Rong Gen Cai

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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)

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Einstein’s Equations (1915): {Geometry matter (energy-momentum)}

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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 :

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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

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

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

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

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

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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

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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

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Friedmann Equations: where:

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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

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

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

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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)

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By using the continuity equation:

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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

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c). Higher derivative theory: Gauss-Bonnet Gravity Gauss-Bonnet Term:

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

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Ansatz:

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This time:

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More General Case: Lovelock Gravity

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Black Hole solution:

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

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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

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(1) Scalar-Tensor Gravity: Consider the action

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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!

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

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

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The Friedmann equations in this theory where

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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.

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

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

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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.

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Thank You !