[1, page 141]. It can be composed of many kinds of constants,
e.g.: elt = N h c [VAsm] with N h = h 2e .
Equation (3) can be paraphrased with some considerations in a further equation (4) , which can be written next to the equations of Coulomb (charges), Newton (gravitation), Rydberg (spectral series) and Schro¨dinger (waves): With
elt = K l2 and according to [1, Fig. 8–1c], elt = 2 "0 it
follows from equation (3):

K = n1 "0 n2

a

b l2

:

(4)

If the relationship

=

1 n

of equation (2) is substi-

tuted in equation (4) and if some more considerations are ex-

amined, the following is obtained:

K = "0

a

b l2

;

(4a)

K

=

"0 0.8

a

b l2

:

(4b)

Thus the following is realised:

1. The meaning of the gigantic numbers n1 and n2 in Newton’s empirical, gravitational constant G analysed
with equation (1) is seen as follows. With the prod-
uct of the inverse of the number ni and of the electri-
cal source ﬂux the minute fraction of the electrical source ﬂux, that is to say , of each “3QT” is described, where is leaving the quarks of a “3QT”. The minute fraction of accounts for the -ﬁeld of a quark or a “3QT”;

2. The quantity of said fraction of the -ﬁeld of a “3QT”

is

1 3:939
=

1018 .

= 2.539 10 19 or inverted 3.939 1018 has the empirical value =1.8095 10

8

[Vm] 2.539 10 19 = 4.594 10 27 [Vm] as absolute

number. These numbers apply to our galactic environ-

ment;

3. The equations (4a) and (4b) signify that the superposition of the -ﬁelds of two quarks or two quark collectives (a and b) produces the gravitational force eﬀect between two quark collectives;
4. These considerations have made the “gravitation” a superposition of physical namely electrical -ﬁelds of highest frequency!
3 Some aspects relating to the -ﬁelds
In Part II it is explained by means of Shapiro’s experiments how electrical ﬁelds and thus the gravitational ﬁelds inﬂuence the photon(-likes). This physical substantiation for example for the reduction of the speed of light (“refractive index of the vacuum”) is to be preferred compared to an substantiation through the geometrical theory of the general relativity.
Gravitational ﬁelds reach inﬁnitely far according to our current ideas. The loci of the quarks (sinus oscillations) of which we and our environment consist, are traversed within 10 20 (electrons) to 10 25 (nucleons) seconds. This means the -ﬁeld of a quark expands into inﬁnity and contracts again within this absurdly short time. The propagation speed of the -ﬁeld is thus inﬁnitely large. (Of course this has an eﬀect on large research projects as e.g. LISA with which the allegedly wave-shaped and light-speed propagation of the gravitational ﬁeld according to the standard physics is to be investigated.)
The inﬁnitely fast propagation of the -ﬁeld has “natural” consequences everywhere. If the composition of the quarks according to Fig. 1 of Part II applies — which is assumed in these models — the electrical ﬁeld enclosing the MMPs also expands at inﬁnite speed. This means the -ﬁelds of the mass-aﬀected particles occur instantaneously. The range of the -ﬁelds is approximately congruent with the range within the Maginpar or the range of the -ﬁelds is congruent with the conﬁnement. The conﬁnement located inside a particle is marked oﬀ from the outer range by a spherical shell around the coordinate centre with approximately the radius of the Maginpar. No causality applies any longer in the small range of the -ﬁeld within the conﬁnement!
The inﬁnitely fast propagation of the -ﬁeld undoubtedly also inﬂuences the uncertainty principle. The latter is valid for the range outside the conﬁnement and therefore for electromagnetic processes. In the outer range with causality —
with t between two events — applies e.g. t E = h or x p = h.
Inside the conﬁnement the ranges for the toroidal magnetic ﬁeld and the electric source ﬁeld are distinguished,
where t = 0 applies because of the instantaneous propa-
gation of the -ﬁeld. Some relation for the interior of the conﬁnement corresponding to the uncertainty principle looks
diﬀerent; the input quantities are certain: N   = Nh. The product from inertia quantum N times frequency N

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corresponds to the impulse p or p and  corresponds to the x or x. (Otherwise N = Nh=c is the deﬁnition equation
for the natural constant N .) Entirely diﬀerent aspects are touched by the inﬁnitely fast
propagation of the -ﬁeld, which are merely mentioned here but not discussed: A) The inﬁnitely fast propagation of the
-ﬁeld revitalises the Mach principle according to which the local behaviour of matter is based on the inﬂuences of the remainder of the universe. B) The universal structure of galactic chains and dark bubbles and the synchronised creation of galaxies are based on the inﬁnitely fast propagation. C) According to the models the centres of the galaxies are quantum objects. The considerations relating to causality and uncertainty also apply to these. D) The Planck length, [1, page 178], is determined through the interaction of MMP and
- -ﬁeld. E) The experiments of A. Zeilinger for teleportation are based on the inﬁnitely fast propagation of the
-ﬁeld in the rapidly enlarging conﬁnement of polarisationentangled photons (12QT).
Submitted on April 05, 2008 Accepted on May 22, 2008
References
1. There in the internet http://www.universum-un.de a book entitled “Models for quarks and elementary particles” will be displayed having a volume of approximately 250 DIN A4 pages.

July, 2008

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Models for Quarks and Elementary Particles — Part IV: How Much Do We Know of This Universe?

Ulrich K. W. Neumann
Tschidererstr. 3, D-86609 Donauwo¨rth, Germany E-mails: Marianne-Dru.Neumann@t-online.de; elgravi@universum-un.de
Essential laws and principles of the natural sciences were discovered at the high aggregation levels of matter such as molecules, metal crystals, atoms and elementary particles. These principles reappear in these models in modiﬁed form at the fundamental level of the quarks. However, the following is probably true: since the principles apply at the fundamental level of the quarks they also have a continuing eﬀect at the higher aggregation levels. In the manner of the law of mass action, eight processes for weak interaction are formulated, which are also called Weak Processes here. Rules for quark exchange of the reacting elementary particles are named and the quasi-Euclidian or complex spaces introduced in Part I associated with the respective particles. The weak processes are the gateway to the “second” strand of this universe which we practically do not know. The particles with complex space, e.g. the neutrino, form this second strand. According to the physical model of gravitation from Part III the particles of both strands have -ﬁelds and are thus subject to the superposition, which results in the attraction by gravity of the particles of both strands. The weak processes (7) and (8) oﬀer a fair chance for the elimination of highly radioactive waste.

1 Introduction

2 The weak interaction

The ﬁrst parts of this series of papers have headline questions which are answered within the scope of the models [1]: I) What is a quark? II) What is mass? III) What is the nature of the gravitational ﬁeld?
Which of the three questions will a physicist representing the current standard model be able to answer positively without hesitation? The standard model of physics combines huge quantities of analyses, conformities with natural laws and theories. However, too many independent quantities that can only be captured empirically still enter the standard model of physics and inconsistencies between individual theories are known. For this reason, theoreticians are looking for new physics especially in the ﬁeld of the strings, loops and branes; however, they have been unable to establish any reference to reality. The standard model of cosmology has the general theory of relativity (GTR) as thread, wherein the GTR is a geometrical and not a physical theory. Despite this deﬁcit the mainstream of cosmologists is absolutely convinced of the big bang model which is based on the GTR, wherein the big bang is a central part of the standard model of cosmology. The physical model of gravitation presented in Part III opens up a new interpretation of our universe. The perspectives of Part III render a Part V for cosmology — the utmost level of organisation — unnecessary. But there is a Part V in preparation concerning the magnetic load, which leads to the undermost level of organisation of our universe. Although many relationships are better recognizable with this model than in the past, there is certainly a lot we do not know of our universe.

The equations of the weak interaction which in the following

are also called ”Weak Processes” are the central content of

the present Part IV. Physics books present equations relating

to the weak interaction. These equations are considered cor-

rect although the authors have no exact idea of what a quark

is, although they are uncertain as to the mass possessed for in-

stance by a neutrino, although they should have doubts in the

uniformity of so-called “elementary particles”, although they

are looking for additional particles that could be included in

the equations.

An often-quoted equation in the literature is formulated

thus:

p+ + e ! n0 + e+:

(1)

According to Table 1 of Part I, each of the four elementary particles involved is a three-quark particle (3QT). If this is used to make a quark equation — which cannot happen in the standard model of physics — according to the models to date equation (1) must read as follows:

|u{uz}jj d + |d{zd}? u ! |d{zd}jj u + |d{dz}? d:

As can be seen, the quarks on both sides do not agree in
number and type. If the left side is correct, an u and an u are missing on the right, instead there are a d and an d too
many on the right; the charge balance would be correct as in
equation (1).
The literature equation (1) cannot be corrected because it
is wrong. To get onto the right track here are some fundamen-
tal remarks concerning equations with particles.

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From [1], Chapter 8.5, page 202: The (A) law of mass

action, the (B) Pauli principle, the (C) superconductivity and

the (D) uncertainty principle were found at higher aggrega-

tion levels of the particle world and applied to (A) molecules,

(B) atoms, (C) metal crystals and (D) elementary particles.

All four can be found again in these models in modiﬁed form

at the fundamental level of the quarks, e.g. in the following

(A) weak processes or with the (B) conﬁgurations of the nu-

clei in [1], Chapter 7.5 or in the (C) “fountain”, Fig. 1 in Part

I, or in the deﬁnition of the natural constant of the (D) inertia quantum N , see penultimate paragraph of Part III. Probably

the eﬀect of such laws and principles has to be seen diﬀer-

ently: Since they apply at the fundamental level they con-

tinue to have an eﬀect also at the higher aggregation levels.

The following is an example using the (B) Pauli’s prin-

ciple. The Pauli principle states for a complete atom — i.e.

for a higher aggregation level — that a shell (K, L, M etc.

with the sub-shells s, p, d etc.) of the atomic shell cannot be

occupied by two electrons.

 In Part I, Table 1 in line A shows the particles |d{zd} jj d

 ee and |u{zu} jj u

for the fundamental level of the quarks.

In addition, Fig. 12 in Part I shows the loci for a |d{dz} Zk.

(A deﬁnition of the “dual-coordination” or brieﬂy “Zk” is

given in Part I, page 74, paragraph 5.) If the locus of a third

d-quark were to be placed in the level of this Zk, either space

I or space III would be occupied with two loci. Such double

occupancy is demanded for the particles ee and by the jj symbol. According to the Pauli principle this means at the fundamental level of the quarks that the particles ee and
are prohibited, see Table 2! Allowed are only the electron
|d{dz} ? d  e and the deldopon |u{uz} ? u  ( ++), where
each quark assumes a diﬀerent position.

Another example relates to the (A) law of mass action.

This law primarily applies to the fundamental quark equa-

tions, but was initially discovered by us by means of the chem-

ical reactions at the high aggregation level. The equation of

a chemical reaction is formulated in the same manner as a

fundamental quark equation. All constituents entering a fun-

damental reaction again come out of the reaction in a changed

composition. Nothing disappears or is added. In this re-

gard, some of the equations for the weak interaction oﬀered

in physics books are totally unsatisfactory, since the particles

on both sides of the equations lack a common basis. This is

also evident from the above equation (1): for the nucleons

there is the quark representation in the standard model, not

for the leptons.

3 The eight weak processes
Reading the following is not easy, the subject however highly interesting for the understanding of our universe. The comments regarding the equations are intended to facilitate this understanding.

Eight processes with the construction

Starting particle ! (Quarkpool) ! Reaction products

p+ + e ! |u{uz} jj d + |d{zd} ? d !

! ! ! n0 + e

(2)

u"ujj d:::dd?d" ! ! |d{zd} jj u+

+ |d{zd} ? u

(2a)

Space type qeR qeR ! ! ! ! qeR koR

n0 + e ! |d{dz} jj u + |d{zd} ? u !

