On solutions to the wave equation on a non-globally hyperbolic manifold

  • I. V. Volovich
  • O. V. Groshev
  • N. A. Gusev
  • E. A. Kuryanovich


We consider the Cauchy problem for the wave equation on a non-globally hyperbolic manifold of special form (the Minkowski plane with a handle) containing closed time-like curves. We prove that the classical solution of the Cauchy problem exists and is unique for initial data satisfying a specific set of additional requirements.


Wave Equation Cauchy Problem STEKLOV Institute Classical Solution Hyperbolic Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Dover Publ., Mineola, NY, 2003).Google Scholar
  2. 2.
    I. Petrowsky, “Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen,” Mat. Sb. 2, 815–870 (1937).zbMATHGoogle Scholar
  3. 3.
    J. Leray, Hyperbolic Differential Equations (Inst. Adv. Study, Princeton, NJ, 1953).Google Scholar
  4. 4.
    V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1971; M. Dekker, New York, 1971).Google Scholar
  5. 5.
    S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge Univ. Press, London, 1973).zbMATHGoogle Scholar
  6. 6.
    M. Visser, Lorentzian Wormholes (Am. Inst. Phys., Woodbury, NY, 1995).Google Scholar
  7. 7.
    J. R. Gott, Time Travel in Einstein’s Universe (Houghton Mifflin, New York, 2001).Google Scholar
  8. 8.
    S. Deser and R. Jackiw, “Time Travel?,” Comments Nucl. Part. Phys. 20, 337–354 (1992); arXiv: hep-th/9206094.Google Scholar
  9. 9.
    I. Ya. Aref’eva, “High Energy Scattering in the Brane-World and Black-Hole Production,” Fiz. Elem. Chastits At. Yadra 31(7a), 169–180 (2000); arXiv: hep-th/9910269.Google Scholar
  10. 10.
    M. Cvetic, G. W. Gibbons, H. Lu, and C. N. Pope, “Rotating Black Holes in Gauged Supergravities; Thermodynamics, Supersymmetric Limits, Topological Solitons and Time Machines,” arXiv: hep-th/0504080.Google Scholar
  11. 11.
    R. J. Gleiser, M. Gürses, A. Karasu, and Ö. Sarıoğlu, “Closed Timelike Curves and Geodesics of Gödel-Type Metrics,” Class. Quantum Grav. 23, 2653–2663 (2006); arXiv: gr-qc/0512037.zbMATHCrossRefGoogle Scholar
  12. 12.
    B. S. Kay, “Quantum Field Theory in Curved Spacetime,” in Encycl. Math. Phys., Ed. by J.-P. Francoise, G. Naber, and T. S. Tsou (Elsevier, Amsterdam, 2006), Vol. 4, pp. 202–212; arXiv: gr-qc/0601008.CrossRefGoogle Scholar
  13. 13.
    A. Ori, “Formation of Closed Timelike Curves in a Composite Vacuum/Dust Asymptotically Flat Spacetime,” Phys. Rev. D 76(4), 044002 (2007); arXiv: gr-qc/0701024.Google Scholar
  14. 14.
    V. M. Rosa and P. S. Letelier, “Stability of Closed Timelike Curves in the Gödel Universe,” Gen. Relativ. Gravit. 39, 1419–1435 (2007); arXiv: gr-qc/0703100.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    I. Ya. Aref’eva and I. V. Volovich, “The Null Energy Condition and Cosmology,” Teor. Mat. Fiz. 155(1), 3–12 (2008) [Theor. Math. Phys. 155, 503–511 (2008)]; arXiv: hep-th/0612098.MathSciNetGoogle Scholar
  16. 16.
    S. Slobodov, “Unwrapping Closed Timelike Curves,” arXiv: 0808.0956.Google Scholar
  17. 17.
