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Einstein's Paper: "Explanation of the Perihelion Motion of Mercury from General Relativity Theory"
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Einstein’s Paper: “Explanation of the Perihelion Motion of Mercury from
General Relativity Theory”
Anatoli Vankov
IPPE, Russia; Bethany College, USA anatolivankov@hotmail.com
2020-05-28
Abstract
The year 1915 of publishing Einstein’s original paper “Explanation of the Perihelion Motion of Mercury from General Relativity Theory” marks the origin of General Relativity. The perihelion advance of Mercury’s orbit was obtained from Astronomical observations by Le Verrier (1859), late Newcomb (1882). Their evaluation of the eﬀect was about 43 arcsec/cen, which coincides with Einstein’s prediction half a Century later derived from his “ﬁeld equations”. Einstein’s original paper and the corresponding Schwarzschild’s letters were published in German. We present their translations of high ﬁdelity and physical accuracy specially made by Professor Roger Rydin from the University of Virginia. The translations are followed with our critical analysis of Einstein’s “approximate” derivation of the Mercury’s perihelion advance from his ﬁeld equations. There is a common belief among Physical Community that Einstein’s prediction is correctly derived, its physical meaning is fully understood, and the value of the GR perihelion advance is conﬁrmed in observations to the high precision, not less than 1 percent. However, from our analysis it follows that every item of the belief is a myth, and Einstein’s prediction is plainly ﬂawed. Schwarzschild expressed a strong objection to Einstein’s teary complete mess in derivation from “ﬁeld equations”. He righteously suggested to derive the eﬀect from his “exact solution”, the metric quadratic form, which should be called “Schwarzschild metric”. But it is drastically diﬀerent from what is known as “Schwarzschild metric”. He purposely derived the metric having no “coordinate” or “central” inﬁnities by introducing “the cut-oﬀ parameter”, in other words, he made the GR Dynamics “renormalizable”. After his premature death, somebody did not like it. So, the metric in his name

was quietly reshaped and acquired the present singularity form. This made hundreds of GR following apprentices and “experts” to work hard and useless on the GR problem of “non-renormalizability” and “non-quatizability”. Unfortunately, Schwarzschild’s derivation of the eﬀect did not make a diﬀerence. A reader can get his own mind on the issues after having been familiarized with historically valuable papers presented here. Maybe, a curiosity awakens about the GR quicksand foundation, the so-called Einstein’s Field Equations, which the whole GR theory is rested upon.
Key words: Einstein, Schwarzschild, General Relativity, Mercury perihelion, ﬁeld equations.
PACS 03.30.+p, 04.20.-g
2

According to general theory of relativity, the elliptical orbit of a planet referred to a Newtonian frame of reference rotates in its own plane in the same direction as the planet moves... The observations cannot be made in a Newtonian frame of reference. They are aﬀected by the precession of the equinoxes, and the determination of the precessional motion is one of the most diﬃcult problems of observational astronomy. It is not surprising that a diﬀerence of opinions could exist regarding the closeness of agreement of observed and theoretical motions... I am not aware that relativity is at present regarded by physicists as a theory that may be believed or not, at will. Nevertheless, it may be of some interest to present the most recent evidence on the degree of agreement between the observed and theoretical motions of the planets.
Gerald Clemence (Rev. Mod. Phys. 19, 361–364, 1947)
3

Erkl¨arung der Perihelbewegung des Merkur aus der allgemeinen Realtivit¨atstheorie Von A. Einstein
According to “The Collected Papers of Albert Einstein” (see our Notes), this is the lecture given to the Prussian Academy of Sciences in Berlin, 18 November 1915 by A. Einstein. Published 25 November 1915 in K¨oniglich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1915): 831-839.
1 Einstein’s paper, 1915
Translation of the paper (along with Schwarzschild’s letter to Einstein) by Roger A. Rydin with the following comments by Anatoli A. Vankov
Explanation of the Perihelion Motion of Mercury from General Relativity Theory Albert Einstein
Introduction In an earlier version of the work appearing in this journal, I have presented the ﬁeld equations of gravity, which are covariant under corresponding transformations having a determinant equals unity. In an Addendum to this work, I have shown that each of the ﬁeld equations is generally covariant when the scalar of the energy tensor of the matter vanishes, and I have thereby shown from the introduction of this hypothesis, through which time and space are robbed of the last vestige of objective reality, that in principle there are no doubts standing against this assertion. 1 In this work, I found an important conﬁrmation of this radical Relativity theory; it exhibits itself namely in the secular turning of Mercury in the
1In a soon to follow manuscript, it will be shown that such a hypothesis is unnecessary. It is only important that one such choice of coordinate system is possible, in which the determinant |gµν| takes the value −1. The following investigation is then independent thereof.
4

course of its orbital motion, as was discovered by Le Verrier. Namely, the approximately 45 per century amount is qualitatively and quantitatively explained without the special hypotheses that he had to assume. 2
Furthermore, it shows that this theory has a stronger (doubly strong) light bending eﬀect in consequence through the gravitational ﬁeld than it amounted to in my earlier investigation.

