In the original calculation by ’t Hooft one considers a minimally-coupled scalar field, which satisfies the Klein–Gordon equation

on the background of a black-hole metric (165) and imposes a brick-wall boundary condition Consider, for simplicity, the four-dimensional case. Expanding the scalar field in spherical coordinates one finds that Eq. (169) becomes One uses the WKB approximation in order to find a solution to this equation. In this approximation one represents , where is a slowly-varying function of while is a rapidly-varying phase. One neglects derivatives of and the second derivative of , one obtains the radial function in the form valid in the region where . The latter condition defines a maximal radius , which is a solution to the equation . For a fixed value of the energy , by increasing the mass of the particle or the angular momentum , the radius approaches so that the characteristic region where the solution (172) is valid is in fact the near horizon region. One imposes an extra Dirichlet condition at so that the one-particle spectrum becomes discrete where is an integer. This relation is used to count the number of one-particle states that correspond to fixed values of energy and angular momentum , Calculating the total number of states, which have the same energy , one has to sum over . This sum can be approximated by an integral where is determined by the condition that .In the near horizon region one approximates the metric function in Eq. (166) by the first two terms in the expansion in powers of ,

where is the inverse Hawking temperature and is the horizon radius. Constant is related to the curvature of spacetime near the horizon. The radial position of the brick wall is , where is the geodesic distance between the brick wall and the horizon. Focusing only on the brick-wall divergent terms, one obtains for the number of states (175)In a thermal ensemble of scalar particles at fixed temperature , each state in the one-particle spectrum can be occupied by any integer number of quanta. One gets for the free energy

or, integrating by parts, Substituting Eq. (177) here and using the integrals one calculates the divergent terms in the free energy and, using equation , the entropy Due to the relations this expression for the entropy can be rewritten in a completely geometric form The leading term proportional to the area was first calculated in the seminal paper of ’t Hooft [214]. The area law in the brick-wall model was also studied in [173, 213, 9, 10, 178].It is an important observation made by Demers, Lafrance and Myers in [62] that the brick-wall divergences are in fact the UV divergences. This can be seen in the Pauli–Villars regularization as was first done in [62]. Applying the Pauli–Villars regularization scheme for the four-dimensional scalar field theory studied here, one introduces five regulator fields of different statistics and masses dependent on the UV cut-off [62]. Together with the original scalar () these fields satisfy two constraints

where for the commuting fields, and for the anticommuting fields. Not deriving the exact expressions for , we just quote here the following asymptotic behavior (where and are some constants), valid in the limit . The total free energy is the sum of all contributions, from the original scalar field and the regulators It is clear that, due to the constraints (185), all brick-wall divergences (with respect to the parameter ) in the free energy (187) and in the entropy cancel. On the other hand, both the free energy and the entropy become divergent if the Pauli–Villars regulator is taken to infinity, thus confirming their identification as UV divergences. For the free energy one finds and for the entropy one hasSeveral remarks are in order.

1) Comparing the entropy calculated in the brick-wall model, (184) or (189), with the entanglement entropy (82) we see that the structure of the UV divergent terms in two entropies is similar. The logarithmic terms in Eqs. (82) and (189) (or (184)) are identical if the black-hole metric has vanishing Ricci scalar, . This is the case, for example, for the Reissner–Nordström black hole considered in [62]. However, the logarithmic terms in the two calculations are different if the Ricci scalar is non-zero. This discrepancy appears to arise due to certain limitations of the WKB approximation. In the exact solutions, known explicitly, for example, for a scalar field in a constant curvature spacetime, the mass always appears in combination . However, this is not seen in the WKB approximation (172). In fact, if one makes this substitution everywhere in the above brick-wall calculation, the Ricci scalar would appear in the brick-wall entropy in a manner, which agrees with the entanglement calculation (82). Moreover, an alternative calculation [104] of the density of states, which does not make use of the WKB approximation, results in an expression for the entropy, which agrees with Eq. (82).

2) The similarity between the two entropies suggests that the UV divergences in the brick-wall entropy (189) can be renormalized by the renormalization of the couplings in the gravitational action in the same way as for the entanglement entropy. That this indeed works was demonstrated in [62].

