A Note on Stress-Tensors, Conservation and Equations of Motion

Some unusual relations between stress tensors, conservation and equations of motion are briefly reviewed.Comment: 4 pages. Invited contribution, A. Peres Festschrift, to be published in Found. Phy

A Note on Matter Superenergy Tensors

We consider Bel-Robinson-like higher derivative conserved two-index tensors H_\mn in simple matter models, following a recently suggested Maxwell field version. In flat space, we show that they are essentially equivalent to the true stress-tensors. In curved Ricci-flat backgrounds it is possible to redefine H_\mn so as to overcome non-commutativity of covariant derivatives, and maintain conservation, but they become model- and dimension- dependent, and generally lose their simple "BR" form.Comment: 3 page

Super-energy and Killing-Yano tensors

In this paper we investigate a class of basic super-energy tensors, namely those constructed from Killing-Yano tensors, and give a generalization of super-energy tensors for cases when we start not with a single tensor, but with a pair of tensors.Comment: 15 pages, typos corrected, references adde

Conserved Matter Superenergy Currents for Hypersurface Orthogonal Killing Vectors

We show that for hypersurface orthogonal Killing vectors, the corresponding Chevreton superenergy currents will be conserved and proportional to the Killing vectors. This holds for four-dimensional Einstein-Maxwell spacetimes with an electromagnetic field that is sourcefree and inherits the symmetry of the spacetime. A similar result also holds for the trace of the Chevreton tensor. The corresponding Bel currents have previously been proven to be conserved and our result can be seen as giving further support to the concept of conserved mixed superenergy currents. The analogous case for a scalar field has also previously been proven to give conserved currents and we show, for completeness, that these currents also are proportional to the Killing vectors.Comment: 13 page

On the Energy-Momentum Density of Gravitational Plane Waves

By embedding Einstein's original formulation of GR into a broader context we show that a dynamic covariant description of gravitational stress-energy emerges naturally from a variational principle. A tensor $T^G$ is constructed from a contraction of the Bel tensor with a symmetric covariant second degree tensor field $\Phi$ and has a form analogous to the stress-energy tensor of the Maxwell field in an arbitrary space-time. For plane-fronted gravitational waves helicity-2 polarised (graviton) states can be identified carrying non-zero energy and momentum.Comment: 10 pages, no figure

Conserved Matter Superenergy Currents for Orthogonally Transitive Abelian G2 Isometry Groups

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.Comment: 15 page

Old and new results for superenergy tensors from dimensionally dependent tensor identities

It is known that some results for spinors, and in particular for superenergy spinors, are much less transparent and require a lot more effort to establish, when considered from the tensor viewpoint. In this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4-dimensional identities by straightforward tensor methods in a signature independent manner. In particular, we consider the quadratic identity for the Bel-Robinson tensor ${\cal T}_{abcx}{\cal T}^{abcy} = \delta_x^y {\cal T}_{abcd}{\cal T}^{abcd}/4$ and also the new conservation laws for the Chevreton tensor, both of which have been obtained by spinor means; both of these results are rederived by {\it tensor} means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and also we are able to conclude that there are no {\it direct} higher dimensional analogues. In addition we demonstrate a simple way to show non-existense of such identities via counter examples; in particular we show that there is no non-trivial Bel tensor analogue of this simple Bel-Robinson tensor quadratic identity. On the other hand, as a sample of the power of generalising dimensionally dependent tensor identities from four to higher dimensions, we show that the symmetry structure, trace-free and divergence-free nature of the four dimensional Bel-Robinson tensor does have an analogue for a class of tensors in higher dimensions.Comment: 18 pages; TeX fil

On the structure of the new electromagnetic conservation laws

New electromagnetic conservation laws have recently been proposed: in the absence of electromagnetic currents, the trace of the Chevreton superenergy tensor, $H_{ab}$ is divergence-free in four-dimensional (a) Einstein spacetimes for test fields, (b) Einstein-Maxwell spacetimes. Subsequently it has been pointed out, in analogy with flat spaces, that for Einstein spacetimes the trace of the Chevreton superenergy tensor $H_{ab}$ can be rearranged in the form of a generalised wave operator $\square_L$ acting on the energy momentum tensor $T_{ab}$ of the test fields, i.e., $H_{ab}=\square_LT_{ab}/2$. In this letter we show, for Einstein-Maxwell spacetimes in the full non-linear theory, that, although, the trace of the Chevreton superenergy tensor $H_{ab}$ can again be rearranged in the form of a generalised wave operator $\square_G$ acting on the electromagnetic energy momentum tensor, in this case the result is also crucially dependent on Einstein's equations; hence we argue that the divergence-free property of the tensor $H_{ab}=\square_GT_{ab}/2$ has significant independent content beyond that of the divergence-free property of $T_{ab}$

The Chevreton Tensor and Einstein-Maxwell Spacetimes Conformal to Einstein Spaces

In this paper we characterize the source-free Einstein-Maxwell spacetimes which have a trace-free Chevreton tensor. We show that this is equivalent to the Chevreton tensor being of pure-radiation type and that it restricts the spacetimes to Petrov types \textbf{N} or \textbf{O}. We prove that the trace of the Chevreton tensor is related to the Bach tensor and use this to find all Einstein-Maxwell spacetimes with a zero cosmological constant that have a vanishing Bach tensor. Among these spacetimes we then look for those which are conformal to Einstein spaces. We find that the electromagnetic field and the Weyl tensor must be aligned, and in the case that the electromagnetic field is null, the spacetime must be conformally Ricci-flat and all such solutions are known. In the non-null case, since the general solution is not known on closed form, we settle with giving the integrability conditions in the general case, but we do give new explicit examples of Einstein-Maxwell spacetimes that are conformal to Einstein spaces, and we also find examples where the vanishing of the Bach tensor does not imply that the spacetime is conformal to a $C$-space. The non-aligned Einstein-Maxwell spacetimes with vanishing Bach tensor are conformally $C$-spaces, but none of them are conformal to Einstein spaces.Comment: 22 pages. Corrected equation (12