Foundations of Physics

, 38:1082 | Cite as

Unwrapping Closed Timelike Curves

  • Sergei Slobodov


Closed timelike curves (CTCs) appear in many solutions of the Einstein equation, even with reasonable matter sources. These solutions appear to violate causality and so are considered problematic. Since CTCs reflect the global properties of a spacetime, one can attempt to extend a local CTC-free patch of such a spacetime in a way that does not give rise to CTCs. One such procedure is informally known as unwrapping. However, changes in global identifications tend to lead to local effects, and unwrapping is no exception, as it introduces a special kind of singularity, called quasi-regular. This “unwrapping” singularity is similar to the string singularities. We define an unwrapping of a (locally) axisymmetric spacetime as the universal cover of the spacetime after one or more of the local axes of symmetry is removed. We give two examples of unwrapping of essentially 2+1 dimensional spacetimes with CTCs, the Gott spacetime and the Gödel spacetime. We show that the unwrapped Gott spacetime, while singular, is at least devoid of CTCs. In contrast, the unwrapped Gödel spacetime still contains CTCs through every point. A “multiple unwrapping” procedure is devised to remove the remaining circular CTCs. We conclude that, based on the given examples, CTCs appearing in the solutions of the Einstein equation are not simply a mathematical artifact of coordinate identifications. Alternative extensions of spacetimes with CTCs tend to lead to other pathologies, such as naked quasi-regular singularities.


