Quantization of the electromagnetic field outside static black holes
and its application to low-energy phenomena
Luı´s C. B. Crispino
Instituto de Fisica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, Sa˜o Paulo, Sa˜o Paulo, Brazil and Departamento de Fisica, Universidade Federal do Para´, Campus Universita´rio do Guama´, 66075-900, Bele´m, Para´, Brazil
Atsushi Higuchi
Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom George E. A. Matsas
Instituto de Fisica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, Sa˜o Paulo, Sa˜o Paulo, Brazil 共Received 24 January 2001; published 11 May 2001兲
We discuss the Gupta-Bleuler quantization of the free electromagnetic field outside static black holes in the Boulware vacuum. We use a gauge which reduces to the Feynman gauge in Minkowski spacetime. We also discuss its relation with gauges used previously. Then we apply the low-energy sector of this field theory to investigate some low-energy phenomena. First, we discuss the response rate of a static charge outside the Schwarzschild black hole in four dimensions. Next, motivated by string physics, we compute the absorption cross sections of low-energy plane waves for the Schwarzschild and extreme Reissner-Nordstro¨m black holes in arbitrary dimensions higher than three.
DOI: 10.1103/PhysRevD.63.124008 PACS number共s兲: 04.70.Dy, 04.62.⫹v I. INTRODUCTION
The prediction that black holes should thermally evapo-rate关1兴 has sparked much interest in quantum field theory in curved spacetimes. One of the difficulties in studying fields in Schwarzschild 关2兴 and other black hole spacetimes, even when the fields are non-interacting, stems from the fact that the solutions to the field equations are functions whose prop-erties are not well known.1 In the low-frequency regime, however, the situation is much simpler in Schwarzschild spacetime. In this regime, the mode functions of the massless scalar field are well known 关4兴. The present authors and Su-darsky used this fact to find the response rate of a static scalar source关5兴 and an analytic approximation for the emis-sion rate of low-energy particles from classical sources 关6兴 outside the Schwarzschild black hole in closed form. On the other hand, the field equations of the electromagnetic field in a black hole spacetime are not decoupled and are difficult to analyze in the Lorenz gauge. However, if we require the field to be divergence-free on a two-sphere 共the spherical Cou-lomb gauge兲, the equations for the physical modes reduce to decoupled scalar field equations. Furthermore, solutions in terms of familiar special functions can be found in the low-energy regime. These observations enabled us recently to calculate the response rate of a static electric charge outside a Schwarzschild black hole in closed form关7兴.
In this paper we examine free quantum electrodynamics in static spherically symmetric spacetimes of arbitrary di-mensions in a modified Feynman gauge. 共This gauge is closely related to the A0⫽0 gauge used by Cognola and
Lecca关8兴 and reduces to the Feynman gauge in Minkowski
spacetime.兲 Then, we calculate some low energy quantities in electrodynamics outside spherically symmetric black holes. First, we review the calculation of the response rate of a static charge outside the four-dimensional Schwarzschild black hole in the Unruh vacuum 关9兴. Next we calculate the low energy absorption cross sections of photons for the Schwarzschild and extreme Reissner-Nordstro¨m black holes in arbitrary dimensions higher than three, extending some results obtained by Gubser 关10兴 using a method 关11,12兴 based on the Newman-Penrose formalism 关13兴.
The paper is organized as follows. In Sec. II we present the mode functions of the electromagnetic field in the space-time of a spherically symmetric black hole in our modified Feynman gauge. Then we discuss the corresponding quan-tum theory and show how the Gupta-Bleuler condition 共see, e.g., 关14兴兲 is implemented to obtain the physical states. In Sec. III we compare the physical modes in the spherical Cou-lomb gauge关7兴 with the ones obtained in the modified Feyn-man gauge. In Sec. IV we review the calculation of the re-sponse rate of a static charge outside a four-dimensional Schwarzschild black hole. In Secs. V and VI we present the photon absorption cross sections by the Schwarzschild and extreme Reissner-Nordstro¨m black holes of arbitrary dimen-sions higher than three. In Sec. VII we summarize the main results and make some remarks. In Appendix A we compute some components of field-strength two-point function in Minkowski spacetime using spherical polar coordinates and show that they agree with those obtained using Cartesian coordinates. In Appendix B a summation formula for Leg-endre functions used in Sec. IV is derived. In Appendix C, a formula which relates the absorption probability to the ab-sorption cross section is derived in arbitrary dimensions. We use the metric signature (⫹⫺⫺•••⫺) and the natural units with G⫽ប⫽c⫽1 throughout this paper.
1See Ref. 关3兴 for some known properties in the Schwarzschild
II. GUPTA-BLEULER QUANTIZATION IN A MODIFIED FEYNMAN GAUGE
In this section we analyze the field equations for the elec-tromagnetic field in spherically symmetric and static space-times in a modified Feynman gauge. Then we discuss the Gupta-Bleuler quantization in this gauge.
The line element of the spacetime we study is
d2⫽ f共r兲dt2⫺h共r兲dr2⫺r2dsp2, 共2.1兲 where dsp2 is the line element of a unit p-sphere. We assume that f (r) and h(r) are positive for r⬎rHand that both f (r) and h(r)⫺1 have simple zeros or both have double zeros at r⫽rH, where rHis the horizon radius. We also assume that f (r),h(r)⫺1→1 as r→⬁. 共Most results in this section, how-ever, are independent of these assumptions.兲 Let us introduce the Wheeler tortoise coordinate r* by
dr*
dr ⫽
冑
hf. 共2.2兲
Then r*(r) is a monotonic function with domain (rH,⫹⬁) and range in (⫺⬁,⫹⬁). Let us define for any two functions q1(r*) and q2(r*) the inner product
具
q1,q2典
⫽冕
⫺⬁⫹⬁
dr*q1共r*兲q2共r*兲, 共2.3兲
where the overline denotes complex conjugation.
