On the Non-Uniqueness of Continuous Extensions Beyond Event Horizons

We investigate the continuation of spacetimes beyond the null boundary of the maximal globally hyperbolic development arising in gravitational collapse. Using the characteristic initial value formulation of the Einstein equations, we show that
fixing the intrinsic geometry of a smooth null hypersurface does not, in general, determine a unique future extension once C^1 regularity across the hypersurface is relaxed. In particular, allowing the spacetime metric to be merely continuous (C^0) permits inequivalent future extensions solving the Einstein equations in the weak (distributional) sense.

In spherical symmetry, we construct explicit examples of such extensions, including non-focusing interior developments in which curvature invariants remain bounded in a neighbourhood to the future of the null hypersurface. We further show that any asymptotically flat exterior completion compatible with these regular interior developments must be non-vacuum. The analysis is extended to axisymmetric spacetimes, where we identify the corresponding freedom in transversal null data on rotating horizons.

These results demonstrate that the standard singular black-hole interior corresponds to a particular admissible continuation beyond the null boundary of the maximal globally hyperbolic development, rather than a uniquely determined geometric outcome of the Einstein equations. All statements are local in nature and do not address global evolution or dynamical selection.