Electrostatic Equilibrium of Two Spherical Charged Masses in General Relativity

TL;DR

This paper presents an exact solution in general relativity for two charged bodies in electrostatic equilibrium, showing that balance can occur without tension for non-critical charges and clarifying the physical parameters involved.

Contribution

The paper derives and analyzes an exact Einstein-Maxwell solution for two charged bodies in equilibrium, correcting previous misunderstandings about physical parameters.

Findings

Balance without tension is possible for non-critical charges.

Correct physical parameters for mass and charge are identified.

The solution's multipole structure is analyzed.

Abstract

Approximate solutions representing the gravitational-electrostatic balance of two arbitrary point sources in general relativity have led to contradictory arguments in the literature with respect to the condition of balance. Up to the present time, the only known exact solutions which can be interpreted as the non-linear superposition of two spherically symmetric (Reissner-Nordstrom) bodies without an intervening strut has been for critically charged masses, $M^2_i = Q^2_i$. In the present paper, an exact electrostatic solution of the Einstein-Maxwell equations representing the exterior field of two arbitrary charged Reissner-Nordstrom bodies in equilibrium is studied. The invariant physical charge for each source is found by direct integration of Maxwell's equations. The physical mass for each source is invariantly defined in a manner similar to which the charge was found. It is shown through numerical methods that balance without tension or strut can occur for non-critically charged bodies. It is demonstrated that other authors have not identified the correct physical parameters for the mass and charge of the sources. Further properties of the solution, including the multipole structure and comparison with other parameterizations, are examined.