Penrose junction conditions with $\Lambda$: Geometric insights into low-regularity metrics for impulsive gravitational waves

TL;DR

This paper reviews Penrose's geometric construction of impulsive gravitational waves, extending it to include a cosmological constant, and introduces a new visualization method using global null geodesics for better understanding low-regularity metrics.

Contribution

It provides a pedagogical review of Penrose's junction conditions with a cosmological constant and introduces a novel visualization technique involving global null geodesics.

Findings

Generalization of Penrose's construction to non-zero $$ in (anti-)de Sitter space.

Development of a visualization using global null geodesics.

Connection between distributional and continuous metric forms.

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 nonexpanding waves which later have been generalized to impulses traveling in all constant-curvature backgrounds, i.e., 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 recently a comparably powerful visualization and understanding has been missing in the case with a cosmological constant $\Lambda\not=0$. Here we review the original Penrose construction and its generalization to non-vanishing $\Lambda$ in a pedagogical way, as well as the recently established visualization: A special family of global null geodesics defines an appropriate comoving coordinate system that allows to relate the distributional to the continuous form of the metric.