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General Relativity and Gravitation

, Volume 43, Issue 9, pp 2409–2420 | Cite as

Geometry and observables in (2+1)-gravity

  • C. MeusburgerEmail author
Research Article
  • 81 Downloads

Abstract

We review the geometrical properties of vacuum spacetimes in (2+1)-gravity with vanishing cosmological constant. We explain how these spacetimes are characterised as quotients of their universal cover by holonomies. We explain how this description can be used to clarify the geometrical interpretation of the fundamental physical variables of the theory, holonomies and Wilson loops. In particular, we discuss the role of Wilson loop observables as the generators of the two fundamental transformations that change the geometry of (2+1)-spacetimes, grafting and earthquake. We explain how these variables can be determined from realistic measurements by an observer in the spacetime.

Keywords

General relativity Quantum gravity Three-dimensional gravity Teichmüller geometry Wilson loops 

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References

  1. 1.
    Carlip S.: Quantum Gravity in 2+1 Dimensions. Cambridge University Press, Cambridge (1998)zbMATHCrossRefGoogle Scholar
  2. 2.
    Benedetti, R., Bonsante, F.: Canonical Wick rotations in 3-dimensional gravity. AMS Memoirs 926, 198 (2009)Google Scholar
  3. 3.
    Meusburger C.: Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant. Commun. Math. Phys. 266, 735–775 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Meusburger C.: Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant. Commun. Math. Phys. 273, 705–754 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Meusburger C.: Cosmological measurements, time and observables in (2+1)-dimensional gravity. Class. Quantum Gravit. 26, 055006 (2009)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Mess G.: Lorentz spacetimes of constant curvature, preprint IHES/M/90/28 (1990). Geom. Dedic. 126(1), 3–45 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Andersson L., Barbot T., Benedetti R., Bonsante F., Goldman W.M., Labourie F., Scannell K.P., Schlenker J.-M.: Notes on a paper of Mess. Geom. Dedic. 126(1), 47–70 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Barbot T.: Globally hyperbolic flat spacetimes. J. Geom. Phys. 53, 123–165 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Franzosi R., Guadagnini E.: Topology and classical geometry in (2+1) gravity. Class. Quantum Gravit. 13, 433–460 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Benedetti R., Guadagnini E.: Cosmological time in (2+1)-gravity. Nucl. Phys. B 613, 330–352 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Goldman W.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2), 200–225 (1984)zbMATHCrossRefGoogle Scholar
  12. 12.
    Thurston W.P.: Earthquakes in two-dimensional hyperbolic geometry. In: Epstein, D.B. (eds) Low Dimensional Topology and Kleinian Groups., pp. 91–112. Cambridge University Press, Cambridge (1987)Google Scholar
  13. 13.
    McMullen C.: Complex earthquakes and Teichmüller theory. J. Am. Math. Soc. 11, 283–320 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Meusburger, C.: Global Lorentzian geometry from lightlike geodesics: What does an observer in (2+1)-gravity see? arXiv:1001.1842 [math-ph] (2009)Google Scholar
  15. 15.
    Meusburger C.: Spacetime geometry in (2+1)-gravity via measurements with returning lightrays. AIP Conf. Proc. 1196, 181–189 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department MathematikUniversität HamburgHamburgGermany

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