! ! ! p+ + e

(3)

d"djj u:::dd?u" ! ! |u{uz} jj d+

+ |d{dz} ? d

(3a)

Space type qeR koR ! ! ! ! qeR qeR

The equations (2) and (3) count among the best known of the weak interaction. For the formulations according to the standard model the common basis of the particles mentioned above is absent. As quark equation (2a) and (3a), they correspond to the characteristics of the law of mass action. Details for a “quark pool” are included in the quark equations. This quark pool stands for the physical process of the reaction of the particles involved which requires a ﬁnite time and during which exchange processes take place. The signs within the brackets explain this exchange. During both the above processes a quark from the Zk of the baryon/nucleon involved is exchanged for the singular quark of the lepton, while the
quark from the Zk of the baryon does not belong to the ujjd-
group. It can also be seen that the structure symbols in the equa-
tions are retained. A jj and a ? symbol each are present on
the left and on the right side of the equation. This is to be correlated with the retention of the baryon and lepton number of the standard model. This means there are ﬁxed rules for the reactions during the weak processes.
In each third line for each reaction the space type qeR or koR, see [5], Part I, page 72/73, of the elementary particle is noted. If two particles from “our” quasi-Euclidian space (qeR) react with each other the probability of the reaction substantially depends on a resonance possibility, i.e. the size of the particles MAGINPARs. In addition to this probability for a reaction there is obviously also a second one. This depends on the space type. This means, two particles with the same space type react with each other with far greater probability than particles with diﬀerent space type.
We are aware of this in the case of the hugely plentiful neutrinos with the complex space type koR which hit the particles of earth with the space type qeR with only an extremely

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3QT

jjLocus
( ) to

level of the locus

lseivnegluolafrthqeuZarkkz

parallel

Space type of particle

Locus level of singular quark vertically
(?) on locus level of Zk

dd d
ee z
koR1 z
e

dd u n0
qeR1
e

uu d p+
qeR1 ?+

uu u
z
koR1 z ( ++)

Space type of particle

qeR2 koR2 koR2 qeR2

  koR “complex” space, qeR quasi-Euclidian space.
The number 1 or 2 designates the number of the -rotation levels per particle. zElementary particles prohibited by the Pauli principle.

Table 2: Space structures of the elementary particles

low probability. The probabilities for a reaction are called MAGINPAR and space type probability. All eight weak processes are characterized in that at least one particle of a process has the space type koR. Thus the space type probability applies to the eight here treated processes which is why we talk about the “weak” interaction.

p+ + ?+ ! ! ! ! n + ( ++) (4) |u{uz} jj d + |u{uz} ? d ! u"ujj d:::uu?d" ! ! |d{zd} jj u+

+ |u{zu} ? u

(4a)

Space type qeR koR ! ! ! ! qeR qeR

n + ( ++) ! |d{dz} jj u + |u{uz} ? u !

! ! ! p+ + ?+ (5) d"djj u:::uu?u" ! ! |u{zu} jj d+

+ |u{zu} ? d

(5a)

Space type qeR qeR ! ! ! ! qeR koR

A hypothesis (here the models under consideration) establishes new predictions/expansions unknown to date for the (physical) teaching applicable to that point, which have to be veriﬁed. Such predictions are made by Table 1 and the still to follow Table 2 with some of the particles noted there, which also occur in the equations (4) and (5). For the sake of brevity the particles of the Tables that have not been found yet will not be further commented upon at this point. Reference is only made to the respective exchange of the quarks in the quark pool, which corresponds to the ﬁxed rules for the reactions mentioned above.
To facilitate the association of the space types with the individual elementary particles Table 2 is inserted.

The best known equation to describe the “ -decay” is the

following:

n0 ! p+ + e + e

(6)

Under the aspect of the standard theories such an equation is possible because four totally independent particles are present, the electric charges involved are correct and the n0 has the greatest mass/energy so that it can decay into the three other particles of lower energy. Under the aspect of the models developed here the two sides of this process cannot be brought into line even from the number of the quarks involved. The right side of the equation comprises nine quarks, the left side three quarks. In other words six quarks have to be added to the left side, while a 6QT or boson is obvious.
The following arguments speak for the photon-like
-gamma ( ) as trigger of the process — incompletely
— described with the above non-equation:
1. The particle is not yet known which is why it is not named so far on the left side of the process;
2. Because of its space type koR the particle — based on
e (Table 2) — is diﬃcult to discover;
3. -gamma brings with it the necessary number and type of quarks and of structure signs jj respectively ?.
The almost known “ -decay” according to the standard model as fully formulated weak process according to these models then becomes the following as particle and quark equation:

n0 +

! ! ! ! p+ + e + e (8)

|d{dz}

jj u + |d{zd} ? uu ?

|d{zd}

!


d from n0 ,
u from Bk


!

! |u{uz} jj d + d? |d{dz} +u? |d{dz}

(8a)

Space type qeR koR ! ! !

!qeR qeR koR

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The central part uu of the  within the quark-equation
(8a) is called a “binding coordination”, brieﬂy “Bk”. The ﬁxed rules for the quark reactions need only be mod-
iﬁed slightly for the reaction type (8) relative to the reaction type (2) and (3) or (4) and (5):
• A baryon each reacts with a photon-like 6QT (instead of 3QT lepton).
• From the original particles, a formally singular quark each (not anti-quark) of a lepton and now part of a Bk in the photon-like is exchanged for a Zk-quark (not from
the ujjd-group) of the baryon.
• In addition to the type and number of the quarks in-
volved the type and number of the structure signs jj and ? now agree on both sides of equation (8) as well.
Since equation (8) relative to the non-equation (6) has been explained, equation (7) is now added where e- (our “normal” photon) has to be additionally considered compared with the standard version.

p+ + e

! ! ! ! n0 + e+ + e (7)

|u{zu}

jj d + |d{dz} ? dd?

|d{dz}

!


u from p+ ,
d from Bk


!

! |d{dz} jj u + |d{zd} ?d+ |d{zd} ?u

(7a)

Space type qeR qeR ! ! !

! qeR qeR koR

Since the following equations (9) and (10) contain particles not yet found from the systematic of Table 1 and 2 here they will not be further commented upon. In structure they correspond to the type of the equations (7) and (8) and complete the set of the weak processes according to these models:

p++?

! ! ! ! n0 + ( 2+)+?+ (9)

|u{zu}

jj d + |u{uz} ? dd?

|u{uz}

!


u from p+ d from Bk

,


!

! |d{dz} jj u + |u{zu} ?u + |u{zu} ?d

(9a)

n0 + ( )

! ! ! ! p++?+ + ( 2 ) (10)

|d{zd}

jj u + |u{uz} ? uu ?

|u{uz}

!


d from n0 ,
u from Bk


!

! |u{uz} jj d + d ? |u{uz} +u ? |u{uz}

(10a)

The weak processes are the gateway to the “second” strand of this universe. The particles having a complex space (koR) form this second strand. “Our” particles with quasiEuclidian space (qeR) from the “ﬁrst” strand overlap those from the second strand without problems, which is why the spaces also overlap without problems. (What is a “space” being created in our imagination?) The “spaces” do not interact with each other.

In contrast with this, the physical = electric -ﬁelds
from qeR and koR interact very well with each other so that their superposition results in the mutual attraction, see [1], page 186, line 18. Measured by the undiscovered particles of Tables 1 and 2 there is much to be discovered behind the gate to the “second” strand of this universe. Judging by the ratio of the gravitational eﬀects of the visible matter and the dark matter what can be discovered behind the gate is a multiple of what we already know.
4 The Meaning of the Weak Processes (7) and (8)
Equations (7) and (8) contain some fascinating technical potential. H. Stumpf deals with nuclear reaction rates of the electroweak interaction [2] and at the end of his paper he refers to L. I. Urutskoev and other Russian authors, who perform experiments regarding this item. The potential of those works includes ﬁnding new routes for the elimination of highly radioactive waste. In a few years this waste from hundreds of disused nuclear reactors will pile up in many states of our earth. The ﬁnal storage of this waste is not clariﬁed and costs for a long time storage with e. g. sarcophagus as in Tschernobyl would be enormous. The duration of storage follows from natural -decay half-life periods of diﬀerent elements or their isotopes which can last for up to 1.5 1024 years for 15228Te, [4], page 34, which mankind cannot live to see.
Equation (7) respectively (7a) demonstrates, that the protons of radioactive elements can have resonance and can react with very short waved photons (e ) into neutrons, positrons and neutrinos. Thereby the structure and the therewith combined beat of the photon shown in Part II, page 77, left column, point (2) and the storage of the photon in an electron (resonance), Part II, page 77, right column, penultimate paragraph are called to mind.
Equation (7) is conﬁrmed by two aspects of the above mentioned experiments of L. I. Urutskoev et al. [3]. First aspect: The central incidents of the experiments are electric discharges between metallic foils in vessels ﬁlled with various ﬂuids, [2], page 455. My interpretation is, that by the discharges those short waved e of process (7) are generated, which can have resonance and reaction with the protons of the (radioactive) elements. Second aspect: The possibility of “low-energy nuclear transformation” is reported in [3]. If an electron and a visible photon have a comparable COMPTONwavelength and therefore have resonance, then the photon has an energy of multiplier 105 less than the electron, [1], Chapter 8.2.3, page 163. With weak interaction nuclei emit short waved e in the range of a few keV up to a few MeV. That means nuclei are in the position to have resonance with those short waved e . If such short waved e arrives at a nucleus and hit (a neutron or) a proton then there is the possibility for the “low-energy” exchange of quarks in a quark-pool according to the rules of page 72, middle of right column, and page 74, upper part left column. By the exchange of quarks

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in accordance with equation (7) the proton transforms into a neutron and by this a new element respectively a new isotope takes shape. New elements respectively isotopes were detected by the authors of [3].
Following are comments on the peculiarities of the weak process (7):
1. The Standard Model of Physics treats the -“decay” as statistical phenomenon or as happening by chance. The model under consideration especially the weak process (7) presents a dosed bombardment of protons by e . The transformation of the protons into products of reaction happens not by chance instead the reaction is determined by the eﬃciency of the law of mass action.
2. Without the knowledge of the weak process (7) Urutskoev et al. with exotic experiments strive for the realisation of reactions according to this process. With knowledge of equation (7) diﬀerent experiments are possible: Possibly one could observe the weak process cease when the bombardment of protons by photons, which can have resonance, is prevented completely. Nevertheless the “radioactive decay” of a specimen with an outer screening could continue because a radiation could be released “from the interior” of this specimen. The latter could stem from the less probable but possible opposite
! reaction of equation (7): n0 + e+ + e p+ + e .
The e originating in the interior of the atomic nucleus would be absorbed after ﬂying a very short distance in the specimen because of a high probability of resonance. By this the weak process (7) would be caused “from the interior”.
3. The weak process (7) cannot be observed in nature, [4], S. 38.
Following are comments on the peculiarities of the weak process (8):
1. The very common but not applicable non-equation (6) claims that the neutrons of radioactive elements would “decay” into protons, electrons and anti-neutrinos. As with equation (7) the law of mass action is valid with equation (8);
2. By the exchange of quarks according equation (8) a new element respectively a new isotope takes shape. The problem is, until now we still do not know the
 because of its complex space koR and beyond
this we cannot shield it from the outside or handle it at all. From that point of view we would be dependent on the sun, on space or on nuclear reactors as genera-
tors for  of whatever intensity and wavelength to
shorten half-life periods by chance.
Eventually a possibility on the basis of the opposite reaction
of the weak process (8) will be revealed. Those  origi-
nating from p+ + e + e ! n0 +  would be absorbed

after ﬂying the shortest distance because of a high probability of resonance. By this the weak process (8) would be released “from the interior”. The opposite reaction of the weak process (8) should be reinforced by proper conditions in such a way that the reaction rates are of suﬃcient size.
In summary: Though till now we do not know the  -
radiation so far and, much less, we can control it, there is hope
to transform the neutrons of radioactive elements by
via the opposite reaction of equation (8). The construction of some technical apparatus for short waved e -radiation as e.g. X-rays of 103 to 106 eV is feasible. By the reaction of proton and e (photon) according to the weak process (7) natural, partly very long time half-life periods can be shortened down to seconds using a technical apparatus! The use
of both types of radiation,  and e , would be decisive
steps for the elimination of highly radioactive waste.
Submitted on April 05, 2008 / Accepted on June 19, 2008
References
1. There in the internet http://www.universum-un.de a book entitled “Models for quarks and elementary particles” will be displayed having a volume of approximately 250 DIN A4 pages.
2. Stumpf H. Change of electroweak nuclear reaction rates by CP- and isospin symmetry braking — a model calculation. Zeitschrift fu¨r Naturforschung, 2006, 61a, 439–456.
3. Filippov D. V., Urutskoev L. I. Annales De La Fondation Louis De Broglie, 2004, v. 29, 1187.
4. Scholz W. Radioaktivita¨t. Teubner Verlag, Wiesbaden, 5. Auﬂage, 2005
5. Neumann U. K. W. Models for quarks and elementary particles. Part I: What is a Quark? Progress in Physics, 2008, v. 2, 71–74.

