Interaction of Hawking radiation with static sources in de Sitter and Schwarzschild-de Sitter spacetimes

Interaction of Hawking radiation with static sources in de Sitter

and Schwarzschild–

de Sitter

spacetimes

J. Castin˜eiras,*I. P. Costa e Silva,†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 19 July 2003; published 31 October 2003兲

We study and look for similarities between the response rates RdS(a0,⌳) and RSdS(a0,⌳,M) of a static scalar source with constant proper acceleration a0 interacting with a massless, conformally coupled Klein-Gordon field共i兲 in de Sitter spacetime, in the Euclidean vacuum, which describes a thermal flux of radiation emanating from the de Sitter cosmological horizon and 共ii兲 in Schwarzschild–de Sitter spacetime, in the Gibbons-Hawking vacuum, which describes thermal fluxes of radiation emanating from both the hole and the cosmological horizons, respectively, where⌳ is the cosmological constant and M is the black hole mass. After performing the field quantization in each of the above spacetimes, we obtain the response rates at the tree level in terms of an infinite sum of zero-energy field modes possessing all possible angular momentum quantum numbers. In the case of de Sitter spacetime, this formula is worked out and a closed, analytical form is obtained. In the case of Schwarzschild–de Sitter spacetime such a closed formula could not be obtained, and a numerical analysis is performed. We conclude, in particular, that RdS(a0,⌳) and R

SdS

(a0,⌳,M) do not coincide in general, but tend to each other when⌳→0 or a0→⬁. Our results are also contrasted and shown to agree共in the proper limits兲 with related ones in the literature.

DOI: 10.1103/PhysRevD.68.084022 PACS number共s兲: 04.70.Dy, 04.62.⫹v I. INTRODUCTION

It is well known from classical electrodynamics that ac-celerated electric charges radiate as seen by an inertial ob-server in Minkowski spacetime. However, according to the equivalence principle, a uniformly accelerated charge is seen by a comoving observer as being static in a ‘‘uniform gravi-tational field,’’ and thus it is not expected to radiate. In the classical context, this apparent paradox was worked out in some detail by Rohrlich, Fulton关1兴, and Boulware 关2兴.

The same problem has also been analyzed in a quantum context关3兴, in terms of photon emission rates, using the fact that an observer comoving with a uniformly accelerated charge views the latter as immersed in a Fulling-Davies-Unruh共FDU兲 thermal bath 关4,5兴. More specifically, the inter-action of the static charge 共as computed by comoving ob-servers兲 with the FDU thermal bath results in the absorption and stimulated emission of zero-energy Rindler photons, which, although unobservable, nevertheless exactly account for the usual photon emission described by an inertial ob-server.

A particularly interesting arena to study interactions be-tween sources and radiation is the vicinity of black holes, where the presence of nontrivial classical and quantum ef-fects offers a wealth of conceptual and technical challenges. In this setting, it has recently been shown that the response

RSch(a0, M ) of a pointlike static scalar source with proper

acceleration a₀ outside a Schwarzschild black hole of mass

M interacting with massless scalar particles of Hawking

ra-diation 共associated with the Unruh vacuum兲 is exactly the same as the response RM(a0)⬅q2a0/4␲2 共in natural units,

where q is a small coupling constant兲 of such a source when it is uniformly accelerated with the same proper acceleration in the inertial vacuum of Minkowski spacetime or, equiva-lently, when it is static in the FDU thermal bath of the Rin-dler spacetime 关6兴. This is surprising because structureless static scalar sources can only interact with zero-energy field modes; such modes probe the global geometry of spacetime and are accordingly quite different in Schwarzschild and Rindler spacetimes. Indeed, this equivalence is not verified, e.g., when either 共i兲 the Unruh vacuum is replaced by the Hartle-Hawking vacuum 关6兴, 共ii兲 the black hole is endowed with electric charge 关7兴 or 共iii兲 the massless Klein-Gordon field is replaced with electromagnetic 关8兴 or massive Klein-Gordon 关9兴 ones. It is hitherto unclear whether the equiva-lence found in Ref.关6兴 is only a remarkable coincidence or if there is something deeper behind it. This circumstance has motivated us to study whether or not the equivalence would persist when one includes the presence of a cosmological constant, i.e., by replacing Schwarzschild with Schwarzschild-de Sitter 共SdS兲 spacetime and Minkowski with de Sitter spacetime.

SdS spacetime may be viewed as describing a spherically symmetric black hole immersed in a universe with a positive cosmological constant ⌳⬎0. It has attracted much attention lately on account of recent type Ia supernovae and cosmic microwave background observations关10兴 indicating that the Universe at large scale has共approximately兲 flat spatial geom-etry and is in accelerated expansion. These data suggest the existence of some background form of energy 共‘‘dark en-ergy’’兲 with negative pressure. The most plausible scenarios to describe this energy include the existence of a positive cosmological constant and quintessence fields. Although in the latter case the energy density of the dark energy is al-lowed to change in time, in many models this variation can *Electronic address: jcastin@ift.unesp.br

†_{Electronic address: ivanpcs@ift.unesp.br} ‡_{Electronic address: matsas@ift.unesp.br}

be neglected for astrophysically relevant scales. In other words, when considering objects such as black holes, one can for most purposes assume the presence of an effective, positive cosmological constant. In light of these facts, the interest in considering SdS black holes becomes clear.

