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 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 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 Trümper 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.

By analogy with the Maxwell tensor of an elemomagnetic field. in general relativity the Riemann and Weyl tensors of spacetime meaics can be demmposed into 'electric' and 'magnetic' pans. Purely electric and purely magnetic Weyl tensors are analogous in term of their invariant classification and the geomeby of &e associated principal null directions. It b shown here, however, thal there are significant mathemalid differences bemeen purely elechic and purely magnetic Riemann tensors. In @cular, it is shown hat, unlike the elenric case, for non-vacuum mtrics, the existence of a purely magnetic Weyl tensor is not a necessary wndition for the existence of a purely magnetic Riemann tensor.

In the Petrov classification (1969) of the Weyl tensor (or spinor), the type I or (1111) case is referred to in the literature as non-degenerate; there is, however, a 'degenerate' class of type I metrics in which the four distinct principal null directions (PND) only span a 3-space at each point; the degeneracy refers to the dimension of the space of PND. This subcase is shown to exist when I3/J2)6. Metrics of the Kasner type provide an important example of the two type I cases, and an illustration of the kind of geometrical insight into the structure of spacetime metrics which is afforded by analysis of the space of PND.

A Weyl spinor of Petrov type I has four different principal null directions (PND) at any point-these can be represented by four points on S+, the sphere of intersection of a spacelike plane 'T=1' with the cone of null directions at that point. Penrose and Rindler (1986) have shown that a frame can be chosen so that these four points form the vertices of a disphenoid (a tetrahedron with opposite edges equal in pairs). This disphenoid has degenerate properties when the Weyl spinor (and its invariants I and J) satisfy certain conditions which are given mainly in terms of restrictions on eigenvalues. However, this approach is not well suited to algebraic computing. This presents a simple computer-algebra-adapted method, based on an invariant, M, formed from the Weyl spinor, for finding and analysing degenerate cases; this is an extension of an earlier work by McIntosh and Arianrhod (1990). It also extends results of Penrose and Rindler concerning the relationship between these degenerate cases and the corresponding eigenvalues of the matrix Psi of Weyl spinor components. In addition, it examines some relationships between these degeneracies and the structure of some exact spacetime metrics, in particular vacuum metrics of the Kasner type. The results of this work have been used elsewhere by the authors and Fletcher to investigate the geometrical structure of the Curzon metric by studying the nature of its PND; they are also being used to study Segre types of the Plebanski spinor formed from the trace-free Ricci spinor.

The Curzon metric has an invariantly defined surface, given by the vanishing of the cubic invariant of the Weyl tensor. On a spacelike slice, this surface has the topology of a 2-sphere, and surrounds the singularity of the metric. In Weyl coordinates, this surface is defined by R=m where R= square root ( rho 2+z2). The two points where the z-axis, the axis of symmetry, cuts this surface, z=m and -m, have the interesting property that, at both of them, the Riemann tensor vanishes. The Weyl tensor is of Petrov type D at all non-singular points of the z-axis, rho =0, except at the two points where it is zero. Off the axis, the Weyl tensor is of type I(M+), and the metric asymptotically tends to flat spacetime away from the source. The principal null directions of the Weyl tensor are shown to be everywhere independent of the angular basis vector delta / delta phi , and their projections into a t=constant, phi =constant plane are presented graphically.