Abstract
In this paper, we analyze the complete thermodynamic and phase transition phenomena of a black hole solution in Hořava-Lifshitz gravity in arbitrary space time. Nature of phase transition is studied using geometrothermodynamic and Ehrenfest’s scheme of standard thermodynamics. We analytically check the Ehrenfest’s equations near the critical point, which is the point of divergence in the heat capacity. Our analysis revels that this black hole exhibits a second order phase transition.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206-206] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
J. Gibbs., The collected works. Volume 1: thermodynamics, Yale University Press, New Haven, U.S.A. (1948).
C. Caratheodory, Untersuchungen über die Grundlagen der Thermodynamik, Gesammelte Mathematische Werke Band 2, Munich, Germany (1995).
R. Hermann, Geometry physics and systems, Marcel Dekker, New York U.S.A. (1973).
R. Mrugala, Geometrical formulation of equilibrium phenomenological thermodynamics, Rep. Math. Phys. 14 (1978) 419.
R. Mrugala, Submanifolds in the thermodynamical phase space, Rep. Math. Phys. 21 (1985) 197.
F. Weinhold, Metric geometry of equilibrium thermodynamics I, J. Chem. Phys. 63 (1975) 2479.
F. Weinhold, Metric geometry of equilibrium thermodynamics II, J. Chem. Phys. 63 (1975) 2484.
F. Weinhold, Metric geometry of equilibrium thermodynamics III, J. Chem. Phys. 63 (1975) 2488.
F. Weinhold, Metric geometry of equilibrium thermodynamics IV, J. Chem. Phys. 63 (1975) 2496.
F. Weinhold, Metric geometry of equilibrium thermodynamics V, J. Chem. Phys. 65 (1976) 558.
G. Ruppeiner, Thermodynamics: a Riemannian geometric model, Phys. Rev. A 20 (1979) A1608.
R. Mrugala, On equivalence of two metrics in classical thermodynamics, Physica A 125 (1984) 631.
P. Salamon, J.D. Nulton and E. Ihrig, On the relation between entropy and energy versions of thermodynamic length, J. Chem. Phys. 80 (1984) 436.
H. Janyszek, On the geometrical structure of the generalized quantum Gibbs states, Rep. Math. Phys. 24 (1986) 1.
H. Janyszek and R. Mrugala, Riemannian geometry and the thermodynamics of model magnetic systems, Phys. Rev. A 39 (1989) 6515 [INSPIRE].
E.J. Brody, Applications of the Kakutani metric to real space renormalization, Phys. Rev. Lett. 58 (1987) 179 [INSPIRE].
D. Brody and N. Rivier, Geometrical aspects of statistical mechanics , Phys. Rev. E 51 (1995) 1006 [INSPIRE].
D. Brody and A. Ritz, On the symmetry of real space renormalization, Nucl. Phys. B 522 (1998) 588 [hep-th/9709175] [INSPIRE].
B.P. Dolan, Geometry and thermodynamic fluctuations of the Ising model on a Bethe lattice, Proc. Roy. Soc. London A 454 (1998) 2655 [cond-mat/9706238].
B.P. Dolan, D.A. Johnston and R. Kenna, The information geometry of the one-dimensional Potts model, J. Phys. A 35 (2002) 9025 [cond-mat/0207180] [INSPIRE].
W. Janke, D.A. Johnston and R.P. K.C. Malmini, The information geometry of the Ising model on planar random graphs, Phys. Rev. E 66 (2002) 056119 [cond-mat/0207573] [INSPIRE].
W. Janke, D.A. Johnston and R. Kenna, The information geometry of the spherical model, Phys. Rev. E 67 (2003) 046106 [cond-mat/0210571] [INSPIRE].
W. Janke, D.A. Johnston and R. Kenna, Information geometry and phase transitions, Physica A 336 (2004) 181 [cond-mat/0401092] [INSPIRE].
