Geometric description of lightlike foliations by an observer in general relativity

Details Geometric description of lightlike foliations by an observer in general relativity Geometric description of lightlike foliations by an observer in general relativity

2005-01-31

arxiv.org/abs/gr-qc/0501100v1

Math.Proc.Cambridge Phil.Soc. 139 (2005) 181-192

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

14 pages, 4 figures

Scientific paper

10.1017/S0305004105008571

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 $\Omega_U^-$.

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