Comparison of Vacuum Static Quadrupolar Metrics
General Relativity and Quantum Cosmology
2025-03-13 v1
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 metric, a variant of the Zipoy-Voorhees metric, is the simplest generalization of the Schwarzschild metric, containing a quadrupole parameter.
Cite
@article{arxiv.1704.06734,
title = {Comparison of Vacuum Static Quadrupolar Metrics},
author = {Francisco Frutos-Alfaro and Hernando Quevedo and Pedro Sánchez},
journal= {arXiv preprint arXiv:1704.06734},
year = {2025}
}