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Electromagnetic radiation generated by a charged particle falling radially into aSchwarzschild black hole: A complex angular momentum description

Antoine Folacci∗ and Mohamed Ould El Hadj†

Equipe Physique Theorique,SPE, UMR 6134 du CNRS et de l’Universite de Corse,

Universite de Corse, Faculte des Sciences,BP 52, F-20250 Corte, France

(Dated: June 30, 2020)

By using complex angular momentum techniques, we study the electromagnetic radiation gener-ated by a charged particle falling radially from infinity into a Schwarzschild black hole. We considerboth the case of a particle initially at rest and that of a particle projected with a relativistic velocityand we construct complex angular momentum representations and Regge pole approximations ofthe partial wave expansions defining the Maxwell scalar φ2 and the energy spectrum dE/dω ob-served at spatial infinity. We show, in particular, that Regge pole approximations involving onlyone Regge pole provide effective resummations of these partial wave expansions permitting us (i)to reproduce with very good agreement the black hole ringdown without requiring a starting time,(ii) to describe with rather good agreement the tail of the signal and sometimes the pre-ringdownphase, and (iii) to explain the oscillations in the electromagnetic energy spectrum radiated by thecharged particle. The present work as well as a previous one concerning the gravitational radiationgenerated by a massive particle falling into a Schwarzschild black hole [A. Folacci and M. Ould ElHadj, Phys. Rev. D 98, 064052 (2018), arXiv:1807.09056 [gr-qc]] highlight the benefits of studyingradiation from black holes in the complex angular momentum framework (they obviously appearwhen the approximations obtained involve a small number of Regge poles and have a clear physicalinterpretation) but also to exhibit the limits of this approach (this is the case when it is necessaryto take into account background integral contributions).

CONTENTS

I. Introduction 1

II. Maxwell scalar φ2 and associated quasinormalringdown 2A. Multipole expansion of the Maxwell scalar

φ2 2B. Regge-Wheeler equation and S-matrix 4C. Compact expression for the multipole

expansion of the Maxwell scalar φ2 4D. Regularization of the partial waveform

amplitudes ψω`m and the Maxwell scalar φ2 5E. Two alternative expressions for the multipole

expansion of the Maxwell scalar φ2 5F. Quasinormal ringdown associated with the

Maxwell scalar φ2 6

III. Maxwell scalar φ2, its CAM representations andits Regge pole approximations 6A. Some preliminary remarks concerning

analytic extensions in the CAM plane 6B. Regge poles, Regge modes and associated

excitation factors 7C. CAM representation and Regge pole

approximation of the Maxwell scalar φ2 basedon the Poisson summation formula 9

∗ [email protected]† [email protected]

D. CAM representation and Regge poleapproximation of the Maxwell scalar φ2 basedon the Sommerfeld-Watson transform 10

IV. Comparison of the Maxwell scalar φ2 with itsRegge pole approximations 10A. Computational methods 11B. Results and comments 11

V. Electromagnetic energy spectrum dE/dω and itsCAM representation 11A. Total energy radiated by the particle and

associated electromagnetic energy spectrum 19B. CAM representation based on the Poisson

summation formula 20C. Computational methods 21D. Numerical results and comments 21

VI. Conclusion 25

References 26

I. INTRODUCTION

In a recent article [1], we advocated for an alternativedescription of gravitational radiation from black holes(BHs) based on complex angular momentum (CAM)techniques, i.e., analytic continuation in the CAM planeof partial wave expansions, duality of the S-matrix, ef-fective resummations involving its Regge poles and theassociated residues, Regge trajectories, semiclassical in-

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terpretations, etc. In this previous article as well as inmore recent works [2, 3] where we have provided CAMand Regge pole analyses of scattering of scalar, electro-magnetic and gravitational waves by a Schwarzschild BH,we have justified the interest of such an approach in thecontext of BH physics and we shall not return here on thissubject. We refer the interested reader to these articlesand, more particularly, to the introduction of Ref. [1] aswell as to references therein for other works dealing withthe CAM approach to BH physics.

In the present article, by using CAM techniques, weshall revisit the problem of the electromagnetic radia-tion generated by a charged particle falling radially intoa Schwarzschild BH. We shall consider both the case of aparticle initially at rest and that of a particle projectedwith a relativistic or an ultra-relativistic velocity and weshall construct CAM representations and Regge pole ap-proximations of the partial wave expansions defining theMaxwell scalar φ2 and the energy spectrum dE/dω ob-served at spatial infinity. This work extends our pre-vious work concerning the CAM and Regge pole analy-ses of the gravitational radiation generated by a massiveparticle falling into a Schwarzschild BH [1]. Both high-light the benefits of working within the CAM frameworkand strengthen our opinion concerning the interest of theRegge pole approach for describing radiation from BHs.

Problems dealing with the excitation of a BH by acharged particle and the generation of the associatedelectromagnetic radiation have been considered, since theearly 1970s, in the literature (for pioneering works on thissubject, see the lectures by Ruffini [4] in Ref. [5] and ref-erences therein, as well as Refs. [6–9] for articles directlyrelevant to our study) and, currently, an ever-increasingimportance is given to them. Indeed, such problems areof great interest with the emergence of multimessengerastronomy which combines the detection and analysis ofgravitational waves with those of other types of radia-tion for a better understanding of our “violent Universe”but also in order to test the BH hypothesis and Ein-stein’s general relativity in the strong-field regime (see,e.g., Refs. [10–12]). In this context, it is particularlyinteresting to study the electromagnetic partner of thegravitational radiation generated during the accretion ofa charged fluid by a BH [13, 14].

Our paper is organized as follows. In Sec. II, we firstconstruct the Maxwell scalar φ2 describing the outgoingelectromagnetic radiation at infinity which is generatedby a charged particle falling radially into a SchwarzschildBH. To do this, by using Green’s function techniques, wesolve in the frequency domain the Regge-Wheeler equa-tion for arbitrary (`,m) modes and we proceed to theirregularization. We also extract from the multipole ex-pansion of φ2 the quasinormal ringdown of the BH. InSec. III, we provide two different CAM representationsof the multipolar waveform φ2: the first one is basedon the Poisson summation formula [15] while the secondone is constructed from the Sommerfeld-Watson trans-formation [16–18]. From each of them, we extract, as

approximations of φ2, the Fourier transform of a sumover Regge poles and Regge-mode excitation factors. Itis important to note that, in order to evaluate numeri-cally these two Regge pole approximations, we need theRegge trajectories (i.e., the curves traced out in the CAMplane by the Regge poles and by the associated residuesas a function of the frequency ω). In Sec. IV, we numer-ically compare the multipolar waveform φ2 constructedby summing over a large number of partial modes (thisis particularly necessary for a particle projected with arelativistic or an ultra-relativistic velocity) as well as theassociated ringdown with the Regge pole approximationsobtained in Sec. III. This permits us to emphasize thebenefits of working with these particular approximationsof the Maxwell scalar φ2. In Sec. V, we focus on theelectromagnetic energy spectrum dE/dω radiated by thecharged particle falling into the Schwarzschild BH andwe numerically compare it with its CAM representationobtained from the Poisson summation formula. In theConclusion, we summarize the main results obtained andbriefly discuss some possible extensions of our approach.

Throughout this article, we adopt units such that G =c = ε0 = µ0 = 1, we use the geometrical conventionsof Ref. [19] and we perform the numerical calculationsusing Mathematica [20]. We, furthermore, consider thatthe exterior of the Schwarzschild BH is defined by the lineelement ds2 = −f(r)dt2+f(r)−1dr2+r2dθ2+r2 sin2 θdϕ2

where f(r) = 1−2M/r andM is the mass of the BH whilet ∈] −∞,+∞[, r ∈]2M,+∞[, θ ∈ [0, π] and ϕ ∈ [0, 2π]are the usual Schwarzschild coordinates.

II. MAXWELL SCALAR φ2 AND ASSOCIATEDQUASINORMAL RINGDOWN

In this section, we shall construct the Maxwell scalarφ2 describing the outgoing radiation at infinity due toa charged particle falling radially from infinity into aSchwarzschild BH. Moreover, we shall extract from themultipole expansion of φ2 the associated ringdown wave-form.

A. Multipole expansion of the Maxwell scalar φ2

We consider a charged particle (we denote by m0 itsmass and by q its electric charge) falling radially intoa Schwarzschild BH. The timelike geodesic followed bysuch a particle is defined by the coordinates tp(τ), rp(τ),θp(τ) and ϕp(τ) where τ is its proper time. Without lossof generality, we can consider that this particle moves inthe BH equatorial plane along the positive x axis and inthe negative direction, i.e., we assume that θp(τ) = π/2,ϕp(τ) = 0 and drp(τ)/dτ < 0. The functions tp(τ), rp(τ)as well as the function tp(r) can be then obtained fromthe geodesic equations (see, e.g., Ref. [21])

f(rp)dtpdτ

=Em0

, (1a)

3

and (drpdτ

)2

− 2M

rp=

(Em0

)2

− 1. (1b)

Here, E is the energy of the particle. It is a constant ofmotion which can be related to the velocity v∞ of theparticle at infinity and to the associated Lorentz factorγ by

Em0

=1√

1− (v∞)2= γ. (2)

The electromagnetic radiation generated by this par-ticle can be described by using the gauge-invariant for-malism introduced by Ruffini, Tiomno and Vishveshwarain Ref. [6] (see also Refs. [22, 23]) and by working in theframework of the Newman-Penrose formalism (see, e.g.,Chap. 8 of Ref. [24]). We shall therefore focus on theMaxwell scalar φ2 which can be expressed at spatial in-finity as

φ2 =1

2√

2(Eθ − iEϕ) + i (Bθ − iBϕ) (3)

where Eθ, Eϕ, Bθ and Bϕ denote the components of theelectromagnetic field (E,B) observed for r →∞. Here, itis important to note that we have defined φ2 with respectto the null basis (l, n,m,m∗) which is normalized suchthat the only nonvanishing scalar products involving thevectors of the tetrad are lµnµ = −1 and mµm∗µ = 1 andwhich is given by (our conventions slightly differ fromthose of Ref. [24])

lµ =

(1

f(r), 1, 0, 0

), (4a)

nµ =1

2

(1,− 1

f(r), 0, 0

), (4b)

mµ =1√2r

(0, 0, 1,

i

sin θ

), (4c)

m∗µ =1√2r

(0, 0, 1,

−isin θ

). (4d)

We recall that the radially infalling particle only ex-cites the even (polar) electromagnetic modes of theSchwarzschild BH and that, in the usual orthonormal-ized basis (er, eθ, eϕ) of the spherical coordinate system,the components of the electric field can be expressed interms of the gauge-invariant master functions ψ`m(t, r)and expanded on the (even) vector spherical harmonicsY `mθ (θ, ϕ) and Y `mϕ (θ, ϕ) in the form [25]

E =

∣∣∣∣∣∣∣∣∣∣∣

Er = 0

Eθ = − 1r

+∞∑=1

+∑m=−`

1`(`+1) ∂rψ`m Y

`mθ

Eϕ = − 1r sin θ

+∞∑=1

+∑m=−`

1`(`+1) ∂rψ`m Y

`mϕ ,

(5)

while the magnetic field B can be obtained from theMaxwell-Faraday equation and its components expressedin terms of those of the electric field. Indeed, for r →+∞, we have ∂tψ`m = −∂rψ`m and we can write

B =

∣∣∣∣∣∣∣∣Br = 0

Bθ = −EϕBϕ = +Eθ.

