Low-energy sector quantization of a massless scalar field outside a Reissner-Nordström black hole and static sources

Low-energy sector quantization of a massless scalar field outside a Reissner-Nordstro¨m

black hole and static sources

J. Castin˜eiras and G. E. A. Matsas

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, Sa˜o Paulo, Sa˜o Paulo, Brazil 共Received 22 February 2000; published 31 July 2000兲

We quantize the low-energy sector of a massless scalar field in Reissner-Nordstro¨m spacetime. This allows the analysis of processes involving soft scalar particles occurring outside charged black holes. In particular, we compute the response of a static scalar source interacting with Hawking radiation using the Unruh 共and the Hartle-Hawking兲 vacuum. This response is compared with the one obtained when the source is uniformly accelerated in the usual vacuum of Minkowski spacetime with the same proper acceleration. We show that both responses are in general different in opposition to the result obtained when the Reissner-Nordstro¨m black hole is replaced by a Schwarzschild one. The conceptual relevance of this result is commented on.

PACS number共s兲: 04.70.Dy, 04.62.⫹v

I. INTRODUCTION

We study the canonical quantization of a massless scalar field outside a Reissner-Nordstro¨m black hole. This is not easy to fully accomplish mostly because the explicit form of the positive- and negative-energy modes is unknown in terms of the usual special functions. This has led many re-searchers to use numerical methods to analyze quantum field issues in this and in similar backgrounds 共see, e.g., 关1兴 and references therein兲. Here we follow the procedure developed in Ref. 关2兴 to analytically quantize the low-energy sector of the scalar field in Reissner-Nordstro¨m spacetime. This al-lows the analytic investigation of processes involving soft particles such as, e.g., the synchrotron radiation emitted by scalar sources orbiting charged black holes 关3兴.

We use our results to analyze the following conceptual issue. It was recently found 关4兴 that the responses of 共i兲 a static scalar source in the Schwarzschild spacetime with the Unruh vacuum and of 共ii兲 a uniformly accelerated scalar source in Minkowski spacetime with the usual vacuum are equivalent provided that both sources have the same proper acceleration. It would be interesting to study, thus, whether or not this equivalence is preserved when the Schwarzschild black hole is supplied with some electric charge. Because 共structureless兲 static sources can only interact with

zero-energy particles, we can use our low-zero-energy quantization to

answer this question accurately. Eventually we show that the presence of electric charge in the black hole breaks the above equivalence. This in conjunction with the fact that no equiva-lence is found in the Schwarzschild spacetime when the sca-lar field is replaced by the Maxwell one关5兴 suggests that the equivalence found in 关4兴 is not valid, in general, for other spacetimes and quantum fields. Whether or not there is something deeper behind it remains an open question for us. We will adopt natural unitsប⫽c⫽G⫽kB⫽1 and signature (⫹ ⫺ ⫺ ⫺).

The paper is organized as follows. In Sec. II we quantize the low-energy sector of the massless scalar field outside a Reissner-Nordstro¨m black hole. In Sec. III we compute the response of a static scalar source interacting with Hawking radiation using the Unruh共and the Hartle-Hawking兲 vacuum, and compare the result with the one obtained when the source is uniformly accelerated in Minkowski spacetime

with the usual inertial vacuum. We present our final consid-erations in Sec. IV.

II. QUANTIZATION OF A MASSLESS SCALAR FIELD OUTSIDE A CHARGED BLACK HOLE

The line element of a Reissner-Nordstro¨m black hole with mass M and electric charge Q⭐M can be written as 关6兴

ds2⫽ f共r兲dt2⫺ f共r兲⫺1dr2⫺r2共d␪2⫹sin2␪d␸2兲,

共2.1兲 where

f共r兲⬅共1⫺r_⫹/r兲共1⫺r_⫺/r兲 共2.2兲 and r_⫾⬅M⫾冑M2⫺Q2. Outside the outer event horizon, i.e., for r⬎r_⫹, we have a global timelike isometry generated by the Killing field ⳵t.

