Equations of Motion of a Point Body

mass in the general case. Thus, the tensor of passive gravitational can only be introduced for a narrow class of binary systems.

mass of an extended

body

However, to answer the question we pose regarding the character of motion of the center of mass of an extended body it is inno w a y n e c e s s a r y to invoke the concept of the tensor of passive gravitational mass. The only thing we must do is compare the motion of the center of mass of an extended body with some idealized picture: with the motion of test (point) body in Riemannian space--time having a metric formally equivalent to the metric created by two moving extended bodies. Coincidence of the expressions for the acceleration of the center of mass of the extended body ~(1) and the expression for the acceleration of the point body n 0 will then mean that under the same initial conditions the center of mass of the extended body and body will m o v e along the same trajectory and have the same law of motion. Since a point definition moves along a geodesic of Riemannian space--time, in this case the center of the extended body will also move along a geodesic. If the expressions for ~!)_

the point body by of mass and n 0

differ by post-Newtonian corrections, then the center of mass of the extended body in the general case will not move along a geodesic of Riemannian space--time. Aside from all else, this approach makes it possible to consider in a natural way the contribution of the proper gravitational field of the extended body to the space--time curvature. 29.

Equations

of Motion of a Point Body

We shall consider a Riemannian space--time with a metric which coincides with the metric of the two moving extended bodies considered heretofore. We shall study the motion of the point body in some neighborhood of the point corresponding to the center of mass of the first body. We can obtain an expression for the acceleration of the point body in two ways: either by using the equations of the geodesics of Riemannian space--time

or by computations analogous to those of the acceleration of the center of mass of an extended body while noting that a point is a body of dimensions so small that all quantities characterizing its internal structure and proper gravitational field are negligibly small. In both cases we arrive at the same result. We write out the equations of the geodesics curacy. Using metric (16.1), we obtain

a~ ~-

, o a~.

2 ~ ) ~ ] - 2 (~ + 1) v ~ ) v ~ o ~ u - 0 ~ OV ~

1 ; . ~ ~o

2 (4~+3+~x--~2+h)"-~/--- ~ U OU~O

degree of ac-

dt

ou [(2~ + 1) v ~co) - ~ ( ~ 1

for i = ~ to a post-Newtonian

-

-

(2 9. ~ )

~1!~ OW "~~

-

1 (4y+4+~)V~)[O~V=__O=V~] q_~V~o)[W~OoU__w~O~U]+O(~).

We shall study the motion of the point body in a neighborhood of the point corresponding to the center of mass of the extended body. For this we expand all potentials contained in expression (29.1) in powers of R -1 to order O(R-~). Suppose that the center of mass of the extended body and the point body at some initial time are located at the same point of space and have identical velocities. We call the geodesic of Riemannian space--time along which the point body moves the support geodesic. We shall now compare the values of the accelerations of the center of mass of the extended body and of the point body at the initial time. ~ ) , for the difference in the accelerations of the Since in the present case x ~ = O, V(0 ) = V(I

point body and the center

of mass of the extended m~

in this case

t

~v 1 --3~nenv~o)+~ (4~-q-~--~'3--~1--4~)0

1828

body we have

3 r p/ dx' " q - ~ = j 1x' I~ ((u~x'~)zq-x~ x'~)] +

+ 2 (y @ t) [2Q~ -- V~1)V~DI no § t2v + 2-}- ~2 -3

+V~Jt~

ix, i~

-+-ao-+- R

'

where the acceleration a~ is entirely due to the gravitational field created by the first ~ ) all terms body. It is very remarkable that in subtracting the accelerations n 0 and a(! characterizing

the structure of the second body

(P(2),Q~,17(2), Q~),

vanish.

We are interested in the question of how, in principle, the center of mass of an arbitrary extended body moves: along a geodesic of Riemannian space--time or not. Since different extended bodies differ from one another in the composition and distribution of matter, the distribution of pressure and velocities of internal motion, shape, etc., for different extended bodies the quantities Q(D, Q(D, P(D, ~f, are distinct, and in passing from one body to another they also change relative to one another. Therefore, if we solve the question of the motion of the center of mass in principle for an entire collection of extended bodies, then we must assume that all these quantities are independent at each moment of time. This is the peculiarity of the general formulation of the motion of the center of mass, since by an approximate choice of extended body (changing the shape and distribution of matter so that some multiple moments of mass vanish and others appear which bring the body into rotation, excite pressure and velocity waves in it, etc.) we can vary within broad limits the values of the quantities (19.21) and (28.5). Considering this circumstance, it is easy to see that in no metric theory of gravitation does the difference of accelerations of a point body and the center of mass of an arbitrary extended body (29.2) vanish. Therefore, in no metric theory of gravitation possessi~g conservation laws of the energy--momentum of matter and the gravitational field taken together does the center of mass of an arbitrary extended body in the post-Newtonian approximation move along a geodesic of Riemannian space--time. In connection with this result the question arises of what is the character of the motion of the center of mass of an extended body relative to a support geodesic of Riemannian space--time on the average after a sufficiently large interval of time. To answer this question we need a tensorial virial theorem. We decompose the motion of each element of the volume of the first body into a sum of two motions: motions caused by the action of the gravitational field of the second body and motions caused by the action of other elements of the first body (the action of the gravitational field, the effect of pressure, etc.). We hereby assume that changes of all quantities in time caused by the action of internal forces occur rather quickly so that the characteristic time T during which these changes occur is small as compared with the period T of revolution of the body along the orbit. We now average the expressions obtained over a time interval To which is considerably larger than the characteristic time T but considerably smaller than the period of revolution T. Then by the tensorial virial theorem the averaged values of the quantities (19.21) and (28.5) will not be independent. In particular,

for the first body we have --

Contraction of the tensor indices in expression

-~ ~0) + -~ V o ) V o ) .

(29.3)

(29.3) gives

3p i I ~)(1) -----~ (I) - - ' ~ ~(1) -JC~ V(I)V(I)~ 9

(29.4)

Similar relations hold for the second body. 30.

Averaged Relative Motion of an Extended Body and a Point Body

As follows from the expression (29.2), the difference of the accelerations of a point body and the center of mass of an extended body at the initial time, when their positions coincide and they have the same velocities, is very small: this difference has post-Newtonian order of magnitude. Since the magnitude of this difference does not depend on the displacement 6x ~ of the point body from the center of mass, it might be expected that in the course 1829