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Pathologies in asymptotically Lifshitz spacetimes

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Abstract

There has been significant interest in the last several years in studying possible gravitational duals, known as Lifshitz spacetimes, to anisotropically scaling field theories by adding matter to distort the asymptotics of an AdS spacetime. We point out that the putative ground state for the most heavily studied example of such a spacetime, that with a flat spatial section, suffers from a naked singularity. Furthermore, stringy effects can not resolve this singularity without producing significant quantum corrections to the entire spacetime, including the asymptotic region. We review the reasons one might worry that asymptotically Lifshitz spacetimes are unstable and employ the initial data problem to study the stability of such systems. Rather surprisingly, this question and even the initial value problem itself for these spacetimes turns out to generically not be well-posed. A generic normalizable state will evolve in such a way to violate Lifshitz asymptotics in finite time. Conversely, enforcing the desired asymptotics at all times puts strong restrictions not just on the metric and fields in the asymptotic region but in the deep interior as well. Generically, even perturbations of the matter field of compact support are not compatible with the desired asymptotics.

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References

  1. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [SPIRES].

    Article  MathSciNet  MATH  Google Scholar 

  2. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  3. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].

    MathSciNet  MATH  Google Scholar 

  4. P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [SPIRES].

    Article  ADS  Google Scholar 

  5. S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  6. S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [SPIRES].

    Article  ADS  Google Scholar 

  7. S.A. Hartnoll and P. Kovtun, Hall conductivity from dyonic black holes, Phys. Rev. D 76 (2007) 066001 [arXiv:0704.1160] [SPIRES].

    ADS  Google Scholar 

  8. C.P. Herzog, Lectures on Holographic Superfluidity and Superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [SPIRES].

    Google Scholar 

  9. T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS2, arXiv:0907.2694 [SPIRES].

  10. J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [SPIRES].

    Google Scholar 

  11. A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. V. Balasubramanian, J. de Boer and D. Minic, Mass, entropy and holography in asymptotically de Sitter spaces, Phys. Rev. D 65 (2002) 123508 [hep-th/0110108] [SPIRES].

    ADS  Google Scholar 

  13. A.M. Ghezelbash and R.B. Mann, Action, mass and entropy of Schwarzschild-de Sitter black holes and the de Sitter/CFT correspondence, JHEP 01 (2002) 005 [hep-th/0111217] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  14. R.B. Mann and D. Marolf, Holographic renormalization of asymptotically flat spacetimes, Class. Quant. Grav. 23 (2006) 2927 [hep-th/0511096] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. D. Marolf, A symptotic flatness, little string theory and holography, JHEP 03 (2007) 122 [hep-th/0612012] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  16. R.B. Mann, D. Marolf, R. McNees and A. Virmani, On the Stress Tensor for Asymptotically Flat Gravity, Class. Quant. Grav. 25 (2008) 225019.

    Article  MathSciNet  ADS  Google Scholar 

  17. S. Kachru, X. Liu and M. Mulligan, Gravity Duals of Lifshitz-like Fixed Points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  18. P. Koroteev and M. Libanov, On Existence of Self-Tuning Solutions in Static Braneworlds without Singularities, JHEP 02 (2008) 104 [arXiv:0712.1136] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  19. M. Taylor, Non-relativistic holography, arXiv:0812.0530 [SPIRES].

  20. R.B. Mann, Lifshitz Topological Black Holes, JHEP 06 (2009) 075 [arXiv:0905.1136] [SPIRES].

    Article  ADS  Google Scholar 

  21. G. Bertoldi, B.A. Burrington and A. Peet, Black Holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent, Phys. Rev. D 80 (2009) 126003 [arXiv:0905.3183] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  22. K. Balasubramanian and J. McGreevy, An analytic Lifshitz black hole, Phys. Rev. D 80 (2009) 104039 [arXiv:0909.0263] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  23. U.H. Danielsson and L. Thorlacius, Black holes in asymptotically Lifshitz spacetime, JHEP 03 (2009) 070 [arXiv:0812.5088] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  24. P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [SPIRES].

