Vacuum polarization in the Schwarzschild spacetime and dimensional reduction

13 Pages·0.2 MB·English

by  R. Balbinot

Vacuum polarization in the Schwarzschild spacetime and dimensional reduction
R. Balbinot(a)∗, A. Fabbri(a)†, P. Nicolini(a)‡, V. Frolov(b)§, P. Sutton(c)∗∗, and A. Zelnikov(b,d)††
1 (a)Dipartimento di Fisica dell’Universit`a di Bologna and INFN sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
0 (b)Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2J1
0 (c)Department of Physics, The Pennsylvania State University, State College, PA, USA 16802-6300
2 (d)P.N. Lebedev Physics Institute, Leninsky pr. 53, Moscow 117924, Russia

n
a
J AmasslessscalarfieldminimallycoupledtogravityandpropagatingintheSchwarzschildspace-
time is considered. After dimensional reduction under spherical symmetry the resulting 2D field
3
2 theoryiscanonicallyquantizedandtherenormalizedexpectationvalueshTabioftherelevantenergy-
momentumtensoroperatorareinvestigated. Asymptoticbehavioursandanalyticalapproximations
2 aregivenfor hTabiintheBoulware, UnruhandHartle-Hawkingstates. Specialattentionisdevoted
v to theblack-hole horizon region where theWKB approximation breaks down.
8
4 PACS number(s): 04.62.+v, 11.10.Gh, 11.10.Kk
0
2
1
0 I. INTRODUCTION
0
/
h In Quantum Field Theory the dimensional reduction of a system obeying some symmetries, such as spherical
t
- symmetry, is obtained by decomposing the field operators in harmonics in the symmetrical subspace. In the case
p of spherical symmetry, decomposing in terms of spherical harmonics effectively reduces a 4D theory to a set of 2D
e
theories characterizedby different values of the angular momentum.
h
Two-dimensionaltheoriesareoftenregardedasusefultoolsforinferringgeneralfeaturesofsystemswhosebehaviour
:
v is sophisticated and difficult to analyze in the physical 4D spacetime. In some spherically symmetric systems the
i
X main physical effects come from the “s-wave sector” – the l = 0 mode. Truncation of higher momentum modes is
then obtained by integrating over the “irrelevant” angular variables. This is the spirit which pervades most of the
r
a vastliterature on2D blackholes,thoughthis s-waveapproximationis notalwaysaccurateenough. These models are
believed to describe the s-wave sector of physical 4D black holes.
Within this perspective, a model of 2D conformally invariantmatter fields interacting with 2D dilaton gravity has
attracted considerable interest recently. The action for this theory is
1
S = d2x√ ge−2φgab∂ ϕ∂ ϕ, (1.1)
a b
−2 −
Z
where ϕ is the scalar field, φ the dilaton, g the 2D background metric and a,b=1,2.
ab
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
§Electronic address: [email protected]
∗∗Electronic address: [email protected]
††Electronic address: [email protected]
1
The reason for this interest lies in the following: the action (1.1) can be obtained by the dimensional reduction of
the 4D action for a massless scalar field minimally coupled to 4D gravity,
1
S(4) = d4x g(4)gµν∂ ϕ∂ ϕ, (1.2)
µ ν
−8π −
Z
p
under the assumption of spherical symmetry. Decomposing the 4D spacetime as
ds2 =g(4)dxµdxν =g dxadxb+e−2φ(xa)dΩ2, (1.3)
µν ab
where dΩ2 is the metric on the unit two-sphere, one obtains the 2D action (1.1) by inserting the decomposition
(1.3) into the action (1.2), imposing ϕ = ϕ(xa), and integrating over the angular variables. Therefore the model
based on the action (1.1) seems more appropriate for discussing the quantum properties of black holes in the s-wave
approximation than other 2D models based on the Polyakov action (describing a minimally coupled 2D massless
scalar field), whose link with the real 4D world is missing. For this reason the efforts of many authors were devoted
to finding the effective action which describes at the quantum level the above2D dilaton gravitytheory ([1]; see also
[6] and [2]) . This effective action, once derived, would allow one to go beyond the fixed background approximation
usually assumed in the studies of the quantum black-hole radiation discovered by Hawking [3]. Such an effective
actionwillgivein fact T foranarbitrary2Dspacetimewhichcouldthenbe usedto study self-consistently,within
ab
h i
this 2D approach, the backreaction of an evaporating black hole, its evolution, and its final fate. Unfortunately the
effective actions so far proposed for the model of eq. (1.1) have serious problems in correctly reproducing Hawking
radiation even in a fixed Schwarzschild spacetime (see the discussion in Ref. [5]; see also [6] for a different point of
view). In any case before embarking on ambitious backreaction calculations and taking seriously puzzling results
(such as antievaporation [4]) one should check for any candidate of the effective action that leads, at least for the
Schwarzschildblackhole,tothecorrectresults. Butwhatarethe exact T forascalarfielddescribedbythe action
ab
h i
(1.1) propagating in a 2D Schwarzschild spacetime that the relevant effective action should predict? The aim of this
paper is to partially answer this question.
