Strong cosmic censorship in $T^{3}$-Gowdy spacetimes

Abstract

Einstein’s vacuum equations can be viewed as an initial value problem, and given initial data there is one part of spacetime, the so-called maximal globally hyperbolic development (MGHD), which is uniquely determined up to isometry. Unfortunately, it is sometimes possible to extend the spacetime beyond the MGHD in inequivalent ways. Consequently, the initial data do not uniquely determine the spacetime, and in this sense the theory is not deterministic. It is then natural to make the strong cosmic censorship conjecture, which states that for generic initial data, the MGHD is inextendible. Since it is unrealistic to hope to prove this conjecture in all generality, it is natural to make the same conjecture within a class of spacetimes satisfying some symmetry condition. Here, we prove strong cosmic censorship in the class of $T^{3}$-Gowdy spacetimes. In a previous paper, we introduced a set $\mathcal{G}_{i,c}$ of smooth initial data and proved that it is open in the $C^{1}\times C^{0}$-topology. The solutions corresponding to initial data in $\mathcal{G}_{i,c}$ have the following properties. First, the MGHD is $C^{2}$-inextendible. Second, following a causal geodesic in a given time direction, it is either complete, or a curvature invariant, the Kretschmann scalar, is unbounded along it (in fact the Kretschmann scalar is unbounded along any causal curve that ends on the singularity). The purpose of the present paper is to prove that $\mathcal{G}_{i,c}$ is dense in the $C^{\infty}$-topology.

  • [baiw] Go to document B. K. Berger, J. Isenberg, and M. Weaver, "Oscillatory approach to the singularity in vacuum spacetimes with $T^2$ isometry," Phys. Rev. D, vol. 64, iss. 8, p. 084006, 2001.
    @article {baiw, MRKEY = {1858721},
      AUTHOR = {Berger, Beverly K. and Isenberg, James and Weaver, Marsha},
      TITLE = {Oscillatory approach to the singularity in vacuum spacetimes with {$T\sp 2$} isometry},
      JOURNAL = {Phys. Rev. {\rm D}},
      FJOURNAL = {Physical Review. D. Third Series},
      VOLUME = {64},
      YEAR = {2001},
      NUMBER = {8},
      PAGES = {084006, 20},
      ISSN = {0556-2821},
      CODEN = {PRVDAQ},
      MRCLASS = {83C75},
      MRNUMBER = {2002h:83073},
      DOI = {10.1103/PhysRevD.64.084006},
      }
  • [bag] Go to document B. K. Berger and D. Garfinkle, "Phenomenology of the Gowdy universe on $T^3\times\Bbb R$," Phys. Rev. D, vol. 57, iss. 8, pp. 4767-4777, 1998.
    @article {bag, MRKEY = {1682356},
      AUTHOR = {Berger, Beverly K. and Garfinkle, David},
      TITLE = {Phenomenology of the {G}owdy universe on {$T\sp 3\times\bold R$}},
      JOURNAL = {Phys. Rev. {\rm D}},
      FJOURNAL = {Physical Review. D. Third Series},
      VOLUME = {57},
      YEAR = {1998},
      NUMBER = {8},
      PAGES = {4767--4777},
      ISSN = {0556-2821},
      CODEN = {PRVDAQ},
      MRCLASS = {83C15},
      MRNUMBER = {99k:83010},
      DOI = {10.1103/PhysRevD.57.4767},
      }
  • [cac] Go to document M. Chae and P. T. Chruściel, "On the dynamics of Gowdy space-times," Comm. Pure Appl. Math., vol. 57, iss. 8, pp. 1015-1074, 2004.
    @article {cac, MRKEY = {2053758},
      AUTHOR = {Chae, Myeongju and Chru{\'s}ciel, Piotr T.},
      TITLE = {On the dynamics of {G}owdy space-times},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {57},
      YEAR = {2004},
      NUMBER = {8},
      PAGES = {1015--1074},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {83C75 (83C05)},
      MRNUMBER = {2005i:83085},
      MRREVIEWER = {Alan D. Rendall},
      DOI = {10.1002/cpa.20016},
      ZBLNUMBER = {1074.83020},
      }
  • [cbag] Go to document Y. Choquet-Bruhat and R. Geroch, "Global aspects of the Cauchy problem in general relativity," Comm. Math. Phys., vol. 14, pp. 329-335, 1969.
