Friedmann Equations and Thermodynamics of Apparent Horizons
Friedmann Equations and Thermodynamics of Apparent Horizons
Yungui Gong1, 2, ∗ and Anzhong Wang2, †
College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
GCAP-CASPER, Department of Physics, Baylor University, Waco, TX 76798, USA
With the help of a masslike function which has dimension of energy and equals to the Misner-Sharp
mass at the apparent horizon, we show that the first law of thermodynamics of the apparent horizon
dE = TAdSA can be derived from the Friedmann equation in various theories of gravity, including
the Einstein, Lovelock, nonlinear, and scalar-tensor theories. This result strongly suggests that the
relationship between the first law of thermodynamics of the apparent horizon and the Friedmann
equation is not just a simple coincidence, but rather a more profound physical connection.
PACS numbers: 98.80.-k,04.20.Cv,04.70.Dy
The derivation of the thermodynamic laws of black
holes from the classical Einstein equation suggests a deep
connection between gravitation and thermodynamics [1].
The discovery of the quantum Hawking radiation [2] and
black hole entropy which is proportional to the area of
the event horizon of the black hole [3] further supports
this connection and the thermodynamic (physical) inter-
pretation of geometric quantities. The interesting rela-
tion between thermodynamics and gravitation became
manifest when Jacobson derived Einstein equation from
the first law of thermodynamics by assuming the propor-
tionality of the entropy and the horizon area for all local
acceleration horizons [4].
In cosmology, like in black holes, for the cosmologi-
cal model with a cosmological constant (called de Sit-
ter space), there also exist Hawking temperature and en-
tropy associated with the cosmological event horizon, and
thermodynamic laws of the cosmological event horizon
[5]. In de Sitter space, the event horizon coincides with
the apparent horizon (AH). For more general cosmologi-
cal models, the event horizon may not exist, but the AH
always exists, so it is possible to have Hawking tempera-
ture and entropy associated with the AH. The connection
between the first law of thermodynamics of the AH and
the Friedmann equation was shown in [6]. Now, we must
ask if this interesting relation between gravitation and
thermodynamics exists in more general theories of grav-
ity, like Brans-Dicke (BD) theory and nonlinear gravi-
tational theory. In [7], the gravitational field equations
for the nonlinear theory of gravity were derived from the
first law of thermodynamics by adding some nonequi-
librium corrections. In this Letter, we show that equi-
librium thermodynamics indeed exists for more general
theories of gravity, provided that a new masslike function
is introduced.
To show our claim, we begin by reviewing the ther-
modynamics of the AH with the use of the Misner-
Sharp (MS) mass in Einstein and BD theories of gravity,
whereby we find the equilibrium thermodynamics fails
to hold for the BD theory. The Einstein equation can be
rewritten as the mass formulas with the help of the MS
mass M. The energy flow through the AH dE is related
with the MS mass. Since the MS mass M, the Hawking
temperature TA, and the entropy SA of the AH are geo-
metric quantities, the first law of thermodynamics of the
AH can be thought of as a geometric relation. Therefore,
we expect the geometric relation to hold in other gravi-
tational theory if it holds in Einstein theory. To achieve
this, we replace the MS mass M by a masslike function
M which equals to the MS mass M at the AH, we then
show that the connection between the first law of ther-
modynamics of the AH and the gravitational equations
holds in scalar-tensor and nonlinear theories of gravity
without adding nonequilibrium correction.