D.A. Johnston, W. Janke and R. Kenna, Information geometry, one, two, three (and four), Acta Phys. Polon. B 34 (2003) 4923 [cond-mat/0308316] [INSPIRE].
H. Lü, J. Mei and C.N. Pope, Solutions to Hořava gravity, Phys. Rev. Lett. 103 (2009) 091301 [arXiv:0904.1595] [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].
J. Sadeghi, K. Jafarzade and B. Pourhassan, Thermodynamical quantities of Hořava-Lifshitz black hole, Int. J. Theor. Phys. 51 (2012) 3891.
J. Simon, Small black holes versus horizonless solutions in AdS, Phys. Rev. D 81 (2010) 024003 [arXiv:0910.3225] [INSPIRE].
A. Kehagias and K. Sfetsos, The black hole and FRW geometries of non-relativistic gravity, Phys. Lett. B 678 (2009) 123 [arXiv:0905.0477] [INSPIRE].
H. Nastase, On IR solution in Hořava gravity theories, arXiv:0904.3604 [INSPIRE].
E. Kiritsis and G. Kofinas, Hořava-Lifshitz cosmology, Nucl. Phys. B 821 (2009) 467 [arXiv:0904.1334] [INSPIRE].
A.J. Padilla, The good, the bad and the ugly . . . of Hořava gravity, J. Phys. Conf. Ser. 259 (2010) 012033 [arXiv:1009.4074] [INSPIRE].
G. Calcagni, Cosmology of the Lifshitz universe, JHEP 09 (2009) 112 [arXiv:0904.0829] [INSPIRE].
M.-i. Park, A test of Hořava gravity: the dark energy, JCAP 01 (2010) 001 [arXiv:0906.4275] [INSPIRE].
S.W. Wei, Y.X. Liu and H. Guo, Thermodynamic geometry of black hole in the deformed Hořava-Lifshitz gravity, Europhys. Lett. 99 (2012) 20004.
Y.S. Myung, Entropy of black holes in the deformed Hořava-Lifshitz gravity, Phys. Lett. B 684 (2010) 158 [arXiv:0908.4132] [INSPIRE].
Y.S. Myung, Thermodynamics of black holes in the deformed Hořava-Lifshitz gravity, Phys. Lett. B 678 (2009) 127 [arXiv:0905.0957] [INSPIRE].
Y.S. Myung and Y.-W. Kim, Thermodynamics of Hořava-Lifshitz black holes, Eur. Phys. J. C 68 (2010) 265 [arXiv:0905.0179] [INSPIRE].
N. Varghese and V.C. Kuriakose, Evolution of electromagnetic and Dirac perturbations around a black hole in Hořava gravity, Mod. Phys. Lett. A 26 (2011) 1645 [arXiv:1010.0549] [INSPIRE].
M.-i. Park, The black hole and cosmological solutions in IR modified Hořava gravity, JHEP 09 (2009) 123 [arXiv:0905.4480] [INSPIRE].
J. Suresh and V.C. Kuriakose, Area spectrum and thermodynamics of KS black holes in Hořava gravity, Gen. Rel. Grav. 45 (2013) 1877 [arXiv:1307.6438] [INSPIRE].
J. Suresh and V.C. Kuriakose, Thermodynamics and quasinormal modes of Park black hole in Hořava gravity, Eur. Phys. J. C 73 (2013) 2613 [arXiv:1310.2011] [INSPIRE].
H. Quevedo, Geometrothermodynamics, J. Math. Phys. 48 (2007) 013506 [physics/0604164] [INSPIRE].
H. Quevedo, Geometrothermodynamics of black holes, Gen. Rel. Grav. 40 (2008) 971 [arXiv:0704.3102] [INSPIRE].
H. Quevedo and A. Vazquez, The geometry of thermodynamics, in Recent developments in gravitation and cosmology, A. Macias et al. eds., AIP, New York U.S.A. (2008).
