Table Of ContentRelativistic Causal Newton Gravity Law
Yury M. Zinoviev∗
Steklov Mathematical Institute, Gubkin Street 8, 119991, Moscow, Russia,
2
e - mail: [email protected]
1
0
2 Abstract. The equations of the relativistic causal Newton gravity law for the planets of the
n solar system are studied in the approximation when the Sun rests at the coordinates origin
a
J and the planets do not interact between each other.
7
1
1 Introduction
]
h
p
The Newton gravity law requires the instant propagation of the force action. The special
-
h relativity requires that the propagation speed does not exceed the speed of light. If the
t
a propagation speed is independent of the gravitating body speed, then it is equal to that of
m
light. The special relativity requires also the gravity law covariance under Lorentz transfor-
[
mations. Poincar´e [1] tried to find such a modification of the Newton gravity law. (Poincar´e
1
considered two mathematical problems in XX century as principal: ”to create the math-
v
2 ematical basis for the quantum physics and for the relativity theory.”) The gravity forces
5 of two physical points should depend not on its simultaneous positions and speeds but on
4
the positions and the speeds at the time moments which differ from each other in the time
3
. interval needed for light covering the distance between the physical points. The gravity force
1
0 acting on one physical point may depend also on the acceleration of another physical point
2
at the delayed time moment. The relativistic Newton gravity law was proposed in the paper
1
: [2]. This law for the two physical points has the form
v
i
rX d 1−c−2 dxk 2 −1/2 dxµk = −ηµµ 3 c−1dxνkF (x ,x ), (1.1)
a dt dt ! dt dt j;µν k j
(cid:12) (cid:12) νX=0
(cid:12) (cid:12)
(cid:12) (cid:12)
j,k = 1,2, j 6= k, µ = 0,...,3. The world line xµ(t) satisfies the condition x0(t) = ct; c is the
k k
speed of light; the diagonal 4×4 - matrix ηµν = η , η00 = −η11 = −η22 = −η33 = 1; the
µν
strength F (x ,x ) is expressed through the vector potential
j;µν k j
∂A (x ,x ) ∂A (x ,x )
j;ν k j j;µ k j
F (x ,x ) = − , (1.2)
j;µν k j ∂xµ ∂xν
k k
d 3 d −1
A (x ,x ) = η m G xµ(t′) c|x −x (t′)|− (xi −xi(t′)) xi(t′) , (1.3)
j;µ k j µµ j dt′ j ! k j k j dt′ j !
i=1
X
∗
This work was supported in part by the Russian Foundation for Basic Research (Grant No. 12-01-
00094), the Program for Supporting Leading Scientific Schools (Grant No. 4612.2012.1) and the RAS
Program”Fundamental Problems of Nonlinear Mechanics.”
1
t′ = c−1(x0 −|x −x (t′)|), j,k = 1,2, j 6= k;
k k j
the gravitation constant G = (6.673±0.003)·10−11m3kg−1s−2 and m is the j body mass.
j
Foraresting bodyworldline(x0(t) = ctandthevector x (t) isconstant) thevector potential
j j
(1.3) coincides with the Coulomb vector potential
A (x ,x ) = m G|x −x (c−1x0)|−1, A (x ,x ) = 0, i = 1,2,3. (1.4)
j;0 k j j k j k j;i k j
If the velocities of bodies are small enough to neglect their squares compared with the square
of the light speed and it is possible to neglect also the time interval c−1|x − x (t′)|, then
k j
the vector potential (1.3) is nearly equal to the Coulomb vector potential (1.4). The vector
potential (1.3) was proposed by Li´enard (1898) and Wiechert (1900) as the generalization of
the Coulomb vector potential (1.4). The substitution of the Coulomb vector potential (1.4)
into the right-hand side of the equation (1.1) for µ = 1,2,3 yields the right-hand side of
the Newton gravity law equations. The equation (1.1) multiplied by (1−c−2|dx /dt|2)−1/2
k
transforms as the vector. The equations (1.1), (1.2) with the Li´enard - Wiechert vector
potential (1.3) are the relativistic version of the Newton gravity law equations.
Sommerfeld ([3], Sec. 38): ”The question may arise: what is the relativistic form of the
Newton gravity law? If the law is supposed to have a vector form, this question is wrong.
