Advertisement

3D flat holography: entropy and logarithmic corrections

  • Arjun BagchiEmail author
  • Rudranil Basu
Open Access
Article

Abstract

We compute the leading corrections to the Bekenstein-Hawking entropy of the Flat Space Cosmological (FSC) solutions in 3D flat spacetimes, which are the flat analogues of the BTZ black holes in AdS3. The analysis is done by a computation of density of states in the dual 2D Galilean Conformal Field Theory and the answer obtained by this matches with the limiting value of the expected result for the BTZ inner horizon entropy as well as what is expected for a generic thermodynamic system. Along the way, we also develop other aspects of holography of 3D flat spacetimes.

Keywords

Gauge-gravity correspondence Conformal and W Symmetry 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
  2. [2]
    L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
  4. [4]
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    A. Achucarro and P.K. Townsend, A Chern-Simons action for three-dimensional Anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [gr-qc/9302012] [INSPIRE].ADSGoogle Scholar
  10. [10]
    A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    S. Carlip, What we dont know about BTZ black hole entropy, Class. Quant. Grav. 15 (1998) 3609 [hep-th/9806026] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    D. Anninos, De Sitter musings, Int. J. Mod. Phys. A 27 (2012) 1230013 [arXiv:1205.3855] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    A. Bagchi, Correspondence between asymptotically flat spacetimes and nonrelativistic conformal field theories, Phys. Rev. Lett. 105 (2010) 171601.ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    A. Bagchi, The BMS/GCA correspondence, arXiv:1006.3354 [INSPIRE].
  21. [21]
    A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004 [arXiv:0912.1090] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    A. Bagchi, Topologically massive gravity and galilean conformal algebra: a study of correlation functions, JHEP 02 (2011) 091 [arXiv:1012.3316] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    A. Bagchi and R. Fareghbal, BMS/GCA redux: towards flatspace holography from non-relativistic symmetries, JHEP 10 (2012) 092 [arXiv:1203.5795] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    G. Barnich, A. Gomberoff and H.A. Gonzalez, The flat limit of three dimensional asymptotically Anti-de Sitter spacetimes, Phys. Rev. D 86 (2012) 024020 [arXiv:1204.3288] [INSPIRE].ADSGoogle Scholar
  26. [26]
    A. Bagchi, S. Detournay and D. Grumiller, Flat-space chiral gravity, Phys. Rev. Lett. 109 (2012) 151301 [arXiv:1208.1658] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    A. Bagchi, S. Detournay, R. Fareghbal and J. Simon, Holography of 3d flat cosmological horizons, Phys. Rev. Lett. 110 (2013) 141302 [arXiv:1208.4372] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [arXiv:1208.4371] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    G. Barnich, A. Gomberoff and H.A. González, Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory, Phys. Rev. D 87 (2013) 124032 [arXiv:1210.0731] [INSPIRE].ADSGoogle Scholar
  30. [30]
    A. Bagchi, S. Detournay, D. Grumiller and J. Simon, Cosmic evolution from phase transition of 3-dimensional flat space, Phys. Rev. Lett. 111 (2013) 181301 [arXiv:1305.2919] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Higher spin theory in 3-dimensional flat space, Phys. Rev. Lett. 111 (2013) 121603 [arXiv:1307.4768] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    H.A. Gonzalez, J. Matulich, M. Pino and R. Troncoso, Asymptotically flat spacetimes in three-dimensional higher spin gravity, JHEP 09 (2013) 016 [arXiv:1307.5651] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    A. Bagchi and D. Grumiller, Holograms of flat space, Int. J. Mod. Phys. D 22 (2013) 1342003 [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    R.N.C. Costa, Aspects of the zero Λ limit in the AdS/CFT correspondence, arXiv:1311.7339 [INSPIRE].
  35. [35]
    C. Krishnan and S. Roy, Desingularization of the Milne universe, arXiv:1311.7315 [INSPIRE].
  36. [36]
    C. Krishnan, A. Raju and S. Roy, A Grassmann path from AdS 3 to flat space, arXiv:1312.2941 [INSPIRE].
  37. [37]
    R. Fareghbal and A. Naseh, Flat-space energy-momentum tensor from BMS/GCA correspondence, arXiv:1312.2109 [INSPIRE].
  38. [38]
    L. Cornalba and M.S. Costa, A new cosmological scenario in string theory, Phys. Rev. D 66 (2002) 066001 [hep-th/0203031] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    L. Cornalba and M.S. Costa, Time dependent orbifolds and string cosmology, Fortsch. Phys. 52 (2004) 145 [hep-th/0310099] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  40. [40]
    A. Bagchi and I. Mandal, On representations and correlation functions of galilean conformal algebras, Phys. Lett. B 675 (2009) 393 [arXiv:0903.4524] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  41. [41]
    A. Bagchi, Tensionless strings and galilean conformal algebra, JHEP 05 (2013) 141 [arXiv:1303.0291] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    S.W. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  43. [43]
    P. Kraus, Lectures on black holes and the AdS 3 /CF T 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    A. Sen, Logarithmic corrections to Schwarzschild and other non-extremal black hole entropy in different dimensions, JHEP 04 (2013) 156 [arXiv:1205.0971] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    S. Detournay, D. Grumiller, F. Scholler and J. Simon, Variational principle and 1-point functions in 3-dimensional flat space Einstein gravity, to appear.Google Scholar
  46. [46]
    M. Cvetič and F. Larsen, General rotating black holes in string theory: Grey body factors and event horizons, Phys. Rev. D 56 (1997) 4994 [hep-th/9705192] [INSPIRE].ADSGoogle Scholar
  47. [47]
    A. Castro and M.J. Rodriguez, Universal properties and the first law of black hole inner mechanics, Phys. Rev. D 86 (2012) 024008 [arXiv:1204.1284] [INSPIRE].ADSGoogle Scholar
  48. [48]
    S. Detournay, Inner mechanics of 3D black holes, Phys. Rev. Lett. 109 (2012) 031101 [arXiv:1204.6088] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    S. Das, P. Majumdar and R.K. Bhaduri, General logarithmic corrections to black hole entropy, Class. Quant. Grav. 19 (2002) 2355 [hep-th/0111001] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  50. [50]
    W. Li, W. Song and A. Strominger, Chiral gravity in three dimensions, JHEP 04 (2008) 082 [arXiv:0801.4566] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  51. [51]
    S.N. Solodukhin, Holography with gravitational Chern-Simons, Phys. Rev. D 74 (2006) 024015 [hep-th/0509148] [INSPIRE].ADSGoogle Scholar
  52. [52]
    Y. Tachikawa, Black hole entropy in the presence of Chern-Simons terms, Class. Quant. Grav. 24 (2007) 737 [hep-th/0611141] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  53. [53]
    K.A. Moussa, G. Clement and C. Leygnac, The black holes of topologically massive gravity, Class. Quant. Grav. 20 (2003) L277 [gr-qc/0303042] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  54. [54]
    S. Olmez, O. Sarioglu and B. Tekin, Mass and angular momentum of asymptotically AdS or flat solutions in the topologically massive gravity, Class. Quant. Grav. 22 (2005) 4355 [gr-qc/0507003] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  55. [55]
    F. Loran, M. Sheikh-Jabbari and M. Vincon, Beyond logarithmic corrections to Cardy formula, JHEP 01 (2011) 110 [arXiv:1010.3561] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Indian Institute of Science Education and ResearchPashanIndia
  2. 2.Centre for High Energy PhysicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations