Gluing constructions for Lorentzian length spaces

Abstract

We introduce an analogue to the amalgamation of metric spaces into the setting of Lorentzian pre-length spaces. This provides a very general process of constructing new spaces out of old ones. The main application in this work is an analogue of the gluing theorem of Reshetnyak for CAT(k) spaces, which roughly states that gluing is compatible with upper curvature bounds. We formulate the theorem in terms of (strongly causal) spacetimes viewed as Lorentzian length spaces.

Causal completions as Lorentzian pre-length spaces

“…This result can be used to carry out explicit calculations regarding the pre-length structure of the future causal completion. 8 Finally, as an illustrative example we consider the particular case of de Sitter spacetime.…”

“…Their notion of causal space lays at the foundations of the theory of Lorentzian pre-length spaces first introduced by Kunzinger and Sämman [28]. The purpose of this work is to present the future (or past) causal completion of a globally hyperbolic spacetime as a Lorentzian pre-length space, thus adding an interesting source of examples to this rapidly growing field [21,11,8,9,6,22,29].…”

On the space of compact diamonds of Lorentzian length spaces

“…For instance, Toponogov type triangle comparison [6], or the existence of hyperbolic angles and exponential maps [10]. Applications of these results include gluing and amalgamation [9], as well as a splitting theorem that generalizes the landmark result for Lorentzian manifolds with non-positive timelike sectional curvature [11].…”

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

“…Moreover, one can relate Alexandrov curvature bounds of the fibers to timelike curvature bounds of the entire space and vice versa, and a first synthetic singularity theorem was established in this setting. Recently, the gluing of Lorentzian pre-length spaces was developed in [10] and applied to gluing of spacetimes, thereby establishing an analogue of a result by Reshetnyak for CAT(𝐾)-spaces, which roughly states that gluing is compatible with upper curvature bounds.…”