A numerical approach to finding general stationary vacuum black holes

TL;DR

This paper develops numerical methods to find general stationary vacuum black hole solutions in Lorentzian signature, extending previous elliptic approaches to more complex spacetimes with horizons and ergoregions.

Contribution

It extends elliptic numerical techniques to arbitrary cohomogeneity stationary black holes with horizons in Lorentzian signature, including boundary conditions and solution algorithms.

Findings

Successfully constructed 4D rotating black holes in a cavity

Demonstrated effectiveness of Newton's method and Ricci flow for Lorentzian solutions

Extended elliptic problem formulation to spacetimes with horizons and ergoregions

Abstract

The Harmonic Einstein equation is the vacuum Einstein equation supplemented by a gauge fixing term which we take to be that of DeTurck. For static black holes analytically continued to Riemannian manifolds without boundary at the horizon this equation has previously been shown to be elliptic, and Ricci flow and Newton's method provide good numerical algorithms to solve it. Here we extend these techniques to the arbitrary cohomogeneity stationary case which must be treated in Lorentzian signature. For stationary spacetimes with globally timelike Killing vector the Harmonic Einstein equation is elliptic. In the presence of horizons and ergo-regions it is less obviously so. Motivated by the Rigidity theorem we study a class of stationary black hole spacetimes, considered previously by Harmark, general enough to include the asymptotically flat case in higher dimensions. We argue the Harmonic Einstein equation consistently truncates to this class of spacetimes giving an elliptic problem. The Killing horizons and axes of rotational symmetry are boundaries for this problem and we determine boundary conditions there. As a simple example we numerically construct 4D rotating black holes in a cavity using Anderson's boundary conditions. We demonstrate both Newton's method and Ricci flow to find these Lorentzian solutions.