The Lagrangian density of the electromagnetic field in a modified Feynman gauge is
LF⫽
冑
⫺g冋
⫺ 1 4FF ⫺1 2G 2册
共2.4兲 with G⫽ⵜA⫹KA, 共2.5兲 where the vector Kis independent of A. Hence the equa-tions of motion areⵜF⫹ⵜG⫺KG⫽0. 共2.6兲 Here we choose
K⫽„0,f
⬘
/共 f h兲,0,0…, 共2.7兲 in which case Eq. 共2.5兲 is written asG⫽1 f tAt⫺
冑
f h 1 rpr冋
rp冑
f hAr册
⫺ 1 r2ⵜ˜ iA i. 共2.8兲 Here i denotes angular variables on the unit p-sphere Sp with metric ˜i j and inverse metric ˜i j 关with signature (⫹•••⫹)], ⵜ˜i is the associated covariant derivative on Sp and ⵜ˜i⬅˜i jⵜ˜j. This choice for K is convenient because the equation for At decouples from the other ones. The field equations 共2.6兲 become ⫺1f t 2 At⫹冑
f h 1 rpr冋
rp冑
f hrAt册
⫹ 1 r2ⵜ˜ 2At⫽0, 共2.9兲 ⫺1 ft 2Ar⫹1 fr冋
冑
f h f rpr冉
rp冑
f hAr冊
册
⫹ 1 r2ⵜ˜ 2A r ⫹1 f r冉
f r2冊
ⵜ˜ iAi⫽0, 共2.10兲 ⫺1ft 2 Ai⫹r 2⫺p冑
f hr冉
冑
f hr p⫺2 rAi冊
⫺ r2 f hr冉
f r2冊
iAr ⫹ 1 r2关ⵜ ˜j共ⵜ˜ jAi⫺ⵜ˜iAj兲⫹i共ⵜ˜jAj兲兴⫽0, 共2.11兲 whereⵜ˜2⬅˜ i jⵜ˜iⵜ˜j.We shall now describe a complete set of solutions A(n;lm). We assign the value 0 for what we call the non-physical modes, 1 or 2 for the physical modes and 3 for the pure-gauge modes. 共These modes will be given below.兲 The label n distinguishes between modes incoming from the past null infinityJ⫺共denoted with n⫽←) and those coming out from the past horizon H⫺ 共denoted with n⫽→).2 The solutions with A(0n;lm)⫽0 (⫽t), and
At(0n;lm)⫽Rl(0n)共r兲Ylme⫺it 共2.12兲 will be called ‘‘non-physical modes’’ because they satisfy the field equations 共2.9兲–共2.11兲 but not the gauge condition G⫽0. Here Ylm is a scalar spherical harmonic on the unit p-sphere with ⵜ˜2Ylm⫽⫺l(l⫹p⫺1)Ylm, where l ⫽0,1,2, . . . and m denotes a set of p⫺1 integers (m1, . . . ,mp⫺1) satisfying l⭓mp⫺1⭓ . . . ⭓m2⭓兩m1兩.共See
Ref.关15兴 for a concise description of spherical harmonics on the p-sphere.兲 They are normalized as
冕
d⍀pYlmYl⬘m⬘⫽␦ll⬘␦mm⬘, 共2.13兲 where d⍀p is the volume element of the unit p-sphere. The function Rl(0n)(r) satisfies冋
2 f ⫹冑
f h 1 rp d dr冉
rp冑
f h d dr冊
⫺ l共l⫹p⫺1兲 r2册
Rl (0n)⫽0. 共2.14兲 We will determine the normalization of the functions Rl(0n)(r) later. The pure-gauge modes are given asA(3n;lm)⫽ⵜ⌳(nlm), 共2.15兲
2Here we treat only the solutions proportional to e⫺it with
⫽0. Thus, if there is any nonzero static field, we will be
where
⌳(nlm)⫽i
Rl(0n)共r兲Ylme⫺it. 共2.16兲 关As usual, the pure-gauge modes satisfy the field equations 共2.9兲–共2.11兲 and G⫽0.兴 The other independent solutions (⫽1,2), which represent physical degrees of freedom, i.e., which satisfy the field equations and G⫽0 but are not pure gauge, will be chosen to have At⫽0. Those with Ar⫽0 are given as Ar(1n;lm)⫽Rl(1n)共r兲Ylme⫺it, 共2.17兲 where
冋
2 f ⫺ l共l⫹p⫺1兲 r2册
Rl (1n)共r兲 ⫹ 1 r2 d dr冋
冑
f hr 2⫺p d dr冉
rp冑
f hRl (1n)共r兲冊
册
⫽0. 共2.18兲 Note here that the condition G⫽0 cannot be solved if l ⫽0. Hence, we have l⭓1. The corresponding angular com-ponents can be found by solving the condition G⫽0 asAi(1n;lm)⫽ r 2⫺p l共l⫹p⫺1兲
冑
f h d dr冋
rp冑
f h Rl (1n)册
iYlm e⫺it. 共2.19兲 We call these modes ‘‘physical modes I.’’ The other set of physical solutions can be obtained by letting At⫽Ar⫽0 and Ai(2n;lm)⫽Rl(2n)共r兲Yi(lm)e⫺it, 共2.20兲 where冋
2 f ⫹ 1冑
f h rp⫺2 d dr冉
冑
f hr p⫺2 d dr冊
⫺共l⫹1兲共l⫹p⫺2兲 r2册
Rl (2n)共r兲⫽0. 共2.21兲Here, the Yi(lm)are divergence-free vector spherical harmon-ics on the unit p-sphere satisfying
ⵜ˜k共ⵜ˜ kYi (lm)⫺ⵜ˜i Yk(lm)兲⫽⫺共l⫹1兲共l⫹p⫺2兲Yi(lm) 共2.22兲 and
冕
d⍀p˜i jYi(lm)Yj(l⬘m⬘)⫽␦ll⬘␦mm⬘. 共2.23兲 共See, e.g., Refs. 关15,16兴.兲 We call these modes ‘‘physical modes II.’’ Physical modes I and II obtained here 共and re-stricted to four dimensions兲 are identical with those in the A0⫽0 gauge 关8兴.In order to discuss Gupta-Bleuler quantization of this field it is convenient to introduce a generalized Klein-Gordon product of classical solutions of Eq.共2.6兲. We first define
⌸⬅ 1
冑
⫺gLF
关ⵜA兴⫽⫺关F
⫹gG兴, 共2.24兲 where G is given in Eq. 共2.5兲. Note that
冑
⫺g ⌸t is the canonical conjugate momentum of A. We write⌸()⬅⌸兩A
⫽A() 共2.25兲
for any solution A()⬅A(n;lm). For any two 共complex兲 solutions A() and A(⬘) we define
W关A(),A(⬘)兴⬅i关A()⌸(⬘)⫺⌸()A(⬘)兴. 共2.26兲 The field equations ensure that this current is conserved. As a result, the generalized Klein-Gordon inner product defined by
共A(),A(⬘)兲⬅
冕
⌺d⌺W
关A(),A(⬘)兴, 共2.27兲
where d⌺⬅dn, is independent of the Cauchy surface⌺ 关17兴. 共Here d is the volume element of the Cauchy surface ⌺ with a normal unit vector n.兲 Note that (A(),A(⬘))
⫽(A(⬘),A()). This guarantees that the norm defined
through Eq.共2.27兲 is real 共and positive definite for the subset of physical solutions with positive frequency兲. By working explicitly with the definition 共2.27兲, we find in general that on a t⫽const surface 共A(),A(⬘)兲⫽⫺i
冕
dp⫹1x冑
⫺g f⫺1g ⫻共A()tA (⬘)⫺A (⬘) tA ()兲. 共2.28兲It is important to note that pure-gauge modes are orthogo-nal to any mode satisfying G⫽0 and, as a result, ⵜF ⫽0. This can be shown as follows. Suppose that A()
and A(⬘) satisfy the condition G⫽0. Then since ⌸()⫽ ⫺F兩A
⫽A(), and similarly for A
(⬘), the inner product
共2.27兲 can be written as (A(),A(⬘))⫽(A(),A(⬘))
inv where 共A(),A(⬘)兲 inv⬅i
冕
⌺d⌺关A () F(⬘)⫺F()A(⬘)兴. 共2.29兲 Now, let A()⫽ⵜ⌳()be a pure-gauge mode. Then共A(),A(⬘)兲 inv⫽i
冕
⌺d⌺F (⬘)ⵜ ⌳() ⫽i冕
⌺d⌺ⵜ共⌳ () F(⬘)兲 ⫽0 共2.30兲since F(⬘)is anti-symmetric.3Thus (A(),A(⬘))⫽0 if A() is a pure-gauge mode and A(⬘) is a physical or pure-gauge mode, i.e.,
共A(3n;lm),A(3n⬘;⬘l⬘m⬘)兲⫽0, 共2.31兲
共A(3n;lm),A(1n⬘;⬘l⬘m⬘)兲⫽0, 共A(3n;lm),A(2n⬘;⬘l⬘m⬘)兲⫽0.