For a spherically symmetric space-time with the metric
ds2 = gabdx
adxb+ r̃2dΩ2, using the MS mass M = r̃(1−
gabr̃,ar̃,b)/2G [8], the a − b components of the Einstein
equation give the mass formulas [9, 10]
M,a = 4πr̃2(T ba − δ
aT )r̃,b, (1)
where the unit spherical metric is given by dΩ2 = dθ2 +
sin2 θdϕ2 and T = T aa . From now on, all the indices are
raised and lowered by the metric gab and the covariant
derivative is with respect to gab. The AH is
r̃A = arA = (H
2 + k/a2)−1/2. (2)
At the AH, the MS mass M = 4πr̃3Aρ/3, which can be
interpreted as the total energy inside the AH. Now we
use the (approximate) generator ka = (1, −Hr) of the
AH, which is null at the horizon, to project the mass
formulas. Since kar̃,a = 0, at the AH we find that
− dE = −ka∇aMdt = d(r̃A)/G = TAdSA, (3)
where the horizon temperature is TA = 1/(2πr̃A) and the
horizon entropy is SA = πr̃
A/G. On the other hand, us-
ing the mass formulae (1), we get the energy flow through
the AH
− dE = −ka∇aMdt = −4πr̃2T ba r̃,bk
= 4πr̃3AH(ρ+ p)dt. (4)
Therefore, the Friedmann equation gives rise to the first
law of thermodynamics −dE = TAdSA of the AH. From
http://arxiv.org/abs/0704.0793v2
the above definitions, we see that the relation −dE =
TAdSA is a geometrical relation which depends on the
only assumption of the Robertson-Walker metric. To
connect the geometrical quantity dE with the energy flow
through the AH, we need to use the Friedmann equa-
tions. Therefore, for any gravitational theory, if we can
write the gravitational field equation as Gµν = 8πGTµν
and regard the right-hand side as the effective energy-
momentum tensor, then we find the energy flow through
the AH, whereby we derive the first law of thermodynam-
ics of the AH −dE = TAdSA. For example, in the Jordan
frame of the scalar-tensor theory of gravity, if we take the
right-hand side of gravitational field equation as the total
effective energy-momentum tensor, then the Friedmann
equation can be regarded as a thermodynamic identity
at the AH [11].
The connection between the first law of thermodynam-
ics and the Friedmann equation at the AH was also found
for gravity with Gauss-Bonnet term, the Lovelock theory
of gravity [6], and the braneworld cosmology [12]. For
a general static spherically symmetric and stationary ax-
isymmetric space-times, it was shown that Einstein equa-
tion at the horizon give rise to the first law of thermody-
namics [13, 14]. For the Lovelock gravity, the interpreta-
tion of gravitational field equation as a thermodynamic
identity was proposed in [15].
Alternatively, the mass formulae (1) can be written as
the so-called unified first law ∇aM = AΨa + W∇aV
[16, 17], where W = (ρ − p)/2 and Ψa = T ba r̃,b +
Wr̃,a. Projecting the unified first law along the direc-
tion tangent to the AH (or trapping horizon in Hay-
ward’s terminology), the first law of thermodynamics
dM = TdS + WdV can be derived, where the hori-
zon temperature and entropy are given, respectively, by
T = ✷r̃/(4π) and S = A/(4G). Based on this result,
the connection between the Friedmann equation and the
first law of thermodynamics of the AH with the work
term was widely discussed for Einstein gravity, Love-
lock’s gravity, the scalar-tensor theory of gravity, the
nonlinear theory of gravity, and the braneworld scenario
[18, 19, 20, 21, 22, 23].
This connection between the Friedmann equation and
the first law of thermodynamics of the AH suggests the
unique role of the AH in thermodynamics of cosmology.
This may be used to probe the property of dark energy
[10, 24]. For example, if we assume that the temperature
of the dark components is T = bTA, then use the relation
T = (ρ+p)/s = (ρ+p)a3/σ, we find that the total energy
density of the dark components is given by
ρ = ρΛ + ρ0
, (5)
where ρ0 = σ
2b2Ga−60 /(6π), ρΛ = 3Λ/(8πG) is the en-
ergy density of the cosmological constant, σ is the con-
stant comoving entropy density, and s is the physical en-
tropy density. The right-hand side of the above equation
contains three different terms, which correspond to, re-
spectively, the cosmological constant, the stiff fluid, and
the pressureless matter. However, the coefficients of these
terms are not all independent. In fact, the current obser-
vational constraints tell us that the stiff fluid is negligibly
small, for which we must assume ρ0 ≪ 1. This in turn im-
plies that the pressureless matter given by the last term
is also negligibly small. So the pressureless matter in
the last term cannot account for dark matter. In other
words, the dark matter must not be in equilibrium with
the AH.