The Gravitational Field As my last two papers have shown, the gravitational ﬁeld in a vacuum for a suitably chosen system of coordinates has to satisfy the following

α

∂Γαµν ∂xα

+

αβ

Γαµβ Γβνα

=

0

(1)

whereby the quantity Γαµν is deﬁned through

Γαµν = −

µν α

= − gαβ
β

µν β

1 =−

gαβ ∂gµβ + ∂gνβ − ∂gµν

2β

∂xν ∂xµ ∂xα

(2)

Otherwise, we make the same fundamental hypothesis as in the last paper, that the scalar of the energy tensor of the ”material“ always vanishes, so that we have the determinant equation

|gµν| = −1

(3)

We place a point mass (the Sun) at the origin of the coordinate system. The gravitational ﬁeld, which this mass point produces, can be calculated from these equations through successive approximations.
In this regard, one may think that the gµν for the given solar mass is not yet mathematically fully determined through (1) and (3). It follows from it that these equations with the necessary transformation with the determinant equal to unity are covariant. It should be correct in this case to consider that all these solutions through such transformations can be reduced to one another, that they themselves are also (by given boundary conditions) only formally but not physically distinguishable from one another. These overlying considerations allow me to obtain a solution without considering the question whether or not it is the only unique possibility.

2E. Freundlich wrote in an earlier contribution about the impossibility that the anomaly of the motion of Mercury is satisﬁed on the basis of Newtonian theory, (Astr. Nachr. 4803, Vol. 201, June 1915).

5

With the above in mind, we go forward. The gµν is given next in the “zero-th approximation” in accord with the Relativity Theory scheme

 −1 0 0 0 

 0 −1 0 0 





 

0

0

−1

0

 

(4)

0 0 01

Or more compactly

gρσ = −δρσ; gρ4 = g4ρ = 0; g44 = 1

(4a)

Hereby, ρ and σ are the indices 1, 2, 3: the δρσ is the Kronecker delta symbol equal to 1 or 0, that is when either ρ = σ or ρ = σ.
We now set forward the following, that the gµν diﬀer from the values given in (4a) by an amount that is small compared to unity. This deviation we handle as a small magnitude change of ”ﬁrst order“, and functions of n-th degree of this deviation as of “n-th order“. Equations (1) and (3) are set in the condition of (4a), for calculation through successive approximations of the gravitational ﬁeld up to the magnitude n-th order of accuracy. We speak in this sense of the ”n-th approximation“; the equations (4a) are the ”zero-th approximation“.
The following given solutions have the following coordinate system-tied properties:
1. All components are independent of x4. 2. The solution is (spatially) symmetric about the origin of the coordinate system, in the sense that one obtains the same solution if one makes a linear orthogonal (spatial) transformation. 3. The equations gρ4 = g4ρ = 0 are valid exactly (for ρ = 1, 2, 3). 4. The gµν possess at inﬁnity the values given in (4a).

First Approximation

It is easy to verify, that ﬁrst order accuracy of the equations (1) and (3) as well as the above named 4 conditions is satisﬁed through the substitution of

gρσ = −δρσ + α

∂2r − δρσ ∂xρ∂xσ r

=

−δρσ

−

α

xρxσ r3

;

g44

α =1−
r

(4b)

The gρ4 as well as g4ρ are thereby set through condition 3; the r means the magnitude of r = x21 + x22 + x23.

6

That condition 3 in the sense of ﬁrst order is fulﬁlled, one sees at once. In a simple way to visualize that ﬁeld equation (1) in the ﬁrst order approximation is also fulﬁlled, one needs only to observe that the neglect of magnitudes of second and higher orders on the left side of equation (1) can be realized successively through the substitution

∂Γαµν ;

∂ µν

α ∂xα α ∂xα α

whereby α only runs from 1 to 3. As one sees from (4b), our theory brings with it that in the case of a
slowly moving mass the components g11 to g33 already appear to the non-zero magnitude of ﬁrst order. We will see later that hereby there is no diﬀerence between Newton’s law (in the ﬁrst order approximation). However, it gives a somewhat diﬀerent inﬂuence of the gravitational ﬁeld on the light ray as in my previous work; as the light velocity is introduced through the equation

gµνdxµdxν = 0

(5)

By use of the Huygen’s principle, one ﬁnds from (5) and (4b) through a simple calculation, that a light ray from the Sun at distance ∆ undergoes an angular deﬂection of magnitude 2α/∆, while the earlier calculation, by which the Hypothesis Tµµ = 0 was not involved, had given the value α/∆. A corresponding light ray from the surface rim of the Sun should give a deviation of 1.7 (instead of 0.85 ). Herein there is no shift of the spectral lines through the gravitational potential, for which Mr. Freundlich has measured the magnitude against the ﬁxed stars, and independently determined that this only depends on g44.

