Evolution of primordial black hole spin due to Hawking radiation

Alexandre Arbey∗

Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3,IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France

Jeremy Auffinger†

Institut d’Astrophysique de Paris, UMR 7095 CNRS,Sorbonne Universites, 98 bis, boulevard Arago, F-75014, Paris, France

Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3,IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France

Departement de Physique, Ecole Normale Superieure de Lyon, F-69342 Lyon, France

Joseph Silk‡

Institut d’Astrophysique de Paris, UMR 7095 CNRS,Sorbonne Universites, 98 bis, boulevard Arago, F-75014, Paris, France

The Johns Hopkins University, Department of Physics and Astronomy, Baltimore, Maryland 21218, USABeecroft Institute of Particle Astrophysics and Cosmology, University of Oxford, Oxford OX1 3RH, UK

(Dated: May 13, 2020)

Near extremal Kerr black holes are subject to the Thorne limit a < a∗lim = 0.998 in the case ofthin disc accretion, or some generalized version of this in other disc geometries. However any limitthat differs from the thermodynamics limit a∗ < 1 can in principle be evaded in other astrophysicalconfigurations, and in particular if the near extremal black holes are primordial and subject toevaporation by Hawking radiation only. We derive the lower mass limit above which Hawkingradiation is slow enough so that a primordial black hole with a spin initially above some generalizedThorne limit can still be above this limit today. Thus, we point out that the observation of Kerrblack holes with extremely high spin should be a hint of either exotic astrophysical mechanisms orprimordial origin.

I. INTRODUCTION

Primordial Black Holes (PBHs) are appealing candi-dates for solving the long-standing issue of dark matter[1–4]. PBHs could have been created at the end of theinflationary stage of the early Universe, when relativelyhigh density fluctuations ∆ρ/ρ & 1 re-entered the Hubblehorizon. The mass collapsing into a PBH through thismechanism is not subject to the lower Chandrasekharlimit of ∼ 1.4M [5], as this limit is a consequence ofthe stellar origin of Black Holes (BHs). Thus, the detec-tion of a sub-solar BH in the merger events of forthcom-ing gravitational waves detectors such as LISA [6] wouldcertainly point to a primordial origin [7].

Most excitingly, there are powerful observational con-straints, primarily from gravitational microlensing in thesubsolar mass regime, but a substantial window remainsopen for PBHs as dark matter in the mass range thatextends from asteroid mass scales down to the mass setby evaporation limits [8].

In principle, depending on their mechanism of produc-tion at the end of inflation, there is no restriction on theinitial spin of a PBH up to near extremal values a∗ . 1.

∗ Also at Institut Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France; alexandre.arbey@ens-lyon.fr† j.auffinger@ipnl.in2p3.fr‡ joseph.silk@physics.ox.ac.uk

On the other hand, for BHs with astrophysical origin,Thorne has shown that in the case of thin disc accre-tion, there is a limit to the reduced spin a∗lim ≈ 0.998.This limit comes from accretion of the surrounding gason a BH, and its balance with superradiance effects [9].Surprisingly, the same a∗lim ≈ 0.998 is found in [10] forBH mergers. In the case of accretion, this limit has beenrecently generalized to other accretion regimes and discgeometries in [11], based on earlier work [12]; reachingsomewhat higher values depending on the disc parame-ters. The overall state-of-art is that, except for reallyspecific accretion environments, the spin of a BH of as-trophysical origin should not exceed a generalized ver-sion of the Thorne limit [13]. However, the superior limita∗ < 1 holds in any case due to the third law of ther-modynamics [14]. Indeed, a BH with a∗ = 1 would haveT = 0, which is classically forbidden for any statisticalsystem. Moreover, its horizon would have disappeared,thus revealing a naked space-time singularity and violat-ing the Cosmic Censorship Conjecture (a comprehensivediscussion of extremal BHs is given in e.g. [15, 16]). Wewould like to emphasize the fact that, even if the ther-modynamics limit a∗ < 1 remains true, nothing preventsa priori astrophysical BHs from reaching a near extremalspin value a∗ = 0.999999..., yet the mechanisms are stillto be proposed.

