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Thermodynamics in f(R,T) theory of gravity
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J
ournal of Cosmology and Astroparticle Physics
An IOP and SISSA journal
M. Sharif and M. Zubair Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan E-mail:
[email protected],
[email protected] Received January 13, 2012 Revised February 27, 2012 Accepted February 28, 2012 Published March 16, 2012
Abstract. A non-equilibrium picture of thermodynamics is discussed at the apparent horizon of FRW universe in f (R, T ) gravity, where R is the Ricci scalar and T is the trace of the energy-momentum tensor. We take two forms of the energy-momentum tensor of dark components and demonstrate that equilibrium description of thermodynamics is not achievable in both cases. We check the validity of the first and second law of thermodynamics in this scenario. It is shown that the Friedmann equations can be expressed in the form of first law of thermodynamics Th dSh′ + Th d S ′ = −dE ′ + W ′ dV , where d S ′ is the entropy production term. Finally, we conclude that the second law of thermodynamics holds both in phantom and non-phantom phases.
Keywords: modified gravity, quantum black holes, dark energy theory
c 2012 IOP Publishing Ltd and Sissa Medialab srl
doi:10.1088/1475-7516/2012/03/028
JCAP03(2012)028
Thermodynamics in f (R, T ) theory of gravity
Contents 1
2 f (R, T ) gravity
2
3 Laws of thermodynamics 3.1 First law of thermodynamics 3.2 Generalized second law of thermodynamics
4 4 6
4 Redefining the dark components 4.1 First law of thermodynamics 4.2 Generalized second law of thermodynamics
7 8 9
5 Concluding remarks
1
10
Introduction
Cosmic observations from anisotropy of the Cosmic Microwave Background (CMB) [1], supernova type Ia (SNeIa) [2], large scale structure [3], baryon acoustic oscillations [4] and weak lensing [5] indicate that expansion of the universe is speeding up rather than decelerating. The present accelerated expansion is driven by gravitationally repulsive dominant energy component known as dark energy (DE). There are two representative directions to address the issue of cosmic acceleration. One is to introduce the ”exotic energy component” in the context of General Relativity (GR). Several candidates have been proposed [6]–[10] in this perspective to explore the nature of DE. The other direction is to modify the Einstein Lagrangian i.e., modified gravity theory such as f (R) gravity [11]. The discovery of black hole thermodynamics set up a significant connection between |κsg | , where κsg is the gravity and thermodynamics [12]. The Hawking temperature T = 2π A surface gravity, and horizon entropy S = 4G obey the first law of thermodynamics [12]–[14]. Jacobson [15] showed that it is indeed possible to derive the Einstein field equations in Rindler spacetime by using the Clausius relation T dS = δQ and proportionality of entropy to the horizon area. Here δQ and T are the energy flux across the horizon and Unruh temperature respectively, viewed by an accelerated observer just inside the horizon. Frolov and Kofman [16] employed this approach to quasi-de Sitter geometry of inflationary universe with the relation −dE = T dS to calculate energy flux of slowly rolling background scalar field. Cai and Kim [17] derived the Freidmann equations of the FRW universe with any spatial curvature from the first law of thermodynamics for the entropy of the apparent horizon. Later, Akbar and Cai [18] showed that the Friedmann equations in GR can be written in the form dE = T dS + W dV at the apparent horizon, where E = ρV is the total energy inside the apparent horizon and W = 12 (ρ − p) is the work density. The connection between gravity and thermodynamics has been revealed in modified theories of gravity including Gauss-Bonnet gravity [18], Lovelock gravity [19, 20], braneworld gravity [21], non-linear gravity [22]–[27] and scalar-tensor gravity [19, 28, 29]. In f (R) gravity and scalar-tensor theory, non-equilibrium description of thermodynamics is required [22]–[29] so that the Clausius relation is modified ¯ Here, dS¯ is the additional entropy production term. in the form T dS = δQ + dS.
