Viscous Cosmology and Thermodynamics of Apparent Horizon

M. Akbar ***Email address<EMAIL_ADDRESS>†††Email address:
<EMAIL_ADDRESS>
Centre for Advanced Mathematics and Physics
National University of Sciences and Technology
Peshawar Road, Rawalpindi, Pakistan

Abstract

It is shown that the differential form of Friedmann equations of FRW universe
can be recast as a similar form of the first law ,$T_{A}dS_{A}=dE+WdV$, of
thermodynamics at the apparent horizon of FRW universe filled with the viscous
fluid. It is also shown that the generalized second law of thermodynamics
holds at the apparent horizon of FRW universe and preserves dominant energy
condition.

PACS numbers: 04.70.Dy, 97.60.Lf

### Introduction:

In the cosmological setting, one can associate Hawking temperature and entropy
with the apparent horizon analogous to the hawking temperature and entropy
associated with the black hole horizon [1, 2, 3]. In the case of de Sitter
space, the event horizon matches with the apparent horizon of FRW universe
with $k=0$ however for more general cosmological models, the event horizon may
not exist but the apparent horizon associated with the Hawking temperature and
entropy always exists. The thermodynamic properties associated with the
apparent horizon of FRW universe filled with the perfect fluid has been
studied by many authors (see for examples [1, 2, 4, 6, 5, 7, 8]). The
extension of this connection between thermodynamics and gravity has also been
carried out in the braneworld cosmology [9, 10, 11, 12]. For a general static
spherically symmetric and stationary axisymmetric spacetimes, It was shown
that the Einstein field equations can be rewritten [18, 19, 20] as a first law
of thermodynamics. More recently, by considering a masslike function, it has
been shown [13] that the equilibrium thermodynamics should exist in the
extended theories of gravity and the Friedman equations in various theories of
gravity can be rewritten as a first law $TdS=dE$ at the apparent horizon of
FRW universe. More recently, Cai et al [5] has shown that by employing
Clausius relation, $\delta Q=TdS$, to the apparent horizon of a FRW universe,
they are able to derive the modified Friedmann equation by using quantum
corrected entropy-area relation.
The cosmological models with perfect cosmic fluid has been studied widely in
literature however viscous cosmic fluid came much later in the study of the
universe [17]. It has been revealed by Barrow [14](also see [15]) that the
thermodynamic entropy associated with bulk viscosity violates the dominant
energy condition and can decrease very well. This fact was resumed [16] in the
framework of the solutions given in reference [15]. By taking into account the
entropy of the sources only, it has been deduced that the entropy of the bulk
viscous sources cannot decrease since it would violate the second law of
thermodynamics for large cosmic time values. However, in order to state the
generalized second law, one has to include both the entropy connected with the
sources and the entropy associated with background geometry. The most of the
study of the second law in the cosmological context has been carried out at
the event horizon (see for examples [15, 16]) however the validity of second
law within the apparent horizon of FRW universe is of great important to
investigate. In this letter we will also investigate this issue. The
cosmological models with bulk viscosity have been studied in the literature
from various point of view (see for examples [15, 21, 22]).
The purpose of this letter is twofold. The first is to show that the Friedmann
equations of FRW universe pertaining cosmic bulk viscosity can be rewritten as
a similar form of the first law of thermodynamics. The other is to discuss the
generalized second law within the apparent horizon of FRW universe.
Let us start with a FRW universe of metric

$ds^{2}=-dt^{2}+a^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}),$ (1)

where $d\Omega^{2}=d\theta^{2}+sin^{2}\theta d\phi^{2}$ stands for the line
element of 2-dimensional unit sphere and the spatial curvature constant
$k=+1$, 0 and $-1$ represents a closed, flat and open universe, respectively.
The above metric (1) can be rewritten in spherical form

$ds^{2}=h_{ab}dx^{a}dx^{b}+\tilde{r}^{2}d\Omega^{2},$ (2)

