Thermodynamics of Apparent Horizon & Dynamics of FRW Spacetime

Rong-Gen Cai （蔡荣根）

Institute of Theoretical Physics Chinese Academy of Sciences

18

2R g R GT

Einstein’s Equations (1915):

{Geometry matter (energy-momentum)}

Thermodynamics of black holes :

horizon

Schwarzschild Black Hole: Mass M

More general:

Kerr-Newmann Black Holes

M, J, Q

No Hair Theorem

On the other hand, for the de Sitter Space (1917):

+ I

I-

Gibbons and Hawking (1977):

Cosmological event horizons

Schwarzschild-de Sitter Black Holes:

Black hole horizon and cosmological horizon:

First law:

4AGS

Thermodynamics of black hole ：

First law ： dM =TdSQuestions: why? (T. Jacobson, 1995)

（ S.Hawking, 1974, J.Bekenstein, 1973)

Two ansatz in FRW：

First law：

Dynamics of spacetime：

dE =TdS

(R.G. Cai and S.P.Kim, JHEP (2005))

Friedmann-Robertson-Walker Universe:

22 2 2 2 2 2 2 2

2( )( sin )

1

drds dt a t r d r d

kr

1) k = -1 open

2) k = 0 flat

3) k =1 closed

a) From the First Law to the Friedmann Equations

Friedmann Equations:

Where:

Our goal :

Some related works: (1) A. Frolov and L. Kofman, JCAP 0305 (2003) 009 (2) Ulf H. Daniesson, PRD 71 (2005) 023516 (3) R. Bousso, PRD 71 (2005) 064024

22 2 2 2 2 2 2 2

2( )( sin )

1

drds dt a t r d r d

kr

Apparent Horizon in FRW Universe :

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 et al., 1997,1998)

By using the continuity equation:

(Cai and Kim, JHEP 0502 (2005) 050 )

Higher derivative theory: Gauss-Bonnet Gravity

Gauss-Bonnet Term:

Black Hole Solution:

Black Hole Entropy:

(R. Myers,1988, R.G. Cai, 2002, 2004)

Ansatz:

This time:

This also holds for more general Lovelock gravity!

Consider a FRW universe

Apparent horizon

And its surface gravity

which is defined by

b) Friedmann equation and the first law of thermodynmaics

Consider the Einstein field equations with perfect fluid

One has the Friedmann equation and the continuity equation

Multiplying both side hands by a factor

Using the definition

One has

Now consider the energy inside the apparent horizon

(Unified first law of thermodynamics, Hayward, 1998,1999)

The case with a Gauss-Bonnet term?

Black hole has an entropy of form

Consider the Friedmann equation in GB gravity

Once again, multiplying a factor with

Defining

It also holds for Lovelock case !

c) Thermodynamics of apparent horizon in brane world scenario

The unified first law: ( S. Hayward, 1998,1999)

Projecting this along a trapping horizon, one can get the first law of Thermodynamics for a dynamical black hole

(RGC and L.M. Cao, hep-th/0612144)( A. Sheykhi, B. Wang and R.G. Cai, hep-th/0701198) (A. Sheykhi, B. Wang and RGC, hep-th/0701261)

For a non-Einstein theory, one can do as follows.

Then one has

Using the relation one could obtain the expression

of horizon entropy. (RGC and L.M. Cao, gr-qc/0611071.)

Two motivations to study the thermodynamics of AH in brane world scenario:(1) dE = T dS + W dV ?

(2) S = ?

(T. Shiromizu, K.I. Maeda and M. Sasaki, PRD, 2000)

The effective equations on an (n-1)-brane:

In the RSII model

Consider a FRW universe on the brane and suppose the matter on the brane is a perfect fluid with

then

The Friedmann equations and continuity equation:

where

One has

(RGC and L.M. Cao, hep-th/0612144)

Some remarks:

1) In the limit,

2) In the limit,

3) The first law of thermodynamics for the apparent horizon

4) When the bulk Weyl tensor does not vanish?

Bulk geometry and area formula of horizon entropy

We obtain

The apparent horizon for a fixed z,

The function has a simple zero root at z_{max},

The horizon area

And the entropy

d) Corrected entropy-area relation and modified Friedmann

equation

RGC, L.M. Cao and Y.P. Hu JHEP 0808, 090 (2008)

Corrected entropy-area relation:

Loop quantum cosmology:

Friedmann equations

Entropy formula

From corrected entropy-area relation to modified Friedmann equation

Friedmann equations

For a FRW universe with a perfect fluid:

The amount of energy crossing the apparent horizon within dt

where A is the area of the apparent horizon.

Assume the temperature

and the Clausius relation

Loop quantum cosmology

Bouncing universe?

More general case:

further

From modified Friedmann equation to corrected entropy-area relation

Entropy formula

The unified first law

The first law of apparent horizon (R.G. Cai and L.M. Cao, hep-th/0612144)

Rewriting the modified Friedmann equation

It is easy to show

Compare with

e) Hawking radiation of apparent horizon in FRW universe

22 2 2 2 2 2 2 2

2( )( sin )

1

drds dt a t r d r d

kr

We know Hawking radiation is always associated with event horizon of spacetime: (1) Black hole, (2) de Sitter space, (3) Rindler horizon

Question: how about apparent horizon in FRW?

22 2 2 2 2 2 2 2

2( )( sin )

1

drds dt a t r d r d

kr

when k=0, it is quite similar to the Painleve-de Sitter metric (M. Parikh, PLB 546, 189 (2002)

There is a Kodama vector:

Now let us consider a particle with mass m in FRW universe. The Hamilton-Jacobi equation:

By use of the Kodama vector, one could define

Then the action:

Consider the incoming mode, the action has a pole at the apparent horizon

(Parikh and Wilczek,2000)

In WKB approximation, the emission rate Gamma is the square of thetunneling amplitude:

The emission rate can be cast in a form of thermal spectrum

The end

(R.G. Cai et al. arXiv:0809.1554 [hep-th] )

f) Conclusions

1) From dQ=TdS to Friedmann equations, here S=A/4G and

2) The Friedmann equation can be recast to a universal form

3) There is a Hawking radiation for the apparent horizon in FRW universe

4) In Einstein gravity and Lovelock gravity, the expression of S has a same form as the black hole entropy

5) In brane world scenario, that form still holds, and we can obtain an expression of horizon entropy associated with apparent horizon, and expect it also holds for brane world black hole horizon.

1) RGC and S.P. Kim, JHEP 0502, 050 (2005) 2) M. Akbar and RGC, PLB 635 ， 7 (2006); PRD 75, 084003 (2007) ； PLB 648, 243 (2007) 3) RGC and L. M. Cao, PRD 75, 064008 (2007) ； NPB 785, 135 (2007) 4) A. Sheykhi, B. Wang and RGC, NPB 779, 1 (2007), PRD 76, 023515 (2007) 5) R.G. Cai, L. M. Cao and Y.P. Hu, JHEP0808, 090 (2008) 6) R.G. Cai, Prog.Theor.Phys.Suppl.172:100-109,2008. 7) R.G. Cai, L. M. Cao and Y.P. Hu, arXiv: 0809.1554 Hawking Radiation of Apparent Horizon in a FRW Universe

My papers on this subject:

Thank You