The Gödel-Schrödinger spacetime and stringy chronology protection

  • Charles Max Brown
  • Oliver DeWolfeEmail author


We show that the null Melvin map applied to a rotational isometry of global anti-de Sitter space produces a spacetime with properties analogous to both the Gödel and Schrödinger geometries. The isometry group of this Gödel-Schrödinger spacetime is appropriate to provide a holographic dual for the same non-relativistic conformal theory as the ordinary Schrödinger geometry, but defined on a sphere. This spacetime also possesses closed timelike curves outside a certain critical radius. We show that a holographic preferred screen for an observer at the origin sits at an interior radius, suggesting that a holographic dual description would only require the chronologically consistent region. Additionally, giant graviton probe branes experience repulson-type instabilities in the acausal region, suggesting a condensation of branes will modify the pathological part of the geometry to remove the closed timelike curves.


D-branes AdS-CFT Correspondence Supergravity Models 


  1. [1]
    S. Hawking, The Chronology protection conjecture, Phys. Rev. D 46 (1992) 603 [INSPIRE].MathSciNetADSGoogle Scholar
  2. [2]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    C.P. Herzog, Lectures on Holographic Superfluidity and Superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [INSPIRE].Google Scholar
  4. [4]
    J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [INSPIRE].Google Scholar
  5. [5]
    M. Ammon, Gauge/gravity duality applied to condensed matter systems, Fortsch. Phys. 58 (2010) 1123 [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. [6]
    D.T. Son, Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrodinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].MathSciNetADSGoogle Scholar
  7. [7]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    A. Adams, K. Balasubramanian and J. McGreevy, Hot Spacetimes for Cold Atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].MathSciNetADSGoogle Scholar
  12. [12]
    K. Balasubramanian and J. McGreevy, The Particle number in Galilean holography, JHEP 01 (2011) 137 [arXiv:1007.2184] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    M. Guica, K. Skenderis, M. Taylor and B.C. van Rees, Holography for Schrödinger backgrounds, JHEP 02 (2011) 056 [arXiv:1008.1991] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    A. Adams and J. Wang, Towards a Non-Relativistic Holographic Superfluid, New J. Phys. 13 (2011) 115008 [arXiv:1103.3472] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    M. Alishahiha and O.J. Ganor, Twisted backgrounds, PP waves and nonlocal field theories, JHEP 03 (2003) 006 [hep-th/0301080] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    E.G. Gimon, A. Hashimoto, V.E. Hubeny, O. Lunin and M. Rangamani, Black strings in asymptotically plane wave geometries, JHEP 08 (2003) 035 [hep-th/0306131] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    A. Adams, C.M. Brown, O. DeWolfe and C. Rosen, Charged Schrödinger Black Holes, Phys. Rev. D 80 (2009) 125018 [arXiv:0907.1920] [INSPIRE].MathSciNetADSGoogle Scholar
  18. [18]
    E. Imeroni and A. Sinha, Non-relativistic metrics with extremal limits, JHEP 09 (2009) 096 [arXiv:0907.1892] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    J. Sonner, A Rotating Holographic Superconductor, Phys. Rev. D 80 (2009) 084031 [arXiv:0903.0627] [INSPIRE].ADSGoogle Scholar
  20. [20]
    K. Godel, An Example of a new type of cosmological solutions of Einstein’s field equations of graviation, Rev. Mod. Phys. 21 (1949) 447 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis and H.S. Reall, All supersymmetric solutions of minimal supergravity in five- dimensions, Class. Quant. Grav. 20 (2003) 4587 [hep-th/0209114] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. [22]
    T. Harmark and T. Takayanagi, Supersymmetric Godel universes in string theory, Nucl. Phys. B 662 (2003) 3 [hep-th/0301206] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    R. Bousso, A Covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    R. Bousso, Holography in general space-times, JHEP 06 (1999) 028 [hep-th/9906022] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    E.K. Boyda, S. Ganguli, P. Hořava and U. Varadarajan, Holographic protection of chronology in universes of the Godel type, Phys. Rev. D 67 (2003) 106003 [hep-th/0212087] [INSPIRE].ADSGoogle Scholar
  26. [26]
    L. Dyson, Chronology protection in string theory, JHEP 03 (2004) 024 [hep-th/0302052] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    K. Behrndt, About a class of exact string backgrounds, Nucl. Phys. B 455 (1995) 188 [hep-th/9506106] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    R. Kallosh and A.D. Linde, Exact supersymmetric massive and massless white holes, Phys. Rev. D 52 (1995) 7137 [hep-th/9507022] [INSPIRE].MathSciNetADSGoogle Scholar
  29. [29]
    M. Cvetič and D. Youm, Singular BPS saturated states and enhanced symmetries of four-dimensional N = 4 supersymmetric string vacua, Phys. Lett. B 359 (1995) 87 [hep-th/9507160] [INSPIRE].ADSGoogle Scholar
  30. [30]
    C.V. Johnson, A.W. Peet and J. Polchinski, Gauge theory and the excision of repulson singularities, Phys. Rev. D 61 (2000) 086001 [hep-th/9911161] [INSPIRE].MathSciNetADSGoogle Scholar
  31. [31]