In this paper we consider a static, pointlike source inter-acting with a conformally coupled, massless Klein-Gordon field in both de Sitter and SdS spacetimes. Quantum field theory in de Sitter spacetime has been much studied in the literature 关11兴. In this case, we take the field state to be the

Euclidean vacuum共see, e.g., Ref. 关12兴兲 describing a thermal

bath as seen by static observers. Our calculations in de Sitter closely follow those of Higuchi关13兴, but our presentation is somewhat different. In particular, it is useful for our purposes to derive the response rate of our pointlike source at a ge-neric position inside the cosmological radius. In the SdS case, there are two main technical hindrances in considering the quantization of the Klein-Gordon field. The first one is the definition of what we shall call the Gibbons-Hawking

vacuum 关14兴. This state describes a situation in which we

have thermal fluxes emanating from both the hole and the cosmological event horizons. As usually, the related tempera-tures are proportional to the corresponding surface gravities

␬h and ␬c. Because in general␬h⫽␬c, there are technical

difficulties in defining such a state关14,15兴 in the whole SdS spacetime. However, for the region between the horizons one may devise an heuristic prescription to define it, since in realistic situations where black holes are formed by gravita-tional collapse in a de Sitter background, it is natural to expect the emission of thermal radiation from both horizons 共see 关14兴 for an outline and further justification for the men-tioned prescription兲. We shall not dwell on these problems in this paper but simply assume that radiation emanates from both horizons at definite temperatures. The second technical difficulty is related with the quantization of the scalar field in SdS spacetime. Due to the spherical symmetry of the prob-lem, the corresponding Klein-Gordon equation is easily separated, but its radial part, except for the near extremal case关16兴, does not appear to be amenable to analytical treat-ment. Accordingly, we shall proceed to its numerical resolu-tion.

The paper is organized as follows. In Sec. II, we briefly review some geometrical aspects of SdS and de Sitter space-times which will be useful for establishing the setting for our analysis and fixing notation. In Sec. III we present the gen-eral formalism to quantize a massless, conformally coupled scalar field in the background of interest. In Sec. IV we apply the formalism to de Sitter spacetime with the Euclidean vacuum, obtaining a simple, closed, analytical form for the response at the tree level of a static source interacting with the radiation from the cosmological horizon. In Sec. V, we apply the formalism to SdS spacetime with the Gibbons-Hawking vacuum and express the response of the static source in terms of a sum over the normal mode angular mo-menta, which is numerically evaluated. We then compare the behavior of this response with the one obtained in the de Sitter case 共and with the related one obtained in Schwarzs-child spacetime with the Unruh vacuum关6兴兲. Finally, in Sec. VI we finish with some conclusions. Throughout this paper,

we adopt natural units (c⫽G⫽ប⫽kB⫽1), the abstract

in-dex notation关17兴 and spacetime signature (⫹⫺⫺⫺). II. THE BACKGROUNDS: de SITTER AND

SCHWARZSCHILD–de SITTER

Before starting our central discussion, it will be useful to recall some geometrical features of de Sitter and SdS space-times. We shall briefly do so here, confining ourselves to the minimum of information necessary to our ends. For more details, see, e.g., Refs. 关14,18兴.

de Sitter and SdS spacetimes are vacuum solutions of Ein-stein’s field equations with a positive cosmological constant ⌳⬎0. Their line elements can be written as

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

with f (r)哫 f_dS(r)⫽1⫺⌳r2_{/3 for de Sitter spacetime and}

f (r)哫 fSdS(r)⫽1⫺2M/r⫺⌳r2/3 for SdS spacetime 关19兴,

where M denotes the mass of the corresponding black hole. Here, the ‘‘time coordinate’’ t and the ‘‘angular coordinates’’

␪ and ␾ have their usual ranges, ⫺⬁⬍t⬍⫹⬁, 0⭐␪⬍␲, 0⭐␾⬍2␲, and for our purposes the ‘‘radial coordinate’’ r must be restricted to non-negative values for which f (r) ⬎0.

Let us first consider de Sitter spacetime. We begin by defining the de Sitter or cosmological radius at ␣⬅

冑

3/⌳, and by noting that f (r)⬎0 implies 0⭐r⬍␣. The ‘‘singular-ity’’ at r⫽␣ is merely due to a bad choice of coordinates, and with an appropriate reparametrization 关18兴 one can ob-tain the corresponding maximal analytic extension. This spacetime has topology S3⫻R and can be isometrically em-bedded as a one-sheeted hyperboloid in 5-dimensional Minkowski spacetime 共see Fig. 1兲. The coordinates (t,r,␪,␾) cover only part of de Sitter spacetime. The causal structure of de Sitter spacetime can be more readily visual-ized through the Penrose diagram in Fig. 2. The origin of the polar coordinates, r⫽0, and past and future infinities I⫺ and I⫹ are represented by vertical and horizontal border-lines, respectively. We note that the region labeled as I in Fig. 2关covered by the coordinates (t,r,␪,␾)] on which we will focus has a global timelike future-directed Killing field

␰a_{⬅(⳵/⳵t)}a_{. The Killing field ␰}a _{becomes lightlike at r}

FIG. 1. Embedding of de Sitter spacetime in a flat background with two dimensions omitted 共circular cross sections are to be thought of as copies of S3_{). The shaded part represents the region} of de Sitter spacetime covered by the coordinates (t,r,␪,␾). One can pick any normal, timelike geodesic as the origin r⫽0.

