Stationary axisymmetric black holes with matter

We provide a geometric framework for the construction of non-vacuum black holes whose metrics are stationary and axisymmetric. Under suitable assumptions we show that the Einstein equations reduce to an Einstein-harmonic map type system and analyze the compatibility of the resulting equations. This framework is fundamental to our construction [3] of metric-stationary axisymmetric bifurcations of Kerr solving the Einstein–Klein–Gordon system, and as such, we include specializations of all of our formulas to the case of a time-periodic massive scalar field.