**10.4. Conceptual issues in de Sitter thermodynamics**

The analysis in the last few sections
was based on a strictly *static* 4-dimensional spacetime. The
black hole metric, for example, corresponds to an eternal black hole
and the vacuum state which we constructed in
section 10.2 corresponds to the
Hartle-Hawking vacuum
[317]
of the Schwarzschild spacetime, describing a black hole in thermal
equilibrium. There is no net radiation flowing to infinity and the
entropy and temperature obtained in the previous sections were based on
equilibrium considerations.

As we said before, there are two different ways of defining the
entropy. In statistical mechanics, the entropy *S*(*E*) is
related to the degrees of freedom [or phase volume] *g*(*E*) by
*S*(*E*) = ln *g*(*E*). Maximization of the phase
volume for systems which can exchange energy
will then lead to equality of the quantity
*T*(*E*)
( *S* /
*E*)^{-1} for the systems. It is usual to identify this
variable as the thermodynamic temperature. The analysis of BH
temperature based on Hartle-Hawking state is analogous to this
approach.

In classical thermodynamics, on the other hand, it is the *change
in* the entropy which can be operationally defined via
*dS* = *dE*/*T*(*E*). Integrating this equation will
lead to the function *S*(*E*) except for an additive constant
which needs to be determined from additional considerations.
This suggests an alternative point of view regarding thermodynamics of
horizons. The Schwarzschild metric, for example, can be thought of as an
*asymptotic limit* of a metric arising from the collapse of a
body forming a black-hole. While developing the QFT in such a spacetime
containing a collapsing black-hole,
we need not maintain time reversal invariance for the vacuum state and -
in fact - it is more natural to choose a state with purely in-going
modes at early times like the Unruh vacuum state
[318].
The study of QFT in such a spacetime shows that,
at late times, there will exist an outgoing thermal radiation of particles
which is totally independent of the details of the collapse. The
temperature in this case will be
*T*(*M*) = 1/8
*M*, which is the same as the one found in the case of the state of
thermal
equilibrium around an "eternal" black-hole. In the Schwarzschild
spacetime, which is asymptotically flat, it is also possible to
associate an energy *E* = *M* with the black-hole.
Though the calculation was done in a metric with a fixed value of energy
*E* = *M*, it seems reasonable to assume that -
as the energy flows to infinity at late times - the mass of the black hole
will decrease. *If* we make this assumption - that the evaporation
of black hole will lead to a decrease of *M* - *then* one can
integrate the equation *dS* = *dM* / *T*(*M*) to
obtain the entropy of the black-hole to be
*S* = 4
*M*^{2} = (1/4)(*A* /
*L*_{P}^{2}) where *A* =
4(2*M*)^{2}
is the area of the event horizon and *L*_{P} = (*G*
/
*c*^{3})^{1/2} is the Planck length.
^{(2)}
The procedure outlined above is similar in spirit to the approach of
classical thermodynamics rather than statistical mechanics.

Once it is realized that only the asymptotic form of the metric matters,
we can simplify the above analysis by
just choosing a time *asymmetric* vacuum and working with the
asymptotic
form of the metric with the understanding that the asymptotic form arose
due to a time asymmetric process (like gravitational collapse).
In the case of black hole spacetimes this is accomplished - for example
- by choosing the Unruh vacuum
[318].
The question arises as to how our unified approach fares in handling
such a situation which is not time symmetric and the horizon
forms only asymptotically as
*t*
.

There exist analogues for the collapsing black-hole in the case of de
Sitter (and even Rindler)
[293].
The analogue in the case of de Sitter
spacetime will be an FRW universe which behaves like a de Sitter
universe only at late times [like in equation (27); this is indeed the
metric describing our universe if
_{} = 0.7,
_{NR} =
0.3]. Mathematically,
we only need to take *a*(*t*) to be a function which has the
asymptotic form
exp(*Ht*) at late times. Such a spacetime is, in general, time
asymmetric and one can choose a vacuum state at early times in such a
way that a thermal spectrum of particles exists at late times.
Emboldened by the analogy with black-hole spacetimes, one can also
directly construct quantum states (similar to Unruh vacuum of black-hole
spacetimes) which are time asymmetric, even in the exact de Sitter
spacetime, with the understanding that the de Sitter universe came about
at late times through a time asymmetric evolution.

The analogy also works for Rindler spacetime. The coordinate system for
an observer with *time dependent* acceleration will generalize the
standard Rindler spacetime
in a time dependent manner. In particular, one can have an observer who
was inertial (or at rest) at early times and is uniformly accelerating
at late times. In this case an event horizon forms at late times exactly
in analogy with a collapsing black-hole. It is now possible to choose
quantum states which are analogous to
the Unruh vacuum - which will correspond to an inertial vacuum state at
early times and will appear as a thermal state at late times. The study
of different `vacuum' states shows
[293]
that radiative flux exists in the quantum states which are time
asymmetric analogues of the Unruh vacuum state.

A formal analysis of this problem will involve setting up the in and out
vacua of the theory, evolving the modes from *t* =
- to *t* =
+ ,
and computing the Bogoliubov coefficients. It is, however, not necessary
to perform the details of such an analysis because
all the three spacetimes (Schwarzschild, de Sitter and Rindler)
have virtually identical kinematical structure.
In the case of Schwarzschild metric, it is well known that
the thermal spectrum at late times arises because the modes
which reach spatial infinity at late times propagate from near the event
horizon at early times and undergo exponential redshift.
The corresponding result occurs in all the
three spacetimes (and a host of other spacetimes).

