Polish spaces of causal curves

Abstract

We propose and study a new approach to the topologization of spaces of (possibly not all) future-directed causal curves in a stably causal spacetime. It relies on parametrizing the curves "in accordance" with a chosen time function. Thus obtained topological spaces of causal curves are separable and completely metrizable, i.e. Polish. The latter property renders them particularly useful in the optimal transport theory. To illustrate this fact, we explore the notion of a causal time-evolution of measures in globally hyperbolic spacetimes and discuss its physical interpretation.

32 pages

References in corpus (6)

• The causal hierarchy of spacetimes
• Limit curve theorems in Lorentzian geometry
• Global hyperbolicity for spacetimes with continuous metrics
• Causality for nonlocal phenomena
• Causal evolution of wave packets
• Theory of optimal transport for Lorentzian cost functions

Cited by in corpus (10)

• Causality for nonlocal phenomena
• Causal evolution of wave packets
• The noncommutative geometry of Zitterbewegung
• Non-commutative Geometry from Perturbative Quantum Gravity in de Sitter spacetime
• Generally covariant $N$-particle dynamics
• Operational causality in spacetime
• Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime
• Relativistic Causality vs. No-Signaling as the limiting paradigm for correlations in physical theories
• Time functions and $K$-causality between measures
• Causal evolution of probability measures and continuity equation