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Journal of High Energy Physics

, 2012:24 | Cite as

Supertranslations and holographic stress tensor

  • Amitabh Virmani
Article

Abstract

It is well known in the context of four dimensional asymptotically flat space-times that the leading order boundary metric must be conformal to unit de Sitter metric when hyperbolic cutoffs are used. This situation is very different from asymptotically AdS settings where one is allowed to choose an arbitrary boundary metric. The closest one can come to changing the boundary metric in the asymptotically flat context, while maintaining the group of asymptotic symmetries to be Poincaré, is to change the so-called ‘supertranslation frame’ ω. The most studied choice corresponds to taking ω = 0. In this paper we study consequences of making alternative choices. We perform this analysis in the covariant phase space approach as well as in the holographic renormalization approach. We show that all choices for ω are allowed in the sense that the covariant phase space is well defined irrespective of how we choose to fix supertranslations. The on-shell action and the leading order boundary stress tensor are insensitive to the supertranslation frame. The next to leading order boundary stress tensor depends on the supertranslation frame but only in a way that the transformation of angular momentum under translations continues to hold as in special relativity.

Keywords

Gauge-gravity correspondence Classical Theories of Gravity 

References

  1. [1]
    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) 1133] [hep-th/9711200] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  2. [2]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  3. [3]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in anti-de Sitter space-time, Phys. Rev. D 59 (1999) 046003 [hep-th/9805171] [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  7. [7]
    M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    R. Beig and B. Schmidt, Einsteins equations near spatial infinity, Commun. Math. Phys. 87 (1982) 65.MathSciNetADSMATHCrossRefGoogle Scholar
  10. [10]
    A. Ashtekar, L. Bombelli and O. Reula, The covariant phase space of asymptotically flat gravitational fields, in Analysis, geometry and mechanics: 200 years after Lagrange, M. Francaviglia and D. Holm eds., North-Holland, Amsterdam The Netherlands (1991).Google Scholar
  11. [11]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  12. [12]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].MathSciNetADSGoogle Scholar
  13. [13]
    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] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    R.M. Wald and A. Zoupas, A general definition ofconserved quantitiesin general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].MathSciNetADSGoogle Scholar
  15. [15]
    R.B. Mann and D. Marolf, Holographic renormalization of asymptotically flat spacetimes, Class. Quant. Grav. 23 (2006) 2927 [hep-th/0511096] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  16. [16]
    R.B. Mann, D. Marolf and A. Virmani, Covariant counterterms and conserved charges in asymptotically flat spacetimes, Class. Quant. Grav. 23 (2006) 6357 [gr-qc/0607041] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  17. [17]
    R.B. Mann, D. Marolf, R. McNees and A. Virmani, On the stress tensor for asymptotically flat gravity, Class. Quant. Grav. 25 (2008) 225019 [arXiv:0804.2079] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    A. Ashtekar and R. Hansen, A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978) 1542 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    A. Ashtekar and J.D. Romano, Spatial infinity as a boundary of space-time, Class. Quant. Grav. 9 (1992) 1069 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  20. [20]
    A. Ashtekar and A. Magnon, From i 0 to the 3 + 1 description of spatial infinity, J. Math. Phys. 25 (1984) 2682.MathSciNetADSMATHCrossRefGoogle Scholar
  21. [21]
    R. Beig, Integration of Einsteins equations near spatial infinity, Proc. Roy. Soc. Lond. A 391 (1984) 295.MathSciNetADSGoogle Scholar
  22. [22]
    G. Compere, F. Dehouck and A. Virmani, On asymptotic flatness and Lorentz charges, Class. Quant. Grav. 28 (2011) 145007 [arXiv:1103.4078] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    J. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].MathSciNetADSGoogle Scholar
  24. [24]
    S. Deser, R. Arnowitt and C. Misner, Heisenberg representation in classical general relativity, Nuovo Cim. 19 (1961) 668 [INSPIRE].MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    S. Deser, R. Arnowitt and C. Misner, Consistency of canonical reduction of general relativity, J. Math. Phys. 1 (1960) 434 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  26. [26]
    R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, gr-qc/0405109 [INSPIRE].
  27. [27]
    T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Annals Phys. 88 (1974) 286 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  28. [28]
    R. Geroch, Asymptotic structure of space-time, in Proceedings of a Symposium on the asymptotic structure of space-time, University of Cincinnati, Cincinnati U.S.A. (1976), P. Esposito and L. Witten eds., Plenum Press, New York U.S.A. (1977).Google Scholar
  29. [29]
    A. Ashtekar, Asymptotic structure of the gravitational field at spatial infinity, in General relativity and gravitation: one hundred years after the birth of Albert Einstein, A. Held eds., Plenum Press, New York U.S.A. (1980).Google Scholar
  30. [30]