–1–
JCAP03(2012)028
1 Introduction
2
f (R, T ) gravity
The action of f (R, T ) theory of gravity is given by [30] Z f (R, T ) 4√ A = dx −g + L(matter) , 16πG
(2.1)
where L(matter) determines matter contents of the universe. The energy-momentum tensor of matter is defined as [33] √ 2 δ( −gL(matter) ) (matter) . (2.2) Tµν = −√ −g δg µν
We assume that the matter Lagrangian density depends only on the metric tensor components gµν so that 2∂L(matter) (matter) Tµν = gµν L(matter) − . (2.3) ∂g µν Variation of the action (2.1) with respect to the metric tensor yields the field equations of f (R, T ) gravity as 1 Rµν fR (R, T ) − gµν f (R, T ) + (gµν − ∇µ ∇ν )fR (R, T ) 2 (matter) (matter) = 8πGTµν − fT (R, T )Tµν − fT (R, T )Θµν , (2.4)
–2–
JCAP03(2012)028
Recently, Harko et al. [30] generalized f (R) gravity by introducing an arbitrary function of the Ricci scalar R and the trace of the energy-momentum tensor T . The dependence of T may be introduced by exotic imperfect fluids or quantum effects (conformal anomaly). As a result of coupling between matter and geometry motion of test particles is nongeodesic and an extra acceleration is always present. In f (R, T ) gravity, cosmic acceleration may result not only due to geometrical contribution to the total cosmic energy density but it also depends on matter contents. This theory can be applied to explore several issues of current interest and may lead to some major differences. Houndjo [31] developed the cosmological reconstruction of f (R, T ) gravity for f (R, T ) = f1 (R) + f2 (T ) and discussed transition of matter dominated phase to an acceleration phase. In a recent paper [26], Bamba and Geng investigated laws of thermodynamics in f (R) gravity. It is argued that equilibrium description exists in f (R) gravity. The equilibrium description of thermodynamics in modified gravitational theories is still under debate as various alternative treatments [32] have been proposed to reinterpret the non-equilibrium picture. These recent studies have motivated us to explore whether the equilibrium description can be obtained in the framework of f (R, T ) gravity. The study of connection between gravity and thermodynamics in f (R, T ) may provide some specific results which would discriminate this theory from various theories of modified gravity. In this paper, we examine whether an equilibrium description of thermodynamics is possible in such a modified theory of gravity. The horizon entropy is constructed from the first law of thermodynamics corresponding to the Friedmann equations. We explore the generalized second law of thermodynamics (GSLT) and find out the necessary condition for its validity. The paper is organized as follows: In the next section, we review f (R, T ) gravity and formulate the field equations of FRW universe. Section 3 investigates the first and second laws of thermodynamics. In section 4, the Friedmann equations are reformulated by redefining the dark components to explore the possible change in thermodynamics. Finally, section 5 is devoted to the concluding remarks.
where ∇µ is the covariant derivative associated with the Levi-Civita connection of the metric and = ∇µ ∇µ . We denote fR (R, T ) = ∂f (R, T )/∂R, fT (R, T ) = ∂f (R, T )/∂T and Θµν = g αβ δTαβ δg µν .
The choice of f (R, T )≡f (R) results in the field equations of f (R) gravity. The energy-momentum tensor of matter is defined as (matter) Tµν = (ρm + pm )uµ uν + pm gµν ,
(2.5)
where uµ is the four velocity of the fluid. If we take L(matter) = −pm , then Θµν becomes (matter) Θµν = −2Tµν − pm gµν .
(2.6)
1 Rµν fR (R, T ) − gµν f (R, T ) + (gµν − ∇µ ∇ν )fR (R, T ) 2 (matter) = 8πGTµν + Tµν fT (R, T ) + pm gµν fT (R, T ).
(2.7)
The field equations in f (R, T ) gravity depend on a source term, representing the variation of the energy-momentum tensor of matter with respect to the metric. We consider only the nonrelativistic matter (cold dark matter and baryons) with pm = 0, therefore the contribution of T comes only from ordinary matters. Thus, eq. (2.7) can be written as an effective Einstein field equation of the form 1 (d) (matter) Rµν − Rgµν = 8πGeff Tµν + T ′ µν , 2 where Geff
1 = fR (R, T )
fT (R, T ) G+ 8π
(2.8)
is the effective gravitational matter dependent coupling in f (R, T ) gravity and 1 1 ′ (d) gµν (f (R, T ) − RfR (R, T )) + (∇µ ∇ν − gµν )fR (R, T ) T µν = fR (R, T ) 2
(2.9)
is the energy-momentum tensor of dark components. Here, prime means non-equilibrium description of the field equations. The FRW universe is described by the metric ds2 = hαβ dxα dxβ + r˜2 dΩ2 ,
(2.10)
where r˜ = a(t)r and x0 = t, x1 = r with the 2-dimensional metric hαβ = diag(−1, a2 /(1 − kr2 )). Here a(t) is the scale factor, k is the cosmic curvature and dΩ2 is the metric of 2dimensional sphere with unit radius. In FRW background, the gravitational field equations are given by k 1 1 2 ˙ 3 H + 2 = 8πGeff ρm + (RfR − f ) − 3H RfRR , (2.11) a fR 2 1 1 k ˙ RR + Rf ¨ RR + R˙ 2 fRRR . − (RfR − f ) + 2H Rf (2.12) − 2H˙ + 3H 2 + 2 = a fR 2
–3–
JCAP03(2012)028
Consequently, the field equations (2.4) lead to
These can be rewritten as k 3 H + 2 = 8πGeff (ρm + ρ′d ), a k ˙ −2 H − 2 = 8πGeff (ρm + ρ′d + p′d ), a
2
(2.14)
(2.15) (2.16)
(R,T ) . The equation of state (EoS) parameter of dark fluid ωd′ is obtained as Here F = 1 + fT8πG (p′d = ωd′ ρ′d ) ¨ RR + R˙ 2 fRRR − H Rf ˙ RR Rf . (2.17) ωd′ = −1 + 1 ˙ 2 (RfR − f ) − 3H RfRR
The semi-conservation equation of ordinary matter is given by ρ˙ + 3Hρ = q.