    N. Drukker, B. Fiol and J. Simon, Godel’s universe in a supertube shroud, Phys. Rev. Lett. 91 (2003) 231601 [hep-th/0306057] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    C. Herdeiro, Special properties of five-dimensional BPS rotating black holes, Nucl. Phys. B 582 (2000) 363 [hep-th/0003063] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    C.A. Herdeiro, Spinning deformations of the D1 - D5 system and a geometric resolution of closed timelike curves, Nucl. Phys. B 665 (2003) 189 [hep-th/0212002] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    D. Brace, C.A. Herdeiro and S. Hirano, Classical and quantum strings in compactified pp waves and Godel type universes, Phys. Rev. D 69 (2004) 066010 [hep-th/0307265] [INSPIRE].MathSciNetADSGoogle Scholar
  35. [35]
    D. Israel, Quantization of heterotic strings in a Godel/anti-de Sitter space-time and chronology protection, JHEP 01 (2004) 042 [hep-th/0310158] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    E.G. Gimon and P. Hořava, Over-rotating black holes, Godel holography and the hypertube, hep-th/0405019 [INSPIRE].
  37. [37]
    C.V. Johnson and H.G. Svendsen, An Exact string theory model of closed time-like curves and cosmological singularities, Phys. Rev. D 70 (2004) 126011 [hep-th/0405141] [INSPIRE].MathSciNetADSGoogle Scholar
  38. [38]
    W.-H. Huang, Instability of tachyon supertube in type IIA Godel spacetime, Phys. Lett. B 615 (2005) 266 [hep-th/0412096] [INSPIRE].ADSGoogle Scholar
  39. [39]
    M.S. Costa, C.A. Herdeiro, J. Penedones and N. Sousa, Hagedorn transition and chronology protection in string theory, Nucl. Phys. B 728 (2005) 148 [hep-th/0504102] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    L. Dyson, Studies Of The Over-Rotating BMPV Solution, JHEP 01 (2007) 008 [hep-th/0608137] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    T.S. Levi, J. Raeymaekers, D. Van den Bleeken, W. Van Herck and B. Vercnocke, Godel space from wrapped M2-branes, JHEP 01 (2010) 082 [arXiv:0909.4081] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    J. Raeymaekers, D. Van den Bleeken and B. Vercnocke, Relating chronology protection and unitarity through holography, JHEP 04 (2010) 021 [arXiv:0911.3893] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    J. Raeymaekers, D. Van den Bleeken and B. Vercnocke, Chronology protection and the stringy exclusion principle, JHEP 04 (2011) 037 [arXiv:1011.5693] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    U. Theis, Free Lunch from T-duality, arXiv:1002.4758 [INSPIRE].
  45. [45]
    B.S. Kim and D. Yamada, Properties of Schroedinger Black Holes from AdS Space, JHEP 07 (2011) 120 [arXiv:1008.3286] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    N. Banerjee, S. Dutta and D.P. Jatkar, Geometry and Phase Structure of Non-Relativistic Branes, Class. Quant. Grav. 28 (2011) 165002 [arXiv:1102.0298] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    P. Kraus and E. Perlmutter, Universality and exactness of Schrödinger geometries in string and M-theory, JHEP 05 (2011) 045 [arXiv:1102.1727] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    M. Blau, J. Hartong and B. Rollier, Geometry of Schrödinger Space-Times, Global Coordinates and Harmonic Trapping, JHEP 07 (2009) 027 [arXiv:0904.3304] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    M. Blau, J. Hartong and B. Rollier, Geometry of Schrödinger Space-Times II: Particle and Field Probes of the Causal Structure, JHEP 07 (2010) 069 [arXiv:1005.0760] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    S.A. Hartnoll and K. Yoshida, Families of IIB duals for nonrelativistic CFTs, JHEP 12 (2008) 071 [arXiv:0810.0298] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    N. Bobev, A. Kundu and K. Pilch, Supersymmetric IIB Solutions with Schrödinger Symmetry, JHEP 07 (2009) 107 [arXiv:0905.0673] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    A. Donos and J.P. Gauntlett, Solutions of type IIB and D = 11 supergravity with Schrödinger(z) symmetry, JHEP 07 (2009) 042 [arXiv:0905.1098] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    H. Ooguri and C.-S. Park, Supersymmetric non-relativistic geometries in M-theory, Nucl. Phys. B 824 (2010) 136 [arXiv:0905.1954] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    S.W. Hawking, G.F.R. Ellis, The Large scale structure of space-time, Cambridge University Press, Cambridge U.K. (1973).zbMATHCrossRefGoogle Scholar
  55. [55]
    A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008 [hep-th/0011156] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    R. Bousso and L. Randall, Holographic domains of anti-de Sitter space, JHEP 04 (2002) 057 [hep-th/0112080] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    S. Deser, R. Jackiw and G. ť Hooft, Physical cosmic strings do not generate closed timelike curves, Phys. Rev. Lett. 68 (1992) 267 [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  58. [58]
    D. Mateos and P.K. Townsend, Supertubes, Phys. Rev. Lett. 87 (2001) 011602 [hep-th/0103030] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  59. [59]
    R. Emparan, D. Mateos and P.K. Townsend, Supergravity supertubes, JHEP 07 (2001) 011 [hep-th/0106012] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  60. [60]
    J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from Anti-de Sitter space, JHEP 06 (2000) 008 [hep-th/0003075] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    M.T. Grisaru, R.C. Myers and O. Tafjord, SUSY and goliath, JHEP 08 (2000) 040 [hep-th/0008015] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    A. Hashimoto, S. Hirano and N. Itzhaki, Large branes in AdS and their field theory dual, JHEP 08 (2000) 051 [hep-th/0008016] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  63. [63]
    N. Drukker, B. Fiol and J. Simon, Godel type universes and the Landau problem, JCAP 10 (2004) 012 [hep-th/0309199] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsKentucky State UniversityFrankfortU.S.A.
  2. 2.Department of PhysicsUniversity of ColoradoBoulderU.S.A.

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