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

Abstract

We derive the gravitational energy-momentum pseudotensor ${\tau }_{\lambda }^{\sigma }$ 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 Weitzenböck 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 ${\tau }_{\lambda }^{\sigma }|\omega$. 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.

20 pages, accepted for publication in Int.J. Geom. Meth. Mod. Phys