By standard canonical quantization we will be able to give the asymptotic (at infinity and near the black hole
horizon) values of T in the three quantum states relevant for a field in the Schwarzschild spacetime, namely:
ab
h i
the Boulware state (vacuum polarization around a static star), the Unruh state (black hole evaporation), and the
Hartle-Hawkingstate (black hole in thermal equilibrium). We will also obtainapproximate analyticalexpressionsfor
T foreveryvalueoftheradialcoordinate. Anyeffectiveactionforthemodelofeq.(1.1)whichisunabletopredict
ab
h i
at least the above asymptotic values of T is incorrect (or better incomplete) and any result based on it has no
ab
h i
physical support.
II. hTABi: ASYMPTOTIC BEHAVIOUR
Our main goal is the evaluation of the renormalized expectation values of the stress tensor operator for the scalar
field ϕ whose dynamics are given by the action (1.1). Here we will be interested in the asymptotic values (at infinity
andnearthehorizon). The followingderivationisjustareadaptationtoourmodelofsectionVI ofthe seminalpaper
by Christensen and Fulling [7] to which we refer the reader (see also [8]).
The classical stress tensor is defined as
2 δS
T = , (2.1)
ab −√ gδgab
−
hence from eq. (1.1)
1
T =e−2φ ∂ ϕ∂ ϕ g ( ϕ)2 . (2.2)
ab a b ab
− 2 ∇
(cid:20) (cid:21)
The scalar field obeys the field equation
a e−2φ ϕ =0. (2.3)
a
∇ ∇
The quantum field operator ϕˆ is then expanded o(cid:0)n a basis (cid:1)u for the solution of eq. (2.3) in terms of annihilation
j
{ }
and creation operators,
ϕˆ= aˆ u +aˆ†u∗ , (2.4)
j j j j
Xj (cid:16) (cid:17)
2
and computing the mean value 0T 0 we have
ab
h | | i
T = T u ,u∗ , (2.5)
h abi ab j j
j
X (cid:2) (cid:3)
where
T u ,u∗ =e−2φ Re ( u ) u∗ (1/2)g u 2 . (2.6)
ab j j ∇a j ∇b j − ab|∇ j|
(cid:2) (cid:3) n (cid:2) (cid:0) (cid:1)(cid:3) o
Taking as the backgroundgeometry the exterior Schwarzschildsolution
ds2 = (1 2M/r)dt2+(1 2M/r)−1dr2, φ= lnr, (2.7)
− − − −
one finds that a set of normalized basis functions of the field equation (2.3) is given by
→
→u (x)= 1 R(r;w)e−iwt, (2.8)
w
√4πw r
←
←u (x)= 1 R(r;w)e−iwt, (2.9)
w
√4πw r
where the radial functions R(r;w) satisfy the differential equation
d2R 2M
+(1 2M/r) R w2R=0, (2.10)
− dr∗2 − r3 −
(cid:20) (cid:21)
and r∗ is the Regge-Wheeler coordinate
r∗ =r+2Mln(r/2M 1). (2.11)
−
Exact solutions of eq. (2.10) are not known; however,one can find their asymptotic behaviour near the horizon,
→R eiwr∗ +→A(w)e−iwr∗,
∼
←R B←(w)e−iwr∗, (2.12)
∼
and at infinity,
→R B→(w)eiwr∗,
∼
←R e−iwr∗ +←A(w)eiwr∗. (2.13)
∼
A and B are the reflection and transmission coefficients (see Ref. [9]).
The T calculated for these modes corresponds to the so-called Boulware vacuum:
ab
h i
∞
B T b B = dw T b ←u ,←u∗ +T b →u ,→u∗ . (2.14)
| a | unren a w w a w w
(cid:10) (cid:11) Z0 n h i h io
For the Unruh vacuum we have
∞
U T b U = dw T b ←u ,←u∗ +coth(4πMw)T b →u ,→u∗ , (2.15)
| a | unren a w w a w w
(cid:10) (cid:11) Z0 n h i h io
whereas for the Hartle-Hawking state
∞
H T b H = dwcoth(4πMw) T b ←u ,←u∗ +T b →u ,→u∗ . (2.16)
| a | unren a w w a w w
(cid:10) (cid:11) Z0 n h i h io
Astheystandtheseexpressionsareill-definedandneedtoberegularized. However,takingintoaccounttheregularity
of the renormalized expectation values H T H on the horizon and the vanishing of B T B as r , some
ab ab
h | | i h | | i → ∞
3
asymptotic expressions can be obtained without recursion to any regularization procedure. For example for r
→ ∞
we can write
lim H T b H = lim( H T b H B T b B )= lim( H T b H B T b B )
r→∞ | a | r→∞ | a | − | a | r→∞ | a | − | a | unren
(cid:10) (cid:11) = lim 2(cid:10) ∞ d(cid:11)w (cid:10) T b (cid:11)→u ,→u∗ +(cid:10)T b ←u ,(cid:11)←u∗(cid:10) . (cid:11) (2.17)
r→∞ e8πMw 1 a w w a w w
Z0 − n h i h i o
Similarly for the leading term at r 2M we have
→
lim B T b B lim ( B T b B H T b H )= lim ( B T b B H T b H ) . (2.18)
r→2M | a | ∼r→2M | a | − | a | r→2M | a | − | a | unren
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
For the Unruh vacuum we have
lim U T b U lim ( U T b U H T b H )= lim ( U T b U H T b H )
r→2M | a | ∼r→2M | a | − | a | r→2M | a | − | a | unren
(cid:10) (cid:11) = lim(cid:10) 2 ∞(cid:11) (cid:10) dw T(cid:11) b ←u ,←u∗(cid:10) , (cid:11) (cid:10) (cid:11) (2.19)
r→2M(cid:26)− Z0 e8πMw−1 a h w wi(cid:27)
and
lim U T b U = lim( U T b U B T b B )= lim( U T b U B T b B )
r→∞ | a | r→∞ | a | − | a | r→∞ | a | − | a | unren
(cid:10) (cid:11) = lim 2(cid:10) ∞ (cid:11)dw (cid:10) T b →u(cid:11) ,→u∗ . (cid:10) (cid:11) (cid:10) (cid:11) (2.20)
r→∞ e8πMw 1 a w w
Z0 − h i
In deriving the above expressions we used the fact that the differences between unrenormalized and renormalized
quantitiesarethesame. Thisbecausethedivergences,beingultraviolet,arestateindependent,hencethecounterterms
are the same for every state. One sees that the basic quantity entering all the expressions is T [u ,u∗] which using
ab w w
the decomposition eqs. (2.8), (2.9) can be written as
1 0 0 1
Tab[uw,u∗w]=E −0 1 +F 1 −0 , (2.21)
(cid:18) (cid:19) (cid:18) (cid:19)
where
1 dR dR∗ f dR∗ dR f2
E = w2 R2+ R +R∗ + R2 (2.22)
8πwf | | dr∗ dr∗ − r dr∗ dr∗ | | r2
(cid:26)(cid:20) (cid:21) (cid:18) (cid:19) (cid:27)
and
i dR dR∗
F = R∗ R (2.23)
−8πf dr∗ − dr∗
(cid:18) (cid:19)
withf (1 2M/r). Usingtheasymptoticexpansionseqs.(2.12),(2.13)fortheradialfunctionthelimitingbehaviours
≡ −
of T can be evaluated.
