The Komar mass function in the de-Rham-Gabadadze-Tolley non-linear theory of massive gravity

TL;DR

This paper derives the Komar mass function for Schwarzschild-de Sitter black holes within the dRGT massive gravity theory, showing its dependence on St"uckelberg fields and reducing to GR results under specific conditions.

Contribution

It introduces a method to compute the Komar mass in dRGT massive gravity, linking it to St"uckelberg fields and extending the understanding of black hole mass definitions in modified gravity.

Findings

Komar mass function depends on St"uckelberg fields.

Reduces to GR Komar mass when gauge function derivatives vanish.

Provides a new perspective on energy definitions in massive gravity.

Abstract

I derive the Komar mass/function for the Schwarszchild de-Sitter (S-dS) black-hole inside the dRGT non-linear theory of massive gravity by taking the usual notion of time-like Killing vector in unitary gauge. The dRGT Komar function depends on the dynamics of the St\"uckelberg fields through the gauge transformation function. It goes to the standard value obtained in General Relativity (GR) if the spatial derivative of the gauge function vanishes. In such a case, the (gauge) function corresponds to the usual notion of time as in GR.