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.

中文翻译：

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弹性方程与广义相对论之间的类比

摘要-

这篇文章对广义相对论的基本几何概念提出了一种新的解释，根据该概念，引力与其产生的黎曼空间的曲率无关，但与该空间的变形有关。使用这样的公式，广义相对论的线性化方程与线性弹性理论的变形相容方程类似。这两种理论对应的方程的算子必须具有相同的性质，即它们的散度完全为零。在相对论中，方程的这种性质引起了该理论的主要问题之一，即描述引力场的方程组是不完备的，因为这些方程的算子散度的消失减少了相互独立的场方程的数量，结果证明这些方程小于度量张量的未知分量的数量。在广义相对论中，方程的一般形式仍是未知的，需要用场方程来补充才能得到一个完整的系统。然而，在弹性理论中，这样的方程是已知的，即这些方程是平衡方程，它们表示与应变张量线性相关的应力张量的散度为零。建议使用上述类比来获得补充广义相对论中的场方程的方程。球对称问题被视为一个应用。得到的解与众所周知的 Schwarzschild 解不一致。它不是奇异的，它定义了一个临界半径，即重力半径的 2/3。