The interior of axisymmetric and stationary black holes: Numerical and analytical studies

TL;DR

This paper combines numerical and analytical methods to study the interior regions of axisymmetric, stationary black holes, establishing and proving a universal relation between horizon areas and angular momentum.

Contribution

It introduces a novel numerical scheme and rigorously proves a universal area-angular momentum relation for black hole horizons.

Findings

Numerical evidence supports the universal relation $ ext{A}_ ext{p} ext{A}_ ext{m} = (8 ext{π}J)^2$.

The relation between event and Cauchy horizon areas is rigorously proven.

The study advances understanding of black hole interior geometry and horizon properties.

Abstract

We investigate the interior hyperbolic region of axisymmetric and stationary black holes surrounded by a matter distribution. First, we treat the corresponding initial value problem of the hyperbolic Einstein equations numerically in terms of a single-domain fully pseudo-spectral scheme. Thereafter, a rigorous mathematical approach is given, in which soliton methods are utilized to derive an explicit relation between the event horizon and an inner Cauchy horizon. This horizon arises as the boundary of the future domain of dependence of the event horizon. Our numerical studies provide strong evidence for the validity of the universal relation $\Ap\Am = (8\pi J)^2$ where $\Ap$ and $\Am$ are the areas of event and inner Cauchy horizon respectively, and $J$ denotes the angular momentum. With our analytical considerations we are able to prove this relation rigorously.