SCIREA Journal of Physics
ISSN: 2706-8862
http://www.scirea.org/journal/Physics
April 14, 2022
Volume 7, Issue 2, April 2022
https://doi.org/10.54647/physics14418
Four Mistakes in the Original Paper of Einstein in 1915
to Calculate the Precession of Mercury’s Perihelion
Mei Xiaochun
Department of Theoretical Physics and Pure Mathematics, Institute of Innovative Physics in
Fuzhou, Fuzhou, China
Abstract
Based on the gravitational field equations of general relativity, Einstein proposed a formula to
calculate the precession of Mercury’s perihelion in 1915. It is pointed out in this paper that
there were four mistakes in Einstein's original paper. The first was the mistake of integral
calculation discovered by Huadi in 2015. If this mistake did not exist, the precession value of
Mercury’s perihelion was 71.7” a century. The second was the mistake of the expansion
coefficient of integral function. If this mistake did not exist, the precession value of Mercury’s
perihelion was 14.3” a century.The third was to treat the perihelion and aphelion of Mercury’s
elliptical orbit as poles of integrand expansion, resulted in that the correction term of general
relativity should be equal to zero. The fourth was Einstein’s assumption that the constant
terms in the formula to calculate the precession of Mercury’s perihelion was equal to those in
the Newton’s formula of gravity. However, by precise calculations based on the
Schwarzschild metric and the geodesic equation of Riemann geometric, this constant term
must be equal to zero. The result means that general relativity can only describe the parabolic
orbits (with minor corrections) of planets in the solar system. It can not describe the elliptical
41
and hyperbolic orbits of celestial bodies. So Einstein did not prove that general relativity
could explain the precession of Mercury's perihelion with the value of 43” a century. Since the
fourth mistake was fatal which made the equations of planetary motion of general relativity
untenable, Einstein's gravity theory of curved space-time does not hold.
KeyWords: General relativity, Newton's theory of gravity, Mercury's perihelion precession,
Riemann geometry, Geodesic equation, Parabolic orbits, Elliptical orbit, Hyperbolic orbit
1. Introduction
Einstein gave a formula to calculate the precession of Mercury's perihelion and obtained the
result of 43”a century in his original paper in 1915 based on the equation of gravitational field.
This result was considered as one of the most important theoretical achievements of general
relativity.
As early as 2005, Dr. Ji Hao in China found that Einstein's calculation on the precession of
Mercury's perihelion was wrong [1]. In February 2016, Dr. Huadi, a member of the Russian
Academy of Astronautics, sent a manuscript to the author of this paper and said that he found
an integral mistake in the Einstein's original paper to calculate the precession of Mercury’s
perihelion [2]. By the correct calculation, the precession value should be 71.7 a century, not
be 43 . The error rate was up to 67 percent.
So the author examined the Einstein's original paper carefully and found that Huadi's
calculations were correct and that Einstein’s calculation was certainly wrong.
In addition, the author also found other three mistakes in the Einstein's calculation.
1. To be able to do the calculation, Einstein introduced a transform of the integrand function,
but the transformation was wrong. With the correct transformation, the precession value of
Mercury's perihelion became 14.3 a century.
2. Einstein took the perihelion and aphelion of the Mercury ellipse orbit as the poles of
integrand, but this was impossible. If the perihelion and aphelion of the Mercury elliptical
orbit were the poles of the integrand, what described was ellipse orbit, the the correction term
of the motion equation of general relativity could not exist. This was a fundamental
mathematical error, not the problem of physics approximation. Many physicists also had made
42
the same mistake in this problem since Einstein did it [3].
3. Based on the Schwarzschild metric and the geodesic equation of Riemann geometric, the
author proved that the constant term in the Einstein's formula for calculating the precession of
Mercury's perihelion must be equal to zero. In indicated that general relativity can only
describe the parabolic motions of celestial bodies in the solar system (with minor corrections).
It can not describe the elliptic and hyperbolic orbital motions of celestial bodies [3].