(2.18)
The energy-momentum tensor of dark components may satisfy the similar conservation laws ρ˙ d + 3H(ρd + pd ) = qd , ρ˙ tot + 3H(ρtot + ptot ) = qtot ,
(2.19) (2.20)
where ρtot = ρm + ρd , ptot = pd and qtot = q + qd is the total energy exchange term and qd is the energy exchange term of dark components. Substituting eqs. (2.13) and (2.14) in the above equation, we obtain fR 3 k 2 qtot = (H + 2 )∂t . (2.21) 8πG a F The relation of energy exchange term in f (R) gravity can be recovered if F = 1. In GR, qtot = 0 for the choice f (R, T ) = R.
3
Laws of thermodynamics
In this section, we examine the validity of the first and second law of thermodynamics in f (R, T ) gravity for FRW universe. 3.1
First law of thermodynamics
Here we investigate the validity of the first law of thermodynamics in f (R, T ) gravity at the apparent horizon of FRW universe. The dynamical apparent horizon is determined by the relation hαβ ∂α r˜∂β r˜ = 0 which leads to the radius of apparent horizon for FRW universe 1 r˜A = q H2 +
–4–
k a2
.
(3.1)
JCAP03(2012)028
where ρ′d and p′d are the energy density and pressure of dark components 1 1 ′ ˙ (RfR − f ) − 3H RfRR , ρd = 8πGF 2 1 1 ′ 2 ˙ ¨ ˙ pd = − (RfR − f ) + 2H RfRR + RfRR + R fRRR . 8πGF 2
(2.13)
The associated temperature of the apparent horizon is defined through the surface gravity κsg as |κsg | , (3.2) Th = 2π where κsg is given by [17]
(3.3)
In GR, the horizon entropy is given by the Bekenstein-Hawking relation Sh = A/4G, 2 is the area of the apparent horizon [12]–[14]. In the context of modified where A = 4π˜ rA gravity theories, Wald [34] proposed that entropy of black hole solutions with bifurcate Killing horizons is a Noether charge entropy. It depends on the variation of Lagrangian density of modified gravitational theories with respect to Riemann tensor. Wald entropy is equal to quarter of horizon area in units of effective gravitational coupling i.e, Sh′ = A/4Geff [35]. In f (R, T ) gravity, the Wald entropy is expressed as Sh′ =
AfR . 4GF
(3.4)
Taking the time derivative of eq. (3.1) and using (2.14), it follows that 3 fR d˜ rA = 4πG˜ rA (ρ′tot + p′tot )HFdt,
(3.5)
rA is the infinitesimal change in radius of the apparent where ρ′tot = ρ′m + ρ′d and p′tot = p′d . d˜ horizon during a time interval dt. Using eqs. (3.4) and (3.5), we obtain 1 1 r˜A fR r˜A ′ 3 ′ ′ dfR + d . (3.6) dSh = 4π˜ rA (ρtot + ptot )Hdt + 2π˜ rA 2GF 2G F If we multiply both sides of this equation with a factor (1 − r˜˙A /2H r˜A ), it follows that 3 2 Th dSh′ = 4π˜ rA (ρ′tot + p′tot )Hdt − 2π˜ rA (ρ′tot + p′tot )d˜ rA + π˜ r 2 Th f R 1 . d + A G F
2 T df π˜ rA h R GF
(3.7)
Now, we define energy of the universe within the apparent horizon. The Misner-Sharp r˜A energy [36] is defined as E = 2G which can be written in f (R, T ) gravity as [24, 28] E′ =
r˜A . 2Geff
(3.8)
3 , we obtain rA In terms of volume V = 43 π˜
E′ =
3V 8πGeff
k H 2 + 2 = V ρ′tot a
–5–
(3.9)
JCAP03(2012)028
√ 1 1 r˜˙A κsg = √ ∂α −hhαβ ∂β r˜A = − 1− r˜A 2H r˜A 2 −h r˜A k =− 2H 2 + H˙ + 2 . 2 a
which represents the total energy inside the sphere of radius r˜A . It is obvious that E ′ > 0, if Geff = GF fR > 0 so that the effective gravitational coupling constant in f (R, T ) gravity should be positive. It follows from eqs. (2.13) and (3.9) that 1 r˜A dfR r˜A fR ′ 3 ′ ′ 2 ′ dE = −4π˜ rA (ρtot + ptot )Hdt + 4π˜ rA ρtot d˜ rA + + d . (3.10) 2GF 2G F Using eq. (3.10) in (3.7), it follows that Th dSh′
′
1 F
,
(3.11)
which includes the work density W ′ = 12 (ρ′tot − p′tot ) [37]. This can be rewritten as Th dSh′ + Th d Sh′ = −dE ′ + W ′ dV, where d Sh′
r˜A (1 + 2π˜ rA Th )d =− 2GTh
fR F
F(E ′ + Sh′ Th ) =− d Th f R
(3.12)
fR F
.
(3.13)
When we compare the cosmological setup of f (R, T ) gravity with GR, Gauss-Bonnet gravity and Lovelock gravity [18]–[20], we obtain an auxiliary term in the first law of thermodynamics. This additional term d Sh′ may be interpreted as entropy production term developed due to the non-equilibrium framework in f (R, T ) gravity. This result corresponds to the first law of thermodynamics in non-equilibrium description of f (R) gravity [26] for f (R, T ) = f (R). If we assume f (R, T ) = R, then the traditional first law of thermodynamics in GR can be achieved. 3.2
Generalized second law of thermodynamics
Recently, the GSLT has been studied in the context of modified gravitational theories [25]–[28]. It may be interesting to investigate its validity in f (R, T ) gravity. For this purpose, we have to show that [28] ′ S˙ h′ + d S˙ h′ + S˙ tot ≥ 0,
(3.14)
′ where Sh′ is the horizon entropy in f (R, T ) gravity, d S˙ h′ = ∂t (d Sh′ ) and Stot is the entropy due to all the matter and energy sources inside the horizon. The Gibb’s equation including all matter and energy fluid is given by [38] ′ Ttot dStot = d(ρ′tot V ) + p′tot dV,
(3.15)
where Ttot is the temperature of total energy inside the horizon. We assume that Ttot is proportional to the temperature of apparent horizon [25, 28], i.e., Ttot = bTh , where 0 < b < 1 to ensure that temperature being positive and smaller than the horizon temperature. Substituting eqs. (3.12) and (3.15) in eq. (3.14), we obtain 24πΞ ′ S˙ h′ + d S˙ h′ + S˙ tot = ≥ 0, r˜A bR where
b (ρ′tot + p′tot )V˙ Ξ = (1 − b)ρ˙ ′tot V + 1 − 2
–6–
(3.16)
JCAP03(2012)028
rA Th )˜ rA fR (1 + 2π˜ rA Th )˜ rA dfR (1 + 2π˜ + d = −dE + W dV + 2GF 2G ′
is the universal condition to protect the GSLT in modified gravitational theories [28]. Using eqs. (2.13) and (2.14), condition (3.14) is reduced to 12πX bRGF H 2 +
k 2 a2
≥ 0,
(3.17)
where
Thus the condition to satisfy the GSLT is equivalent to X ≥ 0. In flat FRW universe, the GSLT is valid with the constraints ∂t ( fFR ) ≥ 0, H > 0 and H˙ ≥ 0. Also, F and fR are positive in order to keep E > 0. If b = 1, i.e., temperature between outside and inside the horizon remains the same then the GSLT is valid only if k 2 fR ˙ ≥ 0. J = H− 2 a F
(3.18)
For eq. (2.10), the effective EoS is defined as ωeff = −1 − 2 H˙ − ak2 /3 H 2 + ak2 . Here H˙ < ak2 corresponds to quintessence region while H˙ > ak2 represents the phantom phase of the universe. It follows that GSLT in f (R, T ) gravity is satisfied in both phantom and non-phantom phases. This result is compatible with [39] according to which entropy may be positive even at the phantom era. Bamba and Geng [26, 27] also shown that second law of thermodynamics can be satisfied in f (R) and f (T ) theories of gravity.