(6)

It should be noted that the vector spherical harmonicsappearing in Eq. (5) are given in terms of the standardscalar spherical harmonics Y `m(θ, ϕ) by

Y `mθ =∂

∂θY `m and Y `mϕ =

∂

∂ϕY `m (7)

and satisfy the “orthonormalization” relation∫S2

dΩ2

[Y `mθ (θ, ϕ)[Y `

′m′

θ (θ, ϕ)]∗

+1

sin2 θY `mϕ (θ, ϕ)[Y `

′m′

ϕ (θ, ϕ)]∗]

= `(`+ 1)δ``′δmm′ . (8)

Here dΩ2 = sin θ dθ dϕ denotes the area element on theunit sphere S2. We also recall that the gauge-invariantmaster functions ψ`m(t, r) appearing in Eq. (5) can bewritten in the form

ψ`m(t, r) =1√2π

∫ +∞

−∞dω ψω`m(r)e−iωt (9)

where their Fourier components ψω`m(r) satisfy theRegge-Wheeler equation[

d2

dr2∗+ ω2 − V`(r)

]ψω`m(r) = Sω`m(r). (10)

Here, Sω`m(r) is a source term, r∗ denotes the tor-toise coordinate which is defined in terms of the radialSchwarzschild coordinate r by dr/dr∗ = f(r) and is givenby r∗(r) = r + 2M ln[r/(2M)− 1] while

V`(r) = f(r)

(`(`+ 1)

r2

)(11)

denotes the Regge-Wheeler potential.As far as the source term Sω`m(r) appearing in the

right-hand side (r.h.s.) of the Regge-Wheeler equation(10) is concerned, it depends on the components, in thebasis of vector spherical harmonics, of the current asso-ciated with the charged particle [25]. Its expression canbe derived from Eqs. (1) and (2) and we obtain

Sω`m(r) = [Y `m(π/2, 0)]∗Sω(r)e+iωtp(r) (12)

where

4

Sω(r) =q√2πf(r)

[+iω

r

(γ2 − 1)r + 2M− Mγ√r [(γ2 − 1)r + 2M ]

3/2

](13a)

and with

tp(r)

2M= −2

3

( r

2M

)3/2− 2

( r

2M

)1/2+ ln

(√r

2M + 1√r

2M − 1

)+

t02M

(13b)

for the particle starting at rest from infinity (i.e., for γ = 1) and

tp(r)

2M= − γ

(γ2 − 1)3/2

√[(γ2 − 1)

r

2M

] [(γ2 − 1)

r

2M+ 1]

− γ(2γ2 − 3)

(γ2 − 1)3/2ln

[√(γ2 − 1)

r

2M+

√(γ2 − 1)

r

2M+ 1

]+ ln

[γ√

r2M +

√(γ2 − 1) r

2M + 1

γ√

r2M −

√(γ2 − 1) r

2M + 1

]+

t02M

(13c)

for a particle projected with a finite kinetic energy atinfinity (i.e., for γ > 1). In Eqs. (13b) and (13c), t0 is anarbitrary integration constant.

B. Regge-Wheeler equation and S-matrix

The Regge-Wheeler equation (10) can be solved by us-ing the machinery of Green’s functions (see, e.g., Ref. [26]for its use in the context of BH physics). Mutatis mu-tandis, taking into account Eq. (12), the reasoning ofSec. IIC of Ref. [27] permits us to obtain the asymp-totic expression, for r → +∞, of the partial amplitudesψω`m(r). We have

ψω`m(r) = e+iωr∗(r)K[`, ω]

2iωA(−)` (ω)

[Y `m(π/2, 0)]∗ (14a)

with

K[`, ω] =

∫ +∞

2M

dr′

f(r′)φinω,`(r

′) Sω(r′)eiωtp(r′). (14b)

Here, we have introduced the solution φinω,`(r) of the ho-mogeneous Regge-Wheeler equation[

d2

dr2∗+ ω2 − V`(r)

]φinω,` = 0 (15)

which is defined by its behavior at the event horizon r =2M (i.e., for r∗ → −∞) and at spatial infinity r → +∞(i.e., for r∗ → +∞):

φinω,`(r∗) ∼

e−iωr∗ (r∗ → −∞)

A(−)` (ω)e−iωr∗ +A

(+)` (ω)e+iωr∗ (r∗ → +∞).

(16)

The coefficients A(−)` (ω) and A

(+)` (ω) appearing in

Eqs. (14) and (16) are complex amplitudes. By evalu-ating, first for r∗ → −∞ and then for r∗ → +∞, the

Wronskian involving the function φinω` and its complexconjugate, we can show that they are linked by

|A(−)` (ω)|2 − |A(+)

` (ω)|2 = 1. (17)

Moreover, with the numerical calculation of the Maxwellscalar φ2 as well as the study of its properties in mind,it is important to note that

φin−ω,`(r) =[φinω,`(r)

]∗, (18a)

A(±)` (−ω) = [A

(±)` (ω)]∗. (18b)

It is worth pointing out that the boundary conditions(16) for φinω,`(r) and therefore the expression (14) of the

partial amplitudes ψω`m(r) involve the S-matrix definedby (see, e.g., Ref. [28])

S`(ω) =

(1/A

(−)` (ω) A

(+)` (ω)/A

(−)` (ω)

− [A(+)` (ω)]∗/A

(−)` (ω) 1/A

(−)` (ω)

).

(19)

Due to Eq. (18b), this matrix satisfies the symmetryproperty S`(−ω) = [S`(ω)]

∗and, due to Eq. (17), it

is in addition unitary, i.e., it satisfies SS† = S†S = 1.Here, it is interesting to recall that, in Eq. (19), the

term 1/A(−)` (ω) and the term A

(+)` (ω)/A

(−)` (ω) are, re-

spectively, the transmission coefficient T`(ω) and the re-flection coefficient Rin

` (ω) corresponding to the scatter-ing problem defined by Eq. (16). As far as the coefficient

−[A(+)` (ω)]∗/A

(−)` (ω) is concerned, it can be considered

as the reflection coefficient Rup` (ω) involved in the scat-

tering problem defining the modes φupω,`(r) [28].

C. Compact expression for the multipole expansionof the Maxwell scalar φ2

We first insert Eq. (14a) into Eq. (9) and we have

ψ`m(t, r) = ψ`(t, r) [Y `m(π/2, 0)]∗ (20a)

5

where

ψ`(t, r) =1√2π

∫ +∞

−∞dω e−iω[t−r∗(r)]

K[`, ω]

2iωA(−)` (ω)

.

(20b)

We now substitute Eq. (20) into Eq. (5). Furthermore,without loss of generality, we assume that the electro-magnetic radiation is observed in a direction lying in theBH equatorial plane and making an angle ϕ ∈ [0, π] withthe trajectory of the particle (due to symmetry consid-erations, we can restrict our study to this interval). Bythen using the addition theorem for scalar spherical har-monics in the form

+∑m=−`

Y `m(θ, ϕ)[Y `m(π/2, 0)]∗ =2`+ 1

4πP`(sin θ cosϕ)

(21)where P`(x) denotes the Legendre polynomial of degree` [29], we obtain, for r → +∞,

r Eθ(t, r, θ = π/2, ϕ) = 0 (22)

and

r Eϕ(t, r, θ = π/2, ϕ) = − 1√2π

∫ +∞


×

[1

4π

+∞∑`=1

2`+ 1

`(`+ 1)

K[`, ω]

2A(−)` (ω)

W`(cosϕ)

]. (23)

In Eq. (23), we have introduced the angular function

W`(cosϕ) =∂

∂ϕP`(cosϕ). (24)

Finally, taking into account Eq. (6), we can write byinserting Eqs. (22) and (23) into Eq. (3)

√2 r

iφ2(t, r, θ = π/2, ϕ) =

1√2π

∫ +∞


×

[1

4π

+∞∑`=1

2`+ 1

`(`+ 1)

K[`, ω]

2A(−)` (ω)

W`(cosϕ)

](25)

for r → +∞.

D. Regularization of the partial waveformamplitudes ψω`m and the Maxwell scalar φ2

To construct the Maxwell scalar φ2, we need to regu-larize the partial amplitudes ψω`m(r) or, more precisely,K[`, ω]. Indeed, the partial waveforms (14) as integralsover the radial Schwarzschild coordinate are divergent atinfinity. This is due to the behavior of the source (13)for r →∞.

To regularize K[`, ω], we integrate twice by partsand use the homogeneous Regge-Wheeler equation (15).

Then, by dropping intentionally the boundary terms atr →∞ (regularization), we obtain

K[`, ω] = q `(`+ 1)K[`, ω]

iω(26a)

with

K[`, ω] =1√2π

∫ +∞

2M

dr′ φinω,`(r′)eiωtp(r

′)

r′2. (26b)

Now, by inserting Eqs. (26a) and (26b) into Eq. (25),we obtain for the Maxwell scalar√

2 r

iqφ2(t, r, θ = π/2, ϕ) =

1√2π

∫ +∞


×

[+∞∑`=1

2`+ 1

4π

K[`, ω]

2iωA(−)` (ω)

W`(cosϕ)

]. (27)

It is in addition interesting to note that, by insertingEqs. (26a) and (26b) into the partial waveform ampli-tudes (14a), we can recover the amplitude term derivedby Cardoso, Lemos and Yoshida in Ref. [9] working inthe Zerilli gauge [30, 31].

Moreover, with the numerical calculation of theMaxwell scalar φ2 as well as the study of its propertiesin mind, we can observe that

K[`,−ω] =[K[`, ω]

]∗(28a)

and

K[`,−ω]/A(−)` (−ω) =

[K[`, ω]/A

(−)` (ω)

]∗(28b)

as a consequence of Eqs. (18a) and (18b). Due to relation(28b), we can see that the term in square brackets inEq. (27) satisfies the Hermitian symmetry property and,as a consequence, that the Maxwell scalar φ2 is a purelyimaginary quantity. Similarly, it is worth pointing outthat the electromagnetic field is a real quantity.

