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The Global Existence Problem in General Relativity

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The Einstein Equations and the Large Scale Behavior of Gravitational Fields

Abstract

We survey some known facts and open questions concerning the global properties of 3+1-dimensional spacetimes containing a compact Cauchy surface. We consider spacetimes with an l-dimensional Lie algebra of space-like Killing fields. For each l ≤ 3, we give some basic results and conjectures on global existence and cosmic censorship.

Supported in part by the Swedish Natural Sciences Research Council, contract no. F-FU 4873-307, and the NSF under contract no. DMS 0104402.

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References

  1. Miguel Alcubierre, Appearance of coordinate shocks in hyperbolic formalisms of general relativity,Phys. Rev. D (3) 55 (1997), no. 10, 5981–5991.

    Article  MathSciNet  ADS  Google Scholar 

  2. Miguel Alcubierre and Joan Massó, Pathologies of hyperbolic gauges in general relativity and other field theories, Phys. Rev. D (3) 57 (1998), no. 8, R4511–R4515.

    Article  ADS  Google Scholar 

  3. Lars Andersson, Constant mean curvature foliations of flat space-times,Comm Anal. Geom. 10 (2002), no. 5, 1125–1150.

    MathSciNet  MATH  Google Scholar 

  4. Lars Andersson and Piotr T. Chruściel, Hyperboloidal Cauchy data for vacuum Einstein equations and obstructions to smoothness of null infinity, Phys. Rev. Lett. 70 (1993), no. 19, 2829–2832.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Lars Andersson, On “hyperboloidal” Cauchy data for vacuum Einstein equations and ob-structions to smoothness of scri,Comm. Math. Phys. 161 (1994), no. 3, 533–568.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Lars Andersson, Piotr T. Chruściel, and Helmut Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations, Comm. Math. Phys. 149 (1992), no. 3, 587–612.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Lars Andersson, Gregory J. Galloway, and Ralph Howard, A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzian and Riemannian geometry, Comm. Pure Appl. Math. 51 (1998), no. 6, 581–624.

    Article  MathSciNet  Google Scholar 

  8. Lars Andersson and Mirta S. Iriondo, Existence of constant mean curvature hypersurfaces in asymptotically fiat spacetimes,Ann. Global Anal. Geom. 17 (1999), no. 6, 503–538.

    Article  MathSciNet  MATH  Google Scholar 

  9. Lars Andersson and Vincent Moncrief, Future complete vacuum spacetimes, grqc/0303045, in this volume.

    Google Scholar 

  10. Lars Andersson, Elliptic-hyperbolic systems and the Einstein equations,Ann. Henri Poincare 4 (2003), no. 1, 1–34.

    Article  ADS  MATH  Google Scholar 

  11. Lars Andersson, Vincent Moncrief, and Anthony J. Tromba, On the global evolution problem in 2 + 1 gravity,J. Geom. Phys. 23 (1997), no. 3–4, 191–205.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Lars Andersson and Alan D. Rendall, Quiescent cosmological singularities,Comm. Math. Phys. 218 (2001), no. 3, 479–511.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Lars Andersson, Henk van Elst, and Claes Uggla, Gowdy phenomenology in scalefree variables, Classical Quantum Gravity 21 (2004), S29–S57, Spacetime Safari: Essays in Honor of Vincent Moncrief on the Classical Physics of Strong Gravitational Fields, special issue of Classical and Quantum Gravity, eds. J. Isenberg and B. Berger.

    Google Scholar 

  14. HåKan Andréasson, Alan D. Rendall, and Marsha Weaver, Existence of CMC and constant areal time foliations in T 2 symmetric spacetimes with Vlasov matter,Comm. Partial Differential Equations 29 (2004), no. 1–2, 237–262.

    MathSciNet  MATH  Google Scholar 

  15. Abhay Ashtekar, Jiří Bičák, and Bernd G. Schmidt, Asymptotic structure of symmetry-reduced general relativity,Phys. Rev. D (3) 55 (1997), no. 2, 669–686.

    Article  MathSciNet  ADS  Google Scholar 

  16. Abhay Ashtekar, Behavior of Einstein-Rosen waves at null infinity, Phys. Rev. D (3) 55 (1997), no. 2, 687–694.

    Article  MathSciNet  ADS  Google Scholar 

  17. Abhay Ashtekar and Madhavan Varadarajan, Striking property of the gravitational Hamiltonian,Phys. Rev. D (3) 50 (1994), no. 8, 4944–4956.