Consider the propagation of a wave packet centered around a radial null
ray in a spherically symmetric (or Rindler) spacetime which has the form
in equation (162) or (175).
The trajectory of the null ray which goes from the initial position
*r*_{in}
at *t*_{in} to a final position *r* at *t* is
determined by the equation

(186) |

where the ^{ ... } denotes terms arising from the transverse part
containing *dr*^{2} (if any).
Consider now a ray which was close to the horizon initially so that
(*r*_{in} - *l* ) << *l* and propagates to
a region far away from the horizon at late times. (In a black hole
metric *r* >> *r*_{in} and the propagation will
be outward directed; in the de Sitter metric we will have
*r* << *r*_{in} with rays propagating towards
the origin. ) Since we have *f* (*r*)
0 as *r*
*l*, the
integral will be dominated
by a logarithmic singularity near the horizon and the regular term denoted
by ^{ ... } will not contribute.
[This can be verified directly from (162) or (175).] Then we get

(187) |

As the wave propagates away from the horizon its frequency will be
red-shifted by the factor
(1 /
(*g*_{00})^{1/2}) so that

(188) |

where *K* is an unimportant constant. It is obvious that the
dominant behaviour
of (*t*) will be
exponential for any null geodesic starting
near the horizon and proceeding away since all the transverse
factors will be sub-dominant to the diverging logarithmic
singularity arising from the integral of (1/*f* (*r*)) near
the horizon. Since
(*t*)
exp[ ± *gt*]
and the phase (*t*)
of the wave will be vary with time as
(*t*) =
<(*t*) *dt*
exp[ ± *gt*],
the time dependence of the wave at late times will be

(189) |

where *Q* is some constant.
An observer at a fixed *r* will see the wave to have the time
dependence
exp[*i*(*t*)]
which, of course, is not monochromatic. If this wave is decomposed into
different Fourier components with respect to *t*, then the
amplitude at frequency is
given by the Fourier transform

(190) |

(191) |

The standard expressions for the temperature are reproduced for
Schwarzschild (*g* = (4*M*)^{-1}),
de Sitter (*g* = *H*) and Rindler spacetimes.
This analysis stresses the fact that the origin of thermal spectrum lies
in the Fourier transforming of an exponentially red-shifted spectrum.

But in de Sitter or Rindler spacetimes there is *no*
natural notion of "energy source" analogous to the mass of the black-hole.
The conventional view is to assume that: (1) In the case of black-holes,
one considers the collapse scenario as "physical" and the natural
quantum state is the Unruh vacuum. The notions of evaporation, entropy
etc. then follow in a concrete manner. The eternal black-hole (and the
Hartle-Hawking vacuum state) is taken to be just a mathematical
construct not realized in nature. (2) In the case of Rindler, one may
like to think of a time-symmetric vacuum state as natural and treat the
situation as one of thermal equilibrium. This forbids using quantum
states with outgoing radiation which could make the Minkowski spacetime
radiate energy - which seems unlikely.

The real trouble arises for spacetimes which are asymptotically de Sitter. Does such a spacetime have temperature and entropy like a collapsing black-hole? Does it "evaporate" ? Everyone is comfortable with the idea of associating temperature with the de Sitter spacetime and most people seem to be willing to associate even an entropy. However, the idea of the cosmological constant changing due to evaporation of the de Sitter spacetime seems too radical. Unfortunately, there is no clear mathematical reason for a dichotomous approach as regards a collapsing black-hole and an asymptotically de Sitter spacetime, since: (i) The temperature and entropy for these spacetimes arise in identical manner due to identical mathematical formalism. It will be surprising if one has entropy while the other does not. (ii) Just as collapsing black hole leads to an asymptotic event horizon, a universe which is dominated by cosmological constant at late times will also lead to a horizon. Just as we can mimic the time dependent effects in a collapsing black hole by a time asymmetric quantum state (say, Unruh vacuum), we can mimic the late time behaviour of an asymptotically de Sitter universe by a corresponding time asymmetric quantum state. Both these states will lead to stress tensor expectation values in which there will be a flux of radiation. (iii) The energy source for expansion at early times (say, matter or radiation) is irrelevant just as the collapse details are irrelevant in the case of a black-hole.

If one treats the de Sitter horizon as a `photosphere' with temperature
*T* = (*H* / 2) and
area *A*_{H} =
4 *H*^{-2},
then the radiative luminosity will be
(*dE*/*dt*)
*T*^{4} *A*_{H}
*H*^{2}. If we take *E* = (1/2)*H*^{-1},
this will lead to a decay law
[319]
for the cosmological constant of the form:

(192) |

where *k* is a numerical constant and the second proportionality is
for *t*
. It is interesting that
this naive model leads to a late time cosmological constant which is
independent of the initial value
(_{i}).
Unfortunately, its value is still far too large.
These issues are not analyzed in adequate detail in the literature and
might have important implications for the cosmological constant problem.

^{2} This integration can determine the
entropy only
up to an additive constant. To fix this constant, one can make the
additional assumption that *S* should vanish when *M* =
0. One may think that this assumption is eminently reasonable since the
Schwarzschild metric reduces to the Lorentzian metric when *M*
0. But note
that in the same limit of *M*
0, the
temperature of the black-hole diverges !. Treated as a limit of
Schwarzschild spacetime, normal flat spacetime has infinite - rather
than zero - temperature.
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