After we have taken gµν in the ﬁrst order approximation, we can also calculate the components Γαµν of the gravitational ﬁeld in the ﬁrst order approximation. From (2) and (4b) we obtain

Γτρσ = −α

δρσ

xτ r3

−

3xρ xσ xτ 2r5

(6a)

where ρ, σ, τ take on the values 1, 2, 3, and

Γσ44

=

Γ44σ

=

− αxσ 2r3

(6b)

7

whereby σ takes on the values 1, 2, 3. The single components in which the index 4 appears once or three times, vanish.

Second Approximation
It will be shown here that we only need the three components Γσ44 accurate in the magnitude of second order to be able to evaluate the planetary orbit with suﬃcient accuracy. For this, it is enough to use the last ﬁeld equation together with the general conditions, which have led to our general solution. The last ﬁeld equation

σ

∂Γσ44 ∂xσ

+

στ

Γσ4τ Γτ4σ

=0

goes with reconsideration of (6b) by neglect of magnitudes of third and higher

orders over to

∂Γσ44 = − α2

σ ∂xσ

2r4

From here follows, with reconsideration of (6b) and the symmetry properties our solution

Γσ44

=

−

αxσ 2r3

α 1−
r

(6c)

Planetary Motion
From the General Relativity theory motion equations of a material point in a strong ﬁeld, we obtain

d2xν ds2

=

στ

Γνστ

dxσ ds

dxτ ds

(7)

From this equation, it follows that the Newton motion equation is obtained

as a ﬁrst approximation. Namely, when the speed of a point particle is small

with respect to the speed of light, so dx1 , dx2 , dx3 are small against dx4. It

follows that we come to the ﬁrst approximation, in which we take on the right

side always the condition σ = τ = 4. One obtains then with consideration of

(6b)

d2xν ds2

=

Γν44

=

− αx4 ; 2r3

(ν = 1, 2, 3);

d2x4 = 0 ds2

(7a)

8

These equations show that one can take as a ﬁrst approximation s = x4. Then the ﬁrst three equations are accurately Newtonian. This leads one to the planar orbit equations in polar coordinates r, φ, and so leads to the known energy and the Law of Area equations

1 u2 + Φ = A; r2 dφ = B

(8)

2

ds

where A and B are constants of the energy- as well as Law of Areas, whereby the shortened form is inserted.

α Φ=− ;
2r

u2

=

dr2

+ r2dφ2 ds2

(8a)

We now have the Equations (7) evaluated to an accurate magnitude. The last of Equations (7) then leads together with (6b) to

d2x4 ds2

=2

σ

Γ4σ4

dxσ ds

dx4 ds

= − dg44 dx4 ds ds

or in magnitude of ﬁrst order exactly to

dx4 = 1 + α

(9)

ds

r

We now go to the ﬁrst of the three equations (7). The right side becomes

a) for the index combination σ = τ = 4

Γν44

dx4 ) 2 ds

or with reconsideration of (6c) and (9) in magnitude of second order exactly

−

αxν 2r3

α 1+
r

b) with reconsideration thereof for the index combination σ = 4, τ = 4 (which alone still comes into consideration), and the fact that the products

dxν dxτ ds ds

9

with reconsideration of (8) are seen as magnitudes of ﬁrst order, 3 and are likewise accurate to second order, we obtain

−

αxν r3

στ

δστ

−

3xσ xτ 2r2

dxσ dxτ ds ds

The summation gives



− αxν r3

u2

−

3 2

dr 2

ds

In hindsight therefore, one obtains for the equations of motion in magnitude

of second order the exact form



d2xν ds2

= − αxν 2r3

1 +

α r

+ 2u2 − 3

dr ds

2 

(7b)

which together with (9) determines the motion of the point mass. Besides, it should be remarked that (7b) and (9) for the case of an orbital motion give no deviation from Kepler’s third law.
From (7b) next follows the exactly valid form of the equation

r2 dφ = B

(10)

ds

where B means a constant. The Law of Areas is also accurate in the magnitude of second order, when one uses the ”period“ of the planet for the time measurement. To now obtain the secular advance of the orbital ellipse from (7b), one inserts the members of ﬁrst order in the brackets of the right side arranging it to best advantage using (10), and in the ﬁrst term of the equations (8), through which operation the members of second order on the right side are not changed. Through this, the brackets take the form