    @article {cbag, MRKEY = {0250640},
      AUTHOR = {Choquet-Bruhat, Yvonne and Geroch, Robert},
      TITLE = {Global aspects of the {C}auchy problem in general relativity},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {14},
      YEAR = {1969},
      PAGES = {329--335},
      ISSN = {0010-3616},
      MRCLASS = {83.53},
      MRNUMBER = {40 \#3872},
      MRREVIEWER = {H. A. Buchdahl},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103841822},
      ZBLNUMBER = {0182.59901},
      }
  • [chr1] P. T. Chruściel, "On space-times with ${ U}(1)\times { U}(1)$ symmetric compact Cauchy surfaces," Ann. Physics, vol. 202, iss. 1, pp. 100-150, 1990.
    @article {chr1, MRKEY = {1067565},
      AUTHOR = {Chru{\'s}ciel, Piotr T.},
      TITLE = {On space-times with {${\rm U}(1)\times {\rm U}(1)$} symmetric compact {C}auchy surfaces},
      JOURNAL = {Ann. Physics},
      FJOURNAL = {Annals of Physics},
      VOLUME = {202},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {100--150},
      ISSN = {0003-4916},
      CODEN = {ADNYA6},
      MRCLASS = {83C05 (53C80)},
      MRNUMBER = {91h:83007},
      MRREVIEWER = {Robert Bartnik},
      ZBLNUMBER = {0727.53078},
      }
  • [chr2] P. T. Chruściel, On Uniqueness in the Large of Solutions of Einstein’s Equations (“Strong Cosmic Censorship”), Canberra: Australian National University Centre for Mathematics and its Applications, 1991.
    @book {chr2, MRKEY = {1143281},
      AUTHOR = {Chru{\'s}ciel, Piotr T.},
      TITLE = {On Uniqueness in the Large of Solutions of {E}instein's Equations (``Strong Cosmic Censorship'')},
      SERIES = {Proc. Centre for Math. Appl.},
      NUMBER = {27},
      PUBLISHER = {Australian National University Centre for Mathematics and its Applications},
      ADDRESS = {Canberra},
      YEAR = {1991},
      PAGES = {130},
      ISBN = {0-7315-0443-7},
      MRCLASS = {83C05 (53C50 53C80 83C75)},
      MRNUMBER = {93c:83008},
      MRREVIEWER = {J. K. Beem},
      }
  • [cai] Go to document P. T. Chruściel and J. Isenberg, "Nonisometric vacuum extensions of vacuum maximal globally hyperbolic spacetimes," Phys. Rev. D, vol. 48, iss. 4, pp. 1616-1628, 1993.
    @article {cai, MRKEY = {1236815},
      AUTHOR = {Chru{\'s}ciel, Piotr T. and Isenberg, James},
      TITLE = {Nonisometric vacuum extensions of vacuum maximal globally hyperbolic spacetimes},
      JOURNAL = {Phys. Rev. {\rm D}},
      FJOURNAL = {Physical Review. D. Third Series},
      VOLUME = {48},
      YEAR = {1993},
      NUMBER = {4},
      PAGES = {1616--1628},
      ISSN = {0556-2821},
      CODEN = {PRVDAQ},
      MRCLASS = {83C15 (83C75)},
      MRNUMBER = {94f:83007},
      MRREVIEWER = {Alan D. Rendall},
      DOI = {10.1103/PhysRevD.48.1616},
      }
  • [ciam] Go to document P. T. Chruściel, J. Isenberg, and V. Moncrief, "Strong cosmic censorship in polarized Gowdy spacetimes," Class. Quantum Grav., vol. 7, iss. 10, pp. 1671-1680, 1990.
