Gauge-Invariant Localization of Infinitely Many Gravitational Energies from All Possible Auxiliary Structures

TL;DR

This paper presents a covariant, gauge-invariant localization method for infinitely many gravitational energy-momenta, using auxiliary structures and partial gauge fixing, resolving longstanding issues with pseudotensors.

Contribution

It introduces a covariant infinite-component localization framework for gravitational energies, accommodating all pseudotensors and background structures, and clarifies the nature of gravitational energy.

Findings

Provides a covariant expression for gravitational energy-momentum distribution.

Shows how to adapt the method to various pseudotensors and background structures.

Resolves objections to pseudotensors by emphasizing the infinite multiplicity of gravitational energies.

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 rather, all of them) or connection, 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 (Landau-Lifshitz, Goldberg, Papapetrou or the like) 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.