4.2 ’t Hooft’s brick-wall model

In 1985 ’t Hooft [214Jump To The Next Citation Point] 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
ds2BH = − g(r)dt2 + g−1(r)dr2 + r2dωd−2 (165 )
as follows
2 −1 2 2 −2 2 2 −1 2 dsopt = g (r)dsBH = − dt + g (r)dr + r g (r)dω d− 2, (166 )
where dω2 d−2 is the metric of the (d − 2 )-unit sphere. In the optical metric, the near-horizon region, where the metric function in Eq. (165View Equation) can be approximated as
4π 2 g(r) = ---(r − r+ ) + O (r − r+) , βH
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 𝜖 from the actual horizon. Then, the regularized optical volume is divergent when 𝜖 is taken to zero
∫ βd− 1 Vopt = Ωd−2 drrd−2g−d∕2 ∼ A (Σ )-H---, (167 ) r𝜖 𝜖d−2
where A(Σ ) = rd−2Ωd− 2 + is the area of the horizon and ∘ ------------ 𝜖 ∼ βH (r𝜖 − r+) is the invariant distance between the brick wall (r = r𝜖) and the actual horizon (r = r+). The entropy of a gas of massless particles at temperature −1 T = βH confined in volume Vopt in d spacetime dimensions
S ∼ V β1−d ∼ A(Σ-)- (168 ) BW opt H 𝜖d−2
in the optical metric, is proportional to the horizon area. We should note that the universal behavior of the regularized optical volume (167View Equation) in the limit of small 𝜖 and its proportionality to the horizon area in this limit was important in establishing the result (168View Equation).

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

(∇2 − m2 )φ = 0 (169 )
on the background of a black-hole metric (165View Equation) and imposes a brick-wall boundary condition
φ(x) = 0 at r = r𝜖. (170 )
Consider, for simplicity, the four-dimensional case. Expanding the scalar field in spherical coordinates −iωt φ = e Yl,m(𝜃,ϕ)f (r ) one finds that Eq. (169View Equation) becomes
( ) 2 − 1 − 2 ( 2 ) l(l + 1) 2 ω g (r)f (r) + r ∂r r g(r)∂rf (r) − ----2---+ m f(r) = 0. (171 ) r
One uses the WKB approximation in order to find a solution to this equation. In this approximation one represents f(r) = ρ(r)eiS(r), where ρ(r) is a slowly-varying function of r while S (r) is a rapidly-varying phase. One neglects derivatives of ρ (r ) and the second derivative of S (r), one obtains the radial function in the form
┌ ------(--------------)---- ±i∫ r-dr-k(r,l,E ) ││ l(l + 1) f (r) = ρ (r)e g(r) , k(r,l,E ) = ∘ E2 − m2 + ----2--- g (r), (172 ) r
valid in the region where k2(r) ≥ 0. The latter condition defines a maximal radius r ω,l, which is a solution to the equation 2 k (rω,l) = 0. For a fixed value of the energy E, by increasing the mass m of the particle or the angular momentum l, the radius rω,l approaches r+ so that the characteristic region where the solution (172View Equation) is valid is in fact the near horizon region. One imposes an extra Dirichlet condition φ = 0 at r = rω,l so that the one-particle spectrum becomes discrete
∫ rω,l dr 2 ----k (r,ω, l) = 2 πn , (173 ) r𝜖 g(r)
where n is an integer. This relation is used to count the number of one-particle states that correspond to fixed values of energy ω and angular momentum l,
1∫ rω,l dr n(ω, l) = -- ----k(r,ω,l). (174 ) π r𝜖 g(r)
Calculating the total number of states, which have the same energy E, one has to sum over l. This sum can be approximated by an integral
∫ 2 ∫ rω r2 3 ∘ ------------- n(ω) = dl(2l + 1)n(ω,l) = -- --2-- k (r,ω ), k(r,ω) = ω2 − m2g (r), (175 ) 3 r𝜖 g (r)
where rω is determined by the condition that k (r,ω ) = 0.

In the near horizon region one approximates the metric function g(r) in Eq. (166View Equation) by the first two terms in the expansion in powers of (r − r+),

4π g(r) = ---(r − r+) + C (r − r+)2 , (176 ) βH
where βH is the inverse Hawking temperature and r+ is the horizon radius. Constant C is related to the curvature of spacetime near the horizon. The radial position of the brick wall is π𝜖2 r𝜖 = r+ + βH, where 𝜖 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 (175View Equation)
2 3 3 ( 2 2 3( ) 2 2 ) n(ω ) = r+βH-ω--+ r+β-Hω-- β C − 4π- + r+-βH-m-ω- ln𝜖 . (177 ) 24 π4𝜖2 24 π4 H r+ 2 π2

