Synchrotron geodesic radiation in Schwarzschild–de Sitter spacetime

Joao P. B. Brito,1, ∗ Rafael P. Bernar,1, † and Luıs C. B. Crispino1, ‡

1Faculdade de Fısica, Universidade Federal do Para, 66075-110, Belem, PA, Brazil(Dated: June 17, 2020)

We analyze the scalar radiation emitted by a source in geodesic circular orbits around aSchwarzschild–de Sitter black hole. We obtain the emitted power using quantum field theory incurved spacetimes framework at tree level. We compare our results with the scalar synchrotronradiation in Schwarzschild spacetime.

I. INTRODUCTION

The recent detection of gravitational waves [1, 2], emit-ted by binary black hole systems, and the first visualiza-tion of a black hole shadow [3], has drawn increasingattention to black hole physics. Moreover, the stronggravity regime, close to these compact objects, plays animportant role in general relativity (GR) [4] and alter-native theories of gravity [5], yielding a wealthy scenariofor the study of fundamental fields in curved spacetimes,including both their classical and quantum behavior.

On one hand, GR is a very successful classical fieldtheory, both theoretically and experimentally. On theother hand, it is a theory unable to describe the space-time physics near the singularities appearing in manyof its black hole solutions. A quantum theory of grav-ity is believed to be able to circumvent such problems.There have been several attempts to quantize gravity,with varying degrees of success so far (see the reviewin Ref. [6]). In the absence of a final quantum theoryof gravity, a more modest approach is the semiclassicalframework [7, 8], in which one considers quantum fieldspropagating in background spacetimes, which are clas-sical solutions of GR. The quantum field theory (QFT)in curved spacetimes approach has been successful in de-scribing some quantum aspects of gravity, such as theparticle creation in dynamic spacetimes [9] or near blackholes (Hawking radiation) [10]. These findings help in theconnection between gravity and quantum theory, leadingto important issues such as the black hole informationparadox [11] and may even point towards their resolu-tion. QFT in curved spacetimes has provided insightseven in flat spacetime, where the Unruh effect, i.e. thefact that accelerated observers in flat spacetime noticethe Minkowski vacuum as a thermal bath of particles, isthe prime example [12, 13].

The phenomenon of radiation emitted by objects mov-ing along geodesics in a black hole spacetime may beanalyzed using the semiclassical framework. The studyof such phenomena is important, as black holes foundin nature are usually surrounded by accretion disks.

∗Electronic address: joao.brito@icen.ufpa.br†Electronic address: rafael.bernar@icen.ufpa.br‡Electronic address: crispino@ufpa.br

This radiation mechanism was originally investigated inRefs. [14, 15], in which the scalar radiation emitted bysources orbiting a Schwarzschild black hole is studied.The scalar field constitutes a simple model that presentsmany qualitative results similar to the electromagnetic(vector) and gravitational (tensor) fields. When thesource is close to the photon sphere, the radiation is ofthe synchrotron type, the so-called synchrotron geodesicradiation. Using the QFT in curved spacetimes frame-work, this type of scalar radiation in asymptotically flatblack hole spacetimes was investigated in Refs. [16–21],the electromagnetic radiation emission in Ref. [22] andthe gravitational radiation emission in Refs. [23, 24]. Re-garding black holes with nonvanishing cosmological con-stant, geodesic synchrotron radiation was studied usingthe Green function framework in Ref. [25].

The de Sitter (dS) solution is the simplest solution ofGR field equations with a nonvanishing cosmological con-stant [26–29]. The study of phenomena in spacetimesasymptotically dS is of great interest, since there is ex-perimental evidence that our Universe is undergoing anaccelerated expansion [30, 31]. In this more realistic sce-nario, the black hole solutions are asymptotically dS,rather than asymptotically flat, so that a static chargelessblack hole is associated to the Schwarzschild–de Sitter(SdS) spacetime, described by the cosmological constantΛ, additionally to the geometric mass M of the centralSchwarzschild black hole [32–35].

