Relationship between Lorentzian GH-type convergence and intrinsic timed-Hausdorff convergence

Determine the precise relationship between Lorentzian Gromov–Hausdorff convergence and precompactness for Lorentzian pre-length spaces (as developed by Mondino–Sämann and related works on Lorentzian metric spaces), and the intrinsic timed-Hausdorff convergence of timed-metric-spaces introduced by Sakovich–Sormani. In particular, ascertain whether and how the convergence frameworks for Lorentzian pre-length/metric spaces correspond to, imply, or are compatible with the timed-Fréchet based intrinsic timed-Hausdorff convergence, and clarify the mapping of causal and metric structures in the respective limit spaces.

Background

The paper surveys several notions of weak convergence for space-times. Beyond converting space-times into timed-metric-spaces and using the intrinsic timed-Hausdorff distance of Sakovich–Sormani, the authors discuss alternative frameworks such as Lorentzian metric spaces (Minguzzi–Suhr, building on Noldus) and Lorentzian pre-length spaces (Kunzinger–Sämann; Mondino–Sämann) that define Lorentzian Gromov–Hausdorff convergence and precompactness via causal diamonds and Lorentz distances.

These alternative spaces are not metric spaces in the usual sense, whereas the Sakovich–Sormani approach yields limits in the category of timed-metric-spaces via timed-Fréchet embeddings. The authors explicitly state that it is not yet clear how these convergence notions relate, highlighting a gap in understanding across frameworks and motivating further investigation into their compatibility and potential translation between limit structures.