From: OsherD on
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