    Article  MathSciNet  ADS  Google Scholar 

  18. Hajer Bahouri and Jean-Yves Chemin, Équations d’ondes quasi-linéaires et estimations de Strichartz, C. R. Acad. Sci. Paris Ser. I Math. 327 (1998), no. 9, 803–806.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. M. Salah Baouendi and Charles Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kovalevsky theorems, Comm. Partial Differential Equations 2 (1977), no. 11, 1151–1162.

    Article  MathSciNet  Google Scholar 

  20. John D. Barrow, Gregory J. Galloway, and Frank J. Tipler, The closed-universe recollapse conjecture, Mon. Not. R. Astron. Soc. 223 (1986), 835–844.

    ADS  MATH  Google Scholar 

  21. Robert Bartnik, Existence of maximal surfaces in asymptotically flat spacetimes, Comm Math. Phys. 94 (1984), no. 2, 155–175.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Robert Bartnik, Regularity of variational maximal surfaces, Acta Math. 161 (1988), no. 3–4, 145–181.

    Article  MathSciNet  MATH  Google Scholar 

  23. Robert Bartnik, Remarks on cosmological spacetimes and constant mean curvature surfaces, Comm Math. Phys. 117 (1988), no. 4, 615–624.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. John K. Beem, Paul E. Ehrlich, and Kevin L. Easley, Global Lorentzian geometry,second ed., Marcel Dekker Inc., New York, 1996.

    MATH  Google Scholar 

  25. R. Beig and N.Ó. Murchadha, Late time behaviour of the maximal slicing of the Schwarzschild black hole, Phys. Rev. D (3) 57 (1998), 4728–4737, gr-qc/9706046.

    Article  MathSciNet  ADS  Google Scholar 

  26. B.K. Berger and D. Garfinkle, Phenomenology of the Gowdy universe on T 3 x R, Phys. Rev. D57 (1998), 4767–4777, gr-qc/9710102.

    MathSciNet  ADS  Google Scholar 

  27. Beverly K. Berger, Piotr T. Chrugciel, James Isenberg, and Vincent Moncrief, Global foliations of vacuum spacetimes with T 2 isometry, Ann. Physics 260 (1997), no. 1, 117–148.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Beverly K. Berger, David Garfinkle, James Isenberg, Vincent Moncrief, and Marsha Weaver, The singularity in generic gravitational collapse is spacelike, local and oscillatory, Modern Phys. Lett. A 13 (1998), no. 19, 1565–1573.

    Article  MathSciNet  ADS  Google Scholar 

  29. Beverly K. Berger, David Garfinkle, and Vincent Moncrief, Comment on “The Gowdy T 3 cosmologies revisited“, gr-qc/9708050.

    Google Scholar 

  30. Beverly K. Berger, David Garfinkle, and Eugene Strasser, New algorithm for Mix-master dynamics, Classical Quantum Gravity 14 (1997), no. 2, L29–L36.

    Article  MathSciNet  ADS  Google Scholar 

  31. Beverly K. Berger, James Isenberg, and Marsha Weaver, Oscillatory approach to the singularity in vacuum spacetimes with T 2 isometry, Phys. Rev. D (3) 64 (2001), no. 8, 084006, 20.

    Article  MathSciNet  Google Scholar 

  32. Beverly K. Berger and Vincent Moncrief, Evidence for an oscillatory singularity in generic U(1) symmetric cosmologies on T 3 x R Phys. Rev. D (3) 58 (1998), 064023, gr-qc/9804085.

    Article  MathSciNet  ADS  Google Scholar 

  33. Beverly K. Berger, Numerical evidence that the singularity in polarized U(1) symmetric cos-mologies on T 3 x R is velocity dominated, Phys. Rev. D (3) 57 (1998), no. 12, 7235–7240, gr-qc/9801078.

    Google Scholar 

  34. Luc Blanchet and Thibault Damour, Hereditary effects in gravitational radiation, Phys. Rev. D (3) 46 (1992), no. 10, 4304–4319.

    Article  MathSciNet  ADS  Google Scholar 

  35. Dieter Brill and Frank Flaherty, Isolated maximal surfaces in spacetime, Comm. Math. Phys. 50 (1976), no. 2, 157–165.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Gregory A. Burnett and Alan D. Rendall, Existence of maximal hypersurfaces in some spherically symmetric spacetimes, Classical Quantum Gravity 13 (1996), no. 1, 111–123.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. John Cameron and Vincent Moncrief, The reduction of Einstein’s vacuum equations on spacetimes with spacelike U(1)-isometry groups, Mathematical aspects of classical field theory (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1992, pp. 143–169.

