On the geometry of electrovacuum spaces in higher dimensions

TL;DR

This paper classifies higher-dimensional electrovacuum solutions in general relativity, revealing conditions under which these spaces are conformally flat or belong to known classes like Majumdar-Papapetrou or Reissner-Nordström.

Contribution

It provides new classification results for extremal and subextremal electrovacuum spaces under specific geometric and potential conditions, extending understanding of higher-dimensional black hole solutions.

Findings

Extremal electrovacuum spaces are in the Majumdar-Papapetrou class if conformally flat.

Three and four-dimensional extremal spaces are conformally flat.

Subextremal spaces with divergence-free Weyl tensor are warped products with Einstein fibers.

Abstract

A classical question in general relativity is about the classification of regular static black hole solutions of the static Einstein-Maxwell equations (or electrovacuum system). In this paper, we prove some classification results for an electrovacuum system such that the electric potential is a smooth function of the lapse function. In particular, we show that an n-dimensional locally conformally flat extremal electrovacuum space must be in the Majumdar-Papapetrou class. Also, we prove that any three or four dimensional extremal electrovacuum space must be locally conformally flat. Moreover, we prove that an n-dimensional subextremal electrovacuum space with fourth-order divergence free Weyl tensor and zero radial Weyl curvature such that the electric potential is in the Reissner-Nordstr\"om class is locally a warped product manifold with (n-1)-dimensional Einstein fibers. Finally, a three dimensional subextremal electrovacuum space with third-order divergence free Cotton tensor was also classified.