Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 49 (2016)

Revision of the brick wall method for calculating the black holethermodynamic quantities†
F. Lenz,∗1,∗3 K. Ohta,∗2 and K.Yazaki∗3The brick wall method was proposed by t’Hooft1)
for calculating the contributions of a scalar ﬁeld to thethermodynamic quantities of black holes, and it has
become a common tool for studying thermodynamic
properties in the spaces with horizons. We examine
the method in Rindler space, which has been often
used as a near horizon approximation for black holes,
by developing an accurate numerical method for thecalculation. We ﬁnd that the previous works1–3) overestimated the partition function and the entropy byalmost 2 orders of magnitude. The origin of the discrepancy between our results and theirs is clariﬁed byrepeating the calculations of the latter in our framework. We also carry out the analogous studies for thescalar ﬁeld in de Sitter space and conﬁrm that ourmethod is applicable to the important case of spherically symmetric spaces.
Rindler space is deﬁned by the following coordinatetransformation from the Minkowski space.
t, x, x⊥ → τ, ξ, x⊥ :1t(τ, ξ) = eaξ sinh aτ,ax(τ, ξ) =1 aξe cosh aτ, (1)awhere a is the acceleration of a particle at ξ = τ = 0(x = a1 , t = 0) in Minkowski space. In the following
calculation, we use a = 1, measuring all quantities inunits of powers of a. A noninteracting scalar ﬁeld withmass m satisﬁes the equation of motion,(∂τ2 − ∂ξ2 − (∇2⊥ − m2 )e2ξ )ϕ(τ, ξ, x⊥ ) = 0.

(2)

The solution vanishing asymptotically at ξ → ∞ is√
2 eξ ),ϕ(τ, ξ, x⊥ ) = e−iωτ eik⊥ x⊥ Kiω ( m2 + k⊥(3)

where K is the MacDonald function.
In order to calculate the thermodynamic quantities,
we restrict the system in the transverse directions to asquare with area A using the periodic boundary condition in the usual way. The brick wall method uses theDirichlet boundary condition at ξ0 with the distancefrom the horizon l = eξ0 on the order of the Planck
length, lP . The eigenvalue equation for ω is√
2 + m2 eξ0 ) = 0,(4)Kiω ( k⊥which results in discrete spectra ωn (k⊥ ), n = 1, 2, . . .
†∗1
∗2
∗3

Condensed from the article in Phys. Rev. D, Vol.92, 064018(2015)
Institue for Theoretical Physics, University of ErlangenNuernberg
Institute of Physics, University of Tokyo
RIKEN Nishina Center- 128 -for each k⊥ . The partition function is then given by∞ ∫A ∑ ∞k⊥ dk⊥ ln(1−e−βωn (k⊥ ) ),(5)ln Z(β) = −2π n=1 0

where β = 2π at the Unruh temperature.
In the previous works, the eigenvalues of ω are obtained by the WKB approximation and the sum over n
for the partition function (5) is replaced by the integral
from 0 to ∞ under the assumption that many n’s contribute to the sum, leading to the well known closed expressions for the thermodynamic quantities1–3) in thecase of a massless scalar ﬁeld. We have solved theequation for m = 0 numerically with high accuracy
and found that only a small number of n’s contributeto the sum with n = 1 term dominating at the Unruh temperature. The origin of the overestimation inthe previous works is the contribution of the region
0 ≤ n < 1, where the spectrum is absent.
The results of our calculation for the partition function ln Z and the entropy S = (1−β∂β ) ln Z at the Unruh temperature based on the numerical solutions ofeq.(4) are shown in TABLE I together with the closed
expressions of the previous works (WKB). They aregiven in units of A/4l2 which equals the BeckensteinHawking entropy for l = lP . Our result for ln Z with
only the n = 1 term in (5) is also shown, demonstrating the dominance of this term. The previous works
are seen to overestimate ln Z by 68 and S by 37.
TABLE I ln Z and S in units of A/4l2
ln Z (n = 1) ln Z (total)
Seq.(4) 1.27 × 10−5 1.30 × 10−5 9.68 × 10−5
WKB
1/360π1/90πAnother important diﬀerence between our calculation and the previous ones is that ours is sensitive tothe boundary condition at the brick wall because of
the sensitivity of eigenvalues for small n’s while theprevious ones are insensitive owing to the assumptionthat many n’s contribute to the sum in equation (5)The dominance of the n = 1 mode implies that thelongitudinal degrees of freedom are practically frozen
at the Unruh temperature and the thermodynamic
properties are determined by the transverse degreesof freedom.
References
1) G.’t Hooft, Nucl. Phys. B 256, 727 (1985).
2) L. Susskind and J. Uglum, Phys. Rev. D 50, 2700
(1994).
3) S. Mukohyama and W. Israel, Phys. Rev. D 58, 104005(1998).