The gravitational field is not a vector field. It has the incomparably complicated tensor
structure.” The Newton gravity law equations and the equations (1.1) - (1.3) define the
interactions. The body interacts only with another body. If two bodies create the common
gravitationalfieldwiththevector potentialA (x,x )+A (x,x ), anybodyshouldinteract
1;µ 1 2;µ 2
with itself and we obtain the infinity in the equations (1.3), (1.4) at x = x . The notion of
k j
gravitational field with the vector potential A (x,x )+A (x,x ) is not compatible with
1;µ 1 2;µ 2
the Newton gravity law and with the relativistic Newton gravity law (1.1) - (1.3).
The delay c−1|x −x (t′)| in the relation (1.3) provides the causality condition according
k j
towhich someevent inthesystem caninfluence theevolutionofthesystem inthefutureonly
and can not influence the behavior of the system in the past, in the time preceding the given
event. The delay c−1|x −x (t′)| in the relation (1.3) is very important: one celestial body
k j
is a good distance off another celestial body. Poincar´e [1]: ”It turned out to be necessary to
consider this hypothesis more attentively and to study the changes it makes in the gravity
laws in particular. First, it obviously enables us to suppose that the gravity forces propagate
not instantly but at the speed of light.” The general relativity does not take into account
the causality condition and the delay.
99.87% of the total mass of the solar system belongs to the Sun. We consider the
relativistic causal Newton gravity law equations [2] for the planets of the solar system in the
natural approximation when the Sun rests at the coordinates origin and the planets do not
interact between each other. In this approximation the problem of planet relativistic motion
was solved in the paper [2]. The planet orbits were given by the formulas which differ
from the formulas defining the ellipses in the precession coefficients only. The precession
coefficients for the solar system planets are practically equal to one. The similar orbits with
another precession coefficients are considered in the general relativity ([4], Chap. 40, Sec.
40.5, relations (40.17), (40.18)). In the beginning of the XVII century Johannes Kepler by
making use of Tycho Brahe (1546 - 1601) astronomical observations found that the planet
orbits areelliptic in the coordinatesystem where the Sunrests (Nicolaus Copernicus (1543)).
The intensive astronomic observations from the middle of the XIX century and the radio-
location after 1966 discovered the advances of orbit perihelion for different planets.
2
In the general relativity the observed value for the Mercury’s perihelion advance is ob-
tained by means of addition the advance of Mercury’s perihelion ([4], Chap. 40, Sec. 40.5,
Appendix 40.3) calculated in the Newton gravity theory and the advance of Mercury’s per-
ihelion calculated for the orbits ([4], Chap. 40, Sec. 40.5, relations (40.17), (40.18)). The
orbits ([4], Chap. 40, Sec. 40.5, relations (40.17), (40.18)) are the approximate solutions of
the geodesic equation for the chosen metrics ([4], Chap. 40, Sec. 40.1, relation (40.3)). It is
not obvious that we can add the advance of Mercury’s perihelion obtained for the orbits ([4],
Chap. 40, Sec. 40.5, Appendix 40.3) and for the orbits ([4], Chap. 40, Sec. 40.5, relations
(40.17), (40.18)). It seems natural to obtain the advance of Mercury’s perihelion, observed
from the Earth, by making use of the Mercury and Earth orbits ([4], Chap. 40, Sec. 40.5,
relations (40.17), (40.18)) calculated without Newton gravity theory. In order to calculate
the advance of Mercury’s perihelion we need to know also the time dependence of the orbit
([4], Chap. 40, Sec. 40.5, relations (40.17), (40.18)) radius. In this paper we study the
similar orbits of the relativistic causal Newton gravity law [2]. We shall show in this paper
that the value of the Mercury’s perihelion advance, observed from the Earth, depends on
the perihelion angles of the Mercury and Earth orbits. The perihelion angle of the planet’s
orbit depends on the planet’s perihelion point due to the precession coefficient in the planet’s
orbit formula. For the experimental verification of the equations (1.1) - (1.3) the perihelion
angles of the Mercury and Earth orbits are needed.
2 Causal Coulomb and Newton laws
The relativistic Lagrange law is the particular case of the relativistic Newton second law
dt d dt dxµ N 3 dtdxα1 dt dxαk
mc +qc−1 ηµµF (x) ··· = 0, (2.1)
dsdt ds dt ! µα1···αk ds dt ds dt
kX=0 α1,..X.,αk=0
dt −1/2 dxi
= c2 −|v|2 , vi = , i = 1,2,3.
ds dt
(cid:16) (cid:17)
where µ = 0,...,3 and the world line xµ(t) satisfies the condition: x0(t) = ct. The force is the
polynomial of the speed in the equation (2.1). It is necessary to define the series convergence
for the force as an infinite series of the speed. The second relation (2.1) implies the identities
3 dt dxα 2 3 dtdxα dt d dtdxα
η = 1, η = 0. (2.2)
αα αα
ds dt ! ds dt dsdt ds dt !