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The Generalized Conversion Factor in Einstein’s Mass-Energy Equation

Ajay Sharma
Fundamental Physics Society, His Mercy Enclave Post Box 107 GPO Shimla 171001, India ajay.sharmaa@rediﬀmail.com

Einstein’s September 1905 paper is origin of light energy-mass inter conversion equa-

tion (L = mc2) and Einstein speculated E = mc2 from it by simply replacing L

by E. From its critical analysis it follows that L = mc2 is only true under special or

/ / ideal conditions. Under general cases the result is L

mc2 (E

mc2). Conse-

quently an alternate equation E = A c2 M has been suggested, which implies that

energy emitted on annihilation of mass can be equal, less and more than predicted by
E = mc2. The total kinetic energy of ﬁssion fragments of U235 or Pu239 is found experimentally 20–60 MeV less than Q-value predicted by mc2. The mass of parti-

cle Ds (2317) discovered at SLAC, is more than current estimates. In many reactions
including chemical reactions E = mc2 is not conﬁrmed yet, but regarded as true. It implies the conversion factor than c2 is possible. These phenomena can be explained with help of generalized mass-energy equation E = A c2 M .

1 Introduction
Mass energy inter-conversion processes are the oldest in nature and constitute the basis of various phenomena. Before Einstein’s work, Newton [1] stated that “Gross bodies and light are convertible into one another. . . ”. Einstein derived light energy-mass inter-conversion equation for Newton’s
perception as L = mc2. Before Einstein scientists such as
S. Tolver Preston [2] Olinto De Pretto [3], Fritz Hasenohrl [4, 5] Frederick Soddi [6] contributed to the topic.
Einstein’s derivation of L = mc2 (from which Einstein speculated E = mc2), is true under special conditions
(where selective values of variables are taken). Under general conditions (when all possible values of parameters are
taken) equations like L = 0:0011 mc2, L = 0:999988 mc2
/ etc. are obtained i.e. L mc2. Thus conversion factor
other than c2 is possible in Einstein’s derivation. Further the generalized mass–energy equation E = Ac2 M, is derived, and E = mc2 is special case of the former depend-
ing upon value of A (depends upon the characteristics conditions of the process). Thus apart from theoretical limitations,
E = mc2 has experimental limitations e.g. sometimes ex-
perimental results diﬀer from it and in many cases it is not
conﬁrmed. Under such cases E = Ac2 M is widely use-
ful and applicable. The ﬁssion fragments result from U235
and Pu239 have total kinetic energy 20–60 MeV less than Qvalue (200 MeV) of reaction predicted by E = mc2 [7–
9]. Palano [10] has conﬁrmed that mass of particle Ds (2317) has been found more than current estimates based upon
E = mc2. Also E = mc2 does not give consistent
results in explaining the binding energy, as it violates the universal equality of masses of nucleons.
All these facts can be explained by E = Ac2 M with value of A less or more than one. E = mc2 is not con-
ﬁrmed in many processes such chemical reactions, atom

bomb explosions, volcanic reactions etc. Whatever may be
the case E = Ac2 M is capable of explaining the phenomena. Thus conversion factor other than c2 is possible, in
Einstein’s September 1905 derivation and conﬁrmed experi-
mentally also.

2 Einstein’s light energy — mass equation L = mc2
and its hidden aspects

Einstein [11] perceived that let there be a luminous body at
rest in co-ordinate system (x; y; z). The system (, , ) is in uniform parallel translation w.r.t. system (x, y, z); and origin of system (, , ) moves along x-axis with relative velocity v. Let a system of plane light waves have energy ` relative to system (x, y, z), the ray direction makes angle  with xaxis of the system (, , ). The quantity of light measured in system (, , ) has the energy [11, 12].

`

=`

1
q

v c

1

cos
vc22

(1)

Einstein has given Eq. (1) in his paper known as Special The-
ory of Relativity [12] and called Eq. (1) as Doppler principle
for any velocities whatever.
Let E0 and H0 are energies in coordinate system (x, y, z) and system (, , ) before emission of light energy, further E1 and H1 are the energies of body in the both systems after
it emits light energy. Thus Einstein wrote various equations
as Energy of body in system (x, y, z)

E0 = E1 + 0:5L + 0:5L = E1 + L;

(2)

Energy of body in system (, , )

h
H0 = H1 + 0:5 L 1

v c


cos  + 1 +

v c

i
cos

(3)

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where = 1=[1 v2=c2]1=2;

H0 = H1 + L;

(4)

or

(H0 E0) (H1 E1) = L ( 1) :

(5)

Einstein calculated, kinetic energy of body before emis-

sion after

oefmliigsshitoennoefrgliyg,hKt e0n(mergbvy,2=K2)(manadvk2=in2e)taics

energy

of

body

0

1

K0

K =L@ q 1 1

vc22

1A

(6)

Einstein considered the velocity in classical region thus applying binomial theorem,

K0

K

=


L1

+

v2 2c2

+

3v4 8c4

+

+

15v6 48c6

+

105v8 384c8

+

:::


1:

(7)

Further Einstein quoted [16] “Neglecting magnitudes of fourth and higher orders, we may place”

K0

K

=

L

v2 2c2

(8)

Mb

v2 2

Ma

v2 2

=

L

v2 2c2

(9)

or

L = (Mb Ma) c2 = mc2;

(10)

or Mass of body after emission
emission (Mb L=c2).

(Ma)

=

Mass

of

body

before

Now replacing L (light energy) by E (total energy or ev-

ery energy) Einstein wrote

E = (Mb Ma) c2 = mc2

(11)

or Mass of body after emission
emission (Mb E=c2).

(Ma)

=

Mass

of

body

before

Thus Einstein derived that conversion factor between

mass and light energy is precisely equal to c2, this aspect

is elaborated by Fadner [13]. But Einstein’s this derivation

has been critically discussed by many such a Planck [14],

Stark [15], Ives [16], Stanchel [17], Okun [18] and N. Ham-

dan [19] etc. At the same time in some references [20, 21] it

is expressed that Einstein has taken hints to derive equation
E = mc2 and from existing literature without acknowledg-

ing the work of preceding scientists. Max Born [22] has ex-

pressed that Einstein should have given references of existing

literature.

Thus Einstein’s work on the topic has been critically an-

alyzed by scientists since beginning, in views of its scientiﬁc

and procedural aspects.

3 The conversion factor between mass-energy other
than c2 is also supported by Einstein’s derivation
under general conditions

As already mentioned Einstein’s September 1905 derivation
of L = mc2 is true under special or ideal conditions (se-
lected values of parameters is taken) only, this aspect is stud-
ied critically with details by the author [23–36] discussing
those aspects which have not been raised earlier. Thus the
value of conversion factor other than c2 is also supported
from Einstein’s derivation under general conditions (all pos-
sible values of variables). The law or phenomena of inter-
conversion of mass and energy holds good in all cases for all
bodies and energies under all conditions.
In the derivation of L = mc2 there are FOUR vari-
ables e.g.

(a) Number of waves emitted,
(b) l magnitude of light energy, (c) Angle  at which light energy is emitted and (d) Uniform velocity, v.

Einstein has taken special values of parameters and in general for complete analysis the derivation can be repeated with all possible values of parameters i.e. under general conditions taking in account the momentum conservation (which is discussed in next sub-section).

(A) The body can emit large number of light waves but Einstein has taken only TWO light waves emitted by luminous body.
(B) The light waves emitted may have diﬀerent magnitudes but Einstein has taken EQUAL magnitudes
(C) Body may emit large number of light waves of diﬀerent magnitudes of energy making DIFFERENT ANGLES (other than 0 and 180 ) assumed by Einstein.
 (D) Einstein has taken velocity in classical region (v c) has not at all used velocity in relativistic region. If velocity is regarded as in relativistic region (v is comparable with c), then equation for relativistic variation of mass with velocity i.e.

Mrel = qM1restvc22

(12)

is taken in account. It must be noted that before Einstein’s work this equation was given by Lorentz [37,38] and ﬁrstly conﬁrmed by Kaufman [39] and afterwards more convincingly by Bucherer [40]. Einstein on June 19, 1948 wrote a letter to Lincoln Barnett [41] and advocated abandoning relativistic mass and suggested that is better to use the expression for the momentum and energy of a body in motion, instead of relativistic mass.

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It is strange suggestion as Einstein has used relativistic mass in his work including in the expression of relativistic kinetic energy [12] from which rest mass energy is derived.
(E) In addition Einstein has assumed that body remains at rest before and after emission of light energy. But the
body may be at rest i.e. v = 0, velocity may be in clas-
sical region and velocity may be in relativistic region
 (v c), the law of inter-conversion of mass and energy
holds good under all conditions. In electron-positron annihilation, the material particles are in motion before and after annihilation. In materialization of energy, a gamma ray photon is converted to electron positron pair, which move in opposite directions to conserve momentum. In nuclear ﬁssion and fusion particles remain in motion in the process of mass energy inter conversion. The thermal neutron which causes ﬁssion has velocity 2185 m/s.
/ 4 L mc2 is mathematically consistent in Einstein’s derivation, under general conditions
Under general conditions (all possible values of variables) the
value of conversion factor other than c2 can be easily justiﬁed
mathematically in Einstein’s derivation [23–36]. This aspect is not touched by the preceding authors [13–21].

in mass is (0:000038077L=cv + L=c2) thus there is no def-
inite value of decrease in mass in Einstein’s derivation. In this case decrease in mass is more than as predicted by Ein-
stein, hence again the conversion factor other than c2 is con-
/ ﬁrmed i.e. L mc2. Like this there are many examples
of this type.
(b) The central equation in Einstein’s derivation is Eq. (1) and binomial theorem is equally applicable to it at any stage i.e. in the beginning or end. Einstein applied bi-
nomial theorem in the end and obtained L = mc2 ,
but the same equation is not obtained if binomial theorem is applied in the beginning. The binomial theorem is simply a mathematical tool and its application at any stage should not aﬀect results i.e. make or mar equation
L = mc2.
The reason is that typical nature of derivation and Eq. (1) is diﬀerent from other relativistic equations. The energy is scalar quantity and independent of direction but Eq. (1) is di-
rectional in nature due to angle . In contrast if binomial
theorem is applied to Relativistic Kinetic Energy in the beginning or at the end then result is same i.e. classical form
of kinetic energy (mrestv2=2). So there is inconsistency in
applications in this case. Applying binomial theorem to Eq. (1) and repeating the
calculations as Einstein did, altogether diﬀerent results are obtained,

(a) In Einstein’s derivation if one wave is regarded as to form angle 0.5 rather than 0 then


` =` 1

v c

cos


1

+

v2 2c2

+

3v4 8c4

+

:

:

:


:

(16)

H0 = H1 + 0:5 L

h
1

v c

cos 0:5


+


1

v c

cos 180

i
;

(13)

or

H0 = H1 +

L


1

+

0:000018038

v c


;

or

K0

K

=

0:000019038 l L

v c

+

L

v2 2c2

;

Here v=c  1, hence v2=c2 and higher terms can be ne-

glected. Thus


` =` 1

v c

cos


or

(H0 E0) (H1 E1) = 0 ;

or
Kb Ka = 0 ;

m (Mb

Ma)

=

0:000038077

L cv

+

L c2

;

(14)

or

L=

mc2 1141

=

0:000876

mc2;

(15)

L / mc2:
Fbeufrothreere, mMiass(imonasosfalfitgehrteemniesrsgioyn):o0f:l0ig0h0t0e3n8e0r7g7yL) ==cMv =b (Lm=acs2s
in (14). According to Einstein if body emits two light waves of en-
ergy 0:5L each in opposite directions then decrease in mass is given by Eq. (10) i.e. m = L=c2 and in this case decrease

or

1 2

Mb

v2

1 2

Ma

v2

=

0

;

or

Mass of body before emission (Mb) = = Mass of body after emission (Ma):

(17)

Thus light energy is being emitted, but under this condi-
tion Einstein’s this derivation does not provide any relation-
ship (equality or proportionality) between mass annihilated
and energy created. Similar is the situation if velocity v = 0.
Hence Einstein’s derivation gives decrease in mass of body
equal to L=c2 only under certain conditions. Thus in this case
derivation is not valid.

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Sr. No 1 2 3

Values of various parameters
0:5L, 0:5L,  = 0 ,  = 180 0:5L, 0:5L,  = 0:5 ,  = 180 0:5001L, 0:49999L,  = 0 ,  = 180

WhetherL = mc2 or L / mc2
L = mc2
L = mc2=901 or L / mc2 L = 0:9999988 mc2 or L / mc2

4 0:5L, 0:5L,  = 0 ,  = 180 but v = 0

No relation between L and m

5 0:5L, 0:5L,  = 0 ,  = 180
but Binomial Theorem is applied in beginning.

