A local characterisation for static charged black holes

TL;DR

This paper provides a local characterization of all static electrovacuum spacetimes, including black hole solutions like Reissner-Nordström and Majumdar-Papapetrou, by establishing a unique relationship between gravitational and electromagnetic potentials.

Contribution

It introduces a purely local criterion that uniquely identifies key static electrovacuum solutions and refines the characterization of Majumdar-Papapetrou spacetimes.

Findings

Classifies all static electrovacuum solutions without sources.

Shows a functional relationship between gravitational and electromagnetic potentials is always present.

Identifies the asymptotically flat subset of solutions as Reissner-Nordström and Majumdar-Papapetrou.

Abstract

We obtain a purely local characterisation that singles out the Majumdar-Papapetrou class, the near-horizon Bertotti-Robinson geometry and the Reissner-Nordstr\"om exterior solution, together with its plane and hyperbolic counterparts, among the static electrovacuum spacetimes. These five classes are found to form the whole set of static Einstein-Maxwell fields without sources and conformally flat space of orbits, this is, the conformastat electrovacuum spacetimes. The main part of the proof consists in showing that a functional relationship between the gravitational and electromagnetic potentials must always exist. The classification procedure provides also an improved characterisation of Majumdar-Papapetrou, by only requiring a conformally flat space of orbits with a vanishing Ricci scalar of the usual conveniently rescaled 3-metric. A simple global consideration allows us to state that the asymptotically flat subset of the Majumdar-Papapetrou class and the Reissner-Nordstr\"om exterior solution are the only asymptotically flat conformastat electrovacuum spacetimes.