Let us now consider a free massless scalar field⌽(x␮) in this background described by the action

S⫽1

2

冕

d

4_x

冑⫺gⵜ

␮_⌽ⵜ

␮⌽, 共2.3兲

where g⬅det兵g_␮␯其. In order to quantize the field, we look for a complete set of positive-energy solutions of the Klein-Gordon equation,䊐u_␻lm⫽0, in the form

u_␻lm⫽

冑

␻

␲ ␺␻l共r兲

r Ylm共␪,␸兲e

⫺i␻t_{, 共2.4兲}

where ␻⭓0, l⭓0 and m苸关⫺l,l兴 are the frequency and angular momentum quantum numbers. The factor

冑

␻/␲was inserted for later convenience and Ylm(␪,␸) are the spherical harmonics. As a consequence␺_␻l(r) must satisfy

冋

⫺ f共r兲_drd

冉

f共r兲 d

dr

冊

⫹Veff共r兲

册

␺␻l共r兲⫽␻

2_␺

␻l共r兲,

共2.5兲 where the effective scattering potential Veff(r) is given by

Veff共r兲⫽

冉

1⫺ 2 M r ⫹ Q2 r2

冊冉

2 M r3 ⫺ 2Q2 r4 ⫹ l共l⫹1兲 r2

冊

. 共2.6兲 Note that Eq. 共2.5兲 admits two sets of independent solutions which will be labeled by ␺_␻l␣ (r) with␣⫽I,II. As a result, we can expand the scalar field ⌽(x␮) in terms of annihila-tion a_␻lm␣ and creation a_␻lm␣† operators, as usual

⌽共x␮_兲⫽

兺

␣⫽I,II l

兺

⫽0 ⬁

兺

m⫽⫺l m⫽⫹l

冕

0 ⫹⬁ d␻关u_␻lm␣ 共x␮兲a_␻lm␣ ⫹H.c.兴, 共2.7兲 where u_␻lm␣ (x␮) are orthonormalized according to the Klein-Gordon inner product关7兴:

i

冕

⌺t d⌺ n␮共u_␻lm␣ *ⵜ_␮u␣_␻⬘_⬘l⬘m⬘⫺ⵜ_␮u_␻lm␣ *•u_␻␣⬘_⬘l⬘m⬘兲 ⫽␦␣␣_⬘␦ll⬘␦mm⬘␦共␻⫺␻

⬘

兲, 共2.8兲 i

冕

⌺t d⌺ n␮共u_␻lm␣ ⵜ_␮u␣_␻⬘_⬘l⬘m⬘⫺ⵜ_␮u_␻lm␣ •u␣_␻⬘⬘_l⬘m⬘兲 ⫽0. 共2.9兲

Here n␮is the future-pointing unit vector normal to the vol-ume element of the Cauchy surface ⌺t. As a consequence,

a_␻lm␣ and a_␻lm␣† satisfy simple commutation relations 关a␻lm␣ ,a␻_⬘l⬘m⬘

␣_⬘†

兴⫽␦␣␣_⬘␦ll⬘␦mm⬘␦共␻⫺␻

⬘

兲. 共2.10兲 The Boulware vacuum兩0

典

is defined by a_␻lm␣ 兩0

典

⫽0 for ev-ery␣, ␻, l, and m关8兴.

A. Small frequency modes

The general solution of Eq.共2.5兲 in terms of special func-tions is not known. However, this can be found for small frequencies as follows. First let us rewrite Eq.共2.5兲 with␻ ⫽0 as d dz

冋

共1⫺z 2_兲d dz关␺␻l共y兲/y兴

册

⫹l共l⫹1兲关␺␻l共y兲/y兴⫽0, 共2.11兲

where we have defined y⬅r/2M, y_⫾⬅r_⫾/2M , and

z⬅ 2y⫺1

y_⫹⫺y_⫺. 共2.12兲

From the Legendre equation共2.11兲, we obtain the two inde-pendent solutions

␺␻lI 共y兲⬅C␻Iy Ql关z共y兲兴, 共2.13兲 ␺␻lII共y兲⬅C␻IIy Pl关z共y兲兴, 共2.14兲 where Ql(z) and Pl(z) are the Legendre functions, and C_␻ I and C_␻II are normalization constants. In order to determine them, we shall analyze in more detail the solutions of Eq. 共2.5兲 near the horizon and at infinity, which can be normal-ized for arbitrary ␻.