No relation between L and m

6 For other energies than light

Equations not considered

/ Table 1: Einstein’s Sep 1905 derivation gives L = mc2 under certain conditions and L mc2 under general conditions

(c) Let the body emits two light waves of slightly diﬀerent

energies i.e. 0:5001L and 0:4999L in opposite direc-

tions and other parameters remain the same as assumed

by Einstein. In this case


H0 = H1 + 0:4999 L 1

+ 0:5001 L 1

vcvccocos s1080+:

(18)

Now proceeding in the same way as Einstein did

K0

K

=

0:0002 L

v c

+

L

v2 2c2

(19)

or

m = Mass of body before emission(Mb) Mass of body after emission(Ma);

=

0:0004

L cv

+

L c2

(20)

or

Ma

=

0:004

L cv

L c2

+

Mb

or

L=

mc2

0:0004

c v

+

1

:

The velocity v is in classical region, say 10 m/s,

L = mc2 0:000083 ;

(21)

L / mc2:

/ Thus, E mc2. Hence conversion factor other than
c2 follows from Einstein’s derivation under general condi-
tions.

(d) Energy emitted in various reactions. In his September

1905 paper Einstein derived Eq. (10) i.e. L = mc2

and then replaced L (light energy) by E (total energy)

and speculated

E = mc2:

(11)

In Eq. (11) E stands for all possible energies of the uni-
verse e.g.: (i) sound energy, (ii) heat energy, (iii) chemical
energy, (iv) nuclear energy, (v) magnetic energy, (vi) electri-
cal energy, (vii) energy emitted in form of invisible radiations,

(viii) energy emitted in cosmological and astrophysical phe-

nomena, (ix) energy emitted volcanic reactions, (x) energies

co-existing in various forms etc., etc.

Now Eq. (1) i.e.

`

=`

1
q

v c

1

cos
vc22

is put forth for light energy by Einstein in June 1905 paper
(` is light energy in moving frame), it is not meant for other
possible energies as quoted above. Einstein never justiﬁed Eq. (1) for all the energies cited
above. The parameters used in Einstein’s equation are deﬁned for light energy only, not for all the energies. Thus by
this derivation L = mc2 is derived under special conditions for light energy only and replacing L by E in Eq. (10) is not
justiﬁed. There are evidences that Einstein worked hurriedly in
other case also e.g. in theory of static universe the introduction of cosmological constant proved to be incorrect and Einstein accepted the mistake later as quoted by Gamow [42].
/ The various cases when E mc2 is justiﬁed are shown
in Table 1.

5 Conservation of momentum in general cases
The momentum is conserved irrespective of the fact that body remains at rest or recoils or tends to recoil after emission of light energy [43]. The law of conservation of momentum can be used to calculate the velocity of recoil in this case also. Let the body of mass 10 kg emits two waves of en-
ergy in visible region of wavelength 5000A_ it corresponds to energy 7:9512 10 19 J. This energy is emitted in two waves i.e., as obvious, 0:5001L (3:97639512 10 19 J) and 0:4999L (3:97480488 10 19 J). Applying the conservation of mo-
mentum [43] the recoil velocity, recoil momentum and recoil
kinetic energy comes out to be 5:3 10 32 m/s, 5:3 10 31 kg m/s and 1:404 10 62 J respectively. This recoil velocity (Vr) will change the uniform velocity v as Vr + v, but it will
not make any diﬀerence to ﬁnal result of change in mass as in
Eq. (21), due to negligible value of Vr [27]. Hence in the law
of conservation of momentum is obeyed in this case also.

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/ 6 Experimental feasibility with conversion factors other less than predicted by E = mc2 i.e. E mc2 is feasible,

than c2

it is an open possibility unless ruled out.

(a) Dirac [44] was one of the ﬁrst physicists to suggest that, in connection with his theory of large numbers, fundamental dimensional constants may vary in time during the expansion of the universe. The idea of variation of the speed of light is suggested in various cosmological models [45, 46] and has been the subject of attention by physicists in investigations of extra dimensions, strings and branes [47]. Webb [48] has reported variations in ﬁne structure constant over cosmological
time scales and hence variations in c. This suggestion implies
E / mc2.
(b) Einstein has derived L = mc2 (conversion factor between mass and energy is precisely equal to c2) under the ex-
tremely special or ideal conditions , which are even diﬃcult to attain practically. The work of scientists before Einstein
also justiﬁes E / mc2. This discussion does not confront with existing experimental situation but addresses those theoretical and exper-
imental issues for which E = mc2 is not analyzed yet.
The mass energy inter-conversion equation, with conversion
factor equal to c2 i.e. E = mc2 has been conﬁrmed in
nuclear physics and is also basis of nuclear physics. Even elementary units of atomic mass (1 amu) or and energy (eV) are based upon it. Thus it will remain standard in measurements as seven days in a week; its validity in this regard is not doubted at all.
The aim is to discuss experimentally those phenomena in
which E = mc2 is not applied yet. The mass energy con-
version processes are weird in nature and all have not been
at all studied in view of E = mc2. The conversion factor other than c2 is discussed for such elusive cases, not for
those it is already conﬁrmed. Hence there is no confrontation with the established experimental situation at all, but aim is
/ to open a mathematical front ( E mc2) for numerous
experimentally unstudied phenomena in nature. This development can be discussed as below.
7 Most abundant chemical reactions
(i) Unconﬁrmed chemical reactions. When Einstein de-
rived E = mc2, chemical reactions were the most abundant sources of energy in nature. Till date E = mc2 is not con-
ﬁrmed in the chemical reaction and the reason cited for this is that equipments are not enough sensitive [49,50]. Consider burning of 1kg straw or paper or petrol in controlled way i.e. in such a way that masses, ashes, gases and energy produced can be estimated. Even if 0.001 kg or 1gm of matter is annihi-
lated then energy equal to 9 1013 J (can drive a truck of mass 1000 kg to distance of 9 107 km) will be produced. Until
the equation is not conﬁrmed in such reactions, then scien-
tiﬁcally E = mc2 may not be regarded as precisely true in
such cases. It is equally possible that energy emitted may be

Reactions in nuclear physics
(ii) Less eﬃciency: The eﬃciency of the nuclear weapons as well as nuclear reactors is far less than the theoretical value
predicted by E = mc2. Robert Serber (member of ﬁrst
American team entered Hiroshima and Nagasaki in September 1945 to assess loses), has indicated [51] that the eﬃciency of “Little Boy” weapon (U235, 49 kg) that was used against Hiroshima was about 2% only. It is assumed that all the atoms don’t undergo ﬁssion, thus material is wasted. But no such waste material is speciﬁcally measured quantitatively. Thus the waste material (nuclear reactor or weapon) must be measured and corresponding energy be calculated, and it must quantitatively explain that why eﬃciency is less. It may require the measurements of all types of energies (may co-exist in various forms) in the processes and experimental errors. Until such experiments are speciﬁcally conducted and
E = mc2 is conﬁrmed, E / mc2 is equally feasible.
(iii) Less energy: In laboratory it is conﬁrmed [7, 52, 53] that using thermal neutrons the total kinetic energy (TKE) of ﬁssion fragments that result from of U235 and Pu239 is 20–
60 MeV less than Q-value (200 MeV) of reaction predicted by E = mc2. This observation is nearly four decades
old. Bakhoum [7] has explained it on the basis of equation
H = mv2 (energy emitted is less than E = mc2). Hence
here E / mc2 is justiﬁed.
(iv) More mass: Palano [10] has conﬁrmed that mass of particle Ds (2317) has been found more than current estimates
based upon E = mc2. Thus in this case E / mc2 is
justiﬁed. (v) Binding energy and mass defect in deuteron: There are two inherent observations [23, 28, 29] about nucleus: ﬁrstly, masses of nucleons are fundamental constants, i.e. they are the same universally (inside and outside the nucleus in all cases); and secondly nuclei possess Binding Energy
(BE = mc2) owing to a mass defect. To explain these observations, in the case of the deuteron (BE = 2:2244 MeV),
the mass defect of nucleons must be 0.002388 amu or about 0.11854% of the mass of nucleons, i.e., nucleons must be lighter in the nucleus. This is not experimentally justiﬁed, as masses of nucleons are universal constants. Thus observa-
tions and predictions based upon E = mc2 are not justi-
ﬁed, hence E / mc2 is equally feasible.
8 Mathematical form of extended equation
Until E = mc2 is not precisely conﬁrmed experimentally
in ALL CASES , it is equally feasible to assume that the en-
ergy emitted may be less than E = mc2 (or E / mc2).
It does not have any eﬀect on those cases where E = mc2
is conﬁrmed, it simply scientiﬁcally stresses conﬁrmation of
E = mc2 in all cases. Also when reactants are in bulk

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amount and various types of energies are simultaneously emitted and energies may co-exist. Thus both the possibilities are equally probable until one is not speciﬁcally ruled out. In view of weirdness in reactions emitting energy in universe, some theoretical inconsistencies in the derivation and nonavailability of data, one can explore the second possibility even as a postulate. All the equations in science are regarded as conﬁrmed when speciﬁcally justiﬁed in all experiments time and again. The reactions involving inter-conversion of mass and energy are utmost diverse, weird and new phenom-
ena are being added regularly, thus E = mc2 needs to be con-
ﬁrmed in all cases. Thus in general, in view of above proportionality it may be taken in account as
dE / c2dm :

constant, just like universal gravitational constant G and k in
Newton’s Second Law of Motion. The reason is that mass en-

ergy inter-conversion are the bizarre processes in nature and not completely studied.
Now consider the case that when mass is converted into energy. Let in some conversion process mass decreases from
Mi(initial mass) to Mf (ﬁnal mass), correspondingly energy increases from Ei (initial energy) to Ef (ﬁnal energy). The Eq. (22) gives inﬁnitesimally small amount of energy dE created on annihilation of mass dm. To get the net eﬀect the
Eq. (22) can be integrated similarly Einstein has obtained the relativistic form of kinetic energy in June 1905 paper [18]

Z

Z

dE = Ac2 dm ;

The above proportionality dE / c2dm can be changed
into equation by introducing a constant of proportionality. The inception of proportionality constant is consistent with centuries old perception of constant of proportionality in physics since days of Aristotle and Newton. In second law of
motion (F = kma) the value of constant of proportionality, k is always unity (like universal constant) i.e. F = ma. When
more and more complex phenomena were studied or values of constants of proportionality were determined then it showed dependence on the inherent characteristics of the phenomena. In case constant of proportionality varies from one situation to other then it is known as co-eﬃcient of proportionality e.g. co-eﬃcient of thermal conductivity or viscosity etc. Thus re-
moving the proportionality between dE and c2dm, we get

dE = Ac2dm ;

(22)

whereA is (a co-eﬃcient) used to remove that sign of propor-
tionality; it depends upon inherent characteristics of the pro-
cesses in which conversion of mass to energy takes place and
it is dimensionless. It has nature precisely like Hubble’s con-
stant (50 and 80 kilometers per second-Megaparsec, Mpc) or
coeﬃcient of viscosity (1:05 10 3 poise to 19:2 10 6 poise)
or co-eﬃcient of thermal conductivity (0.02 Wm 1K 1 to
400 Wm 1K 1) etc. Thus, in fact Hubble’s constant may be
regarded Hubble’s variable constant or Hubble’s coeﬃcient,
as it varies from one heavenly body to other. If “A” is equal to one, then we will get dE = dmc2 i.e. same as Einstein’s
equation.
In Eq. (22) “A” is regarded as conversion factor as it de-
scribes feasibility and extent of conversion of mass into en-
ergy. For example out of bulk mass, the mass annihilated to
energy is maximum in matter-antimatter annihilation, appar-
ently least in chemical reactions, undetermined in volcanic
reactions and cosmological reactions. It (the co-eﬃcient A)
depends upon the characteristic conditions of a particular pro-
cess. It may be constant for a particular process and varies
for the other depending upon involved parameters or experi-
mental situation. Thus “A” cannot be regarded as universal

Initial limit of mass = Mi, Initial limit of Energy = Ei , Final limit of mass = Mf , Final limit of Energy = Ef .

Initially when mass of body is Mi, then Ei is the initial energy of the system. When mass (initial mass, Mi)
is converted into energy by any process under suitable cir-
cumstances the ﬁnal mass of system reduces to Mf . Consequently, the energy of system increases to Ef the ﬁnal energy. Thus Mf and Ef are the quantities after the conversion.
Hence, Eq. (22) becomes

Ef Ei = Ac2 (Mf Mi)

(23)

or

E = Ac2 m

(24)

Energy evolved = Ac2 (decrease in mass): (25)

If the characteristic conditions of the process permit then
whole mass is converted into energy i.e. after the reaction no
mass remains (Mf = 0)

E = Ac2Mi

(26)

In this case energy evolved is negative implies that energy is created at the cost of annihilation of mass and the process is exo-energic nature (energy is emitted which may be in any form). Energy is scalar quantity having magnitude only, thus no direction is associated with it.
Thus the generalized mass-energy equivalence may be stated as
“The mass can be converted into energy or vice-versa under some characteristic conditions of the process,
but conversion factor may or may not always be c2 (9 1016 m2/s2) or c 2.”