3B2 1 − 2A + r2 √ Finally, one chooses s 1 − 2A as the second variable, and again calls it s, so that one has a slightly changed meaning of the constant B:

d2xν

∂Φ =− ;

α

B2

Φ=− 1+

(7c)

ds2

dxν

2r

r2

3This result we can interpret from the ﬁeld components Γνστ with insertion in equation (6a) of the ﬁrst order approximation.

10

By the determination of the orbital form, one now goes forth exactly as in the Newtonian case. From (7c) one next obtains

dr2 + r2dφ2

ds2

= 2A − 2Φ

One eliminates ds from this equation with the help of (10), and so obtains, in which one designates by x the magnitude 1/r:

dx dφ

2

=

2A B2

+

α B2 x

−

x2

+

αx3

(11)

which equation distinguishes itself from the corresponding Newtonian theory only through the last member on the right side.
That contribution from the radius vector and described angle between the perihelion and the aphelion is obtained from the elliptical integral

α2
φ=
α1

dx

2A B2

+

α B2

x

−

x2

+

αx3

where α1 and α2 are the corresponding ﬁrst roots of the equation

2A +

α x − x2 + αx3 = 0

B2 B2

which means, the very close neighboring roots of the equation corresponding to leaving out the last term member.
Hereby we can with reasonable accuracy replace it with

α

α2

dx

φ = 1 + 2 (α1 + α2) α1 −(x − α1)(x − α2)(1 − αx)

or after expanding of (1 − αx)(−1/2):

α φ = 1 + 2 (α1 + α2)

α2 α1

(1

+

α 2

x)dx

−(x − α1)(x − α2)

The integration leads to

3 φ = π 1 + 4 α(α1 + α2)

11

or, if one takes α1 and α2 as reciprocal values of the maximal and minimal distance from the Sun,

3α

φ = π 1 + 2a(1 − e2)

(12)

For an entire passage, the perihelion moves by

α

= 3π

(13)

a(1 − e2)

in the directional sense of the orbital motion, when we designate by a the major half axis, and by e the eccentricity. This leads one to the period T (in seconds), so one obtains with c as light velocity in cm/sec:

=

24π3

T

a2 2c2(1 −

e2)

(14)

This calculation leads to the planet Mercury to move its perihelion for-

ward by 43 per century, while the astronomers give 45 ±5 , an exceptional

diﬀerence between observation and Newtonian theory. This has great signif-

icance as full agreement.

For Earth and Mars the astronomers give a forward movement of 11 and

9 respectively per century, while our formula gives only 4 and 1 , respec-

tively. It appears however from these results, considering the small eccentric-

ity of the orbits of each planet, a smaller eﬀect is appropriate. Conﬁrmation

for the correctness of these values for the movement of the perihelion is the

product

with

the

eccentricity

(e

dπ dt

).

Mercury Venus Earth Mars

(e

dπ dt

)

8.48 ± 0.43

-0.05 ± 0.25

0.10 ± 0.13

0.75 ± 0.35

One considers for these the magnitudes of the Newcomb given values, which I thank Dr. Freundlich for supplying, so one gains the impression that only the forward movement of the perihelion of Mercury will ever be truly proven. I will however gladly allow professional astronomers a ﬁnal say.

End of the paper

12

2 Schwarzschild’s letter to Einstein
Letter from K Schwarzschild to A Einstein dated 22 December 1915
The letter is presented in English owing to Professor Roger A. Rydin

Honored Mr. Einstein, In order to be able to verify your gravitational theory, I have brought myself nearer to your work on the perihelion of Mercury, and occupied myself with the problem solved with the First Approximation. Thereby, I found myself in a state of great confusion. I found for the ﬁrst approximation of the coeﬃcient gµν other than your solution the following two:

gρσ

=

− βxρxσ r5

+ δρσ

β 3r3

;

g44 = 1

As follows, it had beside your α yet a second term, and the problem was

physically undetermined. From this I made at once by good luck a search

for a full solution. A not too diﬃcult calculation gave the following result:

It gave only a line element, which fulﬁlls your conditions 1) to 4), as well as

ﬁeld- and determinant equations, and at the null point and only in the null

point is singular.