B. Normal modes near the horizon and at infinity First let us note that by making the change of variables,

y→x⫽y⫹共y⫹兲

2_ln兩y⫺y

⫹兩⫺共y⫺兲2ln兩y⫺y⫺兩

y_⫹⫺y_⫺ ,

共2.15兲 Eq. 共2.5兲 takes the form

冋

⫺ d 2 dx2⫹4M 2_V eff关r共x兲兴

册

␺␻l共x兲⫽4M2␻2␺␻l共x兲. 共2.16兲 It is convenient to write the two independent solutions of Eq. 共2.16兲 such that ␺_␻l→_{(x) and ␺}

␻l

←_{(x) are associated with}

purely incoming modes from the past white-hole horizon H⫺ _{and from the past null infinityJ}⫺_{, respectively. These}

modes are orthogonal to each other with respect to the Klein-Gordon inner product 共2.8兲. This can be seen by choosing ⌺t⫽H⫺艛J⫺ in Eq. 共2.8兲 and recalling that ␺_␻l→(x) and ␺␻l←(x) vanish onJ⫺ andH⫺, respectively. Hence, by

not-ing from Eq. 共2.6兲 that close to (x⬍0,兩x兩Ⰷ1) and far away from (xⰇ1) the horizon, the scattering potential becomes

Veff(r)⬇0 and Veff(r)⬇l(l⫹1)/r2, respectively, we write

␺_␻l→_共x兲⬇

再

A␻l共e 2i M␻x_⫹R ␻l →_e⫺2iM␻x_{兲 共x⬍0, 兩x兩Ⰷ1兲,} 2il⫹1A_␻lT_␻l→M␻xh_l(1)共2M␻x兲 共xⰇ1兲, 共2.17兲 and ␺␻l←共x兲⬇

再

B_␻lT_␻l←e⫺2iM␻x 共x⬍0,兩x兩Ⰷ1兲, B_␻l关2共⫺i兲l⫹1M␻xh(1)_l 共2M␻x兲*⫹2il_⫹1R ␻l ←_M␻xh l (1)_{共2M␻x兲兴 共xⰇ1兲.} 共2.18兲

Here h_l(1)(2 M␻x) are the spherical Hankel functions and 兩R_␻l←_兩2_,兩R ␻l →_兩2 _{and 兩T} ␻l ←_兩2_,兩T ␻l

→_兩2 _{are the reflection and} transmission coefficients, respectively, satisfying the usual probability conservation equations 兩R_␻l→兩2⫹兩T_␻l→兩2⫽1 and 兩R_␻l←兩2_⫹兩T ␻l ←_兩2_{⫽1. Note that h} l (1) (x)⬇(⫺i)l⫹1exp(ix)/x for兩x兩Ⰷ1. The normalization constants A_␻land B_␻l are ob-tained 共up to an arbitrary phase兲 by letting normal modes 共2.4兲 in the Klein-Gordon inner product 共2.8兲 and using Eq. 共2.16兲 to transform the integral into a surface term:

1 ␻⫺␻

⬘

冋

␺␻l共x兲 d dx␺␻*⬘l共x兲⫺␺␻*⬘l共x兲 d dx␺␻l共x兲

册

冏

_x→⫺⬁ x→⫹⬁ ⫽2␲_{␻ ␦}M 共␻⫺␻

⬘

兲. 共2.19兲 By using the asymptotic solutions共2.17兲,共2.18兲 in Eq. 共2.19兲, we obtain A_␻l⫽B_␻l⫽(2␻)⫺1.

C. Normalization constants

Now we are able to determine the normalization constants

C_␻I and C_␻II by comparing Eqs. 共2.13兲,共2.14兲 close and far away from the black hole with our normalized functions 共2.17兲,共2.18兲 in the low-frequency regime (2M␻xⰆ1). Let

us begin by noticing that for 2 M␻xⰆ1, we have, near the

horizon关see Eq. 共2.17兲兴,

␺␻l→共x兲⬇Mx

冋

共1⫹R␻l→兲

2 M␻x ⫹i共1⫺R␻l

→_兲

册

_{共x⬍0, 兩x兩Ⰷ1兲.}

共2.20兲 In order that Eq. 共2.20兲 have a good behavior in the low-frequency regime we conclude thatR_␻l→⬇⫺1⫹O(␻). As a consequence, for 2 M␻xⰆ1 we obtain from Eq. 共2.20兲 that

