Einstein Field Equations
and
First Law of Thermodynamics
Rong-Gen Cai （蔡荣根）
Institute of Theoretical Physics
Chinese Academy of Sciences
Einstein’s Equations (1915):
R
1
 g  R  8 GT
2
{Geometry
matter (energy-momentum)}
Contents :
a) Brief Introduction to Black Hole Thermodynamic
b) From the First Law of Thermodynamics to
Einstein equations
c) From the First Law of Thermodynamics to
Friedmann equation of FRW universe
d) To What Extent it holds?
Two Examples: (i) Scalar-Tensor Gravity
(ii) f(R) Gravity
e) Non-equilibrium Thermodynamics of spacetime
f) Revisiting the relation between the first law
and Friedmann equation
a) Brief Introduction to Black
Hole Thermodynamics
Schwarzschild Black Hole: Mass M
horizon
More general:
Kerr-Newmann Black Holes
M, J, Q
No Hair Theorem
Four Laws of Black Hole mechanics:
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
k: surface gravity,
J. Bardeen,B. Carter, S. Hawking, CMP,1973
Four Laws of Black Hole Thermodynamics:
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
Key Points: T = k/2π
S= A/4G
J. Bekenstein, 1973; S. Hawking, 1974, 1975
On the other hand,
for the de Sitter Space (1917):
+I
Gibbons and Hawking (1977):
Cosmological event horizons
I-
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 with
entropy is suggested by the observation that
they hide information!
The causal horizons can be simply a boundary
of 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
(entanglement entropy?)
the variation of area of a cross section of a pencil of generators of the past horizon.
Using the Raychaudhuri equation:
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
c) From the First Law to the Friedmann Equations
Friedmann-Robertson-Walker Universe:
2
dr
2
2
2
2
2
ds  dt  a (t )(

r
d


r
sin

d

)
2
1  kr
2
2
1) k = -1
open
2) k = 0
flat
3) k =1
closed
2
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
2
dr
2
2
2
2
2
ds  dt  a (t )(
 r d  r sin  d )
2
1  kr
2
2
2
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:
（S. A. Hayward, 1997,1998)
Then, the amount of energy crossing the apparent horizon within
the time interval dt
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
1) The first law of thermodynamics
dE=TdS -PdV
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 !