If:

x1 = r cos φ cos θ, x2 = r sin φ cos θ, x3 = r sin θ

R = (r3 + α3)1/3 = r

1 α3 1 + 3 r3 + ...

then the line element becomes:

ds2 =

γ 1−

dt2 −

dR2

− R2(dθ2 + sin2 θdφ2)

R

1

−

γ R

R, θ, φ are not “allowed” coordinates, with which one must build the ﬁeld equations, because they do not have the determinant = 1, however the line element expresses itself as the best.
The equation of the orbit remains exactly as you obtained in the ﬁrst approximation (11), only one must understand for x not 1/r, but 1/R, which

13

is a diﬀerence of the order of 10−12, so it has practically the same absolute validity.
The diﬃculty with the two arbitrary constants α and β, which the First Approximation gave, resolves itself thereby, that β must have a determined value of the order of α4 , so as α is given, so will the solution be divergent by continuation of the approximation.
It is after all the clear meaning of your problem in the best order. It is an entirely wonderful thing, that from one so abstract an idea comes out such a conclusive clariﬁcation of the Mercury anomaly. As you see, it means that the friendly war with me, in which in spite of your considerable protective ﬁre throughout the terrestrial distance, allows this stroll in your fantasy land.
14

2.1 Comments
2.1.1 Historical Docs
Einstein’s paper devoted to the GR prediction of Mercury’s perihelion advance, Doc.24 (see Notes), is the only one among his publications that contains the explanation of the GR eﬀect. In his following paper The Foundations of the General Theory of Relativity, 1916, Doc.30, Einstein presents his new (he called it “correct”) calculation of the bending of light while the Mercury perihelion is only mentioned by referring it as in Doc.24, along with Schwarzshild’s work on “the exact solution”.
Meanwhile, numerous works have been published and continue to appear in press with suggestions of “clariﬁcation”, “improvement”, “radical change” or “refutation” of Einstein’s 1915 prediction of the perihelion advance and bending of light. Most of them, in our view, are results of either confusion or lack of qualiﬁcation. Qualiﬁed works presenting a fresh view of the problem, or related new ideas or concepts are discussed in [V an10] (see Notes). Among them, there is a monograph on General Relativity by Bergmann (1942) with Foreword by A. Einstein who clearly acknowledged his own advisory and authorization role in the book composition. Strangely enough, the fact that given there derivations of the GR predictions are quite diﬀerent from those in Doc.24, is not paid much attention in the literature. In spite of methodological diﬀerences, claimed predictions in the above works, however, remain the same.
As a matter of fact, the GR foundational premises have been subjected to changes and reinterpretations (optional, alternative, or claimed “correct” ones) by Einstein himself, his advocates as well as today’s GR specialists and self-proclaimed “experts”. Among the key issues, the problems of energy and angular momentum conservation along with the properties of stress-energymomentum tensor remain the “hot” (better say, controversial) ones. Possibly, this is one of the reasons why there are numerous publications devoted to the GR perihelion advance eﬀect and the light bending, which are considered controversial or arguable.
2.1.2 Einstein and Schwarzschild “in a friendly war”
One may be critical of Einstein’s work (Doc.24) in many respects, some arguable issues are worth noting here.
1. The diﬀerence from Newton’s physics is an appearance of the GR term
15

in the equation of motion (11), not to speak about diﬀerentials with respect to the proper time τ . In operational terms, the proper time is recorded by a clock attached to the test particle moving along the world line s so that components dxµ/ds = uµ deﬁne a tangential 4-velocity unit vector. In SR, it relates to the proper 4-momentum vector P µ = muµ. In Doc.24, however, the proper time is actually interpreted similarly to the coordinate (“far-away”) time t = γτ , which deﬁnes three components of velocity vi = dxi/dt (i = 1, 2, 3). Thus, the Lorentz factor in the relativistic kinetic energy becomes lost.
2. The equation of motion (11) (the objective of the wo√rk) is obtained at the expense of an arbit√rary replacement of s = cτ with s 1 − 2A where a diﬀerence of the factor 1 − 2A from unity has a magnitude of the order α/r0. The corresponding impact on the solution is of the order of the eﬀect in question. At the same time, the GR conservation laws for total energy and angular momentum become controversial: with τ replaced with t, both laws formally appear in the Newtonian form.
3. Unlike in the paper (Doc.24), in Bergman’s book the GR term becomes responsible for both the perihelion advance and the bending of light; consequently, the derivations of both eﬀects principally changed, ﬁrst of all, the Schwarzschild metric was acknowledged as the theoretical basis for the GR eﬀects evaluation. It should be noted that, while the methodology changed, the angular momentum (“area”) law remained unaﬀected by the GR term. One can argue, however, that the GR perihelion advance eﬀect necessarily requires a relativistic generalization of the classical conservation laws.
The immediate response to Einstein’s work (Doc.24) came from Karl Schwarzschild. His famous work “On the Gravitational Field of a Point-Mass, According to Einstein’s Theory”, Sitzungsberichte der K¨oniglich Preußischen Akademie der Wissenschaften, 189-196, 1916) was published less than a month after Einstein’s work. It should be noted that Schwarzschild derived “the exact solution” in the form to be consistent with the Einstein’s “four conditions”. It has no central divergence and cannot be mixed up with the commonly known “Schwarzschild metric”. The less known fact is that Schwarzschild, before the publication, wrote a letter to Einstein in which he criticized Einstein for mistakes in “the successive approximation approach”, as is seen from the above translated Letter from K Schwarzschild to A. Einstein dated 22 December 1915, in ColP ap, vol. 8a, Doc.169.
Schwarzschild claims that his solution reproduces Einstein’s prediction of
16

