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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)}. Contents :.
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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)}
Contents : • Brief Introduction to Black Hole Thermodynamics • From the First Law of Thermodynamics to • Einstein equations • From the First Law of Thermodynamics to • Friedmann equation of FRW universe • To What Extent it holds? • Two Examples: (i) Scalar-Tensor Gravity • (ii) f(R) Gravity • e)Non-equilibrium Thermodynamics of spacetime • 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: k: surface gravity, J. Bardeen,B. Carter, S. Hawking, CMP,1973
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 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. • 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:
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: 1) k = -1 open 2) k = 0 flat 3) k =1 closed
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
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)
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:
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 • 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
