On the Vanishing of Love Numbers for Kerr Black Holes
IINR-TH-2021-001
On the Vanishing of Love Numbers forKerr Black Holes
Panagiotis Charalambous a Sergei Dubovsky a Mikhail M. Ivanov a,ba Center for Cosmology and Particle Physics, Department of Physics, New York University,New York, NY 10003, USA b Institute for Nuclear Research of the Russian Academy of Sciences,
Abstract:
It was shown recently that the static tidal response coefficients, calledLove numbers, vanish identically for Kerr black holes in four dimensions. In thiswork, we confirm this result and extend it to the case of spin-0 and spin-1 perturba-tions. We compute the static response of Kerr black holes to scalar, electromagnetic,and gravitational fields at all orders in black hole spin. We use the unambiguousand gauge-invariant definition of Love numbers and their spin-0 and spin-1 analogsas Wilson coefficients of the point particle effective field theory. This definition alsoallows one to clearly distinguish between conservative and dissipative response con-tributions. We demonstrate that the behavior of Kerr black holes responses to spin-0and spin-1 fields is very similar to that of the spin-2 perturbations. In particular,static conservative responses vanish identically for spinning black holes. This impliesthat vanishing Love numbers are a generic property of black holes in four-dimensionalgeneral relativity. We also show that the dissipative part of the response does notvanish even for static perturbations due to frame-dragging. [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] F e b ontents A.1 Spherical Harmonics 44A.2 Gamma Function 47A.3 Gauss Hypergeometric Function 47
B Calculation of Maxwell-Newman-Penrose Scalars 48C Spin-1 Magnetic Love Numbers 49D Comment on the Near-Field Approximation 51
D.1 Scalar Field Example 52D.2 Teukolsky Equation in the Near-Field Approximation 53D.3 Comparison with the Low-Frequency Solution 54– 1 –
Introduction
The static response of black holes to external perturbations, captured by the so-called Love numbers , has recently attracted significant attention both from theobservational and theoretical sides. On the one hand, black holes’ Love numbers aremeasurable quantities that can be probed with gravitational wave observations [2, 3].On the other hand, they play an important role in the effective field theory (EFT) ofbinary inspirals [4–7], where they determine Wilson coefficients that describe leadingfinite-size effects.The tidal gravitational Love numbers of non-rotating Schwarzschild black holeshave been independently computed by Fang and Lovelace [8], Damour and Nagar[9], and by Binnington and Poisson [10]. Remarkably, they vanish in four dimensionsin general relativity (GR), which poses a naturalness problem from the EFT pointof view [6], and therefore might hint on the existence of a new symmetry of blackholes. Intriguingly, there exist several physical examples where the black holes’ Lovenumbers do not vanish. In particular, the calculation of Love numbers has beenextended to higher dimensions in Refs. [11, 12], which have shown that their identicalvanishing for all multipoles is a unique result taking pace only in four dimensions.Recently this result has been generalized to the cases of spin-0, spin-1 and spin-2perturbations of different parities in Ref. [7]: the static responses of Schwarzschildblack holes are generally non-zero for all these different types of perturbations, butaccidentally they vanish in four dimensions. Moreover, black holes’ Love numberswere found to be non-zero in certain modified gravity theories [3, 13, 14].As of now, it has been firmly established that the Love numbers of all perturbingfields vanish in four dimensions for Schwarzshield black holes [7–11, 15]. However, theproperties of spinning (Kerr) black holes [16] are still under debate. Tidal deforma-tions of slowly rotating black holes were studied in Refs. [17–22], which have foundthat the Love numbers vanish for axisymmetric perturbations. Moreover, Landryand Poisson (2015) [20] have claimed that the Love numbers vanish for other typesof perturbations at first order in black hole’s spin. However, this result was recentlyquestioned by Le Tiec and Casals [23], who argued that conclusions of Landry andPoisson (2015) might have been affected by an uncertainty introduced by the split ofthe gravitational potential into the source and response parts. Similar concerns havebeen earlier raised in the context of Schwarzschild black holes [11, 24]. To avoid thatambiguity, Kol and Smolkin [11] have used an analytic continuation of the relevantgeneral relativity solutions into higher dimensions, which is effectively equivalent to The Love numbers are named after the mathematician A. E. H. Love who introduced them todescribe the tidal deformation of the Earth in Ref. [1]. – 2 –romoting the orbital mode number (multipolar index) (cid:96) to non-integer values. Usinga similar analytic continuation technique, Le Tiec et al. (2020) [23, 25] have obtainednon-vanishing static response coefficients and have claimed that the Love numbersdo not vanish for general spin-2 (tidal) perturbations around Kerr black holes.Recently, Chia (2020) [26] and Goldberger et al. (2020) [27] have pointed out thatthe “Love numbers” that Le Tiec et.al. (2020) have computed actually correspondto dissipative effects, whereas the conservative tidal response vanishes identically forspinning black holes. Analogous results have also appeared in Refs. [22, 28]. All theseworks imply that the Love numbers defined in the classical sense of conservative tidaldeformability are zero for Kerr black holes.In this work, we compute analogs of the tidal Love numbers produced by spin-0and spin-1 perturbations around Kerr black holes. We will define Love numbersas Wilson coefficients of local operators in the worldline point-particle effective fieldtheory. This will allow us to distinguish between the conservative response to externalfields, which is related to static Love numbers, and the dissipative part of black hole’sresponse. The finite-size local EFT operators are expected to be present in the EFTon general grounds and they should have a tensorial structure dictated by the axialsymmetry of the Kerr background. We introduce these couplings in the EFT forstatic fields and demonstrate how the new tensorial Wilson coefficients are relatedto the response coefficients that we have extracted from the solutions to linearizedspin-0, spin-1, and spin-2 field perturbations in the Kerr background. We showthat the structure of these GR solutions is such that the dissipative parts of scalarand electromagnetic responses do not vanish for the Kerr black holes just like theirspin-2 counterparts. However, the EFT Wilson coefficients that capture the local(conservative) responses of spinning black holes vanish for all bosonic perturbingfields.On the technical side, we demonstrate that the analytic continuation procedureutilized in Refs. [25, 26] allows one to avoid the uncertainty in the source/responsesplit and obtain consistent gauge-independent results for response coefficients in thespin-0 and spin-1 cases. We also give an interpretation of this analytic continuationprocedure in the EFT context. We show that the subleading source corrections,which may overlap with the induced response contributions, are, in fact, producedby interactions between external fields and gravitational degrees of freedom. Thisobservation allows one to unambiguously identify the black hole multipole momentinduced by external fields. Indeed, the graviton corrections to the source solution canbe computed order by order within the EFT. Thus, given a full GR solution, one cansubtract the graviton interaction contributions from it and hence robustly extractthe Love numbers. This procedure is equivalent to using the analytic continuation (cid:96) → R .Our paper is structured as follows. We start with a recap of the Newtonianresponse coefficients in Section 2. Then we focus on the scalar response coefficients– 3 –n Section 3, where we discuss in detail their calculation both in general relativity andin the point-particle EFT. In Section 4 we compute static response of the Kerr blackhole to the external electric field and match this result with the EFT calculation. Werepeat the same procedure for the spin-2 (gravitational) perturbation in Section 5.Finally, we recap the main general relativity calculations for all spins in Section 6and extend them to the case of non-static perturbations. We discuss our main resultsand draw conclusions in Section 7. Some additional material is presented in severalappendices. Appendix A is a brief reference to key mathematical relations andconventions. In Appendix B we give some details on the calculation of the Newman-Penrose-Maxwell scalars, which encapsulate the electromagnetic field around theKerr black hole and which are required to extract spin-1 response coefficients. Weexplicitly compute the spin-1 magnetic response coefficients in Appendix C — theyhappen to identically coincide with the electric ones. Finally, in Appendix D wecomment on the validity of the response coefficients computed in the potential regionapproximation. Conventions.
In what follows we will work with the metric with signature ( − , + , + , +) ; Greek letters (e.g. µ, ν , etc.) will denote the spacetime indices; Latinletters from the middle of the alphabet (e.g. i, j , etc.) will denote the spatial 3-dimensional indices; Latin letter from the beginning of the alphabet (e.g. a, b , etc.)will run over the coordinates on the two-sphere S . We will work in the c = G = 1 units in most of the paper. In this section we review the definition of tidal response in Newtonian theory [10,25, 29], and discuss some important subtleties present for spinning bodies.
Spherical bodies.
Let us consider a non-rotating spherical body of mass M andequilibrium radius r s . Now imagine that we adiabatically apply an external gravi-tational field U ext . It is convenient to place the body in the origin of the coordinatesystem. Then we can characterize the external source in terms of the multipolemoments, E L ( t ) = − (cid:96) − ∂ (cid:104) i ...∂ i (cid:96) (cid:105) U ext (cid:12)(cid:12) r =0 , (2.1)where r is the distance from the origin, (cid:104) ... (cid:105) denotes the symmetric trace-free part,and L ≡ i ...i (cid:96) is the multi-index. The multipole moments E L are symmetric trace-free tensors (STFs) that parametrically depend on time t . In what follows we suppressthis explicit parametric dependence of the tidal field E L on time. In response to theexternal field, the body will deform and develop internal multipole moments I L , I L ≡ (cid:90) R d x ρ ( x ) x (cid:104) L (cid:105) , (2.2)– 4 –here ρ ( x ) is perturbed body’s density and x L ≡ x i ...x i (cid:96) . Summing up the twocontributions, we find the following expression for the total Newtonian gravitationalpotential U = Mr − (cid:88) (cid:96) =2 (cid:20) ( (cid:96) − (cid:96) ! E L x L − (2 (cid:96) − (cid:96) ! I L n L r (cid:96) +1 (cid:21) , (2.3)where n L ≡ n i ...n i (cid:96) is the tensor product of unit direction vectors n i ≡ x i / | x | , and wehave omitted the dipole moment (cid:96) = 1 since it corresponds to trivial center-of-masstranslations. At this point, it is convenient to switch to the spherical coordinates anduse an expansion of the external source and induced multipole moments in terms ofthe spherical harmonics Y (cid:96)m ( θ, φ ) . This way Eq. (2.3) can be rewritten as follows: U = Mr − (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m (cid:20) ( (cid:96) − (cid:96) ! E (cid:96)m r (cid:96) − (2 (cid:96) − (cid:96) ! I (cid:96)m r (cid:96) +1 (cid:21) , (2.4)where the angular harmonic coefficients E (cid:96)m are related to the STF components via n L E L = (cid:96) (cid:88) m = − (cid:96) E (cid:96)m Y (cid:96)m , E (cid:96)m = E L (cid:73) S d Ω n L Y ∗ (cid:96)m , (2.5)and d Ω = sin θdθdφ denotes the measure on the two-sphere S .If the external gravitational field is adiabatic and weak, linear response the-ory dictates that the response multipoles should be proportional to the perturbingmultipole moments [29–33], I L ( t ) = λ (cid:96) E L ( t ) − ν (cid:96) ddt E L ( t ) + ... ≈ λ (cid:96) E L ( t − τ (cid:48) ) , or I (cid:96)m = λ (cid:96) E (cid:96)m − ν (cid:96) ddt E (cid:96)m + ... , (2.6)where we have restored the explicit time-dependence for clarity, and “ ... ” denotes non-linear corrections and contributions with more time derivatives. Here the coefficient λ (cid:96) is referred to as the Newtonian tidal Love number, while ν (cid:96) is the dissipativeresponse coefficient related to body’s kinematic viscosity [29]. This contributioncaptures the fact that dissipation produces a time lag τ (cid:48) between the external fieldand body’s response [33]. Note that non-zero viscosity triggers various dissipativeeffects, such as heating of the body and the transfer of angular momentum betweenthe body and the source, known as “tidal torque,” see e.g. [34]. Using the frequency-space ansatz E L ∝ e − iωt , Eq. (2.6) can be written as I (cid:96)m = λ (cid:96) E (cid:96)m + iν (cid:96) ω E (cid:96)m + ... . (2.7) Note that the coupling between different orbital and azimuthal modes is absent by virtue oflinearity, i.e. weakness of the external perturbations. This is true for small adiabatic perturbations of the body’s equilibrium configuration. In thecase of weak friction this description holds for an arbitrary equation of state. Beyond the weakfriction approximation it was formally derived for a homogeneous incompressible body [33]. – 5 –f the external tidal environment is static in body’s rest frame, the viscosity contri-bution disappears.
Spinning bodies.
