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On the Gravitational Two-Body Problem in Special Relativity.

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A N N A L E N D E R PHYSIK
~
~~~~
~~
~~
-
7.Folge. Band 38.1981. Heft 3, S. 169-2443
On the Gravitational Two-Body Problem
in Special Relativity
By G. N. AFANASIEV
and R. A. ASANOV
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, U.S.S.R.
Abstract. The method of PO IN CAR^ is applied to the consideration of the gravitational two-body
problem in the framework of Special Relativity. The formulation of the theory contains two arbitrary
functions. A specific choice of these functions leads to the correct description of three crucial experiments, supporting General Relativity, and the time delay of radar signals. However, the gyroscope
precession is three times less than that of GR (the realization of this experiment is planned this year).
The expaneion over the inverse powers of the light velocity being performed approximate Lorentz
covariant two-body equations without retardation effects are obtained. These equations are compared
with the E I R two-body equations.
tfker das Zweikorperproblem in der speziellen Relativitiitstheorie
I n h a l tsubersicht. Es wird die POINCAR&
Methode auf daa Problem der Gravitationwechlwikung zweier Korper in der speziellen Relativitiifstheorie angewendet. Die Theorie enthiilt zwei
beliebige Funktionen. Im Potentialgrenzfall erhiilt man bei bestimmter Wahl dieser Funktionen eine
richtige Beschreibung der drei bekannten Effekte in der allgemeinen Relativitiitstheorie und des Zeitverzogerungseffektes von Radarsignalen. Die G r 6 h der Priizession dee Gyrwkops ergibt sich dreimal
kleiner als in der Einsbinschen Theorie (die esperimentelle Uberpriifung dieses Effektes ist fiir
1980 geplant). Mit Hilfe einer inversen Potenzreihenentwiaklungnach der Lichtgesahwindigkeit wurden geniiherte Lorentz-kovariante Gleichungen fiir das Zweikorperproblemerhalten, in denen Retardierungseffekte fehlen. Es wird der Zusemmenhang mit anderen Versuchen, Lorentz-kovariante Gleichungen fur das Zweikorperproblem aufzustellen, er6rtert.
1. Poincare Two-Body Relativistic Problem
The early attempts to formulate the gravitational 2-body problem in the framework
of Special Relativity are due to H . P o m c d [l] and H.MINKOWSKI
[2]. The equations
of motion suggested by H. PO~CARII
are:
The following notation was used in (1).
i2 Ann. Physik. 7. Folge, Bd. 38
G. N. AFANASIEVand R.A. Asmov
170
xkiand t, are the i-th Cartesian coordinate and the time of the k-th particle; teand
sk are the proper time and the invariant interval of the k-th particle:
ask
= ,c
at, = I/c2 at; - (dXk)2,
*
X{
= xli - xzi,
T
r2 = Z$, t, - t2 = -,
c
c is the velocity of light.
A, B, C are the following Lorentz-invariant cornbinations of the relative coordinates
ax,
xi and the particle velocities vk = at,
1 - - v1
C=
-
(2)
v2
C*
1/1-8:.1-8822
All the quantities referring to the first particle are taken at the time t, = t, while those
for the second one at the time t, = t - rlc. The functions fl and f a entering in (1)are
arbitrary functions of the Lorentz invariants A, B, C which have the following asyinptotic
behaviour as c goes to infinity:
Equations (1) are made of in such a way as to reproduce the wellknown Newtonian
1
gravitational equations (up to the order -) in the limit c -+
C'J
00. Note
the difference of
[l],who for the sake of siiiiplicity set
expression (1)from the one given by H. POINCARI~
f, = 0. Even more special cases were considered by H.MINKOWSKI
[ Z ] and in ref. [3].
Using the analogy with electrodynamics as a guideline, they defined the two-body
forces with the aid of the Lienard-Wiechert potentials. In this case f, = O j fl = const.
Clearly, equations (1)may be easily generalized for arbitrary potential dependence upon
the interparticle distance.
