Solutions of Einstein's field equations related to Jacobi's inversion problem

TL;DR

This paper introduces a new class of exact solutions to Einstein's field equations, connecting them to Jacobi's inversion problem for hyperelliptic Abelian integrals, with potential implications for understanding axisymmetric gravitational fields.

Contribution

It presents a novel set of solutions to Einstein's equations linked to hyperelliptic Abelian integrals, expanding the mathematical tools available for axisymmetric spacetime modeling.

Findings

Solutions contain n arbitrary complex parameters

Solutions include one arbitrary real solution of Laplace's equation

Solutions are related to Jacobi's inversion problem

Abstract

A new class of exact solutions to the axisymmetric and stationary vacuum Einstein equations containing n arbitrary complex parameters and one arbitrary real solution of the axisymmetric three-dimensional Laplace equation is presented. The solutions are related to Jacobi's inversion problem for hyperelliptic Abelian integrals.