If the test body is rotating, the definition of response coefficientsis more intricate. The rotating body will generally depart from spherical symmetryand hence it will have internal multipole moments even in the absence of an externalperturbation. For a moment, let us assume that the body’s equilibrium configurationcan be approximated as a rigidly rotating sphere. If the rotation is sufficiently slow,the linear response in the body’s rotation frame takes the same form (2.6) as for thenon-rotating body [29]. However, an important effect appears when we switch to aninertial frame. Let us focus on the leading frequency-dependent contribution ∝ ddt E .Because of rotation, the total time derivative in the body’s rotation frame takes thefollowing form ddt E i ...i (cid:96) = ∂∂t E i ...i (cid:96) − (cid:96) (cid:88) n =1 Ω i n j E i ...i n − ji n +1 ...i (cid:96) ≡ ∂∂t E i ...i (cid:96) + κ L (cid:48) L E L (cid:48) , (2.8)where ∂∂t E is the time derivative in a fixed inertial frame. Here we introduced theangular velocity tensor Ω ij = − Ω ji and defined κ i ...i (cid:96) i (cid:48) ...i (cid:48) (cid:96) ≡ − (cid:96) (cid:88) n =1 Ω i n i (cid:48) n δ i i (cid:48) ...δ i n +1 i (cid:48) n +1 δ i n − i (cid:48) n − ...δ i (cid:96) i (cid:48) (cid:96) . Note that the matrix κ L (cid:48) L is odd w.r.t. L ↔ L (cid:48) . We see that in the case of a spinningbody the dissipative response contribution may not vanish even if the external per-turbation is purely static in a non-rotating frame, i.e. ∂ t E L = 0 . Physically, this canbe interpreted as a result of frame-dragging, i.e. the fact that the static sources areviewed as time-dependent by locally rotating observers [35]. In this case, Eq. (2.6)takes the following form I L = ( λ (cid:96) δ LL (cid:48) − ν (cid:96) ( δ LL (cid:48) ∂ t + κ LL (cid:48) )) E L (cid:48) , where κ LL (cid:48) = − κ L (cid:48) L . (2.9)We see that rotating bodies can generate an antisymmetric tensorial response evenfor static external perturbations. Switching to spherical harmonics and using thefrequency-space ansatz E L ∝ e − iωt , we can recast Eq. (2.9) in the form similar toEq. (2.7) I (cid:96)m = ( λ (cid:96) + iν (cid:96) ( ω − m Ω)) E (cid:96)m , (2.10) See also Ref. [36], which shows that a slowly rotating body affected by a stationary externalfield produces a dynamical response. To get this one has to use that Ω ij = Ω ε ikj ˆ z k , ( ε ijk is the Levi-Civita symbol, ˆ z k = δ k ), thencontract κ L (cid:48) L E L (cid:48) with n (cid:104) L (cid:105) , expand n (cid:104) L (cid:105) and E L (cid:48) over the STF tensor basis and use identities (A.16)from Appendix A. – 6 –here Ω is body’s angular velocity and m is the azimuthal (“magnetic”) harmonicnumber. Note the appearance of the term ω − Ω m is generic for rotating bodies [37,38], it is reminiscent of superradiant scattering [39]. Also note that this term is clearlyof non-conservative origin because it is odd under the time reversal transformation ω → − ω , m → − m . If body’s viscosity is not negligible, the dissipative contributioncan survive even if the external tidal environment if static, i.e. ω = 0 . It is wellknown that ν (cid:96) (cid:54) = 0 for both static and spinning black holes, which manifests itselfe.g. in graviton absorption and superradiance [4, 5, 39–41]. Hence, non-vanishing ofthe dissipative response in Eq. (2.10) for spinning black holes is to be expected.The upshot of our discussion is that a rigidly rotating spherically-symmetricbody develops an antisymmetric tensorial response to weak static external perturba-tions. These responses correspond to the imaginary part of harmonic-space responsecoefficients. We will see that a similar picture also holds in a more general casewhen the body’s equilibrium configuration is not spherically-symmetric. In this sit-uation it is convenient to use the following general ansatz for the static response ina non-rotating frame [25], I L = λ LL (cid:48) E L (cid:48) , I (cid:96)m = λ (cid:96)m E (cid:96)m . (2.11)Using the point-particle EFT [27], it can be shown that the part of λ LL (cid:48) whichis even under L ↔ L (cid:48) corresponds to conservative tidal deformations, whereas theantisymmetric part of λ LL (cid:48) captures non-conservative effects such as tidal dissipation.This will be discussed in detail in Section 3. Using the isomorphism between the STFtensors and spherical harmonics, one can also separate conservative and dissipativeresponses at the level of relevant harmonic coefficients I (cid:96)m . In this case dissipativeeffects are encoded in imaginary parts of harmonic-space tidal response coefficients,which map onto the antisymmetric w.r.t. L ↔ L (cid:48) response tensors, whereas thereal part of harmonic coefficients captures the tidal Love numbers and maps ontothe response tensors that are even w.r.t. L ↔ L (cid:48) . This suggests that it is moreappropriate to call quantities λ (cid:96)m defined in Eq. (2.11) “tidal response coefficients”,and reserve the term “Love numbers” only for its conservative real part.All in all, the aggregate potential produced by an external static perturbation(2.4) takes the following form U pert = − ( (cid:96) − (cid:96) ! (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m E (cid:96)m r (cid:96) (cid:34) k (cid:96)m (cid:18) rr s (cid:19) − (cid:96) − (cid:35) , (2.12)where “pert” means that we have subtracted the body’s internal multipole moments, r s is body’s equilibrium radius (in our context this will be black hole’s Schwarzschild Note that our response coefficients differ from those of Refs. [10, 25] by a factor of . For black holes these multipole moments can be straightforwardly extracted from the Kerrmetric, see e.g. [42]. – 7 –adius r s = 2 M ) and k (cid:96)m are dimensionless tidal response coefficients in the New-tonian approximation, defined as λ (cid:96)m = − k (cid:96)m ( (cid:96) − (cid:96) − r (cid:96) +1 s . (2.13) Relativistic picture.
So far our discussion has been entirely in the realm of theNewtonian approximation, which is only a long-distance approximation to the fullgeneral relativity picture. In this more general case, it is convenient to look atthe temporal metric component g = − h in body’s local asymptotic rest-frame [34, 43], which generalizes the Newtonian potential in general relativity. Inthe case of perturbations around the Schwarzschild and Kerr black hole solutions, itcan be written as [10, 20, 25, 44] h pert00 = − ( (cid:96) − (cid:96) ! (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m E (cid:96)m r (cid:96) (cid:34) (cid:16) a r s r + ... (cid:17) + k (cid:96)m (cid:18) rr s (cid:19) − (cid:96) − (cid:16) b r s r + ... (cid:17) (cid:35) , (2.14)where a and b are some calculable spin-dependent coefficients whose exact expres-sions are omitted for clarity. This expression asymptotes Eq. (2.12) for the Newto-nian potential in the long-distance limit and hence, at first glance, provides us witha practical prescription to extract the tidal response coefficients from a given gravi-tational potential produced by an external source in full general relativity. Indeed,apparently, we only need to Taylor expand this potential at spatial infinity and readoff the coefficient in front of the r − (cid:96) − power in the expansion. In what follows wewill denote this procedure by Newtonian matching . It is important to stress thatthe response coefficients that we have discussed so far, and which can be extractedby means of the Newtonian matching, are referred to as the electric type Love num-bers [10]. There also exist so-called magnetic-type response coefficients, which do nothave counterparts in the Newtonian approximation [10, 20]. We will discuss them inSection 5.Newtonian matching for electric response coefficients was justified in Refs. [10,11], which showed that it gives a gauge-independent result for the Schwarzschild back-ground in four dimensions. However, from the expression (2.14) we see that in thephysical case (cid:96) ∈ N / { } there may be some ambiguity if the subleading correctionsto the source appear to have the same power exponent as the response contribution, This choice is only a matter of convention. We could have equally chosen this scale to be, e.g.the difference between the outer and inner horizons of Kerr black holes r + − r − . This would onlylead to a trivial rescaling of response coefficients by a constant factor. The fact that the gravitational potential of the Kerr metric reduces to the flat space expression(2.12) follows from the fact that the Kerr spacetime is asymptotically flat. – 8 –.e. the source and response series overlap. We will see shortly that this ambiguityindeed takes place in the case of the Kerr background when the perturbations arestudied in the advanced Kerr coordinates. A similar ambiguity takes place for Lovenumbers of Schwarzschild black holes in certain higher dimensions. To avoid thisambiguity, Kol and Smolkin [11] have suggested to use the analytic continuation intohigher dimensions, where generically the overlap between the source and responseseries does not happen [11]. Recently Le Tiec et al. (2020) [25] have applied a similaranalytic continuation in four dimensions. To that end, it is enough to treat (cid:96) as ageneric rational number (cid:96) → R . In our work, we show that this approach indeedallows one to avoid the ambiguity and moreover, is motivated from the EFT point ofview. We will also show that the EFT itself provides us with an unambiguous wayto define the conservative part of the tidal response, i.e. the actual Love numbers.Finally, a comment on the role of the no-hair theorems [45–48] is in order. Inour context these theorems essentially state that external perturbing fields cannotsmoothly generate a non-vanishing static profile on top of the Kerr metric. Thismeans that when the external source is adiabatically turned on, the generalizedgravitational potential in Eq. (2.14) should be uniquely defined by the source itselfand body’s response. In other words, the no-hair theorems guarantee that black holes’response is analytic in the vicinity of ω = 0 , there is no gravitational hysteresis (apartfrom a possible change of black hole’s mass and spin) — once the source is turnedoff, the relevant solution will tend to the Kerr metric again. Thus, it is only becauseof the no-hair theorems that the decomposition (2.14) is unique and the definitionof the response coefficients for black holes is meaningful. In this section we will compute the response of a Kerr black hole to an externalspin-0 perturbation. In analogy with the tidal response, it will be referred to asa scalar response coefficient (SRC). First, we will discuss the definition of SRCs inthe Newtonian approximation and in the point-particle EFT. Then we will computethe SRCs by solving the scalar field equation of motion (i.e. the spin-0 Teukolskyequation [49–52]) in the Kerr background. We will do so in two different coordinatesystems and show that the results agree only if we analytically continue the orbitalnumber of relevant scalar perturbations to non-integer values. We also justify thisprocedure from the EFT point of view. Finally, we present an explicit matching be-tween the EFT and general relativity calculations. We will show that the dissipativeresponse is captured by complex tidal response coefficients, whereas their real partdescribes conservative deformations that are natural to associate with classic Lovenumbers. The scalar response coefficients happen to be purely imaginary, and hencewe can conclude that the scalar Love numbers are identically zero for Kerr blackholes. – 9 – .1 Definition from Newtonian Matching
The response coefficients for a free scalar field can be defined in analogy with theNewtonian gravitation potential outside a generic static body tidally deformed by aweak external gravitational field. Indeed, the fully relativistic Klein-Gordon equationfor a static massless scalar field Φ reduces to the Poisson equation on very large scales.Therefore, in the asymptotic limit r → ∞ the static scalar field takes the standardexpression of the Newtonian potential, which can be split into contributions from anexternal source and body’s response similarly to Eq. (2.12), Φ (cid:12)(cid:12)(cid:12) r →∞ = (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m E (0) (cid:96)m r (cid:96) (cid:34) k (0) (cid:96)m (cid:18) rr s (cid:19) − (cid:96) − (cid:35) . (3.1)Note that unlike the gravitation potential, whose multipole expansion can only startwith (cid:96) = 2 by virtue of the equivalence principle, a generic scalar field is allowed tohave a non-trivial dipole moment. For this reason we have omitted the normalizationfactors used in Eq. (2.12).It is known that the full solution to the static Klein-Gordon equation in theSchwarzschild and Kerr backgrounds factorizes in the spherical coordinates, andhence it can be written in the following form [11, 53, 54] Φ = (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m E (0) (cid:96)m r (cid:96) (cid:34)(cid:16) a r s r + ... (cid:17) + k (0) (cid:96)m (cid:18) rr s (cid:19) − (cid:96) − (cid:16) b r s r + ... (cid:17)(cid:35) , (3.2)where a , b are some calculable constants, which depend on black hole’s spin. Atface value, this gives us a tool to extract the SRCs from the full GR solution: wejust have to Taylor expand this solution in r s /r and read off the coefficient in frontof the r − (cid:96) − term. However, at this point it is not clear if the Newtonian definitionof SLNs is unambiguous. This disadvantage can be avoided if the Love numbers aredefined within the point-particle effective field theory. In what follows we will use the point particle effective field theory that includes ourtest scalar field Φ , along with the long-wavelenght metric field g Lµν , and the positionof the compact object x µ (see [4, 5, 7]) for further details). Our goal here is tofocus on three particular aspects: the EFT definition of Love numbers, a possibleambiguity between the source and response contributions that may appear duringthe matching of the EFT and microscopic (GR) calculations, and the EFT extensionin the presence of spin.The core idea of the point particle EFT is that any object acts like a pointparticle when viewed from large enough scales. As we get closer to that object, or– 10 –s measurement precision becomes better, corrections to the point particle descrip-tion become important. These finite size effects are captured by higher derivativeoperators in the context of EFT.We will start with the non-spinning case, which is sufficient for our purposeto define the Love numbers in a way free of the arbitrariness produced by the re-sponse/source split. We will re-introduce the Planck mass M P in order to keep trackof mass dimensions of the EFT operators. Let us write the following action for astatic scalar field coupled to gravity S = S (2)Φ + S (2) h + S Φ h + S pp + S finite − size , (3.3)where S (2)Φ and S (2) h are the bulk quadratic terms for the scalar field and gravitation, S Φ h describes the leading interaction between them, S pp is the point particle worldlineaction, and the part S finite − size captures finite-size effects. Let us describe each termseparately. Bulk action.
