Coincident gauge for static spherical field configurations in symmetric teleparallel gravity

TL;DR

This paper derives transformation rules to achieve the coincident gauge in symmetric teleparallel gravity for static spherically symmetric configurations, exploring different metric forms and connection choices, and discussing implications for black hole solutions.

Contribution

It provides a systematic method to transform into the coincident gauge for static spherically symmetric spacetimes in symmetric teleparallel gravity, including explicit metric forms and connection considerations.

Findings

Schwarzschild metric can take Cartesian, Kerr-Schild, and diagonal forms in the coincident gauge.

Certain diagonal forms fit the coincident gauge for the BBMB black hole, while Cartesian and Kerr-Schild forms do not.

Different connections lead to different boundary term values, but the connection is not uniquely fixed by the gauge.

Abstract

In symmetric teleparallel gravities, where the independent connection is characterized by nonmetricity while curvature and torsion are zero, it is possible to find a coordinate system whereby the connection vanishes globally and covariant derivatives reduce to partial derivatives -- the coincident gauge. In this paper we derive general transformation rules into the coincident gauge for spacetime configurations where the both the metric and connection are static and spherically symmetric, and write out the respective form of the coincident gauge metrics. Taking different options in fixing the freedom in the connection allowed by the symmetry and the field equations, the Schwarzschild metric in the coincident gauge can take for instance the Cartesian, Kerr-Schild, and diagonal (isotropic-like) forms, while the BBMB black hole metric in symmetric teleparallel scalar-tensor theory a certain diagonal form fits the coincident gauge requirements but the Cartesian and Kerr-Schild forms do not. Different connections imply different value for the boundary term which could in principle be physically relevant, but simple arguments about the coincident gauge do not seem to be sufficient to fix the connection uniquely. As a byproduct of the investigation we also point out that only a particular subset of static spherically symmetric connections has vanishing nonmetricity in the Minkowski limit.