    Chapter  Google Scholar 

  38. Myeongju Chae and Piotr T. Chruściel, On the dynamics of Gowdy space-times, Comm. Pure Appl. Math. 57 (2004), no. 8, 1015–1074.

    Article  MathSciNet  MATH  Google Scholar 

  39. Yvonne Choquet-Bruhat, Future complete Einsteinian space times with U(1) symmetry, the unpolarized case, article in this volume, gr-qc/0305060.

    Google Scholar 

  40. Yvonne Choquet-Bruhat and Robert Geroch, Global aspects of the Cauchy problem in general relativity, Comm Math. Phys. 14 (1969), 329–335.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Yvonne Choquet-Bruhat and Vincent Moncrief, Existence theorem for solutions of Einstein’s equations with 1 parameter spacelike isometry groups, Quantization, nonlinear partial differential equations, and operator algebra (Cambridge, MA, 1994), Proc. Symp. Pure Math., vol. 59, Amer. Math. Soc., Providence, RI, 1996, pp. 6780.

    Google Scholar 

  42. Yvonne Choquet-Bruhat, Future global in time Einsteinian spacetimes with U(1) isometry group, Ann. Henri Poincaré 2 (2001), no. 6, 1007–1064.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Yvonne Choquet-Bruhat and Tommaso Ruggeri, Hyperbolicity of the 3 + 1 system of Einstein equations, Comm. Math. Phys. 89 (1983), no. 2, 269–275.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Yvonne Choquet-Bruhat and James W. York, Mixed elliptic and hyperbolic systems for the Einstein equations, gr-qc/9601030.

    Google Scholar 

  45. Demetrios Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. (2) 149 (1999), no. 1, 183–217.

    Article  MathSciNet  MATH  Google Scholar 

  46. Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  47. Piotr T. Chruściel, On space-times with U(1) x U(1) symmetric compact Cauchy surfaces, Ann. Physics 202 (1990), no. 1, 100–150.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Piotr T. Chruściel, On uniqueness in the large of solutions of Einstein’s equations (“strong cos-mic censorship”), Proceedings of the Centre for Mathematics and its Applications, Australian National University, vol. 27, Australian National University Centre for Mathematics and its Applications, Canberra, 1991.

    Google Scholar 

  49. Piotr T. Chruściel, On uniqueness in the large of solutions of Einstein’s equations (“strong cosmic censorship”), Mathematical aspects of classical field theory (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1992, pp. 235–273.

    Google Scholar 

  50. Piotr T. Chruściel, On completeness of orbits of Killing vector fields, Classical Quantum Gravity 10 (1993), no. 10, 2091–2101, ITP Preprint NSF-ITP-93–44.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. Piotr T. Chruściel and Erwann Delay, Existence of non-trivial, vacuum, asymptotically simple spacetimes, Classical Quantum Gravity 19 (2002), no. 9, L71–L79, Erratum, Classical Quantum Gravity 19 (2002), no. 12, 3389.

    Article  Google Scholar 

  52. Piotr T. Chruściel and Gregory J. Galloway, Horizons non-differentiable on a dense set, Comm. Math. Phys. 193 (1998), no. 2, 449–470.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  53. Piotr T. Chruściel and James Isenberg, Nonisometric vacuum extensions of vacuum maximal globally hyperbolic spacetimes, Phys. Rev. D (3) 48 (1993), no. 4, 1616–1628.

    Article  MathSciNet  ADS  Google Scholar 

  54. Piotr T. Chruściel, James Isenberg, and Vincent Moncrief, Strong cosmic censorship in polarised Gowdy spacetimes, Classical Quantum Gravity 7 (1990), no. 10, 1671–1680.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Piotr T. Chruściel and Alan D. Rendall, Strong cosmic censorship in vacuum space-times with compact,locally homogeneous Cauchy surfaces, Ann. Physics 242 (1995), no. 2, 349–385.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. Piotr T. Chruściel and Jalal Shatah, Global existence of solutions of the Yang-Mills equations on globally hyperbolic four-dimensional Lorentzian manifolds,Asian J. Math. 1 (1997), no. 3, 530–548.

    MathSciNet  MATH  Google Scholar 

  57. Neil J. Cornish and Janna Levin, The mixotaster univese: A chaotic Farey tale, Phys. Rev. D (3) 55 (1997), 7489–7510, gr-qc/9612066.

