21 July 2011, Seminar @ USTC-ICTS

Chiang-Mei Chen 陳江梅

Department of Physics, National Central University

OutlineBlack HoleHolographic Principle

Kerr/CFT Correspondence Reissner-Nordstrom /CFT CorrespondenceKerr‐Newman/CFTs Correspondence

Summary

Black HoleGravitational force: universal and attractiveBlack 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)

Black HoleTwo essential characteristics:

singularity: the curvature diverges a the center of black holehorizon: infinite gravitational red shift; one way path into black hole

Schwarzschild Black Hole

Black HoleNo hair theorem:

Black holes have no hair because they are simple.

A black hole is completely characterized by three physical parameters:

MassAngular MomentumCharge

Black HoleReissner-Nordstrom Black Hole: Mass + Charge

Kerr Black HoleMass + Angular momentum

Kerr-Newman Black HoleMass + Charge + Angular momentum

Extremal Limit:inner and outer horizons degenerate

Black HoleBlack Hole Thermodynamics:

Area increasing law (2nd law): S. Hawking

Energy conservation (1st law): Beckenstein, Smarr, 1972

Surface gravity (0th law): κ is constant on the horizon.

Black Hole.

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

Black HoleHawking Temperature:

Black Hole Entropy:

Black holes are classical solutions which capture two quantum effects.

Quantum Effects

Microscopic origin?

Holographic PrincipleHolographic Principle: G. t’ Hooft 1993, L. Susskind 1994

Gravity in bulk (D dimensions)

Field Theory on boundary(D-1 dimensions)

Holographic PrincipleA 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?

Holographic PrincipleAdS/CFT Correspondence: J. Maldacena 1997IIB superstring theory

on AdS5 × S5

N=4 SYM Theory

“Real conceptual change in our thinking about gravity.”E. Witten, Science 285 (1999) 512

Holographic PrincipleHolographic dual of black holes: (theoretical study)

Inspiring more insights to the foundation of quantum gravityGravity 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.

Kerr/CFT CorrespondenceGuica, Hartman, Song, Strominger: arXiv:0809.4266

Kerr black hole: rotating (stationary) black hole characterized by mass M and angular momentum J.Kerr/CFT schema

GRS 1915+105

Kerr/CFT CorrespondenceNear Horizon geometry of Extremal Kerr (NHEK):

NHEK is a warped AdS3 ( Λ = 1 recovers AdS3 ) Isometries:

AdS3Warped AdS3

Kerr/CFT CorrespondenceAsymptotic Symmetry Group (ASG): Boundary Condition NHEK is not asymptotical flat, so there are no prioriobvious 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)

Kerr/CFT: central chargeDiffeomorphism generators:

Conserved charges:algebra(ASG) Dirac brackets of charges

Virasoro algebra:

central charge:

Kerr/CFT: temperatureTemperature:

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):

RN/CFT CorrespondenceCFT 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:0811.4393Garousi, Ghodsi: arXiv:0902.4387

The temperature of the dual CFT is charge dependent; Electric-magnetic duality (in 4D) is broken in the CFT side.

RN/CFT CorrespondenceNear horizon geometry of extremal RN:  

CMC, Huang, ZouarXiv:1001.2833

CMC, Sun, ZouarXiv:0910.2076

Uplifted to 5-dim

Reduced to 2-dim Central Charge 

(Right moving sector)

Near Extremal

RN/CFT CorrespondenceNear extremal case:

CMC, Huang, Zou, arXiv:1001.2833

Gravity side:Scattering amplitude of a probe scalar field

Field Theory side: Two point function of dual operator

The expression of the central charge is the same as the extremal case.

Hidden Conformal SymmetryFor 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:1004.0996

KN/CFTs: previous resultsExtremal KN:

Central charge: asymptotic symmetryHartman, Murata, Nishioka, Strominger: arXiv:0811.4393

Non-extremal KN:Hidden conformal symmetry

Wang, Liu: arXiv:1004.4661Chen, Long: arXiv:1004.5039

The results are incomplete!

KN/CFTs: Hidden SymmetriesThere are twofold hidden conformal symmetries in the solution space of the radial equation.

J-picture Kerr/CFT Q-picture RN/CFT

Kerr-NewmanJ, Q

KerrJ

RNQJ = 0Q = 0

KN/CFTs: J‐pictureA 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 entropiesagreement of absorption cross section with two point functionagreement of real time correlator at Matsubara frequencies three point correlation function

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)

CMC, Huang, Sun, Wu, Zou; arXiv:1060.4092 [hep-th]

Temperatures:

KN/CFTsThere 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).

SummaryHolographic 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?