First Law of Thermodynamics
and
Friedmann Equations Rong-Gen Cai （蔡荣根） Institute of Theoretical Physics, CAS (based on hep-th/0501055(JHEP 02 (2005) 050) with S.P. Kim)

Brief Introduction to Four Laws of Black Hole
Thermodynamics
From the First Law of Thermodynamics to
Friedmann Equations of FRW Universe in
Einstein Gravity
Friedmann Equations in Gauss-Bonnet Gravity
To What Extent it holds?
Two Examples: (i) Scalar-Tensor Gravity
(ii) f(R) Gravity Contents :

Slide4:

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

Slide5:

Four Laws of Black Hole mechanics: k: surface gravity,
J. Bardeen,B. Carter, S. Hawking, CMP,1973

Slide6:

Four Laws of Black Hole Thermodynamics: Key Points: T = k/2π S= A/4G
J. Bekenstein, 1973; S. Hawking, 1974, 1975

Slide7:

On the other hand,
for the de Sitter Space (1917): + I I- Gibbons and Hawking (1977): Cosmological event horizons

Slide8:

Schwarzschild-de Sitter Black Holes: Black hole horizon and cosmological horizon: First law:

Slide9:

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. Jacobson, Phys. Rev. Lett. 75 (1995) 1260
Thermodynamics of Spacetime: The Einstein Equation of State

Slide11:

Friedmann-Robertson-Walker Universe: 1) k = -1
open
2) k = 0
flat
3) k =1
closed b) From the First Law to the Friedmann Equations

Slide12:

Friedmann Equations: where:

Slide13:

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

Slide15:

Horizons in FRW Universe: Particle Horizon:
Event Horizon:
Apparent Horizon:

Slide16:

Apply the first law to the apparent horizon: Make two ansatzes: The only problem is to get dE

Slide17:

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)

Slide18:

By using the continuity equation:

Slide19:

What does it tell us: Classical General relativity Thermodynamics of Spacetime
Quantum gravity Theory Statistical Physics of Spacetime ? Jacobson, Phys. Rev. Lett. 75 (1995) 1260
Thermodynamics of Spacetime: The Einstein Equation of State

Black Hole Solution: Black Hole Entropy: (R. Myers,1988, R.G. Cai,1999, 2002, 2004)

Slide22:

Ansatz:

Slide23:

This time:

Slide24:

More General Case: Lovelock Gravity

Slide25:

Black Hole solution:

Slide26:

Black Hole Entropy: (R.G. Cai, Phys. Lett. B 582 (2004) 237)

Slide28:

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

Slide29:

(1) Scalar-Tensor Gravity: Consider the action

Slide30:

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!

Slide31:

However, if we still take the ansatz and regard as the source, that is, We are able to “derive” the Friedmann equations.

Slide32:

(2) f(R) Gravity Consider the following action: Its equations of motion:

Slide33:

The Friedmann equations in this theory where

Slide34:

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.

Slide35:

However, we note that can be rewritten as in which acts as the effective matter in the universe

Slide36:

In this new form, we use the ansatz We are able to reproduce the corresponding Friedmann
equations in the f(R) gravity theory.

Slide37:

Conclusion and Discussion: We can derive the Friedmann equations in Einstein gravity,
Guass-Bonnet gravity, and more general Lovelock gravity
using the first law of thermodynamics to the apparent horizon,
but not other horizons.
(2) But it does not always hold, for example, in scalar-tensor theory
and f(R) theory.
So far one only considers the FRW universe, clearly it seems so
difficult to reproduce corresponding dynamical equations for
non-homogenous and non-isotropic universe.

Slide38:

Thank You !

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