    Article  ADS  Google Scholar 

  58. Justin Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm Math. Phys. 214 (2000), no. 1, 137–189.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. Justin Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137–189.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  60. Sergio Dain and Helmut Friedrich, Asymptotically flat initial data with prescribed regularity at infinity, Comm Math. Phys. 222 (2001), no. 3, 569–609.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. T. Damour, M. Henneaux, A.D. Rendall, and M. Weaver, Kasner-like behaviour for subcritical Einstein-matter systems, Ann. Henri Poincaré 3 (2002), no. 6, 1049–1111.

    Article  MathSciNet  MATH  Google Scholar 

  62. Douślas M. Eardley, Edison Liang, and Rainer Kurt Sachs, Velocity dominated singularities in irrotational dust cosmologies, J. Math. Phys. 13 (1972), 99–107.

    Article  ADS  Google Scholar 

  63. Douślas 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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. Douślas M. Eardley, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkow-ski space. II. Completion of proof, Comm Math. Phys. 83 (1982), no. 2, 193–212.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  65. Klaus Ecker and Gerhard Huisken, Parabolic methods for the construction of space-like slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys. 135 (1991), no. 3, 595–613.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. G.F.R. Ellis and M.A.H. MacCallum, A class of homogeneous cosmological models, Comm. Math. Phys. 12 (1969), 108–141.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  67. Arthur Fischer and Vincent Moncrief, Hamiltonian reduction of Einstein’s equations of general relativity, Nuclear Physics B (Proc. Suppl) 57 (1997), 142–161.

    MathSciNet  MATH  Google Scholar 

  68. Arthur E. Fischer, Jerrold E. Marsden, and Vincent Moncrief, The structure of the space of solutions of Einstein’s equations. I. One Killing field, Ann. Inst. H. Poincaré Sect. A (N.S.) 33 (1980), no. 2, 147–194.

    MathSciNet  MATH  Google Scholar 

  69. Helmut Friedrich, On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Comm. Math. Phys. 107 (1986), no. 4, 587–609.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  70. Helmut Friedrich, On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations, J. Differential Geom. 34 (1991), no. 2, 275–345.

    MathSciNet  MATH  Google Scholar 

  71. Helmut Friedrich, Hyperbolic reductions for Einstein’s equations, Classical Quantum Gravity 13 (1996), no. 6, 1451–1469.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  72. Helmut Friedrich, István Rácz, and Robert M. Wald, On the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon,grqc/9811021, 1998.

    Google Scholar 

  73. Simonetta Frittelli and Oscar A. Reula, First-order symmetric-hyperbolic Einstein equations with arbitrary fixed gauge, Phys. Rev. Lett 76 (1996), 4667–4670, grqc/9605005.

    Article  ADS  Google Scholar 

  74. Yoshihisa Fujiwara, Hideki Ishihara, and Hideo Kodama, Comments on closed Bianchi models, Classical Quantum Gravity 10 (1993), no. 5, 859–867, gr-qc/9301019.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  75. Gregory J. Galloway, The Lorentzian splitting theorem without the completeness assumption, J. Differential Geom. 29 (1989), no. 2, 373–387.

    MathSciNet  MATH  Google Scholar 

  76. Gregory J. Galloway, Some rigidity results for spatially closed spacetimes, Mathematics of grav-itation, Part I (Warsaw, 1996), Polish Acad. Sci., Warsaw, 1997, pp. 21–34.

    Google Scholar 

  77. David Garfinkle and Marsha Weaver, High velocity spikes in Gowdy spacetimes, Phys. Rev. D (3) 67 (2003), no. 12, 124009, 5.

    Article  MathSciNet  Google Scholar 

  78. Claus Gerhardt, H -surfaces in Lorentzian manifolds, Comm Math. Phys. 89 (1983), no. 4, 523–553.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  79. Claus Gerhardt, Hypersurfaces of prescribed mean curvature in Lorentzian manifolds,Math. Z. 235 (2000), no. 1, 83–97.

    Article  MathSciNet  MATH  Google Scholar 

  80. Robert H. Gowdy, Gravitational waves in closed universes, Phys. Rev. Lett. 27 (1971), 826–829.

    Article  ADS  Google Scholar 

  81. Robert H. Gowdy, Vacuum spacetimes with two-parameter spacelike isometry groups and com-pact invariant hypersurfaces: topologies and boundary conditions, Ann. Physics 83 (1974), 203–241.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  82. Boro Grubišić and Vincent Moncrief, Asymptotic behavior of the T 3 x R Gowdy space-times,Phys. Rev. D (3) 47 (1993), no. 6, 2371–2382, gr-qc/9209006.

