Thermodynamics of f(r) Gravity with the Disformal Transformation

Transcription

1 Thermodynamics of f(r) Gravity with the Disformal Transformation Jhih-Rong Lu National Tsing Hua University(NTHU) Collaborators: Chao-Qiang Geng(NCTS, NTHU), Wei-Cheng Hsu(NTHU), Ling-Wei Luo(AS)

2 Outline Introduction Thermodynamics in Jordan Frame Non-equilibrium description Equilibrium description Thermodynamics in Einstein Frame Summary

3 Introduction Thermodynamics in Jordan Frame Non-equilibrium description Equilibrium description Thermodynamics in Einstein Frame Summary

4 What is f(r) theory? Einstein equation can be obtained by varying the Einstein-Hilbert action: S EH = 1 2κ d4 x g R + S M g μν, Ψ, δs EH = 0 G μν = R μν 1 2 g μνr = κt μν. General Relativity(GR) becomes f(r) theory by replacing the Lagrangian density of GR, R, with a function of R, f(r): S = 1 2κ d4 x g f R + S M g μν, Ψ, δs = 0 FG μν 1 2 g μν f R FR μ ν F + g μν F = κt μν. (F f, R )

5 Jordan Frame v.s. Einstein Frame f(r) gravity in Jordan frame: S = 1 2κ d4 x g f R Conformal transformation: g μν = Ω 2 (x)g μν, Ω 2 x = F R e 4πG 3 ω f(r) gravity in Einstein frame: S = d 4 x g 1 2κ R 1 2 gμν μ ω ν ω V ω, where V ω = 1 2κ FR f F 2.

6 Disformal Transformation [arxiv: gr-qc/ ] In 1992, Bekenstein proposed a new gravity theory which is a special kind of bimetric theory. One of the metric, g μν, describes the gravitational field of the spacetime, the other, γ μν, describes the particle trajectory in gravitational field. Bekenstein argued that two metrics should be related through disformal transformation in order to satisfy equivalence principle and causality: γ μν = A φ, X g μν + B(φ, X) μ φ ν φ. ( ) matter (physical) metric gravitational metric gravitational scalar field

7 Thermodynamics and Gravity In BH physics, the temperature and entropy are associated with the surface gravity and area of the horizon. [Comm. Math. Phys., Volume 31, Number 2 (1973), ] In 1995, T. Jacobson further showed that Einstein equation can be derived from the thermodynamic behavior of spacetime.[arxiv:gr-qc/ v2] In 2005, Cai and Kim demonstrated that the Friedmann equations can be derived from the first law of thermodynamics on the apparent horizon of the universe.[arxiv:gr-qc/ v2] Connection between thermodynamics and f(r) gravity and other modified gravity has been widely investigated. [arxiv: , ]

8 Introduction Thermodynamics in Jordan Frame Non-equilibrium description Equilibrium description Thermodynamics in Einstein Frame Summary

9 Action: S = 1 2κ g f R d 4 x + σ i S M (i) [γαβ ] γ αβ is related with g αβ by the disformal transformation: γ αβ = A(φ, X)g αβ + B φ, X α φ β φ where X = 1 2 gαβ α φ β φ is the kinetic term of disformal field φ. Under the assumption of Principle of Cosmology : Metric g αβ is given by Robertson-Walker metric: (k=0) ds 2 = dt 2 + a 2 (t)(dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 ) φ depends only on time now, i.e., φ = φ t.

10 Action: S = 1 2κ g f R d 4 x + σ i S M (i) [γαβ ] γ αβ is related with g αβ by the disformal transformation: γ αβ = A(φ, X)g αβ + B φ, X α φ β φ where X = 1 2 gαβ α φ β φ is the kinetic term of disformal field φ. Under the assumption of Principle of Cosmology : Metric g αβ is given by Robertson-Walker metric: (k=0) ds 2 = dt 2 + a 2 (t)(dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 ) φ depends only on time now, i.e., φ = φ t.