E. Two alternative expressions for the multipoleexpansion of the Maxwell scalar φ2

It is important to realize that Eq. (27) can also bewritten as√

2 r

iqφ2(t, r, θ = π/2, ϕ) =

1√2π

∫ +∞


×

[+∞∑`=0

2`+ 1

4π

K[`, ω]

2iωA(−)` (ω)

W`(cosϕ)

]. (29)

Indeed, it is possible to start at ` = 0 the discrete sumover ` by noting that

W0(cosϕ) =∂

∂ϕP0(cosϕ) = 0 (30)

6

and that we have formally

A(−)0 (ω) = 1 and K[0, ω] regular. (31)

These last two results are due to the fact that, for ` = 0,the solution of the problem (15)–(16) is φinω,0(r) = e−iωr∗

because the Regge-Wheeler potential (11) vanishes. Ofcourse, in general, it is more natural to work with themultipole expansion (27) of the Maxwell scalar φ2 but, inSec. III C, we shall take (29) as a departure point becauseit will permit us to use the Poisson summation formulain its standard form.

Similarly, it is important to note that Eq. (29) can berewritten in the form√

2 r

iqφ2(t, r, θ = π/2, ϕ) =

1√2π

∫ +∞


×

[+∞∑`=0

(−1)`2`+ 1

4π

K[`, ω]

2iωA(−)` (ω)

W`(− cosϕ)

]. (32)

Indeed, we can recover Eq. (29) from Eq. (32) by usingthe relation [29]

P`(− cosϕ) = (−1)`P`(cosϕ) (33)

in connection with the definition (24). In Sec. III D, weshall take Eq. (32) as a departure point because it willpermit us to use the Sommerfeld-Watson transform in itsstandard form.

F. Quasinormal ringdown associated with theMaxwell scalar φ2

The quasinormal ringdown φQNM

2 generated by thecharged particle falling radially from infinity into aSchwarzschild BH can be extracted from Eq. (27) by fol-lowing, mutatis mutandis, the reasoning of Sec. II E ofRef. [1]. We then obtain

√2 r

iqφQNM

2 (t, r, θ = π/2, ϕ) = −2√

2π Re

[+∞∑`=1

+∞∑n=1

2`+ 1

4πB`n

K[`, ω`n]

A(+)` (ω`n)

e−iω`n[t−r∗(r)]W`(cosϕ)

](34)

where

B`n =

[1

2ω

A(+)` (ω)

ddωA

(−)` (ω)

]ω=ω`n

(35)

denotes the excitation factor associated with the (`, n)quasinormal mode (QNM) of complex frequency ω`n. Its

expression involves the residue of the function 1/A(−)` (ω)

at ω = ω`n. It should be noted that Eq. (34) has been ob-tained by gathering the contributions of the quasinormalfrequencies ω`n and −ω∗`n taking into account the rela-tions (18b) and (28a) which remain valid in the complex

ω plane. As a consequence, the quasinormal ringdownwaveform φQNM

2 appears clearly as a purely imaginaryquantity.

Let us finally recall that, due to the exponentiallydivergent behavior of the terms e−iω`n[t−r∗(r)] as t de-creases, the ringdown waveform φQNM

2 does not providephysically relevant results at early times. It is thereforenecessary to determine, from physical considerations, astarting time tstart for the BH ringdown. In general,by taking tstart = tp(3M), i.e., the moment the parti-cle crosses the photon sphere, we can obtain physicallyrelevant results.

III. MAXWELL SCALAR φ2, ITS CAMREPRESENTATIONS AND ITS REGGE POLE

APPROXIMATIONS

In this section, we shall derive two exact CAM repre-sentations of the Maxwell scalar φ2, the first one by us-ing the Poisson summation formula [15] and the secondone by working with the Sommerfeld-Watson transfor-mation [16–18]. These representations can be written as(the Fourier transform of) a sum over Regge poles plusbackground integrals along the positive real axis and theimaginary axis of the CAM plane. We shall also considerthe Regge pole part of these representations as approxi-mations of the Maxwell scalar φ2 which can be evaluatednumerically from the Regge trajectories followed by theRegge poles and by the excitation factors of the associ-ated Regge modes.

In order to construct the two CAM representations ofthe Maxwell scalar φ2 and the associated Regge pole ap-proximations, we shall follow, mutatis mutandis, Sec. IIIof Ref. [1].

A. Some preliminary remarks concerning analyticextensions in the CAM plane

The CAM machinery permitting us to derive the CAMrepresentations of the multipolar waveform φ2 requiresus to replace in Eqs. (29) and (32) the angular momen-tum ` ∈ N by the angular momentum λ = ` + 1/2 ∈ Cand therefore to work into the CAM plane. As a con-sequence, we need to have at our disposal the func-

tions Wλ−1/2(cosϕ), Wλ−1/2(− cosϕ), A(±)λ−1/2(ω) and

K[λ − 1/2, ω] which are “the” analytic extensions of

W`(cosϕ), W`(− cosϕ), A(±)` (ω) and K[`, ω] in the com-

plex λ plane. We recall that the uniqueness problemfor such analytic extensions is a difficult problem. Wehave briefly discussed it in Sec. IIIA of Ref. [1] (see alsoChap. 13 of Ref. [18]). Here, in order to construct theseanalytic extensions, we shall adopt minimal prescriptionsthat will be justified by the results we shall obtain inSec. IV.

The angular functions W`(cosϕ), W`(− cosϕ) are de-

7

n = 1

n = 2

n = 3

0

2

4

6

8

10

12

14

Re[λn(ω)]

Regge trajectories (2M = 1)

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

ω

Im[λn(ω)]

0 2 4 6 8 10 12 14

-3

-2

-1

0

1

2

3

Re[λn (ω)]

Im[λn(ω)]

Regge trajectories (2M = 1)

ω→0

ω = 1 ω = 5

ω = -1 ω = -5

n = 3

n = 2

n = 1

FIG. 1. Regge trajectories of the first three Regge poles corresponding to electromagnetism in the Schwarzschild BH (2M = 1).The relation (43) permits us to describe the Regge trajectories for ω < 0 by noting that Re[λn(ω)] and Im[λn(ω)] are,respectively, even and odd functions of ω. We observe, in particular, the migration of the Regge poles in the CAM plane.

fined from the Legendre polynomial P`(z) [see Eq. (24)]of which the analytic extension usually considered is thehypergeometric function [29]

Pλ−1/2(z) = F (1/2− λ, 1/2 + λ; 1; (1− z)/2]. (36)

As a consequence, it is natural to take

Wλ−1/2(± cosϕ)

=∂

∂ϕF (1/2− λ, 1/2 + λ; 1; (1∓ cosϕ)/2] (37)

and it is worth noting that, due to the properties of thehypergeometric function, we have

W−λ−1/2(± cosϕ) = Wλ−1/2(± cosϕ) (38)

and

Wλ−1/2(± cosϕ) = [Wλ∗−1/2(± cosϕ)]∗. (39)

Here, it is crucial to keep in mind that, while the an-gular functions W`(± cosφ) are well defined for ϕ ∈[0, π], this is not the case for their analytic extensionsWλ−1/2(± cosφ). Indeed, due to the pathologic behaviorof Pλ−1/2(z) at z = −1, Wλ−1/2(cosφ) diverges in thelimit ϕ → π and Wλ−1/2(− cosφ) diverges in the limitϕ→ 0. Due to the problems they generate on the Reggepole approximations of φ2, we shall return to these resultslater.

Analytic extensions A(±)λ−1/2(ω) and K[λ − 1/2, ω] of

A(±)` (ω) and K[`, ω] are obtained by assuming that the

function φinω,λ−1/2(r) and the coefficients A(±)λ−1/2(ω) can

be defined by the problem (15)–(16) where now ` ∈ N isreplaced by λ − 1/2 ∈ C. Such prescription permits us,in particular, to extend in the CAM plane the properties(18a), (18b), (28a) and (28b). In the following, we shalltherefore consider that

φin−ω,λ−1/2(r) = [φinω,λ∗−1/2(r)]∗, (40a)

A(±)λ−1/2(−ω) = [A

(±)λ∗−1/2(ω)]∗, (40b)

and that

K[λ− 1/2,−ω] =[K[λ∗ − 1/2, ω]

]∗, (41a)

K[λ− 1/2,−ω]/A(−)λ−1/2(−ω) =[

K[λ∗ − 1/2, ω]/A(−)λ∗−1/2(ω)

]∗. (41b)

B. Regge poles, Regge modes and associatedexcitation factors

In the next two subsections, contour deformationsin the CAM plane will permit us to collect, by usingCauchy’s residue theorem, the contributions from theRegge poles of the S-matrix or, more precisely, from the

8

Re[β1 (ω)]

Im[β1 (ω)]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

β1(ω)

(2M = 1)

Re[β2 (ω)]

Im[β2 (ω)]

-100

-50

0

50

100

150

β2(ω)

Re[β3 (ω)]

Im[β3 (ω)]

0 1 2 3 4 5 6-15000

-10000

-5000

0

5000

10000

15000

ω

β3(ω)

FIG. 2. Regge trajectories of the Regge-mode excitation fac-tors (2M = 1). We consider the Regge modes correspondingto the first three Regge poles of which the behavior has beendisplayed in Fig. 1. The relation (45) permits us to describethe Regge trajectories for ω < 0 by noting that Re[βn(ω)] andIm[βn(ω)] are, respectively, odd and even functions of ω.

poles, in the complex λ plane and for ω ∈ R, of the ma-trix Sλ−1/2(ω). It should be noted that these poles canbe defined as the zeros λn(ω) with n = 1, 2, 3, . . . and

ω ∈ R of the coefficient A(−)λ−1/2(ω) [see Eq. (19)]. They

therefore satisfy

A(−)λn(ω)−1/2(ω) = 0. (42)

The Regge poles corresponding to electromagnetism inthe Schwarzschild BH have been studied in Refs. [32, 33].It should be recalled that, for ω > 0, the Regge poleslie in the first and third quadrants of the CAM plane,symmetrically distributed with respect to the origin Oof this plane. In this article, due to the use of Fouriertransforms, we must be able to locate the Regge poleseven for ω < 0. In fact, from the symmetry relation

(40b), we have

λn(−ω) = [λn(ω)]∗ (43)

and we can see immediately that, for ω < 0, the Reggepoles lie in the second and fourth quadrants of the CAMplane, symmetrically distributed with respect to the ori-gin O of this plane. Moreover, if we consider the Reggetrajectories λn(ω) with ω ∈] −∞,+∞[, we can observethe migration of the Regge poles. More precisely, as ωdecreases, the Regge poles lying in the first (third) quad-rant of the CAM plane migrate in the fourth (second)one.

