Stationary axisymmetric black holes with matter

TL;DR

This paper develops a geometric framework for constructing stationary, axisymmetric black holes with matter, reducing Einstein equations to an Einstein-harmonic map system, facilitating future bifurcation and scalar field studies.

Contribution

It introduces a novel geometric approach that simplifies Einstein equations for non-vacuum black holes with matter, enabling new solutions with matter fields.

Findings

Reduction of Einstein equations to an Einstein-harmonic map system

Compatibility analysis of the resulting equations

Framework applicable to scalar field configurations

Abstract

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 will be fundamental to our forthcoming construction 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.