Cut-and-paste for impulsive gravitational waves with Λ: The geometric picture

Impulsive gravitational waves in Minkowski space were introduced by Penrose at the end of the 1960s and have been widely studied since then. Here we focus on nonexpanding waves which were later generalized to impulses traveling in all constant-curvature backgrounds, which 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 visualization and understanding have been missing in the Lambda not equal 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 reattached 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. This finding also shed light on recent discussions of the memory effect in impulsive waves.