Thus, the detection of BHs with a reduced spin higherthan a generalized Thorne limit a∗ > a∗lim could pointeither towards primordial origin [17, 18] or astrophysical

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origin with an exotic accretion history.In this paper, we focus on the mechanisms allowing

a PBH to have today a spin higher than a generalizedThorne limit. For this purpose, we compute the massand spin loss through Hawking radiation of a Kerr PBHand evaluate the minimum initial mass a PBH shouldhave in order to experience a current spin value above ageneralized Thorne limit.

II. HAWKING RADIATION

A. Theoretical aspects

1. Emission rate

Hawking showed that BHs are not as black as was firstsupposed [19]. Throughout, we use a natural system ofunits where G = c = kB = ~ = 1. Hawking used a semi-classical treatment, that is to say the general relativityKerr (or Schwarzschild) metric for space-time

ds2 =

(1− 2Mr

Σ2

)dt2 +

4aMr sin(θ)2

Σ2dtdφ− Σ2

∆dr2

− Σ2dθ2 −(r2 + a2 +

2a2Mr sin(θ)2

Σ2

)sin(θ)2dφ2 ,

(1)

where M is the BH mass, a ≡ J/M is the BH spin pa-rameter (J is the BH angular momentum), Σ ≡ r2 +a2 cos(θ)2 and ∆ ≡ r2 − 2Mr + a2, and a quantummechanics treatment of Standard Model (SM) particlesthrough a wave function ψ satisfying the Dirac equationfor fermions (spin 1/2)

(i/∂ − µ)ψ = 0 , (2)

where /∂ ≡ γν∂ν is the standard Feynman notation, and

the Proca equation for bosons (spin 0, 1 or 2)

(+ µ2)ψ = 0 , (3)

where µ is the particle rest mass. Setting µ = 0 in theseequations of motion also allows to compute the prop-agation of the massless fields (in the following neutri-nos, photons, gravitons). In these equations, we neglectthe couplings between the fields, since they do not affectthe probability of emission of (primary) SM particles viaHawking radiation, but we consider them to obtain theabundance of the final (secondary) particles at infinity,which come from the hadronization or decay of the pri-mary particles. Solving these equations shows that thereis a net emission of particles of type i called the Hawkingradiation (HR). The number of particles emitted per unittime and energy is

d2NidtdE

=1

2π

∑dof.

Γlmi (E,M, a∗)

eE′/T ± 1, (4)

where T is the Kerr BH Hawking temperature

T ≡ 1

2π

(r+ −Mr2+ + a2

), (5)

and r± ≡ M(

1±√

1− (a∗)2)

are the Kerr horizons

radii; a∗ ≡ a/M is the Kerr dimensionless spin parame-ter, it is 0 for a Schwarzschild – non rotating – BH and1 for a Kerr extremal BH; E′ ≡ E −mΩ is the energy ofthe particle that takes into account the horizon rotationwith angular velocity Ω ≡ a∗/(2r+) on top of the totalenergy E ≡ Ekin. + µ; m is the particle angular momen-tum projection m ∈ [−l,+l]. The sum of Eq. (4) is onthe degrees of freedom (dof.) of the particle considered,that is to say the color and helicity multiplicity as wellas the angular momentum l and its projection m. Thequantity Γlmi (E,M, a∗) is called the greybody factor andhas been extensively studied in the literature (see below).It encodes the probability that a particle of type i withangular momentum l and projection m generated at thehorizon of a BH escapes its gravitational well and reachesspace infinity.

2. Evolution of BHs

After computing the greybody factors Γlmi , it is pos-sible to compute the mass and spin loss rates by inte-grating Eq. (4) over all energies and summing over all– massive and massless – SM particles i (6 quarks + 6antiquarks, 3 neutrinos + 3 antineutrinos, 3 charged lep-tons + 3 charged antileptons, 8 gluons, weak W+, W−

and Z0 bosons, the photon and the Higgs boson), plusthe graviton. We define the (positive) f and g factorsfollowing [20, 21]

f(M,a∗) ≡ −M2 dM

dt(6)

= M2

∫ +∞

0

∑i

∑dof.

