The gravitation energy-momentum pseudotensor: The cases of F(R) and F(T) gravity

We derive the gravitational energy-momentum pseudotensor τσλ in metric f(R) gravity and in teleparallel f(T) 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(R) and f(T) are non-equivalent, but differ for a quantity ω(T,B) containing the torsion scalar T and a boundary term B. It is possible to obtain the field equations for ω(T,B) and the related gravitational energy-momentum pseudotensor τσλ|ω. Finally we show that, thanks to this further pseudotensor, it is possible to pass from f(R)-f(T) and vice versa through a simple relation between gravitational pseudotensors.