    @article {ciam, MRKEY = {1075858},
      AUTHOR = {Chru{\'s}ciel, Piotr T. and Isenberg, James and Moncrief, Vincent},
      TITLE = {Strong cosmic censorship in polarized {G}owdy spacetimes},
      JOURNAL = {Class. Quantum Grav.},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {7},
      YEAR = {1990},
      NUMBER = {10},
      PAGES = {1671--1680},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C75},
      MRNUMBER = {91i:83042},
      MRREVIEWER = {Istv{á}n R{á}cz},
      URL = {http://stacks.iop.org/0264-9381/7/1671},
      ZBLNUMBER={0703.53081},
      }
  • [cal] Go to document P. T. Chruściel and K. Lake, "Cauchy horizons in Gowdy spacetimes," Class. Quantum Grav., vol. 21, iss. 3, p. s153-s169, 2004.
    @article {cal, MRKEY = {2053004},
      AUTHOR = {Chru{\'s}ciel, Piotr T. and Lake, Kayll},
      TITLE = {Cauchy horizons in {G}owdy spacetimes},
      JOURNAL = {Class. Quantum Grav.},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {21},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {S153--S169},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C75},
      MRNUMBER = {2005d:83092},
      MRREVIEWER = {Hans G. Ringstr{ö}m},
      DOI = {10.1088/0264-9381/21/3/010},
      ZBLNUMBER = {1047.83004},
      }
  • [dar] Go to document M. Dafermos and A. D. Rendall, "Inextendibility of expanding cosmological models with symmetry," Class. Quantum Grav., vol. 22, iss. 23, p. l143-l147, 2005.
    @article {dar, MRKEY = {2194327},
      AUTHOR = {Dafermos, Mihalis and Rendall, Alan D.},
      TITLE = {Inextendibility of expanding cosmological models with symmetry},
      JOURNAL = {Class. Quantum Grav.},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {22},
      YEAR = {2005},
      NUMBER = {23},
      PAGES = {L143--L147},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C05 (83F05)},
      MRNUMBER = {2006k:83031},
      MRREVIEWER = {Norbert Noutchegueme},
      DOI = {10.1088/0264-9381/22/23/L01},
      ZBLNUMBER = {1087.83050},
      }
  • [cb] Go to document Y. Fourès-Bruhat, "Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires," Acta Math., vol. 88, pp. 141-225, 1952.
    @article {cb, MRKEY = {0053338},
      AUTHOR = {Four{è}s-Bruhat, Y.},
      TITLE = {Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {88},
      YEAR = {1952},
      PAGES = {141--225},
      ISSN = {0001-5962},
      MRCLASS = {36.0X},
      MRNUMBER = {14,756g},
      MRREVIEWER = {F. John},
      DOI = {10.1007/BF02392131},
      ZBLNUMBER = {0049.19201},
      }
  • [gowdy] R. H. Gowdy, "Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: topologies and boundary conditions," Ann. Physics, vol. 83, pp. 203-241, 1974.
    @article {gowdy, MRKEY = {0339764},
      AUTHOR = {Gowdy, Robert H.},
      TITLE = {Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: topologies and boundary conditions},
      JOURNAL = {Ann. Physics},
      FJOURNAL = {Annals of Physics},
      VOLUME = {83},
      YEAR = {1974},
      PAGES = {203--241},
      ISSN = {0003-4916},
      MRCLASS = {83.53},
      MRNUMBER = {49 \#4521},
      MRREVIEWER = {S. Deser},
      ZBLNUMBER = {0325.53061},
      }
  • [grub] Go to document B. Grubivsić and V. Moncrief, "Asymptotic behavior of the $T^3\times{\bf R}$ Gowdy space-times," Phys. Rev. D, vol. 47, iss. 6, pp. 2371-2382, 1993.
    @article {grub, MRKEY = {1209782},
      AUTHOR = {Grubi{š}i{ć},
      Boro and Moncrief, Vincent},
      TITLE = {Asymptotic behavior of the {$T\sp 3\times{\bf R}$} {G}owdy space-times},
      JOURNAL = {Phys. Rev. {\rm D}},
      FJOURNAL = {Physical Review. D. Third Series},
      VOLUME = {47},
      YEAR = {1993},
      NUMBER = {6},
      PAGES = {2371--2382},
      ISSN = {0556-2821},
      CODEN = {PRVDAQ},
      MRCLASS = {83C05 (35Q75 83C30 83C75)},
      MRNUMBER = {94a:83003},
      MRREVIEWER = {Piotr T. Chru{\'s}ciel},
      DOI = {10.1103/PhysRevD.47.2371},
      }
  • [jam] J. Isenberg and V. Moncrief, "Asymptotic behavior of the gravitational field and the nature of singularities in Gowdy spacetimes," Ann. Physics, vol. 199, iss. 1, pp. 84-122, 1990.
