Principal null directions of the Curzon metric

Abstract

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.