    A. DeBenedictis, R. Garattini, and F. S. N. Lobo, “Phantom Stars and Topology Change,” Phys. Rev. D 78(10), 104003 (2008); arXiv: 0808.0839.Google Scholar
  18. 18.
    L.-F. Li and J.-Y. Zhu, “Averaged Null Energy Condition in Loop Quantum Cosmology,” arXiv: 0812.3532.Google Scholar
  19. 19.
    G. Gibbons and H. Kodama, “Repulsons in the Myers-Perry Family,” arXiv: 0901.1203.Google Scholar
  20. 20.
    J. Friedman, M. S. Morris, I. D. Novikov, F. Echeverria, G. Klinkhammer, K. S. Thorne, and U. Yurtsever, “Cauchy Problem in Spacetimes with Closed Timelike Curves,” Phys. Rev. D 42(6), 1915–1930 (1990).CrossRefMathSciNetGoogle Scholar
  21. 21.
    D. Deutsch, “Quantum Mechanics near Closed Timelike Lines,” Phys. Rev. D 44(10), 3197–3217 (1991).CrossRefMathSciNetGoogle Scholar
  22. 22.
    H. D. Politzer, “Path Integrals, Density Matrices, and Information Flow with Closed Timelike Curves,” Phys. Rev. D 49(8), 3981–3989 (1994).CrossRefMathSciNetGoogle Scholar
  23. 23.
    I. Ya. Aref’eva, I. V. Volovich, and T. Ishiwatari, “Cauchy Problem on Non-globally Hyperbolic Space-Times,” Teor. Mat. Fiz. 157(3), 334–344 (2008) [Theor. Math. Phys. 157, 1646–1654 (2008)].Google Scholar
  24. 24.
    I. Ya. Aref’eva and I. V. Volovich, “Time Machine at the LHC,” Int. J. Geom. Methods Mod. Phys. 5(4), 641–651 (2008); arXiv: 0710.2696.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    A. Mironov, A. Morozov, and T. N. Tomaras, “If LHC Is a Mini-Time-Machines Factory, Can We Notice?,” arXiv: 0710.3395.Google Scholar
  26. 26.
    V. A. Il’in and E. I. Moiseev, “Optimization of Boundary Controls of String Vibrations,” Usp. Mat. Nauk 60(6), 89–114 (2005) [Russ. Math. Surv. 60, 1093–1119 (2005)].MathSciNetGoogle Scholar
  27. 27.
    V. V. Kozlov and I. V. Volovich, “Finite Action Klein-Gordon Solutions on Lorentzian Manifolds,” Int. J. Geom. Methods Mod. Phys. 3(7), 1349–1357 (2006); arXiv: gr-qc/0603111.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    H. B. Nielsen and M. Ninomiya, “Future Dependent Initial Conditions from Imaginary Part in Lagrangian,” arXiv: hep-ph/0612032.Google Scholar
  29. 29.
    B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry: Methods and Applications (Nauka, Moscow, 1979; Springer, New York, 1990).Google Scholar
  30. 30.
    O. Forster, Lectures on Riemann Surfaces (Springer, New York, 1999).Google Scholar
  31. 31.
    A. D. Sakharov, “Cosmological Transitions with Changes in the Signature of the Metric,” Zh. Eksp. Teor. Fiz. 87(2), 375–383 (1984) [JETP 60 (2), 214–218 (1984)].Google Scholar
  32. 32.
    I. Ya. Aref’eva and I. V. Volovich, “Kaluza-Klein Theories and the Space-Time Signature,” Pis’ma Zh. Eksp. Teor. Fiz. 41(12), 535–537 (1985) [JETP Lett. 41, 654–656 (1985)].Google Scholar
  33. 33.
    I. Ya. Aref’eva, B. G. Dragović, and I. V. Volovich, “Extra Time-like Dimensions Lead to a Vanishing Cosmological Constant,” Phys. Lett. B 177, 357–360 (1986).CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  • I. V. Volovich
    • 1
  • O. V. Groshev
    • 2
  • N. A. Gusev
    • 3
  • E. A. Kuryanovich
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudnyi, Moscow oblastRussia

Personalised recommendations