⫽␣, which comprises a bifurcate Killing horizon 共see, e.g., Ref. 关15兴 for a definition兲. The observers following integral curves of the Killing field␰a _{in region I will be called static}

for short. Static observers have 4-velocity ua

⫽(␰c_␰ c)⫺1/2␰ a_{, 4-acceleration} aa⫽ubⵜbua⬅⫺ r ␣2

冉

⳵ ⳵r

冊

a , 共2兲

and proper acceleration

adS⫽

冑

⫺aaaa⫽ r ␣2

冉

1⫺ r2 ␣2

冊

⫺1/2 . 共3兲

It is thus clear that a static observer at r⫽0 follows indeed a geodesic.

Let us now turn our attention to SdS spacetime. We shall assume that M /␣⬍1/

冑

27. The zeroes of fSdS(r) are, then,

found at rc⫽ 2␣

冑

3cos

冉

A 3

冊

, 共4兲 rh⫽ ⫺2␣

冑

3 cos

冉

A⫹␲ 3

冊

, 共5兲 r3⫽⫺共rc⫹rh兲, 共6兲

where A⬅arccos关⫺(27M2/␣2)1/2兴 satisfies ␲/2⬍A⬍␲. Here, rc and rh are associated with the cosmological and

black hole horizons, respectively, and satisfy 0⬍rh⬍rc.

Moreover, fSdS(r)⬎0 for rh⬍r⬍rc. The causal structure of

the SdS spacetime is clear in the Penrose diagram 关14兴 dis-played in Fig. 3. We will be interested in the region I where

␰a_{⫽(⳵/⳵t)}a_{is a global timelike future-directed Killing field.}

In SdS spacetime, static observers have 4-velocity ua ⫽(␰c_␰ c)⫺1/2␰ a_{, 4-acceleration} aa⫽ub“bua⬅

冉

M r2⫺ r ␣2

冊

冉

⳵ ⳵r

冊

a , 共7兲

and proper acceleration

aSdS⫽

冑

⫺aaaa⫽

冏

M r2⫺ r ␣2

冏

冉

1⫺ 2 M r ⫺ r2 ␣2

冊

⫺1/2 . 共8兲

Note that static observers with r⫽(M␣2)1/3follow geodesics (aSdS⫽0), due to a balance between the cosmic repulsion

and the black hole attraction.

III. RESPONSE OF A STATIC SOURCE INTERACTING WITH A SCALAR FIELD

Consider now the quantization of a massless, conformally coupled Klein-Gordon field ⌽(x␮), in the background de-fined by Eq. 共1兲, described by the action

S⫽1

2

冕

d

4_x

冑

_⫺g关“a_⌽“

a⌽⫺共1/6兲R⌽兴, 共9兲

where g⬅det兵gab其, and R⫽4⌳⫽12/␣2 is the scalar

curva-ture for both de Sitter and SdS spacetimes. The associated Klein-Gordon equation is

“a_“

a⌽⫹共1/6兲R⌽⫽0. 共10兲

It is well known that quantum field theory takes a rela-tively simple form in globally hyperbolic, stationary space-times where, in particular, a well defined notion of particle can be given 共see, e.g., Ref. 关20兴, and references therein兲. This is the case for the shaded regions in Figs. 2 (0⭐r ⬍␣) and 3 (r_h⬍r⬍r_c). For each such region, we shall look for a set of positive-frequency modes

u_␻lmi 共x␮兲⫽

冑

␻

␲ ␺␻li 共r兲

r Ylm共␪,␾兲e

⫺i␻t _共11兲

associated with the timelike Killing field␰a⫽(⳵/⳵t)a, where

␻⭓0, l苸Z⫹ _{and m苸关⫺l,l兴艚Z are the frequency and the} angular momentum quantum numbers, respectively, and

Ylm(␪,␾) are the spherical harmonics. The factor

冑

␻/␲ has

been introduced for later convenience. The radial part of Eq. 共10兲 then reads

FIG. 2. Penrose diagram of de Sitter spacetime. The shaded region is the one covered by the coordinates (t,r,␪,␾). Horizontal lines cutting this diagram represent 3 spheres, and the lines labeled as r⫽0 represent the worldlines of the ‘‘north and south pole’’ of these 3 spheres. The solid lines labeled as r⫽⬁ correspond to past and future infinitiesI⫺andI⫹.

FIG. 3. Penrose Diagram of Schwarzschild-de Sitter spacetime. The displayed pattern repeats itself infinitely both to the left and to the right. The shaded region is a static, globally hyperbolic region by itself.

冋

⫺ f共r兲_drd

冉

f共r兲 d dr

冊

⫹Veff共r兲

册

␺␻l i _{共r兲⫽␻}2_␺ ␻l i _共r兲, 共12兲 where the effective scattering potential V_eff(r) is given by

Veff共r兲⫽ f 共r兲

冉

1 r d f dr⫹ l共l⫹1兲 r2 ⫹ 2 ␣2

冊

. 共13兲

Note that Eq.共12兲 admits, in general, two sets of independent solutions which will be labeled with i⫽I, II. The u_␻lmi (x␮) modes are assumed to be orthonormalized with respect to the Klein-Gordon inner product 关12兴:

i

冕

⌺d⌺n a_共u ␻lm i *“au_␻⬘l⬘m⬘ i⬘ ⫺u_␻⬘l⬘m⬘ i⬘ “au_␻lm i *兲 ⫽␦ii⬘␦ll⬘␦mm⬘␦共␻⫺␻

⬘

兲, 共14兲 i

冕

⌺d⌺n a_共u ␻lm i _“ au_␻⬘l⬘m⬘ i⬘ _⫺u ␻_⬘l⬘m⬘ i⬘ “au␻lm i _兲⫽0,

where na is the future-directed unit vector normal to some fixed Cauchy surface ⌺. These modes and their respective complex conjugates form a complete orthonormal basis of the space of solutions of Eq. 共10兲 in the regions of interest. As a result, we can expand the field operator as

