Abstract
Hairy black holes and gravitational solitons in three dimensions for the new (BHT) massive gravity theory are considered at the special case when there is a unique maximally symmetric solution. Following the Brown-York approach with suitable counterterms, it is shown that the soliton possesses a fixed negative mass which coincides with that of AdS spacetime regardless the value of the integration constant that describes it. The soliton is then regarded as a degenerate ground state labeled by a modulus parameter. The Euclidean action is shown to be finite and independent of modulus and hair parameters for both classes of solutions, reproducing the hairy black hole free energy. Modular invariance implies that the gravitational hair becomes determined by the modulus parameter. Cardy formula is shown to agree with the semiclassical entropy provided the modulus parameter of the ground state is spontaneously fixed, suggesting that the hairy black hole is in a broken phase. Indeed, it is found that the critical temperature T c = (2πl)−1 characterizes a first order phase transition between the static hairy black hole and the soliton which, due to the existence of gravitational hair, can take place in the semiclassical regime.
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References
S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982) 975 [SPIRES].
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Ann. Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406 [Ann. Phys. 281 (2000) 409]. [SPIRES].
E.A. Bergshoeff, O. Hohm and P.K. Townsend, Massive gravity in three dimensions, Phys. Rev. Lett. 102 (2009) 201301 [arXiv:0901.1766] [SPIRES].
E.A. Bergshoeff, O. Hohm and P.K. Townsend, More on massive 3D gravity, Phys. Rev. D 79 (2009) 124042 [arXiv:0905.1259] [SPIRES].
M. Blagojevic and B. Cvetkovic, Hamiltonian analysis of BHT massive gravity, JHEP 01 (2011) 082 [arXiv:1010.2596] [SPIRES].
M. Blagojevic and B. Cvetkovic, Extra gauge symmetries in BHT gravity, JHEP 03 (2011) 139 [arXiv:1103.2388] [SPIRES].
M. Sadegh and A. Shirzad, Constraint structure of the three dimensional massive gravity, Phys. Rev. D 83 (2011) 084040 [arXiv:1010.2887] [SPIRES].
B. Tekin, Partially massless spin-2 fields in string generated models, hep-th/0306178 [SPIRES].
S. Deser and R.I. Nepomechie, Gauge invariance versus masslessness in de Sitter space, Ann. Phys. 154 (1984) 396 [SPIRES].
S. Deser and A. Waldron, Gauge invariances and phases of massive higher spins in (A)dS, Phys. Rev. Lett. 87 (2001) 031601 [hep-th/0102166] [SPIRES].
S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [SPIRES].
S. Deser, Ghost-free, finite, fourth order D = 3 (alas) gravity, Phys. Rev. Lett. 103 (2009) 101302 [arXiv:0904.4473] [SPIRES].
J. Oliva, D. Tempo and R. Troncoso, Three-dimensional black holes, gravitational solitons, kinks and wormholes for BHT masive gravity, JHEP 07 (2009) 011 [arXiv:0905.1545] [SPIRES].
G. Clement, Warped AdS 3 black holes in new massive gravity, Class. Quant. Grav. 26 (2009) 105015 [arXiv:0902.4634] [SPIRES].
G. Clement, Black holes with a null Killing vector in new massive gravity in three dimensions, Class. Quant. Grav. 26 (2009) 165002 [arXiv:0905.0553] [SPIRES].
E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, Lifshitz black hole in three dimensions, Phys. Rev. D 80 (2009) 104029 [arXiv:0909.1347] [SPIRES].
E. Ayon-Beato, G. Giribet and M. Hassaine, Bending AdS waves with new massive gravity, JHEP 05 (2009) 029 [arXiv:0904.0668] [SPIRES].
M. Chakhad, Kundt spacetimes of massive gravity in three dimensions, arXiv:0907.1973 [SPIRES].
H. Ahmedov and A.N. Aliev, The general type N solution of new massive gravity, Phys. Lett. B 694 (2010) 143 [arXiv:1008.0303] [SPIRES].
H.L.C. Louzada, U. Camara dS and G.M. Sotkov, Massive 3D gravity big-bounce, Phys. Lett. B 686 (2010) 268 [arXiv:1001.3622] [SPIRES].
