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

Understand

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 $.