Einstein Field Equations

and

First Law of Thermodynamics

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)}

a) Brief Introduction to Black Hole Thermodynamicsb) From the First Law of Thermodynamics to Einstein equationsc) From the First Law of Thermodynamics to Friedmann equation of FRW universed) To What Extent it holds? Two Examples: (i) Scalar-Tensor Gravity (ii) f(R) Gravitye) Non-equilibrium Thermodynamics of spacetimef) Revisiting the relation between the first law and Friedmann equation

Contents :

a) Brief Introduction to Black Hole Thermodynamics

horizon

Schwarzschild Black Hole: Mass M

More general:

Kerr-Newmann Black Holes

M, J, Q

No Hair Theorem

The 0th law k =const.

The 1st law d M=k dA/8πG + J d Ω+Φd Q

The 2nd law d A >0

The 3rd law k ->0

Four Laws of Black Hole mechanics:

k: surface gravity, J. Bardeen,B. Carter, S. Hawking, CMP,1973

The 0th law T=const. on the horizon

The 1st law d M= T d S + J d Ω+Φ d Q

The 2nd law d (SBH +Smatter)>=0

The 3rd law T->0

Four Laws of Black Hole Thermodynamics:

Key Points: T = k/2π S= A/4GJ. Bekenstein, 1973; S. Hawking, 1974, 1975

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:

Why does GR know that a black hole has a temperature proportional to its surface gravity and an entropy proportional to its horizon area?

T. Jacobson is the first to ask this question.

T. Jacobson, Phys. Rev. Lett. 75 (1995) 1260

Thermodynamics of Spacetime: The Einstein Equation of State

b) From the first law of thermodynamics to Einstein equations

The causal horizons should be associated withentropy is suggested by the observation thatthey hide information!

The causal horizons can be simply a boundaryof the past of any set of observers.

The heat flow crossing the horizon:

The temperature of the local Rindler horiozn

Now we assume that the entropy is proportional to the horizon area, so that the entropy variation associated with a piece of the horizon

the variation of area of a cross section of a pencil of generators of the past horizon.

Using the Raychaudhuri equation:

(entanglement entropy?)

Using:

With help of the conservation of energy and momentum

and the Einstein Field equations:

What does it tell us:

Classical General relativity Thermodynamics of Spacetime

Quantum gravity Theory Statistical Physics of Spacetime

?

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

c) 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

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:

(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,1999, 2002, 2004)

Ansatz:

This time:

More General Case: Lovelock Gravity

Black Hole solution:

Black Hole Entropy:

(R.G. Cai, Phys. Lett. B 582 (2004) 237)

d) To what extent it holds? Having given a black hole entropy relation to horizon area in some gravity theory, and using the first law of thermodynamics, can one reproduce the corresponding Friedmann equations?

Two Examples: (1) Scalar-Tensor Gravity (2) f(R) Gravity

(Akbar and Cai, PLB 635 (2006) 7 )

(1) Scalar-Tensor Gravity:

Consider the action

The corresponding Freidmann Equations:

On the other hand, the black hole entropy in this theory

It does work if one takes this entropy formula and temperature!

However, if we still take the ansatz

and regard as the source, that is,

We are able to “derive” the Friedmann equations.

(2) f(R) Gravity

Consider the following action:

Its equations of motion:

The Friedmann equations in this theory

where

In this theory, the black hole entropy has the form

If one uses this form of entropy and the first law of thermodynamics, we fail to produce the corresponding Friedmann equation.

However, we note that

can be rewritten as

in which acts as the effective matter in the universe

In this new form, we use the ansatz

We are able to reproduce the corresponding Friedmann equations in the f(R) gravity theory.

e) Non-equilibrium Thermodynamics of Spacetime

(C. Eling, R. Guedens and T. Jacbson, gr-qc/0602001, PRL 96 (2006) 121301)

How to get the field equations for L(R) gravity by using the first law?

Now consider the case with the entropy density being a constant times a function:

Note that in Einstein gravity, it is a constant as considered previously. In that case,

Expand at the point p,

Using the Raychaudhuri equation and the geodesic equation,

RHS=

It is easy to show

Using the conservation of energy and momentum,

This reveals a contradiction, since the RHS is generally not a gradient of a scalar.

The correct way is to consider an entropy production term

If one takes

Then we arrive at

f) Revisiting the relation between the first law and Friedmann equation

dE=TdS -PdV

1) The first law of thermodynamics

2) The Friedmann equation can be obtained from

dE= TdS

(Akbar and Cai, hep-th/0609128)

Consider a FRW universe

Apparent horizon

And its surface gravity

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 entropy inside the apparent horizon

(Unified first law of thermodynamics, Hayward, 1988,1989)

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 !

What is the relation the case for dE=TdS ?

where the apparent horizon radius is assumed to be fixed, the temperature is therefore

On the other hand,

therefore there is the volume term!

Question?

Can we write the Friedmann equation into the form

for the f(R) gravity and scalar-tensor gravity?

Answer: the entropy production term seems needed!

(Akbar, Cai and Cao, in preparation)

Thank You !