ab
h i
Let us start by discussing what is perhaps the most interesting quantity, namely the Hawking flux for this theory,
whosevaluehasbeentheobjectofalivelydebate. OnlyfortheUnruhstateisthereanonvanishingcomponentofthe
fluxTr∗t. NotealsothattheWronskiancontainedinF isconstantsoitcanbecalculatedforallr fromtheasymptotic
expansion. We find therefore
U|Tr∗t|U = U|Tr∗t|U − B|Tr∗t|B
(cid:10) (cid:11) =(cid:10)( U|Tr∗t|U(cid:11) −(cid:10) B|Tr∗t|B(cid:11) )unren =f−1E˙U, (2.24)
where (cid:10) (cid:11) (cid:10) (cid:11)
1 ∞ wdw
E˙ = B(w)2 (2.25)
U 2π e8πMw 1| |
Z0 −
isthe energyflux atinfinity. Notsurprisingly,this fluxispositive;i.e., thereisnoantievaporationoftheblackholein
this theory. We can calculate the total flux using Page’sresult [10] for the w 0 asymptotics of the greybody factor
B(w)2 for l =0 mode, →
| |
4
B(w)2 =16M2w2. (2.26)
| |
Integration over the frequencies leads to the approximate Hawking flux in this 2D theory:
1
E˙Page = . (2.27)
U 7680πM2
This low-frequency approximation for the transmission amplitude should work quite well since high frequencies will
not contribute to the flux because of the Planckian exponent. Note that the value of the Hawking flux E˙Page is
U
exactly 1/10 of the corresponding value coming from the Polyakov theory (massless minimally coupled 2D scalar
field). This damping is due to the potential barrier present in the radial equation (2.10) which reflects the coupling
of the scalar field with the dilaton. In the Polyakov theory there is no potential barrier, hence B(w)2 1 and
| | ≡
E˙Polyakov =10E˙Page.
U U
Accurate numerical calculations of the greybody factor for l =0 mode and the corresponding Hawking flux give
E˙numerical =CE˙Page, (2.28)
U U
where the coefficient
C 1.62. (2.29)
≈
Itisinterestingtocomparethe2D(s-mode)Hawkingfluxwiththatofthe4Dblackhole. B.S.DeWitt[9]providesan
approximate formula for the transmission coefficient, B(w)2 = 27M2w2, which takes into account the contribution
| |
to the 4D Hawking flux of allmomenta (this gives C =1.69), whereasnumericalcalculations [11]of the 4D Hawking
flux at infinity give E˙4D-numerical 1.79E˙Page
U ≈ U
Usingthe asymptoticexpansionwecanextractthe leadingbehaviourof U T b U nearthe horizonandatinfinity
| a |
(see eqs. (2.19), (2.20)):
(cid:10) (cid:11)
1 1/f 1
U Tb U − , (2.30)
| a| r→2M ∼ 7680πM2 1/f2 1/f
(cid:18) − (cid:19)
(cid:10) (cid:11)
and
1 1 1
U Tb U − − , (2.31)
| a| r→∞ ∼ 7680πM2 1 1
(cid:18) (cid:19)
(cid:10) (cid:11)
where now a,b = r,t. From eq. (2.30) one sees the negative energy flux entering the black hole horizon which
compensates the Hawking radiation at infinity.
Using similar methods one obtains (see eqs. (2.17), (2.18))
1 1 0
B Tb B (2.32)
| a| r→2M ∼ 384πM2f 0 1
(cid:18) − (cid:19)
(cid:10) (cid:11)
and
1 1 0
H Tb H − . (2.33)
| a| r→∞ ∼ 384πM2 0 1
(cid:18) (cid:19)
(cid:10) (cid:11)
ThislastequationshowsclearlythattheHartle-Hawkingstateasymptoticallydescribesathermalbathof2Dradiation
at the Hawking temperature T = (8πM)−1. The prefactor is the expected πT2. This is indeed the leading
H 6 H
contribution (in a 1/r expansion) for the s-mode in flat space (see Appendix).
