The Differentiability of Horizons Along Their Generators

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( tright) ,~tin 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( tright) ,~tin left( alpha ,t_{0}right) but not of class C^{1} and H is exactly of class C^{1} at every point gamma left( tright) ,~tin 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. Chruściel locally coincides with a Cauchy horizon.