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.

Exact teleparallel gravity of binary black holes

“…The vanishing of the quantity between the brackets in (16) is just the Laplace equation ∇ 2 ν = 0 in cylindrical coordinates with an axis of symmetry about z. The field equation (16) gives the linear Newtonian gravity; then the field of two monopoles can be introduced as a superposition of the gravitational field of two singularities as…”

“…Also, the solution shows a peculiar singularity behaviour, whereas it exhibits a directional singularity allows the particles approaching the singularity along the axis of symmetry to be geodesically complete accessing to some new region of the space-time [14]. This leads some to conclude that the singularities in Curzon solution are actually rings rather point-like particles [15][16][17]. Interestingly, for the superposition of a ring and another body, a membrane-like singularity appears besides the strut one [18,19].…”

Visualizing spacetime curvature via gradient flows. II. An example of the construction of a Newtonian analogue

“…The "ring" has finite radius but infinite circumference [10]. Rather remarkably late was the computation of w2R in [13], a work which gave visual information on the CC metric based on the principal null directions. This procedure gives much less information than the visualization procedure considered here.…”

Slowly rotating curzon-chazy metric

“…Every light ray emitted from it becomes infinitely redshifted, so that it is effectively invisible [10]. Studying the principal null directions, it was found that this spacetime has an invariantly hypersurface ρ 2 + z 2 = M, on which the Weyl invariant J (determinant of the Weyl five complex scalar functions) vanishes [1,10]. Furthermore, this metric is Petrov type D, except at two points (z = ±M) that intersect the axis ρ = 0, where it is of Petrov type O [1].…”