    L. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B 195 (1982) 76 [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    S. Deser and B. Tekin, Gravitational energy in quadratic curvature gravities, Phys. Rev. Lett. 89 (2002) 101101 [hep-th/0205318] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    S. Deser and B. Tekin, Energy in generic higher curvature gravity theories, Phys. Rev. D 67 (2003) 084009 [hep-th/0212292] [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    S. Deser and B. Tekin, New energy definition for higher curvature gravities, Phys. Rev. D 75 (2007) 084032 [gr-qc/0701140] [INSPIRE].MathSciNetADSGoogle Scholar
  34. [34]
    G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  36. [36]
    R.D. Sorkin, Conserved Quantities as Action Variations, in AMSs Contemporary Mathematics series. Vol. 71: Mathematics and General Relativity, J.W. Isenberg eds., American Mathematical Society Press, Providence U.S.A. (1988) pp. 23-37.Google Scholar
  37. [37]
    A. Virmani, Asymptotic flatness, Taub-NUT and variational principle, Phys. Rev. D 84 (2011) 064034 [arXiv:1106.4372] [INSPIRE].ADSGoogle Scholar
  38. [38]
    D. Marolf, Asymptotic flatness, little string theory and holography, JHEP 03 (2007) 122 [hep-th/0612012] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    D. Marolf and A. Virmani, Holographic renormalization of gravity in little string theory duals, JHEP 06 (2007) 042 [hep-th/0703251] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    G. Compere and F. Dehouck, Relaxing the parity conditions of asymptotically flat gravity, Class. Quant. Grav. 28 (2011) 245016 [arXiv:1106.4045] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    G. Barnich and P.-H. Lambert, A note on the Newman-Unti group, arXiv:1102.0589 [INSPIRE].
  43. [43]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 [arXiv:1102.4632] [INSPIRE].
  44. [44]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    A. Ashtekar, J. Engle and D. Sloan, Asymptotics and Hamiltonians in a first order formalism, Class. Quant. Grav. 25 (2008) 095020 [arXiv:0802.2527] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    J. Le Witt and S.F. Ross, Asymptotically plane wave spacetimes and their actions, JHEP 04 (2008) 084 [arXiv:0801.4412] [INSPIRE].CrossRefGoogle Scholar
  47. [47]
    J. Le Witt and S.F. Ross, Black holes and black strings in plane waves, JHEP 01 (2010) 101 [arXiv:0910.4332] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    S. Hollands, A. Ishibashi and D. Marolf, Counter-term charges generate bulk symmetries, Phys. Rev. D 72 (2005) 104025 [hep-th/0503105] [INSPIRE].MathSciNetADSGoogle Scholar
  49. [49]
    R.B. Mann and R. McNees, Boundary terms unbound! Holographic renormalization of asymptotically linear dilaton gravity, Class. Quant. Grav. 27 (2010) 065015 [arXiv:0905.3848] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    T. Wiseman and B. Withers, Holographic renormalization for coincident Dp-branes, JHEP 10 (2008) 037 [arXiv:0807.0755] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    S.F. Ross, Holography for asymptotically locally Lifshitz spacetimes, Class. Quant. Grav. 28 (2011) 215019 [arXiv:1107.4451] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    M. Baggio, J. de Boer and K. Holsheimer, Hamilton-Jacobi renormalization for Lifshitz spacetime, JHEP 01 (2012) 058 [arXiv:1107.5562] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    R.B. Mann and R. McNees, Holographic renormalization for asymptotically Lifshitz spacetimes, JHEP 10 (2011) 129 [arXiv:1107.5792] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    J. de Boer and S.N. Solodukhin, A holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    G. Arcioni and C. Dappiaggi, Exploring the holographic principle in asymptotically flat space-times via the BMS group, Nucl. Phys. B 674 (2003) 553 [hep-th/0306142] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    E. Alvarez, J. Conde and L. Hernandez, Goursats problem and the holographic principle, Nucl. Phys. B 689 (2004) 257 [hep-th/0401220] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    J.L. Barbon and C.A. Fuertes, Holographic entanglement entropy probes (non)locality, JHEP 04 (2008) 096 [arXiv:0803.1928] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    W. Li and T. Takayanagi, Holography and entanglement in flat spacetime, Phys. Rev. Lett. 106 (2011) 141301 [arXiv:1010.3700] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    G. Compere, P. McFadden, K. Skenderis and M. Taylor, The holographic fluid dual to vacuum Einstein gravity, JHEP 07 (2011) 050 [arXiv:1103.3022] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  60. [60]
    J.M. Martín-García, xAct: Efficient tensor computer algebra, http://www.xact.es/.
  61. [61]
    J.M. Martín-García, R. Portugal and L. Manssur, The Invar tensor package, Computer Physics Communications 177 (2007) 640 [arXiv:0704.1756].ADSMATHCrossRefGoogle Scholar
  62. [62]
    J.M. Martín-García, D. Yllanes and R. Portugal, The Invar tensor package: differential invariants of Riemann, Comput. Phys. Commun. 179 (2008) 586 [arXiv:0802.1274] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  63. [63]
    J.M. Martín-García, xPerm: fast index canonicalization for tensor computer algebra, Computer Physics Communications 179 (2008) 597 [arXiv:0803.0862].ADSMATHCrossRefGoogle Scholar
  64. [64]
    D. Brizuela, J.M. Martín-García and G.A. Mena Marugan, xPert: computer algebra for metric perturbation theory, Gen. Rel. Grav. 41 (2009) 2415 [arXiv:0807.0824] [INSPIRE].ADSMATHCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)GolmGermany

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