α=0 α=0
X X
The equation (2.1) and the second identity (2.2) imply
N 3 dt dxα1 dtdxαk+1
F (x) ··· = 0. (2.3)
α1···αk+1 ds dt ds dt
kX=0α1,...,Xαk+1=0
Let the functions F (x) satisfy the equation (2.3). Then three equations (2.1) for
α1···αk+1
µ = 1,2,3 are independent
d N
m (1−c−2|v|2)−1/2vi −qc−1 (c2 −|v|2)−(k−1)/2
dt
(cid:16) (cid:17) kX=0
3 dxα1 dxαk
× F (x) ··· = 0, i = 1,2,3. (2.4)
iα1···αk dt dt
α1,..X.,αk=0
3
The following lemma is proved in the paper [2].
Lemma. Let there exist a Lagrange function L(x,v,t) such that for any world line xµ(t),
x0(t) = ct, the relation
d ∂L ∂L d N
− = m (1−c−2|v|2)−1/2vi −qc−1
dt∂vi ∂xi dt
(cid:16) (cid:17) kX=0
3 dxα1 dxαk
(c2 −|v|2)−(k−1)/2 F (x) ··· (2.5)
iα1···αk dt dt
α1,..X.,αk=0
holds for any i = 1,2,3. Then the Lagrange function has the form
3
L(x,v,t) = −mc2(1−c−2|v|2)1/2 +q A (x,t)c−1vi +qA (x,t) (2.6)
i 0
i=1
X
and the coefficients F (x) in the equations (2.4) are
iα1···αk
F (x) = 0, k 6= 1, i = 1,2,3, α ,...,α = 0,...,3, (2.7)
iα1···αk 1 k
∂A (x,t) ∂A (x,t)
j i
F (x) = − , i,j = 1,2,3,
ij ∂xi ∂xj
∂A (x,t) 1∂A (x,t)
0 i
F (x) = − , i = 1,2,3. (2.8)
i0
∂xi c ∂t
We define the coefficients
F (x) = 0, F (x) = −F (x), i = 1,2,3. (2.9)
00 0i i0
Then the identity
3 dt dxα dtdxβ
F (x) = 0 (2.10)
αβ
ds dt ds dt
α,β=0
X
of the type (2.3) holds. By making use of the second identity (2.2) and the relations (2.8) -
(2.10) we can rewrite the equation (2.4) with the coefficients (2.7), (2.8) as the relativistic
Newton second law with Lorentz force
dt d dtdxµ 3 dtdxν
mc = −qηµµ F (x)c−1 ,
µν
dsdt ds dt ! ds dt
ν=0
X
∂A (x,t) ∂A (x,t)
ν µ
F (x) = − , µ,ν = 0,...,3. (2.11)
µν
∂xµ ∂xν
For the relativistic Lagrange law the interaction is defined by the product of the charge q
and the external vector potential A (x,t).
µ
Let a distribution e (x) ∈ S′(R4) with support in the closed upper light cone be a
0
fundamental solution of the wave equation
∂ 2 3 ∂ 2
−(∂ ,∂ )e (x) = δ(x), (∂ ,∂ ) = − . (2.12)
x x 0 x x ∂x0! ∂xi!
i=1
X
We prove the uniqueness of the equation (2.12) solution in the class of distributions with
(1)
supports in the closed upper light cone. Let the equation (2.12) have two solutions e (x),
0
4
(2)
e (x). Since its supports lie in the closed upper light cone, the convolution is defined. Now
0
the convolution commutativity
d4xd4ye(2)(x−y)e(1)(y)φ(x) = d4xd4ye(1)(x)e(2)(y)φ(x+y) (2.13)
0 0 0 0
Z Z
implies these distributions coincidence:
e(j)(x) = −(∂ ,∂ ) d4ye(k)(x−y)e(j)(y), (2.14)
0 x x 0 0
Z
j,k = 1,2, j 6= k. Due to the book ([5], Sect. 30) this unique causal distribution is
e (x) = −(2π)−1θ(x0)δ((x,x)), (2.15)
0
3 1, x ≥ 0,
(x,y) = x0y0 − xkyk, θ(x) =
(0, x < 0.
