Congruence Convergence in pp-wave Spacetime

TL;DR

This paper explores how the geodesic completeness of pp-wave spacetimes is affected when considering extended congruences, highlighting issues with non-geodesic congruences and analyzing null expansion via a generalized Raychaudhuri equation.

Contribution

It introduces a generalized Raychaudhuri equation to analyze congruence convergence in pp-wave spacetimes, challenging the traditional view of geodesic completeness.

Findings

Geodesic completeness can be disregarded for extended congruences.

Non-geodesic congruences exhibit diverse convergence issues.

A generalized Raychaudhuri equation is formulated for null congruences.

Abstract

We argue that the well-known geodesic completeness property of pp-waves, can be disregarded once the geodesics are extracted as being extended along sets of Brinkmann coordinates. This issue is investigated in the more general context of congruence convergence and we show that the problem leads to diverse issues for non-geodesic congruences. The discussion is mostly based on the null congruence expansion and a generalized Raychaudhuri equation is also provided.