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Journal of High Energy Physics

, 2016:129 | Cite as

Charged black hole solutions in Gauss-Bonnet-massive gravity

  • S. H. Hendi
  • S. Panahiyan
  • B. Eslam Panah
Open Access
Regular Article - Theoretical Physics

Abstract

Motivated by high interest in the close relation between string theory and black hole solutions, in this paper, we take into account the Einstein-Gauss-Bonnet Lagrangian in the context of massive gravity. We examine the possibility of black hole in this regard, and discuss the types of horizons. Next, we calculate conserved and thermodynamic quantities and check the validity of the first law of thermodynamics. In addition, we investigate the stability of these black holes in context of canonical ensemble. We show that number, type and place of phase transition points may be significantly affected by different parameters. Next, by considering cosmological constant as thermodynamical pressure, we will extend phase space and calculate critical values. Then, we construct thermodynamical spacetime by considering mass as thermodynamical potential. We study geometrical thermodynamics of these black holes in context of heat capacity and extended phase space. We show that studying heat capacity, geometrical thermodynamics and critical behavior in extended phase space lead to consistent results. Finally, we will employ a new method for obtaining critical values and show that the results of this method are consistent with those of other methods.

Keywords

Classical Theories of Gravity Black Holes 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]
    C. Brans and R.H. Dicke, Mach’s principle and a relativistic theory of gravitation, Phys. Rev. 124 (1961) 925 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    R.V. Wagoner, Scalar tensor theory and gravitational waves, Phys. Rev. D 1 (1970) 3209 [INSPIRE].ADSGoogle Scholar
  3. [3]
    C. Barrabes and G.F. Bressange, Singular hypersurfaces in scalar-tensor theories of gravity, Class. Quant. Grav. 14 (1997) 805 [gr-qc/9701026] [INSPIRE].
  4. [4]
    R.-G. Cai and Y.S. Myung, Black holes in the Brans-Dicke-Maxwell theory, Phys. Rev. D 56 (1997) 3466 [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    S. Capozziello and A. Troisi, PPN-limit of fourth order gravity inspired by scalar-tensor gravity, Phys. Rev. D 72 (2005) 044022 [astro-ph/0507545] [INSPIRE].
  6. [6]
    T.P. Sotiriou, f(R) gravity and scalar-tensor theory, Class. Quant. Grav. 23 (2006) 5117 [gr-qc/0604028] [INSPIRE].
  7. [7]
    J.W. Moffat, Scalar-tensor-vector gravity theory, JCAP 03 (2006) 004 [gr-qc/0506021] [INSPIRE].
  8. [8]
    M.H. Dehghani, J. Pakravan and S.H. Hendi, Thermodynamics of charged rotating black branes in Brans-Dicke theory with quadratic scalar field potential, Phys. Rev. D 74 (2006) 104014 [hep-th/0608197] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    V. Faraoni, de Sitter space and the equivalence between f(R) and scalar-tensor gravity, Phys. Rev. D 75 (2007) 067302 [gr-qc/0703044] [INSPIRE].
  10. [10]
    K.-i. Maeda and Y. Fujii, Attractor Universe in the Scalar-Tensor Theory of Gravitation, Phys. Rev. D 79 (2009) 084026 [arXiv:0902.1221] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    S.H. Hendi and R. Katebi, Rotating black branes in Brans-Dicke theory with a nonlinear electromagnetic field, Eur. Phys. J. C 72 (2012) 2235 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    M. Sharif and S. Waheed, Noether symmetries of some homogeneous universe models in curvature corrected scalar-tensor gravity, JCAP 02 (2013) 043 [arXiv:1403.0556] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    E.F. Eiroa and C.M. Sendra, Strong deflection lensing by charged black holes in scalar-tensor gravity, Eur. Phys. J. C 74 (2014) 3171 [arXiv:1408.3390] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    K. Karami, T. Najafi and A. Abdolmaleki, Generalized second law of thermodynamics in scalar-tensor gravity, Phys. Rev. D 89 (2014) 104041 [arXiv:1401.7549] [INSPIRE].ADSGoogle Scholar
  15. [15]
    F. Darabi and A. Parsiya, Cosmology with non-minimal coupled gravity: inflation and perturbation analysis, Class. Quant. Grav. 32 (2015) 155005 [arXiv:1312.1322] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M. Akbar and R.-G. Cai, Thermodynamic Behavior of Field Equations for f(R) Gravity, Phys. Lett. B 648 (2007) 243 [gr-qc/0612089] [INSPIRE].
