Abstract
Euclidean gravity method has been successful in computing logarithmic corrections to extremal black hole entropy in terms of low energy data, and gives results in perfect agreement with the microscopic results in string theory. Motivated by this success we apply Euclidean gravity to compute logarithmic corrections to the entropy of various non-extremal black holes in different dimensions, taking special care of integration over the zero modes and keeping track of the ensemble in which the computation is done. These results provide strong constraint on any ultraviolet completion of the theory if the latter is able to give an independent computation of the entropy of non-extremal black holes from microscopic description. For Schwarzschild black holes in four space-time dimensions the macroscopic result seems to disagree with the existing result in loop quantum gravity.
Similar content being viewed by others
References
S.N. Solodukhin, The Conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51 (1995) 609 [hep-th/9407001] [INSPIRE].
S.N. Solodukhin, On ’Nongeometric’ contribution to the entropy of black hole due to quantum corrections, Phys. Rev. D 51 (1995) 618 [hep-th/9408068] [INSPIRE].
D.V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51 (1995) 5352 [hep-th/9412161] [INSPIRE].
N. Mavromatos and E. Winstanley, Aspects of hairy black holes in spontaneously broken Einstein Yang-Mills systems: Stability analysis and entropy considerations, Phys. Rev. D 53 (1996) 3190 [hep-th/9510007] [INSPIRE].
R.B. Mann and S.N. Solodukhin, Conical geometry and quantum entropy of a charged Kerr black hole, Phys. Rev. D 54 (1996) 3932 [hep-th/9604118] [INSPIRE].
R.B. Mann and S.N. Solodukhin, Universality of quantum entropy for extreme black holes, Nucl. Phys. B 523 (1998) 293 [hep-th/9709064] [INSPIRE].
S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].
T. Govindarajan, R. 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].
K.S. Gupta and S. Sen, Further evidence for the conformal structure of a Schwarzschild black hole in an algebraic approach, Phys. Lett. B 526 (2002) 121 [hep-th/0112041] [INSPIRE].
A. Medved, A Comment on black hole entropy or does nature abhor a logarithm?, Class. Quant. Grav. 22 (2005) 133 [gr-qc/0406044] [INSPIRE].
D.N. Page, Hawking radiation and black hole thermodynamics, New J. Phys. 7 (2005) 203 [hep-th/0409024] [INSPIRE].
R. Banerjee and B.R. Majhi, Quantum Tunneling Beyond Semiclassical Approximation, JHEP 06 (2008) 095 [arXiv:0805.2220] [INSPIRE].
R. Banerjee and B.R. Majhi, Quantum Tunneling, Trace Anomaly and Effective Metric, Phys. Lett. B 674 (2009) 218 [arXiv:0808.3688] [INSPIRE].
B.R. Majhi, Fermion Tunneling Beyond Semiclassical Approximation, Phys. Rev. D 79 (2009) 044005 [arXiv:0809.1508] [INSPIRE].
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].
R. Aros, D. Diaz and A. Montecinos, Logarithmic correction to BH entropy as Noether charge, JHEP 07 (2010) 012 [arXiv:1003.1083] [INSPIRE].
S.N. Solodukhin, Entanglement entropy of round spheres, Phys. Lett. B 693 (2010) 605 [arXiv:1008.4314] [INSPIRE].
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].
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].
A. Sen, Logarithmic Corrections to N = 2 Black Hole Entropy: An Infrared Window into the Microstates, arXiv:1108.3842 [INSPIRE].
S. Ferrara and A. Marrani, Generalized Mirror Symmetry and Quantum Black Hole Entropy, Phys. Lett. B 707 (2012) 173 [arXiv:1109.0444] [INSPIRE].
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].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
J. Breckenridge, R.C. Myers, A. Peet and C. Vafa, D-branes and spinning black holes, Phys. Lett. B 391 (1997) 93 [hep-th/9602065] [INSPIRE].
I. Mandal and A. Sen, Black Hole Microstate Counting and its Macroscopic Counterpart, Nucl. Phys. Proc. Suppl. 216 (2011) 147 [arXiv:1008.3801] [INSPIRE].
S. Bhattacharyya, B. Panda and A. Sen, Heat Kernel Expansion and Extremal Kerr-Newmann Black Hole Entropy in Einstein-Maxwell Theory, JHEP 08 (2012) 084 [arXiv:1204.4061] [INSPIRE].
S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].
D.V. Fursaev and S.N. Solodukhin, On one loop renormalization of black hole entropy, Phys. Lett. B 365 (1996) 51 [hep-th/9412020] [INSPIRE].
R.K. Kaul and P. Majumdar, Quantum black hole entropy, Phys. Lett. B 439 (1998) 267 [gr-qc/9801080] [INSPIRE].
R.K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84 (2000) 5255 [gr-qc/0002040] [INSPIRE].
S. Das, R.K. Kaul and P. Majumdar, A New holographic entropy bound from quantum geometry, Phys. Rev. D 63 (2001) 044019 [hep-th/0006211] [INSPIRE].
A. Ghosh and P. Mitra, A Bound on the log correction to the black hole area law, Phys. Rev. D 71 (2005) 027502 [gr-qc/0401070] [INSPIRE].
M. Domagala and J. Lewandowski, Black hole entropy from quantum geometry, Class. Quant. Grav. 21 (2004) 5233 [gr-qc/0407051] [INSPIRE].