2. The Potential Limit of the Two-Body Equations
Leaving the discussion of the approximate solution methods of (1)and of the known
exact particular solutions for later consider here the potential liinit (M, -+00). In this
case particle 2 undergoes a uniform straight line motion. The Lorentz transforination to
the rest franie of the particle 2 being made the following equations for the motion of the
particle 1 are easily obtained:
Change the proper time in (3) to the coordinate time t
On the Gravitational Two-BodyProblem in Special Relativity
171
Now compare equations (4) with those for the test body motion in General Relativity
(GR) in Schwarzschild metric:
@Xi
axi
-a-.a~ p * (x . v , ) - xi
ap ( x . v1)2 p .v:+
.(1(5)
at2
2r dr
3 F).
+
[-.-
Here
CM
m = - ca
1
2m
p=13,2112,
I--
r
The equations of motion (4) and (5) are exactly the same if the following choice of the
functions f l and f z is made:
2m
Turning now to the two-particle Eqs. (1)one may ask whether it is possible to reconstruct complete two-particle functions fl(A, B, C) and /&A, B, C) entering into
eq. (1)out of the single-particle ones (6) ? Here we give one of the possible receipts. One
must replace in (6) one-particle variables r2, v:, (x vl)through two-particle ones A, B, C
using the following substitution rule :
-
r + B,
2m
+
2 * (7. ( M , Ma)
B
The Lorentz-covariant two-particle eqs. (1)with the functions just defined pass in the
potential limit into the exact equations describing the motion of the test particle in GR.
The same coincidence with Glt in the potential limit could also be obtained in the
framework of the manifest covariant 2-body formalism [4]. I n this formalism only those
parts of particles trajectories interact which have the same time in a given Lorentz frame.
The equations of motion are:
1--
r
+I-
+
= (xv - Y1 * V l v ) * f
(V2” - Y4 * U l ” ) * 4,
w2v = - (xv - Yz v 2 v ) * F
( W l v - Y4 * vzv) * C
Wlv
-
+
-
Here
xfi = X l , W
- x2,(t),
viv and wiv are the four-velocity and acceleration of the i-th partile: yl = (x - vl),
yz = (x wa), y3 = (x x), y4 = (vl vz). f , g, F, Care the functions of invaryants yl, yz, yajy4
The condition of the Lorentz-covariance of the preceding equations leads to a system
of four nonlinear differential equations for the functions f , g, F, C [4].I n the potential
limit one of the particles (say, 2) moves with a constant velocity. Then C = F = 0
and te above mentioned system of equations reduces to the following linear one:
-
-
D - f = 0, D . 9
la.
+ f = 0,
G. N. AFAXTASIEVand R. A. ASANOV
172
where D is the differential operator:
These equations are easily solved. The result is: f is a n arbitrary function of two invariant variab1es.y: - y3 and yl - y4. yz. g is equal t o
where g1 is also a n arbitrary function of the same invariant variables.
From this it is evident that these arbitrary functions can be chosen as t o obtain the
equation identical to that of test particle motion in GR.
3. Comparison with General Relativity
The identity of (4) and (5) means that numerical values of those effects which may
be described with these three equations are the same both within the treated Lorentzcovariant theory and the GR. Fortunately this is true for the three decisive experiments
supporting the GR. We prove this in short.
First, calculate the angular momenta integrals. It follows from (4) or (5):
l--
(7)
T
The dot means the coordinate time derivation. The energy integral is equal to :
It follows from (7) that the test particle always moves in a plane, which may be chosen
as (XU)one. Then (7) and (8) takes a simpler form
1--
2m
=L,
T
Excluding time from (9) one easily obtains the orbital equation for motion of the test
particle :
u;
+ (1 - 2mu)
*
uz =
2mu
(CZ
LZ
,
(u= lp).
Differentiate (10) with respect t o tp:
+
upp, u = 3muz
m
+ -(cz
La
- E).
- &)
+ La
-&
On ths Gravitational Two-Body Problem in Special Relativity
173
Equation (11) does not differ from that of describing the test particle motion in GR.
Because of this the advance of planetary perihelia is the same in both cases. Further, as
for the photon the value of E (= energy per the unit of mass) is equal to c2, one has
upp u -l- 3mu2
that coincides with the light propagation equation in GR. Thus the bending of light is
also the same.
The time delay of a radar signal is also the same. This becomes evident if one takes
into account that this effect may be described with three eqs. (5)that are the same in both
theories. Identifying half of the first two terms in the energy integral (8) as the kinetic
energy contribution (this follows from zero gravitationlimit) andequating it to the photon
energy hv, one obtains the correct value of the red shift :
GM
-+
V N - .