  17. [17]
    J.C.C. de Souza and V. Faraoni, The phase space view of f(R) gravity, Class. Quant. Grav. 24 (2007) 3637 [arXiv:0706.1223] [INSPIRE].ADSzbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    C. Corda and H.J. Mosquera Cuesta, A spherically symmetric and stationary universe from a weak modification of general relativity, Europhys. Lett. 86 (2009) 20004 [arXiv:0903.3645] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. de la Cruz-Dombriz, A. Dobado and A.L. Maroto, Black Holes in f(R) theories, Phys. Rev. D 80 (2009) 124011 [Erratum ibid. D 83 (2011) 029903] [arXiv:0907.3872] [INSPIRE].
  20. [20]
    G. Cognola, E. Elizalde, S. Nojiri, S.D. Odintsov, L. Sebastiani and S. Zerbini, A class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion, Phys. Rev. D 77 (2008) 046009 [arXiv:0712.4017] [INSPIRE].ADSGoogle Scholar
  21. [21]
    T.P. Sotiriou and V. Faraoni, f(R) Theories Of Gravity, Rev. Mod. Phys. 82 (2010) 451 [arXiv:0805.1726] [INSPIRE].
  22. [22]
    S. Nojiri and S.D. Odintsov, Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models, Phys. Rept. 505 (2011) 59 [arXiv:1011.0544] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    J.L. Said and K.Z. Adami, The Generalized Uncertainty Principle in f(R) Gravity for a Charged Black Hole, Phys. Rev. D 83 (2011) 043008 [arXiv:1102.3553] [INSPIRE].ADSGoogle Scholar
  24. [24]
    Y.S. Myung, T. Moon and E.J. Son, Stability of f(R) black holes, Phys. Rev. D 83 (2011) 124009 [arXiv:1103.0343] [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    D.-i. Hwang, B.-H. Lee and D.-h. Yeom, Mass inflation in f(R) gravity: A conjecture on the resolution of the mass inflation singularity, JCAP 12 (2011) 006 [arXiv:1110.0928] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    S.H. Hendi, B.E. Panah and S.M. Mousavi, Some exact solutions of F(R) gravity with charged (a)dS black hole interpretation, Gen. Rel. Grav. 44 (2012) 835 [arXiv:1102.0089] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    S.H. Mazharimousavi, M. Halilsoy and T. Tahamtan, Constant curvature f(R) gravity minimally coupled with Yang-Mills field, Eur. Phys. J. C 72 (2012) 1958 [arXiv:1109.3655] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    J. Man and H. Cheng, Thermodynamic quantities of a black hole with an f(R) global monopole, Phys. Rev. D 87 (2013) 044002 [arXiv:1301.2739] [INSPIRE].ADSGoogle Scholar
  29. [29]
    Y.-G. Miao, F.-F. Yuan and Z.-Z. Zhang, Thermodynamic approach to field equations in Lovelock gravity and f (R) gravity revisited, Int. J. Mod. Phys. D 23 (2014) 1450093 [arXiv:1407.1698] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    S.H. Hendi, R.B. Mann, N. Riazi and B. Eslam Panah, Eguchi-Hanson like space-times in F(R) gravity, Phys. Rev. D 86 (2012) 104034 [arXiv:1210.3629] [INSPIRE].ADSGoogle Scholar
  31. [31]
    S.H. Hendi, (2+1)-Dimensional Solutions in F (R) Gravity, Int. J. Theor. Phys. 53 (2014) 4170 [arXiv:1410.7527] [INSPIRE].
  32. [32]
    S.H. Hendi, B. Eslam Panah and C. Corda, Asymptotically Lifshitz black hole solutions in F(R) gravity, Can. J. Phys. 92 (2014) 76 [arXiv:1309.2135] [INSPIRE].CrossRefGoogle Scholar
  33. [33]
    D. Ida, Brane world cosmology, JHEP 09 (2000) 014 [gr-qc/9912002] [INSPIRE].
  34. [34]
    J.M. Cline and H. Firouzjahi, Brane world cosmology of modulus stabilization with a bulk scalar field, Phys. Rev. D 64 (2001) 023505 [hep-ph/0005235] [INSPIRE].
