Canonical superenergy and angular supermomentum complexes in general relativity and some of their applications

TL;DR

This paper introduces and applies canonical superenergy and angular supermomentum complexes in general relativity, providing new tools for analyzing various spacetime models and comparing with previous tensor-based approaches.

Contribution

It defines the canonical superenergy and angular supermomentum complexes in GR and demonstrates their application to different spacetime scenarios, offering a simpler alternative to tensor averaging methods.

Findings

Applied complexes to analyze closed systems, radiative spacetimes, and Friedmann universes.

Compared complexes with previous tensor approaches, highlighting advantages.

Provided a new framework for studying energy and momentum in GR.

Abstract

Many years ago we have introduced into general relativity, {\bf GR}, the canonical superenergy tensors, $S_i^{~k}$, and the canonical angular supermomentum tensors, $S^{ikl}=(-)S^{kil}$, matter and gravitation. We have obtained these tensors by special averaging of the differences of the canonical energy-momentum and canonical angular momentum. The averaging was performed in Riemann normal coordinates, {\bf RNC(P)}; {\bf P} is beginning of these coordinates. About four years ago we have observed that these tensors can also be obtained in other, simpler way, by using the canonical superenergy and angular super momentum complexes, $_K S_i^{~k}$, and, $_K S~^{ikl}=(-)_K S^{kil}$, respectively. Such complexes can be introduced into {\bf GR} in a natural way starting from canonical energy-momentum and angular momentum complexes. In this paper, at first, we define the canonical superenergy and angular supermomentum complexes in {\bf GR} and then, we apply them to analyze of a closed system, {\bf CS}, Trautman's radiative spacetimes, {\bf TRS}, and Friedman universes, {\bf FU}. Finally, we compare these complexes and the results obtained with their help with the canonical superenergy and angular supermomentum tensors and results obtained with them in past. In Appendix, for convenience, we summarize our old approach to canonical superenergy and angular supermomentum tensors.