The Differentiability of Horizons Along Their Generators

Abstract

AbstractLet H be a (past directed) horizon in a time-oriented Lorentz manifold and $$\gamma :[\left( \alpha ,\beta \right) \rightarrow H$$ γ : [ α , β → H a past directed generator of the horizon, where $$[\left( \alpha ,\beta \right) $$ [ α , β is $$[\alpha ,\beta )$$ [ α , β ) or $$\left( \alpha ,\beta \right) $$ α , β . It is proved that either at every point of $$\gamma \left( t\right) ,~t\in \left( \alpha ,\beta \right) $$ γ t , t ∈ α , β the differentiability order of H is the same, or there is a so-called differentiability jumping point $$\gamma \left( t_{0}\right) ,~t_{0}\in \left( \alpha ,\beta \right) $$ γ t 0 , t 0 ∈ α , β such that H is only differentiable at every point $$\gamma \left( t\right) ,~t\in \left( \alpha ,t_{0}\right) $$ γ t , t ∈ α , t 0 but not of class $$C^{1}$$ C 1 and H is exactly of class $$C^{1}$$ C 1 at every point $$\gamma \left( t\right) ,~t\in \left( t_{0},\beta \right) $$ γ t , t ∈ 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.