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Gluing constructions for Lorentzian length spaces.

Tobias Beran1, Felix Rott1

  • 1Faculty of Mathematics, University of Vienna, Vienna, Austria.

Manuscripta Mathematica
|January 8, 2024
PubMed
Summary
This summary is machine-generated.

We developed a general method for constructing new Lorentzian pre-length spaces from existing ones. This new technique is applied to create a gluing theorem for spacetimes, preserving upper curvature bounds.

Keywords:
51F9951K10 (secondary)53B3053C23 (primary)53C50

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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.