The gravitational energy-momentum pseudotensor: the cases of $f(R)$ and $f(T)$ gravity

Abstract

We derive the gravitational energy-momentum pseudotensor $ \tau^{\sigma}_ {\phantom {\sigma} \lambda} $ in metric $ f\left (R \right) $ gravity and in teleparallel $ f\left (T\right) $ gravity. In the first case, $R$ is the Ricci curvature scalar for a torsionless Levi-Civita connection; in the second case, $T$ is the curvature-free torsion scalar derived by tetrads and Weitzenb\"ock connection. For both classes of theories the continuity equations are obtained in presence of matter. $ f \left (R \right) $ and $ f \left (T \right) $ are non-equivalent but differ for a quantity $ \omega \left (T, B \right) $ containing the torsion scalar $T$ and a boundary term $ B $. It is possible to obtain the field equations for $ \omega \left (T, B \right) $ and the related gravitational energy-momentum pseudotensor $ \tau^{\sigma}_{\phantom {\sigma}\lambda} \vert \omega $. Finally we show that, thanks to this further pseudotensor, it is possible to pass from $ f \left (R \right) $ to $ f \left ( T \right) $ and viceversa through a simple relation between gravitational pseudotensors.