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 generalise 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 antisymmetrising 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. * Electronic address: bredg@mai.liu.se † Electronic address: anhog@mai.liu.se

In the last few years renewed interest in the 3-tensor potential L abc proposed by Lanczos for the Weyl curvature tensor has not only clarified and corrected Lanczos's original work, but generalized the concept in a number of ways. In this paper we first of all carefully summarize and extend some aspects of these results, make some minor corrections, and clarify some misunderstandings in the literature.The following new results are also presented. The (computer checked) complicated second-order partial differential equation for the 3-potential, in arbitrary gauge, for Weyl candidates satisfying Bianchi-type equations is given-in those n-dimensional spaces (with arbitrary signature) for which the potential exists; this is easily specialized to Lanczos potentials for the Weyl curvature tensor. It is found that it is only in four-dimensional spaces (with arbitrary signature) that the nonlinear terms disappear and that certain awkward second-order derivative terms cancel; for fourdimensional spacetimes (with Lorentz signature), this remarkably simple form was originally found by Illge, using spinor methods. It is also shown that, for most fourdimensional vacuum spacetimes, any 3-potential in the Lanczos gauges which satisfies a simple homogeneous wave equation must be a Lanczos potential for the non-zero Weyl curvature tensor of the background vacuum spacetime. This result is used to prove that the form of a possible Lanczos potential recently proposed by Dolan & Kim for a class of vacuum spacetimes is in fact a genuine Lanczos potential for these spacetimes.

We prove that a Lanczos potential L abc for the Weyl candidate tensor W abcd 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 L abc and W abcd , then this system's integrability conditions should be checked; and so on. When we find a non-trivial condition involving only W abcd and its derivatives, then clearly Weyl candidate tensors failing to satisfy that condition cannot be written in terms of a Lanczos potential L abc .

The classical Rainich(-Misner-Wheeler) theory gives necessary and sufficient conditions on an energy-momentum tensor T to be that of a Maxwell field (a 2-form) in four dimensions. Via Einstein's equations these conditions can be expressed in terms of the Ricci tensor, thus providing conditions on a spacetime geometry for it to be an Einstein-Maxwell spacetime. One of the conditions is that T 2 is proportional to the metric, and it has previously been shown in arbitrary dimension that any tensor satisfying this condition is a superenergy tensor of a simple p-form. Here we examine algebraic Rainich conditions for general p-forms in higher dimensions and their relations to identities by antisymmetrisation. Using antisymmetrisation techniques we find new identities for superenergy tensors of these general (non-simple) forms, and we also prove in some cases the converse; that the identities are sufficient to determine the form. As an example we obtain the complete generalisation of the classical Rainich theory to five dimensions.

In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n ≥ 7.