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Cauchy problem for the wave equation on non-global hyperbolic manifolds
- Authors: Groshev O.V1
- Affiliations:
- Steklov Mathematical Institute, Russian Academy of Sciences
- Issue: Vol 15, No 1 (2011)
- Pages: 42-46
- Section: Articles
- URL: https://journals.eco-vector.com/1991-8615/article/view/21079
- Cite item
Abstract
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.
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
Email: groshev@mi.ras.ru
аспирант, отд. математической физики; Математический институт им. В. А. Стеклова РАН; Steklov Mathematical Institute, Russian Academy of Sciences
References
- Petrowsky I. G. Uber das Cauchysche Problem fur Systeme von partiellen Differentialgleichungen // Mat. Sb., 1937. Vol. 2(44), no. 5. Pp. 815-870.
- Лере Ж. Гиперболические дифференциальные уравнения. М.: Наука, 1984. 208 с.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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].
- 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.
- 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.
- 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
- Грошев О. В. О существовании и единственности классических решений задачи Коши на неглобально гиперболических многообразиях // ТМФ, 2010. Т. 164, № 3. С. 441-446
- 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].