For the BD theory [25]
L = −
φR + ωgµν
∂µφ∂νφ
, (6)
the BD scalar φ plays the role of the gravitational con-
stant. The MS mass is [26]
M = φr̃(1− gabr̃,ar̃,b)/2. (7)
At the AH, M = φr̃A/2. The horizon entropy is SA =
πr̃2Aφ, so we get
TAdSA = r̃Adφ/2 + φdr̃A. (8)
On the other hand, we have
− dE = −ka∇aMdt = −r̃Adφ/2 + φdr̃A. (9)
Comparing Eqs. (8) with (9), we find that the equilibrium
thermodynamics −dE = TAdSA fails to hold for the BD
theory. Similarly, it can be shown that −dE = TAdSA
does not hold in the nonlinear and scalar-tensor theories
of gravity. It is exactly because of this that it was argued
nonequilibrium treatment might be needed.
As mentioned above, the mass, temperature and en-
tropy of the AH are all geometrical quantities, and the
first law of thermodynamics of the AH can be regarded
as a geometric relation. Now, the important question is
whether a mass function exists that serves as the bridge
between the Friedmann equation and the first law of ther-
modynamics of the AH without nonequilibrium correc-
tion. In the following, we show that the answer is affir-
mative. It has exactly the dimension of energy, and is
equal to the MS mass at the AH. To distinguish it with
the MS mass, we call it the masslike function.
To show our above claim, let us write the a − b com-
ponents of the Einstein equation as
M,a = −4πr̃2(T ba − δ
aT )r̃,b + r̃,a, (10)
where the mass-like function M is defined as
(1 + gabr̃,ar̃,b). (11)
At the AH, gabr̃,ar̃,b = 0 and the masslike function
M = r̃A/2G, which is equal to the MS mass. For
the Robertson-Walker metric we have gtt = −1, grr =
a2/(1 − kr2) and r̃ = ar. Then, the mass formulas (10)
yield the Friedmann equations
ρ, (12)
(ρ+ 3p). (13)
Combining Eqs. (12) and (13), we can derive the energy
conservation law ρ̇ + 3H(ρ + p) = 0. Thus, the mass
formulas (10) give rise to the full set of the cosmological
equations.
At the AH, the masslike function M = 4πr̃3Aρ/3, which
is the total energy inside the AH. The energy flow is
dE = ka∇aMdt = d(r̃A)/G = TAdSA. (14)
On the other hand, using the mass formulas (10), we get
the energy flow through the AH
dE = ka∇aMdt = −4πr̃2T ba r̃,bk
= 4πr̃3AH(ρ+ p)dt. (15)
Therefore, the Friedmann equation gives rise to the first
law of thermodynamics dE = TAdSA of the AH. While
this result is the same as that obtained by using the MS
mass, we show below that the equilibrium thermodynam-
ics can be derived for BD and nonlinear gravities by using
our newly defined masslike function, although it cannot
be done by using the MS mass, as shown above.
For the BD theory, the mass-like function is defined as
M ≡ φr̃(1 + gabr̃,ar̃,b)/2. (16)
At the AH, it reduces to the MS mass, M = M = φr̃A/2.
The a− b components of the gravitational field equation
become
M,a =− 4πr̃2(T ba − δ
aT )r̃,b + 2πr̃
+ (φr̃),a
ω + 2
r̃2φ,aφ,br̃
r̃2r̃,aφ,bφ
;b − r̃r̃,ar̃,bφ;b
r̃2φ;abr̃
r̃2✷r̃φ,a −
φ,a✷φ.