k=1
X
The relativistic causal Coulomb law is given by the equations of the type (2.11)
m d 1−c−2 dxk 2 −1/2 dxµk = −q ηµµ 3 c−1dxνkF (x ,x ), (2.16)
k k j;µν k j
dt dt ! dt dt
(cid:12) (cid:12) νX=0
(cid:12) (cid:12)
(cid:12) (cid:12)
j,k = 1,2, j 6= k, where the strength F (x ,x ) is given by the relation (1.2) with the
j;µν k j
Li´enard - Wiechert vector potential of the type (1.3)
3 dxν(t)
A (x ,x ) = −4πq K η dte (x −x (t)) j =
j;µ k j j µν 0 k j
dt
ν=0 Z
X
d 3 d −1
−q Kη xµ(t) c|x −x (t)|− (xi −xi(t)) xi(t) , (2.17)
j µµ dt j ! k j k j dt j ! (cid:12)
iX=1 (cid:12)t=t(0)
(cid:12)
x0k −ct(0) = |xk −xj(t(0))|. (cid:12)(cid:12)
Here K is the constant of the causal electromagnetic interaction for two particles with the
charges q . The support of the distribution (2.15) lies in the upper light cone boundary. The
j
interaction speed is equal to that of light. It is easy to prove the second relation (2.17) by
making change of the integration variable
x0 −ct(r) = (|x −x (t(r))|2 +r)1/2. (2.18)
k k j
For r = 0 the relation (2.18) coincides with the third relation (2.17).
The equations (2.16), (1.2), (2.17) are the relativistic causal version of the Coulomb law.
The Lorentz invariant distribution (2.15) defines the delay. The Lorentz invariant solutions
of the equation (2.12) are described in the paper [2]. By making use of these solutions
it is possible to describe the Lorentz covariant equations of the type (2.16), (1.2), (2.17).
The equations (2.16), (1.2), (2.17) are Lorentz covariant and causal due to the distribution
(2.15). The quantum version of the equations (2.16), (1.2), (2.17) is defined in the paper [6].
The solutions of these causal equations do not contain the diverging integrals similar to the
diverging integrals of the quantum electrodynamics.
5
For a world line xµ(t) we define the vector proportional to −ηµµ(∂ ,∂ )A (x,x )
j x x j;µ j
dxµ(t) dxµ(t)
Jµ(x,x ) = −(∂ ,∂ ) dte (x−x (t)) j = dtδ(x−x (t)) j =
j x x 0 j j
dt dt
Z Z
d
xµ c−1x0 δ x−x c−1x0 ,µ = 0,...,3. (2.19)
dx0 j ! j
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
The condition x0(t) = ct implies the continuity equation
j
∂ 3 d ∂
J0(x,x ) = − xi c−1x0 δ x−x c−1x0 ,
∂x0 j dx0 j ! ∂xi j k
iX=1 (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
∂ d ∂
Ji(x,x ) = xi c−1x0 δ x−x c−1x0 , i = 1,2,3, (2.20)
∂xi j dx0 j ! ∂xi j
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
3 ∂
Jµ(x,x ) = 0. (2.21)
∂xµ j
µ=0
X
The integration of the relation
dxµ(t) dxµ(t)
e (x−x (t)) j = d4ye (x−y)δ(y −x (t)) j (2.22)
0 j 0 j
dt dt
Z
along the world line xµ(t) yields
j
dxµ(t)
dte (x−x (t)) j = d4ye (x−y)Jµ(y,x ). (2.23)
0 j 0 j
dt
Z Z
The relations (2.21), (2.23) imply the gauge condition for the vector potential (2.17)
3 ∂
ηµµ A (x,x ) = 0. (2.24)
∂xµ j;µ j
µ=0
X
Due to the gauge condition (2.24) the tensor (1.2), (2.17) satisfies Maxwell equations with
the current proportional to the current (2.19).
The substitution K = −G and two positive or two negative gravitational masses q =
1
±m , q = ±m into the equations (2.16), (1.2), (2.17) yields the relativistic causal Newton
1 2 2
gravity law (1.1) - (1.3). By changing the constants K = −G, q = ±m , q = ±m
1 1 2 2
in the equations from the paper [6] we have the quantum version of the equations (1.1) -
(1.3). The substitution K = −G and also one positive and one negative gravitational masses
q = ±m , q = ∓m into the equations (2.16), (1.2), (2.17) yields the galaxies scattering
1 1 2 2
with an acceleration. Einstein [7]: ”The theoretical physicists studying the problems of the
general relativity can hardly doubt now that the gravitational and electromagnetic fields
should have the same nature.”
