Conserved Matter Superenergy Currents for Orthogonally Transitive Abelian G2 Isometry Groups

TL;DR

This paper demonstrates that in Einstein-Maxwell and related theories, superenergy currents associated with electromagnetic and scalar fields are conserved and aligned with symmetries of spacetimes possessing orthogonally transitive Abelian G2 isometry groups.

Contribution

It extends previous results on conserved superenergy currents to more general symmetry groups and matter fields, including electromagnetic and scalar fields, in four-dimensional spacetimes.

Findings

Superenergy currents are conserved and lie in the orbits of the symmetry group.

Results apply to electromagnetic, scalar, and gravitational superenergy tensors.

Conservation holds in Einstein-Maxwell, Einstein-Klein-Gordon, and related theories.

Abstract

In a previous paper we showed that the electromagnetic superenergy tensor, the Chevreton tensor, gives rise to a conserved current when there is a hypersurface orthogonal Killing vector present. In addition, the current is proportional to the Killing vector. The aim of this paper is to extend this result to the case when we have a two-parameter Abelian isometry group that acts orthogonally transitive on non-null surfaces. It is shown that for four-dimensional Einstein-Maxwell theory with a source-free electromagnetic field, the corresponding superenergy currents lie in the orbits of the group and are conserved. A similar result is also shown to hold for the trace of the Chevreton tensor and for the Bach tensor, and also in Einstein-Klein-Gordon theory for the superenergy of the scalar field. This links up well with the fact that the Bel tensor has these properties and the possibility of constructing conserved mixed currents between the gravitational field and the matter fields.