One of the remarkable features of classical gravity is that it can wrap
up regions of spacetime thereby producing surfaces which act as one way
membranes. The classic example is that of Schwarzschild black hole of
mass *M* which has a compact spherical surface of radius *r*
= 2*M* that act as a horizon. Since the horizon can hide
information - and information is deeply connected with entropy - one
would expect a fundamental relationship between gravity and thermodynamics.
[There is extensive literature in this subject and our citation will be
representative rather than exhaustive;
for a text book discussion and earlier references, see
[288];
for a recent review, see
[289].]
As we saw in the last section,
the de Sitter universe also has a horizon which
suggests that de Sitter spacetime will have non trivial thermodynamic
features
[290].

This result can be demonstrated mathematically in many different ways of
which the simplest procedure is based on the relationship between
temperature and the Euclidean extension of the spacetime. To see this
connection, let us recall that the mean value of some dynamical variable
*f* (*q*) in quantum statistical mechanics can be expressed in
the form

(148) |

where
_{E}(*q*) is the stationary state eigen
function of the Hamiltonian with *H*
_{E}
= *E*
_{E},
= (1 /
*T*) is the inverse temperature and
*Z*() is
the partition function.
This expression calculates the mean value < *E*| *f*|
*E* > in a given energy state and then averages over a Boltzmann
distribution of energy states with the weightage *Z*^{-1}
exp(- *E*).
On the other hand, the quantum mechanical kernel
giving the probability amplitude for the system to go from the state
*q* at time *t* = 0 to the state *q'* at time *t* is
given by

(149) |

Comparing (148) and (149) we find that the thermal average in (148) can be obtained by

(150) |

in which we have done the following: (i) The time coordinate has been
analytically continued to imaginary values with *it* =
. (ii) The system is assumed
to exhibit periodicity in the imaginary time
with period
in the sense
that the state variable *q* has the same
values at = 0 and at
=
. These
considerations continue to hold even
for a field theory with *q* denoting the field configuration at a
given time. If the system, in particular the Greens functions describing
the dynamics, are periodic with a period *p* in imaginary time,
then one can attribute a temperature *T* = (1/*p*) to the
system. It may be noted that the partition function
*Z*()
can also be expressed in the form

(151) |

The first equality is the standard definition for
*Z*();
the second equality follows from (149) and the normalization of
_{E}(*q*); the last equality
arises from the standard path integral expression for the kernel in the
Euclidean sector (with *A*_{E} being the Euclidean action)
and imposing the periodic boundary conditions. (It is assumed that the
path integral measure
*q* includes an
integration over *q*.) We shall have occasion to use this result
later. Equations (150) and (151) represent the relation between the
periodicity in Euclidean time and temperature.

Spacetimes with horizons possess a natural analytic continuation from
Minkowski signature to the Euclidean signature with
*t*
= *it*. If the metric
is periodic in , then one
can associate a natural notion of a temperature to such spacetimes.
For example, the de Sitter manifold with the metric (134)
can be continued to imaginary time arriving at the metric

(152) |

which is clearly periodic in
with the period
(2 / *H*). [The original
metric was a 4-hyperboloid in the 5-dimensional space while equation (152)
represents a 4-sphere in the 5-dimensional space.]
It follows that de Sitter spacetime has a natural notion of temperature
*T* = (*H* / 2)
associated with it.

It is instructive to see how this periodicity arises in the static form of the metric in (141). Consider a metric of the form

(153) |

where *dL*_{}^{2} denotes the transverse 2-dimensional
metric and *f* (*r*) has a simple zero at *r* =
*r*_{H}. Near *r* = *r*_{H}, we can
expand *f* (*r*) in a Taylor series and obtain
*f* (*r*)
*B*(*r* - *r*_{H}) where
*B*
*f'*(*r*_{H}). The structure of the metric in (153)
shows that there is a horizon at *r* =
*r*_{H}. Further, since the general relativistic metric
reduces to
*g*_{00}
(1 +
2_{N})
in the Newtonian limit, where
_{N} is
the Newtonian gravitational potential, the quantity

(154) |

can be interpreted as the gravitational attraction on the surface of the
horizon - usually called the surface gravity. Using the form
*f* (*r*)
2(*r* -
*r*_{H}) near the horizon and shifting to the coordinate
[2^{-1}(*r*
- *r*_{H})]^{1/2}
the metric near the horizon becomes

(155) |

The Euclidean continuation
*t*
= *it* now leads to
the metric

(156) |

which is essentially the metric in the polar coordinates in
the -
plane. For this
metric to be well defined near the origin,
should behave like an angular coordinate
with periodicity
2. Therefore,
we require all well defined physical quantities defined in this spacetime
to have a periodicity in
with the period
(2 /
||). Thus, all metrics of
the form in (153) with a simple zero for *f* (*r*) leads to a
horizon with temperature
*T* = || /
2 =
| *f'*(*r*_{H}) |
/ 4. In the case of de Sitter
spacetime, this gives
*T* = (*H* / 2); for
the Schwarzschild metric, the corresponding analysis gives the well known
temperature
*T* = (1 / 8 *M*)
where *M* is the mass of the black-hole.