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 by9.
In flat spacetime the quantum (one-loop) thermodynamic and statistical entropies coincide as was shown by Allen  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  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.
Living Rev. Relativity 14, (2011), 8
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