Overview
This research investigates the implications of intrinsic timed-Hausdorff convergence within the framework of timed-metric spaces. The study specifically demonstrates that this form of convergence necessitates both (timeless) Gromov-Hausdorff convergence and big bang convergence. Additionally, it highlights related implications for other future-developed convergence criteria.
Research Context
The work draws upon prior notions of distance between certain classes of Lorentzian manifolds, which were introduced by Sakovich and Sormani. These established distances utilize the Hausdorff and Gromov-Hausdorff distances. Consequently, these distance metrics are naturally extendable to a broader spectrum of spaces than their initial application.
Approach
The research examines the relationships between different convergence types in timed-metric spaces. The core approach involves analyzing the consequences of intrinsic timed-Hausdorff convergence. By establishing this foundational convergence, the study then explores its necessary implications for other recognized convergence methods.
Findings
- Intrinsic timed-Hausdorff convergence, when applied to timed-metric-spaces, implies (timeless) Gromov-Hausdorff convergence.
- This same intrinsic timed-Hausdorff convergence also implies big bang convergence.
- The implications further extend to various other related convergences that are anticipated to be developed in the future.