11 Equation of Motion where FG μν = κ σ i T μν i + t μν i L i φ + ( αφ)(4 φ A 2X φ B) 2X ( 4A, X +2B+2XB, X ) T μν i t μν i T μν d = 2 g = 2 g = 1 κ δl i δg μν, + κ T μν d, F f R / R L i ( α φ) = 0 Ag μν 2A, X μ φ ν φ+xb, X μ φ ν φ 2X 4A, X +2B+2XB, X ( α φ) 1 2 g μν f R FR + μ ν F g μν F L i ( α φ),

12 ҧ ሷ With the perfect fluid assumption, one is able to express induced matter in terms of ordinary matter: ρ i (in) = λρi, P i in = λw (in) ρ i where λ = (1 a2 )(3aሶ φ ሶ + a 3 φ) 3aሶ φሶ and w (in) P in i = a φሷ in ρ 3aሶ φ+a ሶ 3 ሷ. φ i Assume that ordinary matter only contain non-relativistic matter(m) and radiation(r). ρ i = ρ m + ρ r ρҧ M ρ i (in) = ρm (in) + ρr (in) = λ ρm + ρ r λρҧ M P i = P r (in) (in) (in) P i = Pm + Pr = λw (in) (ρ m +ρ r ) λw (in) ρ M

13 Modified Friedmann Equations (Non-equilibrium case) H 2 = 8πG 3F തρ M + ρ d + λ തρ M 8πG 3F ሶ H = 4πG F [( തρ M+ ρ d +P r + P d ) + λ 1 + w in തρ M where 4πG F ( ρ t + P t ), ρ d = 1 8πG P d = 1 8πG 1 2 FR f 3H ሶ F, F ሷ + 2HFሶ 1 2 FR f ρ t, are the dark energy density and pressure, respectively.

14 1st & 2nd Law of Thermodynamics (Non-equilibrium case) ሶ ρ t + 3H ρ t + P t = 3H2 ሶ F 8πG Temp. on the horizon First law: Td መS + Td i መS = d E + WdV, entropy production term in non-equil. thermodynamics መS: Horizon entropy in f(r) Second law: d መS dt + d i መ S dt + d መS t = 12πF Hሶ 2 dt GRH 3 0. Entropy change rate of ordinary matter and induced matter within the horizon W: Work density in non-equil. picture

15 Introduction Thermodynamics in Jordan Frame Non-equilibrium description Equilibrium description Thermodynamics in Einstein Frame Summary

16 Modified Friedmann Equation (Equilibrium case) where H 2 = 8πG ρҧ 3 M + ρ d + λρҧ M 8πG ρ 3 t H ሶ = 4πG[( ρҧ M +ρ d +P r +P d ) + λ 1 + w in 4πG(ρ t + P t ) P d = 1 8πG ρ d = 1 8πG 1 2 FR f 3H ሶ F + 3H 2 (1 F), F ሷ + 2HFሶ 1 2 FR f (1 F)(2 H ሶ + 3H 2 ) ρҧ M ] are the dark energy density and pressure in equilibrium picture, respectively.

17 1st & 2nd Law of Thermodynamics (Equilibrium case) ρ t + 3H ρ t + P t = 0 Horizon entropy in equil. description First law: TdS = de + WdV W: Work density in equil. picture Temp. on the horizon E = ρ t V(total energy in equil. picture) Entropy change rate of ordinary matter and induced matter within the horizon Second law : d S + S dt t = 12π Hሶ 2 0. GRH 3

18 Thermodynamics in Jordan Frame Non-equilibrium case First law: Second law: d መS dt + d i መ S dt + d መS t Equilibrium case First law: Second law: Td መS + Td i መS = d E + WdV = 12πF Hሶ 2 dt GRH 3 TdS = de + WdV d S + S dt t = 12π Hሶ 2 0. GRH 3 0 provided F>0.

