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

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.