S. Brian Edgar

Dimensionally dependent tensor identities by double antisymmetrization

Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in n-dimensional space of a pair of fundamental identities involving trace-free (p,p)-forms where 2p⩾n. We generalize Lovelock’s results, and by using the fact that associated with any tensor in n-dimensional space there is associated a fundamental tensor identity obtained by antisymmetrizing over n+1 indices, we establish a very general “master” identity for all trace-free (k,l)-forms. We then show how various other special identities are direct and simple consequences of this master identity; in particular we give direct application to Maxwell, Lanczos, Ricci, Bel, and Bel-Robinson tensors, and also demonstrate how relationships between scalar invariants of the Riemann tensor can be investigated in a systematic manner.

Killing tensors and conformal Killing tensors from conformal Killing vectors

Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition, we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate that it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors, and hence all the Killing tensors (which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors.

All conformally flat pure radiation metrics

The complete class of conformally flat, pure radiation metrics is given, generalizing the metric given recently by Wils.

Integration in the GHP Formalism III: Finding Conformally Flat Radiation Metrics as an Example of an “Optimal Situation”

Held has proposed an integration procedure within the GHP formalism built around four real, functionally independent, zero-weighted scalars. He suggests that such a procedure would be particularly simple for the “optimal situation”, when the formalism directly supplies the full quota of four scalars of this type; a spacetime without any Killing vectors would be such a situation. Wils has recently obtained a conformally flat, pure radiation metric, which has been shown by Koutras to admit no Killing vectors, in general. In order to present a simple illustration of the ghp integration procedure, we obtain systematically the complete class of conformally flat, pure radiation metrics, which are not plane waves. Our result shows that the conformally flat, pure radiation metrics are a larger class than Wils has obtained.

Nonexistence of the Lanczos potential for the Riemann tensor in higher dimensions

The existing refutal, in four-dimensional spacetimes, of the conjecture that the Lanczos tensor can be used as a potential for the Riemann tensor, is derived in a much simpler manner which is valid for dimension n ≥ 4 and any signature.

Curvature-free asymmetric metric connections and Lanczos potentials in Kerr–Schild spacetimes

The set of spinor equations linking the curvature spinors of a Riemann–Cartan space with the curvature spinors of the corresponding Riemann space is derived, and these are used to establish a simple relationship between Ψ-flat connections of the Riemann–Cartan space and Lanczos potentials of the Riemann space. This not only yields, very easily, a recent result of Bergqvist for the Kerr metric, but also enables Bergqvist’s result to be generalized; specifically we show that a curvature-free connection, associated with a class of Kerr–Schild metrics, can be identified as a Lanczos potential for the Weyl conformal curvature spinor of these spaces.

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 TabcxTabcy=δxy TabcdTabcd/4 and also the new conservation law for the Chevreton tensor, both of which have been obtained by spinor means; both of these results are rederived by tensor means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and, moreover, we are able to conclude that there are no direct higher dimensional analogs. In addition we demonstrate a simple way to show the nonexistense of such identities via counter examples; in particular we show that there is no nontrivial Bel ten...

The Lanczos potential for Weyl-candidate tensors exists only in four dimensions

We prove that a Lanczos potential Labc for the Weyl candidate tensor Wabcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations; if the integrability conditions yield another non-trivial differential system for Labc and Wabcd, then this systems integrability conditions should be checked, and so on. When we find a non-trivial condition involving onlyWabcdand its derivatives, then clearly Weyl candidate tensors failing to satisfy that condition cannot be written in terms of a Lanczos potential Labc.

Conditions for a symmetric connection to be a metric connection

A recent result−in ‘general circumstances’ for a four‐dimensional space–time−giving algebraic conditions on a curvature tensor (of a symmetric connection) so that the connection be metric, is shown to be a special case of a more general result; both of these results are shown formally to be of a generic nature. In this new result the conditions are imposed on a tensor of a more general character than the curvature tensor. In addition it is shown that once the symmetric connection is known to be metric, the metric is uniquely defined (up to a constant conformal factor). For those special curvature tensors which are excluded from the original result, supplementary conditions are suggested, which, alongside the original conditions are sufficient to ensure that most of these excluded curvature tensors are also Riemann tensors.

Integration in the GHP Formalism II: An Operator Approach for Spacetimes with Killing Vectors, with Applications to Twisting Type N Spaces

Held has proposed a coordinate- and gauge-free integration procedure within the ghp formalism built around four functionally independent zero-weighted scalars constructed from the spin coefficients and the Riemann tensor components. Unfortunately, a spacetime with Killing vectors (and hence cyclic coordinates in the metric, and in all quantities constructed from the metric) may be unable to supply the full quota of four scalars of this type. However, for such a spacetime additional scalars may be supplied by the components of the Killing vectors. As an illustration we investigate the vacuum type N spaces admitting a Killing vector and a homothetic Killing vector. In a direct manner, we reduce the problem to a pair of ordinary differential operator “master equations”, making use of a new zero-weighted ghp operator. In two different ways, we show how these master equations can be reduced to one real third-order operator differential equation for a complex function of a real variable—but still with the freedom to choose explicitly our fourth coordinate. It is then easy to see there is a whole class of coordinate choices where the problem reduces essentially to one real third-order differential equation for a real function of a real variable. It is also outlined how the various other differential equations, which have been derived previously in work on this problem, can be deduced from our master equations.