Of there mistakes, last one is fatal. It not only shows that Einstein did not prove that
precession value of Mercury's perihelion is 43 " a century, but also shows that the planetary
motion equation of general relativity is wrong. Because describing the elliptical orbit motion
of planets is the minimum requirement for a qualified theory of gravity, general relativity can
not describe the elliptical orbit motion, which means that Einstein's gravity theory of curved
space-time is invalid.
2. The mistake of integral calculation
The calculation in this section is taken from Huadi's unpublished online papers [2]. When
Einstein calculated the precession of Mercury's perihelion based on his gravitational field
equation in 1915, the Schwarzschild solution had not yet been discovered. By calculating the
motion of a particle along geodesics, Einstein obtained the following formula [4]
2
dx
2A
2 2 x x 2 x 3
B
B
d
（1）
Here x 1 / r ， 2GM / c 2 , M is the mass of the sun , A and B are integral constants
determined by the conservation formulas of energy and angle momentum.
Set
being the increased angle while the Mercury moves from the aphelion to the
perihelion of elliptical orbit, the integral of Eq.(1) is
2
1
dx
2
2 A / B ( / B 2 ) x x 2 x 3
（2）
Here 1 1 / r1 and 2 1 / r2 , r1 was the aphelion of orbit and r2 was the perihelion of
orbit . So 1 and 2 are the roots of following equation
f ( x)
2A
x x 2 x 3 0
B2 B2
43
（3）
Then, Einstein wrote Eq.(2) as
2
[1 (1 2 ]
1
dx
( x 1 )( x 2 )(1 x)
（4）
Due to x / r 1 , so (1 x) 1 2 （1 x / 2) , Eq.(4) became
2
1 (1 2 )
1
(1 x / 2)dx
( x 1 )( x 2 )
（5）
The calculation result of Einstein was
3
1 (1 2 )
4
For
the
Mercury’s
r2 4.6004 1010 m and
precision,
take
r1 6.9818 1010 m
（6）
,
1 1.4323 10 11 / m
,
2 2.1737 10 11 / m . The mass of the sun is M 1.9892 1030 Kg ,
the gravity constant G 6.6732 10 11 N m 2 / Kg 2 and the speed of light is c 2.9979 108 m / s .
According to Eq.(6), the precession angle for the Mercury moves a circle around the sun (one
Mercury year is equal to 88 Earth days) is
3
3GM
2( ) (1 2 )
2
c2
1 1
5.0197 10 7 rad
r1 r2
（7）
The precession angle of Mercury's perihelion every 100 Earth years is
5.0350 10 7 415.28 2.0846 10 4 rad 43
（8）
The result was deemed to be consistent with actual observations, and Einstein's theory of
gravity was accepted by physicists due to this result. For a hundred years, physicists have not
doubts for Einstein’s calculation.
However, Wadi pointed out that Einstein's calculation was wrong and the correct result should
be [2]
1
5
5
1 (1 2 ) 2 (1 2 ) 2 1 (1 2 )
4
4
4
（9）
According to Eq. (9), the precession angle of Mercury's perihelion is 71.7 a century, rather
than 43 ". The correct calculation is below
44
2 x (1 2 )
(1 x)( 2 x)
dx
dx
( x 1 )( x 2 )
1 2 (1 2 ) x x 2
2 x (1 2 )
( 2 )
xdx
arcsin
( x 1 )( x 2 ) 1
2
( x 1 )( x 2 )
( x 1 )( x 2 )
arc sin
（10）
（11）
So it can be obtained from Eq.(5)
2
1
1
2
(1 x / 2) dx
arcsin 1
1 (1 2 ) arcsin 2
2 1
2 1
( x 1 )( x 2 ) 4
1 (1 2 ) arcsin 1 arcsin( 1) 1 (1 2 )
4
4
（12）
Eq.(6) becomes
5
1 (1 2 ) 1 (1 2 ) 1 (1 2 )
4
4
（13）
Eq.(13) is different from Eq.(6), and Einstein's calculation is obviously wrong. Eq. (7)
becomes:
2 ( )
5
(1 2 ) 8.3662 10 7 rad
2
（14）
The precession angle of Mercury's perihelion every 100 Earth years is
8.3662 10 7 415.28 3.4743 10 4 rad 71.7
（15）
It is 1.67 times greater than Einstein's calculation.
