4.1 Euclidean path integral and thermodynamic entropy

In 1977, Gibbons and Hawking [119] developed a method based on the Euclidean path integral for studying the thermodynamics of black holes. In this method one obtains what may be called a thermodynamic entropy. One deals with metrics, which satisfy the gravitational field equations, and thus avoid the appearance of metrics with conical singularities. The entanglement entropy, on the other hand, has a well-defined statistical meaning. In ordinary systems, the thermodynamic entropy and the statistical (microscopic) entropies coincide. For black holes the exact relation between the two entropies can be seen from the following reasoning8 [198].

Consider a gravitationally coupled system (gravity plus quantum matter fields) at some arbitrary temperature T = (β )−1. 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 2π β 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 Σ (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 M and the electric charge Q of the gravitational configuration are fixed, one has to specify the fall-off of the metrics for large values of r. Thus, the Euclidean path integral is

∫ ∫ Z (β,M, Q ) = 𝒟g μν 𝒟 ψe−Wgr[g]+Wmat[ψ,g], (158 )
where the integral is taken over β-periodic fields ψ (τ,xi) = ψ(τ + β,x ) and over metrics, which satisfy the following conditions:

i) g μν possesses an abelian isometry with respect to the Killing vector ∂τ;

ii) there exists a surface Σ (horizon) where the Killing vector ∂τ becomes null;

iii) asymptotic fall-off of metric gμν at large values of radial coordinate r is fixed by the mass M and electric charge Q of the configuration.

Since the inverse temperature β and mass M in the path integral are two independent parameters, the path integral (158View Equation) is mostly over metrics, which have a conical singularity at the surface Σ. The integration in Eq. (158View Equation) can be done in two steps. First, one computes the integral over matter fields ψ on the background of a metric, which satisfies conditions i), ii) and iii). The result of this integration is the quantity (15View Equation) used in the computation of the entanglement entropy,

∫ 𝒟 ψe− Wmat[ψ,g] = e−W [β,g]. (159 )
Semiclassically, the functional integration over metrics in Eq. (158View Equation) can be performed in a saddle-point approximation,
−Wtot[β,g(β)] Z(β, Q,M ) = e , (160 )
where metric g (β) μν is a solution to the saddle-point equation
δW [β, g] ----tot-----= 0, Wtot = Wgr [β, g] + W [β, g], (161 ) δg
with the inverse temperature β kept fixed. The solution of this equation is a regular (without conical singularities) metric g (β ) μν. This is the on-shell metric, which incorporates the quantum corrections due to the vacuum polarization by the matter fields. It can also be called an equilibrium configuration, which corresponds to the fixed temperature β− 1. In the saddle point approximation there is a constraint relating the charges at infinity M and Q and the inverse temperature β: β = β(M, Q ).

The thermodynamic entropy is defined by the total response of the free energy −1 F = − β ln Z(β ) to a small change in temperature,

STD = β2dβF = (βd β − 1 )Wtot (β,gβ) (162 )
and involves, in particular, the derivative of the equilibrium configuration gμν(β) with respect to β
dβWtot = ∂βWtot[β,g] + δWtot[β,g]-δgμν. (163 ) δgμν δβ
For an equilibrium configuration, satisfying Eq. (161View Equation), the second term in Eq. (163View Equation) vanishes and thus the total derivative with respect to β coincides with a partial derivative.

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 β. Second, consider this off-shell entropy for an equilibrium configuration, which solves Eq. (161View Equation). Since for the classical gravitational action (112View Equation) one finds (β ∂β − 1)Wgr[β,g] = S (GB, ciB) (116View Equation) and for the quantum effective action one obtains the entanglement entropy (β ∂β − 1)W [β,g] = Sent, the relation between the entanglement entropy and thermodynamic entropy is given by

STD = S(GB, ciB) + Sent. (164 )
Therefore, the entanglement entropy constitutes only a (quantum) part of the thermodynamic entropy of the black hole. The thermodynamic entropy is defined for equilibrium configurations satisfying the quantum corrected Einstein equations (161View Equation). Thus, these configurations are not classical solutions to the Einstein equations but incorporate the quantum (one-loop) corrections. These configurations are regular metrics without conical singularities. The UV divergences in the free energy for these configurations are renormalized in a standard way and thus for the thermodynamic entropy the renormalization statement discussed above holds automatically9.

In flat spacetime the quantum (one-loop) thermodynamic and statistical entropies coincide as was shown by Allen [2Jump To The Next Citation Point] due to the fact that the corresponding partition functions differ by terms proportional to β. In the presence of black holes the exact relation between the two entropies has been a subject of some debate (see, for example, [86, 199Jump To The Next Citation Point]). However, the analysis made in [104Jump To The Next Citation Point] 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.

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