  35. [35]
    P. Brax and C. van de Bruck, Cosmology and brane worlds: A review, Class. Quant. Grav. 20 (2003) R201 [hep-th/0303095] [INSPIRE].ADSzbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    S. Mizuno, S.-J. Lee and E.J. Copeland, Cosmological evolution of general scalar fields in a brane-world cosmology, Phys. Rev. D 70 (2004) 043525 [astro-ph/0405490] [INSPIRE].
  37. [37]
    T. Nihei, N. Okada and O. Seto, Neutralino dark matter in brane world cosmology, Phys. Rev. D 71 (2005) 063535 [hep-ph/0409219] [INSPIRE].
  38. [38]
    L.A. Gergely, Brane-world cosmology with black strings, Phys. Rev. D 74 (2006) 024002 [hep-th/0603244] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    M. Demetrian, False vacuum decay in a brane world cosmological model, Gen. Rel. Grav. 38 (2006) 953 [gr-qc/0506028] [INSPIRE].
  40. [40]
    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    D. Lovelock, The four-dimensionality of space and the Einstein tensor, J. Math. Phys. 13 (1972) 874 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    N. Deruelle and L. Farina-Busto, The Lovelock Gravitational Field Equations in Cosmology, Phys. Rev. D 41 (1990) 3696 [INSPIRE].ADSMathSciNetGoogle Scholar
  43. [43]
    S.H. Hendi, B. Eslam Panah and S. Panahiyan, Magnetic brane solutions of Lovelock gravity with nonlinear electrodynamics, Phys. Rev. D 91 (2015) 084031 [arXiv:1510.08557] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    K.S. Stelle, Classical Gravity with Higher Derivatives, Gen. Rel. Grav. 9 (1978) 353 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    J.W. Maluf, Conformal Invariance and Torsion in General Relativity, Gen. Rel. Grav. 19 (1987) 57 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    M. Farhoudi, On higher order gravities, their analogy to GR and dimensional dependent version of Duff ’s trace anomaly relation, Gen. Rel. Grav. 38 (2006) 1261 [physics/0509210] [INSPIRE].
  47. [47]
    D.G. Boulware and S. Deser, String Generated Gravity Models, Phys. Rev. Lett. 55 (1985) 2656 [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    B. Zumino, Gravity Theories in More Than Four-Dimensions, Phys. Rept. 137 (1986) 109 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    Y.M. Cho, I.P. Neupane and P.S. Wesson, No ghost state of Gauss-Bonnet interaction in warped background, Nucl. Phys. B 621 (2002) 388 [hep-th/0104227] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev. D 65 (2002) 084014 [hep-th/0109133] [INSPIRE].ADSMathSciNetGoogle Scholar
  51. [51]
    R.C. Myers, Superstring Gravity and Black Holes, Nucl. Phys. B 289 (1987) 701 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  52. [52]
    C.G. Callan Jr., R.C. Myers and M.J. Perry, Black Holes in String Theory, Nucl. Phys. B 311 (1989) 673 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B 156 (1985) 315 [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    D.J. Gross and E. Witten, Superstring Modifications of Einstein’s Equations, Nucl. Phys. B 277 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    D.J. Gross and J.H. Sloan, The Quartic Effective Action for the Heterotic String, Nucl. Phys. B 291 (1987) 41 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    R.R. Metsaev and A.A. Tseytlin, Two loop β-function for the generalized bosonic σ-model, Phys. Lett. B 191 (1987) 354 [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    R.R. Metsaev and A.A. Tseytlin, Order alpha-prime (Two Loop) Equivalence of the String Equations of Motion and the σ-model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor, Nucl. Phys. B 293 (1987) 385 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    R.R. Metsaev and A.A. Tseytlin, Curvature Cubed Terms in String Theory Effective Actions, Phys. Lett. B 185 (1987) 52 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  59. [59]
    J.E. Kim, B. Kyae and H.M. Lee, Effective Gauss-Bonnet interaction in Randall-Sundrum compactification, Phys. Rev. D 62 (2000) 045013 [hep-ph/9912344] [INSPIRE].
  60. [60]
    Y.M. Cho and I.P. Neupane, Anti-de Sitter black holes, thermal phase transition and holography in higher curvature gravity, Phys. Rev. D 66 (2002) 024044 [hep-th/0202140] [INSPIRE].ADSMathSciNetGoogle Scholar
  61. [61]
    C. Charmousis and J.-F. Dufaux, General Gauss-Bonnet brane cosmology, Class. Quant. Grav. 19 (2002) 4671 [hep-th/0202107] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  62. [62]
    G. Kofinas, R. Maartens and E. Papantonopoulos, Brane cosmology with curvature corrections, JHEP 10 (2003) 066 [hep-th/0307138] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  63. [63]
    R.-G. Cai and Q. Guo, Gauss-Bonnet black holes in dS spaces, Phys. Rev. D 69 (2004) 104025 [hep-th/0311020] [INSPIRE].ADSMathSciNetGoogle Scholar
  64. [64]
    A. Barrau, J. Grain and S.O. Alexeyev, Gauss-Bonnet black holes at the LHC: Beyond the dimensionality of space, Phys. Lett. B 584 (2004) 114 [hep-ph/0311238] [INSPIRE].
  65. [65]
    K.-i. Maeda and T. Torii, Covariant gravitational equations on brane world with Gauss-Bonnet term, Phys. Rev. D 69 (2004) 024002 [hep-th/0309152] [INSPIRE].ADSMathSciNetGoogle Scholar
  66. [66]
    C. de Rham and A.J. Tolley, Mimicking Lambda with a spin-two ghost condensate, JCAP 07 (2006) 004 [hep-th/0605122] [INSPIRE].CrossRefGoogle Scholar
  67. [67]
    G. Dotti, J. Oliva and R. Troncoso, Exact solutions for the Einstein-Gauss-Bonnet theory in five dimensions: Black holes, wormholes and spacetime horns, Phys. Rev. D 76 (2007) 064038 [arXiv:0706.1830] [INSPIRE].ADSMathSciNetGoogle Scholar
  68. [68]
    R.A. Brown, Brane universes with Gauss-Bonnet-induced-gravity, Gen. Rel. Grav. 39 (2007) 477 [gr-qc/0602050] [INSPIRE].
  69. [69]
    H. Maeda, V. Sahni and Y. Shtanov, Braneworld dynamics in Einstein-Gauss-Bonnet gravity, Phys. Rev. D 76 (2007) 104028 [Erratum ibid. D 80 (2009) 089902] [arXiv:0708.3237] [INSPIRE].
  70. [70]
    C. Charmousis, Higher order gravity theories and their black hole solutions, Lect. Notes Phys. 769 (2009) 299 [arXiv:0805.0568] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    S.H. Hendi and B.E. Panah, Thermodynamics of rotating black branes in Gauss-Bonnet-nonlinear Maxwell gravity, Phys. Lett. B 684 (2010) 77 [arXiv:1008.0102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  72. [72]
    M. Bouhmadi-López, Y.-W. Liu, K. Izumi and P. Chen, Tensor Perturbations from Brane-World Inflation with Curvature Effects, Phys. Rev. D 89 (2014) 063501 [arXiv:1308.5765] [INSPIRE].ADSGoogle Scholar
  73. [73]
    S.H. Hendi, S. Panahiyan and E. Mahmoudi, Thermodynamic analysis of topological black holes in Gauss-Bonnet gravity with nonlinear source, Eur. Phys. J. C 74 (2014) 3079 [arXiv:1406.2357] [INSPIRE].ADSCrossRefGoogle Scholar
  74. [74]
    Y. Yamashita and T. Tanaka, Mapping the ghost free bigravity into braneworld setup, JCAP 06 (2014) 004 [arXiv:1401.4336] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  75. [75]
    A. Maselli, P. Pani, L. Gualtieri and V. Ferrari, Rotating black holes in Einstein-Dilaton-Gauss-Bonnet gravity with finite coupling, Phys. Rev. D 92 (2015) 083014 [arXiv:1507.00680] [INSPIRE].ADSGoogle Scholar
  76. [76]
    D.P. Jatkar, G. Kofinas, O. Mišković and R. Olea, Conformal mass in Einstein-Gauss-Bonnet AdS gravity, Phys. Rev. D 91 (2015) 105030 [arXiv:1501.06861] [INSPIRE].ADSMathSciNetGoogle Scholar
  77. [77]
    Y.-Z. Li, S.-F. Wu and G.-H. Yang, Gauss-Bonnet correction to Holographic thermalization: two-point functions, circular Wilson loops and entanglement entropy, Phys. Rev. D 88 (2013) 086006 [arXiv:1309.3764] [INSPIRE].ADSGoogle Scholar
  78. [78]
    X.-X. Zeng, X.-M. Liu and W.-B. Liu, Holographic thermalization with a chemical potential in Gauss-Bonnet gravity, JHEP 03 (2014) 031 [arXiv:1311.0718] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    S.-J. Zhang, B. Wang, E. Abdalla and E. Papantonopoulos, Holographic thermalization in Gauss-Bonnet gravity with de Sitter boundary, Phys. Rev. D 91 (2015) 106010 [arXiv:1412.7073] [INSPIRE].ADSMathSciNetGoogle Scholar
  80. [80]