The kinetic term for the bulk scalar field in flat space is given by S (2)Φ = − (cid:90) d x ∂ µ Φ ∂ µ Φ = − (cid:90) d x ( ∂ i Φ) , (3.4)where in the last equality we took the static limit. Now let us focus on the gravi-tational sector. It is described by expanding the Einstein-Hilbert action in gravitonperturbations around the flat background (see e.g. [55]), g µν = η µν + 2 h µν . For the purposes of this section it will be enough to work in the gauge that reproducesthe Schwarzschild solution from GR. To that end, we consider the following isotropicperturbations g = − (1 + 2 H ( t, r )) g rr = (1 + 2 H ( t, r )) , (3.5)with all other components given by the unperturbed Minkowski metric. The kineticterm can be extracted directly from the Einstein-Hilbert action S (2) h = M P (cid:90) d x √− gR = M P (cid:90) dtdφd cos θdr (cid:0) H − r ( ∂ r H ) H (cid:1) . (3.6)The leading interaction term between the gravitons and the scalar field stems fromthe scalar field kinetic term, S Φ h = 12 (cid:90) dtdφd cos θdr r ( H − H ) ( ∂ r Φ) ⊂ − (cid:90) d x √− gg µν ∂ µ Φ ∂ ν Φ . (3.7)– 11 – oint particle action. Finally, we include the worldline action for the black hole.It starts with the point particle part S pp ≡ − M (cid:90) ds = − M (cid:90) dτ ( g µν ˙ x µ ˙ x ν ) / = − M (cid:90) d x (cid:90) dτ (1 + H ) δ (4) ( x − x ( τ )) , (3.8)where M is the black hole mass, ds is the infinitesimal proper worldline interval, and τ is the worldline parameter (proper time); the overdot denotes d/dτ . Finite-size effects.
As far as the finite size effects are concerned, it is instructiveto recall some details of linear response theory [4, 5, 27, 40, 41]. The worldline actiondescribing the coupling of a source multipole I L and the tidal field E L ( x ) (which canbe either the gravitational tidal field or its scalar field analog E L ∝ ∂ (cid:104) L (cid:105) Φ ) is givenby [5] S I E = 12 (cid:90) dτ (cid:90) d xδ (4) ( x − x ( τ )) I L ( τ ) E L ( x ) . (3.9)The tidal field E L acts like a source for I L . Hence, in the linear approximation wecan write (cid:104) I L ( τ ) (cid:105) = (cid:90) dτ (cid:48) G ret .L (cid:48) L ( τ, τ (cid:48) ) E L (cid:48) ( x ( τ (cid:48) )) , (3.10)where (cid:104) ... (cid:105) denotes ensemble-averaging w.r.t. internal degrees of freedom and short-scale modes, and we have introduced the retarded Green’s function as follows: G ret L (cid:48) L ( τ, τ (cid:48) ) = − i (cid:104) [ I L ( τ ) , I L (cid:48) ( τ (cid:48) )] (cid:105) θ ( τ − τ (cid:48) ) , (3.11)where θ ( x ) is the Heaviside theta-function. Now we switch to frequency space anduse that causal Green’s function are analytic around ω = 0 . Then, the sphericalsymmetry of the problem dictates the following general expression for the causalGreen’s function G ret L (cid:48) L ( ω ) = (cid:88) n =0 ω n (cid:16) ˆ λ loc. n L (cid:48) L + i ˆ λ non-loc. n +1 L (cid:48) L ω (cid:17) , (3.12)where the tensors ˆ λ loc./non-loc. p L (cid:48) L must be symmetric under exchange L ↔ L (cid:48) , i.e. ˆ λ loc./non-loc. p L (cid:48) L = const · δ (cid:104) L (cid:105)(cid:104) L (cid:48) (cid:105) . (3.13)The terms in the expansion (3.12) that are symmetric under time-reversal symmetry(i.e., are even under ω → − ω ) are dubbed local (“loc.”). We will see shortly that they For clarity, we have omitted the background multipole moments. These are absent forSchwarschild black holes, but are present for Kerr black holes. These moments can be easilytaken into account (see e.g. [41]), but they do not contribute to the tidal response and hence areirrelevant for our discussion. – 12 –orrespond to local terms in the effective action for small-wavelength fluctuations.These terms are manifestly time reversal invariant, and hence they correspond toconservative dynamics. However, the terms dubbed non-local (“non-loc.”) are nottime reversal invariant and hence they describe dissipative effects. Importantly, in thecase of the Schwarzschild metric they disappear in the limit of static perturbations ω → .The physical response also receives contributions from local operators in theworldline action. Effectively, this leads to a renormalization of the conservativeresponse coefficients. All in all, the total conservative response can be described bya set of the following local wordline operators involving only the long-wavelenghtdegrees of freedom, S eff I E = 12 (cid:96) ! (cid:90) dτ d xδ (4) ( x − x ( τ )) (cid:16) λ (cid:96) E L E L + λ (cid:96) ( ω ) ˙ E L ˙ E L + ... (cid:17) . (3.14)This action can be rewritten in the covariant form by using covariant derivativesalong body’s 4-velocity v µ = dx µ dτ and the projector onto directions orthogonal to v µ P µν = δ µν + v µ v ν , D ≡ v µ ∇ µ . (3.15)As a result, the leading worldline interaction term describing finite-size effects in thestatic limit is given by [5, 7] S Love = λ (cid:96) (cid:96) ! (cid:90) d x (cid:90) dτ δ (4) ( x − x ( τ )) (cid:104) P ν (cid:104) µ ...P ν (cid:96) µ (cid:96) (cid:105) ∂ ν ...ν (cid:96) Φ (cid:105) (cid:2) P (cid:104) µ σ ...P µ (cid:96) (cid:105) σ (cid:96) ∂ σ ...σ (cid:96) Φ (cid:3) , (3.16)where we have introduced the multi-derivative operator ∂ ν ...ν (cid:96) ≡ (cid:81) (cid:96)i =1 ∂ ν i , whereas (cid:104) ... (cid:105) stands for the symmetrized traceless component. The leading non-static effectsare captured by the following action S ω = λ (cid:96) ( ω ) (cid:96) ! (cid:90) dτ D (cid:104) P ν (cid:104) µ ...P ν (cid:96) µ (cid:96) (cid:105) ∂ ν ...ν (cid:96) Φ (cid:105) D (cid:2) P (cid:104) µ σ ...P µ (cid:96) (cid:105) σ (cid:96) ∂ σ ...σ (cid:96) Φ (cid:3) (cid:12)(cid:12)(cid:12) x = x ( τ ) . (3.17)The coupling λ (cid:96) will be referred to as the EFT Love number in what follows.Eq. (3.16) can be viewed as a gauge-independent definition of the Love numbers,as the corresponding worldline operator is manifestly covariant. We will see momen-tarily that it is precisely this operator that generates the r − (cid:96) − term in the Newtonianexpansion. Importantly, we will see that the coupling λ (cid:96) does not exhibit logarith-mic running in general relativity in four dimensions. This will guarantee that resultof the Newtonian matching is meaningful. The situation is different for frequency-dependent Love numbers, which generically depend on distance, and hence introducesome ambiguity in the direct application of the Newtonian matching. We will discussthis in Section 6. – 13 –n what follows we will work in the body’s rest frame where v µ = δ µ , hence P µ = 0 , P ij = δ ij , which removes all operators with time derivatives in Eq. (3.16).This gives us the following action relevant for the study of local static response S Love = λ (cid:96) (cid:96) ! (cid:90) d x (cid:90) dτ δ (4) ( x − x ( τ )) ∂ (cid:104) i ...i (cid:96) (cid:105) Φ ∂ (cid:104) i ...i (cid:96) (cid:105) Φ . (3.18) Static response in the EFT.
Our goal here is to compute the static scalar fieldprofile Φ in the presence of interactions with gravitons and an external source ¯Φ . Asa first step, we need to compute the leading order graviton field, which describes thegravitational potential of the black hole on large scales. In the EFT context, theblack hole solution is recovered perturbatively order by order in the long-distanceexpansion controlled by M/ ( M P r ) . Let us start with the first order. The equationsof motion for the graviton modes are given by H = r∂ r H , ∆ H = M M P (cid:90) dτ δ (4) ( x − x ( τ )) . (3.19)Let us work in the black hole’s rest frame, where the unperturbed center-of-massposition is given by x i ( τ ) = 0 , x = τ , such that ∆ H = M M P δ (3) ( x ) ⇒ H = − M πM P r = − r s r , (3.20)where we have introduced the Schwarzschild radius r s = M/ (4 πM P ) . The secondmetric perturbation is given by H = − H = r s r . (3.21)Now we have to compute the scalar field profile. The total static equation of motionfor the scalar field, which includes the leading interaction with gravity and the tidalresponse is given by ∆Φ − r ∂ r ( r H ∂ r )Φ + ( − (cid:96) λ (cid:96) (cid:96) ! ∂ (cid:104) i ...i (cid:96) (cid:105) (cid:0) ∂ (cid:104) i ...i (cid:96) (cid:105) Φ δ (3) ( x ) (cid:1) = 0 . (3.22)In order to compute the black hole response, we introduce an external scalar fieldsource ¯Φ , satisfying the free Poisson equation ∆ ¯Φ = 0 in the r → ∞ asymptotic.Assuming that ¯Φ has an orbital number (cid:96) , we find ∆ ¯Φ = 0 ⇒ ¯Φ = E (0) i ...i (cid:96) x i ...x i (cid:96) , (3.23)where E (0) i ...i (cid:96) is a symmetric trace-free tensor. Note that the solution (3.23) corre-sponds to ¯Φ = (cid:80) (cid:96)m E (0) (cid:96)m r (cid:96) Y (cid:96)m ( θ, φ ) in the spherical coordinates. We want to solveEq. (3.22) perturbatively expanding in r s /r , but keep the explicit λ (cid:96) -dependence, Φ = ¯Φ + Φ (1) h + Φ (1)Love + ... . (3.24)– 14 –e will formally retain corrections linear in λ (cid:96) , but λ (cid:96) itself does not need to besmall. Let us first compute the correction to the source coming from the interactionwith the graviton. We have ∆Φ (1) h = 1 r ∂ r ( r H ∂ r ) ¯Φ . (3.25)Expanding Φ (1) h over spherical harmonics and using the solution from Eq. (3.21) weobtain (cid:18) ∂ r + 2 r ∂ r − (cid:96) ( (cid:96) + 1) r (cid:19) Φ (1) h (cid:96)m = r s r ∂ r ( r∂ r ¯Φ) . (3.26)Note that in the above expression we can lift all restrictions on (cid:96) and treat it as ageneric number. Plugging our source from Eq. (3.23), this equation can be easilysolved. The full solution including the source plus the leading graviton correction isgiven by Φ = (cid:96) (cid:88) m = − (cid:96) E (0) (cid:96)m Y (cid:96)m r (cid:96) (cid:18) − (cid:96) r s r (cid:19) = E (0) i ...i (cid:96) x i ...x i (cid:96) (cid:18) − (cid:96) r s r (cid:19) . (3.27)We observe that the interaction with the graviton induces the sub-leading correctionsto the source. Note that this correction is calculable, i.e. its strength is fixed inthe EFT itself. Importantly, because the graviton propagator scales like /r , thesecorrections are naturally organized as the following power series Φ ⊃ r (cid:96) (cid:18) c r s r + c (cid:16) r s r (cid:17) + ... (cid:19) , (3.28)for any (cid:96) ∈ R . This gives an interpretation of the subleading source corrections:these are just generated by the coupling between the source and perturbative gravity.Our result also justifies the use of the analytic continuation (cid:96) → R for the source-response split, because this indeed allows one to isolate the series Eq. (3.28) andavoid a possible overlap with corrections induced by finite-size effects.Now let us compute the correction to the source coming from the Love interac-tion, which corresponds to induced multipoles. We have ∆Φ (1)Love = − ( − (cid:96) λ (cid:96) E (0) i ...i (cid:96) ∂ i ...i (cid:96) δ (3) ( x ) . (3.29)This equation can be easily solved in Fourier space, Φ (1)Love = ( − i ) (cid:96) λ (cid:96) E (0) i ...i (cid:96) (cid:90) d k (2 π ) e i k · x k i ...k i (cid:96) k = B (cid:96) λ (cid:96) E (0) i ...i (cid:96) x i ...x i (cid:96) r (cid:96) +1 , (3.30)where we have introduced the following normalization constant B (cid:96) ≡ ( − (cid:96) (cid:96) − π / Γ(1 / − (cid:96) ) . (3.31)– 15 –he total solution including linear order correction in λ (cid:96) and r s /r is given by Φ = E (0) i ...i (cid:96) x i ...x i (cid:96) (cid:124)(cid:123)(cid:122)(cid:125) source − (cid:96) r s r (cid:124)(cid:123)(cid:122)(cid:125) graviton interaction + B (cid:96) λ (cid:96) r (cid:96) +1 (cid:124) (cid:123)(cid:122) (cid:125) induced multipole . (3.32)Comparing this to (3.2), we can see that the EFT provides a tool to define the re-sponse coefficients and avoid ambiguity in the source/response split. The subleadingcorrections in the source expansion, which scale as ( r s /r ) n × r (cid:96) , naturally correspondto diagrams produced by the interaction between the source field and the graviton.These diagrams are fixed by the structure of Einstein-Hilbert and the Klein-Gordonactions. Thus, all graviton corrections can be unambiguously computed by expand-ing the Einstein-Hilbert equation in higher order operators involving the gravitonfield. Hence, in principle, during the matching procedure, one can identify all cor-rections coming from the graviton vertices and subtract them from the microscopicsolution obtained in GR. The remaining piece will correspond to the response coef-ficients. In practice, however, the number of diagrams to be computed can be verylarge. From the practical point of view, it is more convenient to do the analyticcontinuation (cid:96) → R , which allows one to easily achieve the same goal of isolatingnon-linear corrections to the source from those generated by the finite-size effects. Inclusion of spin.
