A convenient gauge for virial identities in axial symmetry

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.