Depressing de Sitter in the Frozen Future
ABSTRACT In this paper we focus on the gravitational thermodynamics of the far future.
Cosmological observations suggest that most matter will be diluted away by the
cosmological expansion, with the rest collapsing into supermassive black holes.
The likely future state of our local universe is a supermassive black hole
slowly evaporating in an empty universe dominated by a positive cosmological
constant. We describe some overlooked features of how the cosmological horizon
responds to the black hole evaporation. The presence of a black hole depresses
the entropy of the cosmological horizon by an amount proportional to the
geometric mean of the entropies of the black hole and cosmological horizons. As
the black hole evaporates and loses its mass in the process, the total entropy
increases obeying the second law of thermodynamics. The entropy is produced by
the heat from the black hole flowing across the extremely cold cosmological
horizon. Once the evaporation is complete, the universe becomes empty de Sitter
space that (in the presence of a true cosmological constant) is the maximum
entropy thermodynamic equilibrium state. We propose that flat Minkowski space
is an improper limit of this process which obscures the thermodynamics. The
cosmological constant should be regarded not only as an energy scale, but also
as a scale for the maximum entropy of a universe. In this context, flat
Minkowski space is indistinguishable from de Sitter with extremely small
cosmological constant, yielding a divergent entropy. This introduces an
unregulated infinity in black hole thermodynamics calculations, giving possibly
arXiv:1201.1298v1 [gr-qc] 5 Jan 2012
Depressing de Sitter in the Frozen Future
Andrew P. Lundgren,1, ∗Mihai Bondarescu,2, †and Ruxandra Bondarescu3, ‡
1Albert-Einstein-Institut, Callinstr. 38, 30167 Hannover, Germany
2University of Mississippi, Oxford, MS, USA
3Institute for Theoretical Physics, University of Zurich, Switzerland
In this paper we focus on the gravitational thermodynamics of the far future. Cosmological observations
suggest that most matter will be diluted away by the cosmological expansion, with the rest collapsing into
supermassive black holes. The likely future state of our local universe is a supermassive black hole slowly
evaporating in an empty universe dominated by a positive cosmological constant. We describe some over-
looked features of how the cosmological horizon responds to the black hole evaporation. The presence of
a black hole depresses the entropy of the cosmological horizon by an amount proportional to the geomet-
ric mean of the entropies of the black hole and cosmological horizons. As the black hole evaporates and
loses its mass in the process, the total entropy increases obeying the second law of thermodynamics. The
entropy is produced by the heat from the black hole flowing across the extremely cold cosmological hori-
zon. Once the evaporation is complete, the universe becomes empty de Sitter space that (in the presence
of a true cosmological constant) is the maximum entropy thermodynamic equilibrium state. We propose
that flat Minkowski space is an improper limit of this process which obscures the thermodynamics. The
cosmological constant should be regarded not only as an energy scale, but also as a scale for the maximum
entropy of a universe. In this context, flat Minkowski space is indistinguishable from de Sitter with ex-
tremely small cosmological constant, yielding a divergent entropy. This introduces an unregulated infinity
in black hole thermodynamics calculations, giving possibly misleading results.
Introduction. Our universe appears to have a small posi-
tivecosmologicalconstant[1–3]. Althoughthis hashadlit-
tle effect on the past evolution of the universe, it will com-
pletely dominate the cosmological future . Matter will
stream outwards through the cosmological horizon, leav-
ing behind only the gravitationally bound part of our local
universe. In due time, nothingwill be left but supermassive
black holes, slowly evaporating by Hawking radiation .
The future will asymptote to de Sitter space, the spacetime
of a universe containing nothing but a positive cosmologi-
This bleak future is actually very interesting from the
perspective of gravitational thermodynamics.
mological horizon absorbs the matter and radiation that
streams through it, and grows in response. The entropy
of the horizon is given by its area [6, 7] , so the cosmologi-
cal horizon is producing entropy by absorbing the contents
of the universe. If the cosmological constant is a true con-
stant, then de Sitter spacetime is the final equilibrium state
of our universe. The emptying of the universe is the ap-
proach to this equilibrium state.
