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On the existence of charged electrostatic black holes in arbitrary topology

Martin Reiris

Abstract

The general classification of 3+1-static black hole solutions of the Einstein equations, with or without matter, is central in General Relativity and important in geometry. In the realm of S1-symmetric vacuum spacetimes, a recent classification proved that, without restrictions on the topology or the asymptotic behavior, black hole solutions can be only of three kinds: (i) Schwarzschild black holes, (ii) Boost black holes, or (iii) Myers-Korotkin-Nicolai black holes, each one having its distinct asymptotic and topological type. In contrast to this, very little is known about the general classification of S1-symmetric static electrovacuum black holes although examples show that, on the large picture, there should be striking differences with respect to the vacuum case. A basic question then is whether or not there are charged analogs to the static vacuum black holes of types (i), (ii) and (iii). In this article we prove the remarkable fact that, while one can `charge' the Schwarzschild solution (resulting in a Reissner-Nordström spacetime) preserving the asymptotic, one cannot do the same to the Boosts and to the Myers-Korotkin-Nicolai solutions: the addition of a small or large electric charge, if possible at all, would transform entirely their asymptotic behavior. In particular, such vacuum solutions cannot be electromagnetically perturbed. The results of this paper are consistent but go far beyond the works of Karlovini and Von Unge on periodic analogs of the Reissner-Nordström black holes. The type of result as well as the techniques used are based on comparison geometry a la Bakry-Émery and appear to be entirely novel in this context. The findings point to a complex interplay between asymptotic, topology and charge in spacetime dimension 3+1, markedly different from what occurs in higher dimensions.

On the existence of charged electrostatic black holes in arbitrary topology

Abstract

The general classification of 3+1-static black hole solutions of the Einstein equations, with or without matter, is central in General Relativity and important in geometry. In the realm of S1-symmetric vacuum spacetimes, a recent classification proved that, without restrictions on the topology or the asymptotic behavior, black hole solutions can be only of three kinds: (i) Schwarzschild black holes, (ii) Boost black holes, or (iii) Myers-Korotkin-Nicolai black holes, each one having its distinct asymptotic and topological type. In contrast to this, very little is known about the general classification of S1-symmetric static electrovacuum black holes although examples show that, on the large picture, there should be striking differences with respect to the vacuum case. A basic question then is whether or not there are charged analogs to the static vacuum black holes of types (i), (ii) and (iii). In this article we prove the remarkable fact that, while one can `charge' the Schwarzschild solution (resulting in a Reissner-Nordström spacetime) preserving the asymptotic, one cannot do the same to the Boosts and to the Myers-Korotkin-Nicolai solutions: the addition of a small or large electric charge, if possible at all, would transform entirely their asymptotic behavior. In particular, such vacuum solutions cannot be electromagnetically perturbed. The results of this paper are consistent but go far beyond the works of Karlovini and Von Unge on periodic analogs of the Reissner-Nordström black holes. The type of result as well as the techniques used are based on comparison geometry a la Bakry-Émery and appear to be entirely novel in this context. The findings point to a complex interplay between asymptotic, topology and charge in spacetime dimension 3+1, markedly different from what occurs in higher dimensions.
Paper Structure (5 sections, 11 theorems, 77 equations, 1 figure)

This paper contains 5 sections, 11 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The reduced electrostatic equations and a Bonnor-type transformation.
  3. Proof of the decay estimates
  4. ${\rm S}^{1}_{z}\times {\rm S}^{1}_{\varphi}$-symmetric electrostatic charged data
  5. The proof of the main Theorem

Key Result

Theorem 1

Consider an ${\rm S}^{1}$-symmetric charged electrostatic or magnetostatic spacetime metric in Weyl coordinates, and on the $4$-manifold ${\bf M}=\mathbb{R}_{t}\times [\rho_{0},\infty)_{\rho}\times {\rm S}^{1}_{z}\times {\rm S}^{1}_{\varphi}$, ($\rho_{0}>0$). If the spatial Riemannian $3$-manifold is metrically complete and the norm of the ${\rm S}^{1}$-Killing field $\partial_{\varphi}$ is bound

Figures (1)

  • Figure 1: The grey region is the spatial manifold of the MKN solutions, topologically an open solid 3-torus minus the horizons, one in this case. The only boundary of the manifold is the horizon.

Theorems & Definitions (20)

  • Theorem 1
  • Corollary 2
  • Conjecture 3
  • Proposition 4
  • proof
  • Corollary 5
  • Corollary 6
  • proof
  • Proposition 7
  • proof
  • ...and 10 more