Gluing constructions for Lorentzian length spaces

Area of Science:

• Differential Geometry
• General Relativity
• Topology

Background:

• Metric spaces and CAT(k) spaces are fundamental in geometry.
• Lorentzian geometry is crucial for understanding spacetimes in general relativity.
• Existing methods for constructing spaces have limitations in Lorentzian settings.

Purpose of the Study:

• To introduce a novel amalgamation process for Lorentzian pre-length spaces.
• To develop a gluing theorem analogous to Reshetnyak's theorem for CAT(k) spaces within a Lorentzian context.
• To extend the understanding of spacetime construction and curvature properties.

Main Methods:

• Generalizing the amalgamation of metric spaces to Lorentzian pre-length spaces.
• Formulating a gluing theorem for strongly causal spacetimes viewed as Lorentzian length spaces.
• Addressing the absence of spacelike distance in Lorentzian pre-length spaces.

Main Results:

• A general method for constructing new Lorentzian pre-length spaces is established.
• An analogue of Reshetnyak's gluing theorem is successfully formulated for spacetimes.
• The compatibility of gluing operations with upper curvature bounds in Lorentzian settings is demonstrated.

Conclusions:

• The introduced amalgamation process provides a versatile tool for creating Lorentzian spaces.
• The developed gluing theorem offers new insights into the structure and properties of spacetimes.
• This work bridges concepts from metric geometry and Lorentzian geometry, enabling new research avenues.