19 Introduction Thermodynamics in Jordan Frame Non-equilibrium description Equilibrium description Thermodynamics in Einstein Frame Summary

20 Thermodynamics in Einstein Frame g αβ (x) = Ω 2 x g αβ x The action becomes S = න d 4 x g 1 R 2κ 1 2 gμν μ ω ν ω V ω The RW metric becomes The EoMs dsǁ 2 = dtǁ 2 + a 2 (i) + S M [ω, gμν, Ψ M ] ǁ t dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 G μν = κ T μν i d ǁ t = Ω dt a(t) = Ω a(t) i + t μν ǁ + κ T μν, i 1 g μ L i φ + തV α g g μν ν ω V ω + i L i ( α φ) = 0, ( 1 g δl i δω 1 α gμν ǁ t μν i ) = 0.

21 Modified Friedmann equations H 2 = 8πG ρҧ 3 M + ρ ω + ሚλ ρҧ M 8πG 3 t H = 4πG ρҧ M + ρ ω + P r + P ω + ሚλ 1 + ω in ҧ 4πG( ρ t + P t ) with ρҧ M = ρ m + ρ r, ρ ω = 1 2 ω 2 + V ω, and P ω = 1 2 ω 2 V ω. ρ M It can be shown that the total energy and pressure obeys the continuity equation: ρ t + 3 H ρ t + P t = 0. Thermodynamics in Einstein frame can be considered as an equilibrium description.

22 1 st & 2 nd Law of Thermodynamics (in Einstein frame) ሚS: Horizon entropy in Einstein frame W: work density in Einstein frame First law: Td ሚS = d E + Wd V T: Temperature in Einstein frame E = ρ t V(Total energy within the horizon in Einstein frame) Second law: d d ሚt ሚS + ሚS t = 12π H 2 G R H 3 0 d ሚS t :Entropy change rate of ordinary matter and induced matter within the horizon

23 Summary Consider the physical metric directly coupled to matter. Consider the simple case: γ αβ = η αβ. Interpret the effects of f(r) deviated from GR as dark energy. Verify the first and second laws of thermodynamics in f(r) gravity with disformal transformation in both Jordan and Einstein frames. THANK YOU FOR YOUR ATTENTION!

24 Back-up slides

25 Consider only simple case : γ αβ = η αβ = diag 1, +1, +1, +1. Thus, the disformal transformation becomes η αβ = A(φ, X)g αβ + B φ, X α φ β φ δη αβ = 0 δ β φ = തV β A μν δg μν + തV β δφ where തV β = ( β φ) 4 φ A 2X φ B (2X)( 4A, X +2B+2XB, X ), A μν = Ag μν A, X μ φ ν φ+xb, X μ φ ν φ. 4 φ A 2X φ B

26 1st & 2nd Law of Thermodynamics (Non-equilibrium case) ሶ ρ t + 3H ρ t + P t = 3H2 ሶ F 8πG መS = FA 4G First law: Td መS + Td i መS = de + WdV, W = 1 2 ( ρ t P t ) Second law: d መS dt + d i መ S dt + d መS t = 12πF Hሶ 2 dt GRH 3 0.

27 1st & 2nd Law of Thermodynamics (Equilibrium case) ሶ ρ t + 3H ρ t + P t = 0 S = A 4G W = 1 2 ( ρ t P t ) First law: TdS = de + WdV Second law of thmodynamics: d S + S dt t = 12π Hሶ 2 0. GRH 3

28 1 st & 2 nd Law of Thermodynamics (in Einstein frame) ሚS = ሚ A 4G W = 1 2 ( ρ t P t ) First law: Td ሚS = d E + Wd V T = 1 r 1 ሶ A ǁ 2πr A ǁ 2 H ǁ r A E = ρ t V Second law: d d ሚt ሚS + ሚS t = 12π H 2 G R H 3 0 d ሚS t = 1 T d ρ t V + P t d V,