The differentiablity of horizons along their generators

TL;DR

This paper investigates the differentiability properties of horizons in Lorentzian manifolds, showing that along their generators the differentiability is either uniform or exhibits a single jump, with implications for the structure of such horizons.

Contribution

It proves a dichotomy in the differentiability of horizons along generators and relates horizons to Cauchy horizons in Lorentzian geometry.

Findings

Differentiability along generators is either uniform or has a single jump.

Existence of a differentiability jump point where the horizon's regularity changes.

Mathematical horizons locally coincide with Cauchy horizons.

Abstract

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