Consider a gravitationally coupled system (gravity plus quantum matter fields) at some arbitrary
temperature . A standard way to describe a thermal state of a field system is to use an
Euclidean path integral over all fields in question defined on manifold with periodicity
along the
time-like Killing vector. Suppose that it is a priori known that the system includes a black hole. Thus, there
exists a surface
(horizon), which is a fixed point of the isometry generated by the killing vector. This
imposes an extra condition on the possible class of metrics in the path integral. The other condition to be
imposed on metrics in the path integral is the asymptotic behavior at infinity: provided the
mass
and the electric charge
of the gravitational configuration are fixed, one has to
specify the fall-off of the metrics for large values of
. Thus, the Euclidean path integral is
i) possesses an abelian isometry with respect to the Killing vector
;
ii) there exists a surface (horizon) where the Killing vector
becomes null;
iii) asymptotic fall-off of metric at large values of radial coordinate
is fixed by the mass
and
electric charge
of the configuration.
Since the inverse temperature and mass
in the path integral are two independent parameters,
the path integral (158
) is mostly over metrics, which have a conical singularity at the surface
. The
integration in Eq. (158
) can be done in two steps. First, one computes the integral over matter
fields
on the background of a metric, which satisfies conditions i), ii) and iii). The result of
this integration is the quantity (15
) used in the computation of the entanglement entropy,
The thermodynamic entropy is defined by the total response of the free energy to a
small change in temperature,
Thus, in order to compute the thermodynamic entropy, one may proceed in two steps. First, for a
generic metric, which satisfies the conditions i), ii) and iii) compute the off-shell entropy using the replica
method, i.e., by introducing a small conical singularity at the horizon. This computation is done by taking a
partial derivative with respect to . Second, consider this off-shell entropy for an equilibrium
configuration, which solves Eq. (161
). Since for the classical gravitational action (112
) one finds
(116
) and for the quantum effective action one obtains the entanglement
entropy
, the relation between the entanglement entropy and thermodynamic
entropy is given by
In flat spacetime the quantum (one-loop) thermodynamic and statistical entropies coincide
as was shown by Allen [2] due to the fact that the corresponding partition functions differ
by terms proportional to
. In the presence of black holes the exact relation between the
two entropies has been a subject of some debate (see, for example, [86, 199
]). However, the
analysis made in [104
] shows that in the presence of black hole the Euclidean and statistical free
energies coincide, provided an appropriate method of regularization is used to regularize both
quantities.
http://www.livingreviews.org/lrr-2011-8 |
Living Rev. Relativity 14, (2011), 8
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