    @article {jam, MRKEY = {1048674},
      AUTHOR = {Isenberg, James and Moncrief, Vincent},
      TITLE = {Asymptotic behavior of the gravitational field and the nature of singularities in {G}owdy spacetimes},
      JOURNAL = {Ann. Physics},
      FJOURNAL = {Annals of Physics},
      VOLUME = {199},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {84--122},
      ISSN = {0003-4916},
      CODEN = {ADNYA6},
      MRCLASS = {83C05 (58D30 83C30 83C75)},
      MRNUMBER = {91d:83007},
      MRREVIEWER = {Piotr T. Chru{\'s}ciel},
      }
  • [kar1] Go to document S. Kichenassamy and A. D. Rendall, "Analytic description of singularities in Gowdy spacetimes," Class. Quantum Grav., vol. 15, iss. 5, pp. 1339-1355, 1998.
    @article {kar1, MRKEY = {1623091},
      AUTHOR = {Kichenassamy, Satyanad and Rendall, Alan D.},
      TITLE = {Analytic description of singularities in {G}owdy spacetimes},
      JOURNAL = {Class. Quantum Grav.},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {15},
      YEAR = {1998},
      NUMBER = {5},
      PAGES = {1339--1355},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C75 (83C15 83C20)},
      MRNUMBER = {99d:83077},
      MRREVIEWER = {Simonetta Frittelli},
      DOI = {10.1088/0264-9381/15/5/016},
      ZBLNUMBER = {0949.83050},
      }
  • [mon] V. Moncrief, "Global properties of Gowdy spacetimes with $T^{3}\times {\bf R}$ topology," Ann. Physics, vol. 132, iss. 1, pp. 87-107, 1981.
    @article {mon, MRKEY = {615961},
      AUTHOR = {Moncrief, Vincent},
      TITLE = {Global properties of {G}owdy spacetimes with {$T\sp{3}\times {\bf R}$} topology},
      JOURNAL = {Ann. Physics},
      FJOURNAL = {Annals of Physics},
      VOLUME = {132},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {87--107},
      ISSN = {0003-4916},
      CODEN = {ADNYA6},
      MRCLASS = {83C05 (58D25 83C20)},
      MRNUMBER = {83e:83007},
      }
  • [r] Go to document A. D. Rendall, "Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity," Class. Quantum Grav., vol. 17, iss. 16, pp. 3305-3316, 2000.
    @article {r, MRKEY = {1779512},
      AUTHOR = {Rendall, Alan D.},
      TITLE = {Fuchsian analysis of singularities in {G}owdy spacetimes beyond analyticity},
      JOURNAL = {Class. Quantum Grav.},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {17},
      YEAR = {2000},
      NUMBER = {16},
      PAGES = {3305--3316},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C75 (83C05 83C20)},
      MRNUMBER = {2002e:83066},
      MRREVIEWER = {Simonetta Frittelli},
      DOI = {10.1088/0264-9381/17/16/313},
      ZBLNUMBER = {0967.83021},
      }
  • [raw] Go to document A. D. Rendall and M. Weaver, "Manufacture of Gowdy spacetimes with spikes," Class. Quantum Grav., vol. 18, iss. 15, pp. 2959-2975, 2001.
    @article {raw, MRKEY = {1847897},
      AUTHOR = {Rendall, Alan D. and Weaver, Marsha},
      TITLE = {Manufacture of {G}owdy spacetimes with spikes},
      JOURNAL = {Class. Quantum Grav.},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {18},
      YEAR = {2001},
      NUMBER = {15},
      PAGES = {2959--2975},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C15},
      MRNUMBER = {2002e:83010},
      MRREVIEWER = {Sharmanthie M. Fernando},
      DOI = {10.1088/0264-9381/18/15/310},
      ZBLNUMBER = {0995.83010},
      }
  • [jagg] Go to document H. Ringström, "On Gowdy vacuum spacetimes," Math. Proc. Cambridge Philos. Soc., vol. 136, iss. 2, pp. 485-512, 2004.
