Anatomy of malicious singularities

Abstract

As well known, the b-boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation ρ, defined on the Cauchy completed total spaceĒ of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class [p 0 ] related to the singularity remains in close contact with all other equivalence classes, i.e., if p 0 ∈ cl[p] for every p ∈ E. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra onĒ, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense)

Fundamental Problems in the Unification of Physics

“…As we have seen above, the quantum sector of our model is essentially given by an algebra of random operators. In [11,23] we have demonstrated that the random character of the noncommutative regime makes singularities (even the strongest ones) probabilistically irrelevant. Computations for the closed Friedman world model [22] confirm that the modular evolution in this model goes through singularities without noticing them.…”

“…We have shown that malicious singularities can be studied with the help of this von Neumann algebra (see below Sect. 10, and for details [18,23]). …”

“…This is what happened to the present authors. When we started constructing a mathematical structure unifying general relativity and quantum mechanics [16,17,19] we hardly treated this as something more than just another attempt, but its later versions [20,21,23,31,32] have revealed a surprisingly rich conceptual architecture that suggested unexpectedly new answers to some of old questions. In fact, the above list of "fundamental questions" has been composed basing on suggestions coming from our model.…”

General Relativity on Random Operators

“…Let us notice, however, that if the frame bundle π M : E → M is not locally trivial, the isomorphism does not occur. This happens, for instance, when a singular boundary ∂M is attached to M [2,7].…”

Imprimitivity theorem for groupoid representations

“…To construct the quantum sector of the model we have used a regular representation of a noncommutative convolutive algebra on this groupoid in the bundle of Hilbert spaces. In paper ( [6]) we have applaied this groupoid representation to investigate spacetime singulariies, and in [10] the representation of the fundamental groupoid to the gravitational Aharonov-Bohm effect.…”