In this paper, we use QFT in curved spacetime attree level to investigate the scalar radiation emitted bya source in geodesic circular motion around a SdS blackhole. The remaining of this paper is organized as follows.In Sec. II, we review some features of the SdS space-time, including the circular geodesic analysis. In Sec. III,we revisit some aspects of the scalar field theory in thiscurved background, including the field quantization inthe static patch of the SdS spacetime. In Sec. IV, us-ing lowest order perturbation theory and numerically ob-tained solutions for the Klein-Gordon equation, we com-pute the one-particle-emission amplitude to obtain thepower emitted by the source. In the Sec. VI, we presentour final remarks. We adopt geometrized units in whichc = G = ~ = 1 and the signature (−,+,+,+) for thespacetime metric.

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II. SCHWARZSCHILD–DE SITTER BLACKHOLES

In this section, we review some important features ofthe SdS spacetime, which is a spherically symmetric vac-uum solution of GR field equations with a positive cos-mological constant Λ > 0 and a black hole with mass M.In static coordinates, the SdS line element can be writtenas [33–35]

ds2 = −fΛ(r)dt2 +dr2

fΛ(r)+ r2(dθ2 + sin2 θdφ2), (1)

with

fΛ(r) ≡ 1− 2M

r− Λ

3r2. (2)

We note that the spacetime described by Eq. (2) hasthe Killing vectors ∂t, associated to translations alongt, and ∂φ, as well as K1 ≡ cosφ∂θ − cot θ sinφ∂φ andK2 ≡ − sinφ∂θ−cot θ cosφ∂φ, associated to rotations onthe 2−sphere.

The SdS black hole spacetime presents a cosmological(outer) horizon (Hc) and an event (inner) horizon (Hh).The radial positions of these hypersurfaces, rc and rh,respectively, are obtained by solving

fΛ(r) = 0. (3)

For a black hole solution, we must consider the cosmo-logical constant values in the interval,

0 ≤ Λ < 1/9M2. (4)

In this case, there are up to three real solutions of Eq. (3),two of them are positive (corresponding to the hori-zons’ radial positions, rh and rc) and one is negative[r− = −(rh + rc)]. We obtain the Schwarzschild solutionin the limit Λ → 0, for which rh → 2M and rc → +∞.We obtain the dS solution in the limit M → 0, for whichrh → 0 and rc →

√3/Λ (dS radius). As the Λ term in-

creases from zero (Schwarzschild solution), the two hori-zons get closer, until they degenerate at the radial posi-tion rh = rc = 3M, when Λ = Λext = 1/9M2 (extremecase). The behavior of the function fΛ(r) is illustrated inFig. 1. The spacetime is static in the region rh < r < rc.For Λ > Λext, the spacetime is dynamic for all r > 0 [33].

We shall consider a scalar source rotating around theSdS black hole. In the next subsection, we analyze cir-cular geodesics in the SdS spacetime.

A. Circular geodesics

The equations governing the geodesic trajectories inSdS spacetime can be derived from the Lagrangian,

L =1

2gµν x

µxν , (5)

0 2 3 5 10 15 20-10

-5

0

5

10

r/M

f Λ

M2Λ = 0

M 2Λ = 9 -1

0 rh 3M rmax rc

0

r

f Λ

FIG. 1: Top: The function fΛ(r), given by Eq. (2), for twodifferent choices of the cosmological constant Λ, as indicated.The shaded region encompasses all values of Λ in the interval0 < Λ < 1/9M2. Bottom: The function fΛ(r) for a given valueof Λ > 0, from the interval (4), with its maximum occurringat the radial position rmax.

where the metric components gµν can be obtained fromEq. (1), and the overdot denotes differentiation withrespect to an affine parameter (for timelike geodesics,we identify the affine parameter with the free particle’sproper time).

The Lagrangian, given by Eq. (5), is independent of tand φ, so that we have the following integrals of motion:

pt = −∂L∂t

= fΛ(r)t ≡ E, (6)

pφ =∂L∂φ

= r2φ ≡ L. (7)

Without loss of generality, we shall consider the motionin the equatorial plane (θ = 0 and θ = 0). Noting that2L ≡ ε = −1 (0) for timelike (null) geodesics and usingEqs. (6) and (7), we find that the particle motion is en-tirely described by the energy-balance equation, writtenas

r2 = E2 − 2VΛ(r), (8)

with the central potential,

VΛ(r) ≡ 1

2fΛ(r)

(−ε+

L2

r2

). (9)

We note that the potential given by Eq. (9) vanishes atboth cosmological and event horizons. For massive par-ticles, there are points of minimum and maximum of the

3

FIG. 2: The function F (Λ, r), given by Eq. (12). The intersec-tion with the hatched plane marks the zeros of the function.

potential (9), corresponding to stable and unstable cir-cular orbits, respectively. For massless particles, the po-tential (9) has a maximum at the radial position r = 3M.

