21 July 2011, USTC-ICTS. Chiang-Mei Chen 陳江梅 Department of Physics, National Central University

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1 21 July 2011, USTC-ICTS Chiang-Mei Chen 陳江梅 Department of Physics, National Central University

2 Outline Black Hole Holographic Principle Kerr/CFT Correspondence Reissner-Nordstrom /CFT Correspondence Kerr Newman/CFTs Correspondence Summary

3 Black Hole Gravitational force: universal and attractive Black hole: (J. Wheeler, 1967) nothing (including light) can escape from it. Newtonian mechanism: Michell 1783, Laplace 1796 Energy conservation: Escape velocity: (E = 0) Radius of horizon: (v = c)

4 Black Hole Two essential characteristics: singularity: the curvature diverges a the center of black hole horizon: infinite gravitational red shift; one way path into black hole Schwarzschild Black Hole

5 Black Hole No hair theorem: Black holes have no hair because they are simple. A black hole is completely characterized by three physical parameters: Mass Angular Momentum Charge

6 Black Hole Reissner-Nordstrom Black Hole: Mass + Charge Kerr Black Hole Mass + Angular momentum Extremal Limit: inner and outer horizons degenerate Kerr-Newman Black Hole Mass + Charge + Angular momentum

7 Black Hole Black Hole Thermodynamics: Area increasing law (2 nd law): S. Hawking Energy conservation (1 st law): Beckenstein, Smarr, 1972 Surface gravity (0 th law): κ is constant on the horizon.

8 Black Hole.

9 Black Hole Hawking radiation: After considering the quantum effect, Hawking completely changed his mind. Black holes can actually radiate due to quantum effect. An intuitive picture: near the horizon, particle-anti-particle pairs can be created so that one escapes and the other falls in.

10 Black Hole Hawking Temperature: Black Hole Entropy: Quantum Effects Microscopic origin? Black holes are classical solutions which capture two quantum effects.

11 Holographic Principle Holographic Principle: G. t Hooft 1993, L. Susskind 1994 Gravity in bulk (D dimensions) Field Theory on boundary (D-1 dimensions)

12 Holographic Principle A preliminary hint: symmetry 1. Black Hole: anti de Sitter (AdS) space appears in the near horizon geometry of extremal black holes Symmetry group of AdSn+1: SO(n, 2) 2. Conformal field theory (CFT): Symmetry group of CFTn: SO(n, 2) First Step: Is there a holographic dual CFT for extremal black hole?

13 Holographic Principle AdS/CFT Correspondence: J. Maldacena 1997 IIB superstring theory on AdS 5 S 5 N=4 SYM Theory Real conceptual change in our thinking about gravity. E. Witten, Science 285 (1999) 512

14 Holographic Principle Holographic dual of black holes: (theoretical study) Inspiring more insights to the foundation of quantum gravity Gravity dual to CFT: (application) At the critical points, notable rescaling symmetries emerge, e.g. in condensed matter, superconductor etc. The AdS/CFT correspondence provides a remarkable approach to study strong coupled phenomena. Entropic force: Gravity is an emergence of entropic force.

15 Kerr/CFT Correspondence Guica, Hartman, Song, Strominger: arxiv: Kerr black hole: rotating (stationary) black hole characterized by mass M and angular momentum J. Kerr/CFT schema GRS

16 Kerr/CFT Correspondence Near Horizon geometry of Extremal Kerr (NHEK): NHEK is a warped AdS3 ( Λ = 1 recovers AdS3 ) Isometries: AdS3 Warped AdS3

17 Kerr/CFT Correspondence Asymptotic Symmetry Group (ASG): Boundary Condition NHEK is not asymptotical flat, so there are no priori obvious boundary conditions. Different boundary condition may reveal different physical context. Strong BC rules out all interesting excitations. Weak BC generates ill-defined results. Appropriate BC (admitting Virasoro generators)

18 Kerr/CFT: central charge Diffeomorphism generators: Conserved charges: algebra(asg) Dirac brackets of charges Virasoro algebra: central charge:

19 Kerr/CFT: temperature Temperature: There are no everywhere time-like Killing vector in NHEK, therefore no desired vacuum. Frolov-Thorne vacuum: time-like Killing vector in the region from horizon to the speed of light surface. Boltzmann factor: CFT temperatures: CFT Entropy (Cardy Formula):

20 RN/CFT Correspondence CFT dual of extremal RN black hole: Warped AdS3/CFT2 description The U(1) bundle of warped AdS3 was recovered from the gauge field potential by uplifting the RN black hole into 5D gravity. Hartman, Murata, Nishioka, Strominger: arxiv: Garousi, Ghodsi: arxiv: The temperature of the dual CFT is charge dependent; Electric-magnetic duality (in 4D) is broken in the CFT side.

21 RN/CFT Correspondence Near horizon geometry of extremal RN: CMC, Huang, Zou arxiv: Uplifted to 5-dim Near Extremal CMC, Sun, Zou arxiv: Reduced to 2-dim Central Charge (Right moving sector)

22 RN/CFT Correspondence Near extremal case: Gravity side: Scattering amplitude of a probe scalar field Field Theory side: Two point function of dual operator CMC, Huang, Zou, arxiv: The expression of the central charge is the same as the extremal case.

23 Hidden Conformal Symmetry For a general non-extremal black holes the near horizon geometric AdS structure is broken. However, a probe scalar, with certain limits, is able to exhibit a conformal symmetry. This hidden conformal symmetry is expected inherit from the symmetry of the CFT dual to the considered black hole. Castro, Maloney, Strominger: arxiv:

24 KN/CFTs: previous results Extremal KN: Central charge: asymptotic symmetry Hartman, Murata, Nishioka, Strominger: arxiv: Non-extremal KN: Hidden conformal symmetry Wang, Liu: arxiv: Chen, Long: arxiv: The results are incomplete!

25 KN/CFTs: Hidden Symmetries There are twofold hidden conformal symmetries in the solution space of the radial equation. J-picture Kerr/CFT Q-picture RN/CFT Kerr J Q = 0 Kerr-Newman J, Q J = 0 RN Q

26 KN/CFTs: J picture A neutral probe scalar field doesn t couple with the background gauge field, thus it can only reveal the angular momentum part of the dual CFT (J-picture) related to the Kerr/CFT duality. Supporting evidences: match of BH and CFT entropies agreement of absorption cross section with two point function agreement of real time correlator at Matsubara frequencies three point correlation function

27 KN/CFTs: Q picture Non-rotating scalar field: mode to operator: charge maps to an operator acting on the internal U(1) space (with a normalization factor) Temperatures: CMC, Huang, Sun, Wu, Zou; arxiv: [hep-th]

28 KN/CFTs There are two different individual 2D CFTs holographically dual to KN black holes. This is an expectable result since, from the gravity side, the KN black hole will reduces to the Kerr when Q = 0 while to the RN when J = 0. microscopic hair theorem : Each CFT is associated with a hair of black hole (except the mass).

29 Summary Holographic duality is a real conceptual change in our thinking about gravity. It provides a very powerful technique to analysis strong coupled phenomena at critical point. Holographic principle could be another crazy idea that might actually be true. What is the origin of the holographic principle?