Metric for two arbitrary Kerr sources

Abstract

The full metric describing a stationary axisymmetric system of two arbitrary Kerr sources, black holes or hyperextreme objects, located on the symmetry axis and kept apart in equilibrium by a massless strut is presented in a concise explicit form involving five physical parameters. The binary system composed of a Schwarzschild black hole and a Kerr source is a special case not covered by the general formulas, and we elaborate the metric for this physically interesting configuration too.

Thermodynamics of two black holes

“…The metric describing two charged black holes looks very complicated, nevertheless very simple formulae for the horizon areas and their surface gravities have been found [38]. Recently a series of papers appeared, where these results have been generalized to a pair of charged rotating black holes [39][40][41][42].…”

Thermodynamics of Many Black Holes

“…The black holes are aligned along an axis, and are static in the sense of possessing a time-like Killing isometry in the region between the black hole and acceleration horizons. Though an Israel-Khan-like solution for two rotating black holes is known analytically [37,38], exact solutions for three or more Kerr black holes remain elusive 1 . To make the investigation of the system's thermodynamics accessible, we sidestep any discussion of rotation here.…”

Enhanced Ehlers transformation and the Majumdar-Papapetrou-NUT spacetime

“…This feature can be exploited to build, for the first time, a coalescence of a binary system composed by two rotating regular black holes at equilibrium. While some binary system describing two Kerr sources at equilibrium have been found [12], [5], [4] [11], it can be shown that these sources can never describe a legit couple of black holes because one of the two sources have to be hyper-extremal [18]. If one insists in keeping both Kerr sources under the extremal limit, non-removable conical singularities between the two black holes, not covered by any horizon, appears.…”

“…Thanks to the coordinates transformation (5.20) it is easy to see that the f functions in eqs (5.14) and (5.16) coincide. Moreover the electric field of (5.10) can be cast as A t = F/E − , while the remaining seed 11 More details about this transformation and the differential operators associated to the new set of coordinates can be found in appendix A. structure functions of the LWP metric are ω(x, y) = 0 and γ(x, y) = 0. Hence the seed Ernst potentials for the Majumdar-Papapetrou solution are…”

“…But generally these solutions are neither everywhere regular 1 nor the two sources are causally connected, as in the case of accelerating and rotating metrics belonging to the Plebanski-Demianki family [10]. Between these kind of metrics the more physical are the ones where the conical singularities are only between the two sources, avoiding to have conical defects extending to spacial infinity, which remains a globally well defined asymptotic region [11]. The purpose of this article is to enrich this scenario furnishing an analytical method to add rotation to static solutions, including multi black hole ones, in the standard theory of general relativity, without the cosmological constant, coupled to Maxwell electromagnetism.…”