共2.32兲 Next let us examine the inner product of the non-physical solutions. By letting
Rl(0n)共r兲⬅
冑
f r⫺p/2l(0n)共r兲, 共2.33兲 we find from Eq.共2.14兲冉
2⫹ d 2dr*2⫺V0共r*兲
冊
l(0n)⫽0, 共2.34兲
where r* is given in Eq.共2.2兲 and
V0关r*共r兲兴⫽ fl共l⫹p⫺1兲 r2 ⫺ f h
冋
p共2⫺p兲 4r2 ⫺ f⬘
2 2 f2 ⫹ p 4r冉
f⬘
f ⫹ h⬘
h冊
⫹ f⬙
2 f⫺ f⬘
h⬘
4 f h册
.For spacetimes where f (r) and h(r)⫺1 have simple zeros at r⫽rH, we find V0关r*(rH)兴⫽0 in general. If V0关r*(rH)兴 ⬎0, then the non-physical modes coming out from the horizon have frequencies satisfying 2⬎V0关r*(rH)兴. In particular, for Reissner-Nordstro¨m spacetime in ( p⫹2)dimensions 关18兴, where f共r兲⫽h共r兲⫺1⫽
冋
1⫺冉
r⫹ r冊
p⫺1册冋
1⫺冉
r⫺ r冊
p⫺1册
with r⫾p⫺1⫽M⫾
冑
M2⫺Q2 共so that rH⫽r⫹), we find V0关r*共rH兲兴⫽ 共p⫺1兲2 共2r⫹兲2冋
1⫺冉
r⫺ r⫹冊
p⫺1册
2 . As expected, we have具
l(0→), ⬘l (0←)典
⫽0, i.e., the solutions of Eq. 共2.34兲 incoming from H⫺ and those incoming fromJ⫺ are orthogonal to one another with respect to the inner product defined by Eq.共2.3兲. By normalizing these solutions so that
l(0→)⬇
冑
/˜共ei˜r*⫹Rl(0→)e⫺i˜r*兲 共r*→⫺⬁兲, 共2.35兲 l(0←)⬇e⫺ir*⫹R(0l←)eir* 共r*→⫹⬁兲,共2.36兲
up to a phase factor, where ˜2⫽2⫺V0关r*(rH)兴⭓0 and
Rl(0→)
and Rl(0←) are constants 共with 兩Rl(0→)兩⫽兩Rl(0←)兩), we have
具
l(0←),⬘l (0←)典
⫽具
l(0→),⬘l (0→)典
⫽2␦共⫺⬘
兲. 共2.37兲 Now, by using Eq.共2.28兲 we find共A(0n;lm),A(0n⬘;⬘l⬘m⬘)兲⫽⫺2
具
l (0n) ,(0n⬘l⬘⬘)典
␦nn⬘␦ll⬘␦mm⬘ ⫽⫺4␦nn⬘␦ll⬘␦mm⬘␦共⫺⬘
兲 共2.38兲 where we have used ␦(˜⫺˜⬘
)⫽(˜ /)␦(⫺⬘
). Noting that A(0n;lm) and A(3n;lm) have the same t-components, we immediately obtain from Eqs.共2.28兲 and 共2.31兲 that共A(0n;lm),A(3n⬘;⬘l⬘m⬘)兲⫽⫺4␦
nn⬘␦ll⬘␦mm⬘␦共⫺
⬘
兲. 共2.39兲 Next let us examine the physical modes. By lettingRl(1n)共r兲⬅
冑
l共l⫹p⫺1兲 共 f h兲1/2r⫺p/2⫺1l(1n)共r兲, 共2.40兲 we find from Eq. 共2.18兲
冉
2⫹ d 2 dr*2⫺V1共r*兲冊
l (1n)⫽0, 共2.41兲 where V1关r*共r兲兴⫽ fl共l⫹p⫺1兲 r2 ⫹ p共p⫺2兲 4r2 f h ⫺共p⫺2兲 4 f hr冉
f⬘
f ⫺ h⬘
h冊
. 共2.42兲 共We note that V1→0 as r→r⫹ for Reissner-Nordstro¨mspacetime unlike V0.兲 We first note that the modes A(1n;lm)
are orthogonal to both the non-physical and pure-gauge modes. Next we note
共A(1n;lm),A(1n⬘;⬘l⬘m⬘)兲⫽2
具
l (1n) ,(1n⬘l⬘⬘)典
␦nn⬘␦ll⬘␦mm⬘. 共2.43兲 Thus, by normalizingl(1n)(r) as l(1→)⬇eir*⫹Rl(1→)e⫺ir* 共r*→⫺⬁兲, 共2.44兲 l(1←)⬇e⫺ir*⫹R(1l←)eir* 共r*→⫹⬁兲, 共2.45兲 we have 共A(1n;lm),A(1n⬘;⬘l⬘m⬘)兲⫽4␦ nn⬘␦ll⬘␦mm⬘␦共⫺⬘
兲. 共2.46兲 3Note that Eq. 共2.30兲 would hold even if ⌳()was an arbitrary function. This shows that (A(),A(⬘))invis gauge invariant.
Finally, we let
Rl(2n)共r兲⬅r⫺(p⫺2)/2l(2n)共r兲. Then from Eq. 共2.21兲, we have
冉
2⫹ d 2 dr*2⫺V2共r*兲冊
l (2n)⫽0, 共2.47兲 where V2关r*共r兲兴⫽ f 共l⫹1兲共l⫹p⫺2兲 r2 ⫹ 共p⫺2兲共p⫺4兲 4r2 f h ⫹共p⫺2兲 4 f hr冉
f⬘
f ⫺ h⬘
h冊
. 共2.48兲共We again note that V2→0 as r→r⫹in Reissner-Nordstro¨m
spacetime.兲 Notice that V1⫽V2 if p⫽2. Hence, in the four-dimensional case l(1n)(r) and l(2n)(r) satisfy the same equation. These modes can easily be shown to be orthogonal to the modes previously discussed. By normalizing the func-tions l(2n)(r) in the same way as l(1n)(r) 关see Eqs. 共2.44兲,共2.45兲兴, we have
共A(2n;lm),A(2n⬘;⬘l⬘m⬘)兲⫽4␦
nn⬘␦ll⬘␦mm⬘␦共⫺
⬘
兲. 共2.49兲 In order to quantize the field A, we impose the equal-time commutation relations on the field Aˆ and momentum ⌸ˆt operators: 关Aˆ共t,x兲,Aˆ共t,x⬘
兲兴⫽关⌸ˆt共t,x兲,⌸ˆt共t,x⬘
兲兴⫽0, 共2.50兲 关Aˆ共t,x兲,⌸ˆt共t,x⬘
兲兴⫽ i␦冑
⫺g␦ p⫹1共x⫺x⬘
兲, 共2.51兲 where x and x⬘
represent all spatial coordinates. The field Aˆ can be expanded using the modes we have obtained before:Aˆ共t,x兲⫽
兺
冕
⫺⬁ ⫹⬁ d冑
4兩兩A () 共t,x兲a, 共2.52兲where A() is proportional to e⫺it, A(⫺)⬅A(), a⫺ ⬅a†
and labels discrete quantum numbers. The commu-tation relations 共2.50兲,共2.51兲 are equivalent to the following commutation relations in the ‘‘symplectically smeared’’ form as is the case for scalar fields 关19兴:
关共A(),Aˆ兲,共Aˆ,A(⬘)兲兴⫽共A(),A(⬘)兲. 共2.53兲
Since the inner product must be t independent, the inner product of A() and A(⬘⬘) can be nonzero only if ⫽
⬘
. Thus, we can write共A(),A(⬘⬘)兲⫽M⬘␦共⫺
⬘
兲. 共2.54兲By using Eq. 共2.54兲 in Eq. 共2.53兲 one finds M12关a
2,a⬘3 †
兴M34⫽4M14␦共⫺
⬘
兲.共2.55兲 Then, since M⬘ is invertible in our case, we find
关a,a⬘⬘
†
兴⫽4共M⫺1兲⬘␦共⫺
⬘
兲. 共2.56兲 Note here that we immediately have 关a,a⬘⬘兴⫽0 for ,⬘
⬎0 by letting ⬘
→⫺⬘
in Eq. 共2.56兲 . By using the Klein-Gordon inner products computed above, we find the following commutators: 关alm(3n),a⬘l⬘m⬘ (3n⬘)† 兴⫽⫺关alm(0n),a⬘l⬘m⬘ (3n⬘)† 兴 ⫽␦nn⬘␦ll⬘␦mm⬘␦共⫺⬘
兲, 共2.57兲 关alm(1n),a⬘l⬘m⬘ (1n⬘)† 兴⫽关alm(2n),a⬘l⬘m⬘ (2n⬘)† 兴 ⫽␦nn⬘␦ll⬘␦mm⬘␦共⫺⬘
兲 共2.58兲 with all other commutators vanishing. The Gupta-Bleuler condition关14兴 requires that any physical state 兩phys典
satisfy Gˆ(⫹)兩phys典
