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Quantum gravity of Kerr-Schild spacetimes and the logarithmic correction to Schwarzschild black hole entropy

  • Basem Kamal El-Menoufi
Open Access
Regular Article - Theoretical Physics

Abstract

In the context of effective field theory, we consider quantum gravity with minimally coupled massless particles. Fixing the background geometry to be of the Kerr-Schild type, we fully determine the one-loop effective action of the theory whose finite non-local part is induced by the long-distance portion of quantum loops. This is accomplished using the non-local expansion of the heat kernel in addition to a non-linear completion technique through which the effective action is expanded in gravitational curvatures. Via Euclidean methods, we identify a logarithmic correction to the Bekenstein-Hawking entropy of Schwarzschild black hole. Using dimensional transmutation the result is shown to exhibit an interesting interplay between the UV and IR properties of quantum gravity.

Keywords

Black Holes Effective field theories Models of Quantum Gravity 

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]
    J.F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett. 72 (1994) 2996 [gr-qc/9310024] [INSPIRE].
  2. [2]
    J.F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].
  3. [3]
    N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, Phys. Rev. D 67 (2003) 084033 [Erratum ibid. D 71 (2005) 069903] [hep-th/0211072] [INSPIRE].
  4. [4]
    J.F. Donoghue and B.R. Holstein, Low Energy Theorems of Quantum Gravity from Effective Field Theory, J. Phys. G 42 (2015) 103102 [arXiv:1506.00946] [INSPIRE].CrossRefADSGoogle Scholar
  5. [5]
    N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum corrections to the Schwarzschild and Kerr metrics, Phys. Rev. D 68 (2003) 084005 [Erratum ibid. D 71 (2005) 069904] [hep-th/0211071] [INSPIRE].
  6. [6]
    J.F. Donoghue and B.K. El-Menoufi, QED trace anomaly, non-local Lagrangians and quantum Equivalence Principle violations, JHEP 05 (2015) 118 [arXiv:1503.06099] [INSPIRE].CrossRefADSGoogle Scholar
  7. [7]
    J.F. Donoghue and B.K. El-Menoufi, Nonlocal quantum effects in cosmology: Quantum memory, nonlocal FLRW equations and singularity avoidance, Phys. Rev. D 89 (2014) 104062 [arXiv:1402.3252] [INSPIRE].ADSGoogle Scholar
  8. [8]
    J.F. Donoghue and B.K. El-Menoufi, Covariant non-local action for massless QED and the curvature expansion, JHEP 10 (2015) 044 [arXiv:1507.06321] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  9. [9]
    A.O. Barvinsky and G.A. Vilkovisky, The generalized Schwinger-De Witt technique and the unique effective action in quantum gravity, Phys. Lett. B 131 (1983) 313 [INSPIRE].CrossRefADSGoogle Scholar
  10. [10]
    A.O. Barvinsky and G.A. Vilkovisky, The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity, Phys. Rept. 119 (1985) 1 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  11. [11]
    A.O. Barvinsky and G.A. Vilkovisky, Beyond the Schwinger-Dewitt Technique: Converting Loops Into Trees and In-In Currents, Nucl. Phys. B 282 (1987) 163 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  12. [12]
    A.O. Barvinsky and G.A. Vilkovisky, Covariant perturbation theory. 2: Second order in the curvature. General algorithms, Nucl. Phys. B 333 (1990) 471 [INSPIRE].
  13. [13]
    A.O. Barvinsky, Y.V. Gusev, G.A. Vilkovisky and V.V. Zhytnikov, Asymptotic behaviors of the heat kernel in covariant perturbation theory, J. Math. Phys. 35 (1994) 3543 [gr-qc/9404063] [INSPIRE].
  14. [14]
    I.G. Avramidi, The Covariant Technique for Calculation of One Loop Effective Action, Nucl. Phys. B 355 (1991) 712 [Erratum ibid. B 509 (1998) 557] [INSPIRE].
  15. [15]
    A. Codello and O. Zanusso, On the non-local heat kernel expansion, J. Math. Phys. 54 (2013) 013513 [arXiv:1203.2034] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  16. [16]
    D. Espriu, T. Multamaki and E.C. Vagenas, Cosmological significance of one-loop effective gravity, Phys. Lett. B 628 (2005) 197 [gr-qc/0503033] [INSPIRE].
  17. [17]
    J.A. Cabrer and D. Espriu, Secular effects on inflation from one-loop quantum gravity, Phys. Lett. B 663 (2008) 361 [arXiv:0710.0855] [INSPIRE].CrossRefADSGoogle Scholar
  18. [18]
    R.P. Woodard, Perturbative Quantum Gravity Comes of Age, Int. J. Mod. Phys. D 23 (2014) 1430020 [arXiv:1407.4748] [INSPIRE].CrossRefADSMATHGoogle Scholar
  19. [19]
    N.C. Tsamis and R.P. Woodard, A Caveat on Building Nonlocal Models of Cosmology, JCAP 09 (2014) 008 [arXiv:1405.4470] [INSPIRE].CrossRefADSGoogle Scholar
