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Thermodynamics of Apparent Horizon & Dynamics of FRW Spacetime. Rong-Gen Cai ( 蔡荣根 ). Institute of Theoretical Physics Chinese Academy of Sciences. Einstein’s Equations (1915):. { Geometry matter (energy-momentum)}. Thermodynamics of black holes :.

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Thermodynamics of Apparent Horizon & Dynamics of FRW Spacetime

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Thermodynamics of apparent horizon dynamics of frw spacetime

Thermodynamics of Apparent Horizon &

Dynamics of FRW Spacetime

Rong-Gen Cai (蔡荣根)

Institute of Theoretical Physics

Chinese Academy of Sciences


Thermodynamics of apparent horizon dynamics of frw spacetime

Einstein’s Equations (1915):

{Geometry matter (energy-momentum)}


Thermodynamics of apparent horizon dynamics of frw spacetime

Thermodynamics of black holes :

Schwarzschild Black Hole: Mass M

horizon

More general:

Kerr-Newmann Black Holes

M, J, Q

No Hair Theorem


Thermodynamics of apparent horizon dynamics of frw spacetime

On the other hand,

for the de Sitter Space (1917):

+ I

Gibbons and Hawking (1977):

Cosmological event horizons

I-


Thermodynamics of apparent horizon dynamics of frw spacetime

Schwarzschild-de Sitter Black Holes:

Black hole horizon and cosmological horizon:

First law:


Thermodynamics of apparent horizon dynamics of frw spacetime

Thermodynamics of black hole:

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

First law:dM =TdS

Questions: why? (T. Jacobson, 1995)


Thermodynamics of apparent horizon dynamics of frw spacetime

dE =TdS

First law:

Dynamics of spacetime:

Two ansatz in FRW:

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


Thermodynamics of apparent horizon dynamics of frw spacetime

a) From the First Law to the Friedmann Equations

Friedmann-Robertson-Walker Universe:

1) k = -1

open

2) k = 0

flat

3) k =1

closed


Thermodynamics of apparent horizon dynamics of frw spacetime

Friedmann Equations:

Where:


Thermodynamics of apparent horizon dynamics of frw spacetime

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


Thermodynamics of apparent horizon dynamics of frw spacetime

Apparent Horizon in FRW Universe:


Thermodynamics of apparent horizon dynamics of frw spacetime

Apply the first law to the apparent horizon:

Make two ansatzes:

The only problem is to get dE


Thermodynamics of apparent horizon dynamics of frw spacetime

Suppose that the perfect fluid is the source, then

The energy-supply vector is: The work density is:

(S. A. Hayward et al., 1997,1998)

Then, the amount of energy crossing the apparent horizon within

the time interval dt


Thermodynamics of apparent horizon dynamics of frw spacetime

By using the continuity equation:

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


Thermodynamics of apparent horizon dynamics of frw spacetime

Higher derivative theory:

Gauss-Bonnet Gravity

Gauss-Bonnet Term:


Thermodynamics of apparent horizon dynamics of frw spacetime

Black Hole Solution:

Black Hole Entropy:

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


Thermodynamics of apparent horizon dynamics of frw spacetime

Ansatz:


Thermodynamics of apparent horizon dynamics of frw spacetime

This time:

This also holds for more general Lovelock gravity!


Thermodynamics of apparent horizon dynamics of frw spacetime

b) Friedmann equation and the first law of thermodynmaics

Consider a FRW universe

Apparent horizon

And its surface gravity

which is defined by


Thermodynamics of apparent horizon dynamics of frw spacetime

Consider the Einstein field equations with perfect fluid

One has the Friedmann equation and the continuity equation

Multiplying both side hands by a factor


Thermodynamics of apparent horizon dynamics of frw spacetime

Using the definition

One has

Now consider the energy inside the apparent horizon

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


Thermodynamics of apparent horizon dynamics of frw spacetime

The case with a Gauss-Bonnet term?

Black hole has an entropy of form

Consider the Friedmann equation in GB gravity


Thermodynamics of apparent horizon dynamics of frw spacetime

Once again, multiplying a factor with

Defining

It also holds for Lovelock case !


Thermodynamics of apparent horizon dynamics of frw spacetime

c) Thermodynamics of apparent horizon

in brane world scenario

(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)

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


Thermodynamics of apparent horizon dynamics of frw spacetime

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.)


Thermodynamics of apparent horizon dynamics of frw spacetime

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)


Thermodynamics of apparent horizon dynamics of frw spacetime

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

In the RSII model


Thermodynamics of apparent horizon dynamics of frw spacetime

Consider a FRW universe on the brane and suppose the matter on the

brane is a perfect fluid with

then


Thermodynamics of apparent horizon dynamics of frw spacetime

The Friedmann equations and continuity equation:


Thermodynamics of apparent horizon dynamics of frw spacetime

where

One has


Thermodynamics of apparent horizon dynamics of frw spacetime

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


Thermodynamics of apparent horizon dynamics of frw spacetime

Some remarks:

1) In the limit,

2) In the limit,


Thermodynamics of apparent horizon dynamics of frw spacetime

3)The first law of thermodynamics for the apparent horizon

4) When the bulk Weyl tensor does not vanish?


Thermodynamics of apparent horizon dynamics of frw spacetime

Bulk geometry and area formula of horizon entropy

We obtain


Thermodynamics of apparent horizon dynamics of frw spacetime

The apparent horizon for a fixed z,


Thermodynamics of apparent horizon dynamics of frw spacetime

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


Thermodynamics of apparent horizon dynamics of frw spacetime

The horizon area

And the entropy


Thermodynamics of apparent horizon dynamics of frw spacetime

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:

Friedmann equations

Loop quantum cosmology:

Entropy formula


Thermodynamics of apparent horizon dynamics of frw spacetime

From corrected entropy-area relation to modified Friedmann equation

Friedmann equations

For a FRW universe with a perfect fluid:


Thermodynamics of apparent horizon dynamics of frw spacetime

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


Thermodynamics of apparent horizon dynamics of frw spacetime

Bouncing universe?

Loop quantum cosmology


Thermodynamics of apparent horizon dynamics of frw spacetime

More general case:

further


Thermodynamics of apparent horizon dynamics of frw spacetime

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)


Thermodynamics of apparent horizon dynamics of frw spacetime

Rewriting the modified Friedmann equation


Thermodynamics of apparent horizon dynamics of frw spacetime

It is easy to show

Compare with


Thermodynamics of apparent horizon dynamics of frw spacetime

e) Hawking radiation of apparent horizon in FRW universe

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?


Thermodynamics of apparent horizon dynamics of frw spacetime

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

PLB 546, 189 (2002)

There is a Kodama vector:


Thermodynamics of apparent horizon dynamics of frw spacetime

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:


Thermodynamics of apparent horizon dynamics of frw spacetime

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

(Parikh and Wilczek,2000)


Thermodynamics of apparent horizon dynamics of frw spacetime

In WKB approximation, the emission rate Gamma is the square of the

tunneling 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] )


Thermodynamics of apparent horizon dynamics of frw spacetime

f) Conclusions

  • 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.


Thermodynamics of apparent horizon dynamics of frw spacetime

My papers on this subject:

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


Thermodynamics of apparent horizon dynamics of frw spacetime

Thank You


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