␺_␻l→_{共x兲⬇2iMx 共x⬍0, 兩x兩Ⰷ1兲. 共2.21兲}

Now, we recall that in the low-frequency regime ␺_␻l→(x) is mostly reflected by the scattering potential back to the hori-zon and thus cannot be associated with␺_␻lII(x) which grows asymptotically 关see Eq. 共2.14兲 and recall that Pl(z)⬃zl as

zⰇ1 (rⰇr_⫹)兴. This is not so for ␺_␻lI (x) which decreases asymptotically and indeed fits␺_␻l→(x). This can be shown as follows. Let us first note that, for z⬇1 (r⬇r_⫹),

Ql共z兲⬇ 1 2ln

冏

z⫹1 z⫺1

冏

⫺k

兺

⫽1 l 1 k

⬇关⫺x⫹y⫹⫹ln共y⫹⫺y⫺兲兴共y⫹⫺y⫺兲 2 y_⫹2 ⫺k

兺

⫽1

l 1

k,

共2.22兲 where we have used Eqs. 共2.12兲 and 共2.15兲. Thus, close to the horizon, we obtain from Eq.共2.13兲 that

␺␻lI 共x兲⬇⫺C␻I 共y⫹

⫺y_⫺兲

2y_⫹ x 共x⬍0, 兩x兩Ⰷ1兲. 共2.23兲

Comparing Eqs. 共2.23兲 and 共2.21兲 we find the normalization constant

C_␻I⫽⫺4iMy_⫹/共y_⫹⫺y_⫺兲. 共2.24兲 Therefore, we write, from Eq. 共2.13兲,

␺␻lI 共x兲⫽

⫺4iMy⫹y Ql关z共y兲兴

y_⫹⫺y_⫺ , 共2.25兲

and from Eq.共2.4兲, we obtain the corresponding normalized low-frequency modes 共up to an arbitrary phase兲:

u_␻lmI 共x␮兲⫽ 2y⫹␻

1/2

␲1/2_共y

⫹⫺y⫺兲

Ql关z共x兲兴Ylm共␪,␸兲e⫺i␻t. 共2.26兲 Now we fit ␺_␻lI (x) and ␺_␻l→(x) asymptotically to determine the low-frequency transmission coefficient 兩T_␻l→兩2 _{关see Eq.} 共2.17兲兴. For xⰇ1, Eq. 共2.25兲 becomes

␺_␻lI _共x兲⬇⫺2iM共l!兲 2_y

⫹共y⫹⫺y⫺兲lx⫺l

共2l⫹1兲! 共2M␻xⰆ1兲,

共2.27兲 where we have used that, in this region,

Ql关2y/共y⫹⫺y⫺兲兴⬇ 共l!兲2_共y

⫹⫺y⫺兲l⫹1y⫺l⫺1

2共2l⫹1兲! . Now, from Eq.共2.17兲, we have in the low-frequency regime and for xⰇ1 that

␺␻l→共x兲⬇

il共2l兲!T_␻l→x⫺l

22l⫹1l! Ml␻l⫹1 共2M␻xⰆ1兲, 共2.28兲

where we have used that

h_l(1)共2M␻x兲⫽ jl共2M␻x兲⫹inl共2M␻x兲 共2.29兲 and the fact that the spherical Bessel and Newman functions satisfy †see Eq. 共11.156兲 of Ref. 关9兴‡

j_l共2M␻x兲⬇ 2 l_l! 共2l⫹1兲! 共2 M␻x兲l 共2.30兲 and nl共2M␻x兲⬇⫺ 共2l兲! 2l_l! 共2M␻x兲 ⫺(l⫹1)_{, 共2.31兲}

respectively, for 2 M␻xⰆ1. Thus Eqs. 共2.27兲 and 共2.28兲

co-incide provided that

T_␻l→_⫽22l⫹2共⫺i兲l⫹1y⫹共y⫹⫺y⫺兲l共l!兲3共M␻兲l⫹1

共2l⫹1兲!共2l兲! .