  20. [20]
    S. Deser and R.P. Woodard, Observational Viability and Stability of Nonlocal Cosmology, JCAP 11 (2013) 036 [arXiv:1307.6639] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  21. [21]
    S. Deser and R.P. Woodard, Nonlocal Cosmology, Phys. Rev. Lett. 99 (2007) 111301 [arXiv:0706.2151] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  22. [22]
    X. Calmet, D. Croon and C. Fritz, Non-locality in Quantum Field Theory due to General Relativity, Eur. Phys. J. C 75 (2015) 605 [arXiv:1505.04517] [INSPIRE].CrossRefADSGoogle Scholar
  23. [23]
    M. Maggiore, Dark energy and dimensional transmutation in R 2 gravity, arXiv:1506.06217 [INSPIRE].
  24. [24]
    S. Jhingan, S. Nojiri, S.D. Odintsov, M. Sami, I. Thongkool and S. Zerbini, Phantom and non-phantom dark energy: The cosmological relevance of non-locally corrected gravity, Phys. Lett. B 663 (2008) 424 [arXiv:0803.2613] [INSPIRE].CrossRefADSGoogle Scholar
  25. [25]
    S. Nojiri and S.D. Odintsov, Modified non-local-F(R) gravity as the key for the inflation and dark energy, Phys. Lett. B 659 (2008) 821 [arXiv:0708.0924] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  26. [26]
    J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].MathSciNetADSGoogle Scholar
  27. [27]
    S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
  28. [28]
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  29. [29]
    A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  30. [30]
    C. Rovelli, Black hole entropy from loop quantum gravity, Phys. Rev. Lett. 77 (1996) 3288 [gr-qc/9603063] [INSPIRE].
  31. [31]
    T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
  32. [32]
    T. Jacobson and R.C. Myers, Black hole entropy and higher curvature interactions, Phys. Rev. Lett. 70 (1993) 3684 [hep-th/9305016] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  33. [33]
    J.F. Donoghue, The effective field theory treatment of quantum gravity, AIP Conf. Proc. 1483 (2012) 73 [arXiv:1209.3511] [INSPIRE].CrossRefADSGoogle Scholar
  34. [34]
    C.P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel. 7 (2004) 5 [gr-qc/0311082] [INSPIRE].
  35. [35]
    R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963) 237 [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  36. [36]
    R.P. Kerr and A. Schild, Some algebraically degenerate solutions of Einstein’s gravitational field equations, Applications of nonlinear partial differential equations in mathematical physics, Proceedings of symposia in applied mathematics, Vol. XVII, Amer. Math. Soc., Providence, R.I., U.S.A. (1965), pg. 199.Google Scholar
  37. [37]
    G.C. Debney, R.P. Kerr and A. Schild, Solutions of the Einstein and Einstein-Maxwell Equations, J. Math. Phys. 10 (1969) 1842 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  38. [38]
    S. Chandrasekhar, The mathematical theory of black holes, Clarendon, Oxford, U.K. (1985).MATHGoogle Scholar
  39. [39]
    H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press, Cambridge U.K.Google Scholar
  40. [40]
    D.V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51 (1995) 5352 [hep-th/9412161] [INSPIRE].MathSciNetADSGoogle Scholar
  41. [41]
    S. Banerjee, R.K. Gupta and A. Sen, Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function, JHEP 03 (2011) 147 [arXiv:1005.3044] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  42. [42]
    S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Logarithmic Corrections to N = 4 and N =8 Black Hole Entropy: A One Loop Test of Quantum Gravity, JHEP 11 (2011) 143 [arXiv:1106.0080] [INSPIRE].CrossRefADSMATHGoogle Scholar
  43. [43]
    A. Sen, Logarithmic Corrections to Rotating Extremal Black Hole Entropy in Four and Five Dimensions, Gen. Rel. Grav. 44 (2012) 1947 [arXiv:1109.3706] [INSPIRE].MathSciNetCrossRefADSMATHGoogle 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].MathSciNetCrossRefADSGoogle Scholar
  45. [45]
    S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].
  46. [46]
    T.R. Govindarajan, R.K. Kaul and V. Suneeta, Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole, Class. Quant. Grav. 18 (2001) 2877 [gr-qc/0104010] [INSPIRE].
  47. [47]
    R. Banerjee and B.R. Majhi, Quantum Tunneling Beyond Semiclassical Approximation, JHEP 06 (2008) 095 [arXiv:0805.2220] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  48. [48]
    R. Banerjee and B.R. Majhi, Quantum Tunneling, Trace Anomaly and Effective Metric, Phys. Lett. B 674 (2009) 218 [arXiv:0808.3688] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  49. [49]
