Abstract
The paper is concerned with the spherically symmetric static problem of the General Relativity Theory (GRT). The classical interior solution of this problem found in 1916 by K. Schwarzschild for a fluid sphere is generalized for a linear elastic isotropic solid sphere. The GRT equations are supplemented with the equation for the stresses which is similar to the compatibility equation of the theory of elasticity and is derived using the principle of minimum complementary energy for an elastic solid. Numerical analysis of the obtained solution is undertaken.
Highlights
Theory of Elasticity SolutionTo introduce the proposed approach to General Relativity Theory (GRT) problem for elastic solid, consider the problem of the classical theory of elasticity for a sphere whose gravitational field is described by the Newton theory
The GRT equations are supplemented with the equation for the stresses which is similar to the compatibility equation of the theory of elasticity and is derived using the principle of minimum complementary energy for an elastic solid
The solution of the Schwarzchild spherically symmetric static problem for a fluid sphere is generalized for a linear elastic sphere
Summary
To introduce the proposed approach to GRT problem for elastic solid, consider the problem of the classical theory of elasticity for a sphere whose gravitational field is described by the Newton theory. For a solid sphere with constant density μ and radius R, the Newton gravitational potential φ is the solution of the Poisson equation ( ) 1. For the external (r ≥ R) space, μ = 0 and Equation (1) has the following well known solution: φe. Is the mass of a homogeneous solid sphere whose internal space is Euclidean. For the internal (0 ≤ r ≤ R) space the solution of Equation (1) which satisfies the regularity condition at the sphere center is φi. The theory of elasticity equilibrium equation for the sphere element can be presented as apr.ccsenet.org
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