the GR perihelion advance. But it does not, – because the concept of energy is gone, and coeﬃcients of the equation have very diﬀerent meaning there. Let us see it in more details.
Schwarzschild’s original “exact unique solution” to Einstein’s ﬁeld equations in vacuum for a point source is the squared 4-world line element, which in the polar coordinates is given by

ds2 =

α 1−

dt2 −

α 1−

−1
dR2 − R2dφ2

R

R

(2.1)

where

R = (r3 + α3)1/3

The α is the so-called Schwarzschild’s radius equal to the doubled gravitational radius rg, that is, α = 2rg = 2GM/c20;
the r is the coordinate radial distance, not strictly determined but used
as a measure of distance from the source.
the t is the coordinate (“far-away”) time. the R is the approximation of r to the precision of α3 in the original
Schwarzschild’s metric.

At some historical moment, the Schwarzschild metric in literature acquired the form, in which the R was replaced with the r, so that the metric became divergent. The original metric, as mentioned, does not diverge: R → α as r → 0. This means that, strictly speaking, “the black hole” concept does not come out from the General Relativity theory (see Stephen J. Crothers on the Internet and elsewhere). As is known, Einstein himself resisted the BH idea. He also understood, when speculating about the gravitational radiation (particularly, from a BH neighborhood), that a linear perturbative approximation to the metric does not provide it. As concerns the perihelion advance, Einstein seemed to be quite reluctant to appreciate “the exact solution” and the way how Schwarzschild treated it, in spite of the fact that the Schwarzschild metric respects all four Einstein’s conditions.
Schwarzschild traditionally starts with the three standard integrals of particle motion:

α 1−
R

dt2 ds2

α −1 dR2

− 1− R

ds2

− R2

dφ2 ds2

= const = h

(2.2)

R2 dφ = c ds

(2.3)

17

α dt

1−

= const = 1

R ds

(2.4)

The integral (2.3) coincides with Einstein’s expression for the angular momentum. However, (2.4) is not the energy: according to Schwarzschild, it is “the deﬁnition of the unit of time”. This is in the spirit of the General Relativity Theory, in which the energy is not locally deﬁned. Consequently, the constant (2.2) denoted h must be h = 1 by construction.
After that, the equation of motion for x = 1/R takes the form

dx 2 = 1 − h + hα − x2 + αx3

dφ

c2

c2

(2.5)

where the ﬁrst term on the right sight is zero (compare it with Einstein’s equation (11)).

3 Critique

We are going to follow Einstein’s solution step-by-step. Unfortunately, Einstein wrote the paper (Doc.25) very schematic, so that one needs to restore his line of thoughts and, at the same time, correct mistakes (see Notes).
The exact solution to (11) is given by:

x2

dx

φ=

x1 α(x − x1)(x − x2)(x − x3)

(3.1)

Here we denote x1 = 1/r1, x2 = 1/r2, x3 = 1/r3 (instead of denotations α1 = 1/r1, α2 = 1/r2, α3 = 1/r3 in Doc.24), which are real roots of the homogeneous cubic equation of the bounded motion problem (11). Let the third root x3 be due to the GR term. The integral cannot be calculated analytically (though it can be expressed in terms of elliptic functions). Einstein’s obvious idea is to compare (3.1) with the analogous solution in the classical (Newtonian) formulation of the problem, the GR “small” term being considered a perturbation source.
Einstein’s approach to (3.1) is to eliminate x3 using an exact relationship between the three roots:

1 (x1 + x2 + x3) = α

(3.2)

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Recall, the α = 2rg where rg is the gravitational radius. He begins with the approximation to (3.2) by putting x3 = 1/α with the following algebraic approximation to (3.1) (to the precision of order α2) allowing one to split the
integrand into two additive parts: I(x) = I1(x) + I2(x), where I1 is the main part from the classical solution, and I2 is the part due to a perturbation of the classical solution by the GR term:

