Gauge-Invariant Localization of Infinitely Many Gravitational Energies from all Possible Auxiliary Structures, Or, Why Pseudotensors are Okay

TL;DR

This paper presents a covariant, gauge-invariant approach to localizing infinitely many gravitational energy-momenta using auxiliary structures, showing pseudotensors are valid tools for gravitational energy localization.

Contribution

It introduces a method to obtain covariant gravitational energy-momentum expressions from pseudotensors by gauge fixing and auxiliary structures, resolving longstanding objections.

Findings

Infinite-component localization of gravitational energy-momentum is achieved.

Pseudotensors are valid for energy localization when viewed as part of an infinite set.

The approach extends to angular momentum localization and aligns with Noether's theorem.

Abstract

The problem of finding a covariant expression for the distribution and conservation of gravitational energy-momentum dates to the 1910s. A suitably covariant infinite-component localization is displayed, reflecting Bergmann's realization that there are infinitely many conserved gravitational energy-momenta. Initially use is made of a flat background metric or connection (or rather, all of them), because the desired gauge invariance properties are obvious. Partial gauge-fixing then yields an appropriate covariant quantity without any background metric or connection; one version is the collection of pseudotensors of a given type, such as the Einstein pseudotensor, in_every_ coordinate system. This solution to the gauge covariance problem is easily adapted to any pseudotensorial expression or to any tensorial expression built with a background metric or connection. Thus the specific functional form can be chosen on technical grounds such as relating to Noether's theorem and yielding expected values of conserved quantities in certain contexts and then rendered covariant using the procedure described here. The application to angular momentum localization is straightforward. Traditional objections to pseudotensors are based largely on the false assumption that there is only one gravitational energy rather than infinitely many.