    ADS  Google Scholar 

  25. A. Donos and J.P. Gauntlett, Lifshitz Solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  26. J.L. Friedman, K. Schleich and D.M. Witt, Topological censorship, Phys. Rev. Lett. 71 (1993) 1486 [gr-qc/9305017] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  27. G.J. Galloway, K. Schleich, D. Witt and E. Woolgar, The AdS/CFT correspondence conjecture and topological censorship, Phys. Lett. B 505 (2001) 255 [hep-th/9912119] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  28. O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  29. E. Witten, A Simple Proof of the Positive Energy Theorem, Commun. Math. Phys. 80 (1981) 381 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. G.W. Gibbons, C.M. Hull and N.P. Warner, The Stability of Gauged Supergravity, Nucl. Phys. B 218 (1983) 173 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. W. Boucher, Positive energy without supersymmetry, Nucl. Phys. B 242 (1984) 282 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  32. P.K. Townsend, Positive energy and the scalar potential in higher dimensional (super)gravity theories, Phys. Lett. B 148 (1984) 55 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  33. E. Witten, Instability of the Kaluza-Klein Vacuum, Nucl. Phys. B 195 (1982) 481 [SPIRES].

    Article  ADS  Google Scholar 

  34. D. Brill and H. Pfister, States of negative total energy in Kaluza-Klein theory, Phys. Lett. B 228 (1989) 359 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  35. D. Brill and G.T. Horowitz, Negative energy in string theory, Phys. Lett. B 262 (1991) 437 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  36. K. Copsey and R.B. Mann, States of Negative Energy and AdS 5 × S 5 /Z k , JHEP 05 (2008) 069 [arXiv:0803.3801] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  37. G.T. Horowitz and A.R. Steif, Space-Time Singularities in String Theory, Phys. Rev. Lett. 64 (1990) 260 [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. R. Gregory, S.L. Parameswaran, G. Tasinato and I. Zavala, Lifshitz solutions in supergravity and string theory, JHEP 12 (2010) 047 [arXiv:1009.3445] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  39. G.T. Horowitz and S.F. Ross, Naked black holes, Phys. Rev. D 56 (1997) 2180 [hep-th/9704058] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  40. S.S. Gubser, Curvature singularities: The good, the bad and the naked, Adv. Theor. Math. Phys. 4 (2000) 679 [hep-th/0002160] [SPIRES].

    MathSciNet  MATH  Google Scholar 

  41. G.T. Horowitz and R.C. Myers, The value of singularities, Gen. Rel. Grav. 27 (1995) 915 [gr-qc/9503062] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  42. S.F. Ross and O. Saremi, Holographic stress tensor for non-relativistic theories, JHEP 09 (2009) 009 [arXiv:0907.1846] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  43. M.H. Dehghani and R.B. Mann, Thermodynamics of Lovelock-Lifshitz Black Branes, Phys. Rev. D 82 (2010) 064019 [arXiv:1006.3510] [SPIRES].

    ADS  Google Scholar 

  44. R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, gr-qc/0405109 [SPIRES].

  45. M. Henneaux and C. Teitelboim, Hamiltonian treatment of asymptotically anti-de Sitter spaces, Phys. Lett. B 142 (1984) 355 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  46. J.P.S. Lemos, Cylindrical black hole in general relativity, Phys. Lett. B 353 (1995) 46 [gr-qc/9404041] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  47. J.P.S. Lemos, Two-dimensional black holes and planar general relativity, Class. Quant. Grav. 12 (1995) 1081 [gr-qc/9407024] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. S. Aminneborg, I. Bengtsson, S. Holst and P. Peldan, Making Anti-de Sitter Black Holes, Class. Quant. Grav. 13 (1996) 2707 [gr-qc/9604005] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. R.B. Mann, Pair production of topological anti-de Sitter black holes, Class. Quant. Grav. 14 (1997) L109 [gr-qc/9607071] [SPIRES].

