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

TL;DR

This work derives a gravitational energy-momentum pseudo-tensor for $f(Q)$ gravity by applying Noether's theorem to the gravitational sector of the action with global translations, yielding a locally conserved affine current $\tau^{\alpha}_{\lambda|f(Q)}$ that couples to matter through $\mathcal{T}^{\alpha}_{\ lambda}=T^{\alpha}_{\ lambda}+\tau^{\alpha}_{\ lambda}$. In the coincident gauge, the pseudo-tensor simplifies to a form proportional to $f_Q P^{\alpha}_{\mu\nu} g^{\mu u}_{,\lambda}$ plus a potential term $f(Q)\delta^{\alpha}_{\lambda}$, and total conservation follows from $\partial_{\alpha}(\sqrt{-g}\mathcal{T}^{\alpha}_{\lambda})=0$ on-shell. The authors illustrate the framework with a Schwarzschild exterior in STEGR, showing the gravitational energy density scales with $Q$ and leads to a shell energy that diverges at large radius, and they provide a second-order weak-field expansion to enable gravitational-wave power calculations. A key result is the clear analogy between $f(T)$ and $f(Q)$ pseudo-tensors under corresponding gauge choices, offering a robust tool to probe energy exchange in modified gravity and guiding future observational tests of $f(Q)$-type theories.

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 Weitzenböck gauge, and the same object of symmetric teleparallel $f(Q)$ gravity, in the coincident gauge. Furthermore, we perturb the gravitational pseudo-tensor $τ^α_{\phantomαλ}$ 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