9 Applications of generalized mass energy inter conver-
sion equation E = A c2 m
(i) It is already mentioned in section (3) that if 0.001 kg or
1gm of matter is annihilated then energy equal to 9 1013 J

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(can drive a truck of mass 1000 kg to distance of 9 107 km)

will be produced. Such or similar predictions are not experi-

mentally conﬁrmed and energy emitted can be found less than

predictions.
Let the energy observed is 4:5 1013 J corresponding to mass annihilated 0.001 kg , then value of A from E = = Ac2 m will be 0.5 i.e.

A=

c2

E m

=

4:5 9

1013 J 1016

= 0:5:

(27)

Thus in this case mass energy inter-conversion equation

becomes

E = 0:5c2 m :

(28)

(ii) Let the TKE of ﬁssion fragments of U235 and Pu239 is
175 MeV (as experimentally it is observed less), instead of
expected 200 MeV. It can be explained with help of E = = Ac2 m with value of A is equal to 0.875 i.e.

A=

E c2

m

=

175 200

=

0:875

:

(29)

Thus energy of ﬁssion fragments of U235 and Pu239 is

given by

E = 0:875c2 m :

(30)

Thus value of A less than one is justiﬁed experimentally
in this case.

(iii) The anomalous observation of excess mass of
Ds(2317) can be understood with help of E = Ac2 m, as

mass of the observed particle is found more [10] than pre-
dictions of E = mc2. In this case value of A will be less than one. For understanding consider energy equal to 106 J is

converted into mass, then corresponding mass must be

1:11 10 11 kg. We are considering the case that mass is

found more than this. Let the mass be 1:12 10 11 kg. The

value of A this case is 0.992, as calculated from E =

= Ac2 m i.e.

A

=

106 10:8 105

=

0:992

:

(31)

Thus in this mass energy inter conversion equation be-

comes

E = 0:992c2 m or m = 1:008 E : (32)

Thus corresponding to small energy more mass is emitted.
(vi) E = Ac2 m is useful in explaining the binding energy (2.2244MeV or 3:55904 10 13 J), mass defect (0.002388 amu or 2:388 10 3 amu) and universal equality of mass of nucleons (mn = 1.008664 amu, mp = 1.006082 amu). Ob-
viously neutron and protons contribute equally towards the mass defect (0.001194 amu), then mass of neutron inside nucleus must be 1.00747 amu (mass outside nucleus i.e. in Free State is 1.008664 amu). Similarly corresponding mass of proton in the nucleus must be 1.006082 amu (mass of proton outside nucleus 1.007274 amu). But decrease in mass of nucleons inside nucleus is not justiﬁed, as masses of nucleons are universally same [23, 28, 29].

Thus mass defect of deuteron must be inﬁnitesimally

small, only then masses of nucleons are same inside nucleus

and outside nucleus. Also binding energy must be 2.2244

MeV as experimentally observed. Both these experimentally
conﬁrmed facts can be explained with help of E =Ac2 m.

Let in this case the mass defect is negligibly small i.e.
2:388 10 13 amu or 3:9653 10 40 kg. Then value of A

(coeﬃcient of proportionality or mass energy inter conver-
sion coeﬃcient) is 1010 i.e. for annihilation of inﬁnitesimally

small mass exceptionally large amount of energy is liberated.

Thus

A = c2

E m

=

9

3:5634 10 1016 3:9653

13
10

40 = 1010 ;

(33)

E = 1010c2 m :

(34)

(v) Webb [48] has reported results for time variability of the
ﬁne structure constant or Sommerfeld ﬁne structure constant
( ) using absorption systems in the spectra of distant quasars.
The variation in magnitude of alpha has been observed as

= ( then now) = 0:9 10 5: (35)
now

According to CODATA currently accepted value of alpha
( now) is 7:297352 10 3. Hence from Eq. (35),

then = 0:007296 :

(36)

Now corresponding to the reduced value of ( then = = 0.007296) the the speed of light can be determined from

equation

cthen = 2

e2 then"h

(37)

as 2:994 108m/s (where all terms have usual meanings). Cur-

rently accepted value of the speed of light is 2:99729 108m/s.

To explain the energy emitted with this value of the speed
of light is the value A ( E = Ac2 M)

A

=

c2 c2then

=

1:001

:

(38)

Thus in this case mass energy inter conversion equation

becomes

E = 1:001c2 m :

(39)

/ Hence E

mc2 has both experimental and theo-

retical support, with emergence of new experimental data its

signiﬁcance will increase.

Acknowledgements
The author is highly indebted to Stephen J. Crothers, Dr. E. Bakhoum and Dr. N. Hamdan for critical discussions of the topic and providing related materials. Also the author is highly indebted to Dr. T. Ramasami, Secretary DST for arranging discussion on the topic with Physics Advisory Committee and providing ﬁnancial assistance.
Submitted on March 10, 2008 / Accepted on May 19, 2008

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References
1. Newton I. Optics. pp.CVX, 1704; reprinted by Dover Publications, Inc., New York, 1952.
2. Preston S. T. Physics of the ether. E. & F. N. Spon, London, 1875.
3. De Pretto O. Ipotesi dell’etere nella vita dell’universo. Reale Istituto Veneto di Scienze, Lettere ed Arti, Feb. 1904, tomo LXIII, parte II, 439–500.
4. Haseno¨hrl F. Wien Sitzungen IIA, 1904, v. 113, 1039.
5. Haseno¨hrl F. Annalen der Physik, 1905, Bd. 16, 589.
6. Soddy F. Radioactivity: an elementary treatise from standpoint of the disintegration theory. The Electrician, Printing and Publishing, London, 1904.
7. Bakhoum E. G. Physics Essays, 2002, v. 15, 1; arxiv: physics/0206061.
8. Hambsch F. J. et al. Nucl. Phys. A, 1989, 491, 56.
9. Thiereus H. et al. Phys. Rev. C, 1981, v. 23, 2104.
10. Palano A. et al. Phys. Rev. Lett., 2003, v. 90, 242001.
11. Einstein A. Annalen der Physik, 1905, v. 18, 639.
12. Einstein A. Annalen der Physik, 1905, v. 17, 891–921.
13. Fadner W. L. Am. J. Phys., 1988, v. 56, 2, 144.
14. Planck M. Akad. Weiss. Physik. Math. K1asse, 1907, Bd. 13, 542.
15. Stark J. Physikalische Zeitschrift, 1907, Bd. 8, 881.
16. Ives H. J. Opt. Soc. Am., 1952, v. 42, 540.
17. Stachel J. and Torretti R. Am. J. Phys., 1982, v. 50, 750.
18. Okun L. Acta Physica Polonica B, 2006, v. 37, 1327–1332.
19. Hamdan N., Khariri A., and Lo´pez-Bonilla J. Apeiron, 2007, v. 14, 435–453.
20. Bartocci U. Albert Einstein e Olinto De Pretto: la vera storia della formula piu` famosa del mondo. Ed. Andromeda, Bologna, 1999.
21. Bjerknes C. J. Albert Einstein: The Incorrigible Plagiarist. XTX Inc., Dowers Grove, 2002.
22. Born M. Physics in My Generation. Pergamon Press, London, 1956, 193.
23. Sharma A. International Journal of Nuclear Science and Technology, 2007, v. 3, no. 4, 370–377.
24. Sharma A. Galilean Electrodynamics, 2007, v. 18, no. 5, 99– 100.
25. Sharma A. Kurt Godel Society Collegium Logicum Volume I, 2007, 67–71.
26. Sharma A. Journal of Gravitational Physics, 2007, v. I, no. 1, 9–31.
27. Sharma A. Concepts of Physics, 2006, v. III, no. 4, 345–373. http://merlin.ﬁc.uni.lodz.pl/concepts/2006 4/2006 4 351.pdf
28. Sharma A. Abstract Book: 38th European Group of Atomic Systems, Euro Physics Conference, Isachia (Naples), Italy, 2006, 53.

29. Sharma A. Abstract Book: A Century After Einstein Physics 2005, Institute of Physics, Bristol, University of Warwick, UK, 10–14 April, 2005.
30. Sharma A. Abstract Book, Oral presentation (lecture) on 21st September 2005, The 5th British Gravity Meeting, Oxford, England.
31. Sharma A. Presented at the 19th International Conference on the Applications of Accelerators in Research and Industry, Fort Worth, Texas, USA, 20–25 August, 2006.
32. Sharma A. Physics Essays, 2006, v. 17, 195–222.
33. Sharma A. Proc. Int. Conf. on Number, Time, Relativity, United Physical Society of Russian Federation, Moscow, 2004, 81.
34. Sharma A. Acta Ciencia Indica, 1998, v. XXIV P, no. 4, 153– 158.
35. Sharma A. To be published in Galilean Electrodynamics, Massachusetts, USA.
36. Sharma A. Accepted for publication in Concepts of Physics, Institute of Physics, Poland.
37. Lorentz H. A. Proc. Roy. Soc. Amst., 1899, v. 1, 427–442.
38. Lorentz H A. Proc. Roy. Soc. Amst., 1904, v. 6, 809–831.
39. Kaufmann W. Phys. Z., 1902, v. 4, 55, and Gott. Nacchr. Math. Phys., 1903, v. 4, 90.
40. Bucherer A. H. Phys. Z., 1908, v. 9, 755, Ann. Phys., 1909, v. 28, 513.
41. Einstein A. 1948 Letter to Lincoln Barnett, quoted in “The concept of mass” by Lev Okum, Physics Today, June 1989.
42. Gamow G. My World Line, Viking, New York, 1970, 44.
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44. Dirac P. A. M. Nature, 1937, v. 139, 323; Proc. Roy. Soc. A, 1938, v. 165, 199.
45. Albrecht A. and Magueijo J. Phys. Rev. D, 1999, 59, 043516.
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48. Webb J. K. et al. Phys. Rev. Lett., 1999, 82, 884.
49. Beiser A. Concepts of Modern Physics. 4th edition, McGrawHill International Edition, New York, 1987, 25, 27, 420.
50. McGraw-Hill Encyclopedia of Science and Technology. McGraw-Hill, New York, 1987. v. 14, 582, and v. 4, 338.
51. Serber R. The Los Alamos Primer (U.S. Govt. ﬁrst published as LA1, April 1946), declassiﬁed 1965, annotated book, 1992; also Serber R. (editor), The Los Alamos Primer Univ. of California Press, 1992, 38.
52. Hambsch F. J. et al. Nucl. Phys. A, 1989, v. 491, 56.
53. Thiereus H. et al. Phys. Rev. C, 1981, v. 23, 2104.

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On the Necessity of Aprioristic Thinking in Physics

Elmira A. Isaeva
Institute of Physics, Academy of Sciences of Azerbaijan, 33 H. Javida av., Baku, AZ-1143, Azerbaijan E-mail: el max63@yahoo.com; elmira@physics.ab.az
The thinking which encompasses both reasoning-in-itself and reasoning-for-itself, called “aprioristic thinking” by Hegel, is the freest form of thinking. This form of thinking is imparted to the physical sciences by philosophy. Only under this condition can physics obtain deeper scientiﬁc knowledge.