    Y.-B. Wu et al., Magnetic-field effects on p-wave phase transition in Gauss-Bonnet gravity, Int. J. Mod. Phys. A 29 (2014) 1450094 [arXiv:1405.2499] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  81. [81]
    R. Gregory, S. Kanno and J. Soda, Holographic Superconductors with Higher Curvature Corrections, JHEP 10 (2009) 010 [arXiv:0907.3203] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  82. [82]
    A. Buchel, R.C. Myers and A. Sinha, Beyond η/s = 1/4π, JHEP 03 (2009) 084 [arXiv:0812.2521] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    X.O. Camanho and J.D. Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007 [arXiv:0911.3160] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  84. [84]
    A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin, Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  85. [85]
    L.-Y. Hung, R.C. Myers and M. Smolkin, On Holographic Entanglement Entropy and Higher Curvature Gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].ADSCrossRefGoogle Scholar
  86. [86]
    L. Aránguiz, X.-M. Kuang and O. Mišković, Topological black holes in Pure Gauss-Bonnet gravity and phase transitions, arXiv:1507.02309 [INSPIRE].
  87. [87]
    X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, arXiv:1407.5597 [INSPIRE].
  88. [88]
    G. D’Appollonio, P. Vecchia, R. Russo and G. Veneziano, Regge behavior saves String Theory from causality violations, JHEP 05 (2015) 144 [arXiv:1502.01254] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  89. [89]
    M. Fierz, Force-free particles with any spin, Helv. Phys. Acta 12 (1939) 3 [INSPIRE].CrossRefzbMATHGoogle Scholar
  90. [90]
    M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368 [INSPIRE].ADSGoogle Scholar
  92. [92]
    S.F. Hassan and R.A. Rosen, Resolving the Ghost Problem in non-Linear Massive Gravity, Phys. Rev. Lett. 108 (2012) 041101 [arXiv:1106.3344] [INSPIRE].ADSCrossRefGoogle Scholar
  93. [93]
    S.F. Hassan, R.A. Rosen and A. Schmidt-May, Ghost-free Massive Gravity with a General Reference Metric, JHEP 02 (2012) 026 [arXiv:1109.3230] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  94. [94]
    M. Park, Quantum Aspects of Massive Gravity, Class. Quant. Grav. 28 (2011) 105012 [arXiv:1009.4369] [INSPIRE].ADSCrossRefMathSciNetzbMATHGoogle Scholar
  95. [95]
    C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].ADSGoogle Scholar
  96. [96]
    C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].ADSCrossRefGoogle Scholar
  97. [97]
    K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].ADSCrossRefGoogle Scholar
  98. [98]
    Y.-F. Cai, D.A. Easson, C. Gao and E.N. Saridakis, Charged black holes in nonlinear massive gravity, Phys. Rev. D 87 (2013) 064001 [arXiv:1211.0563] [INSPIRE].ADSGoogle Scholar
  99. [99]
    E. Babichev and A. Fabbri, A class of charged black hole solutions in massive (bi)gravity, JHEP 07 (2014) 016 [arXiv:1405.0581] [INSPIRE].ADSCrossRefGoogle Scholar
  100. [100]
    E. Babichev, C. Deffayet and R. Ziour, Recovering General Relativity from massive gravity, Phys. Rev. Lett. 103 (2009) 201102 [arXiv:0907.4103] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  101. [101]
    L. Alberte, A.H. Chamseddine and V. Mukhanov, Massive Gravity: Resolving the Puzzles, JHEP 12 (2010) 023 [arXiv:1008.5132] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  102. [102]
    K. Koyama, G. Niz and G. Tasinato, Analytic solutions in non-linear massive gravity, Phys. Rev. Lett. 107 (2011) 131101 [arXiv:1103.4708] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  103. [103]
    T.M. Nieuwenhuizen, Exact Schwarzschild-de Sitter black holes in a family of massive gravity models, Phys. Rev. D 84 (2011) 024038 [arXiv:1103.5912] [INSPIRE].ADSGoogle Scholar
  104. [104]
    M.S. Volkov, Self-accelerating cosmologies and hairy black holes in ghost-free bigravity and massive gravity, Class. Quant. Grav. 30 (2013) 184009 [arXiv:1304.0238] [INSPIRE].ADSCrossRefMathSciNetzbMATHGoogle Scholar
  105. [105]
    D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE].
