ON GRAVITATIONAL-FIELDS IN RIEMANNIAN SPACES

A rigorous foundation will be laid for methods of constructing gravita tional fields in given pseudo-Riemannian spaces, particularly Minkowsk i spaces. with a natural alternative to the global law of universal gr avitation, while preserving its local consequences by analogy with loc al transitions from properties in Euclidean spaces to the properties o f Riemannian spaces. There will nevertheless be strong differences in the characteristic relations in global senses. The essence of the theo ries developed here is that in topologically equivalent spaces one can use identical fixed coordinate frames of reference with individually defined points. Specially introduced global Lagrangian comoving frames , in space or in mathematically defined model media, defined on approp riate families of time-like world lines L, at each point of which the three-dimensional velocities vanish, are of particular importance. The se are coordinate systems in which all individual points of the model spaces or media are at rest, with changes occurring only in global tim e, which on the world lines of the family is identical with proper tim e. Inertial frames of reference-generally local Cartesian tetrads S wh ich, at each point on L, serve as a basis for the introduction of a va riety of algebraically and differentially defined mechanical character istic quantities and, in particular, the four- and three-vectors of ab solute velocities and the corresponding absolute accelerations are als o of particular importance. These are all fundamental concepts, in ter ms of which one formulates the basic definitions of mathematical and p hysical models. It will be shown that for free gravitational motion of material media, subject only to forces of inertia and body forces-inc luding in particular, gravity-mechanical laws and mechanical phenomena are described in identical terms in comoving coordinates, on the one hand, and in the special frames of reference on the other. In free fli ght in space, internal motion within the astronaut's cabin, which take s place under conditions of weightlessness, is therefore described loc ally in exactly the same way as the analogous mechanical phenomena in inertial systems, where there are no gravitational forces acting on th e particles of the medium, as they are cancelled out by forces of iner tia. Now this is the situation, in the same sense, both in Newtonian m echanics and in alternative relativistic theories, taking potential en ergy into account. In general relativity theory (GRT), changes in pote ntial energy, due to changes in the positions of bodies in curved spac e, are completely eliminated by a suitable choice of a pseudo-Riemanni an space. We shall establish general properties of gravitational field s in pseudo-Riemannian spaces. In particular, we shall show that the d ensity and potential energy of gravitational fields in comoving Lagran gian coordinates along world lines of the family L are constant, thoug h they may differ from one world line of L to another. This is true no t only in Newtonian mechanics but also in pseudo-Riemannian spaces. As we shall see, in relativistic theories it is always necessary to use the law of universal gravitation or some alternative to it due to the additional specification concerning geometrical aspects of pseudo-Riem annian spaces. It should also be stressed that different choices of th e family of world lines L and the presence of point singularities in t he field may well cause some solutions of the tensor equation of GRT i n a region of empty space to clash with the law of universal gravitati on. The same is true of the expressions for the potential energy corre sponding to these solutions. Our main result will be to demonstrate th at one can use theoretical solutions of problems in Newtonian mechanic s in comoving frames of reference (or on the basis of the data of meas urements carried out on instruments mounted on moving objects). Comput ational methods of inertial navigation theory in Riemannian spaces can be used to construct a complete solution of problems involving the de termination of the metric and laws of motion of bodies in given pseudo -Riemannian spaces, for arbitrarily given observers. The agreement bet ween GRT and Newton's theory (in the basic approximation) is due to th e fact that over short time intervals planetary orbits are almost stra ight lines in Euclidean space or Schwarzschild geodesics.