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

Contents: 1.Black Hole Mechanics and Black Hole Thermodynamics 2.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: horizon Schwarzschild Black Hole: Mass M More general: Kerr-Newmann Black Holes M, J, Q No Hair Theorem

The 0 th lawk =const. The 1 st law d M=k dA/8 πG + Ω dJ +Φd Q The 2 nd lawd A > 0 The 3 rd lawK -> 0 Four Laws pf Black Hole mechanics: K: surface gravity (J.M. Bardeen,B. Carter, S. Hawking, CMP,1973)

The 0 th law T=Const. on the horizon The 1 st law d M= T d S + J d Ω+Φ d Q The 2 nd law d (S BH +S matter )>=0 The 3 rd law T->0 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 Hawking Temperature: Bekenstein-Hawking Entropy: (1) Black hole entropy:

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

Holographic Bound:

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

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: (Willem de Sitter, ) Definition:

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

In the global coordinates:

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

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

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:

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

准德西特相 德西特时空 标准大爆炸宇宙模型 引力描写： 全息描写 ： 暴胀 未来 CFT 1 QFT CFT 2 重整化群的流动 我们的宇宙有个全息图吗？

(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/ (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: Then, the amount of energy crossing the apparent horizon within the time interval dt （ S. A. Hayward, 1997,1998)

By using the continuity equation:

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

Thanks !