It should be noted that the solutions of the problem(15)–(16) with ` replaced by λn(ω) − 1/2 are modesthat are purely outgoing at infinity and purely ingoingat the horizon. They are the “Regge modes” of theSchwarzschild BH [32, 33]. Because of the analogy withthe QNMs, it is natural to define excitation factors forthese modes. In fact, they will appear in the CAM repre-sentations of the Maxwell scalar φ2. By analogy with theexcitation factor associated with the (`, n) QNM of com-plex frequency ω`n [see Eq. (35)], we define the excitationfactor of the Regge mode associated with the Regge poleλn(ω) by

βn(ω) =

1

2ω

A(+)λ−1/2(ω)

ddλA

(−)λ−1/2(ω)

λ=λn(ω)

. (44)

Its expression involves the residue of the ma-trix Sλ−1/2(ω) [or, more precisely, of the function

1/A(−)λ−1/2(ω)] at λ = λn(ω). It should be noted that,

due to Eq. (40b), we have

βn(−ω) = −[βn(ω)]∗. (45)

We have displayed the Regge trajectories of the firstthree Regge poles as well as the Regge trajectories of thecorresponding excitation factors in Figs. 1 and 2. Thesenumerical results have been obtained by using, mutatismutandis, the methods that have permitted us to obtain,in Refs. [25, 27], for the electromagnetic field and forgravitational waves, the complex quasinormal frequenciesof the QNMs and the associated excitation factors (see,e.g., Sec. IV A of Ref. [27]).

It is important to point out that, in Refs. [32, 34], wehave established a connection between the Regge modesand the (weakly damped) QNMs of the SchwarzschildBH. It will play a central role in the interpretation ofour results in Sec. IV, and we recall that, for a givenn, the Regge trajectory λn(ω) with ω ∈ R encodes in-formation on the complex quasinormal frequencies ω`nwith ` = 1, 2, 3, . . . In fact, the index n = 1, 2, 3, . . .not only permits us to distinguish between the differ-ent Regge poles but is also associated with the family ofquasinormal frequencies generated by the Regge modes.

9

C. CAM representation and Regge poleapproximation of the Maxwell scalar φ2 based on the

Poisson summation formula

The first CAM representation of the Maxwell scalar φ2can be derived from Eq. (29) by using the Poisson sum-mation formula [15] as well as Cauchy’s residue theorem.This can be achieved by following, mutatis mutandis, thereasoning of Sec. III C of Ref. [1] which has permitted usto construct a CAM representation of the Weyl scalar Ψ4.In fact, it is even possible to avoid repeating in detail thisreasoning: indeed, we can note that Eq. (24) of Ref. [1]defining Ψ4 and which is the departure of the reasoningof Sec. IIIC of Ref. [1] and Eq. (29) of the present article

are related by the correspondences

rΨ4(t, r, θ = π/2, ϕ)←→√

2 r

iqφ2(t, r, θ = π/2, ϕ),

(46a)

iωK[`, ω]

4A(−)` (ω)

←→ K[`, ω]

2iωA(−)` (ω)

, (46b)

Z`(cosϕ)←→W`(cosϕ). (46c)

As a consequence, Eqs. (48) and (49) of Ref. [1] whichprovide a CAM representation of the Weyl scalar Ψ4 canbe translated to obtain directly a CAM representation ofthe Maxwell scalar φ2. We can write

φ2(t, r, θ = π/2, ϕ) = φB (P)

2 (t, r, θ = π/2, ϕ) + φRP (P)

2 (t, r, θ = π/2, ϕ) (47)

where

√2 r

iqφB (P)

2 (t, r, θ = π/2, ϕ) =1√2π

∫ +∞


∫ ∞0

dλλ

2π

K[λ− 1/2, ω]

2iωA(−)λ−1/2(ω)

Wλ−1/2(cosϕ)

− 1

4π

∫ +i∞

0

dλλeiπλ

cos(πλ)

K[λ− 1/2, ω]

2iωA(−)λ−1/2(ω)

Wλ−1/2(cosϕ)

− 1

4π

∫ −i∞0

dλλe−iπλ

cos(πλ)

K[λ− 1/2, ω]

2iωA(−)λ−1/2(ω)

Wλ−1/2(cosϕ)

(48a)

is a background integral contribution and where√

2 r

iqφRP (P)

2 (t, r, θ = π/2, ϕ) =1√2π

∫ +∞


×

−H(ω)

+∞∑n=1

λn(ω)βn(ω)eiπλn(ω)

cos[πλn(ω)]

K[λn(ω)− 1/2, ω]

2A(+)λn(ω)−1/2(ω)

Wλn(ω)−1/2(cosϕ)

+H(−ω)

+∞∑n=1

λn(ω)βn(ω)e−iπλn(ω)

cos[πλn(ω)]

K[λn(ω)− 1/2, ω]

2A(+)λn(ω)−1/2(ω)

Wλn(ω)−1/2(cosϕ)

(48b)

is the Fourier transform of a sum over Regge poles. Inthese expressions, H denotes the Heaviside step functionand we have introduced the analytic extensions discussedin Sec. III A as well as the Regge poles and the associatedexcitation factors considered in Sec. III B.

We can again check that φ2 is a purely imaginary quan-tity by now considering this new expression. Indeed, dueto the relations (39) and (41b), the first term as well asthe sum of the second and third terms within the squarebrackets on the r.h.s. of Eq. (48a) satisfy the Hermitiansymmetry property. Such a property is also satisfied bythe sum of the two terms within the square bracket onthe r.h.s. of Eq. (48b) as a consequence of the relations(39), (40b), (41a), (43) and (45).

Of course, Eqs. (47) and (48) provide an exact rep-resentation for the Maxwell scalar φ2, equivalent to theinitial expression (25). From this CAM representationof φ2, we can extract the contribution denoted by φRP (P)

2

and given by Eq. (48b) which, as a sum over Regge poles,is only an approximation of φ2. In Sec. IV, we shallcompare it with the exact expression (25) of φ2. How-ever, when considering the term φRP (P)

2 alone, we shallencounter some problems due to the pathological behav-ior of Wλn(ω)−1/2(cosϕ) for ϕ → π (see Sec. III A). In

fact, both the Regge pole approximation φRP (P)

2 and thebackground integral contribution φB (P)

2 are divergent inthe limit ϕ → π but it is worth pointing out that theirsum (47) does not present any pathology.

10

D. CAM representation and Regge poleapproximation of the Maxwell scalar φ2 based on the

Sommerfeld-Watson transform

The second CAM representation of the Maxwell scalarφ2 can be derived from Eq. (32) by using the Sommerfeld-Watson transformation [16–18] as well as Cauchy’sresidue theorem. This can be achieved by following, mu-tatis mutandis, the reasoning of Sec. III D of Ref. [1]which has permitted us to construct a CAM represen-tation of the Weyl scalar Ψ4. Here again, we avoid re-

peating in detail this reasoning: we note that Eq. (26)of Ref. [1] defining Ψ4 and which is the departure of thereasoning of Sec. III D of Ref. [1] and Eq. (32) of thepresent article are related by the correspondences (46a),(46b) and

Z`(− cosϕ)←→W`(− cosϕ). (49)

As a consequence, Eqs. (52) and (53) of Ref. [1] whichprovide a CAM representation of the Weyl scalar Ψ4 per-mit us to obtain directly a CAM representation of theMaxwell scalar φ2. We have

φ2(t, r, θ = π/2, ϕ) = φB (SW)

2 (t, r, θ = π/2, ϕ) + φRP (SW)

2 (t, r, θ = π/2, ϕ) (50)

where√

2 r

iqφB (SW)

2 (t, r, θ = π/2, ϕ) =1√2π

∫ +∞


×

− 1

8π

∫ +i∞

−i∞dλ

λ

cos(πλ)

K[λ− 1/2, ω]

ωA(−)λ−1/2(ω)

Wλ−1/2(− cosϕ)

(51a)

is a background integral contribution and where√

2 r

iqφRP (SW)

2 (t, r, θ = π/2, ϕ) =1√2π

∫ +∞


×

+∞∑n=1

λn(ω)βn(ω)

2i cos[πλn(ω)]

K[λn(ω)− 1/2, ω]

A(+)λn(ω)−1/2(ω)

Wλn(ω)−1/2(− cosϕ)

(51b)

is the Fourier transform of a sum over Regge poles.We can again check that φ2 is a purely imaginary quan-

tity by now considering this last expression. Indeed,due to the relations (39) and (41b), the term within thesquare brackets on the r.h.s. of Eq. (51a) satisfies theHermitian symmetry property. Such a property is alsosatisfied by the term within the square brackets on ther.h.s. of Eq. (51b) as a consequence of the relations (39),(40b), (41a), (43) and (45).

It is important to note that Eq. (50) provides an ex-act expression for the Maxwell scalar φ2, equivalent tothe initial expression (25) and to the expression (48) ob-tained from the Poisson summation formula. From thisCAM representation of φ2, we can extract the contribu-tion denoted by φRP (SW)

2 and given by Eq. (51b) which, asa sum over Regge poles, is only an approximation of φ2.In Sec. IV, we shall compare it with the exact expres-sion (25) of φ2 and with the Regge pole approximationφRP (P)

2 obtained in Sec. III C. However, when consideringthe term φRP (SW)

2 alone, we shall encounter some problemsdue to the pathological behavior of Wλn(ω)−1/2(− cosϕ)for ϕ → 0 (see Sec. III A). In fact, both the Regge poleapproximation φRP (SW)

2 and the background integral con-tribution φB (SW)

2 are divergent in the limit ϕ→ 0 but it is

worth pointing out that their sum (50) does not presentany pathology.

IV. COMPARISON OF THE MAXWELLSCALAR φ2 WITH ITS REGGE POLE

APPROXIMATIONS

In this section, we shall construct numerically the mul-tipolar waveform φ2 given by Eq. (25) by summing overa large number of partial modes. This is particularlynecessary for the radially infalling relativistic or ultra-relativistic particle. We shall also construct the associ-ated quasinormal ringdown φQNM

2 given by Eq. (34). Weshall then compare these two waveforms with the Reggepole approximations φRP (P)

2 and φRP (SW)

2 respectively givenby Eqs. (48b) and (51b) and constructed by consideringone or two Regge poles. This will allow us to highlightthe benefits of working with the Regge pole approxima-tions of φ2.

11

A. Computational methods

To construct numerically the Maxwell scalar φ2 as wellas its quasinormal and Regge pole approximations, weuse, mutatis mutandis, the computational methods de-veloped in Refs. [25, 27] which allowed us to describe theelectromagnetic field and the gravitational waves gener-ated by a particle plunging from the innermost stable cir-cular orbit into a Schwarzschild BH (see, e.g., Sec. IV Aof Ref. [25]).

B. Results and comments

We have compared the multipolar waveform φ2 andthe associated quasinormal ringdown with the Regge poleapproximations φRP (P)

2 in Figs. 3–12 and with the Reggepole approximation φRP (SW)

2 in Figs. 13–17. This has beendone for various values of the angle ϕ ∈ [0, π] exclud-ing the cases ϕ = 0 and ϕ = π for which the Maxwellscalar φ2 vanishes. More precisely, we have consideredthe case of (i) a particle initially at rest at infinity [v∞ = 0(γ = 1)], (ii) a particle projected with a relativistic ve-locity at infinity [we have considered the configurationsv∞ = 0.75 (γ ≈ 1.51) and v∞ = 0.90 (γ ≈ 2.29)], and(iii) a particle projected with an ultra-relativistic veloc-ity at infinity [v∞ = 0.99 (γ ≈ 7.09)]. It should be spec-ified that, in order to obtain numerically stable results,the number of partial modes to include in the sum (25)strongly depends on the initial velocity of the particle:the sum over ` has been truncated at ` = 13 for v∞ = 0,at ` = 15 for v∞ = 0.75 and v∞ = 0.90, and at ` = 19for v∞ = 0.99. It should be noted that the terminologywe used in Ref. [1] to describe the different parts of themultipolar waveform Ψ4 is also adopted for the waveformφ2: we shall thus designate by “pre-ringdown phase” theearly time response of the BH and, as usual, we shall re-fer to the ringdown phase and to the tail of the signal forthose parts of the waveform corresponding respectivelyto intermediate timescales and to very late times.

