Equilibrium configurations of two charged masses in general relativity

Abstract

An asymptotically flat static solution of Einstein-Maxwell equations which describes the field of two non-extreme Reissner -Nordström 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.

Stable photon orbits in stationary axisymmetric electrovacuum spacetimes

“…The N = 2 Bretón-Manko-Aguilar class of electrostatic solutions [49][50][51][52][53] is spanned by five parameters: two masses M ± , two charges Q ± and the separation of centres d. We focus on the Reissner-Nordström di-hole sub-class, in which both bodies are black holes (Q ± ≤ M ± ); this sub-class includes the MP di-holes (M ± = Q ± ) and Weyl-Bach [54] diholes (Q ± = 0) as special cases. With the exception of the MP cases, the charged black holes are held in equilibrium by a 'Weyl strut' [54], and their horizons appear as 'rods' of coordinate length 2 M 2 ± − Q 2 ± on the symmetry axis, in coordinate system (2).…”

“…With the exception of the MP cases, the charged black holes are held in equilibrium by a 'Weyl strut' [54], and their horizons appear as 'rods' of coordinate length 2 M 2 ± − Q 2 ± on the symmetry axis, in coordinate system (2). We examined a two-parameter sub-family: the equal-mass, equal-charge black holes with q = Q ± /M ± and M ± = M , held in equilibrium by a Weyl strut imparting a force [52,53] where σ = 1 − q 2 . The coordinate distance between the horizons is d − 2σM .…”

Charged massive particle at rest in the field of a Reissner-Nordström black hole. II. Analysis of the field lines and the electric Meissner effect

“…Recently Belinski and Alekseev [17] obtained an exact two-body solution to the Einstein-Maxwell equations for a Reissner-Nordström black hole in equilibrium with a naked singularity. We have shown in the Appendix that the Belinski-Alekseev solution, once linearized with respect to the mass and charge of the naked singularity, coincides with our solution.…”

“…Recently an important progress has been achieved by Belinski and Alekseev [17]. They have obtained an exact two-body solution of the Einstein-Maxwell equations in explicit analytic form for the system consisting of a ReissnerNordström black hole and a naked singularity, by using the monodromy transform approach [18].…”

Electrostatic induction in Einstein-Maxwell theory

“…In the papers [1][2][3][4][5] the term (1) arose in approximate calculations, but exact solutions are now available to describe the field of two massive charges at rest [9][10][11][12][13][14][15][16], and it should be possible to identify terms like (1) in them. We want here to take one of these solutions [9] and show that it does in fact contain such terms.…”