where $\tilde{r}=a(t)r$, $x^{0}=t$, $x^{1}=r$ and
$h_{ab}=diag(-1,\frac{a^{2}}{1-kr^{2}})$. One can work out the dynamical
apparent horizon from the relation
$h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r}=0$ which turns out
$\frac{1}{\tilde{r}^{2}_{A}}=H^{2}+k/a^{2}$, where $H\equiv\frac{\dot{a}}{a}$
is the Hubble parameter and the dots denote derivatives with respect to the
cosmic time. let the universe be filled with a viscous fluid and
$u^{\mu}=(u^{0},u^{i})$ is the four-velocity of the fluid. In comoving
coordinates $u^{0}=1$ and $u^{i}=0$. Define
$h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$,
$w_{\mu\nu}=h^{\alpha}_{\mu}h^{\beta}_{\nu}u_{[\alpha;\beta]}$ and
$\theta_{\mu\nu}=h^{\alpha}_{\mu}h^{\beta}_{\nu}u_{(\alpha;\beta)}$ as a
projection tensor, rotation tensor and expansion tensor, respectively. The
scalar expansion is $\theta\equiv\theta^{\mu}_{\mu}=u^{\mu}_{;\mu}$ and
$\sigma_{\mu\nu}=\theta_{\mu\nu}-\frac{1}{3}h_{\mu\nu}\theta$ is the shear
tensor. Taking into consideration of metric (1),
$u^{\mu}_{;\nu}=h^{\mu}_{\nu}\frac{\dot{a}}{a}$ which implies that the
rotation and shear tensors vanish, i.e. $w_{\mu\nu}=\sigma_{\mu\nu}=0$. While
the scalar expansion is $\theta=3\frac{\dot{a}}{a}=3H$. Hence the energy-
momentum tensor of viscous fluid in the background of metric (1) finally
becomes [21]

$T_{\mu\nu}=(\rho+P-\zeta\theta)u_{\mu}u_{\nu}+(P-\zeta\theta)g_{\mu\nu},$ (3)

where $\rho$, $P$ and $\zeta$ are the energy density, thermodynamic pressure
and bulk viscosity respectively. The components of energy-momentum tensor are
$T_{00}=\rho$, $T_{0i}=0$ and $T_{ij}=(P-\zeta\theta)g_{ij}$. Therefore, the
total effect of the bulk viscosity is to reduce the pressure $P$ of the
perfect fluid by an amount $\zeta\theta$, so that the effective pressure of
the viscous fluid turns out to be $\tilde{P}=P-\zeta\theta$. The energy
conservation $T^{\mu\nu}_{;\nu}=0$, yields $\dot{\rho}+3H(\rho+\tilde{P})=0$.
Now we turn to discuss the thermodynamic interpretation of Friedman equations
at the apparent horizon of FWR universe. For this purpose, we first define the
surface gravity
$\kappa=\frac{1}{2\sqrt{-h}}\partial_{a}(\sqrt{-h}h^{ab}\partial_{b}\tilde{r})$
at the apparent of FRW universe. By utilizing above relation, one can easily
find that the surface gravity at the apparent horizon of FRW universe yields

$\kappa=\frac{-1}{\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}).$
(4)

When $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\leq 1$ implies the surface
gravity $\kappa\leq 0$ which leads to the temperature $T=\kappa/2\pi\leq 0$.
However, in reference [1], an approximation,
$\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\ll 1$, has been used while
determining horizon temperature. They found $T=\frac{1}{2\pi\tilde{r}_{A}}$ by
approximating surface gravity $|\kappa|=\frac{1}{\tilde{r}_{A}}$. Now we turn
to define energy of the universe within the apparent horizon. We take total
matter energy $E=V\rho$ inside a sphere of radius $\tilde{r}$ which is also
the Miser-Sharp energy [25],
$E=\frac{\tilde{r}}{2}(1-h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r})$,
within the apparent horizon. Since at the apparent horizon
$(h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r}=0)$, so the Mizner-Sharp
energy is in fact total matter energy inside the sphere of radius
$\tilde{r}_{A}$ and is given by

$E=V\rho.$ (5)

The entropy $S=A/4$ is the horizon entropy, where $A=4\pi\tilde{r}_{A}^{2}$ is
the horizon area. Note that we use the units $\hbar=c=G=\kappa_{B}=1$. From
the Einstein field equations $G_{\mu\nu}=8\pi T_{\mu\nu}$, the Friedman
equations for the viscous fluid of stress-energy tensor (3) can be written as

$H^{2}+\frac{k}{a^{2}}=\frac{8\pi}{3}\rho,$ (6)
$\dot{H}-\frac{k}{a^{2}}=-4\pi(\rho+\tilde{P}).$ (7)

In terms of the apparent horizon, the Friedman equation (6) can be rewritten
as

$\frac{1}{\tilde{r}^{2}_{A}}=\frac{8\pi}{3}\rho.$ (8)

Let the apparent horizon surface acts as the boundary of the thermal system of
the FRW universe. In general, the apparent horizon is not constant but changes
with time. Let $d\tilde{r}_{A}$ be an infinitesimal change in radius of the
apparent horizon in time interval dt. This change in the apparent horizon will
cause a small change $dV$ in the volume $V$ of the universe. This constitutes
two spherical system of space-time with radii $\tilde{r}_{A}$ and
$\tilde{r}_{A}+d\tilde{r}_{A}$ with a common source of fluid with non-zero
pressure and energy density near horizon. Each space-time constituting a
thermal system and satisfying Einstein’s equations, differs infinitesimally in
the extensive variables volume, energy and entropy by $dV$, $dE$ and $dS$,
respectively, while having the same values of intensive variables temperature
$T$ and pressure $\bar{P}$. Thus for these two thermal states of the space-
time, there must exist a certain relation connecting these thermodynamic
quantities. To establish connection among the thermal quantities and Friedman
equations, we differentiate equation (8) which implies

