Rong -Gen Cai ( 蔡荣根） Institute of Theoretical Physics Chinese Academy of Sciences

Holography and Black Hole Physics Rong-Gen Cai (蔡荣根） Institute of Theoretical Physics Chinese Academy of Sciences TexPoint fonts used in EMF: AA

Contents: • Black Hole Mechanics and Black Hole Thermodynamics • Bekenstein Bound and D-Bound • 3. Holography in AdS Space and dS Space • 4. Friedmann equations and first law of thermodynamics

1. Black Hole mechanics and Black Hole Thermodynamics Einstein’s Equations (1915): {Geometry matter (energy-momentum)}

Black Holes: Schwarzschild Black Hole: Mass M horizon More general: Kerr-Newmann Black Holes M, J, Q No Hair Theorem

Four Laws pf Black Hole mechanics: K: surface gravity (J.M. Bardeen,B. Carter, S. Hawking, CMP,1973)

Four Laws of Black Hole Thermodynamics: Key Points: T = k/2π S= A/4G (S. Hawking, 1974, J. Bekenstein, 1973)

2. What we learn from black hole thermodynamics: Holography (1) Black hole entropy: Hawking Temperature: Bekenstein-Hawking Entropy:

(2) Bekenstein Bound and Holographic Bound: Energy: E R Bekenstein Bound (Bekenstein 1981): V, A To be consistent with the second Law of thermodynamics Holography Bound: (‘t Hooft,1993, L. Susskind, 1994)

Holographic Bound:

Bekenstein Bound and Geroch Process . E, R Consider a spherically symmetric black hole (R.G. Cai and Y.S. Myung, PLB 559 (2003)60)

Its horizon and Hawking temperature First law of black hole thermodynamics

The red shift factor near the horizon is given by Therefore near the horizon the proper distance R has the relation to the coordinate distance x The absorbed energy is given by and the increased entropy of the black hole

(3) de Sitter Space and D-bound: Definition: (Willem de Sitter,1872-1934)

Topology Another one: A four dimensional de Sitter space is a hyperboloid embedded in a five dimensional Minkowski space!

In the global coordinates:

Cosmological constant In the static coordinates: 1) Cosmological horizon thermodynamics: (G. Gibbons and S. Hawking,1977) 2) Asymptotically de Sitter Space: for example, SdS space

D-Bound (R. Bousso, 2001): • Entropy bound of a system in de Sitter space This is consistent with the Bekenstein Bound！

D-Bound and Bekenstein Bound: Neutral system Charged system in 4 dim. (S. Hod, J. Bekenstein, B. Linet,2000) Consider a charged system in de Sitter space (R.G. Cai, Y.S. Myung and N. Ohta, CQG 18 (2001) 5429)

When M=Q=0, a pure de Sitter space has a cosmological horizon and entropy D-Bound leads to On the other hand, the cosmological horizon obeys

Consider the large cosmological horizon limit: One has, up to the leading order, The D-Bound gives

3. Holography in AdS and dS Spaces AdS Spaces: AdS/CFT Correspondence (J. Maldacena, 1997) A well-known example: IIB superstring on N=4 SYM on the boundary of AdS_5

(E. Witten) (A. Polyakov) (I. Klebanov) (S. Gubser)

AdS/CFT Correspondence:

AdS/CFT Correspondence: where has two interpretations: on the gravity side, these fields correspond to boundary data or boundary values, for the bulk fields which propagate in the AdS space. on the field theory side, these fields correspond to external source currents coupled to various CFT operators.

Holography in dS Space (A. Strominger, 2000) Quantum Gravity in dS Euclidean CFT on the Boundary of dS Space

我们的宇宙有个全息图吗？ 标准大爆炸宇宙模型引力描写：未来暴胀德西特时空准德西特相全息描写： CFT1 QFT CFT2 重整化群的流动

(1) Holography for AdS Black Holes Cardy-Verlinde Formula for higher dimensional CFTs (J. Cardy, 1986, E. Verlinde, 2000) The CFTs reside on

Consider an (n+2)-dimensional Schwarzschild-adS black hole where Some thermodynamic quantities:

The boundary metric: Thus, one has the thermodynamic quantities of CFTs: It is easy to obtain and to verify:

4) Friedmann equations and first law of thermodynamics Schwarzschild-de Sitter Black Holes: Black hole horizon and cosmological horizon: First law:

Friedmann-Robertson-Walker Universe: 1) k = -1 open 2) k = 0 flat 3) k =1 closed

Friedmann Equations: Where:

Our goal : (R.G. Cai and S.P. Kim, hep-th/0501055 (JHEP 02 (2005) 050))

Horizons in FRW Universe: Particle Horizon: Event Horizon: Apparent Horizon:

Apply the first law to the apparent horizon: Make two ansatzes: The only problem is to get dE

Suppose that the perfect fluid is the source, then The energy-supply vector is: The work density is: （S. A. Hayward, 1997,1998) Then, the amount of energy crossing the apparent horizon within the time interval dt

By using the continuity equation:

What does it tell us: Classical General relativity Thermodynamics of Spacetime Quantum gravity Theory Statistical Physics of Spacetime ? • Jacobson, Phys. Rev. Lett. 75 (1995) 1260 • Thermodynamics of Spacetime: The Einstein Equation of State

Thanks !

Rong -Gen Cai ( 蔡荣根） Institute of Theoretical Physics Chinese Academy of Sciences