U.d. Camara and G.M. Sotkov, New massive gravity domain walls, Phys. Lett. B 694 (2010) 94 [arXiv:1008.2553] [SPIRES].
H. Ahmedov and A.N. Aliev, Type D solutions of 3D new massive gravity, Phys. Rev. D 83 (2011) 084032 [arXiv:1103.1086] [SPIRES].
H. Maeda, Black-hole dynamics in BHT massive gravity, JHEP 02 (2011) 039 [arXiv:1012.5048] [SPIRES].
I. Bakas and C. Sourdis, Homogeneous vacua of (generalized) new massive gravity, Class. Quant. Grav. 28 (2011) 015012 [arXiv:1006.1871] [SPIRES].
A. Ghodsi and M. Moghadassi, Charged black holes in new massive gravity, Phys. Lett. B 695 (2011) 359 [arXiv:1007.4323] [SPIRES].
M. Alishahiha and A. Naseh, Holographic renormalization of new massive gravity, Phys. Rev. D 82 (2010) 104043 [arXiv:1005.1544] [SPIRES].
I. Gullu, T.C. Sisman and B. Tekin, c-functions in the Born-Infeld extended new massive gravity, Phys. Rev. D 82 (2010) 024032 [arXiv:1005.3214] [SPIRES].
A. Sinha, On the new massive gravity and AdS/CFT, JHEP 06 (2010) 061 [arXiv:1003.0683] [SPIRES].
D. Grumiller, N. Johansson and T. Zojer, Short-cut to new anomalies in gravity duals to logarithmic conformal field theories, JHEP 01 (2011) 090 [arXiv:1010.4449] [SPIRES].
A. Sinha, On higher derivative gravity, c-theorems and cosmology, Class. Quant. Grav. 28 (2011) 085002 [arXiv:1008.4315] [SPIRES].
Y. Kwon, S. Nam, J.-D. Park and S.-H. Yi, Quasi normal modes for new type black holes in new massive gravity, Class. Quant. Grav. 28 (2011) 145006 [arXiv:1102.0138] [SPIRES].
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [SPIRES].
L.F. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B 195 (1982) 76 [SPIRES].
S. Deser and B. Tekin, Energy in generic higher curvature gravity theories, Phys. Rev. D 67 (2003) 084009 [hep-th/0212292] [SPIRES].
J.D. Brown and J.W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [SPIRES].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [SPIRES].
V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [SPIRES].
O. Hohm and E. Tonni, A boundary stress tensor for higher-derivative gravity in AdS and Lifshitz backgrounds, JHEP 04 (2010) 093 [arXiv:1001.3598] [SPIRES].
G. Giribet and M. Leston, Boundary stress tensor and counterterms for weakened AdS3 asymptotic in new massive gravity, JHEP 09 (2010) 070 [arXiv:1006.3349] [SPIRES].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [SPIRES].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [gr-qc/9302012] [SPIRES].
G. Giribet, J. Oliva, D. Tempo and R. Troncoso, Microscopic entropy of the three-dimensional rotating black hole of BHT massive gravity, Phys. Rev. D 80 (2009) 124046 [arXiv:0909.2564] [SPIRES].
S. Nam, J.-D. Park and S.-H. Yi, Mass and angular momentum of black holes in new massive gravity, Phys. Rev. D 82 (2010) 124049 [arXiv:1009.1962] [SPIRES].
S. Nam, J.-D. Park and S.-H. Yi, AdS black hole solutions in the extended new massive gravity, JHEP 07 (2010) 058 [arXiv:1005.1619] [SPIRES].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [SPIRES].
B. Mirza and Z. Sherkatghanad, Corrected entropy of the rotating black hole solution of the new massive gravity using the tunneling method and Cardy formula, Phys. Rev. D 83 (2011) 104001 [arXiv:1104.0390] [SPIRES].
R. Li, S. Li and J.-R. Ren, Hawking radiation of fermionic field and anomaly in 2 + 1 dimensional black holes, Class. Quant. Grav. 27 (2010) 155011 [arXiv:1005.3615] [SPIRES].
R. Banerjee, S.K. Modak and D. Roychowdhury, Thermodynamics of Hawking-Page phase transition in AdS black holes, arXiv:1106.3877.
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [SPIRES].