1 ≈
−α(x − x1)(x − x2)(x3 − x)

1

+ α(x1 + x2 + x)

(3.3)

−(x − x1)(x − x2) 2 −(x − x1)(x − x2)

As wished, (3.3) does not have the root x3 and is easily integrated analytically.
The ﬁnal crucial step to the GR solution must be an evaluation of the impact of the GR term on the roots from the GR solution in comparison with the corresponding classical roots x˜1, x˜2 describing a “nearly circular” orbit when x˜1 + x˜2 = 2/r˜0. The radius r˜0 is the one of the classical circle to be compared with the corresponding r0 from the GR exact solution. In the zero-th approximation, it is assumed that the impact of the GR term on the Newtonian solution is negligible.
Considering the eﬀective potential in both cases, it is easy to ﬁnd the relationship [V an10]

3

r0 =

r˜0

−

α 2

(3.4)

Here, r0 >> r3, or equivalently x0 = 1/r0 << x3, and r3 = α(1 + 3α/2). It should be immediately noticed that the above radial diﬀerence in a circular and nearly circular motion translates into the corresponding diﬀerence in circumferences, 3πα, that is exactly the GR perihelion advance eﬀect. A fatal mistake in ﬁnal Einstein’s solution in the problem of planetary perihelion advance is the assumption that the impact of the GR term on the roots x1 and x2 is negligible. Let us deliberate the problem in more details.
Let us assume for a moment that r1 = r˜1, r2 = r˜2, consequently, r0 = r˜0. Then the standard integration of I1 over half a period (from r1 to r2) makes π. The similar analytical integration of I2 gives

3πα ∆θ = 4 (x1 + x2)

(3.5)

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The ﬁnal result would be the angular (perihelion) advance per one revolution given by

3 ∆θ = 2 πα(x1 + x2)

(3.6)

or, for a circular orbit, ∆θ = 3πα/r0 = 6πrg/r0. There (x1 + x2) ≈ (2/r0) while the eccentricity e << 1. The approximate analytical solution to the equation (11) is supposed to be obtained for initial conditions analogous to that in the corresponding classical problem and as a result of a small perturbation of physical parameters such as potential, kinetic and total energy as well as the angular momentum. At the same time, a mathematical approximation is made due to the assumption that the impact of the GR term on the classical roots x1 and x2 is negligible. As a result, the solution could be understood as a periodic, not closed, orbit with a period νθ = 2π (see Bergmann): x(θ) = (1/r0)(1 + e cos νθ), or

r(θ) =

r0

(1 + e cos νθ)

(3.7)

where ν ≈ (1 − 3rg/r0) is a factor of the deﬁcit of full angular rotation in one classical revolution. To complete the revolution, a planet needs to rotate through the additional angle 2π(3rg/r0), which should be observed, in Clemence’s terms, in the inertial frame as a non-classical eﬀect of rotation of orbital plane in the direction of planet motion (the claimed GR perihelion advance eﬀect).
The fact of a physical inconsistency of the prediction (3.7) can be veriﬁed by a substitution of the solution into the original Einstein’s equation (11). The latter in the parametric form with geometrical parameters such as eccentricity e and semilatus rectum p is given by

1 ν2

dx dθ

2

=

(1 −

−

e2)

+

2x

−

x2

+

p2

p

2rg x3

(3.8)

where the GR term is framed. One immediately ﬁnds that the solution (3.7) satisﬁes the equation (3.8)
if and only if the GR term is removed from the equation. Once it is removed, any value of ν does perfectly ﬁt the equation.
It is obvious that the eﬀect ∆θ = 2π(3rg/r0) is (falsely) originated because of the radial shift r˜0 − r0 = 3rg not taken into account. Let us consider an

20

almost circular orbit of a radius r˜0 in the classical case and the radius r0 in the GR case under similar conditions. The fact is that the GR term makes a circumference of the circular orbit shorter by 3πα, which in turn makes a deﬁcit of angle of rotation ∆π = 6πrg/r0. Therefore, in (3.3) we have to drop the wrong assumption of the equality r˜0 = r0 and account for the actual shortage of circumference in the ﬁrst (presumably, “classical”) integral. It makes the angle less than the expected value of π in a half a period, namely:

1

r˜2−∆r

3

2 ∆θ1 = r˜1

I1(r)dr

=

π

−

πα 2

(3.9)

The second (perturbation) integral is not sensitive to the radius alternation and gives a result (in agreement with Einstein’s result)

1

3

2 ∆θ2

=

πα 2

(3.10)

which makes, after summation, a total π, that is, a zero angular advance. We state that, if to follow Einstein’s weird derivation of the eﬀect and
correct mistakes, the result would be zero angular advance. Yet, numerous question arise. For example, why one needs to derive the eﬀect by doing clumsy “perturbation” approximations instead of solving the equation of motion and directly calculating the eﬀect. How is it possible to replace the proper time τ by the coordinate time t? Yet, it is inappropriate to give the ﬁnal formula of the eﬀect in an inconvenient way obtained by Paul Gerber in 1998 without mentioning the author (see Wikipedia).