⌽ˆ共x␮_兲⫽

兺

i⫽I,IIl

兺

⫽0 ⬁

兺

m⫽⫺l l

冕

0 ⫹⬁ d␻关u_␻lmi 共x␮兲a_␻lmi ⫹H.c.兴, 共15兲 where a_␻lmi and a_␻lmi † are annihilation and creation opera-tors, respectively, and satisfy the usual commutation rela-tions

关a␻lmi ,a␻_⬘l⬘m⬘

i⬘ †_兴⫽␦

ii⬘␦ll⬘␦mm⬘␦共␻⫺␻

⬘

兲. 共16兲

The ‘‘Boulware’’ vacuum 兩0

典

is defined by a_␻lmi 兩0

典

⫽0 for every i,␻,l and m. This is the state of ‘‘no particles’’ as defined by the static observers following integral curves of

␰a_.

Let us consider now a pointlike static scalar source lying at (r0,␪0,␸0) described by

j共x␮兲⫽共q/

冑

⫺h兲␦共r⫺r0兲␦共␪⫺␪0兲␦共␸⫺␸0兲, 共17兲

where h⫽⫺ f⫺1r4sin2␪is the determinant of the spatial met-ric induced on an equal t-time hypersurface ⌺ and q is a small coupling constant. This source is coupled to the Klein-Gordon field ⌽ˆ(x␮) via the interaction action

SˆI⫽

冕

d4x

冑

⫺g j⌽ˆ. 共18兲

All the calculations will be carried out at the tree level. The total response, i.e., combined particle emission and absorption probabilities per unit proper time of the source, is given by R⬅

兺

i_⫽I,IIl

兺

_⫽0 ⬁

兺

m_⫽⫺l l

冕

0 ⫹⬁ d␻R_␻lmi , 共19兲 where R_␻lmi ⬅␶⫺1兵兩A_␻lmi em兩2关1⫹ni共␻兲兴⫹兩A_␻lmi abs兩2ni共␻兲其 共20兲 and ␶ is the total proper time of the source. Here A_␻lmi em ⬅

具

i␻lm兩SˆI兩0

典

and A_␻lm

i abs_⬅

_具

_0兩Sˆ

I兩i␻lm

典

are the emission

and absorption amplitudes, respectively, of Boulware states 兩i␻lm

典

, and the nifactors depend on the field state chosen in each case.

Structureless static sources described by Eq.共17兲 can only interact with zero-energy modes关3兴 and thus the response of the source in the Boulware vacuum vanishes. However, in the presence of a background thermal bath, the absorption and stimulated emission rates lead to a non-zero response. In order to deal with zero-energy modes, we need a ‘‘regulator’’ to avoid the appearance of intermediate indefinite results共for a more comprehensive discussion on the interaction of static sources with zero-energy modes, see Ref.关3兴兲. For this pur-pose, we let the coupling constant q oscillate with frequency

␻0 by replacing q with q␻₀⬅

冑

2q cos(␻0t) in Eq. 共17兲 and

take the limit ␻0→0 at the end of our calculations. The

factor

冑

2 has been introduced to ensure that the time average

具

兩q␻0(t)兩 2

_典

t⫽q2 since the absorption and emission rates are

functions of q2. Another equivalent regularization procedure is discussed in 关21兴. A straightforward calculation using 兺m⫽⫺l l _兩Y lm(␪0,␸0)兩2⫽(2l⫹1)/4␲ 关22兴 gives R共r0兲⫽ lim ␻0→0i_⫽I,II

兺

兺

l_⫽0 ⬁ q2␻0冑f共r0兲 4␲2r₀2 ⫻共2l⫹1兲兩␺␻0l i _共r 0兲兩2关1⫹2ni共␻0兲兴. 共21兲

IV. RESPONSE RATE IN de SITTER SPACETIME We are now ready to consider the response rate of the static source in de Sitter spacetime. Taking f (r)哫 fdS(r)

⬅(1⫺r2_/␣2_{), the effective potential共13兲 becomes} Veff

dS_共r兲⫽l共l⫹1兲

r2

冉

1⫺

r2

␣2

冊

. 共22兲

We define a new coordinate z⫽␣/r, and use it to reexpress Eq. 共12兲, with potential 共22兲, in the form

冋

d dz

冉

共1⫺z 2_兲d dz

冊

⫹l共l⫹1兲⫹ ␣2_␻2 1⫺z2

册

␺␻l i _{⫽0, 共23兲}

which is just the associated Legendre equation. It has two sets of linearly independent solutions, P_li␣␻(z) and Q_li␣␻(z) 共cf., e.g., Ref. 关22兴兲, but only the latter is regular at r⫽0 (z⫽⬁). Therefore we only consider modes of the form

u_␻lmIdS共x␮兲⫽C_␻lI

冑

␻

␲

Q_li␣␻共␣/r兲

r Ylm共␪,␾兲e

⫺i␻t_{, 共24兲}

where C_␻lI are normalization constants to be fixed by requir-ing that the modes be orthonormal with respect to the Klein-Gordon inner product共14兲. The physical behavior of the nor-mal modes 共24兲 is clear: one may visualize them in the shaded region of Fig. 2 as emanating from the past horizon, scattering off the line r⫽0, and being reflected back to the future horizon. Alternatively, one may think of the modes ‘‘in spatial terms’’ as converging from a 2-sphere at r⫽␣ onto its center at r⫽0 and then spreading out to r⫽␣again. Of course, the modes ‘‘converge isotropically’’ only for l ⫽0, but ‘‘swirl around’’ as they plunge in onto r⫽0 for l ⫽0.