  106. [106]
    S.F. Hassan and R.A. Rosen, On Non-Linear Actions for Massive Gravity, JHEP 07 (2011) 009 [arXiv:1103.6055] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  107. [107]
    R.-G. Cai, Y.-P. Hu, Q.-Y. Pan and Y.-L. Zhang, Thermodynamics of Black Holes in Massive Gravity, Phys. Rev. D 91 (2015) 024032 [arXiv:1409.2369] [INSPIRE].ADSMathSciNetGoogle Scholar
  108. [108]
    J. Xu, L.-M. Cao and Y.-P. Hu, PV criticality in the extended phase space of black holes in massive gravity, Phys. Rev. D 91 (2015) 124033 [arXiv:1506.03578] [INSPIRE].ADSGoogle Scholar
  109. [109]
    S.H. Hendi, S. Panahiyan, B.E. Panah and M. Momennia, Geometrical thermodynamics of phase transition: charged black holes in massive gravity, arXiv:1506.07262 [INSPIRE].
  110. [110]
    R.A. Davison, Momentum relaxation in holographic massive gravity, Phys. Rev. D 88 (2013) 086003 [arXiv:1306.5792] [INSPIRE].ADSGoogle Scholar
  111. [111]
    M. Blake and D. Tong, Universal Resistivity from Holographic Massive Gravity, Phys. Rev. D 88 (2013) 106004 [arXiv:1308.4970] [INSPIRE].ADSGoogle Scholar
  112. [112]
    R.A. Davison, K. Schalm and J. Zaanen, Holographic duality and the resistivity of strange metals, Phys. Rev. B 89 (2014) 245116 [arXiv:1311.2451] [INSPIRE].ADSCrossRefGoogle Scholar
  113. [113]
    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].
  114. [114]
    Y.S. Myung, Thermodynamics of the Schwarzschild-de Sitter black hole: Thermal stability of the Nariai black hole, Phys. Rev. D 77 (2008) 104007 [arXiv:0712.3315] [INSPIRE].ADSMathSciNetGoogle Scholar
  115. [115]
    A. Sheykhi, M.H. Dehghani and S.H. Hendi, Thermodynamic instability of charged dilaton black holes in AdS spaces, Phys. Rev. D 81 (2010) 084040 [arXiv:0912.4199] [INSPIRE].ADSGoogle Scholar
  116. [116]
    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].ADSGoogle Scholar
  117. [117]
    S. Hendi and S. Panahiyan, Thermodynamic instability of topological black holes in Gauss-Bonnet gravity with a generalized electrodynamics, Phys. Rev. D 90 (2014) 124008 [arXiv:1501.05481] [INSPIRE].ADSGoogle Scholar
  118. [118]
    R.-G. Cai, Thermodynamics of Conformal Anomaly Corrected Black Holes in AdS Space, Phys. Lett. B 733 (2014) 183 [arXiv:1405.1246] [INSPIRE].ADSCrossRefGoogle Scholar
  119. [119]
    B.P. Dolan, Thermodynamic stability of asymptotically anti-de Sitter rotating black holes in higher dimensions, Class. Quant. Grav. 31 (2014) 165011 [arXiv:1403.1507] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  120. [120]
    S.H. Hendi, S. Panahiyan and R. Mamasani, Thermodynamic stability of charged BTZ black holes: Ensemble dependency problem and its solution, Gen. Rel. Grav. 47 (2015) 91 [arXiv:1507.08496] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  121. [121]
    D. Kastor, S. Ray and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes, Class. Quant. Grav. 26 (2009) 195011 [arXiv:0904.2765] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  122. [122]
    C.V. Johnson, Holographic Heat Engines, Class. Quant. Grav. 31 (2014) 205002 [arXiv:1404.5982] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  123. [123]
    B.P. Dolan, Bose condensation and branes, JHEP 10 (2014) 179 [arXiv:1406.7267] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  124. [124]
    B.P. Dolan, Black holes and Boyle’s law — The thermodynamics of the cosmological constant, Mod. Phys. Lett. A 30 (2015) 1540002 [arXiv:1408.4023] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  125. [125]
    D. Grumiller, R. McNees and J. Salzer, Cosmological constant as confining U(1) charge in two-dimensional dilaton gravity, Phys. Rev. D 90 (2014) 044032 [arXiv:1406.7007] [INSPIRE].ADSGoogle Scholar
  126. [126]
    S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  127. [127]
    J.D.E. Creighton and R.B. Mann, Quasilocal thermodynamics of dilaton gravity coupled to gauge fields, Phys. Rev. D 52 (1995) 4569 [gr-qc/9505007] [INSPIRE].
