Global structure of the Zipoy–Voorhees–Weyl spacetime and the δ = 2 Tomimatsu–Sato spacetime

Abstract

We investigate the structure of the ZVW (Zipoy–Voorhees–Weyl) spacetime, which is a Weyl solution described by the Zipoy–Voorhees metric, and the δ = 2 Tomimatsu–Sato spacetime. We show that the singularity of the ZVW spacetime, which is represented by a segment ρ = 0, − σ < z < σ in the Weyl coordinates, is geometrically point-like for δ < 0, string-like for 0 < δ < 1 and ring-like for δ > 1. These singularities are always naked and have positive Komar masses for δ > 0. Thus, they provide a non-trivial example of naked singularities with positive mass. We further show that the ZVW spacetime has a degenerate Killing horizon with a ring singularity at the equatorial plane for δ = 2, 3 and δ ⩾ 4. We also show that the δ = 2 Tomimatsu–Sato spacetime has a degenerate horizon with two components, in contrast to the general belief that the Tomimatsu–Sato solutions with even δ do not have horizons.