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. Due to the absence of a notion of spacelike distance in Lorentzian pre-length spaces we can only formulate the theorem in terms of (strongly causal) spacetimes viewed as Lorentzian length spaces.

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

“…Note that $$X$$ was not intrinsic as well. If we intrinsify $$X$$ (see [8, Theorem 1.7.3]), that is, considering $$X$$ with the new time separation function $$\hat{\tau }(p,q):=\sup \lbrace L_\tau (\gamma ): \gamma$$ future‐directed causal from $$p$$ to $$q\rbrace \cup \lbrace 0\rbrace$$, the lengths of geodesics stay the same, but angles will change. We need the other geodesics as well: For $$p,q$$ not both on the cone such that the straight line connection is not contained in $$X$$, the connecting geodesic will be straight outside the cone, will be tangential to the cone where it touches it, and at such points it might switch over to have a part contained in the cone (which is a geodesic in the part of the cone which can be considered as a $$1+1$$‐dimensional Lorentzian manifold).…”

“…Furthermore, as pioneered by Kronheimer and Penrose [30] and Busemann [16], causality theory and (parts of) Lorentzian geometry can be studied decoupled from the manifold structure [32, Subsection 3.5], [1, 15]. A large class of Lorentzian length spaces have been investigated in [3], where warped products of a one‐dimensional base with metric length space fibers are considered.…”

“…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.…”

“…For $$p,q\in \Omega _x$$ we set $$\omega _x(p,q):=\sup \lbrace L_\tau (\gamma ):\, \gamma :[a,b]\rightarrow \Omega _x$$ is $$\le$$‐causal from $$p$$ to $$q\rbrace \cup \lbrace 0\rbrace$$. For $$p<q$$ we require that there is an $$\omega _x$$‐realizer $$\gamma _{p,q}$$ in $$\Omega _x$$ from $$p$$ to $$q$$, and that this makes $$\Omega _x$$ into a Lorentzian pre‐length space; see, for example, [8, Chapter 1.7.2] or [32, Definition 3.16]. (iii)For every $$y\in \Omega _x$$ we have $$I^\pm (y)\cap \Omega _x\ne \emptyset$$. If, in addition, the neighborho...…”

“…[32, Subsection 4.5]. Remark If $$X$$ is strongly causal and has timelike curvature bounded above/below, we can restrict ourselves to comparison neighborhoods $$U$$ where timelike size bounds are automatically satisfied and of the form of timelike diamonds, and there exists a basis of the topology of such neighborhoods; see [8, Remark 2.1.8].…”