Point Charge Self-Energy in the General Relativity

TL;DR

This paper investigates the self-energy of point charges within general relativity, revealing that singularities can be managed with generalized functions, leading to finite energy and angular momentum, and offering new insights into classical charge divergence issues.

Contribution

It introduces a technique using limiting solution sequences to handle complex generalized functions in Einstein equations, showing finite energy and angular momentum for singular solutions.

Findings

Total energy equals mc^2 for singular solutions.

Angular momentum for Kerr solutions is mc * a.

Singularities can be incorporated without violating Einstein equations.

Abstract

Singularities in the metric of the classical solutions to the Einstein equations (Schwarzschild, Kerr, Reissner -- Nordstr\"om and Kerr -- Newman solutions) lead to appearance of generalized functions in the Einstein tensor that are not usually taken into consideration. The generalized functions can be of a more complex nature than the Dirac $\d$-function. To study them, a technique has been used based on a limiting solution sequence. The solutions are shown to satisfy the Einstein equations everywhere, if the energy-momentum tensor has a relevant singular addition of non-electromagnetic origin. When the addition is included, the total energy proves finite and equal to $mc^2$, while for the Kerr and Kerr--Newman solutions the angular momentum is $mc {\bf a}$. As the Reissner--Nordstr\"om and Kerr--Newman solutions correspond to the point charge in the classical electrodynamics, the result obtained allows us to view the point charge self-energy divergence problem in a new fashion.