T
Some precaution is needed, however. Equations (9) form a complete system. So it is
impossible to add to (9) something like
+ T2Q)Z = c2
(12)
as is demanded by Special Relativity for the photon velocity. Obviously, eqs. (9) and
(12) are not consistent. The relation (12) means that light does not interact with gravity.
The fact that the absence of such an interaction leads t o numerous paradoxes and is not
consistent with energy conservation was pointed out by A. EINSTEIN
aa early as 1911
[5]. Thus for the light we do not impose condition (12). Instead we consider the motion
of photons and test bodies on the same footing and defined photons as test bodies with
velocity equal to c in the absence of gravity. Then relation (8) fixes the energy constant E
equal t o c2 for photons. The aforesaid is applied of course only to eq. (9), obtained from
the general eq..(4) with a very specific choice of the functions fl and f2 (which aimed to
reproduce the exact motion equations of the GR). It seems that different choices of fl
and f 2 which reproduce the experimental situation for slow motions and do not disagree
with (12) for photons are also possible.
Elementary calculations show that a simplified electrodynamic version of the twoparticle forces suggested in [2,3] gives in the potential limit a wrong value for the perihelion precession of Mercury.
i.2
4. Comparison with Known Flat Space (fravitation Theories
We want to mention here the pioneering investigations of G. BIRKHOFFet al. [6]
who have tried to reproduce the results of the three crucial experiments of GR in the
frames offlat space-time. The equations they obtained are ofthe type (3) with the following specific choice of f l and fa:
2mv2
mc(ru)
m(ru) c
= -mc2 - -M -mc2 - 2mv2, f 2 -1.3
*
(13)
1-P
+viqF
The authors [6] claim this choice leads to the correct values for the planetary perihelion
precession, the deflection of light, and red shift. Compare the functions fl and fz of the
present treatment (6) with that of BIRKHOFE(13). Neglecting terms of an order higher
than l/ca one has:
G. N. AFAXASIEVma R. A. ASANOV
174
The difference of fi and fp is of the order Ips, that is of the order of effects predicted by
GR. This forces us to analyse the whole situation more carefully.Without lossofgeneralit y one may consider the motion in the X Y plane. The equations of BIRKFIOFF
are:
d2X
mx
m(r u)
s,
p = -13 (c2 + 2wZ)
13
+- -
From these recover the energy and angular momentum integrals :
Excluding from (9') the proper time one obtains the orbit equation:
ut u2 = ezmu[(c E ) eZmu- cz].
Retain in (10') the terms of an order not higer than l/ca:
+
+
(10')
The solution of this equation is:
r=
r0
1 + p . coswpl'
Here :
For the planetary motion E
(14)
< 0, I E I < c2. So
A2
3maca
ro = - p w l - - 2 a s s ' a = 1 - mcz '
A2
In this case the trajectory is close to an elliptical one. The shift of the perihelion per one
revolution equals
A2
IEl
a
A = - 6nm2c2
'2
that is just the value predicted by GR.
The phenomenon of the red shift as a consequence of the energy conservation is
easily obtained from the energy integral (9'). The value of this shift is just that predicted
by GR.
Consider now the deflection of light. Change in (9') the proper time T to the coordinate
one t :
V2
-=
- 2mIr
1-
C2
1 + ElCZ *
Here V is a usual velocity. Try to choose E in (16)so as to get the experimental value Bexp
for the deflection angle of the light beam passing near the Sun at the distance rmin.The
result is
Ca
On the Gravitational Two-BodyProblem in Special Relativity
175
Experiment gives for the Berp value predicted by GR = - with the error estimated
(
from 2 to 15% [7]. From this one obtains inequalities: E > 3c2 and E > 25cm,respectively,
whereas E = 00 if for the deflection angle one takes the GR value.
