Equilibrium configurations of two charged masses in General Relativity

TL;DR

This paper presents an exact static solution in Einstein-Maxwell theory describing two charged masses in equilibrium, revealing conditions under which black holes and naked singularities can be balanced without external supports.

Contribution

It provides a simple analytical form for the equilibrium condition of two charged masses, including cases with a black hole and a naked singularity, expanding understanding of their possible configurations.

Findings

Equilibrium is not possible for two black holes or two naked singularities.

Balance is achievable when one source is a black hole and the other a naked singularity.

A Schwarzschild black hole can be suspended with a naked singularity under certain parameters.

Abstract

An asymptotically flat static solution of Einstein-Maxwell equations which describes the field of two non-extreme Reissner - Nordstr\"om sources in equilibrium is presented. It is expressed in terms of physical parameters of the sources (their masses, charges and separating distance). Very simple analytical forms were found for the solution as well as for the equilibrium condition which guarantees the absence of any struts on the symmetry axis. This condition shows that the equilibrium is not possible for two black holes or for two naked singularities. However, in the case when one of the sources is a black hole and another one is a naked singularity, the equilibrium is possible at some distance separating the sources. It is interesting that for appropriately chosen parameters even a Schwarzschild black hole together with a naked singularity can be "suspended" freely in the superposition of their fields.