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Selecta Mathematica

, Volume 24, Issue 5, pp 4293–4459 | Cite as

Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems

  • Igor Rodnianski
  • Jared Speck
Article
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Abstract

We prove a stable blowup result for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our result applies to small perturbations of the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) solution with topology \((0,\infty ) \times \mathbb {T}^3\). The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards the future and that has a “Big Bang” singularity at \(\lbrace 0 \rbrace \times \mathbb {T}^3\), where various curvature invariants blow up. We place “initial” data on a Cauchy hypersurface \(\Sigma _1'\) that are close, as measured by a Sobolev norm, to the FLRW data induced on \(\lbrace 1 \rbrace \times \mathbb {T}^3\). We then study the asymptotic behavior of the perturbed solution in the collapsing direction and prove that its basic qualitative and quantitative features closely resemble those of the FLRW solution. In particular, for the perturbed solution, we construct constant mean curvature-transported spatial coordinates covering \((t,x) \in (0,1] \times \mathbb {T}^3\) and show that it also has a Big Bang at \(\lbrace 0 \rbrace \times \mathbb {T}^3\), where its curvature blows up. The blowup confirms Penrose’s Strong Cosmic Censorship hypothesis for the “past-half” of near-FLRW solutions. Furthermore, we show that the equations are dominated by kinetic (that is, time-derivative-involving) terms that induce approximately monotonic behavior near the Big Bang. As a consequence of the monotonicity, we also show that various time-rescaled components of the solution converge to regular functions of x as \(t \downarrow 0\). The most difficult aspect of the proof is showing that the solution exists for \((t,x) \in (0,1] \times \mathbb {T}^3\), and to this end, we derive a hierarchy of energy estimates that allow for the possibility of mild energy blowup as \(t \downarrow 0\). To close these estimates, it is essential that we are able to rule out more singular energy blowup, a step that is in turn tied to the most important ingredient in our analysis: a collection of integral identities that, when combined in the right proportions, yield an \(L^2\)-type approximate monotonicity inequality, a key point being that the error terms are controllable up to the singularity for near-FLRW solutions. In a companion article, we derived similar approximate monotonicity inequalities for linearized versions of the Einstein-scalar field equations and used them to prove linear stability results for a family of spatially homogeneous background solutions. The present article shows that the linear stability of the FLRW background solution can be upgraded to a full proof of the nonlinear stability of its singularity.

Keywords

Constant mean curvature Energy currents Kasner solutions Parabolic gauge Spatial harmonic coordinates Stable blowup Transported spatial coordinates 

Mathematics Subject Classification

Primary: 83C75 Secondary: 35A20 35Q76 83C05 83F05 

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Notes

Acknowledgements

The authors thank Mihalis Dafermos for offering enlightening comments that helped them improve the exposition. They also thank David Jerison for providing insights that aided their proof of Proposition 14.4. Finally, they thank the anonymous referee for their careful reading of the manuscript and for providing helpful feedback.