3) For a non–minimally-coupled field we have the same problem as in the case of the entanglement entropy. For metrics with the non-minimal coupling does not show up in the scalar field equation and does not change the density of states, the free energy and entropy. On the other hand, the non-minimal coupling affects the renormalization of Newton’s constant, even if the background metric is Ricci flat. An attempt was made in [200] to modify the Dirichlet boundary condition at the brick wall and replace it by a more sophisticated condition, which would depend on the value of the non-minimal coupling , so that the resulting entropy would have the UV divergences consistent with the renormalization of Newton’s constant. However, this attempt cannot be considered as successful since it does not reproduce the expected behavior of the entropy for large positive values of .

The calculation of the brick-wall entropy for a rotating black hole is more complicated due to the presence of the super- radiance modes in the spectrum. This issue is considered in [45, 92, 155, 41, 149, 224, 132, 148, 153].

For the integration over the proper time in Eq. (203) is regularized for small due to the thermal exponential factor, so that the UV regulator can be removed. Interchanging the sum and the integral one obtains

Only the term with in Eq. (203) contains the UV divergences. This term in the effective action is proportional to the inverse temperature and thus it does not make any contribution to the entropy. On the other hand, the term gives the free energy at zero temperature. Thus, only the zero temperature contribution to the free energy is UV divergent as was shown by Dowker and Kennedy [73]. In fact the usual way to renormalize the free energy is to subtract the term in Eq. (203) or, equivalently, to subtract the zero temperature free energy, . With this regularization and using Eq. (205) one obtains a sort of high temperature expansion of the effective action. In dimensions one finds for the regularized action [74, 75] where, for , the term also contains a logarithmic term . In four dimensions the first two terms in Eq. (206) are the only terms in the effective action, which are divergent when the integration in the optical metric is taken up to the horizon. The first term in Eq. (206) and the respective term in the free energy and entropy is the contribution of a thermal gas in spatial volume at temperature in flat spacetime. The other terms in Eq. (206) are curvature corrections to the flat spacetime result as discussed by Dowker and Schofield [74, 75].Let us focus on the four-dimensional case. Defining the regularized free energy and entropy one finds (provided one imposes the condition , after taking the derivative with respect to )

where we omit terms, which are finite when the volume integration is extended to the horizon. The important observation now is that where and is the scalar curvature of the original black-hole metric. We recall that the latter is conformally related to the optical metric, , so the relation between the scalar curvature in two spacetimes is Using this relation and the form of the potential term one arrives at Eq. (208).Introducing the cut-off as before, , one finds for the volume in the optical metric

and where is the horizon area and is the value of the scalar curvature at the horizon.Putting everything together, one finds that the entropy in the optical metric is

Comparing this result with the entanglement entropy (82) computed earlier, we find complete agreement. Notice that here is in fact the IR regulator, the brick-wall cut-off, which regularizes the integration in the radial direction in the optical metric. As in ’t Hooft’s original calculation, this divergence can be transformed into a UV divergence by introducing the Pauli–Villars regulator fields of characteristic mass . The UV divergences of the entropy when is taken to infinity are then of the same type as in Eq. (212). This was analyzed in [104].The Euclidean path integral approach in the optical metric was considered in [15, 58, 57, 9, 10, 177, 136].

We remind the reader that the entanglement entropy (82) is obtained by using the Euclidean path integral in the original black-hole metric with a conical singularity at the horizon (for ). The black-hole metric and the optical metric are related by a conformal transformation. This transformation is singular at the horizon and in fact produces a topology change: there appears a new boundary at in the optical metric, which was a tip of the cone in the original black-hole metric. Because of this singular behavior of the conformal transformation, the exact relation between the two Euclidean path integrals is more subtle than for a regular conformal transformation. That the UV divergences in the entropy calculated in these two approaches coincide suggests that the equivalence between the two approaches might extend to the UV finite terms. Although in arbitrary dimension this equivalence may be difficult to prove, the analysis in two dimensions [199] shows that the entanglement entropy and the brick-wall entropy are indeed equivalent. This is, of course, consistent with the formal proof outlined in Section 3.6.

Living Rev. Relativity 14, (2011), 8
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