ELECTRIC AND MAGNETIC WEYL TENSORS - CLASSIFICATION AND ANALYSIS

A Weyl tensor of Petrov type I can be decomposed into two parts, an electric and a magnetic part, by any observer with 4-velocity vector u. It is shown here that when a metric is such that there exists an observer who sees the metric's Weyl tensor as purely electric or purely magnetic, then the Weyl tensor is of Petrov type I(M+) in the Arianrhod-McIntosh classification (and thus its four principal null directions are linearly dependent). It is also shown that an observer exists for whom the Weyl tensor is either purely electric or magnetic if and only if the Weyl tensor is of Petrov type I(M+) and the invariant I of the Weyl tensor is real. The magnetic and electric cases are distinguished by the sign of I. In the electric and magnetic cases, the spanning vectors of the principal null directions at each point are u and two other vectors picked out by the geometry; this combines and simplifies results of Trumper and Narain. The results here are formulated in terms of invariants, and are thus easily amenable to computer classification of metrics. Spacetime examples are discussed, and new theoretical results for the Petrov type D subcase are presented.