3 Advance of Mercury’s perihelion
Due to the paper [2] the relativistic causal Newton gravity law for the solar system has the
form
d 1−c−2 dxk 2 −1/2 dxµk = −ηµµ 3 c−1dxνk F (x ,x ). (3.1)
j;µν k j
dt dt ! dt dt
(cid:12) (cid:12) νX=0 j=1,.X..,10,j6=k
(cid:12) (cid:12)
(cid:12) (cid:12)
6
We give the number k = 1 for Mercury, the number k = 2 for Venus, the number k = 3 for
the Earth, the number k = 4 for Mars, the number k = 5 for Jupiter, the number k = 6 for
Saturn, the number k = 7 for Uranus, the number k = 8 for Neptune, the number k = 9 for
Pluto and the number k = 10 for the Sun.
99.87% of the total mass of the solar system belongs to the Sun. We consider the Sun
resting at the coordinates origin (Nicolaus Copernicus (1543)). Substituting the Sun world
line x0 (t) = ct, xi (t) = 0, i = 1,2,3, into the equalities (1.2), (1.3) we have
10 10
F (x;x ) = 0, i,j = 1,2,3, F (x;x ) = −m G|x|−3xi, i = 1,2,3. (3.2)
10;ij 10 10;i0 10 10
Substituting the Sun world line x0 (t) = ct, xi (t) = 0, i = 1,2,3, and the elliptic orbits into
10 10
theexpressions (1.2)and(1.3)itispossible toshowthatthevaluesofstrengthsF (x ;x )
10;i0 k 10
considerably exceed the values of strengths F (x ;x ) for any k,j = 1,...,9, k 6= j. It is
j;iν k j
possible to show also that the values of strengths F (x ;x ) are negligible. We neglect
j;iν 10 j
the action of any planet on all of the other planets and the Sun. Then the Sun rests at the
coordinates origin. Due to the relations (3.2) in the coordinates system where the Sun rests
at the coordinates origin the first nine equations (3.1) have the form
d 1−c−2 dxk 2 −1/2 dxik = −m G|x |−3xi, i = 1,2,3, k = 1,...,9 (3.3)
dt dt ! dt 10 k k
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
It is shown in the paper [2] that the following values
M (x ) = 3 ǫ xi dxjk −xjdxik 1− 1 dxk 2 −1/2, l = 1,2,3, (3.4)
l k ijl k dt k dt ! c2 dt !
iX,j=1 (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
−1/2
E(x ) = c2 1− 1 dxk 2 −m G|x |−1, k = 1,...,9. (3.5)
k c2 dt ! 10 k
(cid:12) (cid:12)
(cid:12) (cid:12)
are conserved for the equations (3.3(cid:12)). T(cid:12)he antisymmetric in all indices tensor ǫ has the
ijl
normalization ǫ = 1. The conservation of the vector (3.4) is the relativistic second Kepler
123
law. The vector x is orthogonal to the constant vector (3.4). We introduce the polar
k
coordinates in the plane orthogonal to the vector (3.4)
x1 (t) = r (t)cosφ (t), x2 (t) = r (t)sinφ (t), k = 1,...,9. (3.6)
⊥;k k k ⊥;k k k
Let the constants (3.4), (3.5) satisfy the inequalities
c2|M(x )|2 −m2 G2 > 0, (3.7)
k 10
|M(x )|2((E(x ))2 −c4)+m2 G2c2 > 0, k = 1,...,9. (3.8)
k k 10
Due to the paper [2] the equations (3.3) have the solutions
p
k
= 1+e cos(γ (φ (t)−φ )) (3.9)
k k k k;0
r (t)
k
where φ is the constant perihelion angle and the constants
k;0
p = (c2|M(x )|2 −m2 G2)(m GE(x ))−1,
k k 10 10 k
e = (c2|M(x )|2((E(x ))2 −c4)+m2 G2c4)1/2(m GE(x ))−1,
k k k 10 10 k
γ = (c2|M(x )|2 −m2 G2)1/2(c|M(x )|)−1, k = 1,...,9. (3.10)
k k 10 k
7
The equations (3.9) are the relativistic first Kepler law. The orbit (3.9) is not periodic in
general. The substitution of the vector (3.6) with r (t) = a , φ (t) = ω (t − t ) in the
