On a peculiar family of static, axisymmetric, vacuum solutions of the Einstein equations

Abstract

The Zipoy-Voorhees family of static, axisymmetric vacuum solutions orms an interesting family in that it contains the Schwarzschild black hole excepting which all other members have naked singularity. We ana- yze some properties of the region near singularity by studying a natural amily of 2-surfaces. We establish that these have the topology of the 2-sphere by an application of the Gauss-Bonnet theorem. By computing he area, we establish that the singular region is ‘point-like’. Isometric mbedding of these surfaces in the three dimensional Euclidean space is used to distinguish the two types of deviations from spherical symmetry.