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

We investigate the structure of the ZVW (Zipoy-Voorhees-Weyl) spacetime, which is a Weyl solution described by the Zipoy-Voorhees metric, and the delta=2 Tomimatsu-Sato spacetime. We show that the singularity of the ZVW spacetime, which is represented by a segment rho=0, -sigma<z<sigma in the Weyl coordinates, is geometrically point-like for delta<0, string-like for 0<delta<1 and ring-like for delta>1. These singularities are always naked and have positive Komar masses for delta>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 delta=2,3 and delta>=4. We also show that the delta=2 Tomimatsu-Sato spacetime has a degenerate horizon with two components, in contrast to the general belief that the Tomimatsu-Sato solutions with even delta do not have horizons.

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