# 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.