  128. [128]
    G.W. Gibbons, R. Kallosh and B. Kol, Moduli, scalar charges and the first law of black hole thermodynamics, Phys. Rev. Lett. 77 (1996) 4992 [hep-th/9607108] [INSPIRE].ADSCrossRefGoogle Scholar
  129. [129]
    B.P. Dolan, The cosmological constant and the black hole equation of state, Class. Quant. Grav. 28 (2011) 125020 [arXiv:1008.5023] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  130. [130]
    B.P. Dolan, Pressure and volume in the first law of black hole thermodynamics, Class. Quant. Grav. 28 (2011) 235017 [arXiv:1106.6260] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  131. [131]
    D. Kubiznak and R.B. Mann, P-V criticality of charged AdS black holes, JHEP 07 (2012) 033 [arXiv:1205.0559] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  132. [132]
    R.-G. Cai, L.-M. Cao, L. Li and R.-Q. Yang, PV criticality in the extended phase space of Gauss-Bonnet black holes in AdS space, JHEP 09 (2013) 005 [arXiv:1306.6233] [INSPIRE].ADSCrossRefGoogle Scholar
  133. [133]
    M.B.J. Poshteh, B. Mirza and Z. Sherkatghanad, Phase transition, critical behavior and critical exponents of Myers-Perry black holes, Phys. Rev. D 88 (2013) 024005 [arXiv:1306.4516] [INSPIRE].ADSGoogle Scholar
  134. [134]
    S. Chen, X. Liu, C. Liu and J. Jing, PV criticality of AdS black hole in f (R) gravity, Chin. Phys. Lett. 30 (2013) 060401 [arXiv:1301.3234] [INSPIRE].ADSCrossRefGoogle Scholar
  135. [135]
    S.H. Hendi and M.H. Vahidinia, Extended phase space thermodynamics and PV criticality of black holes with a nonlinear source, Phys. Rev. D 88 (2013) 084045 [arXiv:1212.6128] [INSPIRE].ADSGoogle Scholar
  136. [136]
    J.-X. Mo and W.-B. Liu, PV criticality of topological black holes in Lovelock-Born-Infeld gravity, Eur. Phys. J. C 74 (2014) 2836 [arXiv:1401.0785] [INSPIRE].ADSCrossRefGoogle Scholar
  137. [137]
    D.-C. Zou, S.-J. Zhang and B. Wang, Critical behavior of Born-Infeld AdS black holes in the extended phase space thermodynamics, Phys. Rev. D 89 (2014) 044002 [arXiv:1311.7299] [INSPIRE].ADSGoogle Scholar
  138. [138]
    W. Xu and L. Zhao, Critical phenomena of static charged AdS black holes in conformal gravity, Phys. Lett. B 736 (2014) 214 [arXiv:1405.7665] [INSPIRE].ADSCrossRefMathSciNetzbMATHGoogle Scholar
  139. [139]
    E. Caceres, P.H. Nguyen and J.F. Pedraza, Holographic entanglement entropy and the extended phase structure of STU black holes, JHEP 09 (2015) 184 [arXiv:1507.06069] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  140. [140]
    F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem. Phys. 63 (1975) 2479.ADSMathSciNetCrossRefGoogle Scholar
  141. [141]
    F. Weinhold, Metric geometry of equilibrium thermodynamics II, J. Chem. Phys. 63 (1975) 2484.ADSMathSciNetCrossRefGoogle Scholar
  142. [142]
    G. Ruppeiner, Thermodynamics: A Riemannian geometric model, Phys. Rev. A 20 (1979) 1608.ADSCrossRefGoogle Scholar
  143. [143]
    G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory, Rev. Mod. Phys. 67 (1995) 605 [Erratum ibid. 68 (1996) 313] [INSPIRE].
