Mass and angular momentum for the Kerr black hole in TEGR and STEGR

TL;DR

This paper investigates the energy-momentum and angular momentum of Kerr black holes within TEGR and STEGR frameworks, using covariant conserved quantities and gauge choices, revealing new insights and a novel gauge for Schwarzschild in STEGR.

Contribution

It introduces a method to define covariant conserved quantities in TEGR and STEGR using external flat connections and explores gauge effects on these quantities, including a new gauge for Schwarzschild in STEGR.

Findings

Expected black hole mass and angular momentum values obtained.

Different gauges yield different conserved quantities.

A novel gauge for Schwarzschild in STEGR was identified.

Abstract

We study the energy-momentum characteristics of the rotating black hole - Kerr solution of general relativity in the Teleparallel Equivalent of General Relativity (TEGR) and the Symmetric Teleparallel Equivalent of General Relativity (STEGR). The previously constructed spacetime covariant and Lorentz invariant expressions for conserved Noether currents, superpotentials and charges are used. The Noether charges describe total energy, momentum or angular momentum of gravitating system depending on a choice of the displacement vector $\xi$. To define covariant and invariant conserved quantities both in TEGR and in STEGR on needs to use external fields which are flat teleparallel connections. To determine the non-dynamical connections in TEGR and STEGR we use the unified ``turning off'' gravity principle. Besides, to analyse the Noether conserved quantities in these theories, we use the concept of ``gauges''. The gauge changing can affect the Noether conserved quantities. We highlight two ways to turn off gravity - by $M \to 0$ and by $M \to 0 , ~ a \to 0$ which gives us different gauges in TEGR and STEGR. In both kind of gauges we get the expected values of black hole mass and angular momentum. Our attempts to find gauges which could lead to a correspondence to Einstein's equivalence principle for the Kerr solution where unsuccessful both in TEGR and STEGR. However, these exercises helped us to find a related gauge for the Schwarzschild solution in STEGR that is a novelty.