共2.32兲 共Eventually this will be also used as a consistency check for our calculations.兲

Now, let us turn our attention to␺_␻l←(x) which should be fitted with ␺_␻lII(x). Note that ␺_␻lI (x) grows close to the ho-rizon and so cannot be associated with low-frequency left-moving modes which must be mostly reflected back to infin-ity by the scattering potential关see Eq. 共2.13兲 and recall that

Q_l(z)⬇⫺log兩z⫺1兩1/2 as z⬇1兴. In order to fit ␺_␻l←(x) and

␺␻lII(x) asymptotically, we must use Eqs. 共2.29兲 and

共2.30兲,共2.31兲 in Eq. 共2.18兲 for xⰇ1. Moreover, it turns out that this compatibility is achieved if and only if R_␻l← ⬇(⫺1)l_{⫹1. As a result we obtain}

␺␻l←共x兲⬇

22l⫹1共⫺i兲l⫹1l!␻l共Mx兲l⫹1

共2l⫹1兲! 共xⰇ1兲 共2.33兲

for 2 M␻xⰆ1. Now, we note that P_l(z)⬇关(2l)!/2l_(l!)2_兴zl for zⰇ1 †see Eqs. 共8.837.2兲 and 共8.339.2兲 of Ref. 关10兴‡. Hence, using Eqs.共2.12兲 and 共2.14兲, we find that

␺_␻lII_共x兲⬇C ␻ II 共2l兲!y l⫹1 共l!兲2_共y ⫹⫺y⫺兲l 共xⰇ1兲. 共2.34兲 Comparing this equation with Eq. 共2.33兲 and recalling that

x⬇y at infinity, we find the normalization constant

C_␻II⫽2 2l⫹1_共⫺i兲l⫹1_共l!兲3_Ml⫹1_共y ⫹⫺y⫺兲l␻l 共2l⫹1兲!共2l兲! . 共2.35兲 Therefore ␺␻lII共x兲⫽

22l⫹1共⫺i兲l⫹1共l!兲3Ml⫹1共y_⫹⫺y_⫺兲l␻ly Pl关z共y兲兴 共2l⫹1兲!共2l兲!

共2.36兲 and the corresponding normalized small frequency modes are 共up to an arbitrary phase兲

u_␻lmII 共x␮兲⫽2

2l_共l!兲3_Ml_共y

⫹⫺y⫺兲l␻l⫹1/2

␲1/2_{共2l⫹1兲!共2l兲!}

⫻Pl关z共x兲兴Ylm共␪,␸兲e⫺i␻t. 共2.37兲 It can be directly verified that by fitting Eq. 共2.36兲 close to the horizon with Eq. 共2.18兲 for 2M␻xⰆ1, we obtain T_␻l←

⫽T_␻l→ _{关see Eq. 共2.32兲兴, as indeed required for consistency.}

Clearly this guaranties that 兩R_␻l←兩⫽兩R_␻l→兩. Note, however, that R_␻l← andR_␻l→ will in general differ by a phase共in con-trast to T_␻l← andT_␻l→).

Equation共2.7兲 in conjunction with Eqs. 共2.26兲 and 共2.37兲 concludes our low-frequency sector quantization.

III. RESPONSE OF A STATIC SCALAR SOURCE INTERACTING WITH HAWKING RADIATION Let us now compute the response of a static source to the Hawking radiation in the Reissner-Nordstro¨m spacetime. We will consider both Unruh and Hartle-Hawking vacua. Let us describe our pointlike scalar source lying at (r0,␪0,␸0) by

j共x␮兲⫽ q

冑⫺h

␦共r⫺r0兲␦共␪⫺␪0兲␦共␸⫺␸0兲, 共3.1兲 where q is a small coupling constant and h⫽⫺ f⫺1r4sin2␪is the determinant of the spatial metric induced over the equal time hypersurface⌺t. Note that Eq.共3.1兲 guarantees that

冕

⌺t

d⌺ j⫽q 共3.2兲

wherever the source lies. Let us now couple our source j (x␮) to a massless scalar field⌽(x␮) as described by the interac-tion acinterac-tion

S_I⫽

冕

d4x

冑⫺g j⌽.

