Conserved quantities in STEGR and applications

TL;DR

This paper derives covariant conservation laws in STEGR using Noether's theorem, enabling calculation of local and global gravitational quantities like energy density and mass, with applications to simple solutions such as Schwarzschild black holes.

Contribution

It introduces a method to construct covariant conserved currents and superpotentials in STEGR, addressing ambiguities in connection choice and their impact on conserved quantities.

Findings

Calculated energy density for Friedman universe and Schwarzschild black hole.

Obtained total mass of Schwarzschild black hole.

Identified ambiguities in connection determination affecting conserved quantities.

Abstract

We derive conservation laws in Symmetric Teleparallel Equivalent of General Relativity (STEGR) with direct application of Noether's theorem. This approach allows us to construct covariant conserved currents, corresponding superpotentials and invariant charges. A necessary component of our constructions is the concept of "turning off" gravity, introduced in the framework of STEGR to define the flat and torsionless connection. By calculating currents, one can obtain local characteristics of gravitational field like energy density. Surface integration of superpotentials gives charges which correspond to global quantities of the system like mass, momentum, etc. To test our results for the obtained currents and superpotentials, we calculate the energy density measured by freely falling observer in the simple solutions (Friedman universe, Schwartzchild black hole) and total mass of the Schwartzchild black hole. We find ambiguities in obtaining the connection, which explicitly affect the values of conserved quantities, and discuss possible solutions to this problem.