In order to reproduce the GR solution in the EFT, one hasto perturbatively recover the Kerr metric at order a /r , where a ≡ J/M is thenormalized black hole’s spin. To that end one has to introduce vector degrees offreedom of metric perturbations and consider their coupling to black hole’s spinvia the Mathisson - Papapetrou/Routhian formalism [5]. This procedure has beenrecently presented in Ref. [27], see also Refs. [56, 57]. In order to obtain the a /r corrections, we need to take into account the cubic interaction between the scalarand vector graviton modes. The details of this calculation are not essential for ourdiscussion. Once we obtain the following graviton perturbation h φφ = a r · r sin θ , (3.33)it can be coupled to the scalar field through the kinetic term, (cid:90) d x √− gg µν ∂ µ Φ ∂ ν Φ ⊃ (cid:90) d xh φφ ( ∂ φ Φ) ∝ (cid:90) dtdφdθdr r sin θ (cid:20) a m r Φ (cid:21) , (3.34)where in the last equation we used the expansion over spherical harmonics Φ ∝ e imφ .Then, we can easily account for perturbations that are produced by the interactions We assume a gauge consistent with the Boyer-Lindquist coordinates [58]. – 16 –etween the source and the gravitons at the leading order in spin. Varying theaction (3.34) over Φ and using the perturbative expansion Φ = ¯Φ + Φ (1) h + Φ (1) a , (3.35)we obtain that the correction due to black hole spin Φ (1) a satisfies the following dif-ferental equation (cid:18) ∂ r + 2 r ∂ r − (cid:96) ( (cid:96) + 1) r (cid:19) Φ (1) a = a r ∂ φ ¯Φ . (3.36)Using our ansatz for the source (3.23), we obtain the following net expression for thesource interacting with the Kerr black hole at the leading orders in black hole’s massand spin Φ = (cid:88) m E (0) (cid:96)m Y (cid:96)m r (cid:96) (cid:18) − (cid:96) Mr + m a (cid:96) − r (cid:19) . (3.37)Iterating this procedure at higher orders in r s /r , we can reconstruct all power seriesresponsible for the interaction between the source and the graviton. Finite size operators in the presence of spin.
The structure of finite-sizeoperators becomes more complicated due to the spin, because now we can producenew tensor structures using the spin vector s i and the Levi-Civita antisymmetricsymbol (cid:15) ijk . Therefore, now we can write the following general expression [27] forthe causal response function introduced in Eq. (3.10) G ret L (cid:48) L ( ω ) = (cid:88) n =0 ω n (cid:16) ˆ λ loc. n L (cid:48) L + i ˆ (cid:15) loc. n +1 LL (cid:48) ω + ˆ (cid:15) non-loc. n L (cid:48) L + i ˆ λ non-loc. n +1 LL (cid:48) ω (cid:17) . (3.38)Let us now specify symmetry properties of different response tensors in this ex-pression. From the axial symmetry of the problem, the tensors that are even andodd w.r.t. L ↔ L (cid:48) must be constructed from even and odd numbers of spin vec-tors, respectively. The local terms in Eq. (3.38) must be symmetric w.r.t. timereversal invariance (which includes the spin flip), which now corresponds to simulta-neous exchange L ↔ L (cid:48) , ω → − ω . These terms correspond to local operators in thepoint-particle EFT in the body’s rotation frame. In contrast, the non-local termscorrespond to internal (gapless) worldline degrees of freedom that capture dissipa-tion [27] and hence they must be odd w.r.t. exchange L ↔ L (cid:48) , ω → − ω . This impliesthat tensors ˆ λ loc ./ non − loc .p L (cid:48) L and ˆ (cid:15) loc ./ non − loc .p L (cid:48) L in Eq. (3.38) must be even and odd w.r.t. L ↔ L (cid:48) , respectively. Going back to proper time we get, (cid:104) I L (cid:105) = (cid:88) n ( − n (cid:18) ˆ λ loc. n L (cid:48) L + ˆ (cid:15) loc. n +1 LL (cid:48) ddτ + ˆ (cid:15) non-loc. n L (cid:48) L + ˆ λ non-loc. n +1 LL (cid:48) ddτ (cid:19) d n dτ n E L (cid:48) . (3.39) The only available tensor structures are δ ij , (cid:15) ijk and s j = sz j , where z j = δ j . Hence, any tensorthat is odd w.r.t. L ↔ L (cid:48) (but is still STF w.r.t. multi-indices L and L (cid:48) ) has to look like (cid:15) ii (cid:48) k s k times a tensor that is even w.r.t. remaining multi-index exchange ( L − ↔ ( L − (cid:48) . – 17 –nlike the Schwazschild black hole, the terms with time derivatives here do notvanish in the static limit (w.r.t. a fixed inertial frame) because of the rotation ofthe body. Hence, if we want to capture the effects of spin to all orders, we need tokeep track of all powers of frequency here. This problem can be solved if we rewriteEq. (3.38) in a fixed inertial frame. To that end we can expand time derivatives inbody’s rotation frame as ddτ E L (cid:48) = ∂ t E L (cid:48) − (cid:96) Ω ( i (cid:48) q E q i (cid:48) ...i (cid:48) (cid:96) ) , (3.40)where ∂ t is the time derivative in the fixed inertial (source) frame, Ω ij is the an-tisymmetric angular velocity tensor, and we have symmetrized the rightmost termw.r.t. its free indices. To simplify the argument, let us neglect terms with partialtime derivatives ∂ t , which is reasonable since we are interested in the static limit.In this limit, applying an even number of time derivative in the body’s localrotating frame will produce a tensor built out of the same even number of the angularvelocity tensors. Multiplying an original response matrix by this tensor will notchange its parity properties w.r.t. multi-index exchange, ˆ λ loc. n L (cid:48) L d n dτ n → λ loc. n L (cid:48) L ( even L ↔ L (cid:48) ) , ˆ (cid:15) non-loc. n L (cid:48) L d n dτ n → (cid:15) non-loc. n L (cid:48) L ( odd L ↔ L (cid:48) ) , (3.41)where λ n and (cid:15) n are response tensors written in the inertial frame. However, apply-ing an odd number of time derivatives produces a tensor that contains the same oddnumber of the angular velocity tensors. This tensor is odd w.r.t. to the exchange ofits multi-indices, and hence it will change the parity of the corresponding responsematrices w.r.t. exchange L ↔ L (cid:48) , ˆ λ non-loc. n +1 L (cid:48) L d n +1 dτ n +1 → λ non-loc. n L (cid:48) L ( odd L ↔ L (cid:48) ) , ˆ (cid:15) loc. n +1 L (cid:48) L d n +1 dτ n +1 → (cid:15) loc. n +1 L (cid:48) L ( even L ↔ L (cid:48) ) , (3.42)where λ n +1 and (cid:15) n +1 are again response tensors in the inertial frame. We see thatafter we have changed the frame, all terms of non-local origin can be collected intoa new antisymmetric response matrix κ L (cid:48) L , whereas all local terms effectively sum upinto a new symmetric matrix λ L (cid:48) L . Importantly, the transition to the inertial framedid not mix the properties of the response matrices written in Eq. (3.38) — the localresponse is still captured by a matrix that is even w.r.t. L ↔ L (cid:48) , ω → − ω . All in all,we can write (cid:104) I L (cid:105) = (cid:16) λ L (cid:48) L + κ L (cid:48) L (cid:17) E L (cid:48) + (cid:16) Λ L (cid:48) L + Λ (cid:48) L (cid:48) L (cid:17) ∂ t E L (cid:48) + ... , where λ L (cid:48) L = λ LL (cid:48) , κ L (cid:48) L = − κ LL (cid:48) , Λ L (cid:48) L = − Λ L (cid:48) L , Λ (cid:48) L (cid:48) L = Λ (cid:48) LL (cid:48) , (3.43)and “ ... ” stands for operators that involve more than one partial time derivatives.We stress that the instantaneous contribution proportional to κ LL (cid:48) , in fact, stems– 18 –rom non-local operators, which was nicely explained in Ref. [27]. The correspondinginduced multipole I L ∝ κ L (cid:48) L E L (cid:48) describes dissipative effects such as tidal torques ormass loss/accretion [25, 27]. In contrast, the part of the response proportional to λ LL (cid:48) in Eq. (3.43) captures the local static deformation produced by external fields, it isindistinguishable from the effect of local operators in the point particle action.All in all, the leading-order local finite-size effects are described by the followinggeneralized response operator (3.16), S a Love = 12 (cid:96) ! (cid:90) d x (cid:90) dτ δ (4) ( x − x ( τ )) λ (0) j ...j (cid:96) i ...i (cid:96) ∂ (cid:104) j ...j (cid:96) (cid:105) Φ ∂ (cid:104) i ...i (cid:96) (cid:105) Φ , (3.44)which accounts for the violation of the spherical symmetry by the spinning black holebackground. The coupling Eq. (3.44) can always be recast in the manifestly covariantform by means of the projection operator (3.15). Note that λ is a symmetric trace-free tensor w.r.t. upper and lower sets of indices, λ (0) j ...j (cid:96) i ...i (cid:96) = λ (0) (cid:104) j ...j (cid:96) (cid:105)(cid:104) i ...i (cid:96) (cid:105) , (3.45)but its trace w.r.t. the contraction of lower and upper indices does not vanish. Wewill refer to λ (0) and its analogs for higher spins as “Love tensor” in what follows.The calculation of the response of the scalar field induced by this operator isidentical to one presented above, and it yields Φ (1)Love = B (cid:96) r − (cid:96) − E (0) L (cid:48) n L λ (0) L (cid:48) L , (3.46)where the constant B (cid:96) is given in Eq. (3.31). To find the SRCs k (0) (cid:96)m , we need to solve the vacuum Klein-Gordon equation in theKerr background, assuming that the scalar field varies very slowly in time. It isinstructive to carry out our microscopic (i.e. general relativity) calculation in twodifferent coordinate systems. Let us start with the advanced Kerr coordinates, whichare manifestly regular at the horizon. Advanced Kerr coordinates.