Applying to the Robertson-Walker metric, the above
equation gives the Friedmann equations
, (18)
(ρ+ 3p)−
. (19)
The mass formulas (17) or Eqs. (18) and (19) are not
sufficient to describe the full dynamics of the BD cosmol-
ogy. In the BD cosmology, we also need the equation of
motion of the BD scalar field φ in addition to Eqs. (18)
and (19), which is given by
φ̈+ 3Hφ̇ =
3 + 2ω
(ρ− 3p). (20)
From the definition of the masslike function (16), at the
AH we find
dE = M,ak
adt = r̃Adφ/2 + φdr̃A = TAdSA, (21)
where the entropy now is SA = πr̃
Aφ. Using the mass
formulas (17), we get the energy flow through the AH
3 + 2ω
r̃3AH [(ω + 2)ρ+ ωp] +
r̃3AH
− 2r̃3AH
r̃Aφ̇,
where we used Eq. (20) in deriving the above equation.
From Eqs. (18)-(20), the right-hand side of Eq. (22) can
be written as 1
r̃Aφ̇+ φ ˙̃rA. Therefore, we see that in BD
theory, the first law of thermodynamics of the AH dE =
TAdSA can be derived from the Friedmann equation.
The thermodynamic prescription can be easily ex-
tended to general scalar-tensor theory of gravity with the
Lagrangian
L = f(φ)R − gµν∂µφ∂νφ/2− V (φ). (23)
In this case, f(φ) plays the role of the gravitational con-
stant, so now we can define the mass-like function as
M ≡ f(φ)r̃
1 + gabr̃,ar̃,b
/2, (24)
and the horizon entropy as SA = πr̃
Af(φ). Then, using
these definitions, we can show that dE = M,ak
adt =
TAdSA.
For the nonlinear theory of gravity f(R), we can define
the masslike function as
f ′(R)r̃
1 + gabr̃,ar̃,b
, (25)
and the horizon entropy SA = πr̃
′(R), where f ′(R) =
df/dR. Again, it is easy to show that dE = M,ak
adt =
TAdSA. Therefore, the thermodynamics of the AH holds
for both the general scalar-tensor theory of gravity and
the nonlinear theory of gravity.
Now we show how to derive the first law of ther-
modynamics of the AH from the Friedmann equation
in the Lovelock gravity. The Lovelock Lagrangian is
n=0 cnLn [27], where
Ln = 2
µ1ν1···µnνn
α1β1···αnβn
Rα1β1µ1ν1 · · ·R
Using the Robertson-Walker metric, we obtain the Fried-
mann equations in N + 1 dimensional space-time
N(N − 1)
ρ, (26)
)i−1 (
N − 1
(ρ+ p),
where ĉ0 = c0/[N(N − 1)], ĉ1 = 1 and ĉi = ci
j=3(N +
1−j) for i > 1. The masslike function can now be defined
N(N − 1)ΩN r̃N
2r̃−2i −
= ΩN r̃
A ρ, (28)
where ΩN is the volume of unit N -dimensional sphere
and the last equality is evaluated at the AH. Note that
although the geometric form is different, the masslike
function at the AH has the same value as that in Ein-
stein theory of gravity, which is the total energy inside
the AH. The entropy of the AH is
i(N − 1)
N − 2i+ 1
ĉir̃
N+1−2i
A . (29)
From Eqs. (28) and (29), we can easily check that dE =
adt = TAdSA holds with the horizon temperature
TA = 1/(2πr̃A). Using the Friedmann Eqs. (26) and
(27), we find the energy flow through the AH is dE =
NΩNHr̃
A (ρ+p), which is the same as that in Einstein’s
gravity.
By properly defining the masslike function in each the-
ory of gravity, we find that the corresponding Friedmann
equations can be written in the form dE = TAdSA of
the first law of thermodynamics at the AH. In other
words, the thermodynamic description of the gravita-
tional dynamics is manifest through the mass formulas.
Therefore, the gravitational dynamics can be considered
as the thermodynamic identity dE = TAdSA. This is
true for a variety of theories of gravity, including the
Einstein, Lovelock, nonlinear, and scalar-tensor theories.
This non-trivial connection between the thermodynamics
of the AH and the Friedmann equation may represent a
generic connection, and it suggests the unique role that
the AH can play in the thermodynamics of cosmology.
