Boundary value problems for the stationary axisymmetric Einstein equations: a disk rotating around a black hole

TL;DR

This paper explicitly solves boundary value problems for stationary axisymmetric Einstein equations involving a rotating dust disk around a black hole, generalizing known solutions like Kerr and Neugebauer-Meinel disk.

Contribution

It provides explicit solutions using theta functions on hyperelliptic Riemann surfaces, extending the class of known solutions in general relativity.

Findings

Solutions reduce to Kerr black hole without disk

Solutions reduce to Neugebauer-Meinel disk without black hole

Explicit theta function representations of the spacetime geometry

Abstract

We solve a class of boundary value problems for the stationary axisymmetric Einstein equations corresponding to a disk of dust rotating uniformly around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. In the absence of a disk, they reduce to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk.