On-Axis scalar absorption cross section of Kerr-Newman black holes: Geodesic analysis, sinc and low-frequency approximations

# On-Axis scalar absorption cross section of Kerr-Newman black holes: Geodesic analysis, sinc and low-frequency approximations

CAROLINA L. BENONE, LUIZ C. S. LEITE AND LUÍS C. B. CRISPINO    SAM R. DOLAN
###### Abstract

We investigate null geodesics impinging parallel to the rotation axis of a Kerr-Newman black hole, and show that the absorption cross section for a massless scalar field in the eikonal limit can be described in terms of the photon orbit parameters. We compare our sinc and low-frequency approximations with numerical results, showing that they are in excellent agreement.

Keywords: Kerr-Newman black holes; null geodesics; scalar absorption.

PACS numbers: 04.50.-h, 04.50.Kd, 04.20.Jb

## 1 Introduction

Geodesics are a major issue of study in general relativity, since they give us the path followed by free particles in a spacetime. They help us to understand the structure of the spacetime, revealing new features. For geometries with rotation, the study of geodesics shows the presence of an ergoregion, i. e. a region where the particle has to rotate in the same sense as the central object. One can also study the formation of black holes by following the horizon generators, which follow null geodesics.

The curvature of the spacetime can act as a lens, making the path of light deviate from a straight line. This effect is related to the first experimental test of general relativity, and to the measurements of the deflection of light by the sun during the eclipse of 29th May 1919 . This gravitational lensing makes it possible to see multiple images from a single source (e.g Einstein cross). For some cases the light of the source can be even lensed in a ring, in which case we have an Einstein ring.

By studying null geodesics one can also determine the shadow of a black hole. This shadow is expected to be seen in the near future with the Event Horizon Telescope . Depending on the form of the shadow, we can differentiate between various kinds of black holes, which can also be used to test the no-hair theorem .

The importance of studying geodesics extrapolates the limit of classical physics, being important also to understand the behavior of quantum fields around black holes. For instance, one can show that the total absorption cross section of quantum fields by black holes presents regular oscillations around the capture cross section, which is given by , where is the critical impact parameter. Sanchez first noticed this feature when she studied the absorption spectrum of a massless scalar field around a Schwarzschild black hole . From this result she proposed a model involving a function, matching the parameters in order to fit the numerical curves.

In Ref. ?, Sanchez’s result was generalized for static black holes with any dimension. The authors used Regge pole techniques to show that the oscillatory pattern of the absorption cross section can be matched through a function, where the parameters involved are related to the light ring. Recently, Macedo and collaborators showed that this result can be extended for the case of absorption by a Kerr black hole . The Regge pole techniques used in Ref. ? were also applied to charged black holes, whose absorption properties have been extensively investigated in the literature .

In a recent paper the absorption cross section for a massless scalar field impinging upon a charged rotating black hole for different incidence angles has been investigated . It has been shown that, for the on-axis case, the total absorption cross section oscillates around the classical limit, but the sinc approximation for this case was still lacking.

Besides the sinc approximation, in the present paper we investigate on-axis null geodesics in a Kerr-Newman black hole to find the high-frequency limit of the scalar absorption cross section. We organize the rest of this paper as follows: In section 2 we consider null geodesics, finding analytical expressions for the on-axis case. In section 3 we use the Regge pole technique to find an expression for the high-frequency limit of the absorption cross section. In section 4 we present the equations for a massless scalar field impinging upon a Kerr-Newman black hole, describing the numerical procedure we used to find the solution, which we compare with the results for the sinc approximation. In section 5 we reobtain the result for the low frequency limit, showing that it goes to the area of the black hole event horizon. In section 6 we present our final remarks.

## 2 Null orbit equations

The Kerr-Newman black hole can be described by the following line element

 ds2 = (1−2Mr−Q2ρ2)dt2−ρ2Δdr2−ρ2dθ2 (1) + 4Marsin2θ−2aQ2sin2θρ2dtdϕ−ξsin2θρ2dϕ2,

where  , , and . This line element corresponds to a black hole with mass , charge and angular momentum , provided that . This black hole has an event horizon located at and a Cauchy horizon at .

For this case, the equations of motion for a light ray are given by

 ρ2 ˙t=(r2+a2)2−aLz(r2+a2)Δ−sin2θ(a2−Lzasin2θ), (2) ρ4 (3) ρ2 ˙ϕ=a(r2+a2)−a2LzΔ−asin2θ−Lzsin2θ, (4) ρ4 ˙θ2=[K+(Lz−a)2]−(asin2θ−Lz)2sin2θ, (5)

where is the azimuthal angular momentum per unit energy and is the Carter’s constant () per unit energy () squared.

We will focus on the case of incidence along the black hole rotation axis, which implies . For this case, the impact parameter is given by

 b=√K+a2. (6)

To find the circular null orbit we must solve and , in order to find the critical impact parameter and the critical radius . For the critical impact parameter, we find

 b2c=2rc(r2c+a2)rc−M. (7)

As for the critical radius, we obtain

 rc=M+2√M2−(a2+2Q2)/3cos[13arccos[M3−(a2+Q2)M(M2−(a2+2Q2)/3)3/2]]. (8)

In Fig. 1 we display a geodesic with critical impact parameter for a black hole with and . The total angular momentum is not conserved for the rotating case, which implies that the geodesic does not stay in the same plane.

The time for a light ray to leave and return to the same pole, that is to undertake a latitudinal angle of , is called latitudinal period and can be found through

 T0=2∫1−1˙t˙zdz, (9)

where . From Eqs. (2) and (5) we find

 ˙t=ρ−2[(r2+a2)2Δ−(1−z2)a2]∣∣∣r=rc, (10)

and

 (11)

which gives us

 T0=4K{KE(−a2K)+[(r2+a2)2Δ−(a2+K)]K(−a2K)}, (12)

where and are the complete elliptic integrals of the first and second kind, respectively.

The Lyapunov coefficient is given by

 Λ=1˙t√12d2Vdr2, (13)

where . We can take an orbital average of this coefficient , given by

 Λ0=2T0∫1−1¯λ˙t˙zdz, (14)

such that we obtain

 Λ0=4√6r2+2a2−b2T0√KK% (−a2K). (15)

## 3 Sinc Approximation

The total massless scalar absorption cross section for a charged rotating black hole is given by

 σ=∞∑l=0l∑m=−l4π2ω2|Sωlm|2Γωlm, (16)

where are the oblate spheroidal harmonics  and are the greybody factors. By using the complex angular momentum technique we are able to find an analytic formula for the absorption cross section in the high-frequency limit in terms of the properties of the photon orbit. We make the transformation in Eq. (16), where is a complex number. Manipulating this equation, following closely Ref. ?, we find

 σ=σgeo−4π2ω2Re[∞∑n=1eiπ(λn−1/2)λnγn|Sωl0|2sin[π(λn−1/2)]], (17)

where , and are the residues for the Regge poles . For the spheroidal harmonics we consider the approximation used in Ref. ?, given by

 |Sωl0|2≈12π2. (18)

A good approximation for the greybody factors in the high-frequency regime is given by

 Γωlm≈11+e−2π(ω−Ω0L)/Λ0, (19)

where is the orbital frequency for the critical orbit and . By finding the poles of Eq. (19), which are given by

 λn=ωΩ0+i(n+12)β+O(1/ω), (20)

we may obtain the corresponding residues, namely

 γn=−β2π, (21)

where

 β≡Λ0/Ω0. (22)

Substituting Eqs. (18)–(21) in Eq. (17), and using also

 ∞∑p=1ei2pπ(z−a)=ieiπ(z−a)2sin[π(z−a)] (23)

for , we find, from the first Regge pole,

 σ≈σgeo[1−8πβe−πβΩ20b2csinc(2πωΩ0)]. (24)

## 4 Numerical results

A massless scalar field obeys the Klein-Gordon equation, given by . To solve this equation we make a separation of variables and write the solution as

 Ψ=+∞∑l=0+l∑m=−lUωlm(r)√r2+a2Sωlm(θ)eimϕ−iωt, (25)

where are the oblate spheroidal harmonics figuring in Eqs. (16)–(18). We need to solve the radial equation, given by

 d2Uωlm(r⋆)dr2⋆ + {(ω−mar2+a2)2+[2Mr−2r2−Δ+3r2r2+a2Δ]Δ(r2+a2)3 (26) − (a2ω2+λlm−2maω)Δ(r2+a2)2}Uωlm(r⋆)=0,

where is the tortoise coordinate, defined as

 r⋆≡∫dr(r2+a2Δ). (27)

In order to solve Eq. (26) we need to impose boundary conditions, which we fix as

 (28)

in which

 UI=e−iωr⋆N∑j=0hjrj, (29)
 UT=e−i(ω−mΩH)r⋆N∑j=0gj(r−r+)j. (30)

These are the appropriate boundary conditions to describe a scattering problem, since we have an incoming and a reflected wave at infinity and a transmitted wave at the horizon. and are constants to be determined by substituting the boundary conditions in the radial equation for the corresponding limits. In order to simplify the equations we choose . Once we have found the radial solution, we manipulate and , and find the reflection coefficient, which we use to compute the total absorption cross section, knowing that .

In Fig. 2 we compare the results for the sinc approximation, given by Eq. (24), with numerical results for different choices of and . The solid lines are the numerical results, while the dashed lines are the corresponding analytical ones. We see that the sinc approximation presents a good fit to the numerical results, even for relatively small values of the frequency ().

## 5 Low-frequency approximation

The low-frequency limit of the absorption cross section for massless scalar fields is a well known result in the literature, having been found for stationary spacetimes with any dimension . In particular, for the Kerr-Newman spacetime it was first obtained by Maldacena and Strominger . Here we obtain analytically this low-frequency approximation in a slightly different way. We will consider three solutions: One for zero frequency (), one for (), and one for (). We will then consider the overlapping between these solutions in order to find the reflection coefficient and, hence, the absorption cross section.

We first consider the case for zero frequency. The eigenvalues of the spheroidal harmonics obey the expansion

 λlm=l(l+1)+∞∑k=1c2k(aω)2k. (31)

Since we are considering the low-frequency limit, we can take . We now solve Eq. (26) making , for which case we obtain

 U0=√r2+a2AP¯ml(y)+√r2+a2BQ¯ml(y), (32)

where and are the associated Legendre functions of the first and second kind, respectively, and are constants and

 ¯m=−iam√M2−(a2+Q2), (33)
 y=(r−M)ϵ, (34)
 ϵ=1√M2−(a2+Q2). (35)

In the limit , . Since, as , we then take , such that our solution is regular at the horizon.

Now we consider the solution for . In this case Eq. (26) reduces to

 d2Uωlm(r⋆)dr2⋆+[ω2−l(l+1)r2]Uωlm(r⋆)=0, (36)

where we neglected terms of and in Eq. (26). The solution of Eq. (36) is given by

 UI=ωr⋆[(−i)l+1Iωlmh∗l(ωr⋆)+il+1Rωlmhl(ωr⋆)]. (37)

The coefficients of Eq. (37) were chosen such that Eq. (37) reduces to Eqs. (28)-(30) with only the term retained. Indeed, in the limit , we have

 hl(ωr⋆)=(−i)l+1eiωr⋆ωr⋆. (38)

Substituting Eq. (38) in Eq. (37), we obtain

 UI=Iωlme−iωr⋆+Rωlmeiωr⋆. (39)

We will now consider the overlapping between Eqs. (32) and (37). In order to do that, we first consider the limit , for which case the associated Legendre functions reduce to

 Pml(x)=(2l)!2liml!(l−m)!xl, (40)

such that Eq. (32) reduces to

 U0=A(2l)!2li¯ml!(l−¯m)!ϵlrl+1, (41)

where .

Considering now the limit of Eq. (37) for , we obtain

 UI = [(−i)l+1Iωlm+il+1Rωlm]2ll!(2l+1)!(ωr⋆)l+1 (42) + i[(−i)l+1Iωlm−il+1Rωlm](2l)!2ll!(ωr⋆)−l,

where we used that , with

 jl(x)=2ll!(2l+1)!xl, (43)
 nl(x)=−(2l)!2ll!x−(l+1). (44)

In order to guarantee that the solution is not divergent in the low-frequency limit, we assume , such that Eq. (42) gives us

 UI=(−i)l+1Iωlm2l+1l!(2l+1)!(ωr⋆)l+1. (45)

Comparing Eq. (41) and Eq. (45), we find that

 A=(−i)l+12l+1+¯m(l!)2(l−¯m)!ωl+1(2l)!(2l+1)!ϵlIωlm, (46)

such that Eq. (32) is given by

 U0=(−i)l+12l+1+¯m(l!)2(l−¯m)!ωl+1(2l)!(2l+1)!ϵlIωlm√r2+a2P¯ml(−Mϵ+ϵr). (47)

Let us now consider the case for . Taking in Eq. (30), we obtain

 UH=Tωlme−i(ω−mΩH)r⋆. (48)

Considering the case for , we find

 UH=Tωlm. (49)

We now consider the limit in Eq. (47). In order to simplify our calculations we consider only the case , for which we have . Substituting this result in Eq. (47), we have

 U0=(−i)l+12l+1(l!)2l!ωl+1(2l)!(2l+1)!ϵlIωl0√r2++a2. (50)

Comparing Eqs. (49) and (50), we find

 Tωl0Iωl0=(−i)l+12l+1(l!)2l!ωl+1(2l)!(2l+1)!ϵl. (51)

Taking , we find

 Tω00Iω00=−i2ω√r2++a2. (52)

Now we have to substitute Eq. (52) in Eq. (16). However, as it can be also inferred from Eq. (31), the difference between the spheroidal harmonics and the spherical harmonics comes from terms which depend on . Thus, in the limit , , with being the scalar spherical harmonics, which can be written as

 Yml(θ)=√(2l+1)4π(l−m)!(l+m)!Pml(cosθ). (53)

Substituting Eq. (52) and Eq. (53) with in Eq. (16), we obtain

 σ=4π(r2++a2), (54)

which is the area of the black hole event horizon.

## 6 Final remarks

We considered null geodesics with incidence along the rotation axis of a charged rotating black hole. We solved the equations for the critical geodesic, finding analytical expressions for the geodesic parameters, such as the Lyapunov coefficient. We used the Regge pole technique to find a closed form for the high-frequency limit of the total scalar absorption cross section for the on-axis case, the so-called sinc approximation. We presented a selection of numerical results, obtaining excellent agreement with the corresponding analytical approximations, even in the case of relatively small frequencies. We also obtained an analytical approximation for the total absorption cross section in the low-frequency limit.

## Acknowledgments

The authors would like to acknowledge Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for partial financial support. S.D. acknowledges financial support from the Engineering and Physical Sciences Research Council (EPSRC) under Grant No. EP/M025802/1 and from the Science and Technology Facilities Council (STFC) under Grant No. ST/L000520/1.

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