Null Distance and Convergence of Lorentzian Length Spaces.

Area of Science:

• General Relativity
• Differential Geometry
• Topology

Background:

• The null distance, developed by Sormani and Vega, captures topological and causal properties of smooth spacetimes.
• Lorentzian causality theory is typically studied on smooth manifolds.

Purpose of the Study:

• To extend the concept of null distance to Lorentzian length spaces.
• To generalize Lorentzian causality theory beyond smooth manifolds.
• To investigate Gromov-Hausdorff convergence using the null distance in warped product Lorentzian length spaces.

Main Methods:

• Definition and analysis of Lorentzian length spaces.
• Application of Gromov-Hausdorff convergence using the null distance.
• Study of warped product Lorentzian length spaces.

Main Results:

• The null distance is successfully extended to Lorentzian length spaces.
• Lorentzian causality theory is generalized beyond the manifold level.
• Initial results demonstrate the compatibility of null distance-based Gromov-Hausdorff convergence with synthetic curvature bounds.

Conclusions:

• Lorentzian length spaces provide a broader framework for studying spacetime topology and causality.
• The null distance is a powerful tool for analyzing convergence properties in generalized Lorentzian spaces.
• This work lays the foundation for further research into curvature and causality in non-smooth spacetimes.