Let us now evaluate the normalization constants C_␻lI . For this purpose we substitute the modes 共24兲 into Eq. 共14兲, where we choose⌺ to be the t⫽0 hypersurface and we use the orthonormality of the spherical harmonics. We are then left with C_␻lI *CI_␻⬘l共␣/␲兲

冑

␻␻

⬘

共␻⫹␻

⬘

兲Il␻␻⬘⫽␦共␻⫺␻

⬘

兲, 共25兲 where Il␻␻⬘⬅

冕

0 1 _{d y} 1⫺y2关Ql i␣␻_共1/y兲兴 *Q_li␣␻⬘共1/y兲. 共26兲

In order to evaluate I_l␻␻⬘, we use formula 8.703 of Ref.关22兴 to write

Q_li␣␻共1/y兲⫽

冑

␲⌫共1⫹l⫹i␣␻兲y f␣␻

l

共y兲

2l⫹1⌫共l⫹3/2兲e␣␻␲ , 共27兲 where we have defined

f_␣␻l 共y兲⬅yl共1⫺y2兲i␣␻/2 ⫻F

冉

2⫹l⫹i₂ ␣␻,1⫹l⫹i␣␻ 2 ;l⫹ 3 2; y 2

冊

_. 共28兲 Here, F(a,b;c;x) denotes a hypergeometric function. By us-ing f_␣␻l (y ), Higuchi关13兴 evaluated the integral

I_l(2)_␻␻⬘⫽

冕

0 1 _{d y} 1⫺y2y 2_{关 f} ␣␻ l _共y兲兴 *f_␣␻l _⬘共y兲 ⫽2_␣␲

冏

⌫共l⫹3/2兲⌫共i␣␻兲 ⌫关共2⫹l⫹i␣␻兲/2兴⌫关共1⫹l⫹i␣␻兲/2兴

冏

2 ⫻␦共␻⫺␻

⬘

兲. 共29兲

We now use Eq. 共29兲 to compute Il␻␻⬘ and substitute the

result into Eq.共25兲. Apart from an unimportant global phase, we get

C_␻lI ⫽2

l_e␣␻␲_{⌫关共2⫹l⫹i␣␻兲/2兴⌫关共1⫹l⫹i␣␻兲/2兴}

冑

␲␻⌫共1⫹l⫹i␣␻兲⌫共i␣␻兲 . 共30兲 Next, we assume that the field is in the so-called ‘‘Euclid-ean’’ vacuum 共also known as ‘‘Bunch-Davies’’ or ‘‘Birrell-Davies’’ vacuum兲, which describes a thermal bath of tem-perature

TdS⫽1/共2␲␣兲 共31兲

as measured by the inertial observer at r⫽0 共see, e.g., Refs. 关12,15,20兴, and references therein for further properties of this state兲. As a consequence, nI(␻)⬅(e␻␤⫺1)⫺1 with

␤⫺1_⬅T

dS.

In order to compute the response共21兲, we recall that in de Sitter spacetime the sum will be restricted to the set of regu-lar modes, i.e., with i⫽I. Then, we use ␺_␻

0l I _(r 0) ⬅C_␻ 0l I Q_li␣␻0₍␣_/r 0), where C_␻ 0l I

is obtained from Eq. 共30兲, and the identity共cf. Eq. 8.332.1 in 关22兴兲

兩x⌫共ix兲兩2_{⫽␲x/sinh共␲x兲.} As a result, we obtain RdS共r0,␣兲⫽ q2␣f共r0兲1/2 4␲3r₀2 l

兺

_⫽0 ⬁ 22l共2l⫹1兲 ⫻

冏

⌫关共l⫹2兲/2兴⌫关共l⫹1兲/2兴 ⌫共l⫹1兲

冏

2

冏

Q_l

冉

␣ r0

冊

冏

2 , 共32兲 where Q_l(z) is the ordinary Legendre function. Now, we use the doubling formula关22兴

⌫共2x兲⫽22x⫺1_␲⫺1/2_{⌫共x兲⌫共x⫹1/2兲,} with x⬅(l⫹1)/2 in Eq. 共32兲: RdS共r0,␣兲⫽ q2␣f共r0兲1/2 4␲2r₀2 l

兺

⫽0 ⬁ 共2l⫹1兲

冏

Ql

冉

␣ r0

冊

冏

2 . 共33兲

Finally, we use the identity关6兴

兺

l_⫽0 ⬁ 兩Ql共s兲兩2共2l⫹1兲⫽ 1 s2⫺1 共34兲

in Eq. 共33兲 to obtain the final response as a function of the source’s position: RdS共r0,␣兲⫽ q2 4␲2␣

冉

1⫺ r₀2 ␣2

冊

⫺1/2 . 共35兲

For r0⫽0 we recover the formula for the response of an

inertial source in de Sitter spacetime, given in Ref.关13兴. It is convenient to invert Eq.共3兲 to write the response in terms of the source’s proper acceleration, which is a coordinate-independent observable in general relativity:

