Unlike stationary black holes the expanding universe with a cosmic curvature has a dynamically
changing apparent horizon with the radius
, where
is a cosmic curvature [108
]
(see also [296]). Even in the FLRW spacetime, however, the Friedmann equation can be written in the
thermodynamical form
, where
is the work density present in the dynamical
background [8]. For matter contents of the universe with energy density
and pressure
, the work
density is given by
[297
, 298
]. This method is identical to the one established by
Jacobson [324], that is,
.
In metric f (R) gravity Eling et al. [228] showed that a non-equilibrium treatment is required such
that the Clausius relation is modified to
, where
is the
Wald horizon entropy [610
] and
is the bulk viscosity entropy production term. Note
that
corresponds to a Noether charge entropy. Motivated by this work, the connections
between thermodynamics and modified gravity have been extensively discussed – including
metric f (R) gravity [6
, 7
, 281
, 431, 619
, 620
, 230, 103, 51, 50
, 157] and scalar-tensor
theory [281
, 619
, 620
, 108
].
Let us discuss the relation between thermodynamics and modified gravity for the following general
action [53]
In the following we discuss the thermodynamical property of the theories given above. The apparent
horizon is determined by the condition , which gives
in the FLRW
spacetime. Taking the differentiation of this relation with respect to
and using Eq. (13.18
), we obtain
In Einstein gravity the horizon entropy is given by the Bekenstein–Hawking entropy ,
where
is the area of the apparent horizon [59, 75, 293]. In modified gravity theories one can
introduce the Wald entropy associated with the Noether charge [610]:
In Einstein gravity the Misner-Sharp energy [428] is defined by . In f (R) gravity and
scalar-tensor theory this can be extended to
[281
]. Using this expression for
theory, we have
The main reason why the non-equilibrium term appears is that the energy density
and the
pressure
defined in Eqs. (13.20
) and (13.21
) do not satisfy the standard continuity equation for
. On the other hand, if we define the effective energy-momentum tensor
as Eq. (2.9
) in
Section 2, it satisfies the continuity equation (2.10
). This correspond to rewriting the Einstein
equation in the form (2.8
) instead of (13.23
). Using this property, [53] showed that equilibrium
description of thermodynamics can be possible by introducing the Bekenstein–Hawking entropy
. In this case the horizon entropy
takes into account the contribution of both
the Wald entropy
in the non-equilibrium thermodynamics and the entropy production
term.
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