III. hTABi: ANALYTICAL APPROXIMATIONS FOR THE BOULWARE AND HARTLE-HAWKING
STATES
To obtain an analytical expression for T valid for every r (2M < r < ) we use point-splitting regularization
ab
h i ∞
followedbyaWKBapproximationforthemodes. Therenormalizedexpression T isthenobtainedbysubtractionof
ab
h i
renormalizationcounterterms T coming from the DeWitt-Schwinger expansionof the Feynman Green function
h abiDS
and removal of the regulator (point separation). This method is nicely explained in the seminal work of Anderson et
5
al.[12]on T insphericallysymmetricstaticspacetimes,towhichwereferthe readerforalldetails. Thissectionis
µν
h i
just an application of their general method to our (much simpler) s-wave case. Here we just outline the main points
of the derivation.
One first analytically continues the spacetime metric into an Euclidean form by letting τ =it:
ds2 =fdτ2+f−1dr2. (3.1)
By the point-splitting method T is calculated by taking derivatives of the quantity ϕ(x)ϕ(x′) and then
letting x′ x. When the pointshaarbeiusnerpeanrated one can show that h i
→
hTabiunren =e−(φ(x)+φ(x′)) 12 gac′GE;c′b+gbc′GE;ac′ − 21gabgcd′GE;cd′ , (3.2)
(cid:20) (cid:16) (cid:17) (cid:21)
where G is the Euclidean Green function satisfying the equation
E
a(e−2φ G (x,x′))= g−1/2(x)δ2(x,x′) , (3.3)
a E
∇ ∇ −
and the quantities gc′ are the bivectors of parallel transport. The integral representation for G (x,x′) used by
a E
Anderson et al. [12] is the following:
G (x,x′)= dµcos[ω(τ τ′)]p (r )q (r ) , (3.4)
E ω < ω >
−
Z
where, for an arbitrary function F,
1 ∞
dµF (ω) dωF (ω)
≡ 4π
Z Z0
if T =0 (Boulware state), whereas for T >0
∞
dµF (ω) 2T F(ω )+TF(0)
n
≡
Z n=1
X
and ω =2πnT .
n
← →
The modes pω and qω are analogous to the radial functions R/r, R/r used in the previous section. They satisfy
the Euclidean version of eq. (2.10), which we write as
d2S 2 M dS ω2
f + 1 S =0, (3.5)
dr2 r − r dr − f
(cid:18) (cid:19)
and the Wronskian condition
dq dp 1
ω ω
C p q = . (3.6)
ω ω dr − ω dr −fr2
(cid:20) (cid:21)
To express these modes we use the WKB approximation:
1 r W(r)
p exp dr ,
ω
≡ r 2W(r) f
(cid:20)Z (cid:21)
1 r W(r)
p
q exp dr . (3.7)
ω
≡ r 2W(r) − f
(cid:20) Z (cid:21)
By this change of variables one sees that thpe Wronskian condition is satisfied by C =1. Substituting eqs. (3.7) into
ω
the mode equation eq. (3.5) one finds that the function W(r) has to satisfy
f d2W df dW 3f dW 2
W2 =ω2+V + f + (3.8)
2W " dr2 dr dr − 2W (cid:18) dr (cid:19) #
where V = f df. This is solved iteratively starting from the zeroth-order solution
rdr
6
W =ω. (3.9)
By this method one obtains an explicit form for the modes p ,q to be inserted in the general expression of G
w w E
(eq. (3.4)). Taking derivatives of the latter quantity as indicated in eq. (3.2) one eventually arrives at the following
expression for T b :
a unren
(cid:10) (cid:11) T t = T r =e−2φ dµcos(ωǫ ) 1gtt′ω2A 1grr′A
t unren −h r iunren τ −2 1− 2 2
Z (cid:20) (cid:21)
(cid:10) (cid:11) +e−2φi dµωsin(ωǫ ) 1grt′A 1gtr′A , (3.10)
τ 3 4
−2 − 2
Z (cid:20) (cid:21)
where
dp dq dp dq
ω ω ω ω
A =p q , A = , A =q , A =p ,
1 ω ω 2 3 ω 4 ω
dr dr dr dr
and ǫ τ τ′. For the sake of convenience the points are split in time only so that r′ =r.
τ
≡ −
The expansion for the bivectors is
gtt′ = 1 f′2ǫ2+O(ǫ4), (3.11)
−f − 8f
gtr′ = gr′t = f′ǫ+O(ǫ3), (3.12)
− − 2
grr′ =f + f′2fǫ2+O(ǫ4), (3.13)
8
where f′ df/dr.