For timelike circular orbits, i.e. r = r = 0, we have thefollowing conserved quantities:

E2 = rfΛ(r)2

r − 3M, L2 = r2M − r3Λ/3

r − 3M. (10)

Since E and L must be real quantities, circular geodesicsexist in the region,

3M < r ≤(

3M

Λ

)1/3

≡ rmax, (11)

where r = rmax denotes the radial position of the max-imum of fΛ(r). We note that, for 0 < Λ < 1/9M2, wehave rh < rmax < rc.

The stability condition for the circular timelikegeodesics is obtained considering small radial perturba-tions on the orbits, as well as by a direct analysis of thepotential VΛ(r). This condition is found to be [33, 36]

F (Λ, r) ≡ −4Λr4 + 15ΛMr3 + 3Mr − 18M2 ≥ 0. (12)

The function F (Λ, r) is illustrated in Fig. 2. The pointsof the surface above the hatched plane select the param-eters Λ and r for which stable circular orbits can occur.For M2Λ ≤ (64/9) × 10−4, we have an innermost stablecircular orbit, at the radial position risco, and an outer-most stable circular orbit, at the radial position rosco (see,e.g., Ref. [37]). In the case of Λ = 0, we have F (Λ, r) ≥ 0in the interval 6M ≤ r <∞.

The orbital angular velocity of the circular timelikegeodesics is given by

Ω =dφ

dt=φ

t=

√M

r3− Λ

3, (13)

which goes to zero as the circular orbit radial positiontends to rmax, defined in Eq. (11). At the radial position

rmax, the gravitational attraction of the central objectis balanced by the contribution from the cosmologicalconstant. We can invert Eq. (13) to obtain r as a functionof Ω and Λ.

Considering ε = 0 in Eqs. (8)–(9), we find that thethe radial position r0 of the lightlike circular geodesic isgiven by the lower limit of the interval in Eq. (11), i.e.r0 = 3M. Note that r0 is independent of the value of thecosmological constant.

In the next section we analyze the massless scalar fieldin the SdS background. The field quantization procedureis very similar to that of a Schwarzschild spacetime [38].

III. SCALAR FIELD QUANTIZATION

The dynamics of the minimally coupled massless scalarfield Φ(x) in SdS spacetime is governed by the action,

S = −1

2

∫d4x√−g∇µΦ(x)∇µΦ(x), (14)

from which the equation of motion is obtained to be

∇µ∇µΦ(x) =1√−g

∂µ(√−ggµν∂νΦ

)= 0, (15)

where g = −r4 sin2 θ is the determinant of the SdS space-time metric. The positive-frequency solutions to Eq.(15), with respect to the timelike Killing vector field ∂t,can be written in the form,

ukωlm(x) =

√ω

π

Ψkωl(r)

rYlm(θ, φ)e−iωt (ω > 0), (16)

in which Ylm(θ, φ) are the scalar spherical harmonics

and√ω/π is a normalization constant. The k index in

Eq. (16) stands for (i) k = up, denoting modes purely in-coming from the past event horizon (H−h ); and (ii) k = in,denoting modes purely incoming from the past cosmolog-ical horizon (H−c ). From Eqs. (15) and (16), we find thatthe function Ψk

ωl(r) must satisfy the following ordinarydifferential equation:(−fΛ(r)

d

dr

(fΛ(r)

d

dr

)+ Veff (r)

)Ψkωl(r) = ω2Ψk

ωl(r),

(17)with the effective potential defined by

Veff (r) ≡ fΛ(r)

(l(l + 1)

r2+

2M

r3− 2Λ

3

). (18)

For l = 0, the potential Veff (r) changes sign at r = rmaxand has a point of minimum in the static region. Forl = 1, the potential (18) is illustrated in Fig. 3. We seethat as the parameter Λ increases, the potential barrierdecreases.

4

M2Λ=0

M2Λ=10-3

M2Λ=150-1

M2Λ=50-1

2 4 6 8 10

0.00

0.05

0.10

0.15

r/M

M2Veff(r)

ℓ = 1

FIG. 3: The effective potential Veff , given by Eq. (18), withl = 1 and different choices of the cosmological constant Λ.

