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From: OsherD on 3 Apr 2010 03:45 From Osher Doctorow Nicolas Franco of U. Namur FUNDP Belgium, "Towards a noncommutative version of gravitation," arXiv: 1003.5407 v1 [math-ph] 29 Mar 2010, introduces a Noncomutative generalization/modification of Connes' Noncommutative Geometry to replace Riemannian geometry/manifolds by Lorentzian geometry/manifolds. See also Wikipedia's "Causal Structure" which describes Lorentzian manifolds and the fact that the latter express spacetime in General Relativity, involving Causal Relations between points. The words "with and without Causation" in the title should probably be replaced by "with Causation", except that Franco attempts in one section to dispense with restricting the algebra to causal functions by using dilatations, the idea being to replace a supremum with an upper bound by an infinmum with an equivalent lower bound. This appears to be a technicality of "words", since in either case inequalities are used to describe what is roughly "monotonicity in time" (with opposite directions, however). There are two especially remarkable aspects of Franco's paper. The first is the fact that he admits that the generalization/modification is only at the beginning stages, and specifies how and to some extent why this is so. The second is a "dictionary" from geometry including integral and differential calculus to algebra. In this dictionary, infinitesmals of geometry correspond to compact operators of algebra, integrals to traces of algebra, and differentials df to components [D, a] of spectral triples which are described in the paper. From the viewpoint of Probable Causation/Influence, Causation is of course key. Readers will also find some of the work of Penrose on this described in the paper. PI does not use a requirement that the speed of light is the maximum possible speed, but otherwise there are similarities between the Lorentzian view and PI at least with regard to the importance of time and time ordering (like t1 < t2 implying f(t1) < f(t2), etc.) . Osher Doctorow |