Energy, angular momentum, superenergy and angular supermomentum in conformal frames

TL;DR

This paper derives the rules for how various energetic and angular quantities related to gravity and matter transform under conformal changes of the spacetime metric, revealing invariants and complexities in these transformations.

Contribution

It provides a comprehensive analysis of conformal transformation rules for energy-momentum, superenergy, and angular supermomentum tensors, including invariants and special cases.

Findings

Superenergy density and combined matter-superenergy density are conformally invariant in pure gravity.

The Bel-Robinson tensor remains invariant under conformal transformations in pure gravity.

Transformation rules for angular supermomentum tensors are complex, especially for gravity.

Abstract

We find the rules of the conformal transformation for the energetic quantities such as the Einstein energy-momentum complex, the Bergmann-Thomson angular momentum complex, the superenergy tensor, and the angular supermomentum tensor of gravitation and matter. We show that the conformal transformation rules for the matter parts of both the Einstein complex and the Bergmann-Thomson complex are fairly simple, while the transformation rules for their gravitational parts are more complicated. We also find that the transformational rules of the superenergy tensor of matter and the superenergy tensor of gravity are quite complicated except for the case of a pure gravity. In such a special case the superenergy density as well as the sum of the superenergy density and the matter energy density are invariants of the conformal transformation. Besides, in that case, a conformal invariant is also the Bel-Robinson tensor which is a part of the superenergy tensor. As for the angular supermomentum tensor of gravity - it emerges that its transformational rule even for a pure gravity is quite complicated but this is not the case for the angular supermomentum tensor of matter. Having investigated some technicalities of the conformal transformations, we also find the conformal transformation rule for the curvature invariants and, in particular, for the Gauss-Bonnet invariant in a spacetime of arbitrary dimension.