The gravitational energy-momentum pseudo-tensor in $f(Q)$ non-metric gravity

TL;DR

This paper derives and analyzes the gravitational energy-momentum pseudo-tensor in $f(Q)$ non-metric gravity, extending previous frameworks, and explores its properties, perturbations, and applications to gravitational waves and Schwarzschild spacetime.

Contribution

It introduces the affine tensor for energy-momentum in $f(Q)$ gravity, compares it with $f(T)$ gravity, and applies it to gravitational wave energy and Schwarzschild spacetime.

Findings

Conservation of energy-momentum complex on-shell is established.

Perturbed pseudo-tensor provides an expression for gravitational wave energy.

Application to Schwarzschild spacetime yields gravitational energy density.

Abstract

We derive the affine tensor associated with the energy and momentum densities of both gravitational and matter fields, the complex pseudo-tensor, for $f(Q)$ non-metric gravity, the straightforward extension of Symmetric Teleparallel Equivalent of General Relativity (STEGR), characterized by a flat, torsion-free, non-metric connection. The local conservation of energy-momentum complex on-shell is satisfied through a continuity equation. An important analogy is pointed out between gravitational pseudo-tensor of teleparallel $f(T)$ gravity, in the Weitzenb\"ock gauge, and the same object of symmetric teleparallel $f(Q)$ gravity, in the coincident gauge. Furthermore, we perturb the gravitational pseudo-tensor $\tau^{\alpha}_{\phantom{\alpha}\lambda}$ in the coincident gauge up to the second order in the metric perturbation, obtaining a useful expression for the power carried by the related gravitational waves. We also present an application of the gravitational pseudotensor, determining the gravitational energy density of a Schwarzschild spacetime in STEGR gravity, adopting the concident gauge. Finally, analyzing the conserved quantities on manifolds, the Stokes theorem can be formulated for generic affine connections