Such a thermodynamic description of the AH can also be
used to probe other physical systems and properties, such
as the nature of dark energy and the thermodynamics of
black holes in each of these theories.
Finally, we would like to note that, although the newly
defined masslike function reduces to the MS mass at the
AH, the corresponding energy flows passing through the
horizon are different. This explains why our masslike
function gives rise to the first law of thermodynamics
in various theories of gravity, while the MS mass does
not. Because of the masslike function, the energy mo-
mentum tensor includes the contribution of gravitational
fields such as BD scalars, or curvature scalars in non-
linear theory of gravity, in addition to the matter fields.
This treatment allows a reinterpretation of the nonequi-
librium correction introduced in [7]. The studies of other
properties of the newly-defined masslike function, includ-
ing the physical and geometrical difference between the
MS mass and it are important and should be reported
somewhere else.
Y.G. Gong is supported by NNSFC under Grants
No. 10447008 and 10605042, CMEC under Grant No.
KJ060502, and SRF for ROCS, State Education Min-
istry. A. Wang’s work was partially supported by a VPR
fund from Baylor University.
∗ gongyg@cqupt.edu.cn
† anzhong˙wang@baylor.edu
[1] J.M. Bardeen, B. Carter and S.W. Hawking, Commun.
Math. Phys. 31, 161 (1973).
[2] S.W. Hawking, Commun. Math. Phys. 43, 199 (1975);
46, 206(E) (1976).
[3] J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
[4] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).
[5] G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15, 2738
(1977).
[6] R.G. Cai and S.P. Kim, J. High Energy Phys. 02, 050
(2005).
[7] C. Eling, R. Guedens and T. Jacobson, Phys. Rev. Lett.
96, 121301 (2006).
[8] C.M. Misner and D.H. Sharp, Phys. Rev 136, B571
(1964).
[9] E. Poisson and W. Israel, Phys. Rev. D 41, 1796 (1990).
[10] Y.G. Gong, B. Wang and A. Wang, J. Cosmol. Astropart.
Phys. 01, 024 (2007).
[11] M. Akbar and R.G. Cai, Phys. Lett. B 635, 7 (2006).
[12] X.-H. Ge, Phys. Lett. B 651, 49 (2007).
[13] T. Padmanabhan, Class. Quantum Grav. 19, 5387
(2002).
[14] D. Kothawala, S. Sarkar and T. Padmanabhan,
gr-qc/0701002.
[15] A. Paranjape, S. Sarkar and T. Padmanabhan, Phys.
Rev. D 74, 104015 (2006).
[16] S.A. Hayward, Class. Quantum Grav. 15, 3147 (1998).
[17] S. A. Hayward, S. Mukohyama and M.C. Ashworth,
Phys. Lett. A 256, 347 (1999).
[18] M. Akbar and R.G. Cai, Phys. Rev. D 75, 084003 (2007).
[19] R.G. Cai and L.M. Cao, Phys. Rev. D 75, 064008 (2007).
[20] R.G. Cai and L.M. Cao, Nucl. Phys. B 785, 135 (2007).
[21] M. Akbar and R.G. Cai, Phys. Lett. B 648, 243 (2007).
[22] A. Sheykhi, B. Wang and R.G. Cai, Nucl. Phys. B 779,
1 (2007).
[23] A. Sheykhi, B. Wang and R.G. Cai, Phys. Rev. D 76,
023515 (2007).
[24] Y.G. Gong, B. Wang and A. Wang, Phys. Rev. D 75,
mailto:gongyg@cqupt.edu.cn
mailto:anzhong_wang@baylor.edu
http://arxiv.org/abs/gr-qc/0701002
123516 (2007).
[25] C. Brans and R.H. Dicke, Phys. Rev. 124, 925 (1961).
[26] N. Sakai and J.D. Barrow, Class. Quantum Grav. 18,
4717 (2001).
[27] D. Lovelock, J. Math. Phys. 12, 498 (1971).
