Removing naked singularities in static axially symmetric spacetimes by patching with the flat spacetimes

TL;DR

This paper constructs non-singular, static, axially symmetric spacetimes by patching Weyl solutions with flat spacetime, analyzing the energy conditions and shell parameters to ensure physical viability.

Contribution

It provides explicit models of non-singular, non-spherically symmetric spacetimes satisfying energy conditions through a novel patching method.

Findings

Lower bounds on shell size for non-singular spacetimes.

Energy conditions are satisfied for all shell sizes.

Dominant energy condition holds for sufficiently large shells.

Abstract

We investigate static, axially symmetric spacetimes without naked singularities that are constructed by patching Weyl class spacetimes with the flat spacetimes. Once the exterior geometry is specified, the junction conditions determine the shape of a thin shell, which is the boundary between the two patched spacetimes, and the distribution of the energy and pressure on this shell, leaving a parameter representing the shell size free. We examine the cases where the exterior of the shell is given by Curzon--Chazy or Zipoy--Voorhees spacetimes. For each case, we find a lower bound on the shell size. Additionally, we find that the weak and null energy conditions are satisfied for any shell size, while the dominant energy condition is satisfied for sufficiently large shells. These results provide concrete examples of non-singular spacetimes with non-spherically symmetric exteriors that respect the energy conditions.