On the Solution of Spherically Symmetric Static Problem for a Fluid Sphere in General Relativity

Abstract

The paper is devoted to the spherically symmetric static problem of General Theory of Relativity (GTR) originally solved by K. Schwarzschild in 1916 for a particular form of the line element. This classical solution specifies the metric tensor for the external and internal semi-Riemannian spaces for a perfect fluid sphere with constant density and includes the so called gravitational radius rg which is associated with the singular behavior of the solution. The Schwarzschild solution for the external space becomes singular if the sphere radius reaches rg which is referred to as the radius of the Black Hole event horizon. The solution for the internal space gives infinitely high fluid pressure at the center of sphere with radius equal to 9/8 rg. In contrast to the classical solution, the solution presented in the paper is based on the general form of line element for spherically symmetric Riemannian space in which the circumferential component of the metric tensor ρ 2 (r) is an arbitrary function of the radial coordinate. As shown, the solution of the static problem exists for a whole class of functions ρ(r). The particular form of this function is determined in the paper under the assumption according to which the gravitation, changing the Euclidean space to the Riemannian space inside the sphere in accordance with GTR equations, does not affect the sphere mass. The solution obtained for the proposed particular form of the line element cannot be singular neither on the sphere surface nor at the sphere center. Direct comparison with the Schwarzschild solution for external and internal spaces is presented.