    Article  MathSciNet  Google Scholar 

  83. Boro Grubišić, Mixmaster spacetime,Geroch’s transformation, and constants of motion, Phys. Rev. D (3) 49 (1994), no. 6, 2792–2800, gr-qc/9309007.

    Article  MathSciNet  ADS  Google Scholar 

  84. S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, London, 1973, Cambridge Monographs on Mathematical Physics, No. 1.

    Book  MATH  Google Scholar 

  85. Oliver Henkel, Global prescribed mean curvature foliations in cosmological space-times. I, II, J. Math. Phys. 43 (2002), no. 5, 2439–2465, 2466–2485.

    MathSciNet  ADS  MATH  Google Scholar 

  86. Simon D. Hem and John M. Stewart, The Gowdy T 3 cosmologies revisited, Classical Quantum Gravity 15 (1998), no. 6, 1581–1593.

    Article  MathSciNet  ADS  Google Scholar 

  87. Conrad G. Hewitt, Joshua T. Horwood, and John Wainwright, Asymptotic dynamics of the exceptional Bianchi cosmologies,gr-qc/0211071, 2002.

    Google Scholar 

  88. David Hobill, Adrian Burd, and Alan Coley (eds.), Deterministic chaos in general relativity, New York, Plenum Press, 1994.

    Google Scholar 

  89. Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, SpringerVerlag, Berlin, 1997.

    MATH  Google Scholar 

  90. 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).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  91. Gerhard Huisken and Tom Ilmanen, The Riemannian Penrose inequality, Internat. Math. Res. Notices 1997 no. 20, 1045–1058.

    Article  MathSciNet  Google Scholar 

  92. Gerhard Huisken, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437.

    MathSciNet  MATH  Google Scholar 

  93. Mirta S. Iriondo, Enzo O. Leguizamón, and Oscar A. Reula, Einstein’s equations in Ashtekar’s variables constitute a symmetric hyperbolic system, Phys. Rev. Lett 79 (1997), 4732–4735, gr-qc/9710004.

    Article  ADS  MATH  Google Scholar 

  94. Mirta S. Iriondo, On the dynamics of Einstein’s equations in the Ashtekar formulation, Adv. Theor. Math. Phys. 2 (1998), no. 5, 1075–1103.

    MathSciNet  MATH  Google Scholar 

  95. James Isenberg, Progress on strong cosmic censorship, Mathematical aspects of classical field theory (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1992, pp. 403–418.

    Google Scholar 

  96. James Isenberg and Satyanad Kichenassamy, Asymptotic behavior in polarized T 2 - symmetric vacuum space-times,J. Math. Phys. 40 (1999), no. 1, 340–352.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  97. James Isenberg, Rafe Mazzeo, and Daniel Pollack, Gluing and wormholes for the Einstein constraint equations,Comm. Math. Phys. 231 (2002), no. 3, 529–568.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  98. James Isenberg, On the topology of vacuum spacetimes, Ann. Henri Poincaré 4 (2003), no. 2, 369–383, gr-qc/0206034.

    MathSciNet  MATH  Google Scholar 

  99. James Isenberg and Vincent Moncrief, The existence of constant mean curvature foliations of Gowdy 3-torus spacetimes, Comm. Math. Phys. 86 (1982), no. 4, 485–493.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  100. James Isenberg, Symmetries of cosmological Cauchy horizons with exceptional orbits, J. Math. Phys. 26 (1985), no. 5, 1024–1027.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  101. James Isenberg, Asymptotic behavior of the gravitational field and the nature of singularities in Gowdy spacetimes, Ann. Physics 199 (1990), no. 1, 84–122.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  102. James Isenberg, Asymptotic behaviour in polarized and half-polarized U(1) symmetric vac-uum spacetimes, Classical Quantum Gravity 19 (2002), no. 21, 5361–5386.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  103. James Isenberg and Alan D. Rendall, Cosmological spacetimes not covered by a constant mean curvature slicing, Classical Quantum Gravity 15 (1998), no. 11, 3679–3688.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  104. Akihiro Ishibashi, Tatsuhiko Koike, Masaru Siino, and Sadayoshi Kojima, Compact hyperbolic universe and singularities, Phys. Rev. D (3) 54 (1996), no. 12, 7303–7310, gr-qc/9605041.