k k k k k;0
equations (3.3) yields the relativistic third Kepler law:
(1−c−2a2ω2)−1/2a3ω2 = m G. (3.11)
k k k k 10
The equations (3.9) define the trajectory of motion but do not define the time dependence of
this trajectory. Let the constants (3.4), (3.5) satisfy the inequality (3.8) and the inequalities
(E(x ))2 < c4, k = 1,...,9. (3.12)
k
Due to the paper [2] the equations (3.3) have the solutions with the constant parameter ξ
k;0
r (ξ ) = m GE(x )(c4 −(E(x ))2)−1
k k 10 k k
× 1+e sin((c4 −(E(x ))2)1/2c−1ξ ) ,
k k k
t (ξ ) =(cid:16)m G(E(x ))2c−1(c4 −(E(x ))2)−3(cid:17)/2
k k 10 k k
× c3(E(x ))−2(c4 −(E(x ))2)1/2(ξ −ξ )
k k k k;0
−e co(cid:16)s((c4 −(E(x ))2)1/2c−1ξ ) , k = 1,...,9. (3.13)
k k k
(cid:17)
Let us express the constants in the equations (3.9), (3.13) trough the astronomical orbit
data. The orbit eccentricities: e = 0.21, e = 0.007, e = 0.017, e = 0.093, e = 0.048,
1 2 3 4 5
e = 0.056, e = 0.047, e = 0.009, e = 0.249. Therefore 0 < e < 1, k = 1,...,9. Let us
6 7 8 9 k
suppose E(x ) > 0, k = 1,...,9. The curve (3.9) is an ellipse with a precession. The focus
k
of this ellipse is the coordinates origin. The major and minor ”semi - axes” are equal to
a = p (1−e2)−1 = m GE(x )(c4 −(E(x ))2)−1, (3.14)
k k k 10 k k
b = a (1−e2)1/2 = (c2|M(x )|2 −m2 G2)1/2(c4 −(E(x ))2)−1/2, k = 1,...,9. (3.15)
k k k k 10 k
The inequalities (3.7) and e2 < 1 imply the inequality (3.12). Hence the Eqs. (3.13) hold.
k
For the parameters ξ = ±(π/2)c(c4 −(E(x ))2)−1/2 we have the extremal radii
k;± k
r (ξ ) = m GE(x )(c4 −(E(x ))2)−1(1±e ), k = 1,...,9. (3.16)
k k;± 10 k k k
Hence, the ”period” of the motion along the ellipse (3.9) is equal to
T = 2|t (ξ )−t (ξ )| = 2πm Gc3(c4 −(E(x ))2)−3/2, k = 1,...,9. (3.17)
k k k;+ k k;− 10 k
Let us define the mean ”angular frequency” ω = 2πT−1. The relation (3.17) implies
k k
ω = (c4 −(E(x ))2)3/2(m Gc3)−1,
k k 10
(E(x ))2 = c2(c2 −(ω m G)2/3), k = 1,...,9. (3.18)
k k 10
The substitution of the expression (3.18) into the equality (3.14) yields
−3/2
m G = ω2a3 2−1(1+σ (1−(2a ω c−1)2)1/2) , σ = ±1, k = 1,...,9. (3.19)
10 k k k k k k
(cid:16) (cid:17)
Let c → ∞. Then m G = ω2a3(2−1(1 + σ ))−3/2. For σ = 1 this expression agrees with
10 k k k k
the third Kepler law m G = ω2a3. Choosing σ = 1 in the relation (3.19) we get the
10 k k k
”relativistic third Kepler law” for the orbit (3.9)
−3/2 3
ω−2a−3m G = 2−1(1+(1−4ω2a2c−2)1/2) ≈ 1+ ω2a2c−2, k = 1,...,9. (3.20)
k k 10 k k 2 k k
(cid:16) (cid:17)
8
According to thebook([4], Chap. 25, Sec. 25.1, Appendix 25.1)thevaluesω2a3c−2 = 1477m
k k
for k = 1,2,3,4,6 (Mercury, Venus, the Earth, Mars and Saturn), the values ω2a3c−2 =
l l
1478m for l = 5,8 (Jupiter and Neptune), the value ω2a3c−2 = 1476m for Uranus, the value
7 7
ω2a3c−2 = 1469m for Pluto; the major semi-axes a = 0.5791·1011m, a = 1.0821·1011m,
9 9 1 2
a = 1.4960 · 1011m, a = 2.2794 · 1011m, a = 7.783 · 1011m, a = 14.27 · 1011m, a =
3 4 5 6 7
28.69·1011m, a = 44.98·1011m, a = 59.00·1011m. The values ω2a2c−2 = a−1 ·ω2a3c−2,
8 9 k k k k k
k = 1,...,9, arenegligibleandthereforetheSunmassvalues(3.20)obtainedintherelativistic
Kepler problem good agrees with values ω2a3 obtained in Kepler problem.
