On well-posedness and algebraic type of the five-dimensional charged rotating black hole with two equal-magnitude angular momenta

TL;DR

This paper investigates the mathematical properties of a five-dimensional charged rotating black hole with two equal angular momenta, focusing on well-posedness, geometric regularity, and algebraic tensor types.

Contribution

It introduces a regular coordinate system, analyzes the boundary value problem, and characterizes the algebraic types of tensors, revealing differences from known black hole solutions.

Findings

Electromagnetic and Ricci tensors are type D outside the horizon.

Weyl tensor is algebraically general outside, type II on the horizon, and type D on the bifurcation sphere.

The metric is incompatible with the Kerr-Schild form, unlike some known black hole solutions.

Abstract

We study various mathematical aspects of the charged rotating black hole with two equal-magnitude angular momenta in five dimensions. We introduce a coordinate system that is regular on the horizon and in which Einstein-Maxwell equations reduce to an autonomous system of ODEs. Employing Bondi and Kruskal-like coordinates, we analyze the geometric regularity of the black hole metric at infinity and the horizon, respectively, and the well-posedness of the corresponding boundary value problem. We also study the algebraic types of the electromagnetic and curvature tensors. While outside the horizon the electromagnetic and Ricci tensors are of type D, the Weyl tensor is algebraically general. The Weyl tensor simplifies to type~II on the horizon and type~D on the bifurcation sphere. These results imply inconsistency of the metric with the Kerr--Schild form with a geodesic Kerr-Schild vector. This feature is shared by the four-dimensional Kerr-Newman metric and the vacuum Myers-Perry or charged Schwarzschild-Tangherlini geometries in arbitrary dimension, but hence not by the black hole we have considered here.