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

Abstract

We derive the gravitational energy–momentum pseudotensor [Formula: see text] in metric [Formula: see text] gravity and in teleparallel [Formula: see text] gravity. In the first case, [Formula: see text] is the Ricci curvature scalar for a torsionless Levi-Civita connection; in the second case, [Formula: see text] 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. [Formula: see text] and [Formula: see text] are non-equivalent, but differ for a quantity [Formula: see text] containing the torsion scalar [Formula: see text] and a boundary term [Formula: see text]. It is possible to obtain the field equations for [Formula: see text] and the related gravitational energy–momentum pseudotensor [Formula: see text]. Finally we show that, thanks to this further pseudotensor, it is possible to pass from [Formula: see text]–[Formula: see text] and vice versa through a simple relation between gravitational pseudotensors.