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The closed Friedman world model with the initial and final singularities as a non-commutative space. (English) Zbl 0887.53062

Chruściel, Piotr T. (ed.), Mathematics of gravitation. Part I: Lorentzian geometry and Einstein equations. Proceedings of the workshop on mathematical aspects of theories of gravitation, Warsaw, Poland, February 29–March 30, 1996. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 41(1), 153-162 (1997).
Summary: The most elegant definition of singularities in general relativity as \(b\)-boundary points [B. G. Schmidt, Gen. Relativ. Gravitation 1, 269-280 (1971; Zbl 0332.53039)] when applied to the closed Friedman world model, leads to the disastrous situation: both the initial and final singularities form the single point of the \(b\)-boundary which is not Hausdorff separated from the rest of spacetime. We apply Alain Connes’ method of non-commutative geometry, defined in terms of a \(C^*\)-algebra, to this case. It turns out that both the initial and final singularities can be analysed as representations of the \(C^*\)-algebra in a Hilbert space. The method does not distinguish points in spacetime, but identifies space slices of the closed Friedman model as states of the corresponding \(C^*\)-algebra.
For the entire collection see [Zbl 0880.00054].

MSC:

53Z05 Applications of differential geometry to physics
83C47 Methods of quantum field theory in general relativity and gravitational theory
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
46L85 Noncommutative topology
46L87 Noncommutative differential geometry

Citations:

Zbl 0332.53039
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