Ernst equation, Fay identities and variational formulas on hyperelliptic curves

TL;DR

This paper develops a unified theta-functional framework for solving the stationary axisymmetric Einstein equations in vacuum, utilizing Fay's identities and variational formulas on hyperelliptic Riemann surfaces.

Contribution

It introduces a novel approach combining Fay identities and variational formulas to explicitly construct solutions to Einstein's equations.

Findings

Derived formulas for metric functions and Ernst potential

Established a connection between Riemann surface theory and Einstein equations

Provided explicit theta-functional solutions for the vacuum case

Abstract

We present a unified approach to theta-functional solutions of the stationary axisymmetric Einstein equations in vacuum. Using Fay's trisecant identity and variational formulas on hyperelliptic Riemann surfaces, we establish formulas for the metric functions, the Ernst potential and their derivatives.