Download PDF - The Group of Large Diffeomorphisms in General RelativityAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

The Group of Large Diffeomorphisms in General Relativity

Authors 1

Affiliations

  1. Fakultät für Physik der Universität Freiburg, Hermann Herder Straße 3, D-79104 Freiburg, Germany

Abstract

We investigate the mapping class groups of diffeomorphisms fixing a frame at a point for general classes of 3-manifolds. These groups form the equivalent to the groups of large gauge transformations in Yang-Mills theories. They are also isomorphic to the fundamental groups of the spaces of 3-metrics modulo diffeomorphisms, which are the analogues in General Relativity to gauge-orbit spaces in gauge theories.

Bibliography

  1. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourão, and T. Thiemann, Quantization of diffeomorphism invariant theories with local degrees of freedom, J. Math. Phys. 36 (1995), 6456-6493; gr-qc/9504018.
  2. H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups (second edition), Springer Verlag, Berlin-Göttingen-Heidelberg-New York, 1965.
  3. A. E. Fischer, Resolving the singularities in the space of Riemannian geometries, J. Math. Phys. 27 (1986), 718-738.
  4. J. Friedman and D. Witt, Homotopy is not isotopy for homeomorphisms of 3-manifolds, Topology 25 (1986), 35-44.
  5. D. Giulini, Asymptotic symmetry groups of long-ranged gauge configurations, Modern Phys. Lett. A 10 (1995), 2059-2070.
  6. D. Giulini, On the configuration space topology in general relativity, Helv. Phys. Acta 68 (1995), 87-111.
  7. D. Giulini, Quantum mechanics on spaces with finite fundamental group, Helv. Phys. Acta 68 (1995), 438-469.
  8. D. Giulini, What is the geometry of superspace?, Phys. Rev. D (3) 51 (1995), 5630-5635.
  9. D. Giulini, 3-Manifolds for relativists, Internat. J. Theoret. Phys. 33 (1994), 913-930.
  10. D. Giulini and J. Louko, Diffeomorphism invariant states in Wittens 2+1 quantum gravity on R×T2, Classical Quantum Gravity 12 (1995), 2735-2745.
  11. D. Giulini and J. Louko, Theta-sectors in spatially flat quantum cosmology, Phys. Rev. D (3) 46 (1992), 4355-4364.
  12. A. Hatcher, Homeomorphisms of sufficiently large P2-irreducible 3-manifolds, Topology 15 (1976), 343-347.
  13. J. Hempel, 3-Manifolds, Ann. of Math. Stud. 86, Princeton University Press, 1976.
  14. H. Hendriks, Application de la théorie d'obstruction en dimension 3, Bull. Soc. Math. France Mém. 53 (1977), 81-195.
  15. H. Hendriks and D. McCullough, On the diffeomorphism group of a reducible 3-manifold, Topology Appl. 26 (1987), 25-31.
  16. D. McCullough, Mappings of reducible manifolds, in: Geometric and Algebraic Topology, Banach Center Publ. 18 (1986), 61-76.
  17. D. McCullough, Topological and algebraic automorphisms of 3-manifolds, in: Groups of Homotopy, Equivalences and Related Topics, R. Piccinini (ed.), Lecture Notes in Math. 1425, Springer, Berlin, 1990, 102-113.
  18. D. McCullough and A. Miller, Homeomorphisms of 3-manifolds with compressible boundary, Mem. Amer. Math. Soc. 344 (1986).
  19. J. Milnor, Introduction to algebraic K-theory, Ann. of Math. Stud. 72, Princeton University Press, 1971.
  20. S. P. Plotnick, Equivariant intersection forms, knots in S4, and rotations in 2-spheres, Trans. Amer. Math. Soc. 296 (1986), 543-575.
  21. D. Witt, Symmetry groups of state vectors in canonical quantum gravity, J. Math. Phys. 27 (1986), 573-592.
  22. D. Witt, Vacuum space-times that admit no maximal slice, Phys. Rev. Lett. 57 (1986), 1386-1389.
Banach Center Publications
1997/39/1
Export to PBN, Opens in new tab
Pages:
303-315
Main language of publication
English
Published
1997
Exact and natural sciences