In the beginning of the last century, the renowned scientist Anri Bergson [1] gave an advanced notice: “We experience now one of the greatest crises; all our thinking, all ethics, all life, all our spiritual and moral existence are in a condition of intellectual fermentation. . . ”. This fermentation, according to the opinion of the known philosopher Edmund Husserl [2], occurs due to installation dominant in positivistic and naturalistic philosophy. This installation of ordinary consciousness contrasts the human consciousness and being to each other, and, therefore, not taking into account consciousness, can lead to more crisis the European sciences. As pointed out by Husserl, the sciences about the nature can be founded only by means of phenomenology, as a strict philosophy, which is oriented towards a ﬁrst-hand experience of consciousness. Though many years have already passed since then, as these scientists have written, resolute turn in this question is not yet present. Even, in spite of the fact that in one of the achievements of modern physics — in quantum physics — the consciousness of the observer has found a place for itself. In the interpretation of quantum mechanics, the most important upshot of this for physicists is that this problem is related to the problem of consciousness — an interdisciplinary problem concerning not only physicists, but also philosophers, psychologists, physiologists and biologists. Its solution will result in deeper scientiﬁc knowledge. But all the same, for some reason, scientists very often in case of scientiﬁc cognition neglect questions of the interaction between our consciousness and the surrounding world. If we wish to reach fuller scientiﬁc knowledge, we should not deal with physical phenomena and thinking (consciousness) itself separately. The well-known physicist Wigner [3] maintains that the separation between our perception and the laws of nature is no more than simpliﬁcation. And though we are convinced that it has a harmless character, to nevertheless merely forget about it should not be the case. It is clear that deeper scientiﬁc knowledge should include in itself a problem of the theory of cognition — a problem of the origin of knowledge and a logical substantiation of the relevant system of knowledge.
In deciding upon this problem, the cognition theory considers the connection between “I”, my consciousness and an external world, and says that the decision is concealed in the interaction between sensuality and reason. Reason transforms our feelings into thoughts and it means that the representa-

tions are replaced with concepts. If science does not wish to be, as it was described by Hegel [4], a simple unit of data then, of course, it should have concepts and should operate with them. But, if science also does not wish to be positivistic (all sciences, except philosophy, are positivistic) then it should have a rational basis and beginning. Only in this case, does the sole purpose (aﬀair) of science become the concept of the concept. (Hegel has distinguished between the sciences as follows: 1) sciences, as a simple unit of data, 2) the extremely positive sciences, 3) positive sciences, 4) philosophy. Positivism of a physical science is that it does not know that its deﬁnitions are ﬁnal).
Physics, certainly, has a rational basis which is intimately connected with philosophy too. But what prevents a physical science from becoming a “mere” philosophy? Hegel has elaborated on the notion of a positivistic side of the sciences. In physics, this positivism is characterized by the lack of knowledge that its deﬁnitions are ﬁnal and therefore there is no transition into the higher sphere. This ﬁniteness is connected with the ﬁniteness of the cognition (feeling, belief, authority of others, and authority of external and internal contemplation).
However, it is perhaps meant so to happen, as described by Hegel, that thoughtful contemplation, lowering casual conditions and organizing everything, will present the general outline before a detailed intellectual exposition. It is clear then that an intellectual physical science will picture a rational science of Nature in the form of an image which is the external image of Nature. This image is called a physical picture of the world, or, as called by Max Planck [5], the world of a physical science. Planck has explained further about it: “. . . We are compelled to recognize behind the sensual world the second, real world which leads independent existence independent of the person, — the world which we not can comprehend directly, but we comprehend via the sensual world, via known symbols which he informs us, as if we would consider a interesting subject only through the glasses, optical properties of which are absolutely unknown for us”.
Thus, according to Planck, there are three worlds: the real world, the sensual world and the world of a physical science or a physical picture of the world. The real world is the world outside us, it exists irrespective of our understanding of its laws, i.e. irrespective of our consciousness and therefore it is the objective world. The sensual world is our world because

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we perceive it through our bodies of perception: eyes, hearing, charm etc., and it is subjective (it is possible to tell that it is illusion). A physical picture of the world is the world in which can be reﬂected both real and the sensual world. This world is a bridge for us with which help we study the world around. Reﬂection of the real world in the world of a physical science is a physical picture of the real world; it is also possible to describe the quantum world and the science studying this world is the quantum physics. The reason why the real world is the quantum world is because the so-called world of atoms and electrons, as Planck has given above, exists independently of the person. Reﬂection of the sensual world in a physical picture of the world is a physical picture of the sensual world (the classical world) and the corresponding science is the classical physics. Thus, only in case of the thoughtful contemplation can the physics can be concerned with the philosophy of nature.
But when will it be possible to tell, whether the physical science is not simply concerned with philosophy, and even enters into it, to a certain extent it? Based on a well-known classiﬁcation of all sciences by Hegel, the nature philosophy is a science about an idea in another-being. Hegel has thus said: “what is real, is reasonable”, referring to understanding in the context of the reality of a reasonable idea. Such a reality is the maintenance of Hegel’s philosophy. Hegel writes that phenomena, being unstable (random) and existing in continuous ﬂuidity, are in contrast to the idea and do not enter into it. Therefore Hegel takes the idea as the maintenance of his philosophy. In the ancient time, Plato too spoke about ideas [2]. He wrote: “In a horse, in the house or in the ﬁne woman there is nothing real. The reality is concluded as a universal type (idea) of a horse, the house, the ﬁne woman” [6]. Plato conﬁrms the continuous ﬂuidity of all existing forms and asks the question: can the philosophy be within continuous and chaotic ﬂuidity? As a result, the human knowledge is possible only under the condition of the existence of steady ideas, and with the help of it, is possible to distinguish between things based on ﬂuid validity and to plan in it any logical order. Hegel understands that an idea will be steady, if it will be the reality of a “reasonable”. After all, only reason is steady, absolute. But this is not only because it is so ingenious to deﬁne ideas in the way Hegel did it. In “Metaphysics”, Aristotle, criticizing Plato, asserts that the idea of a thing explains nothing in the thing itself, even provided that the idea relates to the thing, as found for example, in the fact that whiteness concerns a white subject. Aristotle did not actually deny the independent existence of ideas, but attributed to them the existence within things themselves. Namely, Hegel’s idea — the reality of the “reasonable” — satisﬁes Aristotle’s requirement. Because, in such determination, the idea is taken from the reality itself. But against Hegel’s reality the mind at once acts. The mind says to us that ideas are no existing chimeras. If science does not want to conceptualize its concept then it, of course, will agree with the mind. Then, very ﬁguratively, it

is described by Hegel as follows: just as meal process is ungrateful to the meal (simply eats it, not giving instead of anything), similarly, thinking process will be ungrateful to aposteriori experience, and will simply give nothing in exchange. In order to receive something from thinking process, it is necessary to make the thinking itself by the subject of thinking. Reﬂection transforms our representations into concepts. And further reﬂections of concepts transform concepts into concepts, i.e. it becomes clear as a concept. Only under such conditions can the science understand its concept. However, only in philosophy do we ﬁnd that the subject of thinking is the thinking itself (for example, for the mathematician, it is numbers, spaces etc.). The thinking, opposing with itself to itself, is the reasoning-for-itself. Process thinking nevertheless is inside and consequently it is the reasoning-in-itself. As a result, the “in itself” and “for itself” reasoning is the most substantial form of free thinking and it is deﬁned by Hegel, as aprioristic thinking. Only by aprioristic thinking can the generality and authenticity be found. Namely, in this thinking, philosophy informs the maintenance of empirical sciences. The obligation of the sciences is not to refuse this process, because it is a very noble act for a science to reach the concept of the concept. But the mind, objecting again, speaks to us: “But what it can give to the physics? ”. At all times, there have been physicists who, knowing about the ﬁniteness of the knowledge of their science, have spoken about deeper scientiﬁc knowledge [8–15]. They envision when it will be possible to speak about the physicist and about the consciousness of the observer simultaneously.
Hegel has very interestingly written: “In the physicist we too get acquainted with the general, with essence, the only distinction between physics and the philosophy of nature is that the philosophy of nature leads up us to the comprehension of the true forms of the concept of natural things”. But doesn’t it mean that in deeper scientiﬁc cognition the physical science has transited into a higher circle which is not present in physics because of its positivism? And the answer to this question is, of course, yes, it does. Thus, only under the condition of deeper scientiﬁc knowledge can we claim that the physical science is the philosophy of nature (in the sense that, for example, the apple is a fruit).
Hegel deﬁnes the philosophy of nature, as a science about an idea in its another being. As he writes, in philosophy we do not learn anything else, except ideas, but the ideas exist here as exterior forms. An exterior form of an idea is its another being. Because the being of an idea (reasoning-in-itself and reasoning-for-itself) takes place in the reason itself. Nature receives its exterior, that exterior which we see, in the exterior process of an idea. In fact, Hegel’s slogan “what is reasonable, is real” is conﬁrmed.
Unwittingly, we could as well resolve one more problem. The maintenance of philosophy, as Hegel writes, is an idea which excludes from itself, the phenomenon, chance. But the maintenance of physics is Nature, its phenomena. At the

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same time we may ask, “when can the physical science become the philosophy of nature?” All becomes clear when we agree with Hegel, that Nature is connected with an idea, in the sense that it is an idea in its another being. The laws of Nature, discovered by our thinking about physics, are also ideas – reasonables of reality.
Thus, as in the past, philosophy will continue to play an important role related to the necessity for the sciences to enter a higher level. Only in this case can the sciences avoid the crisis about which Husserl has always warned us. As Bergson continues that which has been said in the beginning of this article: “. . . The new system, more general, wider should become the doctrine for many decades and even centuries. These new principles should direct all our life on a new way on which the mankind will approach to cognition of true and to happiness increase at the Earth”.
Submitted on March 31, 2008 Accepted on June 23, 2008
References
1. Anri Bergson. Modern scientiﬁc Weltanschaung. Uspekhi Fizicheskikh Nauk, 1920, v. 2, no. 1, 1–13 (in Russian).
2. The philosophical encyclopaedic dictionary, Moscow, 1989, 838 pages
3. Wigner E.P. In: Quantum Theory and Measurements, Eds. J. A. Wheeler, W. H. Zurek, Princeton University Press, 1983, 168 pages
4. Hegel G.W.F. Science of logic. Moscow, 1974, 451 pages.
5. Plank M. The picture of the world of modern physics. Uspekhi Fizicheskikh Nauk, 1929, v. 9, no. 4, 407–436 (in Russian).
6. Klein M. Mathematics. Deﬁniteness loss. Moscow, 1984, 437 pages.
7. Losev A.F. and Taho-Godi A. Aristotle. Moscow, 1982, 285 pages.
8. Everett H. In: Quantum Theory and Measurements, Eds. J. A. Wheeler, W. H. Zurek, Princeton University Press, 1983, 168 pages.
9. De Witt B.S. and Graham N. In: The Many-Worlds Interpretation of Quantum Mechanics, Eds. De Witt B.S., Graham N., Princeton University Press, 1973, 196 pages.
10. Laurikainen K.V. Beyond the atom: the philosophical thought of Wolfgang Pauli. Berlin, Springer-Verlag, 1988, 357 pages.
11. Menskii M.B. Quantum Mechanics: new experiments, new applications, new formulations. Physics-Uspekhi, Turpion Publ. Ltd., 2000, v. 43, no. 6, 585–600 (from Uspekhi Fizicheskikh Nauk, 2000, v. 6, no. 170, 631–649).
12. Kadomsev B.B. Dynamics and Information. Moscow, 1999, 427 pages
13. Shro¨dinger E. The selected works on the quantum mechanics. Moscow, 1977, 356 pages
14. Utiyama R. To what the physics has come. Moscow, 1986, 224 pages

July, 2008

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Elmira A. Isaeva. On the Necessity of Aprioristic Thinking in Physics

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Volume 3

Potential Energy Surfaces of the Even-Even 230-238U Isotopes
Sohair M. Diab and Salah A. Eidy
yFFaaccuultlytyoof fEEndguinceaetiroinng, ,PPhhyys.s.DDeeppt.t,.,AAininSShhaammssUUnniviveersristiyt,y,CCaairioro, ,EEggyyppt t
E-mail: mppe2@yahoo.co.uk
Nuclear structure of 230-238U isotopes hav been studied in the frame work of the in-
teracting boson approximation model (IBM 1). The contour plot of the potential energy surfaces, V ( ; ), shows that all nuclei are deformed and have rotational characters, SU (3). Levels energy spectra belonging to the gsb, , bands, electromagnetic transition rates B(E1) and B(E2), quadrupole moment Q0, deformation parameterare 2 and the strength of the electric monopole transitions X(E0=E2) are calculated. The
calculated values are compared with the available theoretical and experimental data and show reasonable agreement.

1 Introduction

The observation of a large quadrupole moments to 230-238U

isotopes had led to the suggestion that these nuclei might be

deformed and have to be conﬁrmed by the measurement of

their nuclear properties as well as the observation of their ro-

tational band structures. It is noticed that the level schemes

of uranium isotopes are characterized by the existence of two

bands of opposite parity and lay in the region of octupole de-

formations. The primary evidence for this octupole deforma-

ton comes from the parity-doublet bands, fast electric tran-

sition (E1) between the negative and positive parity bands

and the low-lying 1
Many authors have

st,ud0i+2eda2n3d0-22+238eUxciistoattioopnesentheergoyretsitcaatellsy.

using diﬀerent models. The relativistic Mean Field Model

has employed [1–4] to obtain the densities of the cluster and

daughter nuclei. Also, a systematic -decay properties of

the even-even heavy and superheavy nuclei have been inves-

tigated. The energy of the deformed nuclei in the actinide

region has been determined in the frame work of the macro-

scopic — microscopic approach. The Yukawa folding proce-

dure has used [5] together with the Liquid Drop Model [6].