In Figs. 3–6, we have compared the multipolar wave-form φ2 generated by a particle initially at rest at in-finity with its Regge pole approximation φRP (P)

2 obtainedfrom the Poisson summation formula. In Figs. 3–5, forϕ = π/6, π/3 and π/2, we can observe that the Reggepole approximation constructed from only one Reggepole is in good or very good agreement with the ex-act waveform, and that an additional Regge pole doesnot really improve this approximation. More precisely,it is interesting to note that the Regge pole approxima-tion matches the ringdown, describes correctly the pre-ringdown phase and roughly the waveform tail. It ismoreover important to note that it provides a descrip-tion of the ringdown that does not necessitate determin-ing a starting time, in contrast to the ringdown wave-form constructed from the QNMs which is exponentiallydivergent as t decreases. In Fig. 6, for ϕ = 3π/4, we canobserve that the Regge pole approximation is no longer

so interesting. Indeed, it only roughly describes the BHresponse. Here, it should be recall that the Regge poleapproximation φRP (P)

2 diverges for ϕ → π and, as a con-sequence, for ϕ = 3π/4 (i.e., for a value of ϕ rather closeto π), it would be necessary to consider the backgroundintegral contribution φB (P)

2 given by Eq. (48a) to correctlydescribe the multipolar waveform φ2.

In Figs. 7–12, we have compared, for ϕ = π/6 andπ/3, the multipolar waveform φ2 generated by a particleprojected with a relativistic or an ultra-relativistic veloc-ity at infinity with the Regge pole approximation φRP (P)

2

obtained from the Poisson summation formula. We canobserve that the whole signal is impressively describedby the Regge pole approximation constructed from onlyone Regge pole and that this approximation is even moreefficient in the ultra-relativistic context.

In Fig. 13, for ϕ = 3π/4, we have compared the multi-polar waveform φ2 generated by a particle initially at restat infinity with the Regge pole approximation φRP (SW)

2

obtained from the Sommerfeld-Watson transformation.We recall that, while φRP (P)

2 constructed from the Pois-son summation formula diverges in the limit ϕ→ π, theRegge pole approximation φRP (SW)

2 is regular in the samelimit (it only diverges for ϕ→ 0). As a consequence, thelatter approximation should provide better results thanthe former one for ϕ close to π. By comparing Fig. 13with Fig. 6, we can see that this seems to be the case ifwe focus on the ringdown phase of the waveform but thatthe pre-ringdown phase is not described at all. In fact,here, to correctly describe the waveform φ2 we shouldtake into account the background integral contributionφB (SW)

2 given by Eq. (51a).In Figs. 14–17, for ϕ = 5π/6, we have displayed the

multipolar waveform φ2 generated by a particle initiallyat rest at infinity and by a particle projected with a rela-tivistic or an ultra-relativistic velocity, and we have com-pared it with the Regge pole approximation ΨRP (SW)

4 ob-tained from the transformation of Sommerfeld-Watson.Here again, the Regge pole approximation constructedfrom a single Regge pole does not describe the pre-ringdown phase of the Maxwell scalar φ2, but it matchesa large part of the ringdown phase and approximates thetail rather correctly.

V. ELECTROMAGNETIC ENERGYSPECTRUM dE/dω AND ITS CAM

REPRESENTATION

In this section, we shall focus on the electromagneticenergy spectrum dE/dω observed at infinity which is gen-erated by the charged particle falling radially into theSchwarzschild BH. We shall provide its CAM representa-tion from the Poisson summation formula and Cauchy’stheorem and discuss the interest of this representationand of the corresponding Regge pole approximation.

12

ϕ2

ϕ2RP (P) (Regge pole n = 1)

ϕ2QNM (n = 1)

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

(r/iq)ϕ2(t,r,θ=π/2,φ)

(a) v∞ = 0, φ = π/6 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-10

10-8

10-6

10-4

10-2(b) v∞ = 0, φ = π/6 (2M = 1)

ϕ2

ϕ2RP (P) (Regge poles n = 1 and 2)

-10 0 10 20 30 40 50

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

t-r*(r)


(c) v∞ = 0, φ = π/6 (2M = 1)

ϕ2


0 20 40 60 80 100

10-10

10-7

10-4

t-r*(r)

(d) v∞ = 0, φ = π/6 (2M = 1)

FIG. 3. The Maxwell scalar φ2 and its Regge pole approximation φRP (P)

2 for v∞ = 0 (γ = 1) and ϕ = π/6. (a) The Reggepole approximation constructed from only one Regge pole is in very good agreement with the Maxwell scalar φ2 constructedby summing over the first thirteen partial waves. The associated quasinormal response φQNM

2 obtained by summing over the(`, n) QNMs with n = 1 and ` = 1, . . . , 13 is also displayed. At intermediate timescales, it matches very well the Regge poleapproximation. (b) Semilog graph corresponding to (a) and showing that the Regge pole approximation describes very wellthe ringdown, correctly the pre-ringdown phase and roughly the waveform tail. (c) and (d) Taking into account an additionalRegge pole does not improve the Regge pole approximation.

ϕ2


ϕ2QNM (n = 1)

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006


(a)v∞ = 0, φ = π/3 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-10

10-7

10-4

(b) v∞ = 0, φ = π/3 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.006

-0.004

-0.002

0.000

0.002

0.004

t-r*(r)


(c) v∞ = 0, φ = π/3 (2M = 1)

ϕ2


0 20 40 60 80 100

10-10

10-7

10-4

t-r*(r)

(d) v∞ = 0, φ = π/3 (2M = 1)


2 for v∞ = 0 (γ = 1) and ϕ = π/3. (a) The Reggepole approximation constructed from only one Regge pole is in very good agreement with the Maxwell scalar φ2 constructedby summing over the first thirteen partial waves. The associated quasinormal response φQNM

2 obtained by summing over the(`, n) QNMs with n = 1 and ` = 1, . . . , 13 is also displayed. At intermediate timescales, it matches very well the Regge poleapproximation. (b) Semilog graph corresponding to (a) and showing that the Regge pole approximation describes very wellthe ringdown, correctly the pre-ringdown phase and roughly the waveform tail. (c) and (d) Taking into account an additionalRegge pole does not improve the Regge pole approximation.

13

ϕ2


ϕ2QNM (n = 1)

-0.010

-0.005

0.000

0.005


(a) v∞ = 0, φ = π/2 (2M = 1)

ϕ2


ϕ2QNM (n = 1)10-8

10-6

10-4

10-2(b) v∞ = 0, φ = π/2 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.010

-0.005

0.000

0.005

t-r*(r)


(c)v∞ = 0, φ = π/2 (2M = 1)

ϕ2


0 20 40 60 80 100

10-8

10-6

10-4

10-2

t-r*(r)

(d) v∞ = 0, φ = π/2 (2M = 1)


2 for v∞ = 0 (γ = 1) and ϕ = π/2. (a) The Reggepole approximation constructed from only one Regge pole is in rather good agreement with the Maxwell scalar φ2 constructedby summing over the first thirteen partial waves. The associated quasinormal response φQNM

2 obtained by summing over the(`, n) QNMs with n = 1 and ` = 1, . . . , 13 is also displayed. At intermediate timescales, it matches very well the Regge poleapproximation. (b) Semilog graph corresponding to (a) and showing that the Regge pole approximation describes very well alarge part of the ringdown, correctly the pre-ringdown phase and roughly the waveform tail. (c) and (d) Taking into accountan additional Regge pole does not improve the Regge pole approximation.

ϕ2


ϕ2QNM (n = 1)-0.010

-0.005

0.000

0.005


(a) v∞ = 0, φ = 3π/4 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-10

10-7

10-4

10-1(b) v∞ = 0, φ = 3π/4 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.010

-0.005

0.000

0.005

t-r*(r)


(c) v∞ = 0, φ = 3π/4 (2M = 1)

ϕ2


0 20 40 60 80 100

10-10

10-7

10-4

10-1

t-r*(r)

(d) v∞ = 0, φ = 3π/4 (2M = 1)


2 for v∞ = 0 (γ = 1) and ϕ = 3π/4. (a) The Regge poleapproximation constructed from only one Regge pole does not match correctly the Maxwell scalar φ2 constructed by summingover the first thirteen partial waves. The associated quasinormal response φQNM

2 obtained by summing over the (`, n) QNMswith n = 1 and ` = 1, . . . , 13 is also displayed. The discrepancy with the Regge pole approximation is obvious. (b) Semiloggraph corresponding to (a) and showing that the Regge pole approximation describes correctly a small part of the ringdown.(c) and (d) Taking into account an additional Regge pole does not improve the Regge pole approximation.

14

ϕ2


ϕ2QNM (n = 1)

-0.004

-0.002

0.000

0.002

0.004

0.006


(a) v∞ = 0.75, φ = π/6 (2M = 1)

ϕ2


ϕ2QNM (n = 1)10-8

10-6

10-4

10-2(b)

v∞ = 0.75, φ = π/6 (2M = 1)

ϕ2

ϕ2RP (P) (Regge pole n = 1 and 2)

-10 0 10 20 30 40 50-0.004

-0.002

0.000

0.002

0.004

0.006

t-r*(r)


(c) v∞ = 0.75, φ = π/6 (2M = 1)

ϕ2


0 20 40 60 80 100

10-8

10-6

10-4

10-2

t-r*(r)

(d)v∞ = 0.75, φ = π/6 (2M = 1)


2 for v∞ = 0.75 (γ ≈ 1.51) and ϕ = π/6. (a) The Reggepole approximation constructed from only one Regge pole is in very good agreement with the Maxwell scalar φ2 constructedby summing over the first fifteen partial waves. The associated quasinormal response φQNM

2 obtained by summing over the(`, n) QNMs with n = 1 and ` = 1, . . . , 15 is also displayed. At intermediate timescales, it matches very well the Regge poleapproximation. (b) Semilog graph corresponding to (a) and showing that the Regge pole approximation describes very well thepre-ringdown and ringdown phases and correctly approximates the waveform tail. (c) and (d) Taking into account an additionalRegge pole does not improve the Regge pole approximation.