⫽0, 共2.59兲 where Gˆ(⫹) is the positive-frequency part of Gˆ⫽ⵜAˆ ⫹KAˆ. Since this quantity is nonvanishing only for A(0n;lm), this condition is equivalent to
alm(0n)兩phys
典
⫽0 for all 共n,,l,m兲 共with ⬎0兲. 共2.60兲 共We let ⬎0 below.兲 The Boulware vacuum 兩0典
关20兴 is defined by requiring that it be annihilated by all alm(n) opera-tors (⫽0,1,2,3). Note that the states obtained by applying any number of creation operators excluding alm(3n)† are all physical states. Any state of the form alm(3n)†兩phys典
is un-physical becausea(0n⬘l⬘⬘m) ⬘alm(3n)†兩phys
典
⫽⫺␦nn⬘␦ll⬘␦mm⬘␦共⫺⬘
兲兩phys典
⫽0. Note also that the physical states of the form alm(0n)†兩phys典
have zero norm and are orthogonal to any physical states. Thus, as is well known, a physical state 兩phys1典
can bere-garded as equivalent to any state of the form 兩phys1
典
⫹alm(0n)†兩phys2典
. We can take as the representative elementsthe states obtained by applying alm(n)†, ⫽1,2, on 兩0
典
. Unphysical particles created by alm(3n)† will be in thermal equilibrium in the Hartle-Hawking vacuum 关21兴 for a static black hole if we require the gauge-fixed two-point function be non-singular on the horizons as in the scalar case 关22兴. There will also be a flux of unphysical particles in the Unruh vacuum. Therefore, technically speaking, the Hartle-Hawking and Unruh vacua are unphysical if we impose the Gupta-Bleuler condition using the positive-frequency notion in the Boulware vacuum. Fortunately, the Becchi-Rouet-Stora-Tyutin 共BRST兲 quantization does not suffer from thisproblem since the physical state condition does not refer to any notion of positive frequency 关23兴. This issue, however, will not concern us in the following, since all our calcula-tions will be at the tree level.
Mode expansions in spherical polar coordinates are not very widely used and are quite different from the ordinary method using Cartesian coordinates. For this reason, we compute in Appendix A some components of field-strength two-point function in four-dimensional Minkowski space-time using spherical polar coordinates. We find that they agree with the standard results obtained in Cartesian coordi-nates, as they should.
III. THE PHYSICAL MODES IN THE SPHERICAL COULOMB GAUGE
As we have seen, the modified Feynman gauge defined by Eq. 共2.7兲 is useful in finding explicit mode functions. We also noted that this gauge results in the same physical modes as in the A0⫽0 gauge. There is another convenient gauge for
finding physical modes, i.e., the spherical Coulomb gauge, ⵜ˜iAi⫽0. 共One can readily show that any vector A
is gauge-equivalent to a vector satisfying this gauge condition.兲 Let us discuss the relation of the physical modes obtained in this gauge and those found in the previous section.4
It is clear that the physical modes A(2n;lm) in the previ-ous section are also in the spherical Coulomb gauge. The other independent physical modes must have Ai⫽0 in this gauge. First we note that if At and Ar are assumed to be spherically symmetric, then we find from the equations ⵜF
t⫽ⵜFr⫽0 that Ftr is t-independent and (rp/
冑
f h)Ftr is r-independent. Hence, if in addition the modes are assumed to be proportional to e⫺it with⫽0, they must be pure gauge because Ftr⫽0. 共A time-independent solution has Ftr⬀冑
f h/rp. This will represent a static Coulomb field, which we will disregard here.兲 To find the solutions which are not spherically symmetric we let At,Ar⬀Ylme⫺it. From the equationⵜFi⫽0 we find⫺iAt⫽
冑
f h 1 rp⫺2r冋
冑
f hr p⫺2A r册
. 共3.1兲 By substituting this and Ai⫽0 in ⵜFr⫽0, we find that Ar(1⬘n;lm)⫽Rl(1⬘n)(r)Ylme⫺it 共here we use primed indices to refer to the spherical Coulomb gauge兲, where冋
2 f ⫺ l共l⫹p⫺1兲 r2册
Rl (1⬘n)共r兲 ⫹1f drd冋
冑
hf 1 rp⫺2 d dr冉
冑
f hr p⫺2R l (1⬘n)共r兲冊
册
⫽0. 共3.2兲The component At(1⬘n;lm) can be found from Eq. 共3.1兲. 共Then, one can readily show that ⵜF
t⫽0.兲 Indeed one can verify that these modes are related to the ones in the modi-fied Feynman gauge by a gauge transformation:
A(1n;lm)→A(1⬘n;lm)⫽A(1n;lm)⫺ⵜ⌽(1n;lm), 共3.3兲 where ⌽(1n;lm)⫽ r 2⫺p l共l⫹p⫺1兲
冑
f h d dr冉
rp冑
f hRl (1n)冊
Ylme⫺it. 共3.4兲 This gives us At(1⬘n;lm)⫽ ir 2⫺p l共l⫹p⫺1兲冑
f h d dr冉
rp冑
f hRl (1n)冊
Ylme⫺it, 共3.5兲 Ar(1⬘n;lm)⫽ 2r2 l共l⫹p⫺1兲 1 f Rl (1n) Ylme⫺it. 共3.6兲IV. RESPONSE RATE OF A STATIC CHARGE OUTSIDE A FOUR-DIMENSIONAL SCHWARZSCHILD
BLACK HOLE
Here we briefly review, in the context of the previous two sections, the calculation of response rate in the Unruh vacuum 关9兴 of a static electric charge in Schwarzschild spacetime performed in Ref. 关7兴.