    Article  MathSciNet  ADS  Google Scholar 

  105. Ronald Kantowski and Rainer Kurt Sachs, Some spatially homogeneous anisotropic relativistic cosmological models,J. Mathematical Phys. 7 (1966), 443–446.

    Article  MathSciNet  ADS  Google Scholar 

  106. Michael Kapovich, Deformations of representations of discrete subgroups of SO(3, 1), Math. Ann. 299 (1994), no. 2, 341–354.

    Article  MathSciNet  MATH  Google Scholar 

  107. Satyanad Kichenassamy and Alan D. Rendall, Analytic description of singularities in Gowdy spacetimes, Classical Quantum Gravity 15 (1998), no. 5, 1339–1355, preprint at http://www.aei-potsdam.mpg.de.

  108. S. Klainerman, On the regularity of classical field theories in Minkowski space-time R 3+1 Nonlinear partial differential equations in geometry and physics (Knoxville, TN, 1995), Birkhäuser, Basel, 1997, pp. 29–69.

    Chapter  Google Scholar 

  109. Sergiu Klainerman and Matei Machedon, Finite energy solutions of the Yang-Mills equations in R 3+1, Ann. of Math. (2) 142 (1995), no. 1, 39–119.

    Article  MathSciNet  MATH  Google Scholar 

  110. Sergiu Klainerman and Francesco Nicolò, On local and global aspects of the Cauchy problem in general relativity, Classical Quantum Gravity 16 (1999), R73–R157.

    Article  ADS  MATH  Google Scholar 

  111. Sergiu Klainerman, The evolution problem in general relativity, Progress in Mathematical Physics, vol. 25, Birkhauser Boston Inc., Boston, MA, 2003.

    Google Scholar 

  112. Sergiu Klainerman, Peeling properties of asymptotically fiat solutions to the Einstein vacuum equations, Classical and Quantum Gravity 20 (2003), no. 14, 3215–3257.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  113. Sergiu Klainerman and Igor Rodnianski, The causal structure of microlocalized Einstein metrics,math.AP/0109174, 2001.

    Google Scholar 

  114. Sergiu Klainerman, Rough solution for the Einstein vacuum equations, math.AP/0109173, 2001.

    Google Scholar 

  115. Sergiu Klainerman and Igor Rodnianski, Ricci defects of microlocalized Einstein metrics, J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 85–113.

    Article  MathSciNet  MATH  Google Scholar 

  116. Sergiu Klainerman and Sigmund Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22 (1997), no. 56, 901–918.

    MathSciNet  MATH  Google Scholar 

  117. Hideo Kodama, Canonical structure of locally homogeneous systems on compact closed 3-manifolds of types E 3,Nil and Sol, Progr. Theoret. Phys. 99 (1998), no. 2, 173–236.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  118. Tatsuhiko Koike, Masayuki Tanimoto, and Akio Hosoya, Compact homogeneous universes, J. Math. Phys. 35 (1994), no. 9, 4855–4888.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  119. Joachim Krieger, Global regularity of wave maps from r 2+1 to h 2, preprint.

    Google Scholar 

  120. E.M. Lifshitz and I.M. Khalatnikov, Investigations in relativistic cosmology, Adv. in Phys. 12 (1963), 185–249.

    Article  MathSciNet  ADS  Google Scholar 

  121. Xue-Feng Lin and Robert M. Wald, Proof of the closed-universe-recollapse conjecture for diagonal Bianchi type-IX cosmologies,Phys. Rev. D (3) 40 (1989), no. 10, 3280–3286.

    Article  MathSciNet  ADS  Google Scholar 

  122. Hans Lindblad, Counterexamples to local existence for quasilinear wave equations, Math. Res. Lett. 5 (1998), no. 5, 605–622.

    MathSciNet  MATH  Google Scholar 

  123. Hans Lindblad and Igor Rodnianski, The weak null condition for Einstein’s equations, C. R. Math. Acad. Sci. Paris 336 (2003), no. 11, 901–906.

    Article  MathSciNet  MATH  Google Scholar 

  124. Jerrold E. Marsden and Frank J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep. 66 (1980), no. 3, 109–139.

    Article  MathSciNet  ADS  Google Scholar 

  125. Vincent Moncrief, Global properties of Gowdy spacetimes with T 3 x R topology, Ann. Physics 132 (1981), no. 1, 87–107.