Eventu≡ally one arrives at the following expression for Tt in the zero temperature case:
h tiunren
1 1 M2 f2
B T t B = B T r B = + + (2γ+ln(4λ2ǫ2)) , (3.14)
| t | −h | r | i 2πf ǫ2 2r4 4r2
(cid:20) (cid:21)
(cid:10) (cid:11)
which shows 1/ǫ2 and lnǫ divergences as ǫ 0 (λ is a lower limit cutoff in the integral over ω and γ is the
→
Euler constant). To obtain the renormalized expressions one needs to subtract from the above expressions the
renormalizationcounterterm T b obtained using the following Green function (see [15] for the details):
a DS
(cid:10) eφ(cid:11)(x)+φ(x′) 1 m2σ R a a
G(1)(x,x′)= [ (γ+ ln( ))(1+( 1)σ)+ 1 +...], (3.15)
2π − 2 2 12 − 2 2m2
where m2 is an infrared cutoff and a is the DeWitt-Schwinger coefficient for the action (1.1),
1
1
a = (R 6( φ)2+62φ). (3.16)
1
6 − ∇
Here R is the Ricci scalar for the 2D metric and σ is one-half of the square of the distance between the points x and
x′ along the shortest geodesic connecting them. For our splitting
f′2
σt =σ;t =ǫ+ ǫ3+O(ǫ5),
24
f′f
σr =σ;r = ǫ2+O(ǫ4), (3.17)
− 4
and σ =σaσ /2. This allows the counterterm to be evaluated in an ǫ expansion:
a
1 1 5 M2 1fM f2
T t = + + + (2γ+ln(m2ǫ2f)) ,
t DS 2πf ǫ2 12 r4 6 r3 4r2
(cid:20) (cid:21)
(cid:10) (cid:11) 1 1 5 M2 fM f2
T r = + (2γ+ln(m2ǫ2f)) . (3.18)
h r iDS 2πf −ǫ2 − 12 r4 2r3 − 4r2
(cid:20) (cid:21)
7
The renormalized expectation value is then defined as
T =Re lim( T T ) . (3.19)
h abi ǫ→0 h abiunren−h abiDS
h i
In the Boulware state this yields
1 1 M2 1fM f2 m2f
B T t B = ln( ) , (3.20)
| t | WKB 2πf 12 r4 − 6 r3 − 4r2 4λ2
(cid:18) (cid:19)
(cid:10) (cid:11)
1 1 M2 1fM f2 m2f
B T r B = + ln( ) . (3.21)
h | r | iWKB 2πf −12 r4 − 2 r3 4r2 4λ2
(cid:18) (cid:19)
Note that B T B has the correct trace anomaly:
ab
h | | i
a 1 1 d2f 6df M
B Ta B = 1 = (R 6( φ)2+62φ)= + = . (3.22)
h | a| iWKB 4π 24π − ∇ −24π dr2 rdr −3πr3
(cid:18) (cid:19)
Itiseasytoshowthat B T B isnotconserved. Reparametrizationinvarianceoftheaction(1.1)givesthefollowing
ab
h | | i
nonconservation equation ([5], [6])
1 δS
Ta = φ . (3.23)
∇ah bi −√ g δφ∇b
− (cid:28) (cid:29)
A “sourceterm” is present because of the coupling with the dilaton. Eqs.(3.23) arenothing but the 4D conservation
(4)µ
equations T =0 for the minimally coupled massless scalar field of the action (1.2). This allows us to define
µ ν
∇
a “pressure” fDor ourE2D model by rewriting eqs. (3.23) as
M
8πrTθ =∂ Tr + Tr Tt ,
θ r r r2f r − t
∂ Tr =0. (cid:0) (cid:1) (3.24)
r t
Then from eqs. (3.20), (3.21) and (3.24) one has
1 8M 2 4M m2f
B Tθ B = (1 )ln( ) . (3.25)
| θ| 64π2 r5 − r4 − r 4λ2
(cid:20) (cid:21)
(cid:10) (cid:11)
It is rather interesting to note that provided we set m = 2λ the above expressions for B T b B and the pressure
| a |
coincide exactly with the ones derived from the “anomaly induced” effective action for the theory (1.1) [5].
