Comparison of Vacuum Static Quadrupolar Metrics

TL;DR

This paper examines vacuum solutions to Einstein's equations that extend Schwarzschild's metric with a quadrupole parameter, analyzing their properties and showing the q-metric as the simplest such generalization.

Contribution

It provides a detailed comparison of static quadrupolar vacuum metrics, establishing the q-metric as the most straightforward extension of Schwarzschild with a quadrupole.

Findings

All solutions are equivalent up to the quadrupole level.

The q-metric is identified as the simplest quadrupolar generalization.

Solutions are tested for flatness and regularity.

Abstract

We investigate the properties of static and axisymmetric vacuum solutions of Einstein equations which generalize the Schwarzschild spherically symmetric solution to include a quadrupole parameter. We test all the solutions with respect to elementary and asymptotic flatness and curvature regularity. Analyzing their multipole structure, according to the relativistic invariant Geroch definition, we show that all of them are equivalent up to the level of the quadrupole. We conclude that the $q-$metric, a variant of the Zipoy-Voorhees metric, is the simplest generalization of the Schwarzschild metric, containing a quadrupole parameter.