E

2π

Γlmi (E,M, a∗)

eE′/T ± 1dE ,

g(M,a∗) ≡ −Ma∗

dJ

dt(7)

=M

a∗

∫ +∞

0

∑i

∑dof.

m

2π

Γlmi (E,M, a∗)

eE′/T ± 1dE .

Inverting these equations and using the definition of a∗,we obtain the differential equations governing the massand spin of a Kerr BH

dM

dt= −f(M,a∗)

M2, (8)

and

da∗

dt=a∗(2f(M,a∗)− g(M,a∗))

M3. (9)

3

B. Numerical implementation

We solve Eqs. (8) and (9) numerically, using a newcode entitled BlackHawk [22]1. This code contains tabu-lated values of f(M,a∗) and g(M,a∗) obtained throughEqs. (6) and (7). Within BlackHawk, efforts have beenmade to compute the greybody factors Γlmi (E,M, a∗) nu-merically.

Teukolsky and Press [23, 24] have shown that theDirac and Proca equations (2) and (3), once developedin the Kerr metric (1), can be separated into a radialand an angular part with the wave function written asψ(t, r, θ, φ) = R(r)Slm(θ) e−iEt eimφ. In the following,we make the approximation that the particles are mass-less for the greybody factor computations, since the maineffect of the non-zero particle rest mass is to induce a cutin the emission spectra at energies E < µ [25]. This cutin the energy spectra is applied as a post-process and in-cluded in the computation of the integrals (6) and (7).We checked that the approximated spectra only slightlydiffer from the ones with a full massive computation, andthe small differences are smoothed out by the integrationin Eqs. (6) and (7). The radial equation on R(r) for amassless field of spin s reads

1

∆s

d

dr

(∆s+1 dR

dr

)+

(K2 + 2is(r −M)K

∆− 4isEr − λslm

)R = 0 ,

(10)

where λslm is the eigenvalue of the angular part (we usethe polynomial expansion of Dong et al. [21] to computeλslm) and K ≡ (r2 + a∗2M2)E + a∗Mm. Then, Chan-drasekhar and Detweiler [26–29] have shown that throughsuitable function transformation R→ Z and a change ofvariable from the Boyet-Lindquist radial coordinate to ageneralized tortoise coordinate r → r∗, one can trans-form Eq. (10) into a wave equation with a short-rangepotential

d2Z

dr∗2+(E2 − V (r∗)

)Z = 0 . (11)

For details about this transformation, we refer the readerto Appendix A. We solve this wave equation numericallywith Mathematica, starting from an ingoing plane waveat the horizon

Zhor →r∗→−∞

eiEr∗, (12)

and integrating to space infinity where the solution is

Z∞ →r∗→+∞

AeiEr∗

+Be−iEr∗, (13)

we identify the transmission coefficient (greybody factor)

Γ ≡ |A|2 . (14)

This allows us to perform the integrals (6) and (7).BlackHawk uses an Euler-based adaptive time step

method to compute accurately the last stages of the BHlife, when its mass goes down to the Planck mass MP veryquickly. In practice, the time step is decreased when-ever the relative changes in mass or spin are too large(|∆M |/M or |∆a∗|/a∗ > 0.1), leading to a precision of afew percent. When M ∼MP, we consider that Hawkingevaporation terminates.

III. RESULTS

A. Evolution of Kerr BHs

The main difference between Kerr (a∗ 6= 0) andSchwarzschild (a∗ = 0) BHs is that Kerr BHs are axiallysymmetric and not spherically symmetric. This gives afavored axial direction when computing the Hawking ra-diation. The emission of particles with an angular mo-mentum spinning in the same direction as that of the BHis enhanced when a∗ increases. Moreover, for sufficientlysmall energies and high angular momentum

E < ESR ≡a∗m

2r+, (15)

we enter the regime of superradiance (SR), with en-hanced emission. This asymmetry in the Hawking ra-diation causes a net spin loss by the BH (hence the pos-itivity of the g factor defined in Eq. (7)) through theemission of high angular momentum particles. This en-hanced radiation also causes a mass loss larger than inthe Schwarzschild case. Thus, Kerr BHs have a shorterlifetime than Schwarzschild BHs, and it gets shorter andshorter as the initial spin a∗i gets close to 1.

Fig. 1 shows the lifetime of Kerr BHs as a functionof their mass for different initial spins a∗i . The Hawkingradiation computed with BlackHawk for the evaporationincludes all the SM particles (both massive and massless)as well as one massless graviton. We see that the spinindeed reduces the lifetime but the difference is smallcompared to the enormous time range spanned by theBH lifetimes. We verify that the lifetime is approximatelygiven by

tBH ∼M3 , (16)

which can be derived from Eq. (8) by considering thatf(M,a∗) is a constant. This approximate relation stillholds in presence of angular momentum a∗ 6= 0.

Fig. 2 shows an example of the evolution of the KerrBH mass and spin through time. We see that the reducedspin a∗ has a slightly shorter timescale than the mass M .This is easy to understand when looking at Eqs. (8) and

4

109 1010 1011 1012 1013 1014 1015 1016 1017 1018

Mi (g)

101

105

109

1013

1017

1021

1025t B

H(s

)

age of the Universe

a∗i = 0.0000

a∗i = 0.9000

a∗i = 0.9999

FIG. 1. Kerr BH lifetimes tBH as functions of the BH massMBH for different initial spins a∗i = 0, 0.9, 0.9999 (blue,green and red curves respectively). The age of the universe isindicated as a grey horizontal line.

0.0 0.2 0.4 0.6 0.8 1.0t/tBH

0.0

0.2

0.4

0.6

0.8

1.0

M/M

i

0.0

0.2

0.4

0.6

0.8

1.0

a∗

FIG. 2. Kerr BH mass M (plain curve, normalized over theinitial mass Mi = 1016 g) and reduced spin a∗ (dashed curve,starting from an initial spin a∗i = 0.9) as functions of time t(normalized to the BH lifetime tBH).

(9). The first stage of the evolution is a strong decreaseof both mass and spin, corresponding to the Kerr regimewhen the Hawking radiation is enhanced. When we leavethe high-spin region (a∗ . 0.2), the emission becomessimilar to that of a Schwarzschild BH and the mass evo-lution is more monotonic. At the end of the BH life (thelast 10%), a final stage of very fast evaporation occurs,during which the BH loses the major part of its mass(∼ 50%). This is in agreement with the results of Tayloret al. [17]. When reaching the Planck mass, Hawking’stheory does not tell what happens of the BH.

Fig. 3 shows the mass and spin evolutions for the sameinitial mass Mi = 1016 g but different initial spins a∗i =0, 0.9, 0.9999. We see that the lifetime of a Kerr BH

0.0 0.2 0.4 0.6 0.8 1.0t/tS

0.0

0.2

0.4

0.6

0.8

1.0

M/M

i

a∗i = 0.0000

a∗i = 0.9000

a∗i = 0.9999

0.0

0.2

0.4

0.6

0.8

1.0

a∗

FIG. 3. Comparison of Kerr BH mass M (plain curves, nor-malized to the initial mass Mi = 1016 g which is the same forall curves) and spin a∗ (dashed curves) evolutions as func-tions of time t (normalized to the Schwarzschild BH lifetimetS), for different values of the initial spin a∗i ranging (right toleft) from a∗i = 0 (Schwarzschild case) to a∗i = 0.9999 (nearextremal case).

10−4 10−3 10−2 10−1 100

1− a∗i0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

t BH/tS

Mi = 109 g

Mi = 1013 g

Mi = 1018 g

FIG. 4. Kerr BH lifetimes tBH (normalized to theSchwarzschild case tS) for different initial masses Mi =109, 1013, 1018 g (blue, green and red lines, respectively) asfunctions of the initial spin a∗i . The x-axis has been reversedto show 1− a∗i in a logarithmic scale.

can be reduced by almost ∼ 60% when going from theSchwarzschild case a∗i = 0 to the near extremal case a∗i =0.9999. This is compatible with the results of Dong et al.[21]. The higher the initial spin is, the stronger the initialmass loss will be, so the shorter the BH lifetime. We cansee that after most of the spin is radiated away, all curvesshare the same shape as the Schwarzschild one.

Fig. 4 shows the evolution of the lifetime of a Kerr BHas a function of the initial spin a∗i , for different valuesof the initial mass Mi = 109, 1013, 1018 g. We have re-

5

10−4 10−3 10−2 10−1 100

1− a∗i1015

1016

1017

1018

Mi

(g)

ε = 10−1

ε = 10−2

ε = 10−3

ε = 10−4

ε = 10−5

ε = 10−6

ε = 10−7

FIG. 5. Minimum initial mass Mi needed to have a relativespin loss today ε ≡ ∆a∗/a∗i for different values of ε. Thex-axis has been reversed to show 1−a∗i in a logarithmic scale.

versed the x-axis to focus on the near extremal regiona∗i . 1. We see that the lifetime decreases as the initialspin increases, but this saturates as we come closer to theextremal Kerr case a∗i . 1. The decrease of the lifetimerelative to the Schwarzschild case is not the same for allinitial masses since they have a different Hawking emis-sion history: lighter BHs can emit massive particles atthe beginning of their evaporation (in the Kerr regime)while heavier BHs can only emit them at the end of theirevaporation (in the Schwarzschild regime). The differ-ence in the evolution of the lifetimes remains small.

B. Maximum spin

Using these data on the Kerr BH evolution, which is afunction of both mass and spin, we can estimate the max-imum spin a BH can still have today, starting from someinitial spin, and depending on its initial mass. We knowthat some generalized Thorne limit prevents BHs with adisc from having a spin higher than a∗lim & 0.998, due toaccretion and superradiance effects [9, 11, 12]. We alsoknow that the same limit applies to the outcome of BHmergers due to general relativistic dynamics [10]. Onepossibility of overcoming a∗lim, while remaining below thethermodynamics limit a∗ < 1, may be to form a KerrPBH with an initial spin a∗i > a∗lim and to maintain thisspin over time until today. As mentioned in the Intro-duction, the precise value of the Thorne limit can dependon the disc geometry and parameters (accretion regime,viscosity) [11], thus the numerical results presented inthis Section have to be adapted to somewhat higher gen-eralized Thorne limits.

We have seen that the spin decrease time-scale corre-sponds roughly to that of the mass decrease tBH ∼ M3

i .That means that in order to maintain a spin value really

1015 1016 1017 1018

Mi (g)

10−4

10−3

10−2

10−1

100

1−a∗ to

day

Thorne = 0.998

a∗i = 0.9922

a∗i = 0.9994

a∗i = 0.9999

FIG. 6. Value of the spin today a∗today as a functionof the initial mass Mi for different initial spins a∗i =0.9922, 0.9994, 0.9999 (blue, green and red lines, respec-tively). The Thorne limit a∗lim ≈ 0.998 is shown as an hor-izontal dashed line. The y-axis has been reversed to show1− a∗today on a logarithmic scale.

close to the extremal Kerr case, the BH initial mass mustbe sufficiently high. Fig. 5 shows the minimum initialmass needed as a function of the initial spin, for differentvalues of the desired relative spin change ε ≡ ∆a∗/a∗i .As expected, the more we want to have a spin todayclose to the initial one (ε→ 0) the more massive the BHhas to have been originally. As ε → 1 (all initial spinis lost), the minimum mass, for all initial spins, goes toMlim(a∗i ) ∼ 1015 g the mass of the BHs just evaporatingtoday.