M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Black holes and asymptotics of 2 + 1 gravity coupled to a scalar field, Phys. Rev. D 65 (2002) 104007 [hep-th/0201170] [SPIRES].
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [SPIRES].
A. Strominger, Black hole entropy from near-horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [SPIRES].
F. Correa, C. Martinez and R. Troncoso, Scalar solitons and the microscopic entropy of hairy black holes in three dimensions, JHEP 01 (2011) 034 [arXiv:1010.1259] [SPIRES].
S. Carlip, Entropy from conformal field theory at Killing horizons, Class. Quant. Grav. 16 (1999) 3327 [gr-qc/9906126] [SPIRES].
F. Loran, M.M. Sheikh-Jabbari and M. Vincon, Beyond logarithmic corrections to Cardy formula, JHEP 01 (2011) 110 [arXiv:1010.3561] [SPIRES].
J. Oliva, D. Tempo and R. Troncoso, Static spherically symmetric solutions for conformal gravity in three dimensions, Int. J. Mod. Phys. A 24 (2009) 1588 [arXiv:0905.1510] [SPIRES].
N. Cruz and S. Lepe, On the thermal description of the BTZ black holes, Phys. Lett. B 593 (2004) 235 [hep-th/0404218] [SPIRES].
Y. Kurita and M.-a. Sakagami, CFT description of three-dimensional Hawking-Page transition, Prog. Theor. Phys. 113 (2005) 1193 [hep-th/0403091] [SPIRES].
B. Reznik, Thermodynamics and evaporation of the (2 + 1)-dimensions black hole, Phys. Rev. D 51 (1995) 1728 [gr-qc/9403027] [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].
S. Carlip and C. Teitelboim, Aspects of black hole quantum mechanics and thermodynamics in (2 + 1)-dimensions, Phys. Rev. D 51 (1995) 622 [gr-qc/9405070] [SPIRES].
S.W. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [SPIRES].
A. Pérez, D. Tempo and R. Troncoso, Dynamical structure of massive gravity theories in three dimensions, work in progress.
M. Bañados, L.J. Garay and M. Henneaux, The dynamical structure of higher dimensional Chern-Simons theory, Nucl. Phys. B 476 (1996) 611 [hep-th/9605159] [SPIRES].
O. Chandía, R. Troncoso and J. Zanelli, Dynamical content of Chern-Simons supergravity, hep-th/9903204 [SPIRES].
O. Mišković, R. Troncoso and J. Zanelli, Canonical sectors of five-dimensional Chern-Simons theories, Phys. Lett. B 615 (2005) 277 [hep-th/0504055] [SPIRES].
O. Mišković, R. Troncoso and J. Zanelli, Dynamics and BPS states of AdS5 supergravity with a Gauss-Bonnet term, Phys. Lett. B 637 (2006) 317 [hep-th/0603183] [SPIRES].
J. Saavedra, R. Troncoso and J. Zanelli, Degenerate dynamical systems, J. Math. Phys. 42 (2001) 4383 [hep-th/0011231] [SPIRES].
N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [SPIRES].
P.C. Hohenberg, Existence of long-range order in one and two dimensions, Phys. Rev. 158 (1967) 383 [SPIRES].
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [SPIRES].
D. Anninos, S.A. Hartnoll and N. Iqbal, Holography and the Coleman-Mermin-Wagner theorem, Phys. Rev. D 82 (2010) 066008 [arXiv:1005.1973] [SPIRES].
J. Ren, One-dimensional holographic superconductor from AdS 3 /CFT 2 correspondence, JHEP 11 (2010) 055 [arXiv:1008.3904] [SPIRES].
N. Lashkari, Holographic symmetry-breaking phases in AdS 3 /CFT 2, arXiv:1011.3520 [SPIRES].
Y. Liu, Q. Pan and B. Wang, Holographic superconductor developed in BTZ black hole background with backreactions, arXiv:1106.4353 [SPIRES].
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Pérez, A., Tempo, D. & Troncoso, R. Gravitational solitons, hairy black holes and phase transitions in BHT massive gravity. J. High Energ. Phys. 2011, 93 (2011). https://doi.org/10.1007/JHEP07(2011)093
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DOI: https://doi.org/10.1007/JHEP07(2011)093