Electrostatic equilibrium of two spherical charged masses in general relativity

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-Nordström) bodies without an intervening strut has been for critically charged masses,In the present paper, an exact electrostatic solution of the EinsteinMaxwell equations representing the exterior field of two arbitrary charged Reissner-Nordström 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.

Equilibrium configurations of two charged masses in general relativity

“…This simplicity permits us to prove a validity of conjectures of the papers [9] and [16] on exact analytical level. It allows also a direct analytical investigation of the physical properties of the equilibrium state of two non-extreme sources.…”

“…Up to 1997 it remained unknown whether the analogous generalization for the non-extreme bodies can be found. The first solid arguments in favour of existence of a static equilibrium configuration for the "black hole -naked singularity" system was presented in [16]. These results have been obtained thereby numerical calculations and three examples of numerical solutions of the equilibrium equation have been demonstrated.…”

“…For the complete proof it would be necessary to show that such configurations indeed consist of two sources, separated by physically sensible distance between them. However, in [16] it was pointed out that the distance dependence for the equilibrium state is unknown. The authors of [16] also reported that a number of numerical experiments for two black holes and for two naked singularities showed the negative outcomes, i.e.…”

Exact charged two-body motion and the static balance condition in lineal gravity

“…Although it has been conjectured [5] that an exact solution under the condition (2) should exist in general relativity, in the (2nd) post-Newtonian approximation (2) is incompatible with the static balance condition [6], and a test particle analysis [7] suggested that (2) is neither necessary nor sufficient. This suggests a wider range of possibilities for realizing equilibrium in general relativity that do not exist non-relativisitically, and several numerical studies [8,9] have been carried out to this end. Until now no one has yet found -in any relativistic theory of gravity -equilibrium states in which √ 4πGm i > e i for both bodies.…”

“…It is then straightforward to obtain an exact expression for r as a function of τ by inserting the expression for p(τ ) into (8) and solving for r as a function of τ . A number of different types of motion are possible, depending upon a combination of four factors: gravitational attraction, the electric force between charges, the effect of the cosmological constant and relativistic effects.…”