In a thermal ensemble of scalar particles at fixed temperature T = β− 1, each state in the one-particle spectrum can be occupied by any integer number of quanta. One gets for the free energy

∫ ∞ βF = dωdn-(ω) ln (1 − e− βω) (178 ) 0 dω
or, integrating by parts,
∫ ∞ n(ω) βF = − β eβω −-1-dω . (179 ) 0
Substituting Eq. (177View Equation) here and using the integrals
∫ 2 ∫ 3 4 ∞ --dωω--- π--- ∞ -d-ωω--- -π--- 0 eβω − 1 = 6β2 , 0 eβω − 1 = 15β4 , (180 )
one calculates the divergent terms in the free energy
2 3 ( 2 ( ) 2 2 2 ) F = − -r+---βH-− -r+- β C − 4π- βH- + r+m---βH- ln 𝜖 (181 ) 360𝜖2 β4 360 H r+ β4 12 β2
and, using equation 2 S = β ∂βF |β=βH, the entropy
( ( ) ) r2+ r2+ 4π 1 2 2 S = ---2-+ ------ βHC − --- + --r+m ln𝜖. (182 ) 90𝜖 90 βH r+ 6
Due to the relations
∫ ∫ Cr2 − 4πr+- = -1- (R − 2R ) and 4πr2 = 1 (183 ) + βH 8π Σ ii ijij + Σ
this expression for the entropy can be rewritten in a completely geometric form
A(Σ ) 1 ∫ S = ------2 + ----- (Rii − 2Rijij + 30m2 ) ln 𝜖. (184 ) 360π 𝜖 720π Σ
The leading term proportional to the area was first calculated in the seminal paper of ’t Hooft [214Jump To The Next Citation Point]. The area law in the brick-wall model was also studied in [173, 213Jump To The Next Citation Point, 9Jump To The Next Citation Point, 10Jump To The Next Citation Point, 178].

It is an important observation made by Demers, Lafrance and Myers in [62Jump To The Next Citation Point] 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 [62Jump To The Next Citation Point]. Applying the Pauli–Villars regularization scheme for the four-dimensional scalar field theory studied here, one introduces five regulator fields {φi, i = 1,...,5} of different statistics and masses {mi, i = 1,...,5} dependent on the UV cut-off μ [62Jump To The Next Citation Point]. Together with the original scalar φ0 = φ (m0 = m) these fields satisfy two constraints

∑5 ∑5 Δi = 0 m and Δim2i = 0, (185 ) i=0 i=0
where Δi = +1 for the commuting fields, and Δi = − 1 for the anticommuting fields. Not deriving the exact expressions for mi, we just quote here the following asymptotic behavior
5∑ 2 2 2 2 m2- 2 Δim i ln m i = μ b1 + m ln μ2 + m b2, i=0 5∑ 2 m2- Δi ln m i = ln μ2 , (186 ) i=0
(where b1 and b2 are some constants), valid in the limit μ → ∞. The total free energy is the sum of all contributions, from the original scalar field and the regulators
∑5 βF = β ΔiF i. (187 ) i=0
It is clear that, due to the constraints (185View Equation), all brick-wall divergences (with respect to the parameter 𝜖) in the free energy (187View Equation) and in the entropy cancel. On the other hand, both the free energy and the entropy become divergent if the Pauli–Villars regulator μ is taken to infinity, thus confirming their identification as UV divergences. For the free energy one finds
1 β ∑5 1 β3 ∑5 F = − ----Hr2+ ΔiM 2i ln M 2i −------H-r2+C Δi ln M 2i . (188 ) 24 β2 i=0 1440 β4 i=0
and for the entropy one has
∑5 ∫ ∑5 S = -1--A Σ Δim2 ln m2 + ---1-- (Rii − 2Rijij) Δiln m2 . (189 ) 48π i=0 i i 1440 π Σ i=0 i

Several remarks are in order.

1) Comparing the entropy calculated in the brick-wall model, (184View Equation) or (189View Equation), with the entanglement entropy (82View Equation) we see that the structure of the UV divergent terms in two entropies is similar. The logarithmic terms in Eqs. (82View Equation) and (189View Equation) (or (184View Equation)) are identical if the black-hole metric has vanishing Ricci scalar, R = 0. This is the case, for example, for the Reissner–Nordström black hole considered in [62Jump To The Next Citation Point]. 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 m always appears in combination 2 1 m − 6R. However, this is not seen in the WKB approximation (172View Equation). 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 (82View Equation). Moreover, an alternative calculation [104Jump To The Next Citation Point] 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. (82View Equation).

2) The similarity between the two entropies suggests that the UV divergences in the brick-wall entropy (189View Equation) 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 R = 0 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 ξ, 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 ξ.

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, 92Jump To The Next Citation Point, 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. (169View Equation), by including a non-minimal coupling,

2 2 (− ∇ + ξR + m )ϕ = 0 (190 )
in the background of the black-hole metric g μν, which takes the form (165View Equation). The optical metric is conformally related to the black-hole metric, 2σ g¯μν = e gμν, where 2σ e = 1 ∕|gtt| (in the metric (165View Equation) we have that gtt = g(r)). Eq. (190View Equation) can be rewritten entirely in terms of the optical metric ¯gμν (166View Equation) as follows
(∂2t + ˆH2)φopt = 0 ,Hˆ2 = − Δopt + 𝒱 , ( 2 ) 𝒱 = e− 2σ (ξR + m2 ) − (d-−-2)-∇2σ − (d −-2)-(∇σ )2 , (191 ) 2 4
where φ = e(d−42)σφ opt, Δ opt is the Laplace operator for spatial part γopt= ¯g ij ij of the optical metric, while the scalar curvature R and the covariant derivative ∇ are defined with respect to the original metric gμν = e− 2σg¯μν. We notice that, since e− 2σ = g(r), the effective potential 𝒱 in Eq. (191View Equation) 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 ω, which appears in Eq. (171View Equation) in the brick-wall calculation, is thus an eigenvalue of the operator ˆ2 H,
ˆ 2 2 H φω = ω φω . (192 )

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

−1 ∑ − βω F = β n(ω) ln (1 − e ), (193 ) ω
where ω are eigenvalues of the spatial operator Hˆ2 and n (ω ) is the degeneracy of the energy level ω, can be represented in terms of the Euclidean path integral for a field theory with wave operator 2 ˆ2 (∂τ + H ), provided that the Euclidean time τ is a circle with period β. (This property was first clearly formulated by Allen [2].) It order to see this in a rather elementary way, we first notice that
βω ∞∑ ω2β2 ln (1 − e− βω) = −---+ ln βω + ln (1 + --2-2) . (194 ) 2 k=1 4π k
The sum in this expression can be rewritten as a difference of two sums
∑∞ ω2β2 ∞∑ 1 2 4π2k2 ∞∑ 4π2k2 ln(1 + ---2-2) = --ln(ω + ---2--) − ln(---2--) − ln ω (195 ) k=1 4π k k=−∞ 2 β k=1 β
Each of these sums should be understood in terms of the zeta-function regularization. In particular, using the properties of the Riemann ζ-function, we find
∞∑ 2 2 ∞∑ 2 2 ln(4π-k--) = lim d-- ln (4π-k-)− z = − ln β (196 ) k=1 β2 z→0 dzk=1 β2
Collecting together Eqs. (194View Equation), (195View Equation) and (196View Equation) one obtains that
∞ 2 2 F = − 1-∑ n(ω)ω + 1--∑ n(ω ) ∑ ln(ω2 + 4π--k-) (197 ) 2 ω 2β ω k=−∞ β2
The second term in Eq. (197View Equation) 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 t → − iτ. The effective action Wopt then is defined by means of the Euclidean path integral
∫ ∫ e−Wopt = 𝒟 φ e− φopt(−∂2τ+Hˆ2)φopt . (198 ) opt
At finite temperature one closes the Euclidean time by identifying τ and τ + β. Then, the eigenvalues of operator ∂τ then are 2π- iβ k, where k = 0,±1, ±2, ... The effective action can be expressed in terms of the logarithm of the determinant
1 1 ∑ +∑∞ 4 π2k2 Wopt = − -ln det(− ∂2τ + Hˆ2 ) = −-- n(ω ) ln(ω2 + ---2--). (199 ) 2 2 ω k=−∞ β
Comparing this with Eq. (197View Equation) and defining the vacuum energy as
∑ E0 = 1- n (ω)ω (200 ) 2 ω
one arrives at an expression for the free energy (197View Equation)
F = β −1Wopt − E0 . (201 )

Evaluation of the effective action in the optical metric.
The effective action W opt (198View Equation) can be calculated using the heat kernel method. One notes that the heat kernel of the operator 2 ˆ 2 − ∂τ + H takes the form of a product of two heat kernels, for commuting operators ∂2τ and Hˆ2. The heat kernel for operator ∂2τ is computed explicitly, provided the periodicity condition, τ → τ + β, is imposed. One finds for the trace

2 β ∞∑ k2β2 Tres∂τ = √----- e−-4s-. (202 ) 4πs k= −∞
So that the effective action takes the form
1∫ ∞ ds β ∞∑ − k2β2 Wopt = − 2- 2 s-√----- e 4s TrK Hˆ2 . (203 ) 𝜖 4πs k= −∞
The operator ˆH2 = − Δopt + 𝒱 (191View Equation) is defined for the (d − 1)-dimensional metric γopt ij, the spatial part of the optical metric (166View Equation). The trace of the heat kernel of operator ˆ2 H = − Δopt + 𝒱 (191View Equation) can be represented as a series expansion in powers of s,
( ∫ ∫ ( ) ) TrK 2 = ---1----- √ γ---+ s √ γ--- 1R − 𝒱 + O(s2) , (204 ) ˆH (4πs )d−21 opt opt 6 opt
where the integration is taken over the spatial part of the optical metric (166View Equation) and Ropt is the Ricci scalar of (d − 1)-metric γoijpt.