k k
The substitution of the expression (3.20) into the equality (3.18) yields
−1
c−4(E(x ))2 = 1−2ω2a2c−2 1+(1−4ω2a2c−2)1/2 ≈ 1−ω2a2c−2, k = 1,...,9. (3.21)
k k k k k k k
(cid:16) (cid:17)
By making use of the relations (3.10), (3.14), (3.15), (3.20), (3.21) we have
−2 −1/2
γ = 1+4ω2a2c−2(1−e2)−1 1+(1−4ω2a2c−2)1/2 ≈
k k k k k k
(cid:18) (cid:16) (cid:17) (cid:19)
1−2−1ω2a2c−2(1−e2)−1, k = 1,...,9. (3.22)
k k k
The value 2−1ω2a2c−2(1− e2)−1 ≈ 1 −γ is maximal for Mercury: 1− γ ≈ 1.3341· 10−8.
k k k k 1
The precession coefficients (3.22) of the orbits (3.9) are practically equal to one for all
planets. It agrees with Tycho Brahe astronomical observations used by Kepler. For a
hundred years (415 ”periods” of Mercury) the advance of Mercury’s perihelion is nearly
(1−γ )·360·415·3600” ≈ 7”.175. The relations (3.9), (3.22) imply the perihelion angle
1
φ ≈ φ +2πl(1+2−1ω2a2c−2(1−e2)−1), l = 0,±1,±2,.... (3.23)
k;l k;0 k k k
The substitution of the relations (3.14), (3.22) into the equality (3.9) yields
e cos 1−2−1ω2a2c−2(1−e2)−1 (φ (t)−φ ) ≈ a (1−e2)r−1(t)−1, (3.24)
k k k k k k;0 k k k
(cid:16)(cid:16) (cid:17) (cid:17)
k = 1,...,9. In the general relativity the orbits ([4], Chap. 40, Sec. 40.5, relations (40.17),
(40.18)) are the approximate solutions of the geodesic equation for the chosen metrics ([4],
Chap. 40, Sec. 40.1, relation (40.3)). These orbits are the orbits (3.24) with the perihelion
angles φ = 0 and with the precession coefficients 1 − 3ω2a2c−2(1 − e2)−1 instead of the
k;0 k k k
precession coefficients1−2−1ω2a2c−2(1−e2)−1. Itseemsthattheperihelionanglesaremissed
k k k
in [4]. We note that 3ω2a2c−2(1 − e2)−1 · 360· 415· 3600” ≈ 6(1 −γ ) · 360 ·415 · 3600” ≈
1 1 1 1
6 · 7”.175 = 43”.05. Does the orbit (3.24), k = 1, or the orbit ([4], Chap. 40, Sec. 40.5,
relations (40.17), (40.18)) agree with the observed Mercury’s orbit? From the paper ([8],
p. 361) we know: ”Observations of Mercury are among the most difficult in positional
astronomy. They have to be made in the daytime, near noon, under unfavorable conditions
of the atmosphere; and they are subject to large systematic and accidental errors arising
both from this cause and from the shape of the visible disk of the planet. The planet’s path
in Newtonian space is not an ellipse but an exceedingly complicated space-curve due to the
disturbing effects of all of the other planets. The calculation of this curve is a difficult and
laborioustask, andsignificantly different results have been obtained by different computers.”
