Principal Problems of Relativistic Mechanics of Solids

The article deals with a discussion of fundamental problems arising in relativistic mechanics (general relativity) in relation to the determination of stresses generated by gravity in a deformable solid. Three such problems are considered. The first one is associated with the incompleteness of Einstein’s system of equations that includes six mutually independent equations with ten unknown coefficients of the metric tensor. The second one arises when determining the stresses in a solid generated by gravity. For a static problem, three equations of the law of conservation of theory (equilibrium equations) include six unknown stresses, which, unlike Newton’s theory, does not allow determining gravitational stresses. The third problem is related to the reduction of linearized Einstein equations to the equations of Newton’s gravitational theory. Such a reduction turns out to be possible only for empty space and is not valid for a solid body. The noted contradictions can be eliminated by limiting the scope of the theory to a special space, which is Euclidean with respect to spatial coordinates and Riemannian only with respect to time. The discussion is illustrated by a spherically symmetric problem that reduces to ordinary differential equations.