Anatomy of Malicious Singularities

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 space Ē of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class [p0] related to the singularity remains in close contact with all other equivalence classes, i.e., if p0 ∈ 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 on Ē, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense) ∗The correspondence address: ul. Powstańców Warszawy 13/94, 33-110 Tarnów, Poland. E-mail: [email protected]