    Article  ADS  Google Scholar 

  50. D.R. Brill, J. Louko and P. Peldán, Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons, Phys. Rev. D 56 (1997) 3600 [gr-qc/9705012] [SPIRES].

    ADS  Google Scholar 

  51. L. Vanzo, Black holes with unusual topology, Phys. Rev. D 56 (1997) 6475 [gr-qc/9705004] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  52. R.B. Mann, Topological black holes: Outside looking in, gr-qc/9709039 [SPIRES].

  53. J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Lovelock Gravities and Black Holes, JHEP 06 (2010) 008 [arXiv:0912.1877] [SPIRES].

    Article  Google Scholar 

  54. X.O. Camanho and J.D. Edelstein, Causality in AdS/CFT and Lovelock theory, JHEP 06 (2010) 099 [arXiv:0912.1944] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  55. M.H. Dehghani and R.B. Mann, Lovelock-Lifshitz Black Holes, JHEP 07 (2010) 019 [arXiv:1004.4397] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  56. G. Dotti, J. Oliva and R. Troncoso, Exact solutions for the Einstein-Gauss-Bonnet theory in five dimensions: Black holes, wormholes and spacetime horns, Phys. Rev. D 76 (2007) 064038 [arXiv:0706.1830] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  57. E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, Lifshitz Black Hole in Three Dimensions, Phys. Rev. D 80 (2009) 104029 [arXiv:0909.1347] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  58. E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, Analytic Lifshitz black holes in higher dimensions, JHEP 04 (2010) 030 [arXiv:1001.2361] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  59. R.-G. Cai, Y. Liu and Y.-W. Sun, A Lifshitz Black Hole in Four Dimensional R 2 Gravity, JHEP 10 (2009) 080 [arXiv:0909.2807] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  60. D.-W. Pang, On Charged Lifshitz Black Holes, JHEP 01 (2010) 116 [arXiv:0911.2777] [SPIRES].

    Article  ADS  Google Scholar 

  61. R.C. Myers and B. Robinson, Black Holes in Quasi-topological Gravity, JHEP 08 (2010) 067 [arXiv:1003.5357] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  62. M.H. Dehghani and M. Shamirzaie, Thermodynamics of asymptotic flat charged black holes in third order Lovelock gravity, Phys. Rev. D 72 (2005) 124015 [hep-th/0506227] [SPIRES].

    ADS  Google Scholar 

  63. V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  64. M.H. Dehghani and R. Pourhasan, Thermodynamic instability of black holes of third order Lovelock gravity, Phys. Rev. D 79 (2009) 064015 [arXiv:0903.4260] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  65. S. Hollands, A. Ishibashi and D. Marolf, Comparison between various notions of conserved charges in asymptotically AdS-spacetimes, Class. Quant. Grav. 22 (2005) 2881 [hep-th/0503045] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. C. Fefferman and C. Robin Graham, Conformal Invariants, in Elie Cartan et les Mathématiques d’aujourd’hui, Astérisque (1985).

  67. H. Friedrich, Einstein equations and conformal structure - Existence of anti de Sitter type space-times, J. Geom. Phys. 17 (1995) 125 [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. A. Donos, J.P. Gauntlett, N. Kim and O. Varela, Wrapped M5-branes, consistent truncations and AdS/CMT, JHEP 12 (2010) 003 [arXiv:1009.3805] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  69. S.S. Gubser and A. Nellore, Ground states of holographic superconductors, Phys. Rev. D 80 (2009) 105007 [arXiv:0908.1972] [SPIRES].

    ADS  Google Scholar 

  70. S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [SPIRES].

    Article  ADS  Google Scholar 

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  1. Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Keith Copsey & Robert Mann

  2. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

    Keith Copsey

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Copsey, K., Mann, R. Pathologies in asymptotically Lifshitz spacetimes. J. High Energ. Phys. 2011, 39 (2011). https://doi.org/10.1007/JHEP03(2011)039

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