``Nowhere'' differentiable horizons

TL;DR

This paper demonstrates that horizons in general relativity can be non-differentiable on entire regions, challenging the common belief that they are generally smooth hypersurfaces.

Contribution

It constructs explicit examples of Cauchy and black hole event horizons with no differentiable open subsets, showing horizons can be highly irregular.

Findings

Cauchy horizons can lack differentiability on open sets.

Black hole event horizons can also be non-differentiable everywhere.

Challenges the folklore assumption of smoothness of horizons.

Abstract

It is folklore knowledge amongst general relativists that horizons are well behaved, continuously differentiable hypersurfaces except perhaps on a negligible subset one needs not to bother with. We show that this is not the case, by constructing a Cauchy horizon, as well as a black hole event horizon, which contain no open subset on which they are differentiable.