Noting that the effective potential (18) vanishes forboth r = rh and r = rc, we can write the asymptoticsolutions of Eq. (17) in the form,

Ψupωl ≈

Aupωl

(eiωr

∗+Rupωl e−iωr

∗), r & rh,

AupωlTupωl e

iωr∗, r . rc,(19)

Ψinωl ≈

Ainωl

(e−iωr

∗+Rinωleiωr

∗), r . rc,

AinωlT inωl e−iωr∗, r & rh,

(20)

where Akωl are overall normalization constants to be de-termined. The tortoise coordinate r∗ is implicitly definedby dr∗ ≡ fΛ(r)−1dr. Hence, r∗ goes to −∞ (+∞) in thelimit r → rh (r → rc). By considering the Wronskian ofEqs. (19) and (20), one can show that∣∣T kωl∣∣2 +

∣∣Rkωl∣∣2 = 1. (21)

Following the canonical quantization procedure [7, 8,16, 39, 40], we may expand the quantum field operator

Φ(x) in terms of the creation (ak†ωlm) and annihilation

(akωlm) operators, as

Φ(x) =∑k,l,m

∫ ∞0

dω[ukωlm(x)akωlm + uk∗ωlm(x)ak†ωlm

].

(22)To normalize the modes ukωlm, we use the Klein-Gordoninner product [7],

(Φ,Ψ) ≡ i∫

Σ

dΣµ (Φ∗ (∇µΨ)−Ψ (∇µΦ∗)) , (23)

in which dΣµ = dΣnµ, with nµ being a future directedunit vector orthogonal to the Cauchy surface Σ (e.g., the

t = constant hypersurface Σt). Since Φ and Ψ satisfyEq. (15), one can show that the inner product (23) is

independent of the choice of the hypersurface Σ [8, 28].By requiring the orthogonality conditions,(

ukωlm, uk′

ω′l′m′

)= δkk′δll′δmm′δ(ω − ω′) (24)

and (ukωlm, u

k′∗ω′l′m′

)=(uk∗ωlm, u

k′

ω′l′m′

)= 0, (25)

one can show that the creation and annihilation operatorssatisfy the usual nonvanishing commutation relations,[

akωlm, ak†ωlm

]= δkk′δll′δmm′δ(ω − ω′). (26)

The vacuum state is defined as the quantum state an-nihilated by all akωlm [41],

akωlm |0〉 ≡ 0, ∀ (k, ω, l,m), (27)

and the one-particle-state is constructed as

ak†ωlm |0〉 = |k;ωlm〉 . (28)

Using Eqs. (23)–(25) and the differential equation forΨkωl written in terms of the tortoise coordinate, we can

readily obtain (up to a phase) the overall normalizationconstants of Eqs. (19) and (20), namely

Aupωlm = Ainωlm =1

2ω. (29)

In the next section, we consider the scalar field coupledto a classical matter source in a SdS spacetime, perform-ing a geodesic circular orbit around the black hole.

IV. SCALAR RADIATION AND EMITTEDPOWER

We consider the scalar source moving along an equato-rial circular (timelike) geodesic at r = R, with constantangular velocity Ω(R), given by Eq. (13). The source isdescribed by

j(x) =σ√−gu0

δ(r −R)δ(θ − π/2)δ(φ− Ωt), (30)

such that∫dβ(3)j(x) = σ, where β(3) is a hypersurface

orthogonal to the particle’s 4−velocity. The constant σdetermines the magnitude of the source-field interaction.The particle’s 4−velocity is given by

uµ(R) = γ (1, 0, 0,Ω) , (31)

with the normalization factor,

γ =1

(fΛ(R)−R2Ω2)1/2. (32)

5

The source-field coupling is described by the followinginteraction action:

SI =

∫d4x√−gj(x)Φ(x), (33)

where σ can be regarded as a coupling constant that de-termines the magnitude of the interaction between thefield and the source.

Due to the interaction between the field and the source,there exists a nonvanishing probability for the radiationof scalar quanta. To lowest order in perturbation theory,the transition amplitude from the vacuum-state, definedin Eq. (27), to the one-particle-state, with quantum num-bers k, l, m and energy ω, is given by [42]

Ak;ωlmem = 〈k;ωlm| iSI |0〉 = i

∫d4x√−gj(x)uk∗ωlm(x).