RdS共a0,␣兲⫽ q2 4␲2␣共1⫹␣ 2_a 0 2_兲1/2_{. 共36兲}

We note that when ␣a₀Ⰷ1, i.e., when either the source

ap-proaches the cosmological horizon or the cosmological con-stant is small enough (␣ being accordingly large兲, we have

RdS共a0,␣兲⬇ q2a0

4␲2. 共37兲

The right-hand side of Eq. 共37兲 is independent of ␣ and acquires the same form as the response of a static source in the Rindler wedge共i.e., uniformly accelerated in Minkowski spacetime兲 interacting with a massless Klein-Gordon field in the usual inertial vacuum. To see why this occurs, let us first write from Eq. 共3兲 the source’s radial position as

r₀共a₀,␣兲⫽␣关1⫹共␣a₀兲⫺2兴⫺1/2. 共38兲 Thus, for␣a0Ⰷ1, we obtain r0⬇␣. Now, in this region, the

de Sitter line element 共1兲 reduces 共apart from the angular piece兲 to the Rindler form

ds_dS2 ⬇e2␩/␣_dt2_⫺e2␩/␣_d␩2_{, 共39兲}

where ␩⬅␣ln(1⫺r2_/␣2₎1/2 _(⫺⬁⬍␩_{⭐0). Indeed, the}

proper distance between r0 and the cosmological horizon is

␣arctan关1/(␣a0)兴 ——→

␣a0→⬁

1/a0, which is precisely the

proper distance between a static source in the Rindler wedge with proper acceleration a0and its horizon. This observation

combined with the fact that the local temperature at the source共obtained by multiplying temperature 共31兲 by the Tol-man factor 关23兴兲 corresponds to the temperature of the Fulling-Davies-Unruh thermal bath,

T_dSloc⫽ TdS

冑

fdS共r0兲

⬇ a0

2␲, 共40兲

clarifies Eq.共37兲. In particular, the decrease in the tempera-ture TdS⫽1/(2␲␣) 关cf. Eq. 共31兲兴; when ␣ grows large is

perfectly compensated by the source’s approach to the hori-zon关cf. Eq. 共38兲兴.

V. RESPONSE RATE IN SCHWARZSCHILD–de SITTER SPACETIME

We now turn to the field quantization in SdS spacetime and compare the response rate calculated in this case with the one obtained in the previous section关see Eq. 共36兲兴. Similarly, we shall select Klein-Gordon orthonormalized modes of the form共11兲. Equation 共12兲 with the effective scattering poten-tial for SdS spacetime

Veff SdS_共r兲⫽

冉

1⫺2 M r ⫺ r2 ␣2

冊冉

2 M r3 ⫹ l共l⫹1兲 r2

冊

共41兲

admits, now, two sets of linearly independent regular solu-tions ␺_␻li . We shall associate ␺_␻lI ⬅␺_␻l→ and␺_␻lII⬅␺_␻l← with purely ingoing modes emanating from the white hole horizon

Hh⫺共hereafter simply referred as hole horizon兲 and from the

past cosmological horizon H_c⫺, respectively. ␺_␻l→ and ␺_␻l← are Klein-Gordon orthogonal to each other, as can be seen by choosing the Cauchy surface in Eq. 共14兲 to be ⌺ ⫽Hh⫺艛Hc⫺, and then using the fact that␺␻l→ and␺␻l←

van-ish at H_c⫺andH_h⫺, respectively.

For the Gibbons-Hawking vacuum 关14兴, the appropriate thermal factors appearing in Eq. 共20兲 are nI(␻)⬅(e␻␤h

⫺1)⫺1 _{and n}II_(␻)⬅(e␻␤_c⫺1)⫺1_{, where}

␤h⫺1⫽

␬h

2␲ and ␤c ⫺1_⫽␬c

2␲ 共42兲

are the temperatures of the radiation from the hole and the cosmological horizons, respectively, with

␬h⫽ 1 2 d f_SdS dr 兩r⫽rh⫽ 1 2␣2rh 共rc⫺rh兲共rh⫹r¯兲 共43兲 ␬c⫽ 1 2 d fSdS dr 兩r⫽rc⫽ 1 2␣2rc 共rc⫺rh兲共rc⫹r¯兲 共44兲

being the surface gravities of the corresponding horizons, where r¯⬅rh⫹rc. In this case, the response共21兲 can be

writ-ten as RSdS_共r 0, M ,␣兲⫽ q2

冑

f共r0兲 2␲r₀2 _␻lim_0→0i⫽I,II

兺

l

兺

⫽0 ⬁ 共2l⫹1兲 ⫻␻0兩␺_␻0l i _共r 0兲兩2ni共␻0兲. 共45兲

Equation 共45兲 allows one to compute the response provided one has obtained the 共normalized兲 modes. For numerical convenience, let us introduce a new coordinate

x共r兲⫽⫺ 1 2␬_cln

冉

1⫺ r r_c

冊

⫹ 1 2␬_hln

冉

r r_h⫺1

冊

⫹ 1 2␬¯ln

冉

r r ¯⫹1

冊

, 共46兲 where ␬¯⬅(2␣2¯)r ⫺1(rc⫹r¯)(rh⫹r¯). In terms of the new

variable x, Eq.共12兲 is recast in the ‘‘Schro¨dinger-like’’ form

冋

⫺ d 2 dx2⫹Veff SdS_{关r共x兲兴}

册

_␺ ␻l i _共x兲⫽ ␻2_␺ ␻l i _{共x兲. 共47兲}

Near the horizons 关and assuming the realistic case MⰆ␣ 共i.e., rhⰆrc)], we have

V_effSdS共x兲⬃

再

e⫺2␬cxⰆ1 for xⰇ␣

e2␬hxⰆ1 _{for x}⬍0,兩x兩Ⰷ2M. 共48兲

Thus, in these regions the potential becomes exponentially suppressed, and we can approximate Eq. 共47兲 by