The interval of the Kerr spacetime in the advancedKerr coordinates is given by [59] ds = − (cid:18) − M r Σ (cid:19) dv + 2 dvdr − M ra
Σ sin θdvd ˜ φ − a sin θdrd ˜ φ + Σ dθ + (cid:18) r + a + 2 M r Σ a sin θ (cid:19) sin θd ˜ φ , (3.47)where a ≡ J/M is the reduced spin parameter and Σ ≡ r + a cos θ . It is well knownthat in the static case ( ω = 0 ) the Klein-Gordon equation for the massless scalar field– 19 –actorizes into usual scalar spherical harmonics in the Kerr background [53, 60]. Inthe advanced Kerr coordinates this decomposition takes the following form Φ = (cid:88) (cid:96)m E (0) (cid:96)m ˜ R (cid:96)m ( r ) Y (cid:96)m ( θ, ˜ φ ) . (3.48)To match the source boundary condition at infinity, we impose ˜ R (cid:96)m → r (cid:96) at r → ∞ and demand this function to be smooth at the external black hole horizon. Theequation defining the radial mode function ˜ R (cid:96)m takes the following form x (1 + x ) ˜ R (cid:48)(cid:48) (cid:96)m ( x ) + [(1 + 2 x ) + 2 imγ ] ˜ R (cid:48) (cid:96)m ( x ) − (cid:96) ( (cid:96) + 1) ˜ R (cid:96)m ( x ) = 0 , (3.49)where (cid:48) ≡ ∂/∂x and we have defined x ≡ r − r + r + − r − , γ ≡ ar + − r − , r ± = M ± √ M − a . (3.50)Note that r + and r − are the outer and inner horizons of the Kerr black hole, respec-tively. In what follows we will be mostly focusing on r + and we will refer to it simplyas “black hole’s horizon.” We will also use the following notation r sa ≡ r + − r − . (3.51)The solution of Eq. (3.49) regular at the horizon ( x → ) is given by ˜ R (cid:96)m = const · F ( (cid:96) + 1 , − (cid:96) ; 1 + 2 imγ, − x ) , (3.52)where F stands for the Gauss hypergeometric function (see Appendix A for moredetail). In the physical case (cid:96) ∈ N , the function ˜ R (cid:96)m is a polynomial in x ∝ r andhence it does not contain any decaying power of r . Thus, by looking at Eq. (3.2)one may be tempted to conclude that the SRCs are zero for the Kerr background.However, we have to ensure that this is not a result of a cancellation between body’sresponse and the subleading source contributions due to graviton interactions. Tothat end, let us consider an analytic continuation (cid:96) → R , in which case the solutionEq. (3.52) can be Taylor-expanded at spatial infinity as (see Appendix A) ˜ R (cid:96)m = const · (cid:32) Γ(1 + 2 imγ )Γ(2 (cid:96) + 1)Γ( (cid:96) + 1)Γ(1 + (cid:96) + 2 imγ ) x (cid:96) · F (cid:0) − (cid:96), − (cid:96) − imγ, − (cid:96), − x − (cid:1) + Γ(1 + 2 imγ )Γ( − (cid:96) − − (cid:96) )Γ( − (cid:96) + 2 imγ ) x − (cid:96) − · F (cid:0) (cid:96) + 1 , (cid:96) + 1 − imγ, (cid:96) + 2 , − x − (cid:1) (cid:33) −−−→ x →∞ r (cid:96)sa x (cid:96) (cid:18) − (cid:96) − (cid:96) + 1)Γ(1 + (cid:96) + 2 imγ )Γ(2 (cid:96) + 1)Γ( − (cid:96) )Γ( − (cid:96) + 2 imγ ) x − (cid:96) − (cid:19) . (3.53)– 20 –ince the first distinctive contribution in Eq. (3.53) scales as r (cid:96) at infinity, it is naturalto associate it with the external source and its corrections produced by non-lineargravitational interactions. The second distinctive contribution in Eq. (3.53) scalesas r − (cid:96) − at infinity, and hence it is natural to interpret it as black hole’s response.Comparing Eq. (3.53) with the (post-)Newtonian expansion formula (3.2), we findthat for a general multipolar index (cid:96) the response coefficients are given by k (0) (cid:96)m = Γ( − (cid:96) − (cid:96) + 1)Γ(1 + (cid:96) + 2 imγ )Γ(2 (cid:96) + 1)Γ( − (cid:96) )Γ( − (cid:96) + 2 imγ ) (cid:18) r sa r s (cid:19) (cid:96) +1 . (3.54)The scalar tidal response coefficients extracted by means of the analytic continuationdo not vanish even in physical limit (cid:96) → N . In this case Eq. (3.54) can be simplified k (0) (cid:96)m = − imγ ( (cid:96) !) (2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n + 4 m γ ) (cid:18) r sa r s (cid:19) (cid:96) +1 = − imχ (cid:96) !) (2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n (1 − χ ) + m χ ) , (3.55)where in the last line we have introduced the dimensionless spin χ ≡ a/M andused Eq. (A.18). Note that the expression (3.55) vanishes in the limit χ → ,reproducing the well-established result that the scalar response coefficients of non-spinning black holes are zero [7, 11]. Importantly, the response coefficients (3.55) arepurely imaginary. As discussed before, they correspond to dissipative effects and notto the classic conservative static response coefficients which we will refer to as scalarLove numbers.All in all, we have obtained that the radial solution (3.52) in the advanced Kerrcoordinates is a polynomial without any decaying power of r , and, at the same time,the response coefficients are non-zero. The only possibility to reconcile these twofacts is that the GR corrections to the source and the induced response happenedto exactly cancel one another in the advanced Kerr coordinates in the physical limit (cid:96) → N . This is exactly what happens. To see this, we expand the relevant sourcesolution at infinity as follows x (cid:96) F (cid:0) − (cid:96), − (cid:96) − imγ, − (cid:96), − x − (cid:1) = x (cid:96) ∞ (cid:88) n =0 Γ( − (cid:96) + n )Γ( − (cid:96) − imγ + n )Γ( − (cid:96) )Γ( − (cid:96) )Γ( − (cid:96) − imγ )Γ( − (cid:96) + n ) ( − n x − n n ! . (3.56)Let’s focus on the n = 2 (cid:96) + 1 ’th term in the hypergeometric series above. This termscales like r − (cid:96) − just like the response contribution. We have x (cid:96) +12 F (cid:0) − (cid:96), − (cid:96) − imγ, − (cid:96), − x − (cid:1) ⊃ Γ( − (cid:96) − (cid:96) + 1)Γ(1 + (cid:96) − imγ )Γ(2 (cid:96) + 1)Γ( − (cid:96) )Γ( − (cid:96) − imγ ) = imγ ( (cid:96) !) (2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n + 4 m γ ) , (3.57)– 21 –here we took the physicacl limit (cid:96) → N in the second line. This exactly equalsminus the coefficient in front of the response part in Eq. (3.53), hence the subleadingsource contribution exactly cancels the response in the advanced Kerr coordinates.The upshot of our discussion is that the naive identification of the responsecoefficients from the solution to the Klein-Gordon equation may be ambiguous dueto uncertainty in the source/response split. This ambiguity can be removed by meansof the analytic continuation (cid:96) → R [25]. This is explicitly confirmed by a calculationin the Boyer-Lindquist coordinates, to which we proceed now. Boyer-Lindquist coordinates.
The line element of the Kerr spacetime in theBoyer-Lindquist coordinates [58] is given by ds = − (cid:18) − M r Σ (cid:19) dt − (cid:18) M ar sin θ Σ (cid:19) dtdφ + Σ∆ dr + Σ dθ + sin θ (cid:18) r + a + 2 M a r sin θ Σ (cid:19) dφ , (3.58)where ∆ ≡ r − M r + a , Σ ≡ r + a cos θ . The Boyer-Lindquist and advancedKerr coordinates are related via dυ = dt + dr (cid:18) M r ∆ (cid:19) , d ˜ φ = dφ + dr a ∆ . (3.59)The static scalar field equation factorizes in these coordinates as follows [53] Φ = (cid:88) (cid:96)m E (0) (cid:96)m R (cid:96)m ( r ) Y (cid:96)m ( θ, φ ) , (3.60)where the radial mode function R (cid:96)m satisfies the following equation: x ( x + 1) R (cid:48)(cid:48) (cid:96)m ( x ) + (2 x + 1) R (cid:48) (cid:96)m ( x ) + (cid:18) − (cid:0) (cid:96) + (cid:96) (cid:1) + γ m x ( x + 1) (cid:19) R (cid:96)m = 0 . (3.61)We are looking for a solution which is smooth at the black hole’s horizon and has asingularity at spatial infinity. Identifying this solution in the Boyer–Lindquist coor-dinates is not evident, as these coordinates are singular at the Kerr horizon. Nev-ertheless, it can be shown that the regularity at the horizon in the Boyer-Lindquistcoordinates corresponds to the following condition, obtained by Press (1972) [54] andTeukolsky (1973) [50] R (cid:96)m = const · ( r − r + ) + imγ as r → r + . (3.62)The constant should be chosen such that R (cid:96)m /r (cid:96) → at r → ∞ . The solutionsatisfying these boundary conditions is given by R (cid:96)m = const · (cid:18) x x (cid:19) imγ F ( (cid:96) + 1 , − (cid:96), imγ, − x ) , −−−→ x →∞ r (cid:96)sa · (cid:18) x (cid:96) + Γ( − (cid:96) − (cid:96) + 1)Γ(1 + (cid:96) + 2 imγ )Γ(2 (cid:96) + 1)Γ( − (cid:96) )Γ( − (cid:96) + 2 imγ ) x − (cid:96) − (cid:19) , (3.63)– 22 –here in the last step we used an analytic continuation of the hypergeometric functionat spatial infinity and retained only the leading asymptotics. Assuming that (cid:96) ∈ R ,we can extract the SRCs just like in Eq. (3.53) and find the same expression (3.55).However, in contrast to the advanced Kerr coordinates, the part of the solutioncontaining the power r − (cid:96) − does not get canceled by the graviton corrections to thesource even in the physical case (cid:96) → N . This happens due to the presence of theprefactor (cid:0) x x (cid:1) imγ .We see that the coefficient in front of the power r − (cid:96) − depends on a choice ofcoordinates in the physical case (cid:96) → N . If we were to use the naive Newtonianmatching, we would find coordinate-dependent SRCs. The agreement between thedifferent coordinate systems is restored if we use the analytic continuation (cid:96) → R .Finally, we note that non-vanishing of SRCs for Kerr black holes was, in fact,first discovered by Press in 1972 [54], who also argued that they had to be purelyimaginary in order for the solution to satisfy the complex regularity condition atthe black hole horizon. Moreover, Press (1972) has also shown that the SRCscapture the spin down produced by the perturbing scalar field. A similar connectionwas recently discussed in Le Tiec et al. (2020) [25] in the context of gravitationalperturbations. Note that the results presented in that work seem to be affected by an insignificant typoRef. [54] presents the following solution to the radial models of Klein-Gordon equation in the Kerrbackground (their Eq. (10))Press’72: R (cid:96)m = ( r − r − ) − imγ ( r − r + ) imγ F ( (cid:96) + 1 , − (cid:96), imγ ; ( r − r − ) / ( r + − r − )) , (3.64)which, in fact, does not satisfy the radial Teukolsky equation [50]. The actual solution may bewritten in one of the following equivalent forms R (cid:96)m = ( r − r − ) − imγ ( r − r + ) imγ F ( (cid:96) + 1 , − (cid:96), − imγ ; ( r − r − ) / ( r + − r − )) , = ( r − r − ) − imγ ( r − r + ) imγ F ( (cid:96) + 1 , − (cid:96), imγ ; ( r + − r ) / ( r + − r − )) , (3.65)which differs from Eq. (3.64) either by the argument or by the sign in front of the complex partof the third order parameter of the hypergeometric function. This typo has resulted in a signdifference for response coefficients compared to our result. Correcting this typo, Eqs. (10), (15c) ofPress (1972) imply Press’72: Im R (cid:96)m (cid:12)(cid:12)(cid:12) (cid:96) =1 ,m =1 = − a (cid:18) Mr (cid:19) as r → ∞ , (3.66)which coincides with our expression (3.63) at linear order in a . – 23 – .4 Matching to the EFT To match the microscopic and the EFT calculation we need to compare the coeffi-cients in front of the /r (cid:96) +1 power from the two calculations. We have B (cid:96) n (cid:104) L (cid:105) λ L (cid:48) L E L (cid:48) = r (cid:96) +1 s (cid:96) (cid:88) m = − (cid:96) E (0) (cid:96)m Y (cid:96)m k (0) (cid:96)m (3.67)Rewriting the r.h.s. of this equation in the basis of the constant STF tensors Y L(cid:96)m on S as [42] (cid:96) (cid:88) m = − (cid:96) E (0) (cid:96)m Y (cid:96)m k (0) (cid:96)m = (cid:96) (cid:88) m = − (cid:96) k (0) (cid:96)m Y L ∗ (cid:96)m n (cid:104) L (cid:105) π(cid:96) !(2 (cid:96) + 1)!! Y L (cid:48) (cid:96)m E (0) L (cid:48) , (3.68)we arrive at the following equation (cid:32) λ (0) i (cid:48) ...i (cid:48) (cid:96) i ...i (cid:96) − r (cid:96) +1 s B (cid:96) π(cid:96) !(2 (cid:96) + 1)!! (cid:96) (cid:88) m = − (cid:96) k (0) (cid:96)m Y ∗ i ...i (cid:96) (cid:96)m Y i (cid:48) ...i (cid:48) (cid:96) (cid:96)m (cid:33) n (cid:104) i ...n i (cid:96) (cid:105) E (0) i (cid:48) ...i (cid:48) (cid:96) = 0 , (3.69)where we have restored the explicit tensorial indices. In what follows we will refer tothe constant STF tensors Y L(cid:96)m as “Thorne tensors” [42]. They allow us to write thefollowing expression for the scalar response tensor λ (0) i (cid:48) ...i (cid:48) (cid:96) i ...i (cid:96) = r (cid:96) +1 s B (cid:96) π(cid:96) !