Boundary conditions for stationary black holes: Application to Kerr, Martínez-Troncoso-Zanelli, and hairy black holes

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, nonrotating black holes coupled to a real scalar field, in the presence of a negative cosmological constant (the so-called Martinez-Troncoso-Zanelli 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.