P. Salamon, E. Ihrig and R.S. Berry, A group of coordinate transformations which preserve the metric of Weinhold, J. Math. Phys. 24 (1983) 2515.
R. Mrugala, J.D. Nulton, J.C. Schon and P. Salamon, Statistical approach to the geometric structure of thermodynamics, Phys. Rev. A 41 (1990) 3156.
R. Banerjee, S.K. Modak and S. Samanta, Glassy phase transition and stability in black holes, Eur. Phys. J. C 70 (2010) 317 [arXiv:1002.0466] [INSPIRE].
R. Banerjee, S.K. Modak and S. Samanta, Second order phase transition and thermodynamic geometry in Kerr-AdS black hole, Phys. Rev. D 84 (2011) 064024 [arXiv:1005.4832] [INSPIRE].
R. Banerjee and D. Roychowdhury, Critical phenomena in Born-Infeld AdS black holes, Phys. Rev. D 85 (2012) 044040 [arXiv:1111.0147] [INSPIRE].
R. Banerjee, S. Ghosh and D. Roychowdhury, New type of phase transition in Reissner Nordstrom — AdS black hole and its thermodynamic geometry, Phys. Lett. B 696 (2011) 156 [arXiv:1008.2644] [INSPIRE].
R. Banerjee and D. Roychowdhury, Thermodynamics of phase transition in higher dimensional AdS black holes, JHEP 11 (2011) 004 [arXiv:1109.2433] [INSPIRE].
R. Banerjee, S.K. Modak and D. Roychowdhury, A unified picture of phase transition: from liquid-vapour systems to AdS black holes, JHEP 10 (2012) 125 [arXiv:1106.3877] [INSPIRE].
Th. M. Nieuwenhuizen, Ehrenfest relations at the glass transition: solution to an old paradox, Phys. Rev. Lett. 79 (1997) 1317.
Th.M. Nieuwenhuizen, Thermodynamic picture of the glassy state, J. Phys. Condens. Matter 12 (2000) 6543.
J. Jäckle, Models of the glass transition, Rep. Prog. Phys. 49 (1986) 171.
A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60 (1999) 064018 [hep-th/9902170] [INSPIRE].
A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Holography, thermodynamics and fluctuations of charged AdS black holes, Phys. Rev. D 60 (1999) 104026 [hep-th/9904197] [INSPIRE].
M.M. Caldarelli, G. Cognola and D. Klemm, Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories, Class. Quant. Grav. 17 (2000) 399 [hep-th/9908022] [INSPIRE].
S. Nojiri and S.D. Odintsov, Anti-de Sitter black hole thermodynamics in higher derivative gravity and new confining deconfining phases in dual CFT, Phys. Lett. B 521 (2001) 87 [Erratum ibid. B 542 (2002) 301] [hep-th/0109122] [INSPIRE].
R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev. D 65 (2002) 084014 [hep-th/0109133] [INSPIRE].
M. Cvetič, S. Nojiri and S.D. Odintsov, Black hole thermodynamics and negative entropy in de Sitter and Anti-de Sitter Einstein-Gauss-Bonnet gravity, Nucl. Phys. B 628 (2002) 295 [hep-th/0112045] [INSPIRE].
S. Carlip and S. Vaidya, Phase transitions and critical behavior for charged black holes, Class. Quant. Grav. 20 (2003) 3827 [gr-qc/0306054] [INSPIRE].
R.-G. Cai and A. Wang, Thermodynamics and stability of hyperbolic charged black holes, Phys. Rev. D 70 (2004) 064013 [hep-th/0406057] [INSPIRE].
Y.S. Myung, No Hawking-Page phase transition in three dimensions, Phys. Lett. B 624 (2005) 297 [hep-th/0506096] [INSPIRE].
B.M.N. Carter and I.P. Neupane, Thermodynamics and stability of higher dimensional rotating (Kerr) AdS black holes, Phys. Rev. D 72 (2005) 043534 [gr-qc/0506103] [INSPIRE].