(cid:10) (cid:11)
The thermal case is treated similarly. Evaluating the sum over n using the Plana sum formula, one finds that the
stress tensor at finite temperature is obtained from the zero-temperature one by making the substitution
m2f m2β2f
ln( ) 2γ+ln( ) (3.26)
4λ2 → 16π2
(cid:26) (cid:27)
and adding the traceless pure radiation term
π
(T t) = (T r) = , (3.27)
t rad − r rad −6β2f
where β =T−1 .
Summarizing, we find that in the WKB approximation for the Hartle-Hawking state
π 1 1 M2 1fM f2 m2β2f
H T t H = + 2γ+ln( ) , (3.28)
| t | WKB −6β2f 2πf 12 r4 − 6 r3 − 4r2 16π2
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:10) (cid:11)
π 1 1 M2 1fM f2 m2β2f
H T r H = + + 2γ+ln( ) , (3.29)
h | r | iWKB 6β2f 2πf −12 r4 − 2 r3 4r2 16π2
(cid:20) (cid:18) (cid:19)(cid:21)
8
M
H Ta H = B Ta B = , (3.30)
h | a| iWKB h | a| iWKB −3πr3
1 8M 2 4M m2β2f
H P H = (1 ) 2γ+ln( ) , (3.31)
h | | iWKB 64π2 r5 − r4 − r 16π2
(cid:20) (cid:18) (cid:19)(cid:21)
where in this case β =T−1.
H
The analyticexpressionswe haveobtainedfor B T B and H T H havethe correctasymptoticbe-
h | ab| iWKB h | ab| iWKB
haviours at r as inferred in the previous section. B T b B does indeed have the limiting form eq. (2.32)
→∞ | a | WKB
as the horizon is approached, whereas H T b H for large r describes thermal radiation at the Hawking tem-
| a | WKB(cid:10) (cid:11)
perature in agreement with eq. (2.33).
(cid:10) (cid:11)
Inthe Hartle-Hawkingstate the stresstensorshouldbe regularonthe horizon. Thismeansthatonthehorizonthe
leading term of H T b H should be proportionalto the 2D metric, since the manifold of the Euclidean instantonis
| a |
regular and the Hartle-Hawking state respects all its symmetries. But the trace of the stress tensor is known exactly
(cid:10) (cid:11)
because we know the conformal anomaly (3.30) in 2D. So, on the horizon we should obtain
1 1
H T b H = δb H Tc H = δb . (3.32)
| a | r=2M 2 a h | c| i r=2M −48πM2 a
(cid:12) (cid:12)
(cid:10) (cid:11)(cid:12) (cid:12)
In the vicinity of the horizon this prov(cid:12)ides only the leading ter(cid:12)m. Our results eqs. (3.28), (3.29) fulfill this condition.
However, to ensure finiteness of the stress tensor near the horizon in a regular frame one should satisfy the stronger
condition
H Tt H H Tr H
h | t| i−h | r| i =finite. (3.33)
f
This leads to serious concerns regarding the expression we found for the Hartle-Hawking state using the WKB
approximation. The logarithmic term present in eqs. (3.28), (3.29) causes H T b H to be logarithmically
| a | WKB
divergentatthe horizonwhen calculatedin afree-falling frame. This kind oflogarithmicdivergenceis alsopresentin
(cid:10) (cid:11)
the 4D calculation of Anderson et al. for non-vacuum spacetimes like Reissner-Nordstro¨m [12]. However, numerical
computations performed by the same authors give no indication that this divergence actually exists. Similarly, we
suspect that the log term we have in eqs. (3.28), (3.29) is an artifact of the WKB approximation which, as we shall
see in the next section, breaks down near the horizon.
IV. H|TAB|H NEAR THE HORIZON
(cid:10) (cid:11)
From the discussion of the previous section one can see the disappointing fact that in the Hartle-Hawking state
the energy density as measured by a free-falling observer in the WKB approximation diverges logarithmically as one
approaches the horizon r = 2M. On physical grounds we do not expect this to happen, since the Hartle-Hawking
stateisdefinedintermsofmodeswhichareregularatthehorizon. Theoriginofthelogtermin H T b H isin
| a | WKB
the counterterms T b (see eq. (3.18)). The WKB approximation for the modes produces in T b , besides
a DS (cid:10) a unr(cid:11)en
termsoftheformlnǫandand1/ǫ2 whicharecancelledbythecounterterms,onlyamonomialinvolvingf andpowers
(cid:10) (cid:11) (cid:10) (cid:11)
of r. The natural question which arises is whether one can trust the WKB approximation near the horizon.