(2 (cid:96) + 1)!! (cid:96) (cid:88) m = − (cid:96) k (0) (cid:96)m Y ∗ i ...i (cid:96) (cid:96)m Y i (cid:48) ...i (cid:48) (cid:96) (cid:96)m , (3.70)which is valid up to terms antisymmetric in i (cid:48) ...i (cid:48) (cid:96) and i ...i (cid:96) , and up to a Kroneckerdelta in any combination of i p and i (cid:48) q . Note that the expression (3.70) does not imposeany restrictions on the symmetry properties of λ (0) LL (cid:48) w.r.t. multi-index exchange L ↔ L (cid:48) . Hence, our matching procedure based on the expression (3.67) computesboth dissipative and conservative responses.If the Newtonian SRCs k (0) (cid:96)m do not depend on the magnetic number m , i.e. k (0) (cid:96)m = k (0) (cid:96) , which is the case for Schwarzschild black holes, then the sum over theSTF tensors can be explicitly taken, π(cid:96) !(2 (cid:96) +1)!! (cid:80) (cid:96)m = − (cid:96) Y ∗ L(cid:96)m Y L (cid:48) (cid:96)m = δ LL (cid:48) . In this case wereproduce the expression for SLNs obtained in Hui et al. [7] for the Schwarzschildblack holes (upon identification ˆ L → (cid:96) and D → ) λ (0) LL (cid:48) = λ (cid:96) δ LL (cid:48) , λ (cid:96) = ( − (cid:96) π / Γ(1 / − (cid:96) )2 (cid:96) − Γ( − (cid:96) − (cid:96) + 1) Γ(2 (cid:96) + 1)Γ( − (cid:96) ) r (cid:96) +1 s . (3.71)These response coefficients vanish identically in the physical case (cid:96) ∈ N .Let us now explicitly compute the response matrix (3.70) for the (cid:96) = 1 and (cid:96) = 2 sectors. Using formulas from Appendix A, we obtain the following expression for (cid:96) = 1 λ (0) ij = ar s B − , (3.72)– 24 –here for simplicity we have retained only the terms linear in black hole’s spin a .This matrix is antisymmetric, which means that the corresponding dipole worldlinecoupling vanishes because it contracts two gradients of the scalar field. Hence, theEFT Love tensor is zero in this case even though the Newtonian response coefficientsare not. Thus, the Kerr black hole’s response is purely dissipative.Now let us consider the quadrupolar sector (cid:96) = 2 . The corresponding Love tensorcomputed from Eq. (3.70) takes the following form λ (0) ijkl = − (4 π ) χM (cid:104) − χ ) I (1) ij,kl + 5 χ (1 − χ ) I (3) ij,kl + χ I (5) ij,kl (cid:105) , (3.73)where we have introduced the dimensionless black hole spin χ = a/M and used thefollowing real-valued tensors (defined for any n ) I (2 n − L,L (cid:48) ≡ π(cid:96) !(2 (cid:96) + 1)!! (cid:96) (cid:88) m = − (cid:96) m n − Im ( Y ∗ (cid:96)mL Y (cid:96)mL (cid:48) ) = 12 − n − M n − M − M n − M n − M M − M M , (3.74)which are composed of the STF basis matrices given by [23, 25] M = − , M = , M = , M = , (3.75)and is a trivial × matrix. For small spin the response matrix takes the followingsimplified form: λ (0) ijkl = − π γr s I (1) ij,kl + O ( γ ) = − (4 π ) 4 χM I (1) ij,kl + O ( χ ) . (3.76)We observe that the tensor λ (0) ijkl is antisymmetric w.r.t. the upper and lower groupsof indices, i.e. λ (0) ijkl = − λ (0) klij . This means that the local quadrupole worldlineoperator vanishes as well. This result can be extended to higher order multipoles.The antisymmetry of the corresponding matrices stems from the fact that thescalar response numbers are purely imaginary. To see this, let us consider the follow-ing general ansatz for response coefficients consistent with the reality requirementfor the scalar field multipole expansion (3.1) [25]: k (0) (cid:96)m = k (cid:96) + χ ∞ (cid:88) n =1 k (cid:96)n ( χ )( im ) n , (3.77)where k (cid:96)n are real spin-dependent functions. Plugging this into Eq. (3.70) we obtain: λ (0) L (cid:48) L = r (cid:96) +1 s A (cid:96) (cid:32) k (cid:96) δ LL (cid:48) + χ ∞ (cid:88) n =1 ( − n (cid:104) k (cid:96) (2 n − I (2 n − L,L (cid:48) + k (cid:96) (2 n ) R (2 n ) L,L (cid:48) (cid:105) (cid:33) , (3.78)– 25 –here I L,L (cid:48) are the antisymmetric tensors (w.r.t. exchange L ↔ L (cid:48) ) introduced inEq. (3.74), whereas R L,L (cid:48) are new fully symmetric tensors defined as follows [25]: R (2 n ) L,L (cid:48) ≡ π(cid:96) !(2 (cid:96) + 1)!! (cid:96) (cid:88) m = − (cid:96) m n Re ( Y ∗ (cid:96)mL Y (cid:96)mL (cid:48) ) = 12 n M n M M n M − n M M M M . (3.79)Comparing Eq. (3.77) with Eq. (3.78) we see that the real part of the responsecoefficient generates a symmetric part of λ (0) (i.e. even w.r.t. L ↔ L (cid:48) ), and eventuallycontributes to the symmetric Love tensor implying non-trivial local EFT operators.However, the imaginary part of the response coefficients generates an antisymmetricpart of the response, I L (cid:48) ∝ κ L (cid:48) L E L , κ L (cid:48) L = − κ LL (cid:48) , (3.80)which can be identified with a quasi-local contribution given in Eq. (3.43). Since theresponse that we have found is purely imaginary, we conclude that (a) the Love num-bers vanish identically, (b) the tidal response of Kerr black holes to static scalar per-turbations is entirely dissipative. Note that our antisymmetric quadrupolar responsetensor (3.73) coincides (up to a numerical factor) with the tensors that describe theblack hole’s torque obtained in Refs. [25, 27].Finally, we compare the first corrections to the source due to the graviton in-teraction. We will focus on the first r s /r correction and the first non-trivial spincontribution. Taylor expanding Eq. (3.63) we obtain Φ ⊂ r (cid:96) (cid:18) − (cid:96) r s r + a m (4 (cid:96) − r + ... (cid:19) , (3.81)which agrees with the EFT calculation (3.37). In this section we extend our static response calculation to the spin-1 field andcompute the response of a Kerr black hole to a long-wavelength electromagneticperturbation. Similar calculations were done for Schwarzschild black holes in fourdimensions in Ref. [61] and in a general number of dimensions in Ref. [7]. We willstart with the definition of spin-1 response coefficients and Love numbers. Then wewill compute the electromagnetic field around the Kerr black hole by means of theNewton-Penrose formalism [62, 63], and extract the vector response coefficients fromthis solution. Finally, we will match our general relativity calculations to the EFT,which will help us fix the relevant tensor Wilson coefficients. As in the scalar fieldcase, our matching procedure will imply the vanishing of the EFT Love tensor, andhence the spin-1 response will be identified to be purely dissipative.– 26 – .1 DefinitionEFT Love numbers.
The local worldline EFT for the electromagnetic field tozeroth order in metric perturbations is given by the following action [7] S emEFT = S pp − (cid:90) d x F µν F µν + (cid:88) (cid:96) =1 (cid:96) ! (cid:90) d x (cid:90) dτ δ (4) ( x − x ( τ )) λ (1) LL (cid:48) ( ∂ (cid:104) i ...∂ i (cid:96) − E i (cid:96) (cid:105) )( ∂ (cid:104) i (cid:48) ...∂ i (cid:48) (cid:96) − E i (cid:48) (cid:96) (cid:105) )+ (cid:88) (cid:96) =1 (cid:96) ! (cid:90) d x (cid:90) dτ δ (4) ( x − x ( τ ))˜ λ (1) LL (cid:48) (cid:104) ∂ ( i ...∂ i (cid:96) − B i (cid:96) (cid:105) j )( ∂ (cid:104) i (cid:48) ...∂ i (cid:48) (cid:96) − B i (cid:48) (cid:96) (cid:105) j ) , (4.1)where F µν ≡ ∂ [ µ A ν ] is the Maxwell tensor, A µ is the U (1) gauge potential, S pp isthe usual point-particle action (3.8), and we have introduced the electric field vectorand the magnetic tensor as E i = F i , B ij = F ij . (4.2)They can be defined in a manifestly covariant way by means of the body’s 4-velocityand the projector P νµ = δ νµ + v µ v ν , E ν = F µν v µ , B µν = P σµ P ρν F σρ . (4.3)The Wilson coefficients λ (1) LL (cid:48) and ˜ λ (1) LL (cid:48) are the electric and magnetic Love tensors,respectively. In the static limit in the particle’s rest frame E i = F µi v µ = − ∂ i A , (4.4)which implies that the EFT for the electric field only depends on a scalar field A ,and effectively it reduces to the EFT for a massless scalar field that we have studiedin the previous section. In order to study the magnetic field it is convenient to employthe transverse gauge ∂ i A i = 0 , in which case the kinetic term for the electromagneticfield takes the following form in the zero-frequency limit − (cid:90) d xF µν F µν → (cid:90) d x (cid:2) ( ∂ i A ) − ∂ i A j ∂ i A j (cid:3) . (4.5) Definition à la Newtonian matching.
It is also useful to introduce a definition ofelectromagnetic response coefficients in the way similar to the gravitational potentialin the Newtonian approximation. This will prove convenient to extract the responsecoefficients from the general relativity solution that we will obtain in the harmonicspace. To that end we can use the fact that in the static limit the electric field isfixed by A . The equation of motion for A reduces to the Poisson equation in thelong-distance limit, ∇ i A = 0 . (4.6)– 27 –n the static limit A becomes gauge-independent, and hence we can use it to defineresponse coefficients just like in the case of the Newtonian gravitational potential.Since A transforms as a scalar under rotations, it can be written as a series over thescalar spherical harmonics A = (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m α (cid:96)m r (cid:96) (cid:34) k (1) (cid:96)m (cid:18) rr s (cid:19) − (cid:96) − (cid:35) , (4.7)where α (cid:96)m are source harmonic coefficients, which satisfy α ∗ (cid:96)m = ( − m α (cid:96) ( − m ) suchthat A is real. Hence, we can use an analog of the Newtonian matching supplementedwith the analytic continuation (cid:96) → R to extract electric response coefficients. Notethat we have neglected the background electric monopole contribution because weconsider neutral black holes in this paper.One can define the magnetic response using an expansion for the vector part ofthe gauge potential similar to (4.7). It is easiest to do that at the level of the angularcomponent of the electromagnetic tensor [7] F ab = 2 ∇ [ a A b ] = 2 (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) ∇ [ a Y RW b ] (cid:96)m β (cid:96)m (cid:112) (cid:96) ( (cid:96) + 1) r (cid:96) +1 (cid:34) − (cid:96) + 1 (cid:96) ˜ k (1) (cid:96)m (cid:18) rr s (cid:19) − (cid:96) − (cid:35) , (4.8)where [ a, b ] denotes antisymmetrization, β (cid:96)m are magnetic source coefficients, satisfy-ing β ∗ (cid:96)m = ( − m β (cid:96) ( − m ) , and Y RW b (cid:96)m are the Regge-Wheeler transverse vector sphericalharmonics [64], see Appendix A for detail. The normalization factor − ( (cid:96) + 1) /(cid:96) isinserted for convenience. With this factor the multipole expansion for the magneticfield B i = (cid:15) ijk F jk and the electric field E i take very similar forms, e.g. for the radialcomponent we have (cid:18) E r B r (cid:19) = (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) (cid:18) (cid:96)α (cid:96)m β (cid:96)m (cid:19) Y (cid:96)m r (cid:96) − (cid:32) − (cid:96) + 1 (cid:96) (cid:32) k (1) (cid:96)m ˜ k (1) (cid:96)m (cid:33) (cid:16) r s r (cid:17) (cid:96) +1 (cid:33) . (4.9)From this expression our convention becomes natural as we expect the magnetic andelectric response coefficients to coincide in 4 dimensions due to the electric-magneticduality [7, 41], k (1) (cid:96)m = ˜ k (1) (cid:96)m . Hence, we will focus on the electric field in the mainbody of the paper and present an explicit calculation of the magnetic response inAppendix C. Indeed, the relevant U (1) gauge transformation A µ → A µ + ∂ µ α does not alter A in thestationary limit ∂ t → . Recall that A and A r transform as scalars under the SO (3) group transformations, whereasthe vector A a ( a = ( θ, φ )) has two distinctive contributions, which transform as a scalar and as avector under SO (3) [7]. – 28 – .2 Newman-Penrose Formalism To compute A µ in the Kerr black hole background, we will work within the Newman-Penrose (NP) formalism [62, 63]. In this formalism, the electromagnetic tensor isrepresented by 3 complex scalars Φ , Φ , Φ as F µν = 2 (cid:2) Φ ( n [ µ l ν ] + m [ µ m ∗ ν ] ) + Φ l [ µ m ν ] + Φ m ∗ [ µ n ν ] (cid:3) + c.c. , (4.10)where l µ , n µ , m µ ( m ∗ µ is the complex conjugate of m µ ) are the so-called Newman-Penrose null tetrades. Their explicit expressions in the Boyer-Lindquist coordinatesare given in Kinnersley (1969) [65], (cid:96) µ = (cid:18) ( r + a )∆ , , , a ∆ (cid:19) , n µ = (cid:18) r + a , − ∆2Σ , , a (cid:19) m µ = 1 √ r + ia cos θ ) (cid:18) ia sin θ, , , i sin θ (cid:19) . (4.11)The quantities Φ , Φ , Φ will be referred to as Maxwell-Newmann-Penrose (MNP)scalars in what follows. The components that we will need for the Newtonian match-ing are F r and F θφ . They are given by F r = − Re Φ + a sin θ ( a cos θ Re Φ + r Im Φ ) √
