Quantifying closeness of Lorentzian spacetimes

Determine a rigorous and quantitative notion of closeness for pairs of non-isometric Lorentzian spacetimes that captures approximate isometry in Lorentzian geometry and enables well-defined notions of perturbation size and convergence.

Background

The paper emphasizes that, unlike exact isometries, there is ambiguity in formalizing when two Lorentzian spacetimes should be considered close or approximately isometric. This issue is central both to classical problems (e.g., geometrodynamics and convergence) and to quantum gravity approaches where the continuum arises as an approximation to a discrete structure.

While Riemannian geometry admits Gromov–Hausdorff notions mediated by the metric triangle inequality, Lorentzian geometry faces challenges due to the reverse triangle inequality of Lorentzian distance. Various approaches (e.g., Bombelli’s causal set-based closeness, Noldus’s strong metric, Sormani–Vega’s null distance) address aspects of this, but a generally accepted, robust quantitative notion remains unresolved.