Substituting the relations (3.20), (3.21) in the equality (3.13) and introducing the pa-
rameter without physical measure we get
a−1r (τ ) ≈ 1+e sinτ ,
k k k k k
ω t (τ ) ≈ τ −τ −e (1−ω2a2c−2)cosτ ,
k k k k k;0 k k k k
τ = ω a ξ 1+2−1ω2a2c−2 k = 1,...,9. (3.25)
k k k k k k
(cid:16) (cid:17)
9
If we neglect the values ω2a2c−2, then the solutions (3.24), (3.25) of the equations (3.3)
k k
coincide with the solutions of the Kepler problem. Let us define the constant τ in the
k;0
second equality (3.25) by choosing the initial time moment t (0) = 0. Then the equalities
k
(3.25) have the form
a−1r (τ ) ≈ 1+e sinτ ,
k k k k k
ω t (τ ) ≈ τ −e (1−ω2a2c−2)(cosτ −1), k = 1,...,9. (3.26)
k k k k k k k k
Let the direction of the first axis be orthogonal to the vector M(x ). Let the direction of
1
the third axis coincide with the direction of vector M(x ). Then the second axis lies in the
3
plane stretched on the vectors M(x ) and M(x ). Due to the relations (3.6)
1 3
x1(t) = r (t)cosφ (t), x2(t) = −r (t)cosθ sinφ (t), x3(t) = r (t)sinθ sinφ (t),
1 1 1 1 1 1 1 1 1 1 1
x1(t) = r (t)cosφ (t), x2(t) = r (t)sinφ (t), x3 = 0 (3.27)
3 3 3 3 3 3 3
where the inclination of Mercury orbit plane θ = 7o and the values r (t),φ (t), k = 1,3,
1 k k
satisfy the equations (3.24), (3.26). For the definition of Mercury and the Earth trajectories
it is necessary to define the perihelion angles φ , φ in the equations (3.24).
1;0 3;0
”Observations of Mercury do not give the absolute position of the planet in space but
only the direction of a line from the planet to the observer.” ([8], p. 363.) The advance of
Mercury’s perihelion is given by the angle
(x (t (τ ))−x (t (τ )),x (t (τ ))−x (t (τ )))
1 1 1,1 3 3 3,1 1 1 1,2 3 3 3,2
cosα = ,
|x (t (τ ))−x (t (τ ))||x (t (τ ))−x (t (τ ))|
1 1 1,1 3 3 3,1 1 1 1,2 3 3 3,2
c(t (τ )−t (τ )) = |x (t (τ ))−x (t (τ ))|, k = 1,2,
3 3,k 1 1,k 1 1 1,k 3 3 3,k
t (τ )−t (τ ) ≤ 100T ≤ t (τ )−t (τ )+T (3.28)
1 1,2 1 1,1 3 1 1,2 1 1,1 1
where the parameters τ , τ are defined by Mercury’s perihelion points, the parameters
1,1 1,2
τ , τ are the solutions of the second equation (3.28), the numbers T , T are the orbit
3,1 3,2 1 3
”periods” of Mercury and the Earth. The quotient T /T of the Earth and Mercury orbit
3 1
”periods” is approximately equal to 4.15.
By making use of the equations (3.26) we obtain the parameters corresponding to Mer-
cury’s perihelion points:
a−1r (τ ) ≈ 1−e ,
1 1 1,k 1
ω t (τ ) ≈ π(2l +3/2)+e (1−ω2a2c−2),
1 1 1,k k 1 1 1
τ ≈ π(2l +3/2), k = 1,2, (3.29)
1,k k
where l are the integers. The first relation (3.29) coincides with the equality (3.16).
k
Accordingtothebook([4], Chap. 25,Sec. 25.1,Appendix25.1)c−1ω = 275.8·10−17m−1,
1
c−1ω = 66.41·10−17m−1, a = 0.5791·1011m, a = 1.4960·1011m. The substitution of the
3 1 3
second equality (3.29) into the third relation (3.28) yields l −l = 415.
2 1
Due to the second relation (3.28)
x (t (τ )) = x (t (τ ))+c−1|x (t (τ ))−x (t (τ ))|v (t′ ), k = 1,2. (3.30)
3 3 3,k 3 1 1,k 1 1 1,k 3 3 3,k 3 3,k
TheEarthspeedissmall comparedwiththespeedoflight: c−1|v | ≈ c−1ω a ≈ 0.9935·10−4.
3 3 3
We neglect this value (arcsin10−4 ≈ 0o.0057). Then the relations (3.28), (3.30) imply
(x (t (τ ))−x (t (τ )),x (t (τ ))−x (t (τ )))
1 1 1,1 3 1 1,1 1 1 1,2 3 1 1,2
cosα ≈ (3.31)
|x (t (τ ))−x (t (τ ))||x (t (τ ))−x (t (τ ))|
1 1 1,1 3 1 1,1 1 1 1,2 3 1 1,2
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