Cauchy problem for the wave equation on non-global hyperbolic manifolds


We consider Cauchy problem for wave equation on two types of non-global hyperbolic manifolds: Minkowski plane with an attached handle and Misner space. We prove that the classical solution on a plane with a handle exists and is unique if and only if a finite set of point-wise constraints on initial values is satisfied. On the Misner space
the existence and uniqueness of a solution is equivalent to much stricter constraints for the initial data.

About the authors

Oleg V Groshev

Steklov Mathematical Institute, Russian Academy of Sciences

аспирант, отд. математической физики; Математический институт им. В. А. Стеклова РАН; Steklov Mathematical Institute, Russian Academy of Sciences


  1. Petrowsky I. G. Uber das Cauchysche Problem fur Systeme von partiellen Differentialgleichungen // Mat. Sb., 1937. Vol. 2(44), no. 5. Pp. 815-870.
  2. Лере Ж. Гиперболические дифференциальные уравнения. М.: Наука, 1984. 208 с.
  3. Hawking S. W., Ellis G. F. R. The large scale structure of space-time / Cambridge Monographs on Mathematical Physics. Vol. 1. London - New York: Cambridge University Press, 1973. 391 pp.
  4. Politzer H. D. Path integrals, density matrices, and information flow with closed timelike curves // Phys. Rev. D, 1994. Vol. 49, no. 8. Pp. 3981-3989, arXiv: gr-qc/9310027.
  5. Gott J. R. Closed timelike curves produced by pairs of moving cosmic strings: Exact solutions // Phys. Rev. Lett., 1991. Vol. 66, no. 9. Pp. 1126-1129.
  6. Bernal A., Sanchez A. Smoothness of time functions and the metric splitting of globally hyperbolic space-times // Comm. Math. Phys., 2005. Vol. 257, no. 1. Pp. 43-50.
  7. Friedman J., Morris M. S., Novikov I. D., Echeverria F., Klinkhammer G., Thorne K. S., Yurtsever U. Cauchy problem in spacetimes with closed timelike curves // Phys. Rev. D, 1990. Vol. 42, no. 6. Pp. 1915-1930.
  8. Арефьева И. Я., Волович И. В., Ишиватари Т. Задача Коши на неглобально гиперболических многообразиях // ТМФ, 2008. Т. 157, № 3. С. 334-344, arXiv: 0903.0567 [hep-th].
  9. Friedman J. L. The Cauchy problem on spacetimes that are not globally hyperbolic / In: The Einstein equations and the large scale behavior of gravitational fields, 50 Years of the Cauchy problem in general relativity; P. T. Chrusciel et al. New York: Birkhauser, 2004. Pp. 331-346, arXiv: gr-qc/0401004.
  10. Friedman J. L., Morris M. S. Existence and uniqueness theorems for massless fields on a class of spacetimes with closed timelike curves // Comm. Math. Phys., 1997. Vol. 186, no. 3. Pp. 495-530, arXiv: gr-qc/9411033.
  11. Волович И. В., Грошев О. В., Гусев Н. А., Курьянович Э. А. О решениях волнового уравнения на неглобально гиперболическом многообразии / В сб.: Избранные вопросы математической физики и p-адического анализа: Сборник статей / Тр. МИАН, Т. 265. М.: МАИК, 2009. С. 273-287
  12. Грошев О. В. О существовании и единственности классических решений задачи Коши на неглобально гиперболических многообразиях // ТМФ, 2010. Т. 164, № 3. С. 441-446
  13. Aref'eva I. Ya., Volovich I. V. Time Machine at the LHC, // Int. J. Geom. Meth. Mod. Phys., 2008. Vol. 5, no. 4. Pp. 641-651, arXiv: 0710.2696 [hep-th].



Abstract: 67

PDF (Russian): 19

Article Metrics

Metrics Loading ...


  • There are currently no refbacks.

Copyright (c) 2011 Samara State Technical University

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies