Energy-momentum density in small regions: the classical pseudotensors

TL;DR

This paper evaluates classical gravitational pseudotensors in small regions, revealing that most satisfy the equivalence principle at zeroth order, but none in vacuum regions are proportional to the Bel-Robinson tensor, except for specific linear combinations.

Contribution

It identifies a new one-parameter family of linear combinations of classical pseudotensors that are proportional to the Bel-Robinson tensor in vacuum regions.

Findings

Most pseudotensors satisfy the equivalence principle in small regions.

None are proportional to the Bel-Robinson tensor in vacuum regions, except for specific combinations.

A new linear combination of pseudotensors satisfies the Bel-Robinson proportionality criterion.

Abstract

The values for the gravitational energy-momentum density, given by the famous classical pseudotensors: Einstein, Papapetrou, Landau-Lifshitz, Bergmann-Thompson, Goldberg, M{\o}ller, and Weinberg, in the small region limit are found to lowest non-vanishing order in normal coordinates. All except M{\o}ller's have the zeroth order material limit required by the equivalence principle. However for small vacuum regions we find that {\it none} of these classical holonomic pseudotensors satisfies the criterion of being proportional to the Bel-Robinson tensor. Generalizing an earlier work which had identified one case, we found another independent linear combination satisfying this requirement--and hence a one parameter set of linear combinations of the classical pseudotensors with this desirable property.