ϕ2


ϕ2QNM (n = 1)

-0.005

0.000

0.005

0.010


(a) v∞ = 0.90, φ = π/6 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-8

10-6

10-4

10-2 (b) v∞ = 0.90, φ = π/6 (2M = 1)

ϕ2

ϕ2RP (P) (Regge Pole n = 1 and 2)

-10 0 10 20 30 40 50

-0.005

0.000

0.005

0.010

t-r*(r)


(c) v∞ = 0.90, φ = π/6 (2M = 1)

ϕ2


0 20 40 60 80 100

10-8

10-6

10-4

10-2

t-r*(r)

(d) v∞ = 0.90, φ = π/6 (2M = 1)


2 for v∞ = 0.90 (γ ≈ 2.29) and ϕ = π/6. (a) The Reggepole approximation constructed from only one Regge pole is in very good agreement with the Maxwell scalar φ2 constructedby summing over the first fifteen partial waves. The associated quasinormal response φQNM

2 obtained by summing over the(`, n) QNMs with n = 1 and ` = 1, . . . , 15 is also displayed. At intermediate timescales, it matches very well the Regge poleapproximation. (b) Semilog graph corresponding to (a) and showing that the Regge pole approximation describes very wellthe whole signal. (c) and (d) Taking into account an additional Regge pole does not improve the Regge pole approximation.

15

ϕ2


ψ4QNM (n = 1)

-0.02

-0.01

0.00

0.01

0.02

0.03


(a) v∞ = 0.99, φ = π/6 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-9

10-6

10-3

(b) v∞ = 0.99, φ = π/6 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.02

-0.01

0.00

0.01

0.02

0.03

t-r*(r)


(c) v∞ = 0.99, φ = π/6 (2M = 1)

ϕ2


0 20 40 60 80 100

10-9

10-6

10-3

t-r*(r)

(d) v∞ = 0.99, φ = π/6 (2M = 1)


2 for v∞ = 0.99 (γ ≈ 7.09) and ϕ = π/6. (a) The Reggepole approximation constructed from only one Regge pole is in impressive agreement with the Maxwell scalar φ2 constructedby summing over the first nineteen partial waves. The associated quasinormal response φQNM

2 obtained by summing over the(`, n) QNMs with n = 1 and ` = 1, . . . , 19 is also displayed. At intermediate timescales, it matches very well the Regge poleapproximation. (b) Semilog graph corresponding to (a) and showing that the Regge pole approximation describes very wellthe whole signal. (c) and (d) Taking into account an additional Regge pole does not improve the Regge pole approximation.

ϕ2


ϕ2QNM (n = 1)

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006


(a) v∞ = 0.75, φ = π/3 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-8

10-6

10-4

10-2(b) v∞ = 0.75, φ = π/3 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

t-r*(r)


(c) v∞ = 0.75, φ = π/3 (2M = 1)

ϕ2


0 20 40 60 80 100

10-8

10-6

10-4

10-2

t-r*(r)

(d) v∞ = 0.75, φ = π/3 (2M = 1)


2 for v∞ = 0.75 (γ ≈ 1.51) and ϕ = π/3. (a)The Regge pole approximation constructed from only one Regge pole is in very good agreement with the Maxwell scalar φ2

constructed by summing over the first fifteen partial waves. The associated quasinormal response φQNM

2 obtained by summingover the (`, n) QNMs with n = 1 and ` = 1, . . . , 15 is also displayed. At intermediate timescales, it matches very well the Reggepole approximation. (b) Semilog graph corresponding to (a) and showing that the Regge pole approximation describes verywell the ringdown and correctly the pre-ringdown phase and the waveform tail. (c) and (d) Taking into account an additionalRegge pole does not improve the Regge pole approximation.

16

ϕ2

ϕ2RP (P) (Regge Pole n = 1)

ϕ2QNM (n = 1)

-0.005

0.000

0.005


(a) v∞ = 0.90, φ = π/3 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-8

10-5

10-2

(b) v∞ = 0.90, φ = π/3 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.005

0.000

0.005

t-r*(r)


(c) v∞ = 0.90, φ = π/3 (2M = 1)

ϕ2


0 20 40 60 80 100

10-8

10-5

10-2

t-r*(r)

(d) v∞ = 0.90, φ = π/3 (2M = 1)


2 for v∞ = 0.90 (γ ≈ 2.29) and ϕ = π/3. (a)The Regge pole approximation constructed from only one Regge pole is in very good agreement with the Maxwell scalar φ2

constructed by summing over the first fifteen partial waves. The associated quasinormal response φQNM

2 obtained by summingover the (`, n) QNMs with n = 1 and ` = 1, . . . , 15 is also displayed. At intermediate timescales, it matches very well theRegge pole approximation. (b) Semilog graph corresponding to (a) and showing that the Regge pole approximation describesvery well the ringdown, correctly the pre-ringdown phase and roughly approximates the waveform tail. (c) and (d) Taking intoaccount an additional Regge pole does not improve the Regge pole approximation.

ϕ2


ϕ2QNM (n = 1)

-0.03

-0.02

-0.01

0.00

0.01

0.02


(a) v∞ = 0.99, φ = π/3 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-8

10-5

10-2

(b) v∞ = 0.99, φ = π/3 (2M = 1)

ϕ2


-10 0 10 20 30 40 50-0.03

-0.02

-0.01

0.00

0.01

0.02

t-r*(r)


(c) v∞ = 0.99, φ = π/3 (2M = 1)

ϕ2


0 20 40 60 80 100

10-8

10-5

10-2

t-r*(r)

(d) v∞ = 0.99, φ = π/3 (2M = 1)


2 for v∞ = 0.99 (γ ≈ 7.09) and ϕ = π/3. (a) TheRegge pole approximation constructed from only one Regge pole is in very good (and even impressive) agreement with theMaxwell scalar φ2 constructed by summing over the first nineteen partial waves. The associated quasinormal response φQNM

2

obtained by summing over the (`, n) QNMs with n = 1 and ` = 1, . . . , 19 is also displayed. At intermediate timescales, itmatches very well the Regge pole approximation. (b) Semilog graph corresponding to (a) and showing that the Regge poleapproximation describes very well the whole signal. (c) and (d) Taking into account an additional Regge pole does not improvethe Regge pole approximation.

17

ϕ2

ϕ2RP(SW) (Regge pole n = 1)

ϕ2QNM (n = 1)

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020


(a)v∞ = 0, φ = 3π/4 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-10

10-7

10-4

10-1(b) v∞ = 0, φ = 3π/4 (2M = 1)

ϕ2

ϕ2RP(SW) (Regge poles n = 1 and 2)

-10 0 10 20 30 40 50

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

t-r*(r)


(c) v∞ = 0, φ = 3π/4 (2M = 1)

ϕ2


0 20 40 60 80 100

10-10

10-7

10-4

10-1

t-r*(r)

(d) v∞ = 0, φ = 3π/4 (2M = 1)

FIG. 13. The Maxwell scalar φ2 and its Regge pole approximation φRP (SW)

2 for v∞ = 0 (γ = 1) and ϕ = 3π/4. (a) and (b)The pre-ringdown phase of the Maxwell scalar φ2 constructed by summing over the first thirteen partial waves is not describedby the Regge pole approximation constructed from only one Regge pole. However, this approximation matches a large partof the ringdown and roughly approximates the waveform tail. The quasinormal response φQNM

2 obtained by summing over the(`, n) QNMs with n = 1 and ` = 1, . . . , 13 is also displayed. At intermediate timescales, it matches correctly the Regge poleapproximation. (c) and (d) Taking into account an additional Regge pole does not improve the Regge pole approximation.

ϕ2


ϕ2QNM (n = 1)

-0.010

-0.005

0.000

0.005

0.010

0.015


(a) v∞ = 0, φ = 5π/6 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-10

10-7

10-4

10-1(b) v∞ = 0, φ = 5π/6 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

t-r*(r)


(c)v∞ = 0, φ = 5π/6 (2M = 1)

ϕ2


0 20 40 60 80 100

10-10

10-7

10-4

10-1

t-r*(r)

(d) v∞ = 0, φ = 5π/6 (2M = 1)


2 for v∞ = 0 (γ = 1) and ϕ = 5π/6. (a) and (b) Thepre-ringdown phase of the Weyl scalar φ2 constructed by summing over the first thirteen partial waves is not described by theRegge pole approximation constructed from only one Regge pole. By contrast, this approximation matches correctly a largepart of the ringdown and roughly approximates the waveform tail. The quasinormal response φQNM

2 obtained by summing overthe (`, n) QNMs with n = 1 and ` = 1, . . . , 13 is also displayed. At intermediate timescales, it matches correctly the Regge poleapproximation. (c) and (d) Taking into account an additional Regge pole does not improve the Regge pole approximation.

18

ϕ2


ϕ2QNM (n = 1)

-0.02

-0.01

0.00

0.01

0.02


(a) v∞ = 0.75, φ = 5π/6 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-9

10-6

10-3

(b) v∞ = 0.75, φ = 5π/6 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.02

-0.01

0.00

0.01

0.02

t-r*(r)


(c) v∞ = 0.75, φ = 5π/6 (2M = 1)

ϕ2


0 20 40 60 80 100

10-9

10-6

10-3

t-r*(r)

(d) v∞ = 0.75, φ = 5π/6 (2M = 1)


2 for v∞ = 0.75 (γ ≈ 1.51) and ϕ = 5π/6. (a)and (b) The pre-ringdown phase of the Maxwell scalar φ2 constructed by summing over the first fifteen partial waves is notdescribed by the Regge pole approximation constructed from only one Regge pole. By contrast, this approximation matchescorrectly a large part of the ringdown and the waveform tail. The quasinormal response φQNM

2 obtained by summing over the(`, n) QNMs with n = 1 and ` = 1, . . . , 15 is also displayed. At intermediate timescales, it matches very well the Regge poleapproximation. (c) and (d) Taking into account an additional Regge pole does not improve the Regge pole approximation.

ϕ2


ϕ2QNM (n = 1)

-0.04

-0.02

0.00

0.02

0.04


(a) v∞ = 0.90, φ = 5π/6 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-8

10-5

10-2

(b) v∞ = 0.90, φ = 5π/6 (2M = 1)

ϕ2


-10 0 10 20 30 40 50

-0.04

-0.02

0.00

0.02

0.04

t-r*(r)


(c) v∞ = 0.90, φ = 5π/6 (2M = 1)

ϕ2


0 20 40 60 80 100

10-8

10-5

10-2

t-r*(r)

(d) v∞ = 0.90, φ = 5π/6 (2M = 1)


2 for v∞ = 0.75 (γ ≈ 2.29) and ϕ = 5π/6. (a)and (b) The pre-ringdown phase of the Maxwell scalar φ2 constructed by summing over the first fifteen partial waves is notdescribed by the Regge pole approximation constructed from only one Regge pole. By contrast, this approximation matchescorrectly a large part of the ringdown and roughly approximates the waveform tail. The quasinormal response φQNM

2 obtainedby summing over the (`, n) QNMs with n = 1 and ` = 1, . . . , 15 is also displayed. At intermediate timescales, it matches verywell the Regge pole approximation. (c) and (d) Taking into account an additional Regge pole does not improve the Regge poleapproximation.

