The gravitational energy-momentum pseudotensor: the cases of $f(R)$ and $f(T)$ gravity
We derive the gravitational energy-momentum pseudotensor $ τ^σ_ {\phantom σ λ} $ 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 $ ω\left (T, B \right) $ containing the torsion scalar $T$ and a boundary term $ B $. It is possible to obtain the field equations for $ ω\left (T, B \right) $ and the related gravitational energy-momentum pseudotensor $ τ^σ_{\phantom σλ} \vert ω$. 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.