3. The mistake in the expansion of integrand
The second mistake in Einstein's calculation was that the integrand expansion of Eq.(4) did
not hold. Due to (1 2 ) ~ 10 8 1 ,
1 (1 2 )
( x 1 )( x 2 )(1 x)
we write the intergrand of Eq.(4) as
1
1 2 (1 2 ) ( x 1 )( x 2 )(1 x)
（15）
Comparing with Eq.(2), we get
[1 2 (1 2 )]( x 1 )( x 2 )(1 x)
45
2A
2 x x 2 x 3
2
B
B
（16）
By expanding the left side of Eq. (16) , we have
[1 2 (1 2 )]1 2 (1 2 1 2 ) x 1 (1 2 ) x 2 x 3
（17）
Comparing with the right hand side of Eq. (16), we have
2A
[1 2 (1 2 )]1 2
B2
[1 2 (1 2 )](1 2 1 2 )
B2
1 [1 2 (1 2 )][1 (1 2 )] [1 (1 2 ) 2 (1 2 ) 2 ]
[1 2 (1 2 )]
1 2 (1 2 ) 1
或
（18）
（19）
（20）
（21）
To make two sides of Eq.(20) and (21) be equal each other, the items (1 2 ) and
2 (1 2 ) 2
must be omitted. Because ~ 103 ， 1 ~ 2 ~ 10 11 ， (1 2 ) ~ 10 8 1 ，
1 2 ~ 10 19 1 ，these neglects are acceptable. So we get
1 2
B2
2A
1 2
B2
（22）
But the problem is that if the item (1 2 ) is omitted, Eq.(4) becomes
2
1
dx
( x 1 )( x 2 )(1 x)
（23）
Therefore, the factor before the integral sign of Eq.(4) does not exist. If this fact is not
considered, we let
2A
2 x x 2 x 3 ( x 1 )( x 2 )(1 x)
2
B
B
1 2 (1 2 1 2 ) x [1 (1 2 )]x 2 x 3
（23）
By comparing the parameters on the two sides of Eq.(23) and ignoring the high order factors
(1 2 ) and 1 2 , we have
2A
1 2
B2
(1 2 1 2 ) 1 2
B2
46
1 1 (1 2 ) 1
[1 2 (1 2 )]
（24）
The same result are obtained as shown in Eq.(22), so the Einstein’s formula is wrong. The
factor before the integral sign of Eq.(4) should be equal to 1. The result of integral is
2
1
dx
1 (1 2 )
( x 1 )( x 2 )(1 x)
4
（25）
The perihelion precession angle of Mercury moving around the sun for a circle becomes
1
GM
2( ) (1 2 ) 2
2
c
1 1
1.6732 10 7 rad
r1 r2
（26）
The precession angle in a century is
1.6732 10 7 415.28 6.9487 10 5 rad 14.3
（27）
It is only a third of Einstein's calculation.
Anatoli Andrei Vankov, a Russian scholar, wrote an unpublished article in 1947. In this paper,
Vankov points out 17 errors in Einstein's original paper on the calculations of the precession
of Mercury's perihelion, of which the 16th one is that the coefficient 1 (1 2 ) in front of
the integral sign is wrong. It should be replaced by the factor 1 (1 2 ) / 2 . to obtain the
result of 43” a century[5].
But why made this change, Vankov had no any explained. In fact, according to the method of
Vankov, we can also multiply a factor 1 Q (1 2 ) before Eq.(25) and taking arbitrary
value for Q , for example, to take Q 100 , so that the precession angle of Mercury’s
perihelion becomes 4300" , but this is meaningless.