We will model this process by the slow evaporation of a
single non-spinningsupermassiveblackin a deSitter back-
ground. We will present some interesting features that we
believe have been overlooked. In particular, we argue that
de Sitter is a better background than the flat Minkowski
spacetime for understanding gravitational thermodynam-
ics. Minkowski space is typically considered to have zero
entropy, but we take the alternative view that Minkowski
is a limit of de Sitter as the cosmological constant goes to
zero. In this limit, the entropy of the spacetime is infinite,
which introduces an unregulated infinity into the calcula-
In the de Sitter background, the gravitational effect of
The fractional changein the area is small becausethe black
hole horizon is small compared to the cosmological hori-
zon. But the change in the area, and therefore the entropy,
of the cosmological horizon is very large compared to the
entropy of the black hole. Also, the Hawking temperature
of the cosmological horizon is very small even compared
to the black hole temperature, leading to the large produc-
tion of entropy as heat flows across it. The presence of a
black hole depresses the entropy of de Sitter, but the en-
tropy rises as the black hole evaporates into the freezing
cosmological horizon surrounding it. The thermodynam-
ics of black hole evaporationis less clear in flat Minkowski
spacetime, where the decrease in the gravitational entropy
of the black hole forever remains in the outgoing Hawking
radiation. In de Sitter, the increase in entropy can be calcu-
lated directly from the horizon areas, or seen as the result
of heat flowing across the temperature difference between
the two horizons.
Temperature and Entropy in de Sitter. The existence of a
cosmological constant is the simplest explanation for the
accelerating expansion of the universe. In the past, the
density of radiation and matter have dominated that of the
cosmological constant, but the expansion of the universe
dilutes the densities of radiation and matter while not af-
fecting the cosmological constant. The cosmological con-
stant will eventually completely dominate the density of
the universe on cosmological scales, leading to an expo-
nential expansion and a cold, nearly empty universe. The
cosmological metric describing an empty universe with a
cosmological constant is
ds2= −dt2+a2(t)?dr2+ r2(dθ2+ sin2θ dφ2)?
with scale factor a(t) = eHtand H =
the Hubble parameter and Λ is the cosmological constant.
The metric can also be written in static coordinates as
?Λ/3, where H is
+ r2(dθ2+ sin2θ dφ2) .
We will referto r as the radius,althoughit is moreproperly
referred to as the areal radius; it does not measure distance
from the origin, but the area of a sphereconcentric with the
origin is 4πr2.
In the static slicing, it is clear that there is a horizon at
?3/Λ. This has a thermodynamic interpretation; a
horizon has entropy given by one-quarter the horizon area.
For the de Sitter cosmological horizon,
Likewise, the temperature is related to the surface gravity
of the horizon. There is a normalization issue that we will
consider later, but for now we quote the standard result 
Throughout this paper, we will use units where G = ¯ h =
c = kB= 1, but we will give an examplein physicalunits.
The Flat Space Limit. Consider the limit as Λ goes to
zero. The metric becomes the Minkowski metric of flat
spacetime. On physical grounds, flat spacetime is nearly
indistinguishable from one with a very small cosmologi-
cal constant. The entropy and temperature of Minkowski
spacetime should then be those of de Sitter in the limit of
very large horizon size. In this limit, the temperature is
zero, which is not surprising. However, in the same limit
the entropy diverges. This is very different from the typical
assumption that because Minkowski space does not have a
horizon, it has zero entropy.
It is possiblethatinsteadofa truecosmologicalconstant,
tential energyhas thesame effectasΛ. Whenthe field rolls
to a lower potential energy, the effective Λ decreases and
roll down their potentials but not up corresponds to the ex-
pectation that entropy can increase but not decrease. Cos-
mic acceleration in this case is attributed to the fact that the
universe has not reached its true vacuum state for dynam-
ical reasons. If the true cosmological constant is zero, the
end state can be Minkowski space.
If Minkowski space has divergent entropy, then it is not
a good background in which to study black hole thermo-
dynamics. Such calculations would have an unregulated
infinity in them, giving possibly misleading results. We
will now show that the evaporation of a Schwarzschild
black hole in de Sitter space has a satisfying thermody-
namic interpretation and is quite different from the picture
in Minkowski space.
Black Hole Evaporation.
rotating black hole in a universe with a positive cosmolog-
ical constant [11, 12]. The de Sitter-Schwarzschild metric
We now consider a non-
ds2= −f(r)dt2+f(r)−1dr2+r2(dθ2+sin2θdφ2) (5)
f(r) = 1 −2M
For convenience, we use α =
cosmological constant; keep in mind that α grows with de-
creasing Λ. Many of the symmetries of de Sitter spacetime
are broken by the presence of the black hole, but there is
still a time symmetry which is explicit in the static form of
In addition to the cosmological horizon, there is now a
black hole horizon. Each is a sphere where f(r) = 0.
There is a third, negative root which we discard. The hori-
zons are roughly at rBH≈ 2M and rC≈ α. Making the
approximation that M ≪ α, the radii can be expanded as
rBH≈ 2M + 8M3
rC≈ α − M −3
We can now find the entropies associated with each hori-
zon; each is one-quarter the horizon area. These are
?3/Λ to parameterize the
α2+ ... .