Now calculate the time delay of the radar signal. For simplicity consider here only
the propagation of the signal in a radial direction. The total time delay is easily obtained
from (15):
Glt (and present Lorentz covariant theory) gives the following value
4m
ra-2m
4m r
At =-In
-1112
c
r l - 2m
c
r,
which is confirnied experimentally with an accuracy of about 3%. Defining photons as
test particles with the velocity equal to c in the absence of gravity one obtains from (15)
E = 00. For this value of E the time delay of the radar signal equals zero. Also the value
A t B given by (17) with E extracted from (16) is not cosistent with (17'). Thus the Birkhoff theory, explaining three well-known experiments, fails t o reproduce the observed
time delay of the radar signal.
Consider now the gyroscope precession in the gravitational field. The precession
angle of the gyroscope installed on an artificial satellite in the given approach equals
(-
ng)
per one satellite revolution. GR predicts the value
(3i!f)
- [7]. Note that
deriving the SR value one supposes implicitly that consecutive (in time) orientations of
the coordinate system attached t o the gyroscope are linked by an infinitesimal Lorentztransformation (without spatial rotation). The inclusion of the latter allows one to obtain
the GR value given above.
There are known a few more attempts t o simulate the motion equations in GR by
using the specific forces in the framework of SR. It was shown in refs. [S] that if the exact
coincidence in the framework of equations takes place, the force in SR dynamics is
the polynoni of the fourth degree relative to the velocities:
d2XJ
xe
-= F" = 0" dxv a- ax=
- .Q,, ax, a x a a.33
(.5')
arm
-.
at at
dr
-.
-.-.
dr dz dr
I n the case treated the quantities Q& are the differences of the Christoffel symbols I'&,
for the flat (i.e. Lorentz) and curved (i.e. Schwarzshild) spaces. Prove that equations (5')
and (5) are equivalent. I n Cartesian coordinates all the symbols Pwfor the flat space are
zeros. Then, eq. (5') is the motion equation in Schwarzshild metrics, which coincides
with eq. (5)if the parameter z is chosen to be the coordinate time. This means that the
triple suiii in the right-hand side of eq. (5') is converted to the first term of the right-hand
side of eq. (5). So, eqs. (5') and (5) are the same.
6 . Review of the Known Exact 2-Body Solutions
The system of equations (1)which defines the motion of the first particle should be
(
cr)
completedwith the equations for the second one. The choicefor the differencet, - t, =-
niade in (1)corresponds to the retarded action of the particle 2 on the particle 1. An
C
leads to the advanced action (the action from particle 2
176
G. h'. h A X A 4 S I E Vand.R.A. ASANOV
reaches the particle 1 before it leaves particle 2). One may well use instead of the retarded
case (1)the half sum of the retarded and advanced interactions. Then for the electrodynamic case mentioned above it is possible t o find the Lorentz-invariant Lagrangian and t o
recover the integrals of energy, angular and linear momenta [9]. The particular exact
solutions of the SpecialRelativistic two-body problem are known for this case [lo]. The
particles move along the concentric circular orbits with the constant angular velocity.
Exact solutions corresponding t o the similar concentric motion may be found for the
short range potentials too [ll]. If the interaction of 1with 2 is retarded and 2 with 1 is
advanced, then in addition t o the concentric motion one may recover the exact solutions
corresponding to the straight line relativistic 2 particle motion [12].
The exact solutions are unknown if both interactions (1-2 and 2-1) are retarded.
6. Slow Motion Approximation
For the slow motions it is possible to carry out the expansio? of equations (1)on
the inverse powers of l/c. Keeping the terms up t o the order 1/c2one easily obtains:
(18)
where we set :
Note that all the quantities entering into eq. (18) (i.e. coordinate, velocities, accelerations for both particles) are taken at the same time t. The motion of the second particle
satisfies the same equation (with the replacement of indices (1++ 2)). Eqs. (18) are
a system of ordinary differential equations which may be solved by the usual means if
the initial coordinates and velocities of the particles are known.
Using the procedure mentioned a t the end of sect. 2 find p1 and p2. Neglecting
terms of a n orer higher than l/c2 one obtains:
Inserting pl and tp2 into (18) one finds two-body equations different from the wellknown EIH two-body equations. Nevertheless, the potential limit is the same in both
cases. This is partly due t o the fact that the restoration procedure of the two-body functions from the single-particle ones is not unique.