共3.3兲

The total source response, i.e., total particle emission and absorption probabilities per proper time associated with the source, is given by R⬅

兺

␣⫽I,II

兺

l⫽0 ⬁

兺

m⫽⫺l l

冕

0 ⫹⬁ d␻R_␻lm␣ , 共3.4兲 where R_␻lm␣ ⬅␶⫺1兵兩A_␻lm␣ em兩2关1⫹n␣共␻兲兴⫹兩A_␻lm␣ abs兩2n␣共␻兲其 共3.5兲 and␶ is the source’s total proper time.共This is well defined since our source is pointlike.兲 Here A_␻lm␣ em⬅

具

␣␻lm兩SI兩0

典

andA_␻lm␣ abs⬅

具

0兩SI兩␣␻lm

典

are the emission and absorption amplitudes, respectively, of Boulware states 兩␣␻lm

典

, at the tree level. Moreover,

n_U␣共␻兲⬅

再

共e ␻␤_⫺1兲⫺1 _{for ␣⫽I,} 0 for ␣⫽II, 共3.6兲 and n_HH␣ 共␻兲⬅

再

共e ␻␤_⫺1兲⫺1 _{for ␣⫽I,}

共e␻␤⫺1兲⫺1 _{for ␣⫽II,} 共3.7兲

for the Unruh and Hartle-Hawking vacua, respectively, with

␤⫺1_⫽y⫹⫺y⫺

8␲M y_⫹2 . 共3.8兲

We recall that the Unruh vacuum is characterized by a ther-mal flux leaving H⫺ with Hawking temperature␤⫺1 at in-finity given by Eq. 共3.8兲 while the Hartle-Hawking vacuum has in addition a thermal flux coming fromJ⫺characterized by the same temperature at infinity 关11兴.

Let us note that because structureless static sources 共3.1兲 can only interact with zero-energy modes, the total response of this source in the Boulware vacuum vanishes. This is not so, however, in the presence of a background thermal bath since the absorption and共stimulated兲 emission rates render it nonzero. In order to deal with zero-energy modes, we need a

‘‘regulator’’ to avoid the appearance of intermediate indefi-nite results. 共For a more comprehensive discussion on the interaction of static sources with zero-energy modes, see Ref. 关12兴.兲 For this purpose we let the coupling constant q smoothly oscillate with frequency ␻₀, writing Eq. 共3.1兲 in the form j_␻ 0共x ␮_兲⫽ q␻0

冑⫺h

␦共r⫺r0兲␦共␪⫺␪0兲␦共␸⫺␸0兲, 共3.9兲 where q_␻

0⬅冑2q cos(␻0t) and taking the limit␻0→0 at the

end. The factor

冑

2 has been introduced to guarantee that the time average

具

兩q_␻

0(t)兩

2

_典

t⫽q2 since at the tree level the ab-sorption and emission rates are functions of q2. By using Eqs. 共3.9兲 and 共2.7兲 in Eq. 共3.3兲 we obtain the absorption amplitude A_␻lm␣ abs⫽q冑2␲␻0共␺_␻ 0l ␣ _共r 0兲/r0兲f1/2共r0兲 ⫻Ylm共␪0,␸0兲␦共␻⫺␻0兲, 共3.10兲 and we recall that兩A_␻lm␣ em兩⫽兩A_␻lm␣ abs兩. By letting Eq.共3.10兲 into Eq.共3.5兲 we obtain

R_␻lm␣ ⫽q2␻0关兩␺_␻ 0l ␣ _共r 0兲兩2/r0 2_{兴 f}1/2_共r 0兲 ⫻兩Ylm共␪0,␸0兲兩2关1⫹2n␣共␻0兲兴␦共␻⫺␻0兲, 共3.11兲 where it was used that the source’s total proper time is ␶ ⫽2␲f1/2(r₀)lim_␻→0␦(␻).关Here f1/2(r₀) is the gravitational redshift factor.兴

Let us first consider the Unruh vacuum. By using Eqs. 共2.25兲, 共3.6兲, and 共3.11兲 in Eq. 共3.4兲 and making ␻0→0 at the end, we compute the total response

RU⫽

q2a共M⫺Q2/r_⫹兲 4␲2共M⫺Q2/r0兲

共3.12兲 关note that modes u␻lmII (x) do not give any contribution here兴,

where a⫽f ⫺1/2_共r 0兲 2 d f共r₀兲 dr0

is the source’s proper acceleration and we have used

兺

m_⫽⫺l l 兩Ylm共␪0,␸0兲兩2⫽ 2l⫹1 4␲ 共3.13兲 and关4兴

兺

l⫽0 ⬁ 兩Ql共s兲兩2共2l⫹1兲⫽ 1 s2⫺1. 共3.14兲

Next we compare Eq. 共3.12兲 with

RM⫽

q2a

4␲2, 共3.15兲

which is the response associated with our scalar source when it is uniformly accelerated in the usual vacuum of Minkowski spacetime with proper acceleration a. We note that although Eqs.共3.12兲 and 共3.15兲 coincide when Q⫽0, as found in Ref. 关4兴, they do not for Q⫽0. As a result, the presence of electric charge inside the black hole breaks the response equivalence.

