Abstract
The Robinson-Trautman space-times provide solutions of Einstein’s equations with negative cosmological constant, which settle to AdS 4 Schwarzschild black hole at late times. Via gauge/gravity duality they should describe a system out of equilibrium that evolves towards thermalization. We show that the area of the past apparent horizon of these space-times satisfies a generalized Penrose inequality and we formulate as well as provide evidence for a suitable generalization of Thorne’s hoop conjecture. We also compute the holographic energy-momentum tensor and deduce its late time behavior. It turns out that the complete non-equilibrium process on the boundary is governed by Calabi’s flow on S 2. Upon linearization, only special modes that arise as supersymmetric zero energy states of an associated supersymmetric quantum mechanics problem contribute to the solution. We find that each pole of radiation has an effective viscosity given by the eigenvalues of the Laplace operator on S 2 and there is an apparent violation of the KSS bound on η/s for the low lying harmonics of large AdS 4 black holes. These modes, however, do not satisfy Dirichlet boundary conditions, they are out-going and they do not appear to have a Kruskal extension across the future horizon \( \mathcal{H} \) +.
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P.M. Chesler and L.G. Yaffe, Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes, JHEP 07 (2014) 086 [arXiv:1309.1439] [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity Bound Violation in Higher Derivative Gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [INSPIRE].
Y. Kats and P. Petrov, Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory, JHEP 01 (2009) 044 [arXiv:0712.0743] [INSPIRE].
J. Erdmenger, P. Kerner and H. Zeller, Non-universal shear viscosity from Einstein gravity, Phys. Lett. B 699 (2011) 301 [arXiv:1011.5912] [INSPIRE].
A. Rebhan and D. Steineder, Violation of the Holographic Viscosity Bound in a Strongly Coupled Anisotropic Plasma, Phys. Rev. Lett. 108 (2012) 021601 [arXiv:1110.6825] [INSPIRE].
S. Cremonini, The Shear Viscosity to Entropy Ratio: A Status Report, Mod. Phys. Lett. B 25 (2011)1867 [arXiv:1108.0677] [INSPIRE].
I. Robinson and A. Trautman, Spherical Gravitational Waves, Phys. Rev. Lett. 4 (1960) 431 [INSPIRE].
I. Robinson and A. Trautman, Some spherical gravitational waves in general relativity, Proc. Roy. Soc. Lond. A 265 (1962) 463 [INSPIRE].
D. Kramer, H. Stephani, E. Herlt and M. MacCallum, Exact Solutions of Einstein’s Field Equations, Cambridge Unversity Press, Cambridge (1980).
M. Carmeli, Group Theory and General Relativity, McGraw Hill, New York (1977).
J.B. Griffiths and J. Podolsky, Exact Space-Times in Einstein’s General Relativity, Cambridge Unversity Press, Cambridge (2009).
J. Foster and E.-T. Newman, Note on the Robinson-Trautman solutions, J. Math. Phys. 8 (1967) 189.
B. Lukacs, Z. Perjes, J. Porter and A. Sebestyen, Lyapunov functional approach to radiative metrics, Gen. Rel. Grav. 16 (1984) 691.
B.G. Schmidt, Existence of solutions of the Robinson-Trautman equation and spatial infinity, Gen. Rel. Grav. 20 (1988) 65.
A. Rendall, Existence and asymptotic properties of global solutions of the Robinson-Trautman equation, Class. Quant. Grav. 5 (1988) 1339.
D. Singleton, On global existence and convergence of vacuum Robinson-Trautman solutions, Class. Quant. Grav. 7 (1990) 1333.
P. Chrusciel, Semiglobal existence and convergence of solutions of the Robinson-Trautman (two-dimensional Calabi) equation, Commun. Math. Phys. 137 (1991) 289 [INSPIRE].
P.T. Chrusciel, M.A.H. MacCallum and D.B. Singleton, Gravitational waves in general relativity: 14. Bondi expansions and the polyhomogeneity of Scri, Proc. Roy. Soc. Lond. A 436 (1992)299 [gr-qc/9305021] [INSPIRE].
