Inner Cauchy horizon of axisymmetric and stationary black holes with surrounding matter in Einstein-Maxwell theory

TL;DR

This paper demonstrates the existence of a regular inner Cauchy horizon in axisymmetric, stationary black holes with surrounding matter in Einstein-Maxwell theory, revealing a universal relation involving angular momentum, charge, and horizon areas.

Contribution

It establishes the universal relation between horizon areas and derives explicit metric relations for the Cauchy horizon in such black holes.

Findings

Existence of a regular inner Cauchy horizon under specified conditions.

Universal relation $(8 ext{π}J)^2+(4 ext{π}Q^2)^2=A^+A^-$ between horizon areas, angular momentum, and charge.

Explicit metric relation on the Cauchy horizon in terms of the event horizon.

Abstract

We study the interior electrovacuum region of axisymmetric and stationary black holes with surrounding matter and find that there exists always a regular inner Cauchy horizon inside the black hole, provided the angular momentum $J$ and charge $Q$ of the black hole do not vanish simultaneously. In particular, we derive an explicit relation for the metric on the Cauchy horizon in terms of that on the event horizon. Moreover, our analysis reveals the remarkable universal relation $(8\pi J)^2+(4\pi Q^2)^2=A^+ A^-$, where $A^+$ and $A^-$ denote the areas of event and Cauchy horizon respectively.