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# Thermodynamics of Apparent Horizon & Dynamics of FRW Spacetime - PowerPoint PPT Presentation

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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Dynamics of FRW Spacetime

Rong-Gen Cai （蔡荣根）

Institute of Theoretical Physics

Einstein’s Equations (1915):

{Geometry matter (energy-momentum)}

Schwarzschild Black Hole: Mass M

horizon

More general:

Kerr-Newmann Black Holes

M, J, Q

No Hair Theorem

for the de Sitter Space (1917):

+ I

Gibbons and Hawking (1977):

Cosmological event horizons

I-

Black hole horizon and cosmological horizon:

First law:

（S.Hawking, 1974, J.Bekenstein, 1973)

First law：dM =TdS

Questions: why? (T. Jacobson, 1995)

dE =TdS

First law：

Dynamics of spacetime：

Two ansatz in FRW：

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

Friedmann-Robertson-Walker Universe:

1) k = -1

open

2) k = 0

flat

3) k =1

closed

Where:

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

Make two ansatzes:

The only problem is to get dE

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

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

Gauss-Bonnet Gravity

Gauss-Bonnet Term:

Black Hole Entropy:

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

This also holds for more general Lovelock gravity!

Consider a FRW universe

Apparent horizon

And its surface gravity

which is defined by

One has the Friedmann equation and the continuity equation

Multiplying both side hands by a factor

One has

Now consider the energy inside the apparent horizon

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

Black hole has an entropy of form

Consider the Friedmann equation in GB gravity

Defining

It also holds for Lovelock case !

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

Then one has

Using the relation one could obtain the expression

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

in brane world scenario:

(1) dE = T dS + W dV?

(2) S = ?

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

In the RSII model

brane is a perfect fluid with

then

where on the

One has

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

Some remarks: on the

1) In the limit,

2) In the limit,

3) on the The first law of thermodynamics for the apparent horizon

4) When the bulk Weyl tensor does not vanish?

We obtain

The horizon area on the

And the entropy

d) on the 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

Friedmann equations

For a FRW universe with a perfect fluid:

where A is the area of the apparent horizon.

Assume the temperature

and the Clausius relation

Bouncing universe? equation

Loop quantum cosmology

More general case: equation

further

Entropy formula

The unified first law

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

It is easy to show relation

Compare with

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?

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. metric (M. Parikh,

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)

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] )

f) Conclusions of the

• 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

Thank You of the