On canonical pseudotensors, Sparling's form and Noether currents

Abstract

The Lagrangian scenario is applied both to Einstein's rst and Hilbert's second order Lagrangians when the eld variable is the metric and when it is a rigid basis of 1-forms. The canonical spin pseudotensors, as the gravitational counterparts of the canonical spin tensor of the matter elds, are introduced and it is shown that they play the role of superpotentials for the canonical energy-momentum pseudotensors. The canonical spin pseudotensors distinguishes von Freud's and Goldberg's superpotentials for Einstein's, and Mller's superpotential for Hilbert's Lagrangian. (For Hilbert's Lagrangian there is no dierence between the metric and rigid basis descriptions.) Mller's energy-momentum pseudotensor is therefore recovered as the canonical energy-momentum pseudotensor for Hilbert's Lagrangian. The contravariant form of the canonical pseudotensors are shown to satisfy Belinfante{Rosenfeld type equations, furthermore their Belinfante{Rosenfeld combination is always the Einstein tensor. It is shown that, for rst order Lagrangian and rigid basis description, the canonical energy-momentum, the canonical spin and the Noether current are tensorial quantities, and the canonical energy-momentum and spin tensors satisfy the tensorial Belinfante{Rosenfeld equations. These tensorial quantities, however, depend on the basis of 1-