On Freud's superpotential in General Relativity and in Einstein-Cartan theory

TL;DR

This paper revisits Freud's superpotential in general relativity, simplifies its derivation, and extends its validity to Einstein-Cartan and more general gravitational theories, highlighting the persistent nature of Freud's expression.

Contribution

It provides a streamlined derivation of Freud's superpotential and demonstrates its applicability across various gravitational theories including Einstein-Cartan.

Findings

Freud's superpotential can be derived by integrating the gravitational field equation by parts.

The original Freud expression remains valid in Riemann-Cartan and metric-affine spacetimes.

The derivation simplifies understanding of gravitational energy in different theories.

Abstract

The identification of a suitable gravitational energy in theories of gravity has a long history, and it is well known that a unique answer cannot be given. In the first part of this paper we present a streamlined version of the derivation of Freud's superpotential in general relativity. It is found if we once integrate the gravitational field equation by parts. This allows us to extend these results directly to the Einstein-Cartan theory. Interestingly, Freud's original expression, first stated in 1939, remains valid even when considering gravitational theories in Riemann-Cartan or, more generally, in metric-affine spacetimes.