4. The mistake of integrand’s poles.
Einstein was wrong to take the perihelion and aphelion of Mercury's elliptical orbit as the
poles of Eq.(3) . The orbit poles of palatial motions is described by dx / d 0 . By
considering Eq.(16), Eq.(3) becomes
[1 2 (1 2 )]( x 1 )( x 2 )(1 x) 0
（28）
Eq. (28) has three roots. They are x1 1 , x2 2 and x3 1 / . The problem was that
47
Einstein did not calculate the exact values of these three roots by solving the cubic equation (3)
of one variable. In fact, since the constants A and B sum were unknown, Einstein could not
actually obtain the roots of equation (3). However, Einstein assumed that the third root had a
known and definite value x3 1 / , which was simply impossible.
Meanwhile, in his final calculations, Einstein took 1 and 2 as the aphelion and perihelion
of Mercury's elliptical orbit, which was also impossible. Because if the aphelion and
perihelion of Mercury's elliptical orbit were regarded as the roots of equation (3) , what the
equation described was the elliptical orbit of Newtonian gravity, the correction term x 3 of
motion equation of general relativity cannot exist.
In fact, for a quadratic equation of one variable, if its two roots are 1 and 2 , we have
x 2 bx c ( x 1 )( x 2 ) 0
（29）
The relation between roots and the coefficients of equation are
1 2 b
1 2 c
（30）
By ignoring the correction of general relativity, Eq.(3) becomes
x2
2A
x 2 0
2
B
B
（31）
If 1 and 2 are the roots of Eq.(31), it should have
1 2
B2
1 2
2A
B2
（32）
Eq.(32) and Eq.(22) are the same, so the correction item of Eq.(30) must not exist, otherwise
1 and 2 can not be the roots of Eq.(3).
Einstein made a primary mathematical mistake here. This is not a matter of physical
approximation, which made Eq.(4) invalid. Many physicists since Einstein have made the
same mistake in calculating the precession of Mercury's perihelion[ ].
5. The mistake of integral constants in the motion equation of planet
It has been more than 100 years since Einstein proposed his theory of gravity in curved
spacetime, but the constants in the most important motion equations of planetary and light in
48
general relativity have not been rigorously calculated. Einstein directly assumed that the
constant term in Eq.(1) was not equal to zero, and calculated the precession of Mercury's
perihelion based on it.
It is shown below that the constant term on the right-hand side of Eq.(1) should be equal
to zero strictly according to general relativity and the geodesic equations of Riemannian
geometry. Therefore, general relativity can only describe the parabolic orbital motion of
celestial bodies (with minor correction), can not describe the elliptical and hyperbolic orbital
motions, so it cannot be used to calculate the precession of Mercury's perihelion[3]. This was
the most fatal error for general relativity, invalidating Einstein's gravity theory of curved
space time.
By solving the Einstein's equation in the spherically symmetric gravitational field, the
Schwarzschild metric represented in four-dimensional space-time was obtained:
ds 2 c 2 A(r )dt 2 B (r )dr 2 r 2 (d 2 sin 2 d 2 )
（33）
Here
B (r ) 1 / r
1
A(r ) 1 / r
（34）
Einstein assumed that an object moved in a gravitational field along a geodesic and the
geodesic equations were calculated by using Riemann geometry. Therefore, the derivation of
the orbital equations of planets in the solar system required two sets of independent equations,
namely, the Einstein's equation of gravitation field and the geodesic equation of Riemann
geometric.
