ANALOGY BETWEEN THE EQUATIONS OF ELASTICITY AND THE GENERAL THEORY OF RELATIVITY

Abstract

Abstract— The article presents a new interpretation of the basic geometric concept of general theory of relativity, according to which gravity is associated not with the curvature of the Riemannian space generated by it, but with deformations of this space. Using such a formulation, the linearized equations of the general theory of relativity for empty space turn out to be analogous to the compatibility equations of deformations of the linear theory of elasticity. It is essential that the operators of the equations corresponding to these two theories have the same property, namely, their divergence is identically equal to zero. In the theory of relativity, this property of equations gives rise to one of the main problems of the theory, namely, the system of equations describing the gravitational field turns out to be incomplete, since the vanishing of the divergence of the operators of these equations reduces the number of mutually independent field equations that turns out to be less than the number of unknown components of the metric tensor. The general form of the equations, which should be supplemented with the field equations to obtain a complete system, is still unknown in the general theory of relativity. However, in the theory of elasticity, such equations are known, namely, these are the equilibrium equations, which represent the equality to zero of the divergence of the stress tensor linearly related to the strain tensor. It is proposed to use the noted analogy to obtain equations that supplement the field equations in the general theory of relativity. A problem with spherical symmetry is considered as an application. The solution obtained does not coincide with the well-known Schwarzschild solution. It is not singular and defines a critical radius that is 2/3 of the gravitational radius.