  144. [144]
    P. Salamon, J. Nulton and E. Ihrig, On the relation between entropy and energy versions of thermodynamic length, J. Chem. Phys. 80 (1984) 436.ADSCrossRefGoogle Scholar
  145. [145]
    H. Quevedo, Geometrothermodynamics, J. Math. Phys. 48 (2007) 013506 [physics/0604164] [INSPIRE].
  146. [146]
    H. Quevedo and A. Sanchez, Geometrothermodynamics of asymptotically de Sitter black holes, JHEP 09 (2008) 034 [arXiv:0805.3003] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  147. [147]
    H. Quevedo, Geometrothermodynamics of black holes, Gen. Rel. Grav. 40 (2008) 971 [arXiv:0704.3102] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  148. [148]
    S.H. Hendi, S. Panahiyan, B.E. Panah and M. Momennia, A new approach toward geometrical concept of black hole thermodynamics, Eur. Phys. J. C 75 (2015) 507 [arXiv:1506.08092] [INSPIRE].ADSCrossRefGoogle Scholar
  149. [149]
    S.H. Hendi, S. Panahiyan and B.E. Panah, Geometrical method for thermal instability of nonlinearly charged BTZ Black Holes, Adv. High Energy Phys. 2015 (2015) 743086 [arXiv:1509.07014] [INSPIRE].CrossRefGoogle Scholar
  150. [150]
    S. Hendi, A. Sheykhi, S. Panahiyan and B. Eslam Panah, Phase transition and thermodynamic geometry of Einstein-Maxwell-dilaton black holes, Phys. Rev. D 92 (2015) 064028 [arXiv:1509.08593] [INSPIRE].ADSGoogle Scholar
  151. [151]
    S.F. Hassan and R.A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  152. [152]
    N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys. 305 (2003) 96 [hep-th/0210184] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  153. [153]
    S.L. Dubovsky, Phases of massive gravity, JHEP 10 (2004) 076 [hep-th/0409124] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  154. [154]
    A.H. Chamseddine and V. Mukhanov, Higgs for Graviton: Simple and Elegant Solution, JHEP 08 (2010) 011 [arXiv:1002.3877] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  155. [155]
    C. de Rham, G. Gabadadze and A.J. Tolley, Ghost free Massive Gravity in the Stúckelberg language, Phys. Lett. B 711 (2012) 190 [arXiv:1107.3820] [INSPIRE].ADSCrossRefGoogle Scholar
  156. [156]
    J. Kluson, Non-Linear Massive Gravity with Additional Primary Constraint and Absence of Ghosts, Phys. Rev. D 86 (2012) 044024 [arXiv:1204.2957] [INSPIRE].ADSGoogle Scholar
  157. [157]
    J. Kevorkian, Partial Differential Equations, Springer, New York, U.S.A. (2000).zbMATHCrossRefGoogle Scholar
  158. [158]
    L. Alberte and A. Khmelnitsky, Reduced massive gravity with two Stückelberg fields, Phys. Rev. D 88 (2013) 064053 [arXiv:1303.4958] [INSPIRE].ADSGoogle Scholar
  159. [159]
    L. Alberte and A. Khmelnitsky, Stability of Massive Gravity Solutions for Holographic Conductivity, Phys. Rev. D 91 (2015) 046006 [arXiv:1411.3027] [INSPIRE].ADSMathSciNetGoogle Scholar
  160. [160]
    H. Zhang and X.-Z. Li, Ghost free massive gravity with singular reference metrics, arXiv:1510.03204 [INSPIRE].
  161. [161]
    S.H. Hendi, S. Panahiyan and B.E. Panah, PV criticality and geometrothermodynamics of black holes with Born-Infeld type nonlinear electrodynamics, Int. J. Mod. Phys. D 25 (2016) 1650010 [arXiv:1410.0352] [INSPIRE].CrossRefADSGoogle Scholar

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© The Author(s) 2016

Authors and Affiliations

  1. 1.Physics Department and Biruni Observatory, College of SciencesShiraz UniversityShirazIran
  2. 2.Research Institute for Astronomy and Astrophysics of Maragha (RIAAM)MaraghaIran

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