Abstract
We investigate phase transitions and critical phenomena in Kerr-Newman-Anti de Sitter black holes in the framework of the geometry of their equilibrium thermodynamic state space. The scalar curvature of these state space Riemannian geometries is computed in various ensembles. The scalar curvature diverges at the critical point of second order phase transitions for these systems. Remarkably, however, we show that the state space scalar curvature also carries information about the liquid-gas like first order phase transitions and the consequent instabilities and phase coexistence for these black holes. This is encoded in the turning point behavior and the multi-valued branched structure of the scalar curvature in the neighborhood of these first order phase transitions. We re-examine this first for the conventional Van der Waals system, as a preliminary exercise. Subsequently, we study the Kerr-Newman-AdS black holes for a grand canonical and two “mixed” ensembles and establish novel phase structures. The state space scalar curvature bears out our assertion for the first order phase transitions for both the known and the new phase structures, and closely resembles the Van der Waals system.
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References
R.M. Wald, The thermodynamics of black holes, Living Rev. Rel. 4 (2001) 6 [gr-qc/9912119] [SPIRES].
D.N. Page, Hawking radiation and black hole thermodynamics, New J. Phys. 7 (2005) 203 [hep-th/0409024] [SPIRES].
R. Brout, S. Massar, R. Parentani and P. Spindel, A primer for black hole quantum physics, Phys. Rept. 260 (1995) 329 [arXiv:0710.4345] [SPIRES].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [SPIRES].
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] [SPIRES].
P.C.W. Davies, Thermodynamics of black holes, Rep. Prog. Phys 41 (1978) 1313.
P.C.W. Davies, Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space, Class. Quant. Grav. 6 (1989) 1909.
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] [SPIRES].
C.O. Lousto, The fourth law of black hole thermodynamics, Nucl. Phys. B 410 (1993) 155 [Erratum ibid. B 449 (1995) 433] [gr-qc/9306014] [SPIRES].
D. Pavon and J.M. Rubi, Nonequilibrium thermodynamic fluctuations of black holes, Phys. Rev. D 37 (1988) 2052 [SPIRES].
A. Curir, Rotating black holes as dissipative spin-thermodynamical systems, Gen. Rel. Grav. 13 (1981) 417.
C.O. Lousto, The Emergence of an effective two-dimensional quantum description from the study of critical phenomena in black holes, Phys. Rev. D 51 (1995) 1733 [gr-qc/9405048] [SPIRES].
R.-G. Cai and Y.S. Myung, Critical behavior for the dilaton black holes, Nucl. Phys. B 495 (1997) 339 [hep-th/9702159] [SPIRES].
R.-G. Cai, Z.-J. Lu and Y.-Z. Zhang, Critical behavior in 2 + 1 dimensional black holes, Phys. Rev. D 55 (1997) 853 [gr-qc/9702032] [SPIRES].
G. Arcioni and E. Lozano-Tellechea, Stability and critical phenomena of black holes and black rings, Phys. Rev. D 72 (2005) 104021 [hep-th/0412118] [SPIRES].
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] [SPIRES].
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] [SPIRES].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [SPIRES].
L. Tisza, Generalized thermodynamics, MIT Press, Cambridge U.S.A. (1966).
H.B. Callen, Thermodynamics and an introcution to thermostatitics, Wiley & Sons Inc., New York U.S.A. (1985).
F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem Phys. 63 (1975) 2479.
Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations, J. Chem Phys.63 (1975) 2484.
G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory, Rev. Mod. Phys. 67 (1995) 605 [Erratum ibid. 68 (1996) 313] [SPIRES].
S. Ferrara, G.W. Gibbons and R. Kallosh, Black holes and critical points in moduli space, Nucl. Phys. B 500 (1997) 75 [hep-th/9702103] [SPIRES].
J.E. Aman, I. Bengtsson and N. Pidokrajt, Geometry of black hole thermodynamics, Gen. Rel. Grav. 35 (2003) 1733 [gr-qc/0304015] [SPIRES].
R.-G. Cai and J.-H. Cho, Thermodynamic curvature of the BTZ black hole, Phys. Rev. D 60 (1999) 067502 [hep-th/9803261] [SPIRES].
J.-y. Shen, R.-G. Cai, B. Wang and R.-K. Su, Thermodynamic geometry and critical behavior of black holes, Int. J. Mod. Phys. A 22 (2007) 11 [gr-qc/0512035] [SPIRES].
T. Sarkar, G. Sengupta and B. Nath Tiwari, On the thermodynamic geometry of BTZ black holes, JHEP 11 (2006) 015 [hep-th/0606084] [SPIRES].
T. Sarkar, G. Sengupta and B.N. Tiwari, Thermodynamic geometry and extremal black holes in string theory, JHEP 10 (2008) 076 [arXiv:0806.3513] [SPIRES].
L. Diosi, B. Lukacs and A. Racz, Mapping the Van der Waals state space, J. Chem. Phys. 91 (1989) 3061.
L. Diosi and B. Lukacs, Spatial correlation in diluted gases from the viewpoint of the metric of the thermodynamic state space, J. Chem. Phys. 84 (1986) 5081.
G. Ruppeiner, Thermodynamic curvature: origin and meaning, in Advances in thermodynamics. Volume 3, S. Sieniutycz and P. Salamon eds., Taylor & Francis, U.S.A. (1990).
H. Janyszek, Riemannian geometry and stability of thermodynamical equilibrium systems, J. Phys. A 23 (1990) 477.
G. Ruppeiner, Thermodynamics: a Riemannian geometric model, Phys. Rev. A 20 (1979) 1608.
G. Ruppeiner, Thermodynamic critical fluctuation theory?, Phys. Rev. Lett. 50 (1983) 287 [SPIRES].
G. Ruppeiner, Thermodynamic curvature of the multicomponent ideal gas, Phys. Rev. A 41 (1990) 2200.
M. Santoro and A.S. Benight, On the geometrical thermodynamics of chemical reactions, [math-ph/0507026].
H. Janyszek, R. Mrugala, Geometrical structure of the state space in classical statistical and phenomenological thermodynamics, Rep. Math. Phys. 27 (1989) 145.
L. Landau and E.M. Lifshitz, Statistical physics, Pergamon Press, U.K. (1988).
V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [SPIRES].
G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [SPIRES].
S. Carlip and S. Vaidya, Phase transitions and critical behavior for charged black holes, Class. Quant. Grav. 20 (2003) 3827 [gr-qc/0306054] [SPIRES].
G. Ruppeiner, Stability and fluctuations in black hole thermodynamics, Phys. Rev. D 75 (2007) 024037 [SPIRES].
G. Ruppeiner, Thermodynamic curvature and phase transitions in Kerr-Newman black holes, Phys. Rev. D 78 (2008) 024016 [arXiv:0802.1326] [SPIRES].
B. Mirza and M. Zamani-Nasab, Ruppeiner geometry of RN black holes: flat or curved?, JHEP 06 (2007) 059 [arXiv:0706.3450] [SPIRES].
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ArXiv ePrint: 1002.2538
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Sahay, A., Sarkar, T. & Sengupta, G. Thermodynamic geometry and phase transitions in Kerr-Newman-AdS black holes. J. High Energ. Phys. 2010, 118 (2010). https://doi.org/10.1007/JHEP04(2010)118
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DOI: https://doi.org/10.1007/JHEP04(2010)118