The line element of the four-dimensional Schwarzschild spacetime is
d2⫽ f共r兲dt2⫺ f共r兲⫺1dr2⫺r2ds22 共4.1兲 with f (r)⫽1⫺2M/r, where ds22⫽d2⫹sin2d2is the line element of the unit two-sphere. 共The horizon radius is rH ⫽2M.兲 We compute the response rate of the static electric charge to the Hawking radiation present in the Unruh vacuum. The charge is placed at (r,,)⫽(r0,0,0) as
described by the current density
j⫽␦t q
r2sin␦共r⫺r0兲␦共⫺0兲␦共⫺0兲. 共4.2兲 Since this current density is static, it couples to photons with zero energy. It turns out that the rate of response of this current to a single photon vanishes. However, the Bose-Einstein distribution of the thermal photons coming out of the horizon diverges at zero energy. This makes the rates of absorption and stimulated emission indefinite. This ambigu-ity can be resolved by replacing the current densambigu-ity共4.2兲 by j⫽( jt, jr,0,0), where
jt⫽
冑
2 qcos Etr2sin ␦共r⫺r0兲␦共⫺0兲␦共⫺0兲, 共4.3兲 4Again, we only treat solutions proportional to e⫺itwith⫽0.
jr⫽
冑
2qE sin Etr2sin ⌰共r⫺r0兲␦共⫺0兲␦共⫺0兲. 共4.4兲 Here,⌰(x)⫽1 if x⬎0 and ⌰(x)⫽0 if x⭐0. 共The factor of
冑
2 is necessary to make the time average of the squared charge equal to q2.兲 We take the limit E→0 in the end, assuming that the rate is continuous at E⫽0. This continuity has been verified in Minkowski spacetime关24兴.Since the current 共4.2兲 is conserved, it does not interact with pure-gauge particles corresponding to the creation op-erators alm(3n)†. It does interact with the states created by alm(0n)† but these have zero norm and do not contribute to physical probabilities. Hence, as usual, we need to consider only the physical modes. Note also that only physical modes I are relevant because At⫽Ar⫽0 for physical modes II.
Now, the transition amplitude from the Boulware vacuum 兩0
典
to the state兩1n;lm典
⫽alm(1n)†兩0典
by the classical current density j is given byAlmn ⫽i
冕
d4x
冑
⫺g j具
1n;lm兩Aˆ兩0典
. 共4.5兲 Let us recast Eq. 共4.5兲 asAlmn ⬅2iT lm
n ␦共⫺E兲, 共4.6兲 whereTlmn will be calculated later. The corresponding tran-sition rate, i.e. trantran-sition probability per asymptotic proper time, is expressed as
Rlmn ⫽2兩T lm
n 兩2␦共⫺E兲. 共4.7兲
Now we take into account that in the Unruh vacuum there is a thermal flux of photons coming out of the horizon with inverse temperature ⫽8M 共with no particles coming from infinity兲. In the limit E→0, only the absorption and stimulated emission of these thermal photons contribute to the response rate. For this reason, we treat only the modes coming out of the horizon (n⫽→). The total response rate due to these modes with fixed angular momentum, in which absorption, spontaneous and stimulated emissions are all taken into account, is
RElm⫽
冕
0 ⫹⬁ dRlm→冉
2 e⫺1⫹1冊
⫽2coth E 2 兩TElm→ 兩 2. 共4.8兲 Thus, in the limit E→0 we findR0lm⫽ lim
E→0 兩TElm→ 兩
2/共2ME兲. 共4.9兲
Next TElm→ will be calculated. The equation satisfied by the function ql→(r)⬅(2M/r)l(1→)(r) can be obtained from Eq. 共2.41兲 with p⫽2 as d dz
冋
共1⫺z 2兲dql → dz册
⫹冋
l共l⫹1兲⫺ 2 z⫹1 ⫺M 22共z⫹1兲 3 z⫺1册
ql → ⫽0, 共4.10兲where z⬅r/M⫺1. For small , the wave is almost com-pletely reflected back to the horizon. This implies that
l(1→)⬇⫺2 sin关共r*⫺r˜兲兴⬇⫺2r*⫹const 共r⫺rHⰆ2r
H
3
,兩r*兩Ⰶ1兲, 共4.11兲
where r˜are constants. The minus sign has been inserted for later convenience. The Wheeler tortoise coordinate r* de-fined by Eq. 共2.2兲 is written in Schwarzschild spacetime as
r*⫽r⫹2M ln
冉
r2 M⫺1
冊
. 共4.12兲 Thus, Eq.共4.11兲 can be written asl(1→)⬇⫺4Mln共z⫺1兲 共r⫺rHⰆ2rH
3,兩r*兩Ⰶ1兲.
共4.13兲 Now, Eq. 共4.10兲 can be solved explicitly for ⫽0. By choosing the solution for ql→ such that l(1→) ⫽关r/(2M)兴ql→ tends to zero as z→⫹⬁, we find in the small limit l(1→)⫽4M共z⫹1兲
冋
Ql共z兲⫺共z⫺1兲 l共l⫹1兲 dQl共z兲 dz册
, 共4.14兲 where this has been normalized such that Eqs. 共4.13兲 and 共4.14兲 are in agreement at r⬇rH.From gauge invariance of the amplitude 共4.5兲 it is clear that we may use the modes in the spherical Coulomb gauge, A(1⬘→;lm), in place of those in the modified Feynman gauge A(1→;lm). Then the contribution from the r-component will be suppressed in the low energy limit due to extra factors of 关see Eq. 共3.6兲兴. Therefore we need to consider only the t-component in this limit. The t-component can be found in the small limit from Eqs.共2.40兲, 共3.5兲 and 共4.14兲 as At(1⬘→;lm)⫽4i共z⫺1兲
冑
l共l⫹1兲 dQl共z兲 dz Ylm共,兲e ⫺it, 共4.15兲 where we have used the Legendre equation satisfied by Ql(z). By using Eq. 共4.15兲 in Eq. 共4.5兲 and comparing the result with Eq. 共4.6兲, we obtainTlm→ ⫽⫺i
冑
2q共z0⫺1兲冑
l共l⫹1兲 dQl共z0兲 dz0 Y ¯lm共 0,0兲 共4.16兲where z0⫽r0/ M⫺1. Then we find the proper response rate
R0lm
冑
f共r0兲 ⫽ q 2共z 0⫺1兲2 M l共l⫹1兲冑
f共r0兲冋
dQl共z0兲 dz0册
2 兩Ylm共0,0兲兩2, 共4.17兲 where the factor冑
f (r0) appears because here we arecalcu-lating the transition rate per proper time of the charge. We can sum over l and m by using
兺
m⫽⫺l l 兩Ylm共0,0兲兩2⫽共2l⫹1兲/共4兲 共4.18兲 and兺
l⫽1 ⬁ 2l⫹1 l共l⫹1兲冋
dQl共z兲 dz册
2 ⫽ 2Q1共z兲 共z2⫺1兲2. 共4.19兲A derivation of this formula is given in Appendix B. Thus, the total transition probability per proper time of the charge is Rtot⫽
兺
l⫽1 ⫹⬁兺
m⫽⫺l l R0lm冑
f共r0兲 ⫽q 2a共r 0兲 22 Q1共r0/ M⫺1兲, 共4.20兲 where a(r0)⫽Mr0⫺2/冑
f (r0) is the proper acceleration of thecharge.
It is of course possible to work with the modes in the modified Feynman gauge directly and obtain the same result. If we do so, the contribution will come from the r-component of the current because the t-components of physical modes I vanish.
V. LOW ENERGY ABSORPTION CROSS SECTION OF THE SCHWARZSCHILD BLACK HOLE
Here we consider the Schwarzschild black hole in p⫹2 dimensions given by the line element 共2.1兲 with f (r)⫽1 ⫺(rH/r)p⫺1. If the absorption probability of the black hole associated with physical modes I and II arePl(1) andPl(2), respectively, then the absorption cross section is given by
⫽ 共2兲 p p⍀pp
兺
l⫽1 ⬁ 关Ml (1)P l (1)⫹M l (2)P l (2)兴, 共5.1兲where l and are the angular momentum and frequency, respectively, of the modes and ⍀p⫽2( p⫹1)/2/⌫关(p⫹1)/2兴 is the volume of the unit p-sphere. Here, Ml(1) and Ml(2)are the multiplicities of the scalar and divergence-free vector spherical harmonics, respectively, on the p-sphere. These multiplicities are given by 共see, e.g., Refs. 关15,16兴兲
Ml(1)⫽共2l⫹p⫺1兲共l⫹p⫺2兲! 共p⫺1兲!l! , 共5.2兲 and Ml (2)⫽ 共2l⫹p⫺1兲共l⫹p⫺1兲! 共l⫹1兲共l⫹p⫺2兲共p⫺2兲!共l⫺1兲!. 共5.3兲
Equation共5.1兲 is derived in Appendix C.