Fig. 6 gives a reversed view of Fig. 5: starting from aninitial spin a∗i = 0.9922, 0.9994, 0.9999 below or abovethe Thorne limit, it shows the value of the spin todaya∗0 as a function of the initial mass. We see that forsufficiently high initial masses Mi & 1016 − 1017 g, thevalue of the spin has barely changed, as could be alreadyguessed from Fig. 5. For initial masses Mi > M lim

i ≈2 − 3 × 1016 g, the spin value today is still higher thanthe Thorne limit for the two cases where it was higher atthe beginning. That means that a Kerr PBH of initialspin a∗i > a∗lim could still have a spin a∗0 > a∗lim todayif it were sufficiently heavy. The same picture could bedrawn for even higher initial spins a∗i = 0.999999... witha decrease of M lim

i when a∗i increases. Indeed, startingfrom a higher spin, a smaller initial mass is necessary toreach the Thorne limit today through Hawking radiation.

C. Accretion and mergers

The above computation of the PBH spin evolution isrelevant only if the mechanisms leading to the establish-ment of the generalized Thorne limit are avoided, thatis to say accretion of material surrounding the PBH and

6

mergers with other PBHs. The accretion part is clearlynot a problem as accretion is dominated by Hawking ra-diation for sufficiently light PBHs during the radiation-dominated era. During the matter-dominated era, PBHsdo not necessarily evolve in a matter-rich environment asthey do not come from the collapse of a star. Thus, thespin loss is only given by the Hawking radiation, as com-puted with BlackHawk. The merging part should not bebothersome if the PBH merging rate is sufficiently small,which should be the case if PBHs do not contribute toomuch to the dark matter fraction (thus preventing theformation of binaries). At least, some of them shouldhave been isolated until today. Thus, the generalizedThorne limit does not apply to sufficiently rare and lightPBHs.

D. Formation

The question on how to generate such high-spin PBHscan be answered by a profusion of models of inflation andearly Universe cosmology. Every model involving PBHcreation should generate high spin PBHs at least as atail in the distribution [30–32]. In these cases, the obser-vation of high spin BHs should remain a rare event notnecessarily incompatible with an unexpected astrophys-ical origin. However, some models predict a dominationof high spin values for PBHs. We refer to one recentexample, that of PBHs formed by scalar field fragmenta-tion during the matter-dominated period that precedesreheating in an inflationary universe [33, 34]. Precisespin measurements could be accomplished with furtherLIGO-Virgo data [18]. We nevertheless point out thata more realistic scenario in which the transient matter-domination is not complete, taking into account the in-creasing proportion of radiation when this phase ends,predicts somewhat less extreme PBHs, and in fewer quan-tities [35]. One last remark concerns the possible destruc-tion of such extreme PBHs by external perturbations. Ithas been shown in Sorce and Wald [36] that falling mat-ter can not overspin a near extremal Kerr-Newman BH,and a fortiori a near extremal Kerr one, thus the hori-zon should persist and the Hawking radiation paradigmshould hold during the PBH evolution.

IV. CONCLUSIONS

In this paper, we have computed the evolution ofBH spin through Hawking radiation using our new codeBlackHawk. We have seen that a way to presently haveBHs with a spin near the Kerr extremal value a∗ . 1and above some generalized Thorne limit a∗lim & 0.998 isto generate it in the primordial Universe through post-inflationary mechanisms with an initial spin a∗i > a∗lim.Then, if its mass is sufficiently high, Hawking radiationis too slow to drive its spin below the generalized Thornelimit. One interesting result is that the initial PBH mass

needed to retain such a high spin until today is well be-low the mass of the Sun. We conclude that near extremalKerr black holes may exist in nature, if primordial blackholes constitute all of the dark matter in the observation-ally allowed window, or at least some of it in the highermass range. Moreover when such near extremal blackholes enter the galactic environment, accretion of order0.001 of their rest mass would render them sub-extremaland induce Hawking evaporation. Such potential blackhole ”bombs” may make primordial black holes directlydetectable via X-ray or gamma ray emission.