(34)It follows that the probability amplitude, given byEq. (34), is proportional to δ(ω −mΩ), i.e. there is onlyemission of scalar particles with ωm ≡ mΩ. Since ωm andΩ are positive quantities, we have that m ≥ 1.

The emitted power (with fixed k, l and m) is

W k;lmem =

∫ ∞0

dωω

∣∣Ak;ωlmem

∣∣2T

. (35)

Assuming that the source radiates during the whole rangeof coordinate time t, with −∞ < t < ∞, we can writeT =

∫dt = 2πδ(0) [43, 44].

The emitted power (35) is found to be

W k;lmem = 2σ2ω2

m

(fΛ(R)−R2Ω2

) ∣∣∣∣∣Ψkωml

R

∣∣∣∣∣2∣∣∣Ylm (π

2,Ωt)∣∣∣2,

(36)where the total power is obtained by summing over k (inand up), l ≥ 1 and 1 ≤ m ≤ l, namely

Wem =

up∑k=in

∞∑l=1

l∑m=1

W k;lmem . (37)

We note that there is no emission for odd values of l +m, since the time independent quantity |Ylm(π/2,Ωt)|2vanishes in these cases. For even values of l + m, wehave [45]

|Ylm(π/2,Ωt)|2 =2l + 1

4π

(l +m− 1)!!(l −m− 1)!!

(l +m)!!(l −m)!!.

(38)For the computation of the (total) emitted power, we

obtained the quantity∣∣Ψk

ωml

∣∣ by solving Eq. (17) numer-ically [23]. In the next section, we present a selection ofour results.

V. RESULTS

We numerically integrate Eq. (17) for each k = in andk = up modes. The boundary conditions to be satisfied

(Errωin)max

(Errωup)max

0.5 1.0 1.5 2.0 2.5 3.0 3.50

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

(Errωk) max

M2Λ = 150-1, 1≤ℓ≤20

(Errωin)max

(Errωup)max

0.5 1.0 1.5 2.0 2.5 3.00

5.×10-71.×10-61.5×10-62.×10-62.5×10-63.×10-63.5×10-6

(Errωk) max

M2Λ = 50-1, 1≤ℓ≤20

(Errωin)max

(Errωup)max

0.5 1.0 1.5 2.00

2.×10-74.×10-76.×10-78.×10-71.×10-61.2×10-61.4×10-6

ω

(Errωk) max

M2Λ = 15-1, 1≤ℓ≤20

FIG. 4: The quantity(Errkω

)max

, as a function of ω, for

k = up and k = in with M2Λ = 150−1 (top); M2Λ = 50−1

(middle); and M2Λ = 15−1 (bottom). We consider the inter-val 1 ≤ l ≤ 20.

are given by Eqs. (19) and (20), with suitable values ofr. We choose r/M ≥ rh/M + δ for r values near theevent horizon, and r ≤ rc/M − δ, for r values near thecosmological horizon, with

δ = 10−5. (39)

The numerical error is related to the magnitude of δ.In order to obtain the transmission and reflection co-

efficients,∣∣T kωl∣∣2 and

∣∣Rkωl∣∣2, respectively, we compare

the solutions obtained numerically for Ψkωl and d

dr

(Ψkωl

),

with the asymptotic solutions expressed by Eqs. (19) and(20), requiring the usual probability flux conservation,given by Eq. (21), to be satisfied.

As an estimation of the numerical error, we may definethe quantity,

Errkωl ≡∣∣T kωl∣∣2 +

∣∣Rkωl∣∣2 − 1, (40)

which was kept as Errkωl 1.To exemplify the numerical errors, in Fig. 4 we plot(

Errkω

)max

, i.e. the maximum value of the error Errkωl,

considering all values of l in the interval 1 ≤ l ≤ 20, fora given Λ, as a function of ω. We see that this maximumerror is of the order 10−6, for all values of ω. Analogously,

in Fig. 5, we plot(

Errkl

)max

, i.e. the maximum value of

the error Errkωl, considering all values of ω in the interval0 < ω ≤ lΩ(r0), for a given Λ, as a function of l.

