[Paper Review] Gluing constructions for Lorentzian length spaces

Tobias Beran, Felix Rott|PubMed|Jan 24, 2022

Geometric Analysis and Curvature Flows17 references4 citations

TL;DR

This paper introduces a gluing construction for Lorentzian length spaces, extending metric amalgamation to the Lorentzian setting by carefully defining a quotient time separation function. The key result is a Lorentzian analogue of Reshetnyak’s gluing theorem, proving that gluing two strongly causal spacetimes with upper curvature bounds along convex, isometrically compatible boundaries preserves the curvature bound in the resulting space.

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(<i>k</i>) 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.

Motivation & Objective

• To develop a synthetic, metric-based gluing construction for Lorentzian pre-length spaces, analogous to metric space amalgamation.
• To address the challenge of preserving causality and time separation in the quotient space during gluing, which is more complex than in Riemannian or metric settings.
• To establish a Lorentzian version of Reshetnyak’s gluing theorem, ensuring that upper curvature bounds (CAT(k)) are preserved under gluing.
• To provide a framework for constructing new spacetimes from existing ones while maintaining synthetic curvature conditions and causality structure.
• To bridge a gap in synthetic Lorentzian geometry by incorporating fundamental metric constructions like gluing, which are standard in Riemannian and metric geometry.

Proposed method

• Adapting the metric space amalgamation process—disjoint union followed by quotienting under an equivalence relation—to the Lorentzian setting, with special attention to the time separation function.
• Defining a quotient time separation function that respects causality and the Lorentzian structure, ensuring the resulting space remains a Lorentzian pre-length space.
• Introducing a Lorentzian version of Alexandrov’s lemma and a gluing lemma for causal triangles to handle comparison geometry in the glued space.
• Establishing conditions under which the glued space inherits the upper curvature bound (CAT(k)) from the original spacetimes, relying on convexity and isometric compatibility of the gluing boundaries.
• Using the existence of timelike geodesics and the continuity of the time separation function to verify the triangle comparison condition in the glued space.
• Applying a decomposition argument on timelike triangles that cross the gluing boundary, splitting them into sub-triangles within the original spacetimes where curvature bounds are assumed.

Experimental results

Research questions

• RQ1 Can the metric amalgamation construction from Riemannian and metric geometry be generalized to Lorentzian pre-length spaces while preserving causality and curvature bounds?
• RQ2 How can a quotient time separation function be defined in a way that maintains the Lorentzian structure and causal compatibility in the glued space?
• RQ3 Under what conditions does gluing two spacetimes with upper curvature bounds (CAT(k)) result in a spacetime that also satisfies the same curvature bound?
• RQ4 What role does convexity of the gluing boundary and preservation of signed distance play in ensuring the validity of the gluing theorem?
• RQ5 Can the gluing lemma for triangles be formulated and proven in the Lorentzian setting, and how does it support the curvature comparison argument?

Key findings

• The paper successfully constructs a quotient Lorentzian pre-length space from two spacetimes by defining a compatible time separation function on the quotient, ensuring the resulting space remains a well-defined Lorentzian length space.
• A Lorentzian version of the gluing lemma is established, enabling the comparison of causal triangles that cross the gluing boundary by decomposing them into sub-triangles within the original spacetimes.
• The main result, Theorem 5.2.1, proves that if two strongly causal spacetimes with upper curvature bounds (CAT(k)) are glued along convex, isometrically compatible boundaries preserving signed distance, the resulting space also satisfies the same CAT(k) curvature bound.
• The proof relies on decomposing triangles that cross the gluing boundary into sub-triangles lying entirely in one spacetime each, where curvature bounds are assumed, and then reassembling them using the gluing lemma.
• The construction is valid under mild regularity assumptions, and the result is expected to extend to $ C^{1,1} $ metrics based on techniques from prior work.
• The framework provides a synthetic, metric-based method for constructing new spacetimes from existing ones, filling a gap in the theory of Lorentzian length spaces.