Boundary conditions for stationary black holes ; Application to Kerr, Martinez-Troncoso-Zanelli and hairy black holes

TL;DR

This paper develops boundary conditions for numerically modeling stationary black holes in General Relativity, enabling stable simulations of various black hole types including Kerr and hairy black holes.

Contribution

It introduces a novel set of boundary conditions based on apparent horizons for the 3+1 formalism, applicable to diverse stationary black hole solutions.

Findings

Successfully modeled high-spin Kerr black holes.

Obtained solutions for MTZ black holes with scalar fields.

Computed black holes with complex scalar hairs.

Abstract

This work proposes a set of equations that can be used to numerically compute spacetimes containing a stationary black hole. The formalism is based on the 3+1 decomposition of General Relativity with maximal slicing and spatial harmonic gauge. The presence of the black hole is enforced using the notion of apparent horizon in equilibrium. This setting leads to the main result of this paper: a set of boundary conditions describing the horizon and that must be used when solving the 3+1 equations. Those conditions lead to a choice of coordinates that is regular even on the horizon itself. The whole procedure is validated with three different examples chosen to illustrate the great versatility of the method. First, the single rotating black holes are recovered up to very high values of the Kerr parameter. Second, non-rotating black holes coupled to a real scalar field, in the presence of a negative cosmological constant (the so-called MTZ black holes), are obtained. Last, black holes with complex scalar hairs are computed. Eventually, prospects for future work, in particular in contexts where stationarity is only approximate, are discussed.