4
Redefining the dark components
In previous section, we have seen that an additional entropy term d Sh′ is produced in laws of thermodynamics. This can be considered as the result of non-equilibrium description of the field equations. If we redefine the dark components so that the extra entropy production term is vanished, then such formulation is referred as an equilibrium description. It has been seen so far that the equilibrium description does exist in modified theories of gravity [26, 27, 29] and extra entropy production term can be removed. Here, we discuss whether the equilibrium description of f (R, T ) gravity can be anticipated. In fact, we may reduce the entropy production term through this description but it cannot be wiped out entirely. We redefine the energy density and pressure of dark components. The (00) and (11) components of the field equations can be rewritten as
k 3 H + 2 = 8πGeff (ρm + ρd ), a k −2 H˙ − 2 = 8πGeff (ρm + ρd + pd ), a 2
–7–
(4.1) (4.2)
JCAP03(2012)028
2 k k k 2 X = 2(1 − b)H H˙ − 2 H + 2 fR + (2 − b)H H˙ − 2 fR a a a 2 fR k . +(1 − b) H 2 + 2 F∂t a F
) is the effective gravitational coupling, ρd and pd are the energy where Geff = G + fT (R,T 8π density and pressure of dark components given by 1 k 1 2 ˙ , (4.3) ρd = (RfR − f ) − 3H RfRR + 3(1 − fR ) H + 2 8πGF 2 a 1 k 1 2 2 ˙ ¨ ˙ ˙ pd = − (RfR −f )+ 2H RfRR + RfRR + R fRRR −(1−fR ) 2H + 3H + 2 . (4.4) 8πGF 2 a The EoS parameter ωd in this description turns out to be
2
(4.5)
a
In this case the expression of total energy exchange is given by k 1 3 (H 2 + 2 )∂t . qtot = 8πG a F
(4.6)
Since ∂t (fT (R, T )) 6= 0 in f (R, T ) gravity, so that qtot does not vanish. So, we may not establish the equilibrium picture of thermodynamics in this modified gravity. Hence, again we need to consider the non-equilibrium treatment of thermodynamics. This result differs from other modified gravitational theories due to the matter dependence of the Lagrangian density. In f (R) gravity the redefinition of dark components result in local conservation of energy momentum tensor of dark components [26]. It is clear from eqs. (2.17) and (4.5) that the EoS parameter of dark components is not unique in both cases. Thus, one should consider both formulations of the field equations in cosmic discussions. Now we check the validity of the first and second laws of thermodynamics in this scenario. 4.1
First law of thermodynamics
In this representation of the field equations, the time derivative of radius r˜A at the apparent horizon is given by 3 d˜ rA = 4π˜ rA GF(ρtot + ptot )Hdt. (4.7) Since in f (R, T ) gravity, the equilibrium description is not feasible as it can be seen in modified gravitational theories such that f (R), f (T ) and scalar tensor gravity etc. Thus, we use the Wald entropy relation Sh = A/(4Geff ) rather than introducing Bekenstein-Hawking entropy. Using eq. (4.7), the horizon entropy becomes 1 r˜A 1 3 d . (4.8) dSh = 4π˜ rA (ρtot + ptot )Hdt + 2π˜ rA 2G F The associated temperature of the apparent horizon is 1 r˜˙A Th = 1− . 2π˜ rA 2H r˜A
(4.9)
Equations (4.8) and (4.9) imply that 3 2 Th dSh = 4π˜ rA (ρtot + ptot )Hdt − 2π˜ rA (ρtot + ptot )d˜ rA +
–8–
2T π˜ rA h d G
1 F
.
(4.10)
JCAP03(2012)028
¨ RR + R˙ 2 fRRR − H Rf ˙ RR − 2(1 − fR ) H˙ − k2 Rf a ωd = −1 + 1 , ˙ RR + 3(1 − fR ) H 2 + k2 (RfR − f ) − 3H Rf
Introducing the Misner-Sharp energy E=
r˜A = V ρtot , 2GF
we obtain dE =
3 −4π˜ rA (ρtot
+ ptot )Hdt +
2 4π˜ rA ρtot d˜ rA
(4.11) r˜A d + 2G
The total work density is defined as [37]
1 F
.
(4.13)
By combining eqs. (4.10), (4.12) and (4.13), we obtain the following expression of first law of thermodynamics Th dSh + Th d Sh = −dE + W dV, (4.14) where r˜A 1 E 1 d Sh = − = −F (1 + 2π˜ rA Th )d + Sh d 2Th G F Th F 2 2 ˙ π(4H + H + 3k/a ) 1 =− d 2 2 2 2 ˙ F G(H + k/a )(2H + H + k/a )
(4.15)
is the additional term of entropy produced due to the matter contents of the universe. It involves derivative of f (R, T ) with respect to the trace of the energy-momentum tensor. Notice that the first law of thermodynamics Th dSh = −dE + W dV holds at the apparent horizon of FRW universe in equilibrium description of modified theories of gravity [26, 27, 29]. However, in f (R, T ) gravity, this law does not hold due to the presence of an additional term d Sh . This term vanishes if we take f (R, T ) = f (R) which leads to the equilibrium description of thermodynamics in f (R) gravity. 4.2
Generalized second law of thermodynamics
To establish the GSLT in this formulation of f (R, T ) gravity, we consider the Gibbs equation in terms of all matter field and energy contents Ttot dStot = d(ρtot V ) + ptot dV,
(4.16)
where Ttot denotes the temperature of total energy inside the horizon and Stot is the entropy of all the matter and energy sources inside the horizon. In this case, the GSLT can be expressed as S˙ h + d S˙ h + S˙ tot ≥ 0 (4.17) which implies that
12πY bRGF H 2 +
where
k 2 a2
≥ 0,
k k k 2 Y = 2(1 − b)H H˙ − 2 H 2 + 2 + (2 − b)H H˙ − 2 a a a 2 k 1 +(1 − b) H 2 + 2 F∂t . a F
–9–
(4.18)
JCAP03(2012)028
1 1 W = − T (tot)αβ hαβ = (ρtot − ptot ), 2 2
(4.12)
Thus the GSLT is satisfied only if Y ≥ 0. In case of flat FRW universe, the GSLT is met with the conditions ∂t ( F1 ) ≥ 0, H > 0 and H˙ ≥ 0. In thermal equilibrium b = 1, the above condition is reduced to the following form 2 12πH H˙ − ak2 1 ≥ 0, (4.19) B= 2 G H 2 + k2 R F a
5
Concluding remarks
