Physically Realistic Solutions to the Ernst Equation on Hyperelliptic Riemann Surfaces

TL;DR

This paper derives and analyzes hyperelliptic solutions to the Ernst equation using Riemann-Hilbert techniques, revealing physically relevant properties and potential applications to modeling relativistic stars and galaxies.

Contribution

It extends previous work by providing complete proofs and identifying a subclass of solutions with realistic physical properties, including regularity and asymptotic flatness.

Findings

Solutions can describe exterior fields of rotating bodies.

Solutions include limits approaching extreme Kerr black holes.

Explicit formulas are provided for key spacetime regions.

Abstract

We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Neugebauer and Meinel can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical properties of these solutions that was begun in a Physical Review Letter, and supplies complete proofs. We identify a physically interesting subclass where the Ernst potential is everywhere regular except at a closed surface which might be identified with the surface of a body of revolution. The corresponding spacetimes are asymptotically flat and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance a relativistic star or a galaxy. Within this class, one has the freedom to specify a real function and a set of complex parameters which can possibly be used to solve certain boundary value problems for the Ernst equation. The solutions can have ergoregions, a Minkowskian limit and an ultrarelativistic limit where the metric approaches the extreme Kerr solution. We give explicit formulae for the potential on the axis and in the equatorial plane where the expressions simplify. Special attention is paid to the simplest non-static solutions (which are of genus two) to which the rigidly rotating dust disk belongs.