The properties of the states of the alternating parity bands

in actinides are analyzed within the Cluster Model. The

model has been used successfully in calculating levels en-

ergy, quadrupole moments and half-lives of cluster radioac-

tivity. A comparison was mad between the predicted data

[7–13] and the calculated values by other models and show

good agreement.

The band heads, energy spacings within bands and a num-
ber of interband as well as intraband B(E2) transition rates

are well reproduced [14] for all actinide nuclei using the Ex-

actly Separable Davidson (ESD) solution of the Bohr Hamil-

tonian.

The potential energy surfaces are calculated [15] to 230U

using the most advanced asymmetric two-center shell model

that are added to the Yukawa-plus-exponential model. Until now scarce informations are available about the ac-
tinide region in general and this is due to the experimental diﬃculties associated with this mass region. In the present ar-
ticle we used the Interacting Boson Model (IBM 1) which
is a theoretical model and diﬀer than all the previous models used with the actinid nuclei. The aim of the present work is to process calculation for the follows:
1. For the potential energy surfaces, V ( ; ), for all
230-238U nuclei;
2. For levels energy;
3. For the electromagnetic transition rates B(E1) and also calculation for B(E2);
4. For the electric quadrupole moment Q0;
5. For the deformation parameter 2; 6. For the strength of the electric monopole transitions
X(E0=E2).

2 (IBA-1) model 2.1 Level energies

The IBA-1 model was applied to the positive and negative parity low-lying states in even-even 230-238U isotopes. The

proton, , and neutron, , bosons are treated as one boson and

the system is considered as an
and d-bosons. Creation (sydy)

interaction between s-bosons and annihilation (sd~) operat-

ors are for s and d bosons. The Hamiltonian [16] employed

for the present calculation is given as:

H = EP S nd + P AIR (P P ) +

+

1 2

ELL

(L

L)

+

1 2

QQ

(Q

Q) +

(1)

+ 5OCT (T3 T3) + 5HEX (T4 T4) ;

Sohair M. Diab and Salah A. Eid. Potential Energy Surfaces of the Even-Even 230-238U Isotopes

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nucleus 230U 232U 234U 236U 238U

EPS
0.2000 0.2000 0.2000 0.2000 0.2000

P AIR
0.000 0.000 0.0000 0.0000 0.0000

ELL
0.005 0.0050 0.0044 0.0055 0.0057

QQ
0.0150 0.0150 0.0150 0.0150 0.0150

OCT
0.0000 0.0000 0.0000 0.0000 0.0000

HEX
0.0000 0.0000 0.0000 0.0000 0.0000

E2SD(eb)
0.2060 0.1890 0.1782 0.1720 0.1630

Table 1: Table 1: Parameters used in IBA-1 Hamiltonian (all in MeV).

E2DD(eb)
0.6094 0.5591 0.5271 0.5088 0.4822

where

2

P

p

=

1 2

4

n(sysy)(00) n(ss)(00)

pp5(5d~(d~d)y(0d0)yo)(00)o x

3(0)
5;
0

(2)

L L = 10 p3 h(dyd~)(1)x (dyd~)(1)i(0) ;

(3)

0

2

Q

Q

=

p
5

6664

(Syd~+ dys)(2)

(syd~+ +d~s)(2)

p7 p2 7
2


(dyd~)(2) x

(dyd~)(2)

3(0)
7775 ;
0

(4)

T3 T3 =

p
7

h(dyd~)(2)x

(dyd~)(2)i(0)

;

0

(5)

T4

T4

=

3

h(dyd~)(4)x (dyd~)(4)i(0)
0

:

(6)

In the previous formulas, nd is the number of boson; P P , L L, Q Q, T3 T3 and T4 T4 represent pairing, angular mo-
mentum, quadrupole, octupole and hexadecupole interactions

between the bosons; EP S is the boson energy; and P AIR, ELL, QQ, OCT , HEX is the strengths of the pairing, an-

gular momentum, quadrupole, octupole and hexadecupole in-

teractions.

2.2 Transition rates

The electric quadrupole transition operator [16] employed in

this study is given by:

T (E2) = E2SD (syd~+ dys)(2) +

+ p15 E2DD (dyd~)(2) :

(7)

The reduced electric quadrupole transition rates between

Ii ! If states are given by

B (E2; Ii

If )

=

[<

If

k T (E2) k
2Ii + 1

Ii

>]2

:

(8)

3 Results and discussion
3.1 The potential energy surface
The potential energy surfaces [17], V ( , ), for uranium iso-
topes as a function of the deformation parameters and

have been calculated using :

EN N ( ; ) = <NN; jHjNN; > =

= d(NN) 2(1 + 2) + 2(1 + 2) 2

kNN[4 (XX) cos 3 ] +

(9)


+ [XX 2] + N(N

1)


1 10

c0

+

1 7


c2


2;

where

X

=

 2 0:5 7

X

 =  or  :

(10)

The calculated potential energy surfaces, V ( ; ), for ura-
nium series of isotopes are presented in Fig. 1 and Fig. 2. It shows that all nuclei are deformed and have rotational-like characters. The two wells on both oblate and prolate sides are not equal but the prolate is deeper in all nuclei.. The energy and electromagnetic transition rates are calculated cosidering uranium series of isotopes a rotational-like nuclei.

3.2 Energy spectra

IBA-1 model has been used in calculating the energy of the positive and negative parity low -lying levels of uranium series of isotopes. In many deformed actinide nuclei the negative parity bands have been established and these nuclei are considered as an octupole deformed. A simple means to examine the nature of the band is to consider the ratio R which
 for octupole band, R 1, and deﬁned as [18]:

R=

E (I + 3) E (I)

E (I E (I

1)NP B 2)GSB

:

(11)

In the present calculations all values of R for uranium se-
 ries of isotopes are 1, and we treated them as octupole
deformed nuclei. A comparison between the experimental spectra [19–23]
and our calculations, using values of the model parameters given in Table 1 for the ground and octupole bands, are illustrated in Fig. 3. The agreement between the calculated levels energy and their correspondence experimental values for all uranium nuclei are reasonable, but slightly higher especially for the higher excited states. We believe this is due to the change of the projection of the angular momentum which

88

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I+i I+f 01 Exp. 21

230U

232U

234U

236U

238U

9.70(12) 10.0(10) 10.66(20) 11.61(15) 12.09(20)

01 Theor. 21 9.7128 10.0163 10.6479 11.6506 12.1143

21 01

1.9426 2.0033 2.1296 2.3301 2.4229

22 01

0.0107 0.0113 0.0104 0.0095 0.0081

22 02

1.2419 1.3677 1.5411 1.7598 1.8855

23 01

0.0190 0.0131 0.0099 0.0082 0.0066

23 02

0.0027 0.0095 0.0131 0.0144 0.0139

23 03

0.0245 0.0085 0.0031 0.0013 0.0007

24 03

0.7577 0.8679 1.0291 1.2308 1.3730

24 04

0.0508 0.0415 0.1309 0.0710 0.0022

41 21

2.7740 2.8443 3.0182 3.3014 3.4336

41 22

0.0699 0.0480 0.0352 0.0276 0.0213

41 23

0.0046 0.0019 0.0010 0.0007 0.0005

61 41

3.0183 3.0849 3.2707 3.5790 3.7256

61 42

0.0706 0.0532 0.0412 0.0333 0.0260

61 43

0.0128 0.0066 0.0039 0.0026 0.0018

81 61

3.0670 3.1387 3.3335 3.6548 3.8121

81 62

0.0618 0.0503 0.0415 0.0351 0.0381

81 63

0.0201 0.0117 0.0073 0.0049 0.0034

101 81

2.9919 3.0827 3.2910 3.6237 3.7930

101 82

0.0510 0.0439 0.0383 0.0340 0.0280

Table 2: Table 2: Values of the theoretical reduced transition probability, B(E2) (in e2 b2).

Ii I+f 230U 232U 234U 236U 238U 11 01 0.1353 0.1602 0.1824 0.2071 0.2294 11 02 0.0531 0.0512 0.0492 0.0475 0.0449 31 21 0.2509 0.2811 0.3075 —— —— 31 22 0.0811 0.0763 0.0711 —— —— 31 23 0.0013 0.0002 0.0000 —— —— 51 41 0.3628 0.3913 —— —— —– 51 42 0.0862 0.0831 —— —— —– 51 43 0.0020 0.0006 —— —— —– 71 61 0.4809 0.5064 —— —— —— 71 62 0.0816 0.0811 —— —— —— 91 81 0.6043 0.6267 —— —— —— 91 82 0.0736 0.0749 —— —— ——
Table 3: Table 3: Values of the theoretical reduced transition probability, B(E1) (in  e2b).

nucleus 230U 232U 234U 236U 238U

Q0

9.920 10.020 10.340 10.800 11.020

2

0.263 0.264 0.272 0.282 0.286

Table 4: Table 4: The calculated electric quadrupole moment Q0 and deformation parameter 2.

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Fig. 1: Potential energy surfaces for 230-238U nuclei at = 0 (prolate) and 60 (oblate).

Fig. 2: Contour plot of the potential energy surfaces for 230-238U nuclei.

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Fig. 3: Comparison between experimental (Exp.) [19–23] and theoretical (IBA-1) energy levels in 230-238U.

is due to band crossing and octupole deformation. From -

bands [24] octupole deformation deformation has observed
at I = 14 (for 232U), I = 10 (for 234U), I = 15 (for 236U) and I = 10 (for 238U) respectively.

Unfortunately there is no enough measurements of elec-

tromagnetic transition rates B (E2) or B (E1) for these series
of nuclei. The only measured B (E2; 0+1 ! 2+1 )’s are pre-
sented, in Table 2 for comparison with the calculated values.

The parameters E2SD and E2DD used in the present calcu-

lations are displayed in Table 1.

The calculated [equations 12, 13] electric quadrupole mo-

ment Q0 and deformation parameter 2 are given in Table 4.
It is clear that both values are increasing with the increase of

the neutron number of uranium isotopes.

Q0

=

 16B (E2)exp: 1=2 5

;

(12)

2

=

[B (E2)exp:]1=2
3Z4R02

(13)

3.3 Electric monopole transitions
The electric monopole transitions, E0, are normally occur-
ring between two states of the same spin and parity by trans-

ferring energy and zero unit of angular momentum. The
strength of the electric monopole transitions, Xif0f (E0=E2),
[25] are calculated using equations (14, 15) and presented in
Table 5.

Xif 0 f

(E0=E2)

=

B (E0; Ii B (E2; Ii

If ) I0f )

;

(14)

Xif0f (E0=E2) = (2.54 109) A3=4

E5(MeV)
KL

(E

2)

Te(E0; Ii Te(E2; Ii

If I0f

) )

:

(15)

3.4 Conclusions
The IBA-1 model has been applied successfully to 230-238U isotopes and we have got:
1. The ground state and octupole bands are successfully reproduced;
2. The potential energy surfaces are calculated and show rotational behavior to 230-238U isotopes where they are mainly prolate deformed nuclei;
3. Electromagnetic transition rates B (E1) and B (E2)
are calculated;

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I+i I+f I+0f 230U
02 01 21 0.660

232U 234U 0.560 0.001

226U 0.920

238U 1.470

03 01 21 13.370 1.300 15.910 0.282 —–

03 01 22 2.400 0.410 3.000 1.960 212.500

03 01 23 3.510 0.280 2.420 1.240 0.520

03 02 21 0.620 0.590 0.660 0.500 —–

03 02 22 0.110 0.180 0.120 0.001 1.500

03 02 23 0.180 0.130 0.100 0.001 3.720

04 01 22 1.960 7.750 0.001 0.230 7.250

04 01 23 1.320 0.250 —– 04 01 24 32.660 0.330 1.000

0.250 0.170

0.190 0.460

04 02 22 —–

0.020 0.0000 0.060 3.250

04 02 23 —— 0.750 —–

0.070 0.080

04 02 24 —–

—– 0.000 0.100 0.2000

04 03 21 0.330 —– —–

24.000 19.000

04 03 22 0.020 0.080 0.110 0.330 4.750

04 03 23 0.010 2.750 —–

0.360 0.130

04 03 24 0.330 —– 17.000 0.520 0.300

Table 5: Table 5. Theoretical Xif0f (E0/E2) ratios for E0 transitions in Ra isotopes.

4. Electric quadrupole moment Q0 are calculated;
5. Deformation parameter 2 are calculated.
Submitted on May 31, 2008 Accepted on June 23, 2008
References
1. Bhattacharya M. and Angopadhyay G. G. Phys. Rev. C, 2008, v. 77, 027603.
2. Pei J. C., Xu F. R., Lin Z. J. and Zhao E. G. Phys. Rev. C, 2007, v. 76, 044326.
3. Xu F. R. and Pei J. C. Phys. Lett. B, 2006, v. 642, 322. 4. Patra S. K., Gupta R. K., Sharma B. K., Stevenson P. D. and
Greiner W. J. Phys. G, 2007, v. 34, 2073. 5. Bartel J., Dobrowdski A., Pomorski K. Int. J. of Mod. Phys. E,
2007, v. 16, 459. 6. Royer G. and Bonilla C. J. Radioanalytical and Nucl. Chem.,
2007, v. 272, 237. 7. Kuklin S. N., Adamian G. G. and Antonenko N. V. Phys. Rev.
C, 2005, v. 71, 014301. 8. Buck B., Merchant A. C., Perez S. M. and Seals H. E. J. Phys.
G, 2005, v. 31, 1499. 9. Balasubramaniam M., Kumarasamy S. and Arunachalam N.
Phys. Rev. C, 2004, v. 70, 017301. 10. Adamian G. G., Antonenko N. V., Jolos R. V., Palchikov Yu. V.,
Scheid W. and Shneidman T. M. Nucl. Phys. A, 2004, v. 734, 433. 11. Ren Z., Xu C. and Wang Z. Phys. Rev. C, 2004, v. 70, 034304.

12. Buck B., Merchant A. C. and Perez S. M. Phys. Rev. C, 2003, v. 68, 024313.
13. Jolos R. V. and Scheid W. Phys. Rev. C, 2003, v. 67, 014313.
14. Bonatsos D., Mc Cutchan E. A., Minkov N., Casten R. F., Yotov P., Lenis D., Petrellis D. and Yigitoglu I. Phys. Rev. C, 2007, v. 76, 064312.
15. Poenaru D. N., Gherghescu R. A. and Greiner W. Phys. Rev. C, 2006, v. 73, 014608.
16. Feshband H. and Iachello F. Ann. Phys., 1974, v. 84, 211.
17. Ginocchio J. N and Kirson M. W. Nucl. Phys. A, 1980, v. 350, 31.
18. Gross E., de Boer J., Diamond R.M., Stephens F. S. and Tjom P. Phys. Rev. Lett, 1975, v. 35, 565.
19. Akovali Y. A. Nucl. Data Sheets, 1993, v. 69, 155.
20. Browin E. Nucl. Data Sheets, 2006, v. 107, 2579.
21. Browin E. Nucl. Data Sheets, 2007, v. 108, 681.
22. Browin E. Nucl. Data Sheets, 2006, v. 107, 2649.
23. Chukreev V, Makarenko V. V. and Martin M. J. Nucl. Data Sheets, 2002, v. 97, 129.
24. Raduta A. A. and Ionescu D. Phys. Rev. C, 2003, v. 67, 044312.
25. Rasmussen J. O. Theory of E0 transitions of spheroidal nuclei.
Nucl. Phys., 1960, v. 19, 85.

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Volume 3

LETTERS TO PROGRESS IN PHYSICS
A Brief Note on “Un-Particle” Physics

Ervin Goldfain
Photonics Co., Welch Allyn Inc., Skaneateles Falls, NY 13153, USA E-mail: ervingoldfain@gmail.com
The possibility of a hidden sector of particle physics that lies beyond the energy range of the Standard Model has been recently advocated by many authors. A bizarre implication of this conjecture is the emergence of a continuous spectrum of massless ﬁelds with non-integral scaling dimensions called “un-particles”. The purpose of this Letter is to show that the idea of “un-particles” was considered in at least two previous independent publications, prior to its ﬁrst claimed disclosure.

The Standard Model (SM) is a highly successful theoretical framework that describes the relationships among all known elementary particles and the attributes of three of the four forces that act on these particles — electromagnetism, the strong force and the weak force. SM covers an energy range upper limited by the weak interaction scale of approx. 300 GeV. Despite the remarkable success of SM, it seems likely that a much deeper understanding of nature will be achieved as physicists continue to probe the fundamental constituents of matter at increasingly higher energies. Both theory and experiments strongly indicate that new phenomena await discovery beyond the SM range and reaching into the Terascale region. The Large Hadron Collider (LHC) at CERN is based on high energy proton beams and is scheduled to begin operation later this year. Moreover, further exploiting the Terascale physics will be possible in the near future with a new accelerator known as the International Linear Collider (ILC). It is believed that running both LHC and ILC will provide clues on how to go about solving many of the open questions challenging the current SM.
The possibility of a yet-unseen sector that lies in the Terascale range and is weakly coupled to SM has been recently advocated by many authors [1– 6]. A bizarre implication of this conjecture is the emergence of a continuous spectrum of massless states with non-integral scaling dimensions called “un-particles”. In classical physics, the energy, linear momentum and mass of a free point particle are linked
through the relativistic connection (c = 1):

E2 = p2 + m2:

(1)

Quantum mechanics converts (1) into a dispersion rela-
tion for the corresponding quantum waves, with the mass m ﬁxing the low frequency cut-oﬀ ( = 1):

un-particles at either LHC or ILC may show up as “missing” energy in certain decay channels [1–6].
The purpose of this Letter is to set the record straight and point out that the idea of “un-particles”, ﬁrst claimed in [1, 2], was previously considered elsewhere. To the best of our knowledge, there are at least two publications where a similar or identical concept was introduced and discussed:
1. In 2005, Prof. F. Smarandache has launched the term un-matter as part of his novel mathematical framework of Neutrosophy and Fuzzy Logic [7, 8];
2. In 2006, the author has formulated the concept of fractional number of ﬁeld quanta in connection with the development of quantum ﬁeld theory using complex dynamics [9].
It is unfortunate that neither one of [1– 6] have referenced these contributions.
Submitted on March 23, 2008 Accepted on April 07, 2008
References
1. Georgi H. Phys. Rev. Lett., 2007, v. 98, 221601. 2. Georgi H. Phys. Lett. B, 2007, v. 650, 275. 3. Cheung K. et al. Phys. Rev. D, 2007, v. 76, 055003. 4. Bander M. et al. Phys. Rev. D, 2007, v. 76, 115002. 5. Chen C.-H. and Geng C.-Q. Phys. Rev. D, 2007, v. 76, 115003. 6. Zwicky R. Phys. Rev. D, 2008, v. 77, 036004. 7. Smarandache F. Progress in Physics, 2005, v. 1. 8. Smarandache F. Progress in Physics, 2005, v. 2. 9. Goldfain E. Chaos, Solitons and Fractals, 2006, v. 28, 913.

!2 = k2 + m2:

(2)

Unlike (1) or (2), un-particles are conjectured to emerge as streams of fractional objects, something that has never been either imagined or seen before. A possible signal of

Ervin Goldfain. A Brief Note on “Un-Particle” Physics

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LETTERS TO PROGRESS IN PHYSICS

International Injustice in Science
Florentin Smarandache
Chair of Math & Sciences Department, University of New Mexico, Gallup, NM 87301, USA E-mail: smarand@unm.edu
In the scientiﬁc research, it is important to keep our freedom of thinking and not being yoked by others’ theories without checking them, no matter where they come from. Cogito, ergo sum (I think, therefore I am), said Descartes (1596 –1650), and this Latin aphorism became his ﬁrst principle in philosophy.

Inspired by D. Rabounski [1] and M. Apostol [2] I read more articles about injustices in science (for example [3]) and in arts and letters occurring in contemporary societies. The poet Plautus (254-184 B.C.) had once exclaimed that homo homini lupus (man is a wolf for man), so people make problems to people. In this short letter to the editor, I would like to list some inconvenient cases that manifest today:
There exist reviewing and indexing publications and institutes made just for a propagandistic way, and not reviewing all relevant literature on the topics, but reviewing their people and their ideas while ignoring, boycotting, denigrating, or discrediting other people and ideas. They exercise an international traﬃc of inﬂuence by manipulations and falsiﬁcations of information (such as biographies, history of events, etc.), discourage people for working on topics diﬀerent from theirs, and use subversive techniques in their interest of hegemony in science, arts, and letters.
The science, art, and literature of the powerful are like that: If you don’t cite them, it is your fault as if you have not read them. However, if they don’t cite you, it’s your fault too as if you did not deserve to be cited because you have published in so-called by them “obscure publications”, even if these people have “borrowed” your idea without acknowledgement. They categorize as “obscure, unimportant, not by establishment” those journals, publishing houses, cultural centers and researchers or creators that do not obey to them or that dare to be independent thinkers, in order that these people with power positions stigmatize them in the public’s eye (because they can not control these publications). While the publications and centers of research they control they proclaim as “the best”. The science/art & letters establishments continue to ignore or minimalize the research and creation done outside the establishment. It became a common procedure that people who control the so-called “high” publications abuse their power and they “take” ideas from less circulated publications and publish them in these “high” publications without citation, as their own ideas!
There are journals using hidden peer-reviewers that delay the publication until someone else from their house get credit for your paper’s ideas.
Secret groups and services ignore and even boycott per-

sonalities who are independent in thinking and don’t follow the establishment or don’t obey to them; they manipulate national and international awards in science, arts, literature, also they manipulate university positions, high research jobs, funding; they try to conﬁscate the whole planet’s thought by making biased so-called “reference sites” (as the self-called “encyclopedias”, “dictionaries”, “handbooks”, etc.) where they slander independent thinkers, while blocking other sites they don’t like; that’s why the whole human history of science, arts, letters has to be re-written; the search engines bring these “reference sites” amongst the ﬁrst pages in a search, even they are not the most relevant to the search topic, and since most of the hurry readers browse only the beginning pages [they don’t spend time to look at all of them], it is a high probability that the populace is manipulated according to the biased information of these so-called “free” (just because they are not free!) reference sites; these groups try to conﬁscate the Internet at the global scale; always, during history, there were and unfortunately there still are intentions from some secret groups or services to dominate others. . . They try to transform other countries in spiritual colonies by brain washing. Secret groups and services do not only politic, economic, or military espionage, but also scientiﬁc, artistic, literary manipulations in the proﬁt of their people.
Unfortunately, big cultures continue to destroy small cultures and to delete the collective memory of small nations. History is written by winners, says the aphorism, but this is not correct, history should be written by all parts. International organisms are created who unfortunately only serve the interests of a few powers, not of the whole world.
There are people believing they detain the absolute truth, and if somebody dares to have a diﬀerent opinion from them, he or she is blacklisted, slandered, banned from various publications, etc.
The public opinion is provoked, manipulated through propaganda, publicity, dissemination by those who detain the power or control the mass media and the national and international awards, and these awards have been created in purpose to impose some people and ideologies.
There exist scientiﬁc, artistic, literary, or cultural associations/organizations whose hidden goal is to manipulate peo-

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ple in their propagandistic interest and indoctrinate them. The literature they start to send (after collecting your membership money!) reﬂects only their ideas and praise only their people, while ignoring or boycotting others’. Nolens volens (unwilling or willing) the “member” of such association becomes their spiritual slave. Consequently, you are yoked to this association’s propaganda. Better to be independent and not belonging to any association/organization.
The author would like to express his gratitude to V. Christianto, D. Rabounski, M. Apostol, and E. Goldfain for their comments.
Submitted on March 22, 2008 Accepted on March 24, 2008
References
1. Rabounski D. Declaration of Academic Freedom. Progress in Physics, 2006, v. 1, 57–60.
2. Apostol M. Comment on the Declaration of Academic Freedom by D. Rabounski. Progress in Physics, 2007, v. 3, 40.
3. Boerner R. The suppression of inconvenient facts in physics. http://www.suppressedscience.net

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Progress in Physics is an American scientiﬁc journal on advanced studies in physics, registered with the Library of Congress (DC, USA): ISSN 1555-5534 (print version) and ISSN 1555-5615 (online version). The journal is peer reviewed and listed in the abstracting and indexing coverage of: Mathematical Reviews of the AMS (USA), DOAJ of Lund University (Sweden), Zentralblatt MATH (Germany), Scientiﬁc Commons of the University of St. Gallen (Switzerland), Open-J-Gate (India), Referential Journal of VINITI (Russia), etc. Progress in Physics is an open-access journal published and distributed in accordance with the Budapest Open Initiative: this means that the electronic copies of both full-size version of the journal and the individual papers published therein will always be acessed for reading, download, and copying for any user free of charge. The journal is issued quarterly (four volumes per year).
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Editorial board: Dmitri Rabounski (Editor-in-Chief) Florentin Smarandache Larissa Borissova Stephen J. Crothers
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