The inner Cauchy horizon of axisymmetric and stationary black holes with surrounding matter

TL;DR

This paper demonstrates that for axisymmetric, stationary black holes with surrounding matter, the inner Cauchy horizon can be smoothly extended if the black hole has non-zero angular momentum, and establishes a universal relation between horizon areas and angular momentum.

Contribution

It provides an explicit relation for the metric at the inner Cauchy horizon and proves the universal area-angular momentum relation for such black holes.

Findings

Inner Cauchy horizon exists and is regular for non-zero angular momentum.

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

Inner Cauchy horizon becomes singular as angular momentum approaches zero.

Abstract

We investigate the interior of regular axisymmetric and stationary black holes surrounded by matter and find that for non-vanishing angular momentum of the black hole the space time can always be extended regularly up to and including an inner Cauchy horizon. We provide an explicit relation for the regular metric at the inner Cauchy horizon in terms of that at the event horizon. As a consequence, we obtain the universal equality $(8\pi J)^2 = A^+ A^-$ where $J$ is the black hole's angular momentum and $A^-$ and $A^+$ denote the horizon areas of inner Cauchy and event horizon, respectively. We also find that in the limit $J \to 0$ the inner Cauchy horizon becomes singular.