In general, the emitted power of radiation starts in-creasing from zero, at R = rmax, reaches a maximumand then decreases to zero again as R→ 3M. The radialposition of the peak of emission approaches r = 3M aswe increase the multipole number l. The emitted powerassociated to in modes is generally dominant, except for

6

(Errℓin)max

(Errℓup)max

0 5 10 15 200

2.×10-8

4.×10-8

6.×10-8(Errℓk) max

M2Λ = 150-1, 0<ω≤ ℓ Ω(r0)

(Errℓin)max

(Errℓup)max

0 5 10 15 200

5.×10-91.×10-81.5×10-82.×10-8

(Errℓk) max

M2Λ = 50-1, 0<ω≤ ℓ Ω(r0)

(Errℓin)max

(Errℓup)max

0 5 10 15 200

2.×10-94.×10-96.×10-98.×10-91.×10-81.2×10-81.4×10-8

ℓ

(Errℓk) max

M2Λ = 15-1, 0<ω≤ ℓ Ω(r0)

FIG. 5: The quantity(Errkl

)max

, as a function of l, for k = up

and k = in with M2Λ = 150−1 (top); M2Λ = 50−1 (middle);and M2Λ = 15−1 (bottom). We consider the interval 0 <ω ≤ lΩ(r0).

the region close to r = 3M, in which the up modes startto give a significant contribution. We note that, for or-bits close to r = 3M, the major contribution (> 97%)to the emitted power, for a given multipole number l,comes from the l = m mode, similarly to what happensin asymptotically flat spacetimes [46, 47].

In Fig. 6, we plot the emitted power by the orbitingsource as a function of Ω, for a fixed value of l = m, fordifferent choices of Λ. We see that, for l = m = 1, thepeak of emission starts increasing with Λ, but after a cer-tain value of the cosmological constant (Λ ∼ 30−1M−2)the peak starts to decrease. This behavior changes forhigher values of l = m, with the peak of emission mono-tonically decreasing, as the value of Λ is increased.

In Fig. 7, we plot separately the contribution fromthe in and up modes to the emitted power, for differentchoices of the cosmological constant Λ and of the multi-pole numbers l = m. We see that, when M2Λ < 150−1,the behavior of the emitted power, as a function of theangular velocity (Ω) of the source is similar to that ofthe Schwarzschild case [18]. On the other hand, forM2Λ > 150−1, we have an amplification of the poweremitted by modes with lower values of l = m. This effectis more evident for the in modes, i.e., the modes purelyincoming from the cosmological horizon (Hc).

In Fig. 8, we plot the total emitted power, given byEq. (37), for two choices of the cosmological constant.The l summation was truncated at a maximum valuel = lmax. We see that, when the source lies relativelyfar from the black hole (where the l = 1 mode contri-bution is dominant), the emitted power is basically thesame for any choice of lmax, but as the orbit of the sourceapproaches r = 3M, higher multipole modes start to con-tribute significantly, exhibiting a synchrotronic behaviorof the emitted power.

M2Λ=15-1

M2Λ=20-1

M2Λ=50-1

M2Λ=150-1

M2Λ=10-3

M2Λ=0

0 5 10 150.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

102 M Ω

102M2σ-2Wemlm

ℓ=m=1

M2Λ=15-1

M2Λ=20-1

M2Λ=50-1

M2Λ=150-1

M2Λ=10-3

M2Λ=0

0 5 10 150.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

102 M Ω

102M2σ-2Wemlm

ℓ=m=2

M2Λ=15-1

M2Λ=20-1

M2Λ=50-1

M2Λ=150-1

M2Λ=10-3

M2Λ=0

0 5 10 150.000

0.002

0.004

0.006

0.008

0.010

0.012

102 M Ω

102M2σ-2Wemlm

ℓ=m=5

FIG. 6: The emitted power as a function of Ω, given by thesum of the in and up modes, for l = m = 1 (top), l = m = 2(middle) and l = m = 5 (bottom), with different choices ofthe parameter Λ. The curves are plotted up to the values ofMΩ corresponding to the radial position R = 3M (indicatedby the vertical lines).

VI. FINAL REMARKS

In this paper we have used QFT in curved spacetimeat tree level to investigate the scalar radiation emitted bya source orbiting a Schwarzschild–de Sitter (SdS) blackhole. We have presented numerical results for the par-tial (with fixed l and m) and total emitted powers, asfunctions of the angular velocity of the source.