References

  1. 1.
    Andersson, L., Moncrief, V.: Elliptic-hyperbolic systems and the Einstein equations. Ann. Henri Poincaré 4(1), 1–34 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andersson, L., Moncrief, V.: Future complete vacuum spacetimes. In: The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pp. 299–330 (2004)CrossRefGoogle Scholar
  3. 3.
    Andersson, L., Rendall, A.D.: Quiescent cosmological singularities. Commun. Math. Phys. 218(3), 479–511 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Anguige, K., Tod, K.P.: Isotropic cosmological singularities. I. Polytropic perfect fluid spacetimes. Ann. Phys. 276(2), 257–293 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Anguige, K.: A class of plane symmetric perfect-fluid cosmologies with a Kasner-like singularity. Class. Quantum Gravity 17(10), 2117–2128 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Balakrishna, J., Daues, G., Seidel, E., Suen, W.M., Tobias, M., Wang, E.: Coordinate conditions in three-dimensional numerical relativity. Class. Quantum Gravity 13(12), L135–L142 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Baouendi, M.S., Goulaouic, C.: Remarks on the abstract form of nonlinear Cauchy–Kovalevsky theorems. Commun. Partial Differ. Equ. 2(11), 1151–1162 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Barrow, J.D.: Quiescent cosmology. Nature 272, 211–215 (1978)CrossRefGoogle Scholar
  9. 9.
    Bartnik, R.: Existence of maximal surfaces in asymptotically flat spacetimes. Commun. Math. Phys. 94(2), 155–175 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bartnik, R.: Remarks on cosmological spacetimes and constant mean curvature surfaces. Commun. Math. Phys. 117(4), 615–624 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Belinskiĭ, V.A., Khalatnikov, I.M.: Effect of scalar and vector fields on the nature of the cosmological singularity. Ž. Èksper. Teoret. Fiz. 63, 1121–1134 (1972)Google Scholar
  12. 12.
    Beyer, F., LeFloch, P.G.: Second-order hyperbolic Fuchsian systems and applications. Class. Quantum Gravity 27(24), 245012 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bieri, L., Zipser, N. (eds.): Extensions of the Stability Theorem of the Minkowski Space in General Relativity. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  14. 14.
    Choquet-Bruhat, Y., Isenberg, J., Moncrief, V.: Topologically general U(1) symmetric vacuum space-times with AVTD behavior. Nuovo Cimento Soc. Ital. Fis. B 119(7–9), 625–638 (2004)MathSciNetGoogle Scholar
  15. 15.
    Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–335 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Choquet-Bruhat, Y.F.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Christodoulou, D.: The Action Principle and Partial Differential Equations, Annals of Mathematics Studies, vol. 146. Princeton University Press, Princeton (2000)Google Scholar
  18. 18.
    Christodoulou, D.: The Formation of Shocks in 3-Dimensional Fluids, European Mathematical Society (EMS), EMS Monographs in Mathematics, Zürich (2007)Google Scholar
  19. 19.
    Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  20. 20.
    Chruściel, P.T., Galloway, G.J., Pollack, D.: Mathematical general relativity: a sampler. Bull. Am. Math. Soc. (N.S.) 47(4), 567–638 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chruściel, P.T., Isenberg, J.: Nonisometric vacuum extensions of vacuum maximal globally hyperbolic spacetimes. Phys. Rev. D (3) 48(4), 1616–1628 (1993)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Claudel, C.M., Newman, K.P.: The Cauchy problem for quasi-linear hyperbolic evolution problems with a singularity in the time. Proc. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1972), 1073–1107 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Damour, T., Henneaux, M., Rendall, A.D., Weaver, M.: Kasner-like behaviour for subcritical Einstein-matter systems. Ann. Henri Poincaré 3(6), 1049–1111 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Eardley, D., Liang, E., Sachs, R.: Velocity-dominated singularities in irrotational dust cosmologies. J. Math. Phys. 13, 99–106 (1972)CrossRefGoogle Scholar
  25. 25.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)Google Scholar
  26. 26.
    Fischer, A.E., Moncrief, V.: Hamiltonian reduction of Einstein’s equations of general relativity. Nuclear Phys. B Proc. Suppl. 57, 142–161 (1997). Constrained dynamics and quantum gravity 1996 (Santa Margherita Ligure)Google Scholar
  27. 27.
    Fischer, A.E., Moncrief, V.: Reducing Einstein’s equations to an unconstrained Hamiltonian system on the cotangent bundle of Teichmüller space. In: Physics on Manifolds (Paris, 1992), pp. 111–151 (1994)Google Scholar
  28. 28.
    Fischer, A.E., Moncrief, V.: Hamiltonian reduction of Einstein’s equations and the geometrization of three-manifolds. In: International Conference on Differential Equations, vols. 1, 2 (Berlin, 1999), pp. 279–282 (2000)CrossRefGoogle Scholar
  29. 29.
    Fischer, A.E., Moncrief, V.: The reduced Hamiltonian of general relativity and the \(\sigma \)-constant of conformal geometry. In: Mathematical and Quantum Aspects of Relativity and Cosmology (Pythagoreon, 1998), pp. 70–101 (2000)Google Scholar
  30. 30.
