# Geodesic congruences in exact plane wave spacetimes and the memory effect

## TL;DR

This paper studies the evolution of geodesic congruences in exact plane gravitational wave spacetimes, revealing how shear growth causes focusing and correlates with pulse amplitude, thus providing insights into the gravitational memory effect.

## Contribution

It derives analytical expressions for shear and expansion of geodesic congruences in exact plane wave spacetimes using specific pulse profiles, highlighting the connection to the memory effect.

## Key findings

- Shear growth leads to focusing of geodesic congruences.
- Focusing time correlates with pulse amplitude.
- Qualitative results are consistent across different pulse derivatives.

## Abstract

Displacement and velocity memory effects in the exact, vacuum, plane gravitational wave line element have been studied recently by looking at the behaviour of pairs of geodesics or via geodesic deviation. Instead, one may investigate the evolution of geodesic congruences. In our work here, we obtain the evolution of the kinematic variables which characterise timelike geodesic congruences, using chosen pulse profiles (square and sech-squared) in the exact, plane gravitational wave line element. We also analyse the behaviour of geodesic congruences in possible physical scenarios describable using derivatives (first, second and third) of one of the chosen pulses. Beginning with a discussion on the generic behaviour of such congruences and consequences thereof, we find exact analytical expressions for shear and expansion with the two chosen pulse profiles. Qualitatively similar numerical results are noted when various derivatives of the sech-squared pulse are used. We conclude that for geodesic congruences, a growth (or decay) of shear causes focusing of an initially parallel congruence, after the departure of the pulse. A correlation between the `focusing time (or $u$ value, $u$ being the affine parameter)' and the amplitude of the pulse (or its derivatives) is found. Such features distinctly suggest a memory effect, named in recent literature as ${\cal B}$ memory.