According to the standard method, from the Schwarzschild metric of Eq.(33), four geodesic
equations can be obtained. By taking / 2 , both sides of equal sign of one equation are
equal to zero, and the remaining three independent equations are [6] [7] :
2
2
2
d 2 r B dr
r d
A dx 0
0
ds 2 2 B ds B ds 2 B ds
d 2 2 dr d
0
ds 2 r ds ds
d 2 x 0 A dr dx 0
0
ds 2
A ds ds
或
49
（35）
（36）
d dx 0
A
0
ds ds
（37）
Here x 0 ct , B dB (r ) / dr , A dA(r ) / dr . The integrals of Eq.(36) and (37) are
d
J
ds
（38）
dx 0
K
ds
A(r )
（39）
r2
Here J and K
are integral constants. By considering Eq. (38) and (39), the integral of
Eq.(35) is [7]
2
2
2
dr
d K
B r 2
E
A
ds
ds
（40）
or
dr
1
1/ 2
ds B
J2 K2
E 2
r
A
（41）
Based on Eqs.(38), (39) and (40), it can be proved that the time-independent orbit equation of
planet of general relativity is [3]
2
du
K 2 E E
u 2
2 u u 3
2
J
J
d
（42）
The time-dependent orbit equation of planet of general relativity is
2
2
E 2 3E
dr
2 d
2
c 1 2 c 2 2
r
r
K
K
dt
dt
2
c 2 E 3 c 2 J 2
3E
c 1 2 2 2 3 2 3 1
K r
K r
r
K r
2
2
（43）
To determine the integral constant, it is necessary to compare equation (43) with the of
Newton's equation of gravity. Let L be the angle momentum of unit mass with cJ L ~ rV ,
as well as V / c 1 in the weak gravity field of the sun, we have
c 2 J 2 L2 2GMr 2V 2 2GM V 2
3 ~
r3
r
c2r 3
r c2
（44）
By ignoring the the high order correction terms to contain 2 / r 2 and V 2 / c 2 in Eq.(43), we
can obtain
50
1 dr
2 d
r
2 dt
dt
2
2
2
3E
E
GM c
2
1 2
2
2 K
r
K
（45）
If Eq.(45) is the motion equation of Newtonian gravity, the coefficient before the term
GM / r must be equal to zero. Let
3E
2 1
K2
E
1
K2
or
（46）
Eq.(45) becomes
1 dr
2 d
r
2 dt
dt
2
2
GM
0
r
（47）
On the other hand, according to the Newton's theory of gravity, the energy conservation
formula of a celestial body with unit mass in the solar gravity field is
2
2
GM
1 dr
2 d
E
r
r
2 dt
dt
（48）
When the constant (total mechanical energy) E GM / 2b 0 , the celestial body moves in
the elliptical orbit ( b is the major axis of elliptical orbit). When E GM / 2a 0 , the
celestial moves in the hyperbolic orbit ( a is the parameter of hyperbolic orbit). When E 0 ,
the celestial body moves in the parabolic orbit. So Eq.(47) only describes the parabolic orbit
of celestial body, can not describe the elliptical and hyperbolic orbits.
On the other hand, substituting Eqs.(38), (39) and (41) in Eq.(33), we can get ds 2 Eds 2 ,
so constant E 0, 1 [3]. For the motion of an object with mass, it takes E 1 . For the
motion of light, Einstein assumed E 0 . Therefore, according to Eq.(46), we have K 1 .
Up to now, all integral constants in the geodesic equations are determined. Eq.(43) becomes
2
2
c 2 2c 2 2 c 2 3 L2
dr
2 d
3 3 1
r
r
r2
r
r
r
dt
dt
2
（49）
We write Eq.(49) in the standard form with
2
2
1 dr
2 d
U (r ) 0
r
2 dt
dt
The potential energy of unit mass in the gravitational field is
51
（50）
U (r )
GM
r
2
2 2
L2
1
1
r
r 2 c 2r 2
r
（51）
According to Eq.(50) and (51), general relativity can only describe the parabolic orbital
motions (with minor corrections). It can not describe the elliptical and hyperbolic motions.
