EQUATIONS OF MOTION OF A SYSTEM OF HEAVY MASSES TAKING INTO ACCOUNT THEIR INTERNAL STRUCTURE AND ROTATION*

V. A. Fock

Applications of Einstein’s theory of gravitation (the so-called general theory of relativity) may be divided into two classes: cosmological applications and applications to a bounded system of masses immersed in Galilean space. In the present report only bounded systems of masses are considered.

A correct mathematical formulation of a physical problem must always ensure uniqueness of the solution. In addition to the conditions expressing the isolation of the system of masses, this must include conditions fixing the choice of the coordinate system.

In the theory of Galilean space (the so-called special theory of relativity) the problem of choosing a coordinate system in principle also exists, but it is usually not posed there in explicit form, since the coordinates are assumed to be Galilean. However, the requirement to use Galilean coordinates, although natural, is logically independent, and it should, strictly speaking, be formulated explicitly. In Einstein’s theory of gravitation the equations are written from the very beginning in a generally covariant form, and the question of the choice of coordinate system arises by itself. Therefore, in the theory of gravitation it is necessary explicitly to formulate additional conditions for the coordinates. It proves possible to impose such conditions as determine the coordinate system uniquely, up to a Lorentz transformation (harmonic coordinates).

One may ask whether the harmonic coordinate system has fundamental significance or only practical significance, since its use simplifies calculations. With regard to

* A brief account of a report read at a meeting of the Einstein session of the Division of Physical and Mathematical Sciences on December 1, 1955. Since all the formulae mentioned and used in the report are contained in our book (V. A. Fock, The Theory of Space, Time and Gravitation, Gostekhizdat, 1955), we do not reproduce them here.

To the heliocentric system of Copernicus a similar question had already been posed in the time of Copernicus. (Ossiander) The question of the fundamental significance of a harmonic coordinate system in Einstein’s theory must be resolved in the same way as the analogous question of the significance of an inertial coordinate system in Newton’s theory, and both these questions are connected with the question of the Copernican system.

If one holds that in Newton’s theory inertial coordinate systems have fundamental significance, and that this significance is not undermined by the possibility of a generally covariant formulation of Newton’s equations of motion (Lagrange’s equations of the second kind), then it is necessary to acknowledge that in Einstein’s theory as well the fundamental significance of harmonic coordinate systems cannot be undermined by the possibility of a generally covariant formulation of the equations. This point of view leads to recognition of the preferential character of the Copernican system, whereas denial of the fundamental significance of the harmonic coordinate system in Einstein’s theory inevitably leads to denial of the preferential character of the Copernican system.

The fundamental significance of the harmonic coordinate system is based on the fact that the existence of such a system reflects objective properties of the space-time continuum. It goes without saying that recognition of this fact is not connected with any prescription to use precisely the harmonic system and not some other one.

The practical advantages of the harmonic coordinate system, however, are beyond doubt. Its introduction makes it possible, in particular, to formulate unambiguously the approximate equations of motion of a system of masses, taking into account their internal structure and rotation.

In studying the motion of a system of masses, one may distinguish the external and internal problems. The external problem studies the motion of bodies as wholes (the mechanics of a system with a finite number of degrees of freedom), while the internal problem studies the motion inside each body (continuum mechanics).

To obtain the equations of motion of the external problem in the nonrelativistic approximation (Newtonian equations of motion of bodies as wholes), it is sufficient to use in the internal problem the equation of continuity. To derive relativistic corrections to the equations of motion of bodies as wholes, it is necessary in the internal problem to employ all the nonrelativistic equations of motion of a continuous medium. The necessity of this is clear in view of the weightiness of the internal energy of bodies.

In the case of rotating liquid bodies, such a consideration leads, in particular, to the Lyapunov equation determining the shape of the body’s surface.

In the approximate formulation under consideration, the system of moving masses may be regarded as conservative (neglecting—

by radiation). In this case one can indicate the explicit form of all ten classical integrals of motion (including relativistic corrections), namely the integrals of energy (mass) and momentum, and the integrals of angular momentum and of the motion of the center of inertia of the bodies.

The problem of the motion of a system of masses is part of the problem of determining gravitational potentials (the solution of Einstein’s equations). In the case of spherically symmetric nonrotating masses, explicit expressions can be given for the gravitational potentials, valid also inside the system of bodies.

Of special interest are the values of the gravitational potentials at “moderately large” distances from the system of bodies (i.e., at distances that are large in comparison with the dimensions of the system, but small in comparison with the length of the gravitational waves it emits). In the general case of rotating elastic bodies, simple asymptotic expressions exist for the gravitational potentials; moreover, the constant values of all ten integrals of motion of the system of bodies enter into them.

At extremely large distances (in the wave zone), the asymptotic expressions have a somewhat different form. Their leading terms can be obtained with such accuracy that the nonlinearity of Einstein’s equations is already taken into account. These solutions also take into account the spherical waves emitted by the system of masses.