The Noether formalism for constructing conserved quantities in teleparallel equivalents of general relativity

TL;DR

This paper develops a comprehensive Noether formalism for constructing conserved quantities in teleparallel gravity theories, unifying previous approaches and analyzing gauge choices affecting conserved charges.

Contribution

It presents a complete, covariant formalism for conserved quantities in TEGR and STEGR using the Noether method, uniting earlier partial results.

Findings

Conserved currents and charges are covariant under relevant symmetries.

Different gauge choices lead to different conserved quantities.

The formalism is applicable to both TEGR and STEGR theories.

Abstract

This paper brings a methodological character where we present a comprehensive formalism for constructing conserved quantities in the Teleparallel Equivalent of General Relativity (TEGR) and Symmetric Teleparallel Equivalent of General Relativity (STEGR). It was developed in series of our earlier works and, here, we unite it into a complete form. By employing the Noether method within a tensor formalism, conserved currents, superpotentials, and charges are constructed. These are shown to be covariant under coordinate transformations and local Lorentz rotations in TEGR, while in STEGR, they are covariant under coordinate transformations. The teleparallel (flat) connections in both theories are defined using the "turning off gravity" principle. Uniting such defined flat connections with tetrad in TEGR and metric in STEGR a new fruitful in applications notion "gauge" is introduced. The choice of various initial tetrads in TEGR or initial coordinates in STEGR leads to different gauges, what gives different conserved quantities. Finally, we discuss an appropriate choice of gauges from a possible set of them.