2Σ + a sin θ √ a cos θ Re Φ − r Im Φ )∆ ,F θφ = 2 Im Φ (cid:0) a + r (cid:1) sin θ + ar Re Φ sin θ ∆ √ − √ ar Re Φ sin θ − √ a Im Φ sin θ cos θ − a Im Φ sin θ cos θ ∆ √ , (4.12)where ∆ = r − M r + a , Σ = r + a cos θ . The quantities Φ and ˜Φ , defined as ˜Φ ≡ ( r − ia cos θ ) ( r + − r − ) Φ , are separable solutions of the Teukolsky equations for spin weights s = +1 and s = − , respectively [49, 50] (see also Refs. [66–68]). Assuming that the externalsource is located at spatial infinity, the leading asymptotic behaviors of these MNPscalars are given by [50, 66] Φ ∼ Φ ∼ Φ ∼ r (cid:96) − + const · r − (cid:96) − , (4.13)where “const” is some calculable constant. Hence, in the asymptotic limit r → ∞ both the magnetic and electric response coefficients can be extracted from a singleMNP scalar Φ F θφ (cid:12)(cid:12)(cid:12) r →∞ = 2 Im Φ r sin θ , F r (cid:12)(cid:12)(cid:12) r →∞ = − Re Φ . (4.14)– 29 – .3 From Maxwell-Newman-Penrose Scalars to Response Coefficients In order to compute Φ for the Kerr metric, we will follow the algorithm proposedin Bi ˇ cák and Dvo ˇ rák (1976) [66]. We will use the Maxwell equations rewritten interms of the Newman-Penrose quantities, which are presented in Appendix B. As afirst step, we compute ˜Φ . In the Kerr background it factorizes as ˜Φ = ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m R (2) (cid:96)m ( r ) − Y (cid:96)m ( θ, φ ) , (4.15)where − Y (cid:96)m are the spin-weighted spherical harmonics with weight − , and a (cid:96)m arethe source-dependent constant harmonic coefficients. Note that we do not imposeany restrictions on a (cid:96)m – they are generic complex numbers because Φ is complex.The radial function (2) R (cid:96)m ( r ) satisfies the s = − Teukolsky equation [49, 50], (cid:18) − (cid:96) − (cid:96) + γm ( γm − i (2 x + 1)) x ( x + 1) (cid:19) R (2) (cid:96)m ( x ) + x ( x + 1) R (cid:48)(cid:48) (2) (cid:96)m ( x ) = 0 , (4.16)supplemented with the following smoothness boundary condition at the horizon ( r → r + ) [50] R (cid:48)(cid:48) (2) (cid:96)m ( x ) = const · x imγ , as x → . (4.17)The relevant solution is given by R (2) (cid:96)m ( x ) = (cid:18) xx + 1 (cid:19) + iγm x ( x + 1) F ( (cid:96) + 2 , − (cid:96), iγm ; − x ) , (4.18)which can be conveniently written as R (cid:96)m ( x ) ≡ (cid:0) xx +1 (cid:1) − iγm y (cid:96)m ( x ) with y (cid:96)m ≡ (cid:18) xx + 1 (cid:19) iγm x ( x + 1) F ( (cid:96) + 2 , − (cid:96), iγm ; − x ) . (4.19)Now we can compute the second MNP scalar Φ , which also factorizes in the sphericalcoordinates Φ = ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m R (0) (cid:96)m ( r ) +1 Y (cid:96)m ( θ, φ ) , (4.20)where +1 Y (cid:96)m are the spin-weighted spherical harmonics with weight s = +1 . Theradial part R (0) (cid:96)m can be extracted from R (2) (cid:96)m using the Maxwell equations in the NPformalism (see Appendix B for more detail) R (0) (cid:96)m = 2 r sa (cid:96) ( (cid:96) + 1) (cid:18) ddr + iam ∆ (cid:19) R (2) (cid:96)m = (cid:18) x x (cid:19) − iγm (cid:96) ( (cid:96) + 1) d dx y (cid:96)m . (4.21)Once Φ and Φ are determined, we can use the remaining two Maxwell equations inthe NP formalism to extract Φ . This quantity does not fully factorize in the Kerr– 30 –ackground Φ = √ r + − r − ) ( r − ia cos θ ) (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m (cid:96) ( (cid:96) + 1) (cid:18) xx + 1 (cid:19) − iγm · (cid:40) [( (cid:96) + 1) (cid:96) ] / (cid:34) ( r − ia cos θ ) r sa ddx ( y (cid:96)m ) − y (cid:96)m (cid:35) Y (cid:96)m − ia sin θ ddx ( y (cid:96)m ) · +1 Y (cid:96)m (cid:41) , (4.22)where we have neglected the monopole contribution that corresponds to a shift ofblack hole’s charge. The leading asymptotic in the limit x → ∞ , which is relevantfor the Newtonian matching, reads Φ (cid:12)(cid:12)(cid:12) x →∞ = √ x (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m ( (cid:96) ( (cid:96) + 1)) / · (cid:20) x ddx ( y (cid:96)m ) − y (cid:96)m (cid:21) Y (cid:96)m . (4.23)As anticipated, Φ factorizes in the asymptotic limit r → ∞ . Recall that we areeventually interested in A , which is related to Φ via ∂ r A = − F r = 2 Re Φ . At this point we can rewrite a (cid:96)m as, √ (cid:96) ( (cid:96) + 1)) / a (cid:96)m r sa = ( α (cid:96)m + iβ (cid:96)m ) , (4.24)where α (cid:96)m and β (cid:96)m satisfy the reality conditions α ∗ (cid:96)m = ( − m α ∗ (cid:96) ( − m ) and β ∗ (cid:96)m =( − m β ∗ (cid:96) ( − m ) . The harmonic coefficients α (cid:96)m and β (cid:96)m capture electric and magneticparts of the Maxwell tensor, respectively. To obtain A , we integrate Eq. (4.23) asfollows A = 2 Re (cid:90) r dr (cid:48) Φ ( r (cid:48) ) = Re (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) √ a (cid:96)m r sa ( (cid:96) ( (cid:96) + 1)) / Y (cid:96)m (cid:90) x dx (cid:48) x (cid:48) (cid:20) x (cid:48) ddx (cid:48) ( y (cid:96)m ( x (cid:48) )) − y (cid:96)m ( x (cid:48) ) (cid:21) = (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m α (cid:96)m y (cid:96)m x = (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) α (cid:96)m A (0) (cid:96)m ( r ) Y (cid:96)m , (4.25)where in the last line we used that the sum (cid:80) (cid:96)m α (cid:96)m Y (cid:96)m y (cid:96)m is real, which is aconsequence of α ∗ (cid:96)m = α (cid:96) ( − m ) , Y ∗ (cid:96)m = ( − m Y (cid:96) ( − m ) , and y ∗ (cid:96)m = y (cid:96) ( − m ) (see Eq. (4.19)).Expanding the radial mode functions at spatial infinity we obtain A (0) (cid:96)m ( r ) = const · ( x + 1) F ( (cid:96) + 2 , − (cid:96), iγm, − x ) −−−→ x →∞ r (cid:96)sa x (cid:96) (cid:18) − (cid:96) − (cid:96) + 2)Γ( (cid:96) + 2 imγ + 1)Γ(1 − (cid:96) )Γ(2 (cid:96) + 1)Γ( − (cid:96) + 2 imγ ) x − (cid:96) − (cid:19) . (4.26)– 31 –ow we can compare this result with the large-distance approximation (4.7) and readoff the following electromagnetic response coefficients k (1) (cid:96)m = Γ( − (cid:96) − (cid:96) )Γ( (cid:96) + 2 imγ + 1)Γ( − − (cid:96) )Γ(2 (cid:96) + 1)Γ( − (cid:96) + 2 imγ ) (cid:18) r sa r s (cid:19) (cid:96) +1 = imγ ( (cid:96) + 1)!( (cid:96) − (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n + 4 m γ ) (cid:18) r sa r s (cid:19) (cid:96) +1 = imχ (cid:96) + 1)!( (cid:96) − (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n (1 − χ ) + m χ ) , (4.27)where we replaced Γ( (cid:96) + 2) / Γ(1 − (cid:96) ) → Γ( (cid:96) ) / Γ( − (cid:96) − in the first line, then as-sumed the physical values (cid:96) ∈ N , and finally used Eq. (A.18). As in the scalar case,the electromagnetic response coefficients are purely imaginary, which means that thestatic Love numbers must vanish. We will confirm that shortly.It is instructive to take the limit γ → and compare our resulting expressionwith the Schwarzschild black hole electromagnetic Love numbers k S computed inRef. [7]. This work defined electromagnetic Love numbers w.r.t. the scalar mode Ψ S , defined as Ψ S = r (cid:112) (cid:96) ( (cid:96) + 1) ∂ r A (4.28)in four dimensions. This means that we need to differentiate A w.r.t. the radialcoordinate r to obtain Ψ S , which produces an additional factor − ( (cid:96) + 1) /(cid:96) in frontof the Love number. With this factor taken into account, we have k S ≡ − ( (cid:96) + 1) (cid:96) k (1) (cid:96)m (cid:12)(cid:12)(cid:12) γ =0 = − ( (cid:96) + 1) (cid:96) Γ( − (cid:96) − (cid:96) + 2)Γ( (cid:96) + 1)Γ(1 − (cid:96) )Γ(2 (cid:96) + 1)Γ( − (cid:96) )= Γ( − (cid:96) − (cid:96) + 2)Γ( (cid:96) )Γ(1 − (cid:96) )Γ(2 (cid:96) + 1)Γ( − (cid:96) − , (4.29)where we have used Γ( x ) x = Γ( x + 1) . This expression exactly coincides with theelectromagnetic Love numbers given in Ref. [7] after the identification ˆ L → (cid:96) and D → . It vanishes once we take the physical limit (cid:96) → N . In this section, we perform an explicit matching of the worldline point-particle ef-fective field theory that includes electromagnetism to the results of the full GRcalculation. This will allow us to extract the electric polarizability operator in theEFT from the electric response coefficients that we have previously found in thissection. The calculation of the magnetic susceptibilities can be easily performed inthe same fashion. We present this calculation in Appendix C for completeness.– 32 –he calculation of the electric Love numbers is identical to the scalar field Lovenumber matching. Introducing an external background source as ¯ A = ¯ α i ...i (cid:96) x i ...x i (cid:96) , (4.30)where ¯ α i ...i (cid:96) is an STF tensor, and solving the equation of motion for A just like inthe scalar field case we obtain A = (cid:88) (cid:96)m ¯ α (cid:96)m r (cid:96) Y (cid:96)m + λ (1) LL (cid:48) ¯ α L n L (cid:48) ( − (cid:96) +1 (cid:96) − π / Γ(1 / − (cid:96) ) r − (cid:96) − . (4.31)In principle, we could match directly Eq. (4.25) and Eq. (4.31) as A is gauge-independent in the static limit. However, generally it is more appropriate to matchthe components of the electric tensor, such as E r , in order to ensure that the result ingauge-independent. Acting on Eqs. (4.25) and Eq. (4.31) with one derivative w.r.t.the radial coordinate r , matching the two results, and rewriting the sum over thespherical harmonics in terms of the Thorne STF tensors we obtain the followingexpression for the electromagnetic response matrix λ (1) i (cid:48) ...i (cid:48) (cid:96) i ...i (cid:96) = − r (cid:96) +1 s B (cid:96) π(cid:96) !(2 (cid:96) + 1)!! (cid:96) (cid:88) m = − (cid:96) k (1) (cid:96)m Y ∗ i ...i (cid:96) (cid:96)m Y i (cid:48) ...i (cid:48) (cid:96) (cid:96)m , (4.32)where B (cid:96) is a constant given in Eq. (3.31). As in the scalar case, we see that therequirement that λ (1) L (cid:48) L is even w.r.t. exchange L ↔ L (cid:48) has disappeared in theexpression (4.32), and hence we can interpret it as a general expression for responsecoefficients that includes conservative and dissipative effects on the same footing.Plugging γ = 0 , we find that this expression reduces the Love numbers forthe Schwarzschild black holes given in Eq. (5.30) of Ref. [7] (upon identification ˆ L → (cid:96), D → , and up to a sign), λ (1) LL (cid:48) = λ ( E ) (cid:96) δ LL (cid:48) , where λ ( E ) (cid:96) = ( − (cid:96) +1 π / Γ(1 / − (cid:96) )2 (cid:96) − Γ( − (cid:96) − (cid:96) + 2)Γ( (cid:96) + 1)Γ(1 − (cid:96) )Γ(2 (cid:96) + 1)Γ( − (cid:96) ) r (cid:96) +1 s . (4.33)This Wilson coefficient vanishes for physical values of the orbital number (cid:96) ∈ N .Now let us get back to the expression for the electromagnetic response tensor(4.32). As in the scalar case, we see that the electric response tensors are antisym-metric for all (cid:96) ’s as a result of vanishing of the real part of k (1) (cid:96)m . For instance, in thequadrupolar sector we have λ (1) ijkl = − (4 π ) χM
45 12 (cid:104) − χ ) I (1) ij,kl + 5 χ (1 − χ ) I (3) ij,kl + χ I (5) ij,kl (cid:105) = − π γr s I (1) ij,kl + O ( γ ) = − (4 π ) 2 χM I (1) ij,kl + O ( χ ) , (4.34)– 33 –here we used the dimensionless spin χ = a/M and took the γ → limit in the lastline. The STF basis tensors I (1) , I (3) , I (5) are defined in Eqs. (3.74,3.75). We see thatjust like in the scalar case, the local electromagnetic worldline EFT couplings vanisheven though the imaginary electric response coefficients do not. We conclude thatthe spin-1 response is purely dissipative. For completeness, in this section we present the computation of the static responseof Kerr black hole to the external gravitational perturbation. This calculation hasbeen discussed in detail in Refs. [25, 26], and some important technical results werepreviously obtained in Refs. [44, 69]. Our main novel result here will be an explicitmatching of the spin-2 Kerr black hole response coefficients to the worldline EFTWilson coefficients along the lines of the previous sections.