SBH≈ π(4M2+ 32M4
SC≈ π(α2− 2αM − 2M2− 5M3
Stot≈ π(α2− 2αM + 2M2− 5M3
The important feature is that the total entropy Stot in-
creases as the mass of the black hole decreases, because
the −2παM term in the cosmological horizon entropy is
muchlargerthanSBH. We knowthat ablackholewill emit
Hawking radiation and lose mass in the process, eventually
evaporating away to nothing. We see that the total entropy
will increase in this process, as required by the second law
of thermodynamics. We will now consider the thermody-
namics in more detail.
The temperatureof each horizon is T =
gravity κ is defined by
α2) + ...
α) + ... (10)
α) + ... (11)
2π. The surface
kα∇µkβ= κ kβ,
where k is the timelike Killing vector that expresses the
time-invariance of the metric. We write
k = N∂
where N is a normalization constant. The static coordi-
nates we are using have problems at the horizons, so we
must do the calculation in another coordinate system such
as Painlev´ e-Gullstrand; the result is
κ = Nf′(r)
The normalization is fixed by requiring that kαkα= −1
at the observer who measures the temperature [11, 13], so
The normalization accounts for the gravitational redshift
when comparing the temperatures measured by observers
at different radii. In a Minkowski background, N = 1 at
asymptotic infinity. The temperatures are
TBH(r) = −κBH
We must reverse the sign of the surface gravity on the cos-
mological horizon since it is an outer horizon rather than
an inner horizon. Note that the surface gravity is evaluated
at the relevanthorizon,while the redshift is evaluatedat the
radius of interest. These are the temperatures as measured
by static observers, i.e. those that flow along the Killing
vector kµso they remain at a constant value of r.
The black hole always has a higher temperature than the
of M =
but the two horizons merge. We clearly see that this is a
non-equilibrium situation because there are two non-equal
temperatures. Heat will flow from hot to cold, causing the
black hole to evaporate and producing entropy.
We separatelyconsiderthe changesin entropyof the two
horizons. We use the thermodynamic expression
1 − 2M
1 − 16M2
3√3α, at which point the temperatures are equal
relating the change in entropy δS to the heat absorbed or
emitted by the horizon δQ. We model the evaporation of
the black hole as a small change in M, with α of course
remaining fixed. The luminosity of the black hole is small
enough that this quasi-static approximation is reasonable.
For a small change δM in the black hole mass, the heat
where (−gtt)−1/2is a redshift factor to give the heat mea-
sured by an observer at radius r.
Substituting Eq. (19) into Eq. (18) gives
It is simple to evaluate the left and right-hand sides of
Eq. (20) for either the black hole or cosmological hori-
zon, although we omit the algebra. For each horizon, the
changein entropy is exactly what is expectedfrom the heat
crossing the horizon, δS = TδQ. This holds even without
the approximation that M ≪ α, and also holds regardless
of the value of r where it is evaluated, since the redshift
factors (−gtt)−1/2in the heat and the temperature cancel.
The black hole horizon loses a small amount of entropy by
emitting some heat at a relatively high temperature. The
cosmological horizon gains a much larger amount of en-
tropy by absorbing the heat at a lower temperature. It is
significant that the normalization constant is determined
using only one horizon, which then correctly predicts the
entropy production at the other horizon.
We clearly see that black hole evaporation proceeds in
the de Sitter background because it leads to an increase in
entropy. Heat flows from hot to cold, and the entropy pro-
duction at each horizon is exactly what is expected from
Eq. (18). This is clearer than the corresponding situation
in Minkowski space, where the entropy lost by the black
hole is foreverleft in the form of Hawking radiation. There
is of course an intermediate state even in de Sitter where
the Hawking radiation is traveling outward toward the cos-
mological horizon, but this is a short time compared to the
evaporation timescale of a supermassive black hole in our
As an example, we can imagine that the Local Super-
cluster collapses to form a supermassive black hole .
The mass of this black hole will be roughly 1015M⊙,
giving a Schwarzschild radius of 300 light years. The cos-
mological horizon has a radius of 18 billion light years.
The entropies are measured in units of Planck area APl=
G¯ hc−3≈ 3 × 10−102square light years. The temper-
ature is converted from an inverse distance with a factor
¯ hc ≈ 2 × 10−23eV light years. In these units, the black
hole horizon has an entropy of 10107and a temperature1of
10−27eV, while the cosmological horizon has an entropy
10122and a temperature of 10−34eV. The entropy pro-
duced by the evaporation of the black hole is 10114, which
is clearly much larger than the entropy of the black hole
horizon itself (it is nearly the geometric mean of the two
Weyl Curvature Hypothesis. We can now make con-
tact with the Weyl Curvature Hypothesis of Penrose ,
which states that the difference in entropy between the ini-
tial and final states of the universe is related to the growth
of the Weyl curvature. The beginning of the universe is
1These are the temperatures without the g−1/2
redshift factor included.
likely to have very small Weyl curvature but it grows due
to the productionof singularities in the late universe. How-
ever, the de Sitter metric has zero Weyl curvature but ex-
tremely large entropy, so it seems there cannot be a direct
relation between Weyl curvature and gravitational entropy.