19

ϕ2


ϕ2QNM (n = 1)

-0.10

-0.05

0.00

0.05


(a) v∞ = 0.99, φ = 5π/6 (2M = 1)

ϕ2


ϕ2QNM (n = 1)

10-8

10-5

10-2

(b) v∞ = 0.99, φ = 5π/6 (2M = 1)

ϕ2


-10 0 10 20 30 40 50-0.10

-0.05

0.00

0.05

t-r*(r)


(c) v∞ = 0.99, φ = 5π/6 (2M = 1)

ϕ2


0 20 40 60 80 100

10-8

10-5

10-2

t-r*(r)

(d) v∞ = 0.99, φ = 5π/6 (2M = 1)


2 for v∞ = 0.75 (γ ≈ 7.09) and ϕ = 5π/6. (a)and (b) The pre-ringdown phase of the Maxwell scalar φ2 constructed by summing over the first nineteen partial waves is notdescribed by the Regge pole approximation constructed from only one Regge pole. This approximation roughly matches theringdown and the waveform tail. The quasinormal response φQNM

2 obtained by summing over the (`, n) QNMs with n = 1 and` = 1, . . . , 19 is also displayed. At intermediate timescales, it matches very well the Regge pole approximation. (c) and (d)Taking into account an additional Regge pole does not improve the Regge pole approximation.

A. Total energy radiated by the particle andassociated electromagnetic energy spectrum

The electromagnetic power P radiated at spatial infin-ity by the charged particle, i.e., the rate dE/dt at whichthe electromagnetic field generated by this particle car-ries energy to infinity, can be obtained as the flux of thePoynting vector R across a spherical surface S(r) withradius r →∞: we have

P =dE

dt= limr→∞

∫S(r)

R · dS (52)

with R = E ∧ B and dS = r2 sin θ dθ dϕ er. By usingEqs. (5), (6) and (20a) as well as the orthonormalizationrelation (8) for the vector spherical harmonics and theaddition theorem for scalar spherical harmonics (21), weobtain

dE

dt(t) =

1

4π

∑`m

2`+ 1

`(`+ 1)

∣∣∣∂tψ`(t, r → +∞)∣∣∣2 (53a)

or, more explicitly, by using Eqs. (20b) and (26a),

dE

dt(t) =

q2

4π

+∞∑`=1

(2`+ 1)`(`+ 1)

×

∣∣∣∣∣ 1√2π

∫ +∞

−∞dω

e−iω[t−r∗(r)]

2iωA(−)` (ω)

K[`, ω]

∣∣∣∣∣2

(53b)

with r → +∞.

The previous result provides, by integration over t, thetotal energy E radiated by the charged particle during itsfall in the BH. We have

E =

∫ +∞

−∞dtdE

dt(t) (54a)

=q2

4π

+∞∑`=1

(2`+ 1)`(`+ 1)

×∫ +∞

−∞dt

∣∣∣∣∣ 1√2π

∫ +∞

−∞dω

e−iωt

2iωA(−)` (ω)

K[`, ω]

∣∣∣∣∣2

(54b)

[note that the dependence in r now disappears due tothe change of variable t → t + r∗(r)]. We can obtain analternative expression for the total energy E by applyingthe Parseval-Plancherel theorem to Eq. (54b). This givesimmediately

E =q2

4π

+∞∑`=1

(2`+ 1)`(`+ 1)

∫ +∞

−∞dω

∣∣∣∣∣ K[`, ω]

2iωA(−)` (ω)

∣∣∣∣∣2

.

(55)

This new form of E permits us to derive the expressionof the (total) electromagnetic energy spectrum dE/dωradiated by the particle. Indeed, from a physical pointof view, it is defined for ω ≥ 0 and satisfy

E =

∫ +∞

0

dωdE

dω(ω). (56)

20

Then, by using Eq. (28b) in Eq. (55), we obtain

E =q2

2π

+∞∑`=1

(2`+ 1)`(`+ 1)

∫ +∞

0

dω

∣∣∣∣∣ K[`, ω]

2iωA(−)` (ω)

∣∣∣∣∣2

(57)

and by comparing Eq. (57) with Eq. (56) we have

dE

dω(ω) =

+∞∑`=1

dE`dω

(ω) (58a)

where

dE`dω

(ω) =q2

8πω2×(2`+1)`(`+1) Γ`(ω)

∣∣∣K[`, ω]∣∣∣2 (58b)

denotes the partial energy spectrum corresponding to the`th mode. It is very important to note that in Eq. (58b),we have chosen to introduce explicitly the greybody fac-tors

Γ`(ω) =1∣∣A(−)

` (ω)∣∣2 (59)

of the Schwarzschild BH corresponding to the electro-magnetic field. It is worth pointing out that we can writeE given by Eqs. (56) and (58) in the form

E =

+∞∑`=1

E` (60a)

where

E` =

∫ +∞

0

dωdE`dω

(ω) (60b)

denotes the partial energy radiated in the `th mode.Finally, it is important to note that Eq. (58) can also

be written as

dE

dω(ω) =

q2

8πω2

+∞∑`=0

(2`+1)`(`+1) Γ`(ω)∣∣∣K[`, ω]

∣∣∣2. (61)

Indeed, here again, as in Sec. II E, it is possible to startat ` = 0 the discrete sum over ` due to the relations(31). In the next subsection, we shall take Eq. (61) as astarting point because it will permit us to use the Poissonsummation formula in its standard form.

B. CAM representation based on the Poissonsummation formula

In order to start the CAM machinery permitting usto derive a CAM representation of the electromagneticenergy spectrum dE/dω, it is necessary to replace inEq. (61) the angular momentum ` ∈ N by the angularmomentum λ = `+ 1/2 ∈ C and therefore to have at ourdisposal the analytic extensions in the complex λ plane

of all the functions of ` appearing in Eq. (61). In fact,in Sec. III A, we have already discussed the construc-

tion of the analytic extensions of A(±)` (ω) and K[`, ω].

It should be however noted that, here, the situation isa little bit more complicated: indeed, we need the ana-

lytic extensions of Γ`(ω) = 1/∣∣A(−)

` (ω)∣∣2 and

∣∣K[`, ω]∣∣2.

Fortunately, in Sec. II of Ref. [35] where the absorp-tion problem for a massless scalar field propagating in aSchwarzschild BH has been considered, the analytic ex-tension Γλ−1/2(ω) of the greybody factor Γ`(ω) has beendiscussed. Here, we shall adopt the same prescription,i.e., we shall assume that

Γλ−1/2(ω) =1

A(−)λ−1/2(ω) [A

(−)λ∗−1/2(ω)]

∗ . (62)

We recall that this particular extension permits us towork with an even function of λ which is purely real. (Formore details concerning the properties of the greybodyfactor Γλ−1/2(ω), we refer to Sec. II of Ref. [35].) Fur-thermore, we shall adopt an analogous prescription for

the analytic extension of∣∣K[`, ω]

∣∣2 by considering that it

is given by K[λ− 1/2, ω] [K[λ∗ − 1/2, ω]]∗.

In order to derive a CAM representation of the elec-tromagnetic energy spectrum dE/dω, the use of CAMtechniques requires in addition the determination of thesingularities of the analytic extensions in the complex λplane of all the functions of ` appearing in Eq. (61). Here,the only singularities to consider are the simple poles ofthe greybody factor Γλ−1/2(ω). In fact, they have beenalso studied in Sec. II of Ref. [35]. Let us just recall that:

(i) The singularities of the function Γλ−1/2(ω) are theRegge poles λn(ω), i.e., the zeros of the function

A(−)λn(ω)−1/2(ω) [see Eq. (42)], as well their complex

conjugates [λn(ω)]∗, i.e., the zeros of the function

[A(−)λ∗−1/2(ω)]

∗. For ω > 0, the Regge poles λn(ω)

lie in the first and in the third quadrant of theCAM plane, symmetrically distributed with respectto the origin O of this plane and, as a consequence,the Regge poles [λn(ω)]

∗lie in the second and in

the fourth quadrant of this plane.

(ii) The residues of the function Γλ−1/2(ω) at the poles

λn(ω) and [λn(ω)]∗

are complex conjugate of eachother and we have in particular

γn(ω) = Res[Γλ − 1/2(ω)]λ=λn(ω)

=1[(

ddλA

(−)λ−1/2(ω)

)[A

(−)λ∗−1/2(ω)]

∗]λ=λn(ω)

.(63)

We have now at our disposal all the ingredients per-mitting us to obtain a CAM representation of the electro-magnetic energy spectrum dE/dω by using the Poissonsummation formula [15] as well as Cauchy’s residue the-orem. In fact, this can be achieved by following, mutatis

21

mutandis, the reasoning of Sec. II of Ref. [35] where aCAM representation of the absorption cross section ofthe Schwarzschild BH has been derived [we invite thereader to compare Eq. (3) of Ref. [35] with Eq. (61) of the

present article]. Taking into account the previous consid-erations concerning the greybody factor Γλ−1/2(ω), itspoles and the associated residues, we obtain

dE

dω(ω) =

dE

dω

B,Re

(ω) +dE

dω

B,Im

(ω) +dE

dω

RP

(ω) (64)

where

dE

dω

B,Re

(ω) =q2

4πω2

∫ +∞

0

dλλ(λ2 − 1/4)Γλ−1/2(ω)∣∣∣K[λ− 1/2, ω]

∣∣∣2 (65a)

is a background integral contribution along the real axis,

dE

dω

B,Im

(ω) = − q2

4πω2

∫ +i∞

0

dλλ(λ2 − 1/4)Γλ−1/2(ω)∣∣∣K[λ− 1/2, ω]

∣∣∣2 eiπλ

cos(λπ)(65b)

is a background integral contribution along the imaginary axis and

dE

dω

RP

(ω) = − q2

2ω2Re

(+∞∑n=1

eiπ[λn(ω)−1/2]λn(ω)(λn(ω)2 − 1/4

)γn(ω)

sin[π(λn(ω)− 1/2)]

× K[λn(ω)− 1/2, ω][K[[λn(ω)]

∗ − 1/2, ω]]∗)

(66)

is a sum over the Regge poles lying in the first quadrantof the CAM plane. Of course, Eqs. (64), (65) and (66)provide an exact CAM representation of the electromag-netic energy spectrum dE/dω, equivalent to the initialpartial wave expansion (58).

C. Computational methods

To construct numerically the electromagnetic energyspectrum (58) radiated by a charged particle falling radi-ally into the Schwarzschild BH and its CAM representa-tion (64)-(66), we have used the computational methodsthat have allowed us to obtain numerically the Maxwellscalar φ2 and its Regge pole approximations in Sec. IV.It should be noted that here, we have in addition eval-uated the background integral along the real axis (65a)by taking λ ∈ [0, 25] and the background integral alongthe imaginary axis (65b) by taking λ ∈ [0, 6i] (due to theterm eiπλ/ cos[λπ] in the expression of its integrand, thisintegral converges rapidly).

