The differentiablity of horizons along their generators

David Szeghy

paper

The differentiablity of horizons along their generators

arXiv:2109.11793

Abstract

Let $H$ be a (past directed) horizon in a time-oriented Lorentz manifold and $γ : [(α, β) \to H$ a past directed generator of the horizon, where $[(α, β)$ is $[α, β)$ or $(α, β)$ . It is proved that either at every point of $γ (t), t \in (α, β)$ the differentiability order of $H$ is the same, or there is a so-called differentiability jumping point $γ (t_{0}), t_{0} \in (α, β)$ such that $H$ is only differentiable at every point $γ (t), t \in (α, t_{0})$ but not of class $C^{1}$ and $H$ is exactly of class $C^{1}$ at every point $γ (t), t \in (t_{0}, β)$ . We will use in the proof a result which shows, that every mathematical horizon in the sense of P. T. Chruściel locally coincides with a Cauchy horizon.