Ralph Howard
University of South Carolina

Regularity of Horizons in Spacetimes and of Riemannian Distance Functions

Horizons are one of the more important objects in the causality theory of spacetimes: Cauchy horizons are the boundaries where predictability breaks down and event horizons are the boundaries beyond which no return is possible. Much of the classical work in cosmology, such as Hawking's proof of the area theory (sometimes called the second law of black hole thermodynamics), assumes that horizons are piecewise C^2. But, starting with any closed set in a Riemannian manifold, let f be the Riemannian distance from the set. Then the part of the graph where f is nonzero is an horizon in a product spacetime. Therefore the regularity of horizons is no better than the regularity of distance functions which can be non-differentiable on dense sets. We give a regularity theory for horizons, based on Alexandrov's definition of the second derivatives of a convex function, that is sufficient for applications to cosmology and which also leads to new results for Riemannian distance functions and cut loci.