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Particles with spin in stationary flat spacetimes

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Abstract

We construct stationary flat three-dimensional Lorentzian manifolds with singularities that are obtained from Euclidean surfaces with cone singularities and closed one-forms on these surfaces. In the application to (2 + 1)-gravity, these spacetimes correspond to models containing massive particles with spin. We analyse their geometrical properties, introduce a generalised notion of global hyperbolicity and classify all stationary flat spacetimes with singularities that are globally hyperbolic in that sense. We then apply our results to (2 + 1)-gravity and analyse the causality structure of these spacetimes in terms of measurements by observers. In particular, we derive a condition on observers that excludes causality violating light signals despite the presence of closed timelike curves in these spacetimes.

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References

  1. Staruszkiewicz A.: Gravitation theory in three-dimensional space. Acta Phys. Pol. 24, 735–740 (1963)

    MathSciNet  Google Scholar 

  2. ’t Hooft G., Deser S., Jackiw R.: Three-dimensional einstein gravity: dynamics of flat space. Ann. Phys. 152(1), 220–235 (1984)

    Article  MathSciNet  Google Scholar 

  3. Carlip S.: Exact quantum scattering in 2 + 1 dimensional gravity. Nucl. Phys. B 324(1), 106–122 (1989)

    Article  MathSciNet  Google Scholar 

  4. de Sousa Gerbert P.: On spin and (quantum) gravity in 2 + 1 dimensions. Nucl. Phys. B 346(2–3), 440–472 (1990)

    Article  Google Scholar 

  5. Jackiw R., Deser S.: Classical and quantum scattering on a cone. Commun. Math. Phys. 118, 495–509 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. ’t Hooft G.: Canonical quantization of gravitating point particles in 2 + 1 dimensions. Class. Quantum Grav. 10, 1653–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. ’t Hooft G.: The evolution of gravitating point particles in 2 + 1 dimensions. Class. Quantum Gravity 10(5), 1023–1038 (1993)

    Article  MathSciNet  Google Scholar 

  8. ’t Hooft G.: Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness. Class. Quantum Gravity 13(5), 1023–1039 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carlip S.: Quantum Gravity in 2 + 1 Dimensions. Cambridge University Press, New York (2003)

    Google Scholar 

  10. Gott J.R.: Closed timelike curves produced by pairs of moving cosmic strings: exact solutions. Phys. Rev. Lett. 66, 1126–1129 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. ’t Hooft G., Deser S., Jackiw R.: Physical cosmic strings do not generate closed timelike curves. Phys. Rev. Lett. 68(3), 267–269 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deser S.: Physical obstacles to time travel. Class. Quantum Gravity 10, S67–S73 (1993)

    Article  MathSciNet  Google Scholar 

  13. Jackiw R., Deser S.: Time travel?. Comments Nucl. Part. Phys. 20, 337–354 (1992)

    Google Scholar 

  14. Masur H.: Ergodic theory of translation surfaces. In: Hasselblatt, B., Katok, A. (eds) Handbook of Dynamical Systems, vol. 1B, pp. 527–547. Elsevier, Amsterdam (2006)

    Chapter  Google Scholar 

  15. Masur H., Tabachnikov S.: Rational billiards and flat surfaces. In: Hasselblatt, B., Katok, A. (eds) Handbook of Dynamical Systems, vol. 1A, pp. 1015–1089. Elsevier, Amsterdam (2002)

    Chapter  Google Scholar 

  16. Troyanov, M.: On the moduli space of singular euclidean surfaces. In: Handbook of Teichmüller theory, vol. 1. Eur. Math. Soc., pp. 507–540 (2007)

  17. Boileau M., Leeb B., Porti J.: Geometrization of 3-dimensional orbifolds. Ann. Math. 162, 195–290 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cooper D., Hodgson C., Kerckhoff S.: Three-Dimensional Orbifolds and Cone-Manifolds, vol. 5 of MSJ Memoirs. Mathematical Society of Japan, Tokyo (2000)

    Google Scholar 

  19. Benedetti, R., Bonsante, F.: (2 + 1) Einstein spacetimes of finite type. In: Handbook of Teichmüller Theory. vol. II, volume 13 of IRMA Lect. Math. Theor. Phys. pp. 533–609. Eur. Math. Soc., Zürich (2009)

  20. Barbot T., Bonsante F., Schlenker J.-M.: Collisions of point particles in locally AdS spacetimes I. Local description and global examples. Commun. Math. Phys. 308(1), 147–200 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Bonsante F., Schlenker J.-M.: AdS manifolds with particles and earthquakes on singular surfaces. Geom. Funct. Anal. 19(1), 41–82 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Krasnov K., Schlenker J.-M.: Minimal surfaces and particles in 3-manifolds. Geom. Dedicata 126, 187–254 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lecuire, C., Schlenker, J.-M.: The convex core of quasifuchsian manifolds with particles. 0909.4182 (September 2009)

  24. Steif A.R., Deser S.: Gravity theories with lightlike sources in d = 3. Class. Quantum Gravity 9, L153–L160 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Galloway G.: Closed timelike geodesics. Trans. Am. Math. Soc. 285, 379–388 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sánchez M.: On causality and closed geodesics of compact lorentzian manifolds and static spacetimes. Differ. Geom. Appl. 24, 21–32 (2006)

    Article  MATH  Google Scholar 

  27. Troyanov M.: Les surfaces euclidiennes à singularités coniques. Enseign. Math. 32, 79–94 (1986)

    MathSciNet  MATH  Google Scholar 

  28. Thurston W.P.: Shapes of polyhedra and triangulations of the sphere. Geom. Topol. Monog. 1(1), 511–549 (1998)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire d’Analyse Non Linéaire et Géométrie, Université d’Avignon et des pays de Vaucluse, 33, Rue Louis Pasteur, 84 018, Avignon, France

    Thierry Barbot

  2. Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058, Erlangen, Germany

    Catherine Meusburger

Authors
  1. Thierry Barbot
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  2. Catherine Meusburger
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Correspondence to Thierry Barbot.

Additional information

T. Barbot was partially supported by CNRS, ANR GEODYCOS and A.N.R. program “Extensions of Teichmueller-Thurston theories (ETTT)”, ANR-09-BLAN-0116-01. C. Meusburger’s work is supported by the Emmy-Noether fellowship ME 3425/1-1 of the German Research Foundation (DFG). Research visits during which work on this project was undertaken were also supported by the Emmy-Noether fellowship ME 3425/1-1.

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Barbot, T., Meusburger, C. Particles with spin in stationary flat spacetimes. Geom Dedicata 161, 23–50 (2012). https://doi.org/10.1007/s10711-011-9692-y

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