R.-G. Cai, S.P. Kim and B. Wang, Ricci flat black holes and Hawking-Page phase transition in Gauss-Bonnet gravity and dilaton gravity, Phys. Rev. D 76 (2007) 024011 [arXiv:0705.2469] [INSPIRE].
Y.S. Myung, Y.-W. Kim and Y.-J. Park, Thermodynamics and phase transitions in the Born-Infeld-Anti-de Sitter black holes, Phys. Rev. D 78 (2008) 084002 [arXiv:0805.0187] [INSPIRE].
Y.S. Myung, Phase transition between non-extremal and extremal Reissner-Nordstrom black holes, Mod. Phys. Lett. A 23 (2008) 667 [arXiv:0710.2568] [INSPIRE].
G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, Phase transitions in charged topological-AdS black holes, JHEP 05 (2008) 107 [arXiv:0801.4921] [INSPIRE].
M. Cadoni, G. D’Appollonio and P. Pani, Phase transitions between Reissner-Nordstrom and dilatonic black holes in 4D AdS spacetime, JHEP 03 (2010) 100 [arXiv:0912.3520] [INSPIRE].
H. Liu, H. Lü, M. Luo and K.-N. Shao, Thermodynamical metrics and black hole phase transitions, JHEP 12 (2010) 054 [arXiv:1008.4482] [INSPIRE].
A. Sahay, T. Sarkar and G. Sengupta, On the phase structure and thermodynamic geometry of R-charged black holes, JHEP 11 (2010) 125 [arXiv:1009.2236] [INSPIRE].
Q.-J. Cao, Y.-X. Chen and K.-N. Shao, Black hole phase transitions in Hořava-Lifshitz gravity, Phys. Rev. D 83 (2011) 064015 [arXiv:1010.5044] [INSPIRE].
S.-W. Wei, Y.-X. Liu, Y.-Q. Wang and H. Guo, Thermodynamic geometry of black hole in the deformed Hořava-Lifshitz gravity, Europhys. Lett. 99 (2012) 20004 [arXiv:1002.1550] [INSPIRE].
B.R. Majhi and D. Roychowdhury, Phase transition and scaling behavior of topological charged black holes in Hořava-Lifshitz gravity, Class. Quant. Grav. 29 (2012) 245012 [arXiv:1205.0146] [INSPIRE].
W. Kim and Y. Kim, Phase transition of quantum corrected Schwarzschild black hole, Phys. Lett. B 718 (2012) 687 [arXiv:1207.5318] [INSPIRE].
Y.-D. Tsai, X.N. Wu and Y. Yang, Phase structure of Kerr-AdS black hole, Phys. Rev. D 85 (2012) 044005 [arXiv:1104.0502] [INSPIRE].
F. Capela and G. Nardini, Hairy black holes in massive gravity: thermodynamics and phase structure, Phys. Rev. D 86 (2012) 024030 [arXiv:1203.4222] [INSPIRE].
D. Kubiznak and R.B. Mann, P-V criticality of charged AdS black holes, JHEP 07 (2012) 033 [arXiv:1205.0559] [INSPIRE].
S.-W. Wei and Y.-X. Liu, Critical phenomena and thermodynamic geometry of charged Gauss-Bonnet AdS black holes, Phys. Rev. D 87 (2013) 044014 [arXiv:1209.1707] [INSPIRE].
M. Eune, W. Kim and S.-H. Yi, Hawking-Page phase transition in BTZ black hole revisited, JHEP 03 (2013) 020 [arXiv:1301.0395] [INSPIRE].
P.C.W. Davies, Thermodynamics of black holes, Rep. Prog. Phys. 41 (1978) 1313.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1408.0911
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Suresh, J., Tharanath, R. & Kuriakose, V.C. A unified thermodynamic picture of Hořava-Lifshitz black hole in arbitrary space time. J. High Energ. Phys. 2015, 19 (2015). https://doi.org/10.1007/JHEP01(2015)019
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2015)019