Closed timelike curves Gödel universe Gott spacetime 


  1. 1.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1973) MATHGoogle Scholar
  2. 2.
    Cooperstock, F.I., Tieu, S.: Closed timelike curves and time travel: dispelling the myth. Found. Phys. 35, 1497–1509 (2005) MATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Gödel, K.: An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Rev. Mod. Phys. 21, 447–450 (1949) MATHCrossRefADSGoogle Scholar
  4. 4.
    Gott, J.R.I.: Closed timelike curves produced by pairs of moving cosmic strings—exact solutions. Phys. Rev. Lett. 66, 1126–1129 (1991) MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Ellis, G.F.R., Schmidt, B.G.: Singular space-times. Gen. Relativ. Gravit. 8, 915–953 (1977) MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    van Stockum, W.J.: The gravitational field of a distribution of particles rotating around an axis of symmetry. Proc. R. Soc. Edinb. A 57, 135–154 (1937) MATHGoogle Scholar
  7. 7.
    Tipler, F.J.: Rotating cylinders and the possibility of global causality violation. Phys. Rev. D 9, 2203–2206 (1974) CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Stephani, H., et al.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2003) MATHGoogle Scholar
  9. 9.
    Chandrasekhar, S., Wright, J.P.: The geodesics in Godel’s universe. Proc. Natl. Acad. Sci. USA 47, 341–347 (1961) MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Carter, B.: Global structure of the Kerr family of gravitational fields. Phys. Rev. 174, 1559–1571 (1968) MATHCrossRefADSGoogle Scholar
  11. 11.
    Marder, L.: Flat space-times with gravitational fields. Proc. R. Soc. Lond. A 252, 45–50 (1959) MATHADSMathSciNetGoogle Scholar
  12. 12.
    Deser, S., Jackiw, R., ’t Hooft, G.: Three-dimensional Einstein gravity: dynamics of flat space. Ann. Phys. 152, 220–235 (1984) CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Herrera, L., Santos, N.O.: Geodesics in Lewis space-time. J. Math. Phys. 39, 3817–3827 (1998) MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Culetu, H.: On a stationary spinning string spacetime. J. Phys. Conf. Ser. 68, 012036–012040 (2007) CrossRefADSGoogle Scholar
  15. 15.
    Jensen, B., Soleng, H.H.: General-relativistic model of a spinning cosmic string. Phys. Rev. D 45, 3528–3533 (1992) CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Ori, A.: Rapidly moving cosmic strings and chronology protection. Phys. Rev. D 44, 2214–2215 (1991) CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Deser, S., Jackiw, R., ’t Hooft, G.: Physical cosmic strings do not generate closed timelike curves. Phys. Rev. Lett. 68, 267–269 (1992) MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Morris, M.S., Thorne, K.S., Yurtsever, U.: Wormholes, time machines, and the weak energy condition. Phys. Rev. Lett. 61, 1446–1449 (1988) CrossRefADSGoogle Scholar
  19. 19.
    Frolov, V.P., Novikov, I.D.: Physical effects in wormholes and time machines. Phys. Rev. D 42, 1057–1065 (1990) CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Ori, A.: Formation of closed timelike curves in a composite vacuum/dust asymptotically flat spacetime. Phys. Rev. D 76, 044002 (2007) CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Tipler, F.J.: Causality violation in asymptotically flat space-times. Phys. Rev. Lett. 37, 879–882 (1976) CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Hawking, S.W.: Chronology protection conjecture. Phys. Rev. D 46, 603–611 (1992) CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Gott, J.R., Alpert, M.: General relativity in a (2+1)-dimensional space-time. Gen. Relativ. Gravit. 16, 243–247 (1984) CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Schleich, K., Witt, D.M.: Generalized sums over histories for quantum gravity. 1. Smooth conifolds. Nucl. Phys. B 402, 411–491 (1993) MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Mars, M., Senovilla, J.M.M.: Axial symmetry and conformal Killing vectors. Class. Quantum Gravity 10, 1633 (1993). arXiv:gr-qc/0201045 MATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Garcia-Parrado, A., Senovilla, J.M.M.: Causal structures and causal boundaries. Class. Quantum Gravity 22, R1 (2005). arXiv:gr-qc/0501069 MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Geroch, R.P.: Local characterization of singularities in general relativity. J. Math. Phys. 9, 450 (1968) MATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Penrose, R.: Asymptotic properties of fields and space-times. Phys. Rev. Lett. 10, 66 (1963) CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Geroch, R.P., Kronheimer, E.H., Penrose, R.: Ideal points in space-time. Proc. R. Soc. Lond. A 327, 545 (1972) MATHADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Scott, S.M., Szekeres, P.: The abstract boundary: a new approach to singularities of manifolds. J. Geom. Phys. 13, 223 (1994). arXiv:gr-qc/9405063 MATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Vickers, J.A.G.: Generalized cosmic strings. Class. Quantum Gravity 4, 1 (1987) MATHCrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Schmidt, B.G.: A new definition of singular points in general relativity. Gen. Relativ. Gravit. 1, 269–280 (1971) MATHCrossRefADSGoogle Scholar
  33. 33.
    Johnson, R.A.: The bundle boundary in some special cases. J. Math. Phys. 18, 898–902 (1977) MATHCrossRefADSGoogle Scholar
  34. 34.
    Misner, C.W.: Taub-nut space as a counterexample to almost anything In: Ehlers, J. (ed.) Relativity Theory and Astrophysics, vol. 1. Relativity and Cosmology Lectures in Applied Mathematics, vol. 8, American Mathematical Society, Providence (1967) Google Scholar
  35. 35.
    Krasnikov, S.: Unconventional stringlike singularities in flat spacetime. Phys. Rev. D 76, 024010 (2007) CrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Carter, B.: The commutation property of a stationary, axisymmetric system. Commun. Math. Phys. 17, 233 (1970) MATHCrossRefADSGoogle Scholar
  37. 37.
    Banados, M., Teitelboim, C., Zanelli, J.: The black hole in three-dimensional space-time. Phys. Rev. Lett. 69, 1849 (1992). arXiv:hep-th/9204099 MATHCrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Cutler, C.: Global structure of Gott’s two-string spacetime. Phys. Rev. D 45, 487–494 (1992) CrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Carroll, S.M., et al.: Energy-momentum restrictions on the creation of Gott time machines. Phys. Rev. D 50, 6190–6206 (1994) CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Boyda, E.K., et al.: Holographic protection of chronology in universes of the Gödel type. Phys. Rev. D 67, 106003 (2003) CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Brecher, D., et al.: Closed timelike curves and holography in compact plane waves. J. High Energy Phys. 10, 31–49 (2003) CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Astefanesei, D., Mann, R.B., Radu, E.: Nut charged space-times and closed timelike curves on the boundary. J. High Energy Phys. 1, 49 (2005) CrossRefADSMathSciNetGoogle Scholar
  43. 43.
    Bonnor, W.B., Santos, N.O., MacCullum, M.A.H.: An exterior for the Gödel spacetime. Class. Quantum Gravity 15, 357–366 (1998) MATHCrossRefADSGoogle Scholar
  44. 44.
    Geroch, R.: A Method for generating solutions of Einstein’s equations. J. Math. Phys. 12, 918–924 (1971) MATHCrossRefADSMathSciNetGoogle Scholar
  45. 45.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of British ColumbiaVancouverCanada

Personalised recommendations