4 research outputs found
A local potential for the Weyl tensor in all dimensions
In all dimensions and arbitrary signature, we demonstrate the existence of a
new local potential -- a double (2,3)-form -- for the Weyl curvature tensor,
and more generally for all tensors with the symmetry properties of the Weyl
curvature tensor. The classical four-dimensional Lanczos potential for a Weyl
tensor -- a double (2,1)-form -- is proven to be a particular case of the new
potential: its double dual.Comment: 7 pages; Late
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 -space.
The non-aligned Einstein-Maxwell spacetimes with vanishing Bach tensor are
conformally -spaces, but none of them are conformal to Einstein spaces.Comment: 22 pages. Corrected equation (12
Dimensionally Dependent Tensor Identities by Double Antisymmetrisation
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.Comment: 17 pages, 2 figure