S. Chandrasekhar, On algebraically special perturbations of black holes, Proc. Roy. Soc. Lond. A 392 (1984) 1.
G.-Y. Qi and B.F. Schutz, Robinson-Trautman equations and Chandrasekhar’s special perturbation of the Schwarzschild metric, Gen. Rel. Grav. 25 (1993) 1185 [INSPIRE].
W.E. Couch and E.T. Newman, Algebraically special perturbations of the Schwarzschild metric, J. Math. Phys. 14 (1973) 285 [INSPIRE].
O.J.C. Dias and H.S. Reall, Algebraically special perturbations of the Schwarzschild solution in higher dimensions, Class. Quant. Grav. 30 (2013) 095003 [arXiv:1301.7068] [INSPIRE].
T. Regge and J.A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957) 1063 [INSPIRE].
F.J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24 (1970) 737 [INSPIRE].
S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, Oxford (1983).
K.D. Kokkotas and B.G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Rel. 2 (1999) 2 [gr-qc/9909058] [INSPIRE].
V. Cardoso and J.P.S. Lemos, Quasinormal modes of Schwarzschild anti-de Sitter black holes: Electromagnetic and gravitational perturbations, Phys. Rev. D 64 (2001) 084017 [gr-qc/0105103] [INSPIRE].
I.G. Moss and J.P. Norman, Gravitational quasinormal modes for anti-de Sitter black holes, Class. Quant. Grav. 19 (2002) 2323 [gr-qc/0201016] [INSPIRE].
I. Bakas, Energy-momentum/Cotton tensor duality for AdS 4 black holes, JHEP 01 (2009) 003 [arXiv:0809.4852] [INSPIRE].
F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rept. 251 (1995) 267 [hep-th/9405029] [INSPIRE].
K.P. Tod, Analogues of the past horizon in the Robinson-Trautman metrics, Class. Quant. Grav. 6 (1989) 1159.
E. Calabi, Extremal Kähler metrics, in Seminar on Differential Geometry, S.-T. Yau ed., Annals of Mathematics Studies, 102 Princeton University Press, Princeton (1982).
E. Calabi, Extremal Kähler metrics II, in Differential Geometry and Complex Analysis, I. Chavel and H. Farkas eds., Springer-Verlag, Berlin (1985).
A. Futaki, Kähler-Einstein Metrics and Integral Invariants, Lecture Notes in Mathematics, vol. 1314, Springer-Verlag, Berlin (1988).
G. Tian, Canonical Metrics in Kähler Geometry, Lectures in Mathematics, Birkhäuser, Basel (2002).
P.T. Chrusciel and D.B. Singleton, Nonsmoothness of event horizons of Robinson-Trautman black holes, Commun. Math. Phys. 147 (1992) 137 [INSPIRE].
J. Bicak and J. Podolsky, Cosmic no hair conjecture and black hole formation: An Exact model with gravitational radiation, Phys. Rev. D 52 (1995) 887 [INSPIRE].
J. Bicak and J. Podolsky, Global structure of Robinson-Trautman radiative space-times with a cosmological constant, Phys. Rev. D 55 (1997) 1985 [gr-qc/9901018] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [INSPIRE].
T. Aubin, Non-linear Analysis on Manifolds. Monge-Ampére Equations, Springer-Verlag, Berlin (1982).
O. Svitek, Apparent horizons in D-dimensional Robinson-Trautman spacetime, AIP Conf. Proc. 1122 (2009) 404 [arXiv:0812.3345] [INSPIRE].
J. Podolsky and O. Svitek, Past horizons in Robinson-Trautman spacetimes with a cosmological constant, Phys. Rev. D 80 (2009) 124042 [arXiv:0911.5317] [INSPIRE].
E.W.M. Chow and A.W.C. Lun, Apparent horizons in vacuum Robinson-Trautman space-times, gr-qc/9503065 [INSPIRE].