For n ⁄= 0 the integration over the proper time s in Eq. (203View Equation) is regularized for small s due to the thermal exponential factor, so that the UV regulator 𝜖 can be removed. Interchanging the sum and the integral one obtains

∞ ∫ ( )2m −d ( ) ∑ ∞ ds- m− d2 − n24βs2 β- d- 0 s s e = 2 ζ(d − 2m )Γ 2 − m , m = 0, 1, 2, .. (205 ) n=1
Only the term with n = 0 in Eq. (203View Equation) contains the UV divergences. This term in the effective action is proportional to the inverse temperature β and thus it does not make any contribution to the entropy. On the other hand, the n = 0 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 n = 0 term in Eq. (203View Equation) or, equivalently, to subtract the zero temperature free energy, R F = F (T ) − F (T = 0). With this regularization and using Eq. (205View Equation) one obtains a sort of high temperature expansion of the effective action. In d dimensions one finds for the regularized action [74Jump To The Next Citation Point, 75Jump To The Next Citation Point]
β1− d (d ) β3−d ( d ) ∫ √ ---( 1 ) W Ropt = − -----ζ(d)Γ -- Vopt −------ζ(d − 2)Γ --− 1 γopt -Ropt − 𝒱 + O (β4−d),(206 ) πd ∕2 2 4 πd∕2 2 6
where, for d = 4, the term O(β4 −d) also contains a logarithmic term ln β. In four dimensions the first two terms in Eq. (206View Equation) 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. (206View Equation) and the respective term in the free energy and entropy is the contribution of a thermal gas in (d − 1) spatial volume Vopt at temperature T = β− 1 in flat spacetime. The other terms in Eq. (206View Equation) 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 F R = β− 1W Ropt and entropy Sopt = β2 ∂βF R one finds (provided one imposes the condition β = βH, after taking the derivative with respect to β)

2 ∫ ( ) Sopt = 2π--T3Vopt + -1-TH √ γopt 1Ropt − 𝒱 , (207 ) d=4 45 H 12 6
where we omit terms, which are finite when the volume integration is extended to the horizon. The important observation now is that
∫ ( ) ∫ (( ) ) √ γ--- 1-R − 𝒱 = √ γ--e− 2σ 1-− ξ R − m2 , (208 ) opt 6 opt opt 6
where e−2σ = g(r) and R is the scalar curvature of the original black-hole metric. We recall that the latter is conformally related to the optical metric, opt 2σ gμν = e gμν, so the relation between the scalar curvature in two spacetimes is
Ropt = e− 2σ (R − 6∇2 σ − 6(∇ σ)2). (209 )
Using this relation and the form of the potential term 𝒱 one arrives at Eq. (208View Equation).

Introducing the cut-off 𝜖 as before, π𝜖2 r𝜖 = r+ + βH-, one finds for the volume in the optical metric

3 ∫ β-HA-(Σ) βH--- Vopt = 16π3 𝜖2 + 32π3 Σ(Rii − 2Rijij)ln𝜖 + O (𝜖) (210 )
and
∫ √ ---- 1 1 β γopt( -Ropt − 𝒱 = ((--− ξ)R (r+) − m2 )-H-A+ ln 𝜖−1 + O(𝜖), (211 ) 6 6 2π
where A+ = 4πr2+ is the horizon area and R (r+) is the value of the scalar curvature at the horizon.

Putting everything together, one finds that the entropy in the optical metric is

A ∫ ( 1 1 (( 1 ) ) ) Sopdt=4 = ---+---+ -----(Rii − 2Rijij) −---- --− ξ R − m2 ln 𝜖. (212 ) 360π 𝜖2 Σ 720π 24π 6
Comparing this result with the entanglement entropy (82View Equation) computed earlier, we find complete agreement. Notice that 𝜖 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 μ. The UV divergences of the entropy when μ is taken to infinity are then of the same type as in Eq. (212View Equation). This was analyzed in [104].

The Euclidean path integral approach in the optical metric was considered in [15, 58, 57, 9Jump To The Next Citation Point, 10Jump To The Next Citation Point, 177, 136].

We remind the reader that the entanglement entropy (82View Equation) is obtained by using the Euclidean path integral in the original black-hole metric with a conical singularity at the horizon (for β ⁄= βH). 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 r = r+ 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 [199Jump To The Next Citation Point] 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.


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