For either physical modes I or II, we have for large r (⬇r*) that关see Eqs. 共2.41兲 and 共2.47兲兴
再
2⫹d 2 dr2⫺冋
l共l⫹p⫺1兲⫹ p共p⫺2兲 4册
1 r2冎
l (←)共r兲⬇0. 共5.4兲 The asymptotic solution with the correct normalization, i.e. with the coefficient of e⫺ir being one, isl(←)共r兲⬇
冑
2rJl⫹(p⫺1)/2共r兲 共rHⰆr兲. 共5.5兲 Thus, for smallrl(←)共r兲⬇Cl共r/rH兲l⫹p/2 共rHⰆrⰆ⫺1兲, 共5.6兲 where Cl⫽
冑
4 ⌫„l⫹共p⫹1兲/2…冉
rH 2冊
l⫹p/2 . 共5.7兲Now, near the horizon, where VⰆ2 (⫽1,2), these solutions behave like关see Eqs. 共2.41兲 and 共2.47兲兴
l(←)共r兲⬇Dl
()e⫺ir* 共r⫺rHⰆ2r
H
3兲, 共5.8兲
where the constants Dl()will be determined later. Hence, for 兩r*兩Ⰶ1 we have in this region l(←)(r)⬇Dl() and the absorption probability is given byPl()⫽兩Dl()兩2.
In order to find the coefficient Dl(), we solve the equa-tions forl(←)(r) with⫽0. For physical modes I, it turns out that the function rpRl(1←) satisfies a hypergeometric equation with the variable w⫽1⫺(r/rH)p⫺1. Thus, we find in the small limit
l(1←)共r兲⫽Dl(1)共r/rH兲1⫺p/2F„⫺共l⫹p⫺1兲/共p⫺1兲,
l/共p⫺1兲;1;w…. 共5.9兲
This function approaches Dl(1) for r→rH as required. By using the formula 关25兴
F共␣,;␥;w兲⬇⌫共␥兲⌫共⫺␣兲
⌫共兲⌫共␥⫺␣兲 共⫺w兲⫺␣
共␣⬍,⫺wⰇ1兲 共5.10兲 in Eq.共5.9兲, we find asymptotically
l(1←)共r兲⬇Dl(1)El (1)共r/rH兲l⫹p/2 共rHⰆrⰆ⫺1兲, 共5.11兲 where El(1)⫽ ⌫关共2l⫹p⫺1兲/共p⫺1兲兴 ⌫关l/共p⫺1兲兴⌫关共l⫹2p⫺2兲/共p⫺1兲兴. 共5.12兲 By comparing Eqs.共5.6兲 and 共5.11兲, we obtain
Pl
(1)⫽关Cl/E
l
Thus, the absorption probability behaves like 2l⫹p. Similarly, for physical modes II, we find in the small limit
l(2←)共r兲⫽Dl
(2)共r/rH兲( p/2⫺1)F„⫺共l⫹1兲/共p⫺1兲,
共l⫹p⫺2兲/共p⫺1兲;1;w…. 共5.14兲 Then by using Eq. 共5.10兲 we find asymptotically
l(2←)共r兲⬇Dl (2) El(2)共r/rH兲l⫹p/2 共rHⰆrⰆ⫺1兲, 共5.15兲 where El(2)⫽ ⌫关共2l⫹p⫺1兲/共p⫺1兲兴 ⌫关共l⫹p⫺2兲/共p⫺1兲兴⌫关共l⫹p兲/共p⫺1兲兴. 共5.16兲 By comparing Eqs.共5.6兲 and 共5.15兲, we obtain
Pl
(2)
⫽关Cl/El
(2)兴2. 共5.17兲
Again, the absorption probability behaves like2l⫹p. There-fore the physical modes with l⫽1 give the dominant contri-bution to the absorption probability, as expected.
By substitutingP1(1) andP1(2) in Eq.共5.1兲, we obtain the total absorption cross section for low energy as
⫽ 2r H 2A H 2共p⫹1兲
冋
2 p关⌫„p/共p⫺1兲…兴4 关⌫„共p⫹1兲/共p⫺1兲…兴2⫹1册
, 共5.18兲where AH⫽⍀prHp is the area of the black hole.
One would need to solve the differential equations satis-fied byl(←) numerically to find the low-energy absorption cross section for non-extreme Reissner-Nordstro¨m black holes.5However, it is possible to calculate it analytically for the extreme Reissner-Nordstro¨m black hole. We turn to this problem in the next section.
VI. LOW ENERGY ABSORPTION CROSS SECTION
OF THE EXTREME REISSNER-NORDSTRO¨ M
BLACK HOLE
The line element of the extreme Reissner-Nordstro¨m spacetime is given by Eq. 共2.1兲 with f (r)⫽h(r)⫺1⫽关1 ⫺(rH/r)p⫺1兴2. As in the Schwarzschild case, we have l(←)(r)⬇Cl(r/rH)l⫹p/2for rHⰆrⰆ⫺1where Clis given by Eq. 共5.7兲.
Now, let us analyze the physical modes close to the hori-zon. Concerning mode I, we write
冋
2⫹ d 2 dr*2⫺ l共l⫹p⫺1兲 共p⫺1兲2r*2册
l (1n)⬇0 共⫺2r*3ⰇrH兲, 共6.1兲where we have used that the Wheeler tortoise coordinate r*
关see Eq. 共2.2兲兴 behaves in this region as r*⬇⫺共p⫺1兲 ⫺2r H 2 r⫺rH . 共6.2兲 Hence we obtain l(1←)⬇Dl(1)
冑
⫺r* 2 H (1) 共⫺r*兲 共⫺2r*3ⰇrH兲 共6.3兲 with ⫽冋
1 4⫹ l共l⫹p⫺1兲 共p⫺1兲2册
1/2 ⫽1 2⫹ l p⫺1. 共6.4兲 This solution behaves like Dl(1)e⫺ir* 共up to a phase factor兲 very close to the horizon (⫺r*Ⰷ1), where the effectivepotential V1becomes negligible, as expected. Hence, the ab-sorption probability is Pl(1)⫽兩Dl(1)兩2. Note that, for small z
with positive, one has
H(1)共z兲⬇⫺i⌫共兲
冉
2 z冊
. 共6.5兲 Hence, l(1←)⬇Dl(1)Kl (1)冋
共p⫺1兲共r⫺rH兲 rH册
⫺1/2 共共rH兲1/3Ⰶ⫺r*Ⰶ1兲, 共6.6兲 where Kl(1)⫽⫺i⌫共兲冑
冋
2共p⫺1兲 rH册
⫺1/2 . 共6.7兲Again, in order to determine Dl(1), we solve the equation for l(1←)(r) in the small limit:
l(1←)共r兲⫽Dl(1)Kl (1)
冉
r rH冊
1⫺p/2 ⫻冉
1⫺ l l⫹p⫺1w冊
共⫺w兲 l/( p⫺1), 共6.8兲where w⫽1⫺(r/rH)p⫺1, and we note that Eq. 共6.8兲 has been normalized such that it agrees with Eq. 共6.6兲 near the horizon. On the other hand, we find asymptotically
l(1←)⬇ l l⫹p⫺1Dl (1) Kl(1)
冉
r rH冊
l⫹p/2 共rHⰆrⰆ⫺1兲. 共6.9兲 By comparing this with Eq.共5.6兲 we find5It is possible to obtain a result in a closed form if one takes the
Pl
(1)⫽
冏
共l⫹p⫺1兲Cll Kl(1)
冏
2
. 共6.10兲
We note that Pl(1)⬀p⫹2lp/(p⫺1). Hence the dominant con-tribution to the scattering cross section comes from the l ⫽1 term: P1 (1)⫽ 4 2p2 †共p⫺1兲1/( p⫺1)⌫关共p⫹3兲/2兴⌫关共p⫹1兲/共2p⫺2兲兴‡2 ⫻
冉
rH 2冊
2⫹p⫹2/(p⫺1) . 共6.11兲The contribution from physical modes II can be found in a similar manner. By substituting Eq. 共6.2兲 in Eq. 共2.47兲, we have near the horizon
冋
2⫹ d 2 dr*2⫺ 共l⫹1兲共l⫹p⫺2兲 共p⫺1兲2r*2册
l (2n) ⬇0 共⫺2r*3ⰇrH兲. 共6.12兲 Then l(2←) will have the same form as l(1←) in Eq.共6.3兲 except that in Eq.共6.4兲 is replaced by
⬘
⫽冋
14⫹共l⫹1兲共l⫹p⫺2兲 共p⫺1兲2册
1/2
.