We conjecture that there is a more subtle relation between
the Weyl curvature and the gravitational entropy. The Ein-
stein equation relates the Ricci curvature to the cosmolog-
ical constant and local stress-energy tensor Tµνby
This does not determine the full Riemann curvature of the
spacetime; the Weyl tensor is free to evolve except for
the Bianchi identities, which effectively provide boundary
conditionsrelating Weyl to Ricci and therefore to Tµν. The
Ricci curvature then roughly plays the role of macroscopic
observables,while the entropycomesfrom coarse-graining
over states of the Weyl tensor. For instance, gravitational
waves are modes of the Weyl tensor. The effect of the
cosmological constant is to not allow gravitational wave
modes with very long wavelengths. When a black hole
evaporates, the cosmological horizon grows, and it could
be that this allows more gravitational wave modes to exist
and so increases the number of accessible microstates in
the Weyl tensor. This idea needs further development.
Discussion and Conclusions.
the cosmological constant, the maximum entropy of the
Schwarzschild-de Sitter solution is obtained in empty de
Sitter space. The presence of a black hole shrinks the cos-
mological horizon and depresses the total entropy. Both
the black hole and cosmological horizons emit Hawking
radiation, but the black hole is hotter so the net flow of
energy is outward from the black hole towards the cosmo-
logical horizon. The radiation passes through the cosmo-
logical horizon, carrying a small amount of entropy away
from the black hole. When the radiation crosses the cos-
mological horizon, the entropy in the radiation disappears
from the perspective of an observer between the two hori-
zons. However, by Eq. (18), it is clear that this heat flow-
ing across the much colder cosmological horizon produces
much more entropy than was lost by the black hole.
An interesting feature is that the black hole can be re-
placed by a spherical star with the same mass. The metric
external to the star will be the same, so the cosmological
horizon will behave exactly the same. There will be no
black hole horizon, but if the star loses mass in the form
of radiation it will produce entropy as the radiation flows
across the cosmological horizon. This suggests that any
matter within the horizon depresses the entropy relative to
empty de Sitter. Also, the collapse of a massive star to a
black hole does increase the total entropy, since a black
hole horizon forms (and has larger entropy than the star)
but the cosmological horizon is not affected.
Black hole evaporation is a non-equilibrium process, so
it produces entropy; the end of the evaporation should cor-
respond to maximum entropy and to the equilibrium state.
2gµνT) + Λgµν.
For a fixed value of
There is evidence [16, 17] that empty de Sitter space has
the maximum entropy of spacetimes with a fixed cosmo-
logical constant. The equilibrium state should in a sense be
the most generic state. The equilibrium state of a gas in a
box is one where the gas is uniformly distributed and fea-
tureless. de Sitter space is maximally symmetric, meaning
there is a full set of ten symmetries expressed by Killing
vectors (one time and three space translations, three rota-
tions, and three boosts). Minkowski space possesses the
same symmetries but does not have the preferred scale that
thecosmologicalconstantprovides,so it is evenmoresym-
metric. The cosmological constant could equally well be
regarded as an energy scale or as setting the maximum to-
tal entropy of our universe.
If the true cosmological constant is zero, then the maxi-
mum entropy state of the universe is Minkowski. The cos-
mological constant behavior could then be mimicked by a
quintessence scalar field [9, 10] with an ultralight mass of
∼ 10−33eV that slowly rolls down its potential. Another
ultralight scalar field of mass ∼ 10−23eV is a possible
alreadyreachedthe bottom of its potential and formeddark
matter particles that Bose-condensed into the halos around
galaxies. Oncethe darkenergyscalar field reachesthe final
minima of its potential, it could form new particles as well
converting a small fraction of the cosmological constant
entropy into entropy for the particles. The temperature and
entropy of both the dark matter and the quintessence scalar
field particles would be interesting from a thermodynamics
perspective. In addition, black holes are ineffective dark
matteraccretorsdueto thelowviscosityofdarkmatter par-
ticles and it is unclear how scalar field entropy would flow
into the cosmological horizon. Modeling such a scenario is
subject to future work.
Acknowledgements. RB is supported by the Dr. Tomalla
Foundation and the Swiss National Foundation. We thank
Badri Krishnan, Alex Nielsen, and Bjoern Schmekel for
useful discussions and Phillipe Jetzer for encouragement
and support. APL thanks Alex Nielsen for pointing out
that the black hole can be replaced by a spherical star. MB
thanks Edward Witten for useful discussions, and APL is
especially grateful to James York Jr. for introducing him to
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