Freud's Identity of Differential Geometry, the Einstein-Hilbert Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR

TL;DR

This paper rigorously analyzes Freud's identity within differential geometry and clarifies its consistency with Einstein's equations, while highlighting issues in the conventional formulation of energy-momentum conservation in General Relativity.

Contribution

It provides a detailed mathematical clarification of Freud's identity using differential forms and addresses misconceptions about its consistency with Einstein's field equations.

Findings

Freud's identity is mathematically consistent with Einstein's equations.

Common applications of Freud's identity in energy-momentum conservation are flawed.

The paper offers detailed calculations accessible to beginners in differential forms.

Abstract

We reveal in a rigorous mathematical way using the theory of differential forms, here viewed as sections of a Clifford bundle over a Lorentzian manifold, the true meaning of Freud's identity of differential geometry discovered in 1939 (as a generalization of results already obtained by Einstein in 1916) and rediscovered in disguised forms by several people. We show moreover that contrary to some claims in the literature there is not a single (mathematical) inconsistency between Freud's identity (which is a decomposition of the Einstein indexed 3-forms in two gauge dependent objects) and the field equations of General Relativity. However, as we show there is an obvious inconsistency in the way that Freud's identity is usually applied in the formulation of energy-momentum "conservation laws" in GR. In order for this paper to be useful for a large class of readers (even those ones making a first contact with the theory of differential forms) all calculations are done with all details (disclosing some of the "tricks of the trade" of the subject).