THE WAVE EQUATIONS FOR THE WEYL TENSOR/SPINOR AND DIMENSIONALLY DEPENDENT TENSOR IDENTITIES

Abstract

By reconciling the wave equation for the Weyl tensor with the corresponding wave equation for the Weyl spinor, we establish a new tensor identity—involving the sum of terms each consisting of a product of the Weyl and Ricci tensors—valid in four (and only four) dimensions. This enables us to give, for the first time, the correct and simplest form of the wave equation for the Weyl tensor in four-dimensional nonvacuum spacetimes. The wave equation for the Weyl tensor in n(> 4) dimensional nonvacuum spaces is also presented for the first time; we show that there does not exist an analogous n-dimensional tensor identity matching the four-dimensional one, and so it follows that there does not exist an analogous simplification of the Weyl wave equation in the n-dimensional case. It is also shown how our new identity, and some other recently discovered identities, relate to a large class of dimensionally dependent identities found some time ago by Lovelock.