We propose an algebraic formulation of the notion of causality for spectral triples corresponding to globally hyperbolic manifolds with a well defined noncommutative generalization. The causality is given by a specific cone of Hermitian elements respecting an algebraic condition based on the Dirac operator and a fundamental symmetry. We prove that in the commutative case the usual notion of causality is recovered. We show that, when the dimension of the manifold is even, the result can be extended in order to have an algebraic constraint suitable for a Lorentzian distance formula.

Abstract.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.

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We give a new definition of dimension spectrum for non-regular spectral triples and compute the exact (i.e. non only the asymptotics) heat-trace of standard Podleś spheres S 2 q for 0 < q < 1, study its behavior when q → 1 and fully compute its exact spectral action for an explicit class of cut-off functions.

Abstract:We prove smoothness of the domain of outer communications (d.o.c.) of the Black Saturn solutions of Elvang and Figueras. We show that the metric on the d.o.c. extends smoothly across two disjoint event horizons with topology R × S 3 and R × S 1 × S 2 . We establish stable causality of the d.o.c. when the Komar angular momentum of the spherical component of the horizon vanishes, and present numerical evidence for stable causality in general.