Geometric description of lightlike foliations by an observer in general relativity

Abstract

We introduce new concepts and properties of lightlike distributions and foliations (of dimension and co-dimension 1) in a space-time manifold of dimension $n$, from a purely geometric point of view. Given an observer and a lightlike distribution $Ω$ of dimension or co-dimension 1, its lightlike direction is broken down into two vector fields: a timelike vector field $U$ representing the observer and a spacelike vector field $S$ representing the relative direction of propagation of $Ω$ for this observer. A new distribution $Ω_U^-$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $Ω$ and $Ω_U^-$ are integrable, the pair Ω,Ω_U^- $ represents the wave fronts of a stationary wave for the observer $U$. However, we show in an example that the integrability of $Ω$ does not imply the integrability of $Ω_U^-$.