$\frac{1}{\tilde{r}_{A}^{3}}d\tilde{r}_{A}=4\pi(\rho+\tilde{P})Hdt$ (9)

Multiplying both sides of this equation with a factor
$\tilde{r}_{A}^{3}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$, one can
rewrite this equation in a form

$\frac{\kappa}{2\pi}\frac{d(4\pi\tilde{r}_{A}^{2})}{4}=-4\pi\tilde{r}_{A}^{2}(\rho+P)H(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})dt$
(10)

One can recognize that the quantities $\frac{\kappa}{2\pi}$ and
$\frac{4\pi\tilde{r}_{A}^{2}}{4}$ are the temperature $T$ and entropy $S$
respectively. Therefore the above equation can be rewritten as

$TdS=-4\pi\tilde{r}_{A}^{3}(\rho+\tilde{P})H(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})dt$
(11)

Now we turn to the total matter energy (5) and taking differential of it, one
gets

$dE=4\pi\tilde{r}_{A}^{2}\rho
d\tilde{r}_{A}-4\pi\tilde{r}_{A}^{3}(\rho+\tilde{P})Hdt.$ (12)

Using equations (11) and (12) one yields

$dE=TdS+WdV,$ (13)

where $W=\frac{1}{2}(\rho-\tilde{P})$ is the work density. The above equation
is the unified first law [24] of relativistic thermodynamics. Note that the
above thermal identity is obtained from the Friedman equation with viscous
fluid together with the characteristics of the apparent horizon, while the
author of reference [24] studied the thermodynamics of trapping horizon of
dynamical black hole. It has been found that the entropy associated with
apparent horizon is proportional to the horizon area which is originally
initiated from the black hole horizon entropy that satisfies the so-called
area formula [26]. In the case of perfect fluid, $\zeta=0$, one can compare
the above thermal identity with the standard form of the first law, $dE=TdS-
PdV$, of thermodynamics. In fact, the negative pressure term $-P$ in the first
law is taken the place of the work density $W$. Notice that for pure de Sitter
spacetime, $\rho=-P$, then one acquires the standard form $dE=TdS-PdV$. Note
that we are considering the universe as a thermal object with apparent horizon
as its boundary. So it useful to define other thermodynamic quantities like
Enthalpy $H$ and Gibbs free energy $G$. The Enthalpy is defined by the
equation $H=U+PV$, where $U=E$ is the total internal energy which is taken to
be the total matter energy of the universe bounded by the apparent horizon. So
the Enthalpy of the universe bounded by the apparent horizon can be written in
terms of apparent radius $H=\frac{4\pi}{3}(\rho+P)\tilde{r}_{A}^{3}$. So the
heat capacity of the universe enveloped by the apparent horizon at constant
pressure of the perfect is defined as

$C_{P}=(\frac{\partial H}{\partial T})_{P}=(\frac{\partial
H}{\partial\tilde{r}_{A}}\frac{\partial\tilde{r}_{A}}{\partial T})_{P},$ (14)

which yields
$C_{P}=4\pi\tilde{r}_{A}^{2}(\rho+P)(\frac{\partial\tilde{r}_{A}}{\partial
T})_{P}$. In general the temperature,
$T=\kappa/2\pi\equiv\frac{-1}{2\pi\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$.
However, if one considers the approximation,
$\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\ll 1$ (also see [1, 9]), the
temperature $T=\mid\kappa\mid/2\pi\equiv 1/2\pi\tilde{r}_{A}$ which implies
that the heat capacity of the universe is negative provided the dominant
energy condition holds. On the other hand, if one consider general temperature
$T=\frac{-1}{2\pi\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$
associated with the apparent horizon which implies the heat capacity of the
universe is positive definite provided
$\frac{\dot{\tilde{r}}_{A}}{H\tilde{r}_{A}}\leq 1$. It is also interesting to
note that the total matter energy $E=V\rho$, entropy $S=A/4$ and temperature
$T=1/2\pi\tilde{r}_{A}$ satisfy the relation $E=TS$ .