The fact, f (R, T ) gravity is the generalization of f (R) gravity is based on coupling between matter and geometry [30]. This theory can be applied to explore several issues of current interest in cosmology and astrophysics. We have discussed the laws of thermodynamic at the apparent horizon of FRW spacetime in this modified gravity. Akbar and Cai [23] have shown that the Friedmann equations for f (R) gravity can be written into a form of the first law of ¯ where dS¯ is the additional entropy term due to thermodynamics, dE = T dS + W dV + T dS, non-equilibrium thermodynamics. Bamba and Geng [26, 27] established the first and second laws of thermodynamics at the apparent horizon of FRW universe with both non-equilibrium and equilibrium descriptions. We have found that the picture of equilibrium thermodynamics is not feasible in f (R, T ) gravity even if we specify the energy density and pressure of dark components (see section 4). Thus the non-equilibrium treatment is used to study the laws of thermodynamics in both forms of the energy-momentum tensor of dark components. In f (R, T ) gravity, qtot does not vanish so there exists some energy exchange with the horizon. The non-equilibrium description can be interpreted as due to some energy flow between inside and outside the apparent horizon. The first law of thermodynamics is obtained at the apparent horizon in FRW background for f (R, T ) gravity. We observe that the additional entropy term is produced as compared to GR, GaussBonnet gravity [18], Lovelock gravity [19, 20] and braneworld gravity [21]. The equilibrium and non-equilibrium description of thermodynamics in f (R) gravity can be obtained if the term fT (R, T ) vanishes i.e., Lagrangian density depends only on geometric part. We have established the GSLT with the assumption that the total temperature inside the horizon Ttot is proportional to the temperature of the apparent horizon Th and evaluated its validity conditions. The GSLT in f (R) gravity follows from condition (4.4) if F = 1. It is concluded that in thermal equilibrium, GSLT is satisfied in both phantom and non-phantom phases.
References [1] WMAP collaboration, C. Bennett et al., First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Preliminary maps and basic results, Astrophys. J. Suppl. 148 (2003) 1 [astro-ph/0302207] [INSPIRE]; WMAP collaboration, D.N. Spergel et al., First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters, Astrophys. J. Suppl. 148 (2003) 175 [astro-ph/0302209] [INSPIRE];
– 10 –
JCAP03(2012)028
3 and R = 6(H ˙ + 2H 2 + k/a2 ). B ≥ 0 clearly holds when the Hubble parameter for V = 34 π˜ rA and scalar curvature have same signatures. It can be seen that main difference of results of (R,T ) . We remark that in both f (R, T ) gravity with f (R) gravity is the term F = 1 + fT8πG definitions of dark components, the GSLT is valid both in phantom and non-phantom phases of the universe.
WMAP collaboration, D. Spergel et al., Wilkinson Microwave Anisotropy Probe (WMAP) three year results: implications for cosmology, Astrophys. J. Suppl. 170 (2007) 377 [astro-ph/0603449] [INSPIRE].
[3] E. Hawkins et al., The 2dF Galaxy Redshift Survey: Correlation functions, peculiar velocities and the matter density of the universe, Mon. Not. Roy. Astron. Soc. 346 (2003) 78 [astro-ph/0212375] [INSPIRE]; SDSS collaboration, M. Tegmark et al., Cosmological parameters from SDSS and WMAP, Phys. Rev. D 69 (2004) 103501 [astro-ph/0310723] [INSPIRE]; The 2dFGRS collaboration, S. Cole et al., The 2dF Galaxy Redshift Survey: Power-spectrum analysis of the final dataset and cosmological implications, Mon. Not. Roy. Astron. Soc. 362 (2005) 505 [astro-ph/0501174] [INSPIRE]. [4] SDSS collaboration, D.J. Eisenstein et al., Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies, Astrophys. J. 633 (2005) 560 [astro-ph/0501171] [INSPIRE]. [5] B. Jain and A. Taylor, Cross-correlation tomography: measuring dark energy evolution with weak lensing, Phys. Rev. Lett. 91 (2003) 141302 [astro-ph/0306046] [INSPIRE]. [6] V. Sahni and A.A. Starobinsky, The case for a positive cosmological Lambda term, Int. J. Mod. Phys. D 9 (2000) 373 [astro-ph/9904398] [INSPIRE]; V. Sahni, Dark matter and dark energy, Lect. Notes Phys. 653 (2004) 141 [astro-ph/0403324] [INSPIRE]; T. Padmanabhan, Dark Energy and Gravity, Gen. Relativ. Gravit. 40 (2008) 529 [arXiv:0705.2533] [INSPIRE]. [7] R. Caldwell, A Phantom menace?, Phys. Lett. B 545 (2002) 23 [astro-ph/9908168] [INSPIRE]; S. Nojiri and S.D. Odintsov, Quantum de Sitter cosmology and phantom matter, Phys. Lett. B 562 (2003) 147 [hep-th/0303117] [INSPIRE]; S. Nojiri and S.D. Odintsov, de Sitter brane universe induced by phantom and quantum effects, Phys. Lett. B 565 (2003) 1 [hep-th/0304131] [INSPIRE]. [8] B. Feng, X.-L. Wang and X.-M. Zhang, Dark energy constraints from the cosmic age and supernova, Phys. Lett. B 607 (2005) 35 [astro-ph/0404224] [INSPIRE]; Z.-K. Guo, Y.-S. Piao, X.-M. Zhang and Y.-Z. Zhang, Cosmological evolution of a quintom model of dark energy, Phys. Lett. B 608 (2005) 177 [astro-ph/0410654] [INSPIRE]. [9] A.Y. Kamenshchik, U. Moschella and V. Pasquier, An alternative to quintessence, Phys. Lett. B 511 (2001) 265 [gr-qc/0103004] [INSPIRE]; M. Bento, O. Bertolami and A. Sen, Generalized Chaplygin gas, accelerated expansion and dark energy matter unification, Phys. Rev. D 66 (2002) 043507 [gr-qc/0202064] [INSPIRE].
