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Drawing from the theory of optimal transport we propose a rigorous notion of a causal relation for Borel probability measures on a given spacetime. To prepare the ground, we explore the borderland between Lorentzian geometry, topology and measure theory. We provide various characterisations of the proposed causal relation, which turn out to be equivalent if the underlying spacetime has a sufficiently robust causal structure. We also present the notion of the ‘Lorentz–Wasserstein distance’ and study its basic properties. Finally, we outline the possible applications of the developed formalism in both classical and quantum physics.
Annales Henri Poincaré – Springer Journals
Published: Mar 13, 2017
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