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This work proposes a set of equations that can be used to numerically computespacetimes containing a stationary black hole. The formalism is based on the3+1 decomposition of General Relativity with maximal slicing and spatialharmonic gauge. The presence of the black hole is enforced using the notion ofapparent horizon in equilibrium. This setting leads to the main result of thispaper: a set of boundary conditions describing the horizon and that must beused when solving the 3+1 equations. Those conditions lead to a choice ofcoordinates that is regular even on the horizon itself. The whole procedure isvalidated with three different examples chosen to illustrate the greatversatility of the method. First, the single rotating black holes are recoveredup to very high values of the Kerr parameter. Second, non-rotating black holescoupled to a real scalar field, in the presence of a negative cosmologicalconstant (the so-called MTZ black holes), are obtained. Last, black holes withcomplex scalar hairs are computed. Eventually, prospects for future work, inparticular in contexts where stationarity is only approximate, are discussed.

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