On the breakdown criterion in general relativity

Klainerman, Sergiu; Rodnianski, Igor

doi:10.1090/S0894-0347-09-00655-9

On the breakdown criterion in general relativity

Authors: Sergiu Klainerman and Igor Rodnianski
Journal: J. Amer. Math. Soc. 23 (2010), 345-382
MSC (2010): Primary 35J10
DOI: https://doi.org/10.1090/S0894-0347-09-00655-9
Published electronically: December 24, 2009
MathSciNet review: 2601037
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a geometric criterion for the breakdown of an Einstein-vacuum space-time foliated by a constant mean curvature, or maximal, foliation. More precisely we show that the foliated space-time can be extended as long as the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded. We make no restrictions on the size of the initial data. The paper uses heavily the results of the authors' previous papers.

[And] Michael T. Anderson, On long-time evolution in general relativity and geometrization of 3-manifolds, Comm. Math. Phys. 222 (2001), no. 3, 533–567. MR 1888088, https://doi.org/10.1007/s002200100527
[BKM] J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61–66. MR 763762
[Br] Y. Fourès-Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math. 88 (1952), 141–225 (French). MR 53338, https://doi.org/10.1007/BF02392131
[C-K] Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. MR 1316662
[Fried] F. G. Friedlander, The wave equation on a curved space-time, Cambridge University Press, Cambridge-New York-Melbourne, 1975. Cambridge Monographs on Mathematical Physics, No. 2. MR 0460898
[HE] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, London-New York, 1973. Cambridge Monographs on Mathematical Physics, No. 1. MR 0424186
[EM1] Douglas M. Eardley and Vincent Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), no. 2, 171–191. MR 649158
[EM2] Douglas M. Eardley and Vincent Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), no. 2, 171–191. MR 649158
[HKM] Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), no. 3, 273–294 (1977). MR 420024, https://doi.org/10.1007/BF00251584
[Kl] Sergiu Klainerman, PDE as a unified subject, Geom. Funct. Anal. Special Volume (2000), 279–315. GAFA 2000 (Tel Aviv, 1999). MR 1826256, https://doi.org/10.1007/978-3-0346-0422-2_10
[Kl-Ma] S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in 𝐑³⁺¹, Ann. of Math. (2) 142 (1995), no. 1, 39–119. MR 1338675, https://doi.org/10.2307/2118611
[Kl-Ro1] Sergiu Klainerman and Igor Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (2005), no. 3, 437–529. MR 2125732, https://doi.org/10.1007/s00222-004-0365-4
[Kl-Ro2] S. Klainerman and I. Rodnianski, A geometric approach to the Littlewood-Paley theory, Geom. Funct. Anal. 16 (2006), no. 1, 126–163. MR 2221254, https://doi.org/10.1007/s00039-006-0551-1
[Kl-Ro3] S. Klainerman and I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal. 16 (2006), no. 1, 164–229. MR 2221255, https://doi.org/10.1007/s00039-006-0557-8
[Kl-Ro4] Sergiu Klainerman and Igor Rodnianski, On the radius of injectivity of null hypersurfaces, J. Amer. Math. Soc. 21 (2008), no. 3, 775–795. MR 2393426, https://doi.org/10.1090/S0894-0347-08-00592-4
[Kl-Ro5] Sergiu Klainerman and Igor Rodnianski, A Kirchoff-Sobolev parametrix for the wave equation and applications, J. Hyperbolic Differ. Equ. 4 (2007), no. 3, 401–433. MR 2339803, https://doi.org/10.1142/S0219891607001203
[M] Vincent Moncrief, An integral equation for spacetime curvature in general relativity, Surveys in differential geometry. Vol. X, Surv. Differ. Geom., vol. 10, Int. Press, Somerville, MA, 2006, pp. 109–146. MR 2408224, https://doi.org/10.4310/SDG.2005.v10.n1.a5
[Sob] S. Sobolev, Méthodes nouvelle à resoudre le problème de Cauchy pour les équations linéaires hyperboliques normales, Matematicheskii Sbornik, vol. 1 (43) 1936, 31 -79.
[Wang] Q. Wang, Causal geometry of Einstein vacuum space-times. Ph.D. thesis, Princeton University, 2006.