For p⭓3, the contribution from physical mode II is sup-pressed compared to that from physical modes I since
⬘
⬎. For p⫽2, i.e. in four dimensions, Eqs. 共6.1兲 and 共6.12兲 satisfied by l(1n) andl(2n) are the same. Thus, the absorp-tion probabilities of these two types of modes are equal for each l.Hence, we obtain the following results: ⫽4 3共rH兲 4A H for p⫽2, 共6.13兲 ⫽ p共rH兲 2⫹2/(p⫺1)A H 共p⫹1兲关2共p⫺1兲兴2/( p⫺1)†⌫关共p⫹1兲/共2p⫺2兲兴‡2 for p⭓3. 共6.14兲 Equation 共6.13兲 is in agreement with the result derived by Gubser 关10兴 . It is interesting to note that the low energy absorption cross section is proportional to a fractional power of if p⭓4.
VII. CONCLUSIONS
In this paper we showed that the field equations for free electrodynamics can be reduced to decoupled scalar field equations in a modified Feynman gauge in the spacetime of spherically symmetric black hole. 共We also noted that the equations for the physical modes simplify in the spherical Coulomb gauge.兲 Then we examined the Gupta-Bleuler quantization in this modified Feynman gauge.
Next we reviewed the calculation of the response rate of a
static charge outside the four-dimensional Schwarzschild black hole. It is easy to extend our result to the Schwarzs-child black hole in arbitrary dimensions, though we cannot simplify the resulting infinite series. A closed-form expres-sion for the response rate of a static source has been obtained for massless scalar field in four-dimensional Reissner-Nordstro¨m black hole 关26兴. The extension of the results in this paper to the Reissner-Nordstro¨m black hole will also be possible in principle, but the result will probably not be ex-pressible共even兲 as an infinite series of familiar special func-tions in any dimensions.
Finally we calculated the absorption cross sections of low energy photons by the Schwarzschild and extreme Reissner-Nordstro¨m black holes in arbitrary dimensions. The corre-sponding cross section of massless scalar particles is known to be equal to the horizon area关27兴 as long as the black hole is spherically symmetric. No such universality holds for pho-tons as our results and previous results关10兴 show. It is inter-esting that there are two modes, modes I and II, which be-have differently in higher dimensions. It is also intriguing that the absorption cross section scales as a fractional power of the energy , i.e. as 2⫹p⫹2/(p⫺1) in p⫹2 dimensions. Finally, it would be interesting to compare the results here or similar results obtained using our gauge with those in string theory. 共See, e.g., Ref. 关10兴 and references therein for ex-amples of such comparison.兲
ACKNOWLEDGMENTS
L.C. and G.M. would like to acknowledge partial financial support from CAPES through the PICDT program and Con-selho Nacional de Desenvolvimento Cientı´fico e Tecno-lo´gico, respectively.
APPENDIX A: FIELD-STRENGTH TWO-POINT FUNCTION IN SPHERICAL POLAR COORDINATES IN FOUR-DIMENSIONAL MINKOWSKI SPACETIME
Minkowski spacetime is the simplest spherically symmet-ric spacetime. In this appendix we compute some compo-nents of the two-point function
G共x,x
⬘
兲⬅具
0兩Fˆ共x兲Fˆ共x⬘
兲兩0典
共A1兲 in spherical polar coordinates and verify that they agree with those previously computed in Cartesian coordinates in four-dimensional Minkowski spacetime. We need to consider only physical modes because pure-gauge modes do not con-tribute to Fˆ and the coefficient operators of the non-physical modes have zero commutators with all operators appearing on the right-hand side of Eq. 共A1兲.Since there is no horizon, all modes come in from infinity and go out to infinity. Therefore, we do not need here the label n⫽←, → which distinguished between the modes coming out of the horizon and those incoming from infinity. By the remark after Eq. 共2.43兲 we have l(1)(r) ⫽2r jl(r), where jl(x) is the spherical Bessel function of order l, in four-dimensional Minkowski spacetime. Hence for physical modes I we have exactly
Ar(1;lm)⫽2
冑
l共l⫹1兲r jl共r兲Ylm共,兲e
⫺it, 共A2兲
where Ylm(,) are the familiar scalar spherical harmonics on S2. The tr-component of the corresponding field strength is
Ftr(1;lm)⫽⫺i2
冑
l共l⫹1兲r jl共r兲Ylm共,兲e
⫺it, 共A3兲 since the t-component vanishes. These modes are sufficient for computing Gtrtr(x,x