D. Numerical results and comments

We now display and discuss a few results concerningthe electromagnetic energy radiated by the charged par-ticle falling radially into a Schwarzschild BH. Here again,as in Sec. IV B, we have focused our attention on (i) a

particle initially at rest at infinity [v∞ = 0 (γ = 1)],(ii) a particle projected with a relativistic velocity at in-finity [we have considered the configurations v∞ = 0.75(γ ≈ 1.51) and v∞ = 0.90 (γ ≈ 2.29)], and (iii) a parti-cle projected with an ultra-relativistic velocity at infinity[v∞ = 0.99 (γ ≈ 7.09)].

In Fig. 18, we have displayed some partial electromag-netic energy spectra dE`/dω corresponding to the lowestmodes. Our results are in perfect agreement with thosealready obtained in the literature (see Refs. [7, 9] but notein these articles, the authors used Gaussian units whilewe consider electromagnetism in the Heaviside system).In Fig. 19, we have displayed the total electromagneticenergy spectrum dE/dω for the configurations consideredin Fig. 18. It should be noted that, in order to obtain nu-merically stable results, the number of modes to includein the sum (58a) strongly depends on the initial velocityof the particle. This clearly appears if we examine theordinate scales used in the semilog graphs of Fig. 18. Infact, we have truncated the sum over ` at ` = 10 forv∞ = 0, at ` = 15 for v∞ = 0.75 and v∞ = 0.90, and at` = 20 for v∞ = 0.99.

In Table I, we have used Eq. (60) to compute the totalenergy E radiated by the charged particle for the valuesv∞ = 0, 0.75, 0.90 and 0.99 of its velocity at infinity. Asexpected, E increases with v∞ (see also Refs. [7, 9]) whilethe rate of convergence of the series over the partial en-ergies E` which defines it decreases. In other terms, i.e.,from a physical point of view, we can observe that for

22

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

1q2dEℓ/dω

Partial energy spectra (2M = 1)v∞ = 0

ℓ = 1

ℓ = 2

ℓ = 310-18

10-14

10-10

10-6

10-2Partial energy spectra (2M = 1)

v∞ = 0ℓ = 1

ℓ = 5

ℓ = 11

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

1q2dEℓ/dω

Partial energy spectra (2M = 1)v∞ = 0.75

ℓ = 1

ℓ = 2

ℓ = 3

10-11

10-8

10-5

10-2

Partial energy spectra (2M = 1)v∞ = 0.75ℓ = 1

ℓ = 5

ℓ = 15

0.000

0.005

0.010

0.015

0.020

0.025

1q2dEℓ/dω


ℓ = 1

ℓ = 2

ℓ = 3ℓ = 4

10-10

10-7

10-4

10-1 Partial energy spectra (2M = 1)v∞ = 0.90ℓ = 1

ℓ = 5

ℓ = 15

0.5 1.0 1.5 2.00.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

ω

1q2dEℓ/dω


ℓ = 1

ℓ = 2

ℓ = 3ℓ = 4

ℓ = 5

1 2 3 4 5 6 7

10-5

10-4

0.001

0.010

ω


ℓ = 1

ℓ = 5

ℓ = 19

FIG. 18. The partial electromagnetic energy spectra radiated by a charged falling radially into a Schwarzschild BH. Theresults are given for v∞ = 0, 0.75, 0.90 and 0.99.

23

v∞ = 0

v∞ = 0.75

v∞ = 0.90

v∞ = 0.99

0.00

0.02

0.04

0.06

0.081q2dE/dω

Total energy spectrum (2M = 1)

v∞ = 0

v∞ = 0.75

v∞ = 0.90

v∞ = 0.99

0 1 2 3 4

10-10

10-7

10-4

10-1

ω

1q2dE/dω

FIG. 19. The total electromagnetic energy spectrum radi-ated by a charged falling radially into a Schwarzschild BH.The results are given for v∞ = 0, 0.75, 0.90 and 0.99.

v∞ = 0, the ` = 1 mode radiates the largest amountof energy (83.20%), and that summing over the first fivemodes, we reach 99.99% of the total electromagnetic en-ergy radiated; on the other hand, for v∞ = 0.99, the` = 1 mode is responsible for only 16.54% of the totalelectromagnetic energy radiated while the sum over thefirst five modes represents only 63.41% of this energy.

TABLE I: The total energy E radiated by the chargedparticle is considered for the values v∞ = 0, 0.75, 0.90 and0.99 of its velocity at infinity. The percentage of energyradiated in the ` = 1 mode and in the first five modes isalso considered. Here, we have taken 2M = 1.

v∞ (γ) (1/q)2E (1/q)2E1 (1/q)2∑5

`=1 E`

0 (1) 3.4049× 10−3 83.20% 99.99%

0.75 (1.51) 1.0181× 10−2 69.20% 99.83%

0.90 (2.29) 2.3251× 10−2 49.45% 97.68%

0.99 (7.09) 1.0726× 10−1 16.54% 63.41%

In Fig. 20, we have compared the electromagnetic en-ergy spectrum dE/dω given by Eq. (58) with its CAMrepresentation (64)-(66). This permits us to emphasizethe respective role of the background integrals (65a) and(65b) and of the Regge pole sum (66). In particular, wecan observe that, for very low frequencies, in order tomatch the exact energy spectrum, it is necessary to takeinto account these two background integrals and to con-

sider the first two Regge poles in the Regge pole sum.Out of this frequency regime, the exact energy spectrumcan be perfectly described by only considering the back-ground integral along the real axis and a single Reggepole in the Regge pole sum. Here, it is worth pointingout that the Regge pole approximation cannot be used toresum the total electromagnetic energy spectrum becausethe CAM representation is dominated by the backgroundintegrals. However, we can observe in Fig. 21 that it isthe Regge pole approximation which explains the oscilla-tions appearing in the electromagnetic energy spectrum.Due to the connection existing between the Regge modesand the (weakly damped) QNMs of the SchwarzschildBH [32, 34], we can also associate these oscillations withthe quasinormal frequencies of the BH.

24

Exact

Sum over Regge poles (n = 1)

+ background integral (real axis)

Sum over Regge poles (n = 1 and 2)

+ background integral

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

1q2dE/dω

Total energy spectrum (2M = 1)v∞ = 0

Exact





10-11

10-7

10-3


v∞ = 0

Exact





0.000

0.005

0.010

0.015

1q2dE/dω


v∞ = 0.75

Exact





10-11

10-7

10-3

Total energy spectrum (2M = 1)v∞ = 0.75

Exact





0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

1q2dE/dω


Exact





10-11

10-9

10-7

10-5

0.001

0.100 Total energy spectrum (2M = 1)v∞ = 0.90

Exact





0 1 2 3 40.00

0.02

0.04

0.06

0.08

ω

1q2dE/dω


Exact





0 1 2 3 4

10-5

10-4

0.001

0.010

0.100

ω


FIG. 20. The electromagnetic energy spectrum radiated by a charged particle falling radially into a Schwarzschild BHcompared with its CAM representation. The respective roles of the background integrals (65a) and (65b) and of the Reggepole sum (66) clearly appear.

25

Exact

Background integral (real axis)


Exact - Background integral (real axis)

10-6

10-5

10-4

0.001

0.010

1q2dE/dω


v∞ = 0

10-5

10-4

0.001

0.010

1q2dE/dω


v∞ = 0.90

0.0 0.5 1.0 1.5 2.0

10-4

0.001

0.010

0.100

ω

1q2dE/dω


FIG. 21. The oscillations in the electromagnetic energyspectrum radiated by a charged particle falling radially intoa Schwarzschild BH explained by the Regge pole approxima-tion.

VI. CONCLUSION

In this paper, we have revisited the problem of theelectromagnetic radiation generated by a charged par-ticle falling radially into a Schwarzschild BH. We haveobtained a series of results which highlight the benefitsof working within the CAM framework and strengthenour opinion concerning the interest of the Regge pole ap-proach for describing radiation from BHs because theyare fairly close to those previously reported in Ref [1]where we discussed an analogous problem in the contextof gravitational radiation.

We have described the electromagnetic radiation bythe Maxwell scalar φ2 and we have extracted from its

multipole expansion (27) the Fourier transform of a sumover the Regge poles of the BH S-matrix involving, in ad-dition, the excitation factors of the Regge modes. It con-stitutes an approximation of φ2 which can be evaluatednumerically from the Regge trajectories associated withthe Regge poles and their residues. In fact, we have con-structed two different Regge pole approximations of φ2:the first one, which has been obtained from the Poissonsummation formula, is given by Eq. (48b) and providesvery good results (even impressive results for relativis-tic particles) for observation directions in a large angu-lar sector around the particle trajectory; the second one,which has been derived by using the Sommerfeld-Watsontransformation, is given by Eq. (51b) and is helpful in alarge angular sector around the direction opposite to theparticle trajectory. More precisely, it should be notedthat, in general, these two Regge pole approximationscan reproduce with very good agreement the quasinor-mal ringdown (it is worth pointing out that, in contrastto the QNM description of the ringdown, the Regge poledescription does not require a starting time) as well aswith rather good agreement the tail of the signal and thatthe first approximation even describes the pre-ringdownphase. All our results have been achieved by taking intoaccount only one Regge pole. To understand the interestof this fact, it is important to recall that the partial waveexpansion defining φ2 is a slowly convergent series, espe-cially in the case of a particle projected with a relativis-tic or an ultra-relativistic velocity into the BH; its Reggepole approximations are efficient resumations which per-mit us, in addition, to extract the physical information itencodes. It is interesting to recall that, for the analogousproblem in the context of gravitational radiation [1], wehave obtained rather similar results for the Weyl scalarΨ4 but that, in this case, taking into account additionalRegge poles sometimes improves the Regge pole approx-imations. It should be finally noted that we have alsoconsidered the electromagnetic energy spectrum dE/dω(a topic we did not touch on in Ref [1]) and, by using thePoisson summation formula, we have constructed from itsmultipole expansion (58) its CAM representation givenby Eqs. (64)–(66). Unfortunately, here the full CAMrepresentation is necessary to describe the whole electro-magnetic energy spectrum but the corresponding Reggepole approximation (66) is however helpful to understandits oscillations and associate them with QNMs.

In future works, we would like to go beyond the rel-atively simple problems examined in the present paperand in Ref. [1] by revisiting, using CAM techniques, theproblem of the radiation generated by a particle withan arbitrary orbital angular momentum plunging into aSchwarzschild or a Kerr BH. It would be in addition inter-esting to extract asymptotic expressions from the back-ground integral contributions appearing in the variousCAM representations in order to improve the physicalinterpretation of the results. We would also like to gobeyond the case of BHs by considering that of neutronstars and white dwarfs. In this context, the recent CAM

26

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