    R. Aros, D.E. Diaz and A. Montecinos, Logarithmic correction to BH entropy as Noether charge, JHEP 07 (2010) 012 [arXiv:1003.1083] [INSPIRE].CrossRefADSMATHGoogle Scholar
  50. [50]
    R.-G. Cai, L.-M. Cao and N. Ohta, Black Holes in Gravity with Conformal Anomaly and Logarithmic Term in Black Hole Entropy, JHEP 04 (2010) 082 [arXiv:0911.4379] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  51. [51]
    R.K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84 (2000) 5255 [gr-qc/0002040] [INSPIRE].
  52. [52]
    T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].CrossRefADSMATHGoogle Scholar
  53. [53]
    T.D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  54. [54]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516.MathSciNetCrossRefADSGoogle Scholar
  55. [55]
    J.F. Donoghue and T. Torma, Infrared behavior of graviton-graviton scattering, Phys. Rev. D 60 (1999) 024003 [hep-th/9901156] [INSPIRE].ADSGoogle Scholar
  56. [56]
    N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge U.K. (1982).CrossRefMATHGoogle Scholar
  57. [57]
    L. Parker and D. Toms, Quantum Field Theory in Curved Spacetime, Quantum Fields and Gravity, Cambridge University Press, Cambridge U.K. (2009).Google Scholar
  58. [58]
    P.B. Gilkey, The spectral geometry of a Riemannian manifold, J. Diff. Geom. 10 (1975) 601 [INSPIRE].MathSciNetMATHGoogle Scholar
  59. [59]
    B.S. DeWitt, Dynamical theory of groups and fields, Conf. Proc. C 630701 (1964) 585 [Les Houches Lect. Notes 13 (1964) 585].Google Scholar
  60. [60]
    G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].MathSciNetADSGoogle Scholar
  61. [61]
    J.D. Brown and J.W. York, Jr., The microcanonical functional integral. 1. The gravitational field, Phys. Rev. D 47 (1993) 1420 [gr-qc/9209014] [INSPIRE].
  62. [62]
    M. Bañados, C. Teitelboim and J. Zanelli, Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem, Phys. Rev. Lett. 72 (1994) 957 [gr-qc/9309026] [INSPIRE].
  63. [63]
    L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].MathSciNetADSGoogle Scholar
  64. [64]
    S.N. Solodukhin, The conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51 (1995) 609 [hep-th/9407001] [INSPIRE].MathSciNetADSGoogle Scholar
  65. [65]
    D. Garfinkle, S.B. Giddings and A. Strominger, Entropy in black hole pair production, Phys. Rev. D 49 (1994) 958 [gr-qc/9306023] [INSPIRE].
  66. [66]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
  67. [67]
    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].
  68. [68]
    V. Iyer and R.M. Wald, A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-qc/9503052] [INSPIRE].
  69. [69]
    R.C. Myers, Black hole entropy in two-dimensions, Phys. Rev. D 50 (1994) 6412 [hep-th/9405162] [INSPIRE].MathSciNetADSGoogle Scholar
  70. [70]
    R.J. Riegert, A Nonlocal Action for the Trace Anomaly, Phys. Lett. B 134 (1984) 56 [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  71. [71]
    D.J. Gross, M.J. Perry and L.G. Yaffe, Instability of Flat Space at Finite Temperature, Phys. Rev. D 25 (1982) 330 [INSPIRE].MathSciNetADSMATHGoogle Scholar
  72. [72]
    J.W. York, Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082 [INSPIRE].CrossRefADSGoogle Scholar
  73. [73]
    R. Balbinot and A. Fabbri, Two-dimensional black holes and effective actions, Class. Quant. Grav. 20 (2003) 5439 [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  74. [74]
    C.G. Callan Jr., S.B. Giddings, J.A. Harvey and A. Strominger, Evanescent black holes, Phys. Rev. D 45 (1992) 1005 [hep-th/9111056] [INSPIRE].MathSciNetADSGoogle Scholar
  75. [75]
    J.G. Russo, L. Susskind and L. Thorlacius, The endpoint of Hawking radiation, Phys. Rev. D 46 (1992) 3444 [hep-th/9206070] [INSPIRE].MathSciNetADSGoogle Scholar
  76. [76]
    D.A. Lowe, Semiclassical approach to black hole evaporation, Phys. Rev. D 47 (1993) 2446 [hep-th/9209008] [INSPIRE].MathSciNetADSGoogle Scholar
  77. [77]
    V.F. Mukhanov, A. Wipf and A. Zelnikov, On 4-D Hawking radiation from effective action, Phys. Lett. B 332 (1994) 283 [hep-th/9403018] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  78. [78]
    G.A. Campbell and R.A. Matzner, A model for peaking of galactic gravitational radiation, J. Math. Phys. 14 (1973) 1 [INSPIRE].CrossRefADSGoogle Scholar
  79. [79]
    J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar

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© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of MassachusettsAmherstU.S.A.

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