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): {Geometry matter (energy-momentum)}

a) Brief Introduction to Black Hole Thermodynamics 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 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 0 th lawk =const. The 1 st law d M=k dA/8 πG + J d Ω+Φd Q The 2 nd lawd A >0 The 3 rd lawk ->0 Four Laws of Black Hole mechanics: k: surface gravity, J. Bardeen,B. Carter, S. Hawking, CMP,1973

The 0 th law T=const. on the horizon The 1 st law d M= T d S + J d Ω+Φ d Q The 2 nd law d (S BH +S matter )>=0 The 3 rd law T->0 Four Laws of Black Hole Thermodynamics: 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 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 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 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: 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) (3) R. Bousso, PRD 71 (2005)

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/ , PRL 96 (2006) ) 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/ )

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 !