Nonanalytic extensions of the extreme Reissner-Nordstroem metric in terms of weak solutions

TL;DR

This paper constructs a weakly extended, non-smooth version of the extreme Reissner-Nordstroem black hole metric, involving a topological string that breaks spherical symmetry and affects the horizon structure.

Contribution

It introduces a novel weak solution extension of the extreme Reissner-Nordstroem metric incorporating a topological string, breaking spherical symmetry and satisfying Einstein-Maxwell equations in a weak sense.

Findings

Extension is continuous but not smooth at the horizon.

The manifold is topologically incomplete and pierced by a string.

Spherical symmetry is globally broken to axial symmetry.

Abstract

A basic extension of the exterior part of the extreme Reissner-Nordstroem solution in terms of a continuous metric and gauge potential is constructed. This extension is not smooth at the null hypersurface given by the Cauchy-Killing horizon which separates isometric copies of the exterior metric. The Maxwell-Einstein system of equations is satisfied only in a weak sense. The manifold is topologically incomplete and the spherical symmetry is globally broken down to an axial symmetry. This behaviour can be attributed to the effect of a 'topological string', in the sense of a infinitesimally thin closed stringlike object 'sitting on the rim' of the black hole and holding it open by means of an accompanying impulsive gravitational wave. The resulting differentiable manifold and the corresponding horizons are not anymore simply connected, being 'pierced' by the strings.