– 11 –
JCAP03(2012)028
[2] Supernova Cosmology Project collaboration, S. Perlmutter et al., Measurements of the cosmological parameters Omega and Lambda from the first 7 supernovae at z ≥ 0.35, Astrophys. J. 483 (1997) 565 [astro-ph/9608192] [INSPIRE]; Supernova Cosmology Project collaboration, S. Perlmutter et al., Discovery of a supernova explosion at half the age of the Universe and its cosmological implications, Nature 391 (1998) 51 [astro-ph/9712212] [INSPIRE]; Supernova Cosmology Project collaboration, S. Perlmutter et al., Measurements of Omega and Lambda from 42 high redshift supernovae, Astrophys. J. 517 (1999) 565 [astro-ph/9812133] [INSPIRE]; Supernova Search Team collaboration, A.G. Riess et al., Type Ia supernova discoveries at z > 1 from the Hubble Space Telescope: Evidence for past deceleration and constraints on dark energy evolution, Astrophys. J. 607 (2004) 665 [astro-ph/0402512] [INSPIRE]; A.G. Riess, L.-G. Strolger, S. Casertano, H.C. Ferguson, B. Mobasher, et al., New Hubble Space Telescope Discoveries of Type Ia Supernovae at z ≥ 1: Narrowing Constraints on the Early Behavior of Dark Energy, Astrophys. J. 659 (2007) 98 [astro-ph/0611572] [INSPIRE].
[10] T. Padmanabhan, Accelerated expansion of the universe driven by tachyonic matter, Phys. Rev. D 66 (2002) 021301 [hep-th/0204150] [INSPIRE]; A. Sen, Rolling Tachyon, JHEP 04 (2002) 048 [hep-th/0203211] [INSPIRE].
[12] J.M. Bardeen, B. Carter and S. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE]. [13] S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE]. [14] J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE]. [15] T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE]. [16] A.V. Frolov and L. Kofman, Inflation and de Sitter thermodynamics, JCAP 05 (2003) 009 [hep-th/0212327] [INSPIRE]. [17] R.-G. Cai and S.P. Kim, First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe, JHEP 02 (2005) 050 [hep-th/0501055] [INSPIRE]. [18] M. Akbar and R.-G. Cai, Thermodynamic Behavior of Friedmann Equations at Apparent Horizon of FRW Universe, Phys. Rev. D 75 (2007) 084003 [hep-th/0609128] [INSPIRE]. [19] R.-G. Cai and L.-M. Cao, Unified first law and thermodynamics of apparent horizon in FRW universe, Phys. Rev. D 75 (2007) 064008 [gr-qc/0611071] [INSPIRE]. [20] R.-G. Cai, L.-M. Cao, Y.-P. Hu and S.P. Kim, Generalized Vaidya Spacetime in Lovelock Gravity and Thermodynamics on Apparent Horizon, Phys. Rev. D 78 (2008) 124012 [arXiv:0810.2610] [INSPIRE]. [21] A. Sheykhi, B. Wang and R.-G. Cai, Thermodynamical properties of apparent horizon in warped DGP braneworld, Nucl. Phys. B 779 (2007) 1 [hep-th/0701198] [INSPIRE]; A. Sheykhi, B. Wang and R.-G. Cai, Deep connection between thermodynamics and gravity in Gauss-Bonnet braneworld, Phys. Rev. D 76 (2007) 023515 [hep-th/0701261] [INSPIRE]. [22] C. Eling, R. Guedens and T. Jacobson, Non-equilibrium Thermodynamics of Spacetime, Phys. Rev. Lett. 86 (2006) 121301 [gr-qc/0602001] [INSPIRE]. [23] M. Akbar and R.-G. Cai, Thermodynamic Behavior of Field Equations for f(R) Gravity, Phys. Lett. B 648 (2007) 243 [gr-qc/0612089] [INSPIRE]. [24] Y. Gong and A. Wang, The Friedmann equations and thermodynamics of apparent horizons, Phys. Rev. Lett. 99 (2007) 211301 [arXiv:0704.0793] [INSPIRE]. [25] K. Bamba and C.-Q. Geng, Thermodynamics in f (R) gravity with phantom crossing, Phys. Lett. B 679 (2009) 282 [arXiv:0901.1509] [INSPIRE]. [26] K. Bamba and C.-Q. Geng, Thermodynamics in f (R) gravity in the Palatini formalism, JCAP 06 (2010) 014 [arXiv:1005.5234] [INSPIRE]. [27] K. Bamba and C.-Q. Geng, Thermodynamics of cosmological horizons in f (T ) gravity, JCAP 11 (2011) 008 [arXiv:1109.1694] [INSPIRE].
– 12 –
JCAP03(2012)028
[11] S. Nojiri and S.D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, Int. J. Geom. Methods Mod. Phys. 4 (2007) 115 [hep-th/0601213] [INSPIRE]; T.P. Sotiriou and S. Liberati, Metric-affine f(R) theories of gravity, Annals Phys. 322 (2007) 935 [gr-qc/0604006] [INSPIRE]; T.P. Sotiriou and V. Faraoni, f(R) Theories Of Gravity, Rev. Mod. Phys. 82 (2010) 451 [arXiv:0805.1726] [INSPIRE]; A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].
[28] S.-F. Wu, B. Wang, G.-H. Yang and P.-M. Zhang, The generalized second law of thermodynamics in generalized gravity theories, Class. Quantum Grav. 25 (2008) 235018 [arXiv:0801.2688] [INSPIRE]. [29] [K. Bamba, C.Q. Geng and S. Tsujikawa, Equilibrium thermodynamics in modified gravitational theories, Phys. Lett. B 688 (2010) 101. [30] T. Harko, F.S. Lobo, S. Nojiri and S.D. Odintsov, f (R, T ) gravity, Phys. Rev. D 84 (2011) 024020 [arXiv:1104.2669] [INSPIRE]. [31] M. Houndjo, Reconstruction of f(R, T) gravity describing matter dominated and accelerated phases, Int. J. Mod. Phys. D 21 (2012) 1250003 [arXiv:1107.3887] [INSPIRE].
[33] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fileds, Butterworth-Heinemann, Oxford U.K.(2002). [34] R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE]. [35] R. Brustein, D. Gorbonos and M. Hadad, Wald’s entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling, Phys. Rev. D 79 (2009) 044025 [arXiv:0712.3206][INSPIRE]. [36] C.W. Misner and D.H. Sharp, Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Phys. Rev. 136 (1964) B571 [INSPIRE]; D. Bak and S.-J. Rey, Cosmic holography, Class. Quant. Grav. 17 (2000) L83 [hep-th/9902173] [INSPIRE]. [37] S.A. Hayward, Unified first law of black hole dynamics and relativistic thermodynamics, Class. Quant. Grav. 15 (1998) 3147 [gr-qc/9710089] [INSPIRE]; S.A. Hayward, S. Mukohyama and M. Ashworth, Dynamic black hole entropy, Phys. Lett. A 256 (1999) 347 [gr-qc/9810006] [INSPIRE]. [38] G. Izquierdo and D. Pavon, Dark energy and the generalized second law, Phys. Lett. B 633 (2006) 420 [astro-ph/0505601] [INSPIRE]. [39] S. Nojiri and S.D. Odintsov, Inhomogeneous equation of state of the universe: Phantom era, future singularity and crossing the phantom barrier, Phys. Rev. D 72 (2005) 023003 [hep-th/0505215] [INSPIRE].
– 13 –
JCAP03(2012)028
[32] Y. Gong and A. Wang, The Friedmann equations and thermodynamics of apparent horizons, Phys. Rev. Lett. 99 (2007) 211301 [arXiv:0704.0793] [INSPIRE]; C. Eling, Hydrodynamics of spacetime and vacuum viscosity, JHEP 11 (2008) 048 [arXiv:0806.3165] [INSPIRE]; E. Elizalde and P.J. Silva, F(R) gravity equation of state, Phys. Rev. D 78 (2008) 061501 [arXiv:0804.3721] [INSPIRE]; S.-F. Wu, X.-H. Ge, P.-M. Zhang and G.-H. Yang, Dynamical horizon entropy and equilibrium thermodynamics of generalized gravity theories, Phys. Rev. D 81 (2010) 044034 [arXiv:0912.4633] [INSPIRE].
