Static axisymmetric space-times with prescribed multipole moments

TL;DR

This paper presents a method to construct static axisymmetric space-times from any set of multipole moments, providing algebraic and power series solutions, and proves a conjecture related to the existence of such space-times.

Contribution

It introduces a novel method for deriving static axisymmetric space-times from prescribed multipole moments, including a proof of Geroch's conjecture in the axisymmetric case.

Findings

Established a power series expression for finite multipole sets

Proved the existence of static space-times for any multipole set under convergence conditions

Confirmed a conjecture on monopole-quadrupole solutions

Abstract

In this article we develop a method of finding the static axisymmetric space-time corresponding to any given set of multipole moments. In addition to an implicit algebraic form for the general solution, we also give a power series expression for all finite sets of multipole moments. As conjectured by Geroch we prove in the special case of axisymmetry, that there is a static space-time for any given set of multipole moments subject to a (specified) convergence criterion. We also use this method to confirm a conjecture of Hernandez-Pastora and Martin concerning the monopole-quadropole solution.