    Fischer, A.E., Moncrief, V.: The reduced Einstein equations and the conformal volume collapse of 3-manifolds. Class. Quantum Gravity 18(21), 4493–4515 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Fischer, A.E., Moncrief, V.: Hamiltonian reduction and perturbations of continuously self-similar \((n+1)\)-dimensional Einstein vacuum spacetimes. Class. Quantum Gravity 19(21), 5557–5589 (2002)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Friedmann, A.: On the curvature of space. Gen. Relativ. Gravity 31(12), 1991–2000 (1999). Translated from the 1922 German originalMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Friedrich, H.: The conformal structure of Einstein’s field equations. In: Conformal Groups and Related Symmetries: Physical Results and Mathematical Background (Clausthal-Zellerfeld, 1985), pp. 152–161 (1986)Google Scholar
  34. 34.
    Friedrich, H.: On the existence of \(n\)-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun. Math. Phys. 107(4), 587–609 (1986)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Friedrich, H.: On the global existence and the asymptotic behavior of solutions to the Einstein–Maxwell–Yang–Mills equations. J. Differ. Geom. 34(2), 275–345 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Friedrich, H.: Conformal Einstein evolution. In: The Conformal Structure of Space-Time, pp. 1–50 (2002)Google Scholar
  37. 37.
    Friedrich, H.: Sharp asymptotics for Einstein-\(\lambda \)-dust flows. Commun. Math. Phys. 350(2), 803–844 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Friedrich, H., Rendall, A.: The Cauchy problem for the Einstein equations. In: Einstein’s Field Equations and Their Physical Implications, pp. 127–223 (2000)CrossRefGoogle Scholar
  39. 39.
    Garfinkle, D., Gundlach, C.: Well-posedness of the scale-invariant tetrad formulation of the vacuum Einstein equations. Class. Quantum Gravity 22, 2679–2686 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Gerhardt, C.: \(H\)-surfaces in Lorentzian manifolds. Commun. Math. Phys. 89(4), 523–553 (1983)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Griffiths, J., Podolskỳ, J.: Exact Space-Times in Einstein’s General Relativity, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009)zbMATHCrossRefGoogle Scholar
  42. 42.
    Gundlach, C., Martín-García, J.M.: Well-posedness of formulations of the Einstein equations with dynamical lapse and shift conditions. Phys. Rev. D 74, 024016 (2006)CrossRefGoogle Scholar
  43. 43.
    Guo, Y.: Smooth irrotational flows in the large to the Euler–Poisson system in \( {R}^{3+1}\). Commun. Math. Phys. 195(2), 249–265 (1998)zbMATHGoogle Scholar
  44. 44.
    Hawking, S.W.: The occurrence of singularities in cosmology. III. causality and singularities. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 300(1461), 187–201 (1967)zbMATHCrossRefGoogle Scholar
  45. 45.
    Hawking, S.W., Penrose, R.: The singularities of gravitational collapse and cosmology. Proc. R. Soc. Lond. Ser. A 314, 529–548 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Isenberg, J., Kichenassamy, S.: Asymptotic behavior in polarized \(T^2\)-symmetric vacuum space-times. J. Math. Phys. 40(1), 340–352 (1999)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Isenberg, J., Moncrief, V.: Asymptotic behavior of the gravitational field and the nature of singularities in Gowdy spacetimes. Ann. Phys. 199(1), 84–122 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Kasner, E.: Geometrical theorems on Einstein’s cosmological equations. Gen. Relativ. Gravit. 40(4), 865–876 (2008). Reprinted from Am. J. Math. 43 (1921), 217–221, With an editorial comment by John Wainwright and a biography of Kasner compiled by Andrzej KrasińskiGoogle Scholar
  49. 49.
    Kichenassamy, S.: The blow-up problem for exponential nonlinearities. Commun. Partial Differ. Equ. 21(1–2), 125–162 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Kichenassamy, S.: Fuchsian equations in Sobolev spaces and blow-up. J. Differ. Equ. 125(1), 299–327 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Kichenassamy, S.: Nonlinear Wave Equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 194. Marcel Dekker Inc., New York (1996)Google Scholar
  52. 52.
    Kichenassamy, S., Rendall, A.D.: Analytic description of singularities in Gowdy spacetimes. Class. Quantum Gravity 15(5), 1339–1355 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Klainerman, S., Rodnianski, I.: On the breakdown criterion in general relativity. J. Am. Math. Soc. 23(2), 345–382 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Lemaître, A.G.: Contributions to a British Association Discussion on the Evolution of the Universe. Nature 128, 704–706 (1931)CrossRefGoogle Scholar
  55. 55.
    Lindblad, H., Rodnianski, I.: Global existence for the Einstein vacuum equations in wave coordinates. Commun. Math. Phys. 256(1), 43–110 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. 171(3), 1401–1477 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Lloyd, N.G.: Degree Theory. Cambridge University Press, Cambridge (1978). Cambridge Tracts in Mathematics, No. 73Google Scholar
  58. 58.
    Loizelet, J.: Solutions globales des équations d’Einstein–Maxwell. Ann. Fac. Sci. Toulouse Math. (6) 18(3), 565–610 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Lübbe, C., Valiente Kroon, J.A.: A conformal approach for the analysis of the non-linear stability of radiation cosmologies. Ann. Phys. 328, 1–25 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Misner, C.W.: Mixmaster universe. Phys. Rev. Lett. 22, 1071–1074 (1969)zbMATHCrossRefGoogle Scholar
  61. 61.