We note that the equality between Eqs. 共3.12兲 and 共3.15兲 is recovered when r0⬇r⫹. Hence, close to the horizon, a static source in the Unruh vacuum responds as if it were static in the Rindler wedge 共i.e., uniformly accelerated in Minkowski spacetime兲 with the usual inertial vacuum pro-vided that both sources have the same proper acceleration. Moreover, Eq.共3.12兲 can be written in this region in terms of the proper temperature 关13兴 ␤₀⫺1⫽␤⫺1/

冑

f (r₀) on the source’s location as

R_U⬇ q

2 2␲␤0

. 共3.16兲

Equation 共3.16兲 coincides with the response associated with our source when it is at rest in Minkowski spacetime with a background thermal bath characterized by a temperature

␤0⫺1. This result is not surprising because close to the hori-zon the scattering potential vanishes and the zero-energy modes leavingH⫺are completely reflected back towards the horizon.

Now let us turn our attention to the Hartle-Hawking vacuum. An analogous calculation leads us to the following response: RHH⫽ q2a共M⫺Q2/r_⫹兲 4␲2共M⫺Q2/r0兲 ⫹q 2_共M⫺Q2_/r ⫹兲共M⫺Q2/r0兲 4␲2r_⫹2r₀2a , 共3.17兲 where we have used that P0关z(r0)兴⫽1 and Y00⫽1/冑4␲. 关Note that, in this case, only l⫽0 contributes in Eq. 共3.4兲.兴 The first term on the right-hand side of Eq.共3.17兲 is identical to the one obtained with the Unruh vacuum and is associated with the thermal flux leavingH⫺. The second term is asso-ciated with the thermal flux coming fromJ⫺. As a consis-tency check we note that for r→r_⫹, we obtain RHH⫽RU. This should be so because close to the horizon, zero-energy particles coming from J⫺ cannot overpass the scattering barrier. Consequently, in this limit, the second term on the right-hand side of Eq. 共3.17兲 must vanish. Now, when the source is far away from the hole, the second term on the right-hand side of Eq.共3.17兲 dominates because zero-energy particles leaving J⫺ are not able to reach the asymptotic region. Moreover, in this region, Eq.共3.17兲 can be rewritten in the form

RHH⬇

q2

Hence, far away from the hole, the source behaves as if it were in Minkowski spacetime immersed in a thermal bath with temperature␤⫺1, as expected.

IV. DISCUSSIONS

We have quantized the low-energy sector of a massless scalar field in Reissner-Nordstro¨m spacetime. The results ob-tained were used to analyze the response of a static source interacting with Hawking radiation using the Unruh and Hartle-Hawking vacua. We have shown that, in general, static sources outside charged black holes 共with the Unruh vacuum兲 do not behave similarly to uniformly accelerated sources in Minkowski spacetime 共with the usual inertial vacuum兲 as previously found for neutral black holes 关4兴. This in conjunction with the fact that no equivalence is found in the Schwarzschild spacetime when the scalar field is re-placed by the Maxwell one 关5兴 shows that the equivalence

found in关4兴 is not valid, in general, for other spacetimes and quantum fields. Whether or not there is something deeper behind it remains an open question for us. We have also verified that close to and far away from the horizon our source behaves as if it were at rest in a thermal bath in Minkowski spacetime with proper temperature associated with the Unruh and Hartle-Hawking vacua, respectively. The low-energy quantization presented here can be used to ana-lyze other processes occurring outside charged black holes.

ACKNOWLEDGMENTS

We are thankful to L.C.B. Crispino for reading the manu-script. J.C. and G.M. would like to acknowledge full and partial support from the Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo 共FAPESP兲 and Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico 共CNPq兲, respec-tively.

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