# Cut-and-paste for impulsive gravitational waves with $\Lambda$: The geometric picture

## TL;DR

This paper extends Penrose's geometric 'cut-and-paste' method to impulsive gravitational waves in (anti-)de Sitter spaces, providing a visualisation that clarifies their structure, junction conditions, and implications for the memory effect.

## Contribution

It introduces a new geometric picture for impulsive waves in curved backgrounds, generalising Penrose's approach to non-zero cosmological constant scenarios.

## Key findings

- Provides a visualisation of impulsive waves in (anti-)de Sitter space
- Clarifies Penrose junction conditions in curved backgrounds
- Enhances understanding of the memory effect in impulsive waves

## Abstract

Impulsive gravitational waves in Minkowski space were introduced by Roger Penrose at the end of the 1960s, and have been widely studied over the decades. Here we focus on non-expanding waves which later have been generalised to impulses travelling in all constant-curvature backgrounds, that is also the (anti-)de Sitter universe. While Penrose's original construction was based on his vivid geometric `scissors-and-paste' approach in a flat background, until now a comparably powerful visualisation and understanding have been missing in the ${\Lambda\not=0}$ case. In this work we provide such a picture: The (anti-)de Sitter hyperboloid is cut along the null wave surface, and the `halves' are then re-attached with a suitable shift of their null generators across the wave surface. This special family of global null geodesics defines an appropriate comoving coordinate system, leading to the continuous form of the metric. Moreover, it provides a complete understanding of the nature of the Penrose junction conditions and their specific form. These findings shed light on recent discussions of the memory effect in impulsive waves.