Gravitational energy-momentum pseudo-tensor in Palatini and metric $f(R)$ gravity

TL;DR

This paper derives the gravitational energy-momentum pseudo-tensor in both Palatini and metric $f(R)$ gravity, revealing differences in cosmological energy density that could distinguish the two formalisms.

Contribution

It provides the first derivation of the pseudo-tensor in both approaches and highlights a key difference in energy density behavior in cosmology.

Findings

Energy density vanishes in metric $f(R)$ gravity for FLRW spacetime.

Energy density is non-zero in Palatini $f(R)$ gravity.

Potential to discriminate between approaches based on energy density differences.

Abstract

We derive the gravitational energy-momentum pseudo-tensor $\tau^\mu_{\phantom{\mu}\nu}$ in both Palatini and metric approaches to $f(R)$ gravity. We then obtain the related cosmological gravitational energy density. Considering a flat Friedmann-Lema\^itre-Robertson-Walker spacetime, the energy density complex of matter and gravitation vanishes in the metric approach, but results non-vanishing in the Palatini formalism. This feature could be relevant in order to physically discriminate between the two approaches.