A C∗-algebra approach to noncommutative Lorentzian geometry of globally-hyperbolic spacetimes

The causal and metric structure of globally hyperbolic spacetimes is investigated from a C∗-algebra point of view related to part of Connes’ noncommutative geometry programme. No foliation of the spacetime by means of spacelike surfaces is employed, but the complete Lorentzian geometry is considered. Several results are produced. As a first result, Connes’ functional formula of the distance is generalized to the Lorentzian case using the d’Alembert operator and the causal functions of a globally hyperbolic spacetime (continuous functions which do not decrease along future-directed causal curves). The formula concerns the so-called Lorentzian distance of a pair of events. (The Lorentzian distance locally determines the causal part of the Synge world function, satisfies an inverse triangular inequality and completely determines the topology, the differentiable structure, the metric tensor and the temporal orientation of a globally hyperbolic spacetime.) The functional formula for the Lorentzian distance is a consequence of some global regularity results. Afterwards, using a C∗-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures. The generalized spacetime consists of a direct set of of Hilbert spaces and a related class of C∗-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalize the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C∗-algebras. A partialordering relation between pairs of loci generalizes the causal order relation in spacetime and a generalized Lorentz distance of loci is defined by means of a class of densely-defined symmetric operators which play the rôle of a Lorentzian metric. These structures enjoy properties which are essentially similar to those in commutative globally hyperbolic spacetimes. Specializing back the formalism to the usual globally hyperbolic spacetime, it is found that compactly-supported probability measures give rise a non-punctual extension of the concept of events enjoying causal and metric properties similar to the usual ones.