⫺d 2_␺ ␻l i dx2 ⬇␻ 2_␺ ␻l i _{共x兲. 共49兲} 共2003兲

This leads to the asymptotic behavior of the modes ␺␻l→共x兲⬇

再

A_␻l共ei␻x⫹R_␻l→e⫺i␻x兲 共x⬍0,兩x兩Ⰷ2M 兲 A_␻lT_␻l→ei␻x 共xⰇ␣兲 共50兲 and ␺_␻l←_共x兲⬇

再

B␻lT␻l←e⫺i␻x 共x⬍0,兩x兩Ⰷ2M 兲 B_␻l共e⫺i␻x⫹R_␻l←ei␻x兲 共xⰇ␣兲. 共51兲 Here, 兩R_␻l→兩2, 兩R_␻l←兩2 and 兩T_␻l→兩2, 兩T_␻l←兩2 are reflection and transmission coefficients, respectively, for the ‘‘scattering problem’’ defined by Eq.共47兲. They satisfy usual ‘‘conserva-tion of probability’’ laws: 兩R_␻l→兩2⫹兩T_␻l→兩2⫽1 and 兩R_␻l←兩2 ⫹兩T_␻l←_兩2_{⫽1. The normalization constants A}

␻l and B␻l can be obtained by imposing Klein-Gordon orthonormality of the modes ␺_␻l→ and ␺_␻l← with respect to the Klein-Gordon inner product共14兲, where Eq. 共47兲 is used to transform the result-ing integrals into surface terms共see 关7兴 for details兲. Then, by using Eqs.共50兲 and 共51兲, we obtain A_␻l⫽B_␻l⫽(2␻)⫺1.

The modes␺_␻l→ and␺_␻l← can be obtained numerically for small␻ and different l values by evolving Eq.共47兲 with the effective potential 共41兲 and the asymptotic forms 共50兲 and 共51兲. The corresponding total response RSdS_{can be obtained,}

then, from Eq.共45兲. We note that the larger the value of l, the higher the barrier of the scattering potential Veff

SdS

(r) 关24兴 and therefore the main contributions come from modes with small l. How far we must sum over l in Eq. 共45兲 to obtain a satisfactory numerical result will depend on how close to the black hole horizon the source lies. The closer to the horizon the further over l we must sum.

In Fig. 4, we plot the response RSdSas a function of the

source’s proper acceleration a0 for various values of␣. As

originally defined, a0 is not a one-to-one function of the

source’s radial coordinate r0 关cf. Eq. 共8兲兴. In particular, note

from Eq. 共8兲 that a0→⬁ either when r→rh or r→rc. For

the sake of clarity in the result presentation, we circumvent this feature by dropping ‘‘兩 . 兩’’ ’s in Eq.共8兲. This is ‘‘com-pensated’’ by plotting the absolute value of RSdS. After this procedure, the response rate is kept unchanged but a0

ac-quires negative sign between the ‘‘equilibrium’’ point re

⫽(M␣2₎1/3_{共where a}

0⫽0) and rc, becoming then a

one-to-one function of r0. In special, a0→⬁ when r0→rh, but a0

→⫺⬁ when r0→rc. Moreover, we have ‘‘compactified’’

the range of the proper acceleration from (⫺⬁,⫹⬁) to (0,1) by introducing the variable

␹共a0兲⫽关1⫺tanh共␳a0兲兴/2,

which is a monotonic function of the source’s proper accel-eration a0 共and where␳ is a free parameter fixed by

numeri-cal convenience兲. The graph reveals that RSdS_(a

0) tends to RSch(a0)⫽q2兩a0兩/4␲2 as ␣→⬁, i.e., the response for a

source in SdS spacetime approaches the one in Schwarzs-child spacetime共in the Unruh vacuum兲 when the cosmologi-cal constant goes to zero共provided the source keeps the same proper acceleration兲. This result would be quite expected if we assumed that r0 共rather than a0) was kept constant in the

process. This is so because when the cosmological radius␣

→⬁, the geometry approaches Schwarzschild’s spacetime

and the contribution from the cosmological horizon in Eq. 共45兲 becomes negligible both because␬c→0 and because the

low-energy modes emanating from the cosmological horizon have to ‘‘travel a longer distance’’ through the potential bar-rier to reach the source. As a result, these modes are mostly scattered back, and contribute very little to the total response. However, the explanation is much more subtle when we con-sider␣→⬁ with a₀fixed. For 0⬍␹(a₀)⬍1/2, the larger the

␣ the more the influence of the black hole overcomes that of the cosmic expansion. Thus, as␣→⬁, the ‘‘left half’’ of the curves in Fig. 4 should indeed converge to Schwarzschild for the same reasons pointed out above. Nevertheless, for 1/2 ⬍␹(a₀)⬍1 the convergence was not expected a priori be-cause in this region r0⬎(2M␣2)1/3Ⰷ2M⬇rh, i.e., the

larger the␣ the more the influence of the cosmic expansion overcomes that of the black hole. Indeed, by neglecting the terms M /r and M /r2 in Eqs. 共1兲, 共8兲 and 共41兲, we obtain

fSdS(r)⬇ fdS(r), 兩aSdS兩⬇adS and Veff SdS_(r)⬇V

eff

dS_{(r) 共where}

‘‘兩. 兩’’ is used here to comply with our convention according to which aSdS⬍0 in this region兲. Now, in this region, the

zero-energy modes ingoing from the black hole are unable to interact with the source共as confirmed by an explicit numeri-cal numeri-calculation omitted here兲, and thus the modes ingoing from the cosmological horizon dominate in Eq. 共45兲. This indicates, at first sight, that RSdS(a0) approaches RdS(a0)

