### 4.2 ’t Hooft’s brick-wall model

In 1985 ’t Hooft [214] proposed a model, which was one of the first successful demonstrations that an entropy that scales as an area can be associated, in a rather natural way, to a black-hole horizon. The idea of ’t Hooft’s calculation was to consider a thermal gas of Hawking particles propagating just outside the black-hole horizon. The entropy in the canonical description of the system is calculated by means of the Wentzel–Kramers–Brillouin (WKB) approximation. Provided the temperature of the gas is equal to the Hawking temperature, the result of this calculation is unambiguous. However, there is an important subtlety: the density of states of a Hawking particle becomes infinite as one gets closer to the horizon. The reason for this is simple. Close to the horizon all particles effectively propagate in the optical metric. The later is conformally related to the black-hole metric
as follows
where is the metric of the -unit sphere. In the optical metric, the near-horizon region, where the metric function in Eq. (165) can be approximated as
occupies an infinite volume. Clearly, the infinite volume contains an infinite number of states. In order to regularize this infinity ’t Hooft introduced a brick wall, an imaginary boundary at some small distance from the actual horizon. Then, the regularized optical volume is divergent when is taken to zero
where is the area of the horizon and is the invariant distance between the brick wall () and the actual horizon (). The entropy of a gas of massless particles at temperature confined in volume in spacetime dimensions
in the optical metric, is proportional to the horizon area. We should note that the universal behavior of the regularized optical volume (167) in the limit of small and its proportionality to the horizon area in this limit was important in establishing the result (168).

#### 4.2.1 WKB approximation, Pauli–Villars fields

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 has

Several 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].

#### 4.2.2 Euclidean path integral approach in terms of optical metric

##### Field equation in optical metric.
Consider a slightly more general equation than Eq. (169), by including a non-minimal coupling,

in the background of the black-hole metric , which takes the form (165). The optical metric is conformally related to the black-hole metric, , where (in the metric (165) we have that ). Eq. (190) can be rewritten entirely in terms of the optical metric  (166) as follows
where , is the Laplace operator for spatial part of the optical metric, while the scalar curvature and the covariant derivative are defined with respect to the original metric . We notice that, since , the effective potential in Eq. (191) vanishes at the horizon. This is a general feature of wave equations in the black-hole background: the fields become, effectively, massless in the near horizon region. The frequency , which appears in Eq. (171) in the brick-wall calculation, is thus an eigenvalue of the operator ,

##### The canonical free energy and Euclidean path integral.
The canonical free energy (178)

where are eigenvalues of the spatial operator and is the degeneracy of the energy level , can be represented in terms of the Euclidean path integral for a field theory with wave operator , provided that the Euclidean time is a circle with period . (This property was first clearly formulated by Allen [2].) It order to see this in a rather elementary way, we first notice that
The sum in this expression can be rewritten as a difference of two sums
Each of these sums should be understood in terms of the zeta-function regularization. In particular, using the properties of the Riemann -function, we find
Collecting together Eqs. (194), (195) and (196) one obtains that
The second term in Eq. (197) can be expressed in terms of the Euclidean path integral. This can be seen as follows. In the Euclidean formulation one first makes a Wick rotation of time . The effective action then is defined by means of the Euclidean path integral
At finite temperature one closes the Euclidean time by identifying and . Then, the eigenvalues of operator then are , where . The effective action can be expressed in terms of the logarithm of the determinant
Comparing this with Eq. (197) and defining the vacuum energy as
one arrives at an expression for the free energy (197)

##### Evaluation of the effective action in the optical metric.
The effective action  (198) can be calculated using the heat kernel method. One notes that the heat kernel of the operator takes the form of a product of two heat kernels, for commuting operators and . The heat kernel for operator is computed explicitly, provided the periodicity condition, , is imposed. One finds for the trace

So that the effective action takes the form
The operator  (191) is defined for the -dimensional metric , the spatial part of the optical metric (166). The trace of the heat kernel of operator  (191) can be represented as a series expansion in powers of ,
where the integration is taken over the spatial part of the optical metric (166) and is the Ricci scalar of -metric .

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.