The Euclidean modes Y =( rp , rq ) (see eq. (3.7)) satisfy a Schr¨odinger-like equation
ω ω
d2Y 2M 2M
U(r∗)Y =0 , U(r∗(r))=ω2+V , V = f , f = 1 . (4.1)
dr∗2 − r3 − r
(cid:18) (cid:19)
Solving iteratively the equation for W2 (see eq. (3.8)),
1 d2(W2) 5 d(W2) 2
W2 =ω2+V + , (4.2)
4W2 dr∗2 − 16 W4 dr∗
(cid:18) (cid:19)
we get
9
W2 =(W2) +(W2) +(W2) +... , (4.3)
0 1 2
(W2) =ω2, (4.4)
0
(W2) =V , (4.5)
1
1 d2V 5 dV 2
(W2) = , (4.6)
2 4(ω2+V)dr∗2 − 16 (ω2+V)2 dr∗
(cid:18) (cid:19)
(4.7)
wNhotileetthhaetnVex∼t “fc,orarsecdtoioanl”l i(tWs d2)eriivsatailvreesad∂yrk∗fiVn.iteF.orTωhis=in0dtichaetefisrstthatetrtmhse(WWK2B)0aapnpdro(xWim2)a1tiovanncisahnantotthweohrkornizeoanr
2
the horizon for the zero-frequency mode. For the modes with non-zero ω =ω =(4M)−1n we have
n
1 1 1
W2 = n2+f 1+ +O(n−4) +O(f2). (4.8)
(2M)2 4 n2
(cid:20) (cid:18) (cid:19) (cid:21)
One can see that the convergence of the WKB series implies that n is at least greater then 1. Evaluation of the
correspondingseries for ϕˆ2 and the stress tensor H T b H near the horizonleads to exactly the same conclusion:
| a |
(cid:10) (cid:11) (cid:10)n 1. (cid:11) (4.9)
≫
Clearly, the standard WKB approximation can not be applied for the calculation of the contribution of the n = 0
andn=1modestoquantumaveragesnearthehorizon. Toobtainamorereliableanalyticalexpressionfor H T b H
| a |
near the horizon we need a better approximation for the Green function for these modes.
(cid:10) (cid:11)
In Ref. [13] it was demonstrated that a more accurate calculation of the contribution of the n = 0 mode cures
the analogous logarithmic divergence in the total ϕˆ2 . Here we follow a similar approach to analyze the stress
WKB
tensor (see also [14]).
(cid:10) (cid:11)
One can decompose the thermal Euclidean Green function for the Y modes as
1 +∞ cosw (τ τ′)
G (τ,r;τ′,r′)= n − G (r,r′), (4.10)
E β [f(r)f(r′)]1/4 n
n=−∞
X
wherewewritew forthefrequencyinsteadofjustwasbeforetomakethedependenceonnmoreclear(w =2πn/β).
n n
Near the horizon the function G (r,r′) satisfies the following differential equation (with r =r′):
n
6
α2 4n2 1
∂2G + − +O(f) G =0, (4.11)
L n− M2 4L2 n
(cid:18) (cid:19)
where L is defined by
dr
dL= (4.12)
f1/2
and
1 n2
α2 = + . (4.13)
6 12
The differential equation (4.11) admits solutions in terms of Bessel functions of imaginary argument,
αL αL
G (r,r′)=(LL′)1/2I ( <)K ( >). (4.14)
n n n
M M
One can show that this solution obeys the derivative condition resulting from integrating the differential equation
(3.3) for G acrossthe delta function singularity at τ =τ′,r=r′. Using the above Green function one can calculate
E
the corresponding contribution to the stress tensor for each n near the horizon.
For a contribution to the Green function of the form
e−iwn(t−t′)F (r,r′) (4.15)
n
the corresponding contribution to the unrenormalized stress tensor in the Hartle-Hawking state is
10