Interior Weyl-type Solutions of the Einstein-Maxwell Field Equations

TL;DR

This paper explores static, spherically symmetric solutions to the Einstein-Maxwell equations with a focus on Weyl-type relationships between electric and gravitational potentials, presenting new exact solutions and analyzing their physical properties.

Contribution

It extends previous work by deriving new exact solutions for charged matter distributions and analyzing conditions under which the Majumdar condition applies.

Findings

Existence of relationships between matter density, electric field density, and charge density.

New exact solutions generalizing the Schwarzschild interior solution.

Solutions satisfy energy and regularity conditions and match Reissner-Nordström metrics.

Abstract

Static solutions of the electro-gravitational field equations exhibiting a functional relationship between the electric and gravitational potentials are studied. General results for these metrics are presented which extend previous work of Majumdar. In particular, it is shown that for any solution of the field equations exhibiting such a Weyl-type relationship, there exists a relationship between the matter density, the electric field density and the charge density. It is also found that the Majumdar condition can hold for a bounded perfect fluid only if the matter pressure vanishes (that is, charged dust). By restricting to spherically symmetric distributions of charged matter, a number of exact solutions are presented in closed form which generalise the Schwarzschild interior solution. Some of these solutions exhibit functional relations between the electric and gravitational potentials different to the quadratic one of Weyl. All the non-dust solutions are well-behaved and, by matching them to the Reissner-Nordstr\"{o}m solution, all of the constants of integration are identified in terms of the total mass, total charge and radius of the source. This is done in detail for a number of specific examples. These are also shown to satisfy the weak and strong energy conditions and many other regularity and energy conditions that may be required of any physically reasonable matter distribution.