If the motion is not slow, but the masses of particles are essentially different, then
one may use the method suggested in [13] for solving equations (1).Namelx, in first
approximation the motion of the greater mass M , is supposed to be uniform and straight lined. For the given motion of M2equations(1)are ordinary differential equations. Solve
them and recover the motion M,. So the motion of M , is known. Inserting it in the equations for M , one again obtains the equation for M2 but with the corrections of a n order
of MJM,. This procedure may be continued further and if it is convergent (no proof is
known) then one has a definite answer.
I n both approximation thes knowledge of the initial positions and velocities of the
particles is sufficient for recovering the future particles story. But for the exact system
of equations (1)(plus those for the second particle) it seems impossible to formulate a
well-defined initial value problem: Due to the finite propagation of the interaction one
On the Gravitational Two-Body Problem in Special Relativity
177
must either specify initial conditions on the finite part of the particle trajectories, or
specify at a given point not only the initial coordinates and velocitiesbut all higher derivatives as well [14]. Thus the approximate solutions mentioned above fill a very narrow
gap in a total variety of exact solutions. A possible way to overcome these difficulties,
which is greatly appreciated, is to formulate a suitable heuristic principle [15] which
permits one to restrict the above mentioned variety of solutions. However, the relation
of this principle to experiment is unclear.
I n ref. El61 a general form of the approximately invariant relativistic 2-body Lagrangian was found. By comparing eq. (16) with that obtained from an approximately
invariant relativistic Lagrangian [16], one easily restores the Lagrangian corresponding
to eq. (18). This in turn gives motion integrals corrrsponding to energy, angular and linear
momenta.
7. Galilean Covariant Gravitational Equations
Now return once more t o the single-particle equations (4).As we mentioned the choice
(6) of the f i and f 2 results in the correct description of three main experiments of GR.
Eqs. (4)are related to the rest fame of the particle 2, so information about the motion
of this particle is lost. This in turn implies that eqs. (4)contain information neither on
the motion of a two-particle system as a whole nor on the symmetry properties of the
two-particle system. It is possible to defreeze the degreesof freedom of the second particle
in a variety of ways. One may require for example the Lorentz covariance of the twoparticle system. This is a case considered earlier. On the other hand, one may consider
eqs. (4) as a potential limit of the two-body equations covariant under the Galilean
transformations. Such two-body eqs. may be put in the form:
(Xi = Xlj - x,i, vi = V l j - Vzi).
Here F, and F2are functions of three Galilean invariants
r2= Cx;,
v2 = C v t , (r u) = Cxpi.
In order to obtaian in the potential limit the test particle motion in Schwarzschild
metric one may choose Fland F2as follows :
-
Clearly this choice leads to the correct description of the main experiments of GR.
Attempts to modify the Newtonian law of gravitation have a long history (GAUSS,
HELMHOLTZ,
NEUMA",R~EMA",WEBEB).The possibility of the approximate description of the experiments mentioned (in the framework of Newtonian mechanics) was studied extensively by H. TEEDER[17]. As distinct from the latter eqs. (19) reproduce exactly in the potential limit the equations of motion in GR, and, as a consequence of this,
three main experiments of GR (plus radar signal retardation). Insufficiency of all these
attempts (including the present one) to construct the Galilean covariant two-particle
theory is due t o the simple fact, that these theories describing the majority of GR tests
do not describe the special relativistic effects (contraction of space and time intervals
etc.).
178
0. N. AFANASIEVand R. A. ASANOV
8. Conclusions
The approach adopted here is rather phenomenological. I n fact, the unknown functions may be approximately fixed by comparing either with an experiment or with GR.
This contrasts t o the fundamentality of the EINSTEIN
approach in GR, where the only
information which is needed is the distribution of matter. But until now there is no
satisfactory solution of the 2-body problem in GR. On the other hand, special relativity
suggests the interesting possibilities which were mentioned above.
We are very thankful t o Prof. N. A. CHERNIKOV
and Dr. N. S. SHAVOKHINA
for useful
discussions.
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'
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~~N.
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Bei der Redaktion eingegangen am 30. Juli 1980, revidiertes Yanriskript 9. Oktober 1980.
Anschr. d. Verf.: Dr. G. N. AFANASIEVand Dr. R. A. ASANOV
Laboratory of Theoretical Physics
Joint Institute for Nuclear Research
Dubna, 141980
U.S.S.R.
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