Cut-and-paste for impulsive gravitational waves with Λ: the mathematical analysis.

Impulsive gravitational waves are theoretical models of short but violent bursts of gravitational radiation. They are commonly described by two distinct spacetime metrics, one of local Lipschitz regularity and the other one even distributional. These two metrics are thought to be 'physically equivalent' since they can be formally related by a 'discontinuous coordinate transformation'. In this paper we provide a mathematical analysis of this issue for the entire class of nonexpanding impulsive gravitational waves propagating in a background spacetime of constant curvature. We devise a natural geometric regularisation procedure to show that the notorious change of variables arises as the distributional limit of a family of smooth coordinate transformations. In other words, we establish that both spacetimes arise as distributional limits of a smooth sandwich wave taken in different coordinate systems which are diffeomorphically related.

Null Distance and Convergence of Lorentzian Length Spaces.

The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov-Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.

A note on the Gannon-Lee theorem.

We prove a Gannon-Lee theorem for non-globally hyperbolic Lorentzian metrics of regularity C 1 , the most general regularity class currently available in the context of the classical singularity theorems. Along the way, we also prove that any maximizing causal curve in a C 1 -spacetime is a geodesic and hence of C 2 -regularity.

The future is not always open.

We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open and may differ from the corresponding sets defined via piecewise C 1 -curves. By refining the notion of a causal bubble from Chrusciel and Grant (Class Quantum Gravity 29(14):145001, 2012), we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (push-up). The phenomena described here are, in particular, relevant for recent synthetic approaches to low-regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.

The Singularity Theorems of General Relativity and Their Low Regularity Extensions.

On the occasion of Sir Roger Penrose's 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to explore the nature of the singularities predicted by the classical theorems. Aiming at the more mathematically minded reader, we give a pedagogical introduction to the classical theorems with an emphasis on the analytical side of the arguments. We especially concentrate on focusing results for causal geodesics under appropriate geometric and initial conditions, in the smooth and in the low regularity case. The latter comprise the main technical advance that leads to the proofs of C1-singularity theorems via a regularisation approach that allows to deal with the distributional curvature. We close with an overview on related lines of research and a future outlook.

Penrose junction conditions with Λ : geometric insights into low-regularity metrics for impulsive gravitational waves.

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 Λ ≠ 0 . Here we review the original Penrose construction and its generalization to non-vanishing 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.