If it is not, assuming that the Newtonian approximation of general relativity can describe the
elliptical orbital motion, the constant term on the right side of Eq.(48) should be
E
GM
2
2
K
cb
1
3E
3GM
2 1 2
2
K
cb
or
（52）
Eq.(45) becomes
2
2
3GM GM
1 dr
GM
2 d
1 2
r
2 dt
2b
cb r
dt
（53）
It means that when a planet moves in an elliptical orbit, the Newtonian gravity becomes
3GM
F 1 2
cb
GM r
3
r
（54）
If a celestial body moves in a hyperbolic orbit, the constant term on the right side of Eq.(48)
should be
1
E GM
K 2 c 2a
3E
3GM
2 1 2
2
K
c a
or
（55）
Eq.(45) becomes
1 dr
2 d
r
2 dt
dt
2
2
3GM
1 2
c a
GM GM
2a
r
（56）
The Newtonian gravity becomes
3GM GM r
F 1 2
c a r3
（57）
The formulas (54) and (57) are obviously impossible, because according to the Newtonian
theory of gravity, the basic form of gravity is the same whether a body moves in parabolic,
elliptical or hyperbolic orbits. The basic formula of gravity can only contain the basic constant
of physics. It can not contain the special factors a and b of individual orbits. The factors
52
3GM /(c 2b) and 3GM /(c 2 a ) can not exist in Eq.(54) and (57), otherwise physics would
have no regularity and unity.
Substituting E 1 and K 1 in Eq.(42) we get
2
du
2 u u 2 u 3
J
d
（58）
Let x u and B J in Eq.(58) and comparing it with Eq.(1), it can be seen that Eq.(1) has
an extra constant term 2 A / B 2 which was added by Einstein artificially and not exist actually
according the strict calculation. Since Eq. (58) does not describe an elliptical orbit, it is
impossible using it to calculate the precession of Mercury's perihelion.
Take the derivatives of Eqs.( (1) and (58) with respect to ( x u ), the result are the same
with
2 3 2
d 2u
u 2
u
2
2
d
J
（59）
Physicists after Einstein used Eq.(59) to calculate the precession of Mercury's perihelion and
obtained the same value 43 " . If the correction of general relativity is not considered, Eq.(59)
becomes
d 2u
2
u 2
2
d
J
（60）
Eq.(60) describes the conic orbit motion with the solution
u
2 / J 2
1 e cos( 0 )
In Eq.(61), e C / J 2 is the the eccentricity in which C
（61）
is an integral constant. When
e 1 , Eq.(61) describes elliptical orbit. When e 1 , it describers parabolic orbit and when
e 1 , it describes
hyperbolic orbit. Exactly what kind of motion is made depends on the
total mechanical energy of celestial body. According to Eq. (47), we should take e 1 . Thus
Eq.(59) of general relativity can only describes the parabolic orbit (with minor corrections),
not the elliptic and hyperbolic orbits.
53
6. Conclusion
Einstein proposed a formula to calculate the precession of Mercury perihelion and obtained
the result of 43 " a century based on the gravitational field equation of curved space-time. The
result was considered was considered to be one of the most important theoretical support for
general relativity.
In this paper, four mistakes in Einstein's original paper are pointed out. One was a
miscalculation of the integral, discovered by Mr. Wadi. The other three are found by the
author of this paper, including the calculation error of the coefficient of integral expansion,
the error of taking the aphelion and perihelion of mercury's elliptical orbit as the poles of the
integrand, and the calculation error of the constant term in the equation of planetary motion in
general relativity.
Among them, the calculation error of the constant term in the planetary motion equation is the
most fatal and irreparable. Because the constant term should be zero, general relativity can
describe only the parabolic orbital motions of objects in the solar system (with minor
corrections), not the elliptical and hyperbolic orbital motions of objects. Since describing the
elliptical orbits of planets is the minimum requirement for a theory of gravity, Einstein's
gravity theory of curved space-time can only be considered wrong.
In fact, Einstein's curved space-time gravity theory also leads to a number of serious problems,
such as the definition of energy momentum tensor of gravitational field, the singularity of
space-time, the interchangeability of time and space in black holes and so on [11, 12, 13].
Non-baryonic dark matter and dark energy in cosmology are also the results of regarding
gravity as the curvature of space-time [14,15].
Therefore, the conclusion of this paper is that modern physics must completely abandon the
geometrical description of gravity in curved space-time and return to the dynamic description
in flat space-time through reinventing the Newton's theory of gravity.
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54
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56
Mei Xiaochun