⬘
) because physical modes II have At⫽Ar⫽0. We obtain with x⫽(t,r,,) and x⬘
⫽(t⬘
,r⬘
,⬘
,⬘
) Gtrtr共x,x⬘
兲⫽ 1 rr⬘
l兺
⫽1 ⬁ l共l⫹1兲兺
m⫽⫺l l Ylm共,兲Y¯lm共⬘
,⬘
兲 ⫻冕
0 ⬁ djl共r兲jl共r⬘
兲e⫺i(t⫺t⬘) ⫽ 1 82共rr⬘
兲2 l兺
⫽1 ⬁ l共l⫹1兲共2l⫹1兲 ⫻Pl共cos␥兲Ql冋
⫺共t⫺t⬘
⫺i⑀兲 2⫹r2⫹r⬘
2 2rr⬘
册
, 共A4兲 where we have used in the last step the formulas兺
m⫽⫺ll
Ylm共,兲Y¯lm共
⬘
,⬘
兲⫽2l⫹14 Pl共cos␥兲, with cos␥⫽coscos
⬘
⫹sinsin⬘
cos(⫺⬘
), and 关25兴冕
0 ⬁ djl共r兲jl共r⬘
兲e⫺i(t⫺t⬘) ⫽ 1 2rr⬘
Ql冋
⫺共t⫺t⬘
⫺i⑀兲2⫹r2⫹r⬘
2 2rr⬘
册
.We will simplify Eq. 共A4兲 later. Next we consider the component
G共x,x
⬘
兲⫽sinsin⬘
4 ˜⑀i j˜⑀i⬘j⬘G
i ji⬘j⬘共x,x
⬘
兲, where the anti-symmetric tensor on S2,˜⑀i j, has the compo-nent ˜⑀⫽(sin)⫺1. 共This definition differs from that of Regge and Wheeler 关28兴 by a minus sign.兲 Only the modes A(2;lm) contribute here. First, we have l(2)(r) ⫽2r jl(r). Then since Rl(2)(r)⫽l(2)(r) for p⫽2, we findAi(2;lm)⫽2r jl共r兲Yi(lm)共,兲e⫺it, 共A5兲
where the transverse vector spherical harmonics Yi(lm) are given by
Yi(lm)⫽˜⑀i jⵜ˜ jY
lm
冑
l共l⫹1兲. 共A6兲The corresponding field strength is given by
1 2˜⑀
i jF i j
(2;lm)⫽2
冑
l共l⫹1兲r jl共r兲Ylm共,兲e⫺it.共A7兲 Then in exactly the same way as for Gtrtr, we find
G共x,x
⬘
兲⫽sinsin⬘
82 l兺
⫽1 ⬁ l共l⫹1兲共2l⫹1兲 ⫻Pl共cos␥兲Ql冋
⫺共t⫺t⬘
⫺i⑀兲 2⫹r2⫹r⬘
2 2rr⬘
册
. 共A8兲 On the other hand one can calculate G␣␥␦(x,x⬘
) using plane wave modes. In the Cartesian coordinate system this is given in components as共see, e.g., Ref. 关29兴兲Gtata(C)共x,x
⬘
兲⫽⫺4兵1⫹2关共a兲2⫺共t兲2兴⫺2其G6⫹共x,x⬘
兲, 共A9兲 Gtatb(C)共x,x⬘
兲⫽⫺8ab⫺2G6⫹共x,x⬘
兲, 共A10兲 Gtaab (C)共x,x⬘
兲⫽Gabta(C) 共x,x⬘
兲⫽⫺8 tb⫺2G6⫹共x,x⬘
兲, 共A11兲 Gabab(C) 共x,x⬘
兲⫽4兵1⫹2关共a兲2⫹共b兲2兴⫺2其G 6 ⫹共x,x⬘
兲, 共A12兲 Gacbc(C) 共x,x⬘
兲⫽8ab⫺2G6⫹共x,x⬘
兲, 共A13兲where a,b,c take values x, y ,z with a, b and c being all distinct 共no summation convention is used here兲 and ␣ ⫽x␣⫺x
⬘
␣. We have defined G6
⫹(x,x
⬘
)⫽(434)⫺1, where2⫽共t⫺t
⬘
⫺i⑀兲2⫺共x⫺x⬘
兲2⫺共y⫺y⬘
兲2⫺共z⫺z⬘
兲2⫽共t⫺t
⬘
⫺i⑀兲2⫺r2⫺r⬘
2⫹2rr⬘
cos␥. 共A14兲The components Gtrtr in spherical polar coordinates can be obtained from the Cartesian ones共A9兲–共A13兲 by using stan-dard tensor transformation:
Gtrtr共x,x
⬘
兲⫽4 rr⬘
共共xx⬘
⫹yy⬘
⫹zz⬘
兲 2 ⫹2关共xy⬘
⫺x⬘
y兲2⫹共xz⬘
⫺x⬘
z兲2 ⫹共yz⬘
⫺y⬘
z兲2兴兲G6 ⫹共x,x⬘
兲 2 , ⫽ 1 26兵关共t⫺t⬘
⫺i⑀兲 2⫺r2⫺r⬘
2兴 ⫻cos␥⫹2rr⬘
其. 共A15兲 Similarly, we find G共x,x⬘
兲⫽r 2r⬘
2sinsin⬘
26 ⫻兵关共t⫺t⬘
⫺i⑀兲2⫺r2⫺r⬘
2兴cos␥⫹2rr⬘
其. 共A16兲 Equations 共A15兲 and 共A4兲 and Eqs. 共A16兲 and 共A8兲 can be shown to agree by using the formula兺
l⫽1 ⬁l共l⫹1兲共2l⫹1兲Pl共t兲Ql共z兲⫽2共tz⫺1兲
共z⫺t兲3 , 共A17兲
which can be proved by applying (d/dt)关(t2⫺1)d/dt兴 on both sides of the well-known formula
兺
l⫽0 ⬁共2l⫹1兲Pl共t兲Ql共z兲⫽1/共z⫺t兲 共A18兲 and using the Legendre equation satisfied by Pl(t).
APPENDIX B: A DERIVATION OF EQ.„4.19…
Define F共z兲⬅
兺
l⫽1 ⬁ 2l⫹1 l共l⫹1兲冋
d dzQl共z兲册
2 . 共B1兲By the Legendre equation satisfied by Ql(z), we have d dz
再
共z 2⫺1兲2冋
d dzQl共z兲册
2冎
⫽2l共l⫹1兲共z2⫺1兲Ql共z兲d dzQl共z兲. 共B2兲 Hence, d dz关共z 2⫺1兲2F共z兲兴⫽共z2⫺1兲d dz再
l兺
⫽1 ⬁ 共2l⫹1兲关Ql共z兲兴2冎
. 共B3兲 Now recall that关5兴兺
l⫽0 ⬁共2l⫹1兲关Ql共z兲兴2⫽ 1
z2⫺1. 共B4兲 关This can be obtained by squaring both sides of Eq. 共A18兲 and integrating over the variable t from ⫺1 to 1.兴 By sub-stituting Eq. 共B4兲 in Eq. 共B3兲, we obtain
d dz关共z 2⫺1兲2F共z兲兴⫽共z2⫺1兲d dz
冋
1 z2⫺1⫺ 1 4冉
ln z⫹1 z⫺1冊
2册
⫽⫺ 2z z2⫺1⫹ln z⫹1 z⫺1, 共B5兲where we have used Q0共z兲⫽
1 2 ln
z⫹1
z⫺1. 共B6兲
Integration of Eq. 共B5兲 leads to 共z2⫺1兲2F共z兲⫽z lnz⫹1
z⫺1 ⫺2⫹C, 共B7兲 where C is a constant. Recalling that Ql(z)⬃z⫺l⫺1 as z
→⬁, we find that the left-hand side tends to zero in this
limit. Therefore C⫽0. Hence,
F共z兲⫽ 1 共z2⫺1兲2
冉
z ln z⫹1 z⫺1 ⫺2冊
⫽ 2Q1共z兲 共z2⫺1兲2. 共B8兲APPENDIX C: ABSORPTION CROSS SECTION IN SPHERICAL POLAR COORDINATES:
A DERIVATION OF EQ.„5.1…
We consider the divergence-free vector eigenfunctions of the Laplacian on the ( p⫹1)-dimensional Euclidean space. That is, we examine the solutions of the following equations:
ⵜb共ⵜbBa⫺ⵜaB
b兲⫽⫺2Ba 共C1兲 and
ⵜaBa⫽0. 共C2兲
Let x⫽(x1,x2, . . . ,xp⫹1) be the Cartesian coordinates. There are plane wave solutions of the form
Bj
(C;kˆ⑀ˆ)⫽⑀ˆ
jeik•x 共C3兲 with⬎0, where kˆ is a unit vector and k⫽kˆ, and⑀ˆ is a polarization unit vector orthogonal to kˆ. We define
共B(1),B(2)兲 E⬅
冕
dp⫹1xB(1)
aB
(2)a, 共C4兲
for any vectors Ba(1) and Ba(2)on this space. Then 共B(C;kˆ⑀ˆ),B(C;⬘kˆ⬘⑀ˆ⬘)兲
E⫽共2兲p⫹1⑀ˆ•⑀ˆ
