Geometric description of lightlike foliations by an observer in general relativity

TL;DR

This paper introduces a geometric framework for analyzing lightlike distributions and foliations in space-time, focusing on how an observer perceives wave fronts and their relative directions.

Contribution

It defines new concepts of lightlike distributions relative to an observer and explores their properties, including the conditions for integrability and the behavior of opposite propagation directions.

Findings

A new distribution $ ext{Omega}_U^-$ is introduced to represent opposite propagation directions.

The paper shows that integrability of a lightlike distribution does not imply integrability of its opposite.

Conditions for wave fronts of stationary waves are characterized in terms of integrability.

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 \Omega ,\Omega_U^- $ represents the wave fronts of a stationary wave for the observer $U$. However, we show in an example that the integrability of $\Omega $ does not imply the integrability of $\Omega_U^-$.