=

24π3

T

a2 2c2(1 −

e2)

This work should be read in the context of contemporary literature devoted to critical analysis of General Relativity validity, outside of “the mainstream”. There are too many stumbles, puzzles, and misconceptions there, all leading to GR foundations, the so-called Einstein’s Field Equations, which are usually hidden from ordinary reader’s eyes.

21

A Einstein’s original paper, 1915
Erkl¨arung der Perihelbewegung des Merkur aus der allgemeinen Realtivit¨atstheorie
Von A. Einstein
22

23

24

25

26

27

28

29

30

31

Notes
The original text in German of the translated Einstein’s paper was included without translation into Volume 6 of The Collected Papers of Albert Einstein (further ColP ap for short); Volume 6 (1996), Doc.24. A. J. Knox, M. J. Klein, and R. Schulmann, Editors. Princeton University Press. These papers are assigned numbers Doc. namber. The paper under discussion is referred to as Doc.24. It was provided with Editor’s Notes numbered on the paper margins. We retain these numbers in the attached original paper (Doc.24) for making our N otes and comments on the selected spots in addition to our other comments, sometimes, with a reference to our work “General Relativity Problem of Mercury’s Perihelion Advance Revisited”, arXiv, physics. gen-ph, 1008.1811v1, Aug. 2010 (further [V an10] for short).
As a matter of fact, we found another edition of ColP ap in which Vol. 6 (1997) contains the above Einstein’s paper, Doc.24, translated into English by Brian Doyle, and reprinted from A Source Book in Astronomy and Astrophysics, 1900 - 1975, edited by Kenneth R. Lang and Owen Gingerich. There are Editors’ Notes numbers on pages but actual Notes are not provided, and mistakes made in the original publication are not corrected in the translation.

In the following N otes, corrected mistakes are listed.

1, 2. Einstein refers to the paper, Doc.21 in Vol. 6, “On the General

Theory of Relativity”, 1915. There he makes a further reference to the similar

paper (1914) in the Sitzungsberichte, Doc.9. He is concerned about a return

to the General Covariance Principle after partly abandoning it in works with

Grossman (the Entwurf theory). Here he states that the hypothesis about

the vanishing trace of “matter” tensor having the metric determinant being

equal to unity is in accord with the General Covariance of ﬁeld equations.

Next Einstein refers to the Addendum which is titled “On the General Theory

of Relativity, Addendum”, 1915, Doc.22. There he says: “In the following

we assume the conditions

ν ν

Tνν

=

0

really

to

be

generally

true”

and

“Then,

however, w√e are also entitled to add to our previous ﬁeld equation the limiting condition −g = 1. See also the paper “The Field Equations of Gravitation”,

November 25, 1915, Doc.25.

3. Here (in Einstein’s footnote) is the statement about the change of the

above theoretical premises.

4, 5. See the historical note by Editors of ColP ap, also [V an10],

6. In the original, a minus sign on the r.h.s. of the ﬁrst equation is

32

missing (corrected).

7. In the original, ν should be r (corrected).

8. See the note by Editors of ColP ap about historical discussion of the

issue in Norton, 1984.

9, 10. See the note by Editors of CalP ap concerning the calculations,

reference to Doc.30, 1916, The Foundations of the General Theory of Rela-

tivity, and the condition

ν ν

Tνν

=

0

,

Doc.25,

retracted.

Also

about

history

of red-shift observations.

12. In the original, r2 in the ﬁrst term on the r.h.s. should be r3 (cor-

rected).

13. In the original, a minus sign is missing on the r.h.s. of the equation

(corrected).

14. In the original, xr should be xτ (corrected). 15. In the second equation, r is missing: the factor 2 should be 2r

(corrected).

16. In the ﬁrst two equations for φ, the factor α in front of the integral

should be α/2 (corrected).

17. See the historical note by Editors of ColP ap about the data source

Newcomb 1895.

B Schwarzschild’s letter to Einstein
Letter from K Schwarzschild to A Einstein dated 22 December 1915
(ColP ap, vol. 8a, Doc.169, posted on Internet)

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