MAGNETIC CURVATURES

By analogy with the Maxwell tensor of an electromagnetic field, in general relativity the Riemann and Weyl tensors of spacetime metrics can be decomposed into 'electric' and 'magnetic' parts. Purely electric and purely magnetic Weyl tensors are analogous in terms of their invariant classification and the geometry of the associated principal null directions. It is shown here, however, that there are significant mathematical differences between purely electric and purely magnetic Riemann tensors. In particular, it is shown that, unlike the electric case, for non-vacuum metrics, the existence of a purely magnetic Weyl tensor is not a necessary condition for the existence of a purely magnetic Riemann tensor.