Appendix A: From the Teukolsky equation to aSchrodinger-like wave equation

In this Appendix we will briefly present the analyticalmethod used by Chandrasekhar and Detweiler [26–29] totransform the Teukolsky radial equation Eq. (10) intothe Schrodinger-like wave equation Eq. (11). The firstchange consists in moving from the Boyet-Lindquist co-ordinate r to a generalized tortoise coordinate r∗ relatedto r through

dr∗

dr=ρ2

∆, (A1)

where ∆ ≡ r2 − 2Mr + a2, ρ ≡ r2 + α2 and α2 ≡ a2 +am/E. This differential change of variable can be solvedto give

r∗(r) = r +rSr+ + am/E

r+ − r−ln

(r

r+− 1

)− rSr− + am/E

r+ − r−ln

(r

r−− 1

), (A2)

where rS ≡ 2M is the Schwarzschild radius. The in-verse relation has no analytical expression and must becomputed numerically by solving the differential equation(A1). On top of this change, one changes the functionfrom R(r) to Z(r∗) by imposing that the final result isa Schrodinger-like wave equation. Surprisingly, this isalways something one can do for the values of the spins = 0, 1, 2, 1/2 and for both Schwarzschild (a∗ = 0) andKerr (a∗ 6= 0) BHs. The precise transformations aregiven in the papers by Chandrasekhar and Detweiler andare of the form

Z(r∗) = A(r∗)R(r(r∗)) +B(r∗)dR

dr∗, (A3)

dZ

dr∗= C(r∗)R(r(r∗)) +D(r∗)

dR

dr∗. (A4)

Imposing that the equation governing Z is Eq. (11) whileR satisfies Eq. (10) gives a system of equations that thefunctions A, B, C and D must fulfill. Solutions of thissystem give the form of the potential Vs(r(r

∗)). Thesepotentials are, for a field of spin s

V0(r) =∆

ρ4

(λ0 lm +

∆ + 2r(r −M)

ρ23r2∆

ρ4

), (A5)

7

V1/2,±(r) = (λ1/2 lm + 1)∆

ρ4∓√

(λ1/2,l,m + 1)∆

ρ4

×(

(r −M)− 2r∆

ρ2

), (A6)

V1,±(r) =∆

ρ4

((λ1 lm + 2)− α2 ∆

ρ4∓ iαρ2 d

dr

(∆

ρ4

)),

(A7)

V2(r) =∆

ρ8

(q − ρ2

(q − β∆)2

((q − β∆)

(ρ2∆q′′ (A8)

− 2ρ2q − 2r(q′∆− q∆′))

+ ρ2(κρ2 − q′

+ β∆′)(q′∆− q∆′)))

. (A9)

The different potentials for a given spin lead to the sameresults. In the potential for spin 2 particles, the followingquantities appear

q(r) = νρ4 + 3ρ2(r2 − a2)− 3r2∆ , (A10)

q′(r) = r((4ν + 6)ρ2 − 6(r2 − 3Mr + 2a2)

), (A11)

q′′(r) = (4ν + 6)ρ2 + 8νr2 − 6r2 + 36Mr − 12a2 ,(A12)

q′∆− q∆′ =− 2(r −M)νρ4 + 2ρ2(2νr∆− 3M(r2 + a2)

+ 6ra2) + 12r∆(Mr − a2) , (A13)

β± = ±3α2 , (A14)

κ± = ±√

36M2 − 2ν(α2(5ν + 6)− 12a2) + 2βν(ν + 2) ,(A15)

q − β+∆ = ρ2(νρ2 + 6Mr − 6a2) , (A16)

q − β−∆ = νρ4 + 6r2(α2 − a2) + 6Mr(r2 − α2) , (A17)

where ν ≡ λ2 lm + 4.

More details on how we numerically solve theSchrodinger-like wave equation (11) with the potentialsEqs. (A5) to (A9) are presented in the BlackHawk manual[22].

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