We have found that the emitted power strongly de-

7

l=m=1l=m=2l=m=3l=m=4l=m=5l=m=10l=m=20

0 5 10 150.000

0.002

0.004

0.006

0.008

102 M Ω

102M2σ-2Wemin;lm

M2Λ = 0

l=m=1l=m=2l=m=3l=m=4l=m=5l=m=10l=m=20

0 5 10 150.000

0.002

0.004

0.006

0.008

102 M Ω

102M2σ-2Wemup;lm

M2Λ = 0

l=m=1l=m=2l=m=3l=m=4l=m=5l=m=10l=m=20

0 5 10 150.000

0.002

0.004

0.006

0.008

102 M Ω

102M2σ-2Wemin;lm

M2Λ = 150-1

l=m=1l=m=2l=m=3l=m=4l=m=5l=m=10l=m=20

0 5 10 150.000

0.002

0.004

0.006

0.008

102 M Ω

102M2σ-2Wemup;lm

M2Λ = 150-1

l=m=1l=m=2l=m=3l=m=4l=m=5l=m=10l=m=20

0 5 10 150.000

0.002

0.004

0.006

0.008

102 M Ω

102M2σ-2Wemin;lm

M2Λ = 50-1

l=m=1l=m=2l=m=3l=m=4l=m=5l=m=10l=m=20

0 5 10 150.000

0.002

0.004

0.006

0.008

102 M Ω

102M2σ-2Wemup;lm

M2Λ = 50-1

l=m=1l=m=2l=m=3l=m=4l=m=5l=m=10l=m=20

0 2 4 6 8 10 120.000

0.002

0.004

0.006

0.008

102 M Ω

102M2σ-2Wemin;lm

M2Λ = 15-1

l=m=1l=m=2l=m=3l=m=4l=m=5l=m=10l=m=20

0 2 4 6 8 10 120.000

0.002

0.004

0.006

0.008

102 M Ω

102M2σ-2Wemup;lm

M2Λ = 15-1

FIG. 7: The emitted power, as a function of Ω, given by Eq. (36), with different choices of l = m, for in (left) and up (right)modes. We consider the black hole with different choices of the parameter Λ, as indicated.

8

lmax=5lmax=10lmax=15lmax=20

M2Λ = 150-1

0 5 10 150.00

0.05

0.10

0.15

0.20

102 M Ω

102M

2σ-2W

em

(a)

lmax=5lmax=10lmax=15lmax=20

M2Λ = 15-1

0 2 4 6 8 10 120.00

0.05

0.10

0.15

0.20

102 M Ω

102M

2σ-2W

em

(b)

FIG. 8: The total emitted power, given by Eq. (37), as a function of Ω, in the SdS spacetime with (a) M2Λ = 150−1 and (b)M2Λ = 15−1. The summations in l were truncated at l = lmax, as indicated.

pends on the value of the cosmological constant whenΛ > 150−1M−2. We have also shown that the emittedpower associated to lower values of the multipole num-ber l is amplified as Λ increases. In the Schwarzschild–anti–de Sitter geometry (for which Λ < 0), for sufficientlyhigher values of |Λ|, an enhancement in the emitted powerassociated to higher values l = m has been reported [25].

The scalar radiation considered in this paper havequalitative features similar to more realistic scenarios,as the ones of electromagnetic and gravitational radia-tion. Nevertheless, it is known that, in asymptoticallyflat spacetimes, the contribution of the high multipolesto the emitted power depends on the spin of the radia-tion field [23, 24, 46, 48]. Thus, a similar investigation offields with nonzero spin in asymptotically dS solutions,such as the SdS spacetime, will reveal the high multipolesbehavior and their contribution to the emitted power, to-gether with their relation to the black hole size. It will

also be interesting to extend this work to more generalblack hole spacetimes that are asymptotically dS, charac-terized by additional parameters, such as electric chargeand angular momentum.

Acknowledgments

The authors would like to thank Coordenacao deAperfeicoamento de Pessoal de Nıvel Superior (CAPES,Brazil) — Finance Code 001, and Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico (CNPq, Brazil)for partial financial support. This research has also re-ceived funding from the European Union’s Horizon 2020research and innovation programme under the H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740.

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