    Article  MathSciNet  ADS  Google Scholar 

  126. Vincent Moncrief, Reduction of Einstein’s equations for vacuum space-times with spacelike U(1) isometry groups, Ann. Physics 167 (1986), no. 1, 118–142.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  127. Vincent Moncrief, Reduction of Einstein’s equations for cosmological spacetimes with spacelike U(1)-isometry groups, Physique quantique et geometric (Paris, 1986), Hermann, Paris, 1988, pp. 105–117.

    Google Scholar 

  128. Vincent Moncrief, Reduction of the Einstein-Maxwell and Einstein-Maxwell-Higgs equations for cosmological spacetimes with spacelike U(1) isometry groups, Classical Quantum Gravity 7 (1990), no. 3, 329–352.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  129. Vincent Moncrief and Douślas M. Eardley, The global existence problem and cosmic censorship in general relativity,Gen. Relativity Gravitation 13 (1981), no. 9, 887–892.

    Article  MathSciNet  ADS  Google Scholar 

  130. Vincent Moncrief and James Isenberg, Symmetries of cosmological Cauchy horizons,Comm Math. Phys. 89 (1983), no. 3, 387–413.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  131. Peter Orlik and Prank Raymond, On 3-manifolds with local SO(2) action, Quart. J. Math. Oxford Ser. (2) 20 (1969), 143–160.

    Article  MathSciNet  MATH  Google Scholar 

  132. R. Penrose, Techniques of differential topology in relativity, SIAM, Philadelphia, PA., 1972.

    Book  MATH  Google Scholar 

  133. R. Penrose, Some unsolved problems in classical general relativity, Seminar on Differential Geometry, Princeton Univ. Press, Princeton, N.J., 1982, pp. 631–668.

    Google Scholar 

  134. Alan D. Rendall, Global properties of locally spatially homogeneous cosmological models with matter, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 3, 511–526.

    Article  MathSciNet  MATH  Google Scholar 

  135. Alan D. Rendall, Constant mean curvature foliations in cosmological space-times, Hely. Phys. Acta 69 (1996), no. 4, 490–500, Journées Relativistes 96, Part II (Ascona, 1996), gr-qc/9606049.

    MathSciNet  ADS  MATH  Google Scholar 

  136. Alan D. Rendall, Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry, Comm Math. Phys. 189 (1997), no. 1, 145–164.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  137. Alan D. Rendall, Global dynamics of the mixmaster model, Classical Quantum Gravity 14 (1997), no. 8, 2341–2356.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  138. Alan D. Rendall, Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity, Classical Quantum Gravity 17 (2000), no. 16, 3305–3316.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  139. Alan D. Rendall, Theorems on existence and global dynamics for the Einstein equations, Living Rev. Relativ. 5 (2002), 2002–6, 62 pp. (electronic).

    MathSciNet  Google Scholar 

  140. Alan D. Rendall and Marsha Weaver, Manufacture of Gowdy spacetimes with spikes, Classical Quantum Gravity 18 (2001), no. 15, 2959–2975.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  141. Hans Ringström, Curvature blow up in Bianchi VIII and IX vacuum spacetimes, Classical Quantum Gravity 17 (2000), no. 4, 713–731.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  142. Hans Ringström, The Bianchi IX attractor, Ann. Henri Poincaré 2 (2001), no. 3, 405–500.

    Article  ADS  MATH  Google Scholar 

  143. Hans Ringström, Asymptotic expansions close to the singularity in Gowdy spacetimes, Clas-sical Quantum Gravity 21 (2004), S305–S322.

    Article  ADS  MATH  Google Scholar 

  144. Hans Ringström, On Gowdy vacuum spacetimes, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 2, 485–512.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  145. Peter Scott, The geometries of 3-manifolds, Bull.London Math. Soc. 15 (1983), no. 5, 401–487.

    Article  MathSciNet  MATH  Google Scholar 

  146. Jalal Shatah, The Cauchy problem for harmonic maps on Minkowski space,Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 1126–1132.

    Chapter  Google Scholar 

  147. Stephen T.C. Siklos, Occurrence of whimper singularities, Comm. Math. Phys. 58 (1978), no. 3, 255–272.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  148. Larry Smarr and Jr. York James W., Kinematical conditions in the construction of spacetime,Phys. Rev. D (3) 17 (1978), no. 10, 2529–2551.