K.A. Meissner, Black hole entropy in loop quantum gravity, Class. Quant. Grav. 21 (2004) 5245 [gr-qc/0407052] [INSPIRE].
A. Ghosh and P. Mitra, An Improved lower bound on black hole entropy in the quantum geometry approach, Phys. Lett. B 616 (2005) 114 [gr-qc/0411035] [INSPIRE].
A. Ghosh and P. Mitra, Counting black hole microscopic states in loop quantum gravity, Phys. Rev. D 74 (2006) 064026 [hep-th/0605125] [INSPIRE].
J. Engle, A. Perez and K. Noui, Black hole entropy and SU(2) Chern-Simons theory, Phys. Rev. Lett. 105 (2010) 031302 [arXiv:0905.3168] [INSPIRE].
R. Basu, R.K. Kaul and P. Majumdar, Entropy of Isolated Horizons revisited, Phys. Rev. D 82 (2010) 024007 [arXiv:0907.0846] [INSPIRE].
J. Engle, K. Noui, A. Perez and D. Pranzetti, Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons, Phys. Rev. D 82 (2010) 044050 [arXiv:1006.0634] [INSPIRE].
J. Engle, K. Noui, A. Perez and D. Pranzetti, The SU(2) Black Hole entropy revisited, JHEP 05 (2011) 016 [arXiv:1103.2723] [INSPIRE].
R.K. Kaul, Entropy of quantum black holes, SIGMA 8 (2012) 005 [arXiv:1201.6102] [INSPIRE].
R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
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].
M. Duff, Observations on Conformal Anomalies, Nucl. Phys. B 125 (1977) 334 [INSPIRE].
S. Christensen and M. Duff, New Gravitational Index Theorems and Supertheorems, Nucl. Phys. B 154 (1979) 301 [INSPIRE].
S. Christensen and M. Duff, Quantizing Gravity with a Cosmological Constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].
M. Duff and P. van Nieuwenhuizen, Quantum Inequivalence of Different Field Representations, Phys. Lett. B 94 (1980) 179 [INSPIRE].
S. Christensen, M. Duff, G. Gibbons and M. Roček, VANISHING ONE LOOP β-function IN GAUGED N ¿ 4 SUPERGRAVITY, Phys. Rev. Lett. 45 (1980) 161 [INSPIRE].
N.D. Birrel and P.C.W. Davis, Quantum Fields in Curved Space, Cambridge University Press, New York U.S.A. (1982).
P.B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, Publish or Perish Inc. U.S.A. (1984)
D. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
M. Duff and S. Ferrara, Generalized mirror symmetry and trace anomalies, Class. Quant. Grav. 28 (2011) 065005 [arXiv:1009.4439] [INSPIRE].
T. Jacobson, Renormalization and black hole entropy in Loop Quantum Gravity, Class. Quant. Grav. 24 (2007) 4875 [arXiv:0707.4026] [INSPIRE].
G. Gibbons and S. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
R.C. Henry, Kretschmann scalar for a Kerr-Newman black hole, astro-ph/9912320 [INSPIRE].
C. Cherubini, D. Bini, S. Capozziello and R. Ruffini, Second order scalar invariants of the Riemann tensor: Applications to black hole space-times, Int. J. Mod. Phys. D 11 (2002) 827 [gr-qc/0302095] [INSPIRE].
S. Mukherji and S.S. Pal, Logarithmic corrections to black hole entropy and AdS/CFT correspondence, JHEP 05 (2002) 026 [hep-th/0205164] [INSPIRE].
A. Chatterjee and P. Majumdar, Universal canonical black hole entropy, Phys. Rev. Lett. 92 (2004) 141301 [gr-qc/0309026] [INSPIRE].
M.-I. Park, Testing holographic principle from logarithmic and higher order corrections to black hole entropy, JHEP 12 (2004) 041 [hep-th/0402173] [INSPIRE].
A. Majhi and P. Majumdar, Charged Quantum Black Holes: Thermal Stability Criterion, Class. Quant. Grav. 29 (2012) 135013 [arXiv:1108.4670] [INSPIRE].
L. Smolin, Linking topological quantum field theory and nonperturbative quantum gravity, J. Math. Phys. 36 (1995) 6417 [gr-qc/9505028] [INSPIRE].
K.V. Krasnov, On Quantum statistical mechanics of Schwarzschild black hole, Gen. Rel. Grav. 30 (1998) 53 [gr-qc/9605047] [INSPIRE].
A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Quantum geometry and black hole entropy, Phys. Rev. Lett. 80 (1998) 904 [gr-qc/9710007] [INSPIRE].
A. Ashtekar, J.C. Baez and K. Krasnov, Quantum geometry of isolated horizons and black hole entropy, Adv. Theor. Math. Phys. 4 (2000) 1 [gr-qc/0005126] [INSPIRE].
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].
P. Kraus and F. Larsen, Microscopic black hole entropy in theories with higher derivatives, JHEP 09 (2005) 034 [hep-th/0506176] [INSPIRE].
R.C. Myers and M. Perry, Black Holes in Higher Dimensional Space-Times, Annals Phys. 172 (1986) 304 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1205.0971
Rights and permissions
About this article
Cite this article
Sen, A. Logarithmic corrections to Schwarzschild and other non-extremal black hole entropy in different dimensions. J. High Energ. Phys. 2013, 156 (2013). https://doi.org/10.1007/JHEP04(2013)156
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2013)156