    @article {jagg, MRKEY = {2042268},
      AUTHOR = {Ringstr{ö}m, Hans},
      TITLE = {On {G}owdy vacuum spacetimes},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {136},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {485--512},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {83C05 (35B40 35L70 83C15 83C75)},
      MRNUMBER = {2005h:83018},
      MRREVIEWER = {Simonetta Frittelli},
      DOI = {10.1017/S0305004103007321},
      ZBLNUMBER = {1050.83020},
      }
  • [jag2] Go to document H. Ringström, "Asymptotic expansions close to the singularity in Gowdy spacetimes," Class. Quantum Grav., vol. 21, iss. 3, p. s305-s322, 2004.
    @article {jag2, MRKEY = {2053013},
      AUTHOR = {Ringstr{ö}m, Hans},
      TITLE = {Asymptotic expansions close to the singularity in {G}owdy spacetimes},
      JOURNAL = {Class. Quantum Grav.},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {21},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {S305--S322},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C05 (83C20 83C75)},
      MRNUMBER = {2005e:83011},
      MRREVIEWER = {Piotr T. Chru{\'s}ciel},
      DOI = {10.1088/0264-9381/21/3/019},
      ZBLNUMBER = {1040.83032},
      }
  • [jag3] Go to document H. Ringström, "On a wave map equation arising in general relativity," Comm. Pure Appl. Math., vol. 57, iss. 5, pp. 657-703, 2004.
    @article {jag3, MRKEY = {2032917},
      AUTHOR = {Ringstr{ö}m, Hans},
      TITLE = {On a wave map equation arising in general relativity},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {57},
      YEAR = {2004},
      NUMBER = {5},
      PAGES = {657--703},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {58J45 (83C05)},
      MRNUMBER = {2005h:58049},
      DOI = {10.1002/cpa.20015},
      ZBLNUMBER = {1059.83011},
      }
  • [jag4] Go to document H. Ringström, "Existence of an asymptotic velocity and implications for the asymptotic behavior in the direction of the singularity in $T^3$-Gowdy," Comm. Pure Appl. Math., vol. 59, iss. 7, pp. 977-1041, 2006.
    @article {jag4, MRKEY = {2222842},
      AUTHOR = {Ringstr{ö}m, Hans},
      TITLE = {Existence of an asymptotic velocity and implications for the asymptotic behavior in the direction of the singularity in {$T\sp 3$}-{G}owdy},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {59},
      YEAR = {2006},
      NUMBER = {7},
      PAGES = {977--1041},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {83C75 (83C05 83C30)},
      MRNUMBER = {2007d:83111},
      MRREVIEWER = {Alan D. Rendall},
      DOI = {10.1002/cpa.20105},
      ZBLNUMBER = {1114.83010},
      }
  • [jag5] Go to document H. Ringström, "Curvature blow up on a dense subset of the singularity in $T^3$-Gowdy," J. Hyperbolic Diff. Equations, vol. 2, iss. 2, pp. 547-564, 2005.
    @article {jag5, MRKEY = {2151121},
      AUTHOR = {Ringstr{ö}m, Hans},
      TITLE = {Curvature blow up on a dense subset of the singularity in {$T\sp 3$}-{G}owdy},
      JOURNAL = {J. Hyperbolic Diff. Equations},
      FJOURNAL = {Journal of Hyperbolic Differential Equations},
      VOLUME = {2},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {547--564},
      ISSN = {0219-8916},
      MRCLASS = {83C75 (35Q75 83C20)},
      MRNUMBER = {2006f:83081},
      MRREVIEWER = {Piotr T. Chru{\'s}ciel},
      DOI = {10.1142/S021989160500052X},
      ZBLNUMBER = {02202331},
      }

Authors

Hans Ringström

Department of Mathematics
Royal Institute of Technology
S-100 44 Stockholm
Sweden