A convenient gauge for virial identities in axial symmetry

TL;DR

This paper introduces a new gauge choice for axial symmetry in General Relativity that simplifies the calculation of virial identities, demonstrated through Kerr black holes with scalar hair, aiding in physical insights and numerical accuracy.

Contribution

The paper proposes a novel convenient gauge for axial symmetry that simplifies virial identity calculations, extending tools previously limited to spherical symmetry.

Findings

A new gauge simplifies the gravitational action contribution in axial symmetry.

Application of the gauge to Kerr black holes with scalar hair demonstrates its effectiveness.

Facilitates better physical understanding and numerical analysis of axially symmetric systems.

Abstract

Virial identities are a useful mathematical tool in General Relativity. Not only have they been used as a numerical accuracy tool, but they have also played a significant role in establishing no-go and no-hair theorems while giving some physical insight into the considered system from an energy balance perspective. While the calculation of these identities tends to be a straightforward application of Derrick's scaling argument~\cite{derrick1964comments}, the complexity of the resulting identity is system dependent. In particular, the contribution of the Einstein-Hilbert action, due to the presence of second-order derivatives of the metric functions, becomes increasingly complex for generic metrics. Additionally, the Gibbons-Hawking-York term needs to be taken into account \cite{Herdeiro:2021teo}. Thankfully, since the gravitational action only depends on the metric, it is expected that a ``convenient'' gauge that trivializes the gravitational action contribution exists. While in spherical symmetry such a gauge is known (the $m-\sigma$ parametrization), such has not been found for axial symmetry. In this letter, we propose a ``convenient'' gauge for axial symmetry and use it to compute an identity for Kerr black holes with scalar hair.