The closed Friedman world model with the initial and final singularities as a non-commutative space

Michael Heller; Wiesław Sasin

Banach Center Publications (1997)

  • Volume: 41, Issue: 1, page 153-161
  • ISSN: 0137-6934

Abstract

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The most elegant definition of singularities in general relativity as b-boundary points, 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 space-time. 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 space-time, but identifies space slices of the closed Friedman model as states of the corresponding C*-algebra.

How to cite

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Heller, Michael, and Sasin, Wiesław. "The closed Friedman world model with the initial and final singularities as a non-commutative space." Banach Center Publications 41.1 (1997): 153-161. <http://eudml.org/doc/252197>.

@article{Heller1997,
abstract = {The most elegant definition of singularities in general relativity as b-boundary points, 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 space-time. 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 space-time, but identifies space slices of the closed Friedman model as states of the corresponding C*-algebra.},
author = {Heller, Michael, Sasin, Wiesław},
journal = {Banach Center Publications},
keywords = {singularities; Friedman cosmology; non-commutative geometry; closed Friedman model; -boundary points},
language = {eng},
number = {1},
pages = {153-161},
title = {The closed Friedman world model with the initial and final singularities as a non-commutative space},
url = {http://eudml.org/doc/252197},
volume = {41},
year = {1997},
}

TY - JOUR
AU - Heller, Michael
AU - Sasin, Wiesław
TI - The closed Friedman world model with the initial and final singularities as a non-commutative space
JO - Banach Center Publications
PY - 1997
VL - 41
IS - 1
SP - 153
EP - 161
AB - The most elegant definition of singularities in general relativity as b-boundary points, 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 space-time. 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 space-time, but identifies space slices of the closed Friedman model as states of the corresponding C*-algebra.
LA - eng
KW - singularities; Friedman cosmology; non-commutative geometry; closed Friedman model; -boundary points
UR - http://eudml.org/doc/252197
ER -

References

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  14. [14] J. L. Koszul, Fibre bundles and differential geometry, Tata Institute of Fundamental Research, Bombay, 1960. 
  15. [15] J. Madore, An Introduction to Noncommutative Differential Geometry and Its Physical Applications, Cambridge University Press, Cambridge, 1995. Zbl0842.58002
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  17. [17] J. Renault, A groupoid approach to C*-algebras, Lecture Notes in Math. 793, Springer, Berlin - Heidelberg - New York, 1980. Zbl0433.46049
  18. [18] W. Sasin, Differential spaces and singularities in differential space-times, Demonstratio Mathematica 24 (1991), 601-634. Zbl0786.58004
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  20. [20] B. G. Schmidt, A new definition of singular points in general relativity, Gen. Rel. Grav. 1 (1971), 269-280. Zbl0332.53039

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