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 $\Omega$ 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 $\Omega$ for this observer. A new distribution ${\Omega }_U^{-}$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $\Omega$ and ${\Omega }_U^{-}$ are integrable, the pair Ω,Ω_U^- $representsthewavefrontsofastationarywavefortheobserver$U$.However,weshowinanexamplethattheintegrabilityof$Ω$doesnotimplytheintegrabilityof$Ω_U^-$.

14 pages, 4 figures