Axially symmetric rotating black hole with regular horizons

TL;DR

This paper analyzes the general form of axially symmetric rotating black hole metrics near horizons, classifying conditions for regularity based on curvature invariants and tensor components across different extremality cases.

Contribution

It derives a general metric form near horizons characterized by integers p and q, ensuring regularity under various curvature boundedness conditions.

Findings

Regular metrics depend on asymptotic expansions with parameters p and q.

Conditions for regularity differ between nonextremal, extremal, and ultraextremal horizons.

Comparison of curvature invariants and tensor components provides criteria for horizon regularity.

Abstract

We consider the metric of an axially symmetric rotating black hole. We do not specify the concrete form of a metric and rely on its behavior near the horizon only. Typically, it is characterized (in the coordinates that generalize the Boyer-Lindquist ones) by two integers $p$ and $q$ that enter asymptotic expansions of the time and radial metric coefficients in the main approximation. For given $p,$ $q$ we find a general form for which the metric is regular, and how the expansions of the metric coefficients look like. We compare two types of requirement: (i) boundedness of curvature invariants, (ii) boundedness of separate components of the curvature tensor in a free falling frame. Analysis is done for nonextremal, extremal and ultraextremal horizons separately.