### 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 *r _{g}* which is associated with the singular behavior of the solution. The Schwarzschild solution for the external space becomes singular if the sphere radius reaches

*r*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

_{g}*r*.

_{g}*f*

^{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

*f*(

*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

This work is licensed under a Creative Commons Attribution 3.0 License.

Applied Physics Research ISSN 1916-9639 (Print) ISSN 1916-9647 (Online)

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