    Article  MathSciNet  ADS  Google Scholar 

  149. Larry Smarr, Radiation gauge in general relativity, Phys. Rev. D (3) 17 (1978), no. 8, 1945–1956.

    Article  MathSciNet  ADS  Google Scholar 

  150. Hart F. Smith and Christopher D. Sogge, On Strichartz and eigenfunction estimates for low regularity metrics,Math. Res. Lett. 1 (1994), no. 6, 729–737.

    MathSciNet  MATH  Google Scholar 

  151. Hart F. Smith and Daniel Tataru, Sharp local well-posedness results for the nonlinear wave equation, http://www.math.berkeley.edu/~tataru/nlw.html/~tataru/nlw.html, 2001.

    Google Scholar 

  152. Chrisopher D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis, vol. II, International Press, Cambridge, MA, 1995.

    MATH  Google Scholar 

  153. Christopher D. Sogge, On local existence for nonlinear wave equations satisfying variable coefficient null conditions,Comm. Partial Differential Equations 18 (1993), no. 11, 1795–1821.

    Article  MathSciNet  MATH  Google Scholar 

  154. Christopher D. Sogge, Fourier integral operators and nonlinear wave equations, Mathematics of gravitation, Part I (Warsaw, 1996), Polish Acad. Sci., Warsaw, 1997, pp. 91–108.

    Google Scholar 

  155. Masayuki Tanimoto, New varieties of Gowdy space-times, J. Math. Phys. 39 (1998), no. 9, 4891–4898.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  156. Terence Tao, Global regularity of wave maps. II. Small energy in two dimensions,Comm Math. Phys. 224 (2001), no. 2, 443–544.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  157. Daniel Tataru, Nonlinear wave equations, Proceedings of the ICM, Beijing 2002, vol. 3, 2003, pp. 209–220, math.AP/0304397.

    Google Scholar 

  158. Daniel Tataru, Rough solutions for the wave-maps equation, http://math.berkeley.edu/~tataru/nlw.html/~tataru/nlw.html, 2003.

    Google Scholar 

  159. Daniel Tataru, The wave maps equation, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 2, 185–204 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  160. William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton University Press, Princeton, NJ, 1997, Edited by Silvio Levy.

    MATH  Google Scholar 

  161. Claes Uggla, Henk van Elst, John Wainwright, and George F.R. Ellis, The past attractor in inhomogenous cosmology, Phys. Rev. D 68 (2003), 103–502.

    Article  Google Scholar 

  162. Juan Antonio Valiente Kroon, A new class of obstructions to the smoothness of null infinity, Comm. Math. Phys. 244 (2004), no. 1, 133–156.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  163. John Wainwright and George F.R. Ellis (eds.), Dynamical systems in cosmology,Cambridge University Press, Cambridge, 1997, Papers from the workshop held in Cape Town, June 27-July 2, 1994.

    Google Scholar 

  164. John Wainwright and Lucas Hsu, A dynamical systems approach to Bianchi cosmologies: orthogonal models of class A, Classical Quantum Gravity 6 (1989), no. 10, 1409–1431.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  165. Robert M. Wald, General relativity, University of Chicago Press, Chicago, III., 1984.

    MATH  Google Scholar 

  166. Robert M. Wald, Gravitational collapse and cosmic censorship, Black holes, gravitational radiation and the universe, Fund. Theories Phys., vol. 100, Kluwer Acad. Publ., Dordrecht, 1999, pp. 69–85.

    Google Scholar 

  167. Marsha Weaver, Dynamics of magnetic Bianchi VI0 cosmologies, Classical Quantum Gravity 17 (2000), no. 2, 421–434, gr-qc/9909043.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  168. Marsha Weaver, James Isenberg, and Beverly K. Berger, Mixmaster behavior in inhomogeneous cosmological spacetimes, Phys. Rev. Lett 80 (1998), 2984–2987, grqc/9712055.

    Article  ADS  Google Scholar 

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  1. Department of Mathematics, University of Miami, Coral Gables, FL, 33124, USA

    Lars Andersson

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  1. Laboratoire de Mathématiques et de Physique Théorique, CNRS UMR 6083, Avenue Grammont, 37200, Tours, France

    Piotr T. Chruściel

  2. Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14467, Golm, Germany

    Helmut Friedrich

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Andersson, L. (2004). The Global Existence Problem in General Relativity. In: Chruściel, P.T., Friedrich, H. (eds) The Einstein Equations and the Large Scale Behavior of Gravitational Fields. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7953-8_3

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  • DOI: https://doi.org/10.1007/978-3-0348-7953-8_3

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