The local worldline EFT of gravitational perturbations isbuilt out of various operators constructed from the Weyl tensor [5, 7]. In four di-mensions this tensor has two distinctive components, E µσ = C µνσρ v ν v ρ , B µνσ = P µ (cid:48) µ P ν (cid:48) ν P σ (cid:48) σ C ρµ (cid:48) ν (cid:48) σ (cid:48) v ρ . (5.1)In the body’s rest frame these components reduce to E (2) ij ≡ C i j , B (2) ijk ≡ C ijk (5.2)Note that magnetic tensor can also be dualized as B µσ = (cid:15) µαβν C αβσρ v ν v ρ , but herewe will not do that in order to match the convention of Ref. [7]. The most genericquadratic action for E (2) ij and B (2) ijk is given by S gravEFT = S pp + (cid:90) d x h D h + (cid:88) (cid:96) =2 (cid:96) ! (cid:90) d x (cid:90) dτ δ (4) ( x − x ( τ )) λ (2) L (cid:48) L ∂ (cid:104) i ...∂ i (cid:96) − E (2) i (cid:96) − i (cid:96) (cid:105) ∂ (cid:104) i (cid:48) ...∂ i (cid:48) (cid:96) − E (2) i (cid:48) (cid:96) − i (cid:48) (cid:96) (cid:105) + (cid:88) (cid:96) =2 (cid:96) ! (cid:90) d x (cid:90) dτ δ (4) ( x − x ( τ ))˜ λ (2) L (cid:48) L ∂ (cid:104) i ...∂ i (cid:96) − B (2) i (cid:96) − i (cid:96) (cid:105) j ∂ (cid:104) i (cid:48) ...∂ i (cid:48) (cid:96) − B (2) i (cid:48) (cid:96) − i (cid:48) (cid:96) (cid:105) j , (5.3)where (cid:82) d x h D h denotes the graviton kinetic term, whose explicit expression canbe found e.g. in Refs. [7, 55]. The tensorial Wilson coefficients λ (2) L (cid:48) L and ˜ λ (2) L (cid:48) L willbe referred to as the electric and magnetic spin-2 Love tensors, respectively.– 34 – esponse coefficients in the Newtonian limit. The general spin-2 tidal re-sponse coefficients can be related to the harmonic expansion of the Newtonian po-tential in the large distance limit. Their calculation relies on the curvature Weylscalar, defined as ψ ≡ C αβγδ l α m β l γ m δ , (5.4)where C αβγδ is the Weyl tensor projected onto the Newman-Penrose null tetrades [62,63]. In the Newtonian limit the Weyl scalar takes the following form, ψ = − m i m j ∇ i ∇ j U , (5.5)where U is the Newtonian potential, and ∇ i is covariant derivative of the 3-dimensionaleuclidean spacetime. Plugging the expression for the Newtonian potential (2.12) intoEq. (5.5), we find ψ (cid:12)(cid:12)(cid:12) r →∞ = ∞ (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) (cid:115) ( (cid:96) + 2)( (cid:96) + 1) (cid:96) ( (cid:96) − r (cid:96) − E (cid:96)m (cid:20) k (cid:96)m (cid:16) r s r (cid:17) (cid:96) +1 (cid:21) +2 Y (cid:96)m ( θ, φ ) , (5.6)where +2 Y (cid:96)m denotes the s = +2 spin-weighted spherical harmonics. In the relativis-tic regime this expression can be generalized as follows [25, 44, 69]: ψ (cid:12)(cid:12)(cid:12) r →∞ = ∞ (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) (cid:115) ( (cid:96) + 2)( (cid:96) + 1) (cid:96) ( (cid:96) − r (cid:96) − (cid:18) E (cid:96)m + i (cid:96) + 13 B (cid:96)m (cid:19) (cid:20) k (cid:96)m (cid:16) r s r (cid:17) (cid:96) +1 (cid:21) +2 Y (cid:96)m ( θ, φ ) , (5.7)where E (cid:96)m , B (cid:96)m are the spherical harmonic coefficients of the electric-type and magnetic-type tidal tensors, defined by means of the Weyl tensor as follows: E L ≡ (cid:96) − ∇ (cid:104) i ...i (cid:96) C | i | | i (cid:105) , B L ≡ (cid:96) − (cid:96) + 1)! ∇ (cid:104) i ...i (cid:96) (cid:15) jk | i C i | jk (cid:105) . (5.8)The electric-type tidal tensor is a relativistic generalization of the Newtonian tidaltensor discussed in Section 2. Note that the response coefficients are the same formagnetic-type and electric-type perturbations as a consequence of the gravitationalelectric-magnetic duality, which takes place for fluctuations around Kerr black holesin four dimensions [41]. Eq. (5.7) can be used to extract the Newtonian response coefficients from the fullgeneral relativity calculation. Indeed, the Weyl scalar ψ factorizes in the Kerrbackground as [25] ψ = ∞ (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) (cid:115) ( (cid:96) + 2)( (cid:96) + 1) (cid:96) ( (cid:96) − (cid:18) E (cid:96)m + i (cid:96) + 13 B (cid:96)m (cid:19) R s =+2 (cid:96)m ( r ) +2 Y (cid:96)m ( θ, φ ) , (5.9)– 35 –here the radial function R s =+2 (cid:96)m satisfies the following differential equation (cid:20)(cid:18) − (cid:96) − (cid:96) + 6 + γm ( γm + i x + 1)) x ( x + 1) (cid:19) + 3(2 x + 1) ddx + x ( x + 1) d dx (cid:21) R s =+2 (cid:96)m ( x ) = 0 . (5.10)The solution smooth at the black hole horizon must satisfy the following boundarycondition in the Boyer-Lindquist coordinates [50], R s =+2 (cid:96)m = const · x − imγ as r → r + ( x → . (5.11)The desired radial function is given by R s =+2 (cid:96)m = const · (1 + x ) − imγ − x imγ − F ( − (cid:96) − , (cid:96) − − iγm ; − x ) . (5.12)Taylor-expanding this function at spatial infinity we find R s =+2 (cid:96)m = x (cid:96) − r (cid:96) − sa (cid:18) x − (cid:96) − Γ( − (cid:96) − (cid:96) − (cid:96) + 2 γim + 1)Γ( − (cid:96) − iγm − (cid:96) )Γ(2 (cid:96) + 1) (cid:19) , (5.13)which provides us with the following gravitational response coefficients k (cid:96)m ≡ k (2) (cid:96)m = Γ( − (cid:96) − (cid:96) − (cid:96) + 2 γim + 1)Γ( − (cid:96) − iγm − (cid:96) )Γ(2 (cid:96) + 1) (cid:18) r sa r s (cid:19) (cid:96) +1 . (5.14)Note that this expression coincides with Eqs. (4.29, 4.40) of Ref. [25]. For the physicalcase (cid:96) ∈ N we have k (2) (cid:96)m = − imγ ( (cid:96) − (cid:96) + 2)!(2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n + 4 m γ ) (cid:18) r sa r s (cid:19) (cid:96) +1 = − imχ (cid:96) − (cid:96) + 2)!(2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n (1 − χ ) + m χ ) , (5.15)where in the last line we expressed the result in terms of the dimensionless black holespin χ = a/M . Importantly, the spin-2 response coefficients are purely imaginaryjust like their spin-0 and spin-1 counterparts. This means that the tidal spin-2 Lovetensors must vanish identically. We will focus on the electric part of the spin-2 perturbations captured by E (2) ij ≡ C i j in what follows. The calculation of the magnetic part can be carried out in a similarfashion. To match the electric-type Love numbers, it is sufficient to consider onlythe following scalar graviton modes g = − h , g ij = δ ij (1 + 2˜ h ) , (5.16)– 36 –hich corresponds to Newtonian gauge. In this gauge the gravity kinetic term takesthe following form (cid:90) d x h D h = 116 π (cid:90) d x (cid:104) h ∆˜ h − h ∆˜ h (cid:105) . (5.17)The field ˜ h does not appear in S pp at zeroth order in particle’s displacement fromthe center of mass position. Thus, in this approximation it can be integrated outfrom the action by means of its equation of motion ˜ h = h , which gives us (cid:90) d x h D h = 18 π (cid:90) d xh ∆ h . (5.18)Then the electric part of the Weyl tensor takes the following form E (2) ij = − ∂ i ∂ j h . (5.19)All in all, in the static limit the EFT takes the same form as the EFT for a scalarfield, modulo a factor π in the graviton kinetic term S gravEFT = S pp + 18 π (cid:90) d x h ∆ h + (cid:88) (cid:96) =2 (cid:96) ! (cid:90) d x (cid:90) dτ δ (4) ( x − x ( τ )) λ (2) i ...i (cid:96) i (cid:48) ...i (cid:48) (cid:96) ( ∂ (cid:104) i ...∂ i (cid:96) (cid:105) h )( ∂ (cid:104) i (cid:48) ...∂ i (cid:48) (cid:96) (cid:105) h ) , (5.20)where S pp is the standard point-particle action (3.8). Repeating the scalar fieldcalculation for a fixed multipolar index (cid:96) , we can easily obtain the following staticresponse h = (cid:96) (cid:88) m = − (cid:96) ¯ E (cid:96)m r (cid:96) Y (cid:96)m + λ (2) LL (cid:48) n L (cid:48) ¯ E L · ( − (cid:96) π (cid:96) − π / Γ(1 / − (cid:96) ) r − (cid:96) − , (5.21)where ¯ E (cid:96)m are spherical modes of the background source and ¯ E L is the correspondingconstant STF tensor. In order to be rigorous and ensure that the result for responsecoefficients is gauge-independent, we need to match gauge invariant quantities fromboth sides. The simplest such quantity is the rr component of the electric part ofthe Weyl tensor [7], C r r = E (2) rr = − ∂ r h . (5.22)Taking two derivatives w.r.t. r in Eq. (5.21) and in the formula for the Newtonianpotential (2.12), and matching the two expressions we obtain λ (2) i (cid:48) ...i (cid:48) (cid:96) i ...i (cid:96) = 2 r (cid:96) +1 s πB (cid:96) π(cid:96) !(2 (cid:96) + 1)!! (cid:96) (cid:88) m = − (cid:96) k (2) (cid:96)m Y ∗ i ...i (cid:96) (cid:96)m Y i (cid:48) ...i (cid:48) (cid:96) (cid:96)m , (5.23)– 37 –here k (2) (cid:96)m are given in Eq. (5.15), B (cid:96) a constant is given in Eq. (3.31). Plugging γ = 0 , we find that expression (5.23) coincides with the Love numberfor the Schwarzschild black holes given in Eq. (5.50) of Ref. [7] for generic (cid:96) ∈ R (uponidentification ˆ L → (cid:96), D → , and modulo the conventional factor π ). However, thisexpression vanishes in the physical case (cid:96) ∈ N / { } , λ (2) i (cid:48) ...i (cid:48) (cid:96) i ...i (cid:96) = λ ( C E ) (cid:96) δ (cid:104) i (cid:48) ...i (cid:48) (cid:96) (cid:105)(cid:104) i ...i (cid:96) (cid:105) ,λ ( C E ) (cid:96) = 28 π ( − (cid:96) π / Γ(1 / − (cid:96) )2 (cid:96) − Γ( − (cid:96) − (cid:96) − (cid:96) + 1)Γ( − (cid:96) − − (cid:96) )Γ(2 (cid:96) + 1) r (cid:96) +1 s = 0 if (cid:96) ∈ N / { } . (5.24)Using explicit formulas for the Thorne tensors from Appendix A, we obtain thefollowing expression for the EFT quadrupolar worldline tensor coupling in terms ofthe dimensionless spin parameter χ = a/M : λ (2) ijkl = − χM (cid:104) − χ ) I (1) ij,kl + 5 χ (1 − χ ) I (3) ij,kl + χ I (5) ij,kl (cid:105) , (5.25)where the STF basis tensors I (1) , I (3) , I (5) are defined in Eqs. (3.74,3.75). For smallspin this expression simplifies as, λ (2) ijkl = − π π ) γr s I (1) ij,kl + O ( χ ) = − χM I (1) ij,kl + O ( χ ) . (5.26)Just like in the case of spin-0 and spin-1 perturbations, the worldline finite-sizeoperators vanish even though the Newtonian response coefficients do not. Note thatour expressions for the response matrices (5.25,5.26) coincide with those presentedin Le Tiec et al. (2020) [25] and those obtained in Goldberger et al. (2020) [27] (upto a conventional numerical factor). The antisymmetric response captured by thesematrices is responsible for the dissipative effect of tidal torques. In this section we demonstrate that black hole’s response coefficients for any per-turbing boson field can be extracted directly from Teukolsky equations for relevantNewman-Penrose scalars. Then we will present the response coefficients for time-dependent perturbations. Note that the same result can be obtained by a direct matching of h from (5.21) and theNewtonian potential (2.12) because our choice of Newtonian gauge (5.16) is precisely the one thatreproduces the Newtonian limit at large distances. – 38 – .1 Static Responses An important observation is that the Kerr black hole response coefficients for allfields can be extracted directly from the solution to the radial Teukolsky equationfor a generic spin weight s [50], (cid:34) s + s − (cid:96) − (cid:96) + ( γm ) + iγms (2 x + 1) x ( x + 1)+ ( s + 1)(2 x + 1) ddx + x ( x + 1) d dx (cid:35) R ( x ) = 0 . (6.1)This solution needs to satisfy the following boundary condition at the future horizon R = const × ( r − r + ) iγm − s , as