    Newman, R.P.A.C.: On the structure of conformal singularities in classical general relativity. Proc. R. Soc. Lond. Ser. A 443(1919), 473–492 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Newman, R.P.A.C.: On the structure of conformal singularities in classical general relativity. II. Evolution equations and a conjecture of K. P. Tod. Proc. R. Soc. Lond. Ser. A 443(1919), 493–515 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Oliynyk, T.A.: Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant. Commun. Math. Phys. 346(1), 293–312 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Penrose, R.: Gravitational collapse: the role of general relativity. Gen. Relativ. Gravity 34(7), 1141–1165. Reprinted from Rivista del Nuovo Cimento 1969, Numero Speziale I, 252–276 (2002)Google Scholar
  65. 65.
    Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 57–59 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Penrose, R.: Singularities and time-asymmetry. In: General Relativity: An Einstein Centenary Survey, pp. 581–638 (1979)Google Scholar
  67. 67.
    Reiris, M.: On the asymptotic spectrum of the reduced volume in cosmological solutions of the Einstein equations. Gen. Relativ. Gravity 41(5), 1083–1106 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Rendall, A.D.: Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity. Class. Quantum Gravity 17(16), 3305–3316 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Rendall, A.D.: Theorems on existence and global dynamics for the Einstein equations. Living Rev. Relativ. 8, 6 (2005)zbMATHCrossRefGoogle Scholar
  70. 70.
    Ringström, H.: Curvature blow up in Bianchi VIII and IX vacuum spacetimes. Class. Quantum Gravity 17(4), 713–731 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Ringström, H.: The Bianchi IX attractor. Ann. Henri Poincaré 2(3), 405–500 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Ringström, H.: Future stability of the Einstein-non-linear scalar field system. Invent. Math. 173(1), 123–208 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Ringström, H.: Power law inflation. Commun. Math. Phys. 290(1), 155–218 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Rodnianski, I., Speck, J.: The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant. J. Eur. Math. Soc. (JEMS) 15(6), 2369–2462 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Rodnianski, I., Speck, J.: A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation. Ann. Math. (2) 187(1), 65–156 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Shao, A.: Breakdown criteria for nonvacuum Einstein equations (2010). (English) 2066657291; Shao, Arick; 520417454; Copyright ProQuest, UMI Dissertations Publishing 2010; 9781124046952; 2010; 3410986; 66569; 50824791; English; M1: Ph.D.; M3: 3410986Google Scholar
  77. 77.
    Shao, A.: On breakdown criteria for nonvacuum Einstein equations. Ann. Henri Poincaré 12(2), 205–277 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Sideris, T.C.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. 101(4), 475–485 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Speck, J.: The non-relativistic limit of the Euler–Nordström system with cosmological constant. Rev. Math. Phys. 21(7), 821–876 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Speck, J.: Well-posedness for the Euler–Nordström system with cosmological constant. J. Hyperbolic Differ. Equ. 6(2), 313–358 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Speck, J.: The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant. Sel. Math. 18(3), 633–715 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Speck, J.: The stabilizing effect of spacetime expansion on relativistic fluids with sharp results for the radiation equation of state. Arch. Ration. Mech. Anal. 210(2), 535–579 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Speck, J.: The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates. Anal. PDE 7(4), 771–901 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Speck, J., Strain, R.M.: Hilbert expansion from the Boltzmann equation to relativistic fluids. Commun. Math. Phys. 304(1), 229–280 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Ståhl, F.: Fuchsian analysis of \(S^2\times S^1\) and \(S^3\) Gowdy spacetimes. Class. Quantum Gravity 19(17), 4483–4504 (2002)MathSciNetGoogle Scholar
  86. 86.
    Strauss, W.A.: Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, vol. 73. Published for the Conference Board of the Mathematical Sciences, Washington (1989)Google Scholar
  87. 87.
    Taylor, M.E.: Partial Differential Equations. III, Applied Mathematical Sciences, vol. 117. Springer, New York (1997). Nonlinear Equations, Corrected reprint of the 1996 originalGoogle Scholar
  88. 88.
    Tod, K.P.: Isotropic singularities and the \(\gamma =2\) equation of state. Class. Quantum Gravity 7(1), L13–L16 (1990)MathSciNetzbMATHGoogle Scholar
  89. 89.
    Tod, K.P.: Isotropic singularities and the polytropic equation of state. Class. Quantum Gravity 8(4), L77–L82 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Tod, K.P.: Isotropic cosmological singularities. In: The Conformal Structure of Space-Time, pp. 123–134 (2002)Google Scholar
  91. 91.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)zbMATHCrossRefGoogle Scholar
  92. 92.
    Wang, Q.: Improved breakdown criterion for Einstein vacuum equations in CMC gauge. Commun. Pure Appl. Math. 65(1), 21–76 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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