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Nonanalytic extensions of the extreme Reissner–Nordström metric in terms of weak solutions

A basic extension of the exterior part of the extreme Reissner–Nordstrom 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 an infinitesimally thin closed string-like 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 no longer simply connected, being 'pierced' by the strings.