R. Penrose, Gravitational collapse: The role of general relativity, Riv. Nuovo Cim. 1 (1969) 252 [INSPIRE].
G.W. Gibbons, Some comments on gravitational entropy and the inverse mean curvature flow, Class. Quant. Grav. 16 (1999) 1677 [hep-th/9809167] [INSPIRE].
H.L. Bray and P.T. Chrusciel, The Penrose inequality, gr-qc/0312047 [INSPIRE].
K.P. Tod, More on Penrose’s quasi-local mass. Class. Quant. Grav. 3 (1986) 1169.
K.S. Thorne, Non-spherical gravitational collapse: A short review, in Magic Without Magic, J. Klauder ed., Freeman, San Francisco (1972).
G.W. Gibbons, Birkhoff ’s invariant and Thorne’s Hoop Conjecture, arXiv:0903.1580 [INSPIRE].
M. Cvetič, G.W. Gibbons, C.N. Pope, G.W. Gibbons and C.N. Pope, More about Birkhoff ’s Invariant and Thorne’s Hoop Conjecture for Horizons, Class. Quant. Grav. 28 (2011) 195001 [arXiv:1104.4504] [INSPIRE].
K.P. Tod, The hoop conjecture and the Gibbons-Hawking construction of trapped surfaces, Class. Quant. Grav. 9 (1992) 1581.
G.D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917)199.
P.M. Pu, Some inequalities in certain non-orientable Riemannian manifolds, Pacific J. Math. 2 (1952)55.
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
M. Henningson and K. Skenderis, Holography and the Weyl anomaly, Fortsch. Phys. 48 (2000) 125 [hep-th/9812032] [INSPIRE].
V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
K. Skenderis, Asymptotically Anti-de Sitter space-times and their stress energy tensor, Int. J. Mod. Phys. A 16 (2001) 740 [hep-th/0010138] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
J.D. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, second edition, Pergamon Press, New York (1987).
P.K. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].
P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].
D.T. Son and A.O. Starinets, Viscosity, Black Holes and Quantum Field Theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 95 [arXiv:0704.0240] [INSPIRE].
E. Berti and K.D. Kokkotas, Quasinormal modes of Reissner-Nordstrom-anti-de Sitter black holes: Scalar, electromagnetic and gravitational perturbations, Phys. Rev. D 67 (2003) 064020 [gr-qc/0301052] [INSPIRE].
V. Cardoso, R. Konoplya and J.P.S. Lemos, Quasinormal frequencies of Schwarzschild black holes in anti-de Sitter space-times: A Complete study on the asymptotic behavior, Phys. Rev. D 68 (2003) 044024 [gr-qc/0305037] [INSPIRE].
A.S. Miranda and V.T. Zanchin, Quasinormal modes of plane-symmetric anti-de Sitter black holes: A Complete analysis of the gravitational perturbations, Phys. Rev. D 73 (2006) 064034 [gr-qc/0510066] [INSPIRE].
G. Michalogiorgakis and S.S. Pufu, Low-lying gravitational modes in the scalar sector of the global AdS 4 black hole, JHEP 02 (2007) 023 [hep-th/0612065] [INSPIRE].
S. Bhattacharyya, S. Lahiri, R. Loganayagam and S. Minwalla, Large rotating AdS black holes from fluid mechanics, JHEP 09 (2008) 054 [arXiv:0708.1770] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, R. Loganayagam, G. Mandal, S. Minwalla et al., Local Fluid Dynamical Entropy from Gravity, JHEP 06 (2008) 055 [arXiv:0803.2526] [INSPIRE].
P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 102 (2009) 211601 [arXiv:0812.2053] [INSPIRE].
G.B. de Freitas and H.S. Reall, Algebraically special solutions in AdS/CFT, JHEP 06 (2014) 148 [arXiv:1403.3537] [INSPIRE].
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Bakas, I., Skenderis, K. Non-equilibrium dynamics and AdS 4 Robinson-Trautman. J. High Energ. Phys. 2014, 56 (2014). https://doi.org/10.1007/JHEP08(2014)056
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DOI: https://doi.org/10.1007/JHEP08(2014)056