Point massive particle in General Relativity

TL;DR

This paper demonstrates that the Schwarzschild solution in isotropic coordinates can be interpreted as the exact solution of Einstein's equations with a delta-type source representing a point particle, clarifying its mathematical and topological properties.

Contribution

It proves that the Schwarzschild solution in isotropic coordinates is the asymptotically flat solution with a delta-type energy-momentum tensor for a point particle, using a generalized function approach.

Findings

Schwarzschild solution in isotropic coordinates corresponds to a point particle source.

Solution is mathematically defined as a generalized function after integration.

Global differences include a topologically trivial structure and a change from attraction to repulsion near the particle.

Abstract

It is well known that the Schwarzschild solution describes the gravitational field outside compact spherically symmetric mass distribution in General Relativity. In particular, it describes the gravitational field outside a point particle. Nevertheless, what is the exact solution of Einstein's equations with $\delta$-type source corresponding to a point particle is not known. In the present paper, we prove that the Schwarzschild solution in isotropic coordinates is the asymptotically flat static spherically symmetric solution of Einstein's equations with $\delta$-type energy-momentum tensor corresponding to a point particle. Solution of Einstein's equations is understood in the generalized sense after integration with a test function. Metric components are locally integrable functions for which nonlinear Einstein's equations are mathematically defined. The Schwarzschild solution in isotropic coordinates is locally isometric to the Schwarzschild solution in Schwarzschild coordinates but differs essentially globally. It is topologically trivial neglecting the world line of a point particle. Gravity attraction at large distances is replaced by repulsion at the particle neighbourhood.