Penrose junction conditions with $$\Lambda $$: geometric insights into low-regularity metrics for impulsive gravitational 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 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 $${\Lambda \not =0}$$ Λ ≠ 0 . Here we review the original Penrose construction and its generalization to non-vanishing $$\Lambda $$ Λ 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.

Cut-and-paste for impulsive gravitational waves with $$\Lambda $$: the mathematical analysis

“…[39, Fig. 2], and [40,Fig. 6]) in a way that directly generalises Penrose's original illustration for the = 0-case in [31, Fig.…”

On the space of compact diamonds of Lorentzian length spaces

“…The seminal work of Kunzinger and Sämann on Lorentzian length spaces [28] has paved the way to introduce such methods in the realm of Relativity, in which other promising approaches are currently being developed, as is the case of Lorentzian metric spaces [36] or almost Lorentzian length spaces [38] just to mention a few. These developments are suitable candidates to provide useful tools for the analysis of physical scenarios where the standard regular (at least C 2 ) framework is no longer assumed, either from the theoretical onset-for instance, when considering C 1,1 [19,21,30,31], C 1 [20,27] or even C 0 [15,17,18,22,44] metrics-or in the interest of further our understanding of the most recent observation of black hole merging and gravitational waves [16,32], in light of the theory of impulsive waves [42,43].…”

Gluing of Lorentzian length spaces and the causal ladder

“…Recently, the original construction by Penrose has also been studied in other constant curvature spacetimes, see e.g. [29,30]. Finally, matching also finds applications in alternative theories of gravity, cf [31].…”