Conserved matter superenergy currents for orthogonally transitive Abelian G ₂ isometry groups

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.

On the conservation of superenergy and its applications

“…To test this possibility one needs to construct conserved quantities involving the Bel--Robinson tensor and the Chevreton tensor. This possibility has been explored in [2,8,9,16] assuming the existence of Killing vectors with certain geometric properties.…”

f-SYMBOLS, KILLING TENSORS AND CONSERVED BEL-TYPE CURRENTS

“…the first index divergence vanishes if the trace of the f-symbol is a constant. In this particular case, since the metric tensor is covariantly constant and trivially satisfies equation (10), we can substract the trace part from the f-symbol and the remaining trace free part is itself a f-symbol.…”

“…The case of a four-dimensional Einstein-Maxwell theory with a source-free electromagnetic field is discussed in [9,10]. While in [9] it is shown that for electromagnetic fields inheriting the symmetry of the spacetime the Chevreton superenergy currents are individually conserved if there are hypersurface orthogonal Killing vector fields, in [10] is treated the case when there is a two-parameter Abelian isometry group that acts orthogonally transitive on non-null surfaces and it is shown that the corresponding superenergy currents lies in the orbits of the group and are conserved.…”

On the energy content of electromagnetic and gravitational plane waves through super-energy tensors

“…the gravitational potential). General properties of the Chevreton tensor and conservation laws have been studied only recently in [24,[32][33][34][35][36][37] with some applications to explicit spacetime solutions.…”