关see Eq. 共36兲兴 rather than RSch_(a

0). The reason why RSdS(a0) also approaches RSch(a0) in this region can be

un-derstood by recalling that for large enough ␣, RdS(a0)

ap-proaches the response for a uniformly accelerated source in Minkowski spacetime RM⫽q2a0/4␲2 共see Sec. III兲, which is

in turn equivalent to RSch_(a

0)关6兴. We conclude, thus, that the

0 0.2 0.4 0.6 0.8 1 χ(a₀) 0 0.0005 0.001 0.0015 0.002 0.0025 q-2RSdS_α = 20 q-2RSdS_{α = 40} q-2RSdS_{α = 80} q-2RSdS_{α = 160} q-2RSch

FIG. 4. RSdS_{is plotted vs␹(a}

0)共which is a monotonic function of a0) for various values of␣ 关with M⫽2, ␻0⫽10⫺4, ␳⫽80 and the sum over l is performed up to l⫽8 共inclusive兲兴. In particular, ␹(a0)⫽0, ␹(a0)⫽1/2, and ␹(a0)⫽1 correspond to the cases where the source is at r0⫽rh, geodesic at r0⫽(M␣2)1/3, and at

r0⫽rc, respectively. Note that RSdS approaches RSch 共bottom

fact that RSdS(a0)→ RSch(a0) everywhere when␣→⬁ is a

consequence of the共nontrivial兲 equivalence between the re-sponses in Rindler and Schwarzschild spacetimes.

In Fig. 5 we compare RSdS with RdS 关cf. Eq. 共36兲兴, as a function of a₀. We use the same convention as in Fig. 4, but now, it is convenient to introduce another compactification variable 关25兴: ␨(a0)⫽(r0⫺rh)/(rc⫺rh), where r0⫽r0(a0)

is the source’s radial position as a function of the proper acceleration obtained by inverting Eq. 共8兲 共without ‘‘兩 . 兩’’兲. Note that, unlike␹, the variable␨depends on both M and␣, which are fixed in Fig. 5. In this figure, the various graphs correspond to the various lmaxvalues of the maximum l used

to do the sum 共45兲. Note that the sum converges very fast away from the horizons, but less so for the regions near them. Figure 5 clearly suggests that RSdScoincides with RdS near both horizons. That they should coincide near the cos-mological horizon could be inferred from our discussion in Fig. 4, but that they coincide in both horizons can be broadly understood from the fact that the SdS spacetime is isometric to Rindler spacetime near them. This becomes manifest after the change of coordinates r哫␩h⬅(2␬h)⫺1ln关fSdS(r)兴 and r哫␩_c⬅(2␬_c)⫺1ln关f_SdS(r)兴, which allows one to recast the SdS line element near the black hole and cosmological hori-zons 共apart from the angular piece兲 as

dsSdS 2 _⬇e2␬h␩h_dt2_⫺e2␬h␩h_d␩ h 2 _共52兲 and ds_SdS2 ⬇e2␬c␩cdt2⫺e2␬c␩cd␩_c2, 共53兲 respectively. Now, we recall that the same happens for de Sitter spacetime close to its horizon 关cf. Eq. 共39兲兴. As a re-sult, RSdS(a₀) turns out to approach RdS(a₀) and both ap-proach RM(a0)⫽q2a0/(4␲2) 关cf. Eqs. 共36兲 and 共37兲兴 at the

horizons.

VI. CONCLUSIONS

We have quantized here a massless, conformally coupled scalar field in de Sitter and Schwarzschild–de Sitter space-times. In both cases, the field interacts with a static scalar source. We have computed the response rates of this source interacting with the Hawking radiation described by the Eu-clidean vacuum 共in the de Sitter case兲 and by the Gibbons-Hawking vacuum 共in Schwarzschild–de Sitter spacetime兲. The comparison of the responses 共as functions of the proper acceleration of the source兲 shows, in particular, that the equivalence obtained in关6兴 between the responses of a static source outside a Schwarzschild black hole 共with the Unruh vacuum兲 and of a uniformly accelerated source in Minkowski spacetime共with the inertial vacuum兲 was not re-produced here, i.e., the introduction of a cosmological con-stant breaks the original equivalence.

Although the responses in de Sitter and SdS spacetimes 共with the respective vacua兲 do not coincide, in general, very near the black hole and cosmological horizons they are in-deed equivalent. It is so because in both these regions the source is expected to behave as if it were uniformly acceler-ated in Minkowski spacetime, in the inertial vacuum. We have also recovered everywhere the response of a static source in Schwarzschild spacetime共with the Unruh vacuum兲 from our response in SdS spacetime as the cosmological con-stant vanishes. We have shown that this is a direct conse-quence of the nontrivial equivalence found in 关6兴.

ACKNOWLEDGMENTS

J.C. and I.S. would like to acknowledge full support from Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo 共FAPESP兲. G.M. is thankful to FAPESP and Conselho Na-cional de Desenvolvimento Cientı´fico e Tecnolo´gico for par-tial support.

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