The averaged tensors of the relative energy-momentum and angular momentum in general relativity and some their applications

TL;DR

This paper introduces a new averaging method for energy-momentum and angular momentum tensors in general relativity, resulting in tensorial quantities with proper dimensions that depend on a fundamental length, and applies these to analyze gravitational energy in cosmological models.

Contribution

It presents a novel averaging technique that yields tensorial energy-momentum and angular momentum quantities with correct dimensions, extending previous superenergy tensor approaches.

Findings

Averaged tensors depend on a fundamental length L>0.

Positive definite energy density in all Friedman universes.

Application to vacuum gravitational energy and cosmological models.

Abstract

There exist at least a few different kind of averaging of the differences of the energy-momentum and angular momentum in normal coordinates {\bf NC(P)} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [1-8] giving the {\it canonical superenergy and angular supermomentum tensors}. In this paper we present another averaging of the differences of the energy-momentum and angular momentum which gives tensorial quantities with proper dimensions of the energy-momentum and angular momentum densities. But these averaged relative energy-momentum and angular momentum tensors, closely related to the canonical superenergy and angular supermomentum tensors, {\it depend on some fundamental length $L>0$}. The averaged relative energy-momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to {\it coordinate independent} analysis (local and in special cases also global) of this field. We have applied the averaged relative energy-momentum tensors to analyze vacuum gravitational energy and momentum and to analyze energy and momentum of the Friedman (and also more general) universes. The obtained results are very interesting, e.g., the averaged relative energy density is {\it positive definite} for the all Friedman universes.