### Generalized Second Law:

Recently a lot of attention has been granted to the generalized second law of
thermodynamics in the accelerating universe handled by the dark energy [27].
Using a particular model for dark energy, the generalized second law has been
studied in reference [28] with the boundaries of universe enveloped by the
apparent as well event horizon. It is important to investigate the generalized
second law as defined in the region of the universe bounded by the apparent
horizon in more general context.
Let us now consider a region of FRW universe bounded by the apparent horizon
filled by a perfect fluid of energy density $\rho$ and pressure $P$. We assume
that the region bounded by the apparent horizon acts as a thermal system with
boundary defined by the apparent horizon. Since the apparent horizon is not
constant but varies with time. As the apparent radius changes , the volume
enveloped by the apparent horizon will also change, however the thermal system
bounded by the apparent horizon remains in equilibrium when it moves from one
state to another so that the temperature of the system must be uniform and the
same as the temperature of its surroundings. This requires that the
temperature of total energy inside the apparent horizon should be in
equilibrium with the temperature associated with the apparent horizon because
we are not considering the flow of energy through the horizon.
In order to state that a given cosmological solution satisfies/violates the
second law of thermodynamics one should define first what is the
generalization of the second law to the case of FRW universe. It should
include both the entropy due to sources and the entropy associated with the
background geometry. So the generalized second law can be expressed as

$\dot{S}_{A}+\dot{S}_{m}\geq 0,$ (15)

where $S_{A}=A/4G$ is the Bekenstein-Hawking entropy associated with the
apparent horizon of FRW universe and $S_{m}$ is the entropy due to the matter
sources inside the apparent horizon. From the Friedmann equations (6) and (7),
it can easily be shown that
$\dot{\tilde{r}}_{A}=4\pi\tilde{r}_{A}^{3}H(\rho+P)\geq 0$ provided the
universe driven by the source of perfect fluid preserves dominant energy
condition. Since we are assuming the region of a FRW universe enveloped by the
apparent horizon as thermal system so that the change of energy $dE$ of the
universe from one state to another must be connected through the Friedmann
equations together with the characteristics of the apparent horizon. It has
been shown [9, 3, 10] that the Friedmann equations at apparent horizon of FRW
universe filled with perfect fluid satisfied the thermal identity
$dE=T_{A}dS_{A}+WdV$ instead of satisfying the standard first law
$T_{A}dS_{A}=dE+PdV$, where $T_{A}$ is the temperature associated with the
apparent horizon. Since we are assuming the local equilibrium so that the
change of matter energy $dE_{m}$ satisfies the first law of thermodynamics,
$dE_{m}=T_{A}dS_{m}-PdV$, where $T_{A}=|\kappa|/2\pi$ is the temperature
associated with the apparent horizon and is given by

$T=\frac{1}{2\pi\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}),$
(16)

where $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\leq 1$ ensures that the
temperature is positive. From the first law, $dE_{m}=T_{A}dS_{m}-PdV$, one can
obtain

$T_{A}\dot{S}_{m}=4\pi\tilde{r}_{A}^{2}\dot{\tilde{r}}_{A}(\rho+P)-4\pi\tilde{r}_{A}^{3}H(\rho+P).$
(17)

Now we turn to find out the entropy change $\dot{S}_{H}$ associated with the
apparent horizon of FRW universe which yields

$T_{A}\dot{S}_{A}=4\pi\tilde{r}_{A}^{3}H(\rho+P)-2\pi\tilde{r}_{A}^{2}(\rho+P)\dot{\tilde{r}}_{A}.$
(18)

Adding equations (17) and (18), one gets the expression for the generalized
second law

$T_{A}(\dot{S}_{m}+\dot{S}_{H})=2\pi\tilde{r}_{A}^{2}\dot{\tilde{r}}_{A}(\rho+P).$
(19)

It is obvious from the above equation that $\dot{S}_{m}+\dot{S}_{H}\geq 0$
implies that the generalized second law holds provided $\rho+P\geq 0$.

### Conclusion:

It is shown that the differential form of Friedmann equations of FRW universe
filled with a viscous fluid can be rewritten as a similar form of the first
law ,$TdS=dE+WdV$, of thermodynamics at the apparent horizon of FRW universe.
It can easily be seen that the similar identity also holds for the perfect
fluid when bulk viscosity of the fluid is zero. It is also shown that the heat
capacity of the universe filled with perfect fluid is negative if one
considers the approximate temperature $T=|\kappa|/2\pi=1/2\pi\tilde{r}_{A}$.
However if one utilizes the general expression
$T=\frac{-1}{2\pi\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$,
the heat capacity of the universe is positive definite provided
$\frac{\dot{\tilde{r}}_{A}}{H\tilde{r}_{A}}\leq 1$ which implies that the
universe within the apparent horizon of FRW universe is thermodynamically
stable. We also verify that the generalized second law of thermodynamics at
the apparent horizon of FRW universe holds provided that the matter source
satisfies the dominant energy condition. It is interesting to investigate the
generalized second law for the extended theories of gravity in a region
bounded by the apparent horizon of FRW universe. The work in this respect is
under progress.

## Acknowledgments

I would like to thank Rong-Gen Cai for his useful comments. The work is
supported by a grant from National university of Sciences and Technology.

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