New Representation of some Static and Axisymmetric Vacuum Solutions

TL;DR

This paper introduces a new, simplified representation of static, axisymmetric vacuum solutions to Einstein's equations using a different coordinate system, characterized by a specific analytic function.

Contribution

It presents a novel coordinate system and a compact form for solutions, linking metric functions to a Cauchy-Newman problem and gauge choices.

Findings

Existence of a class of solutions with proper asymptotic behavior

Solutions can be expressed in a simple compact form

Relationship between coordinate choice and metric functions established

Abstract

We solve the Einstein vacuum-equations for the case of static and axisymmetric solutions in a system of coordinates different from the Weyl standard one. We prove that there exists a class of solutions with the appropriate asymptotical behaviour which can be written in a simple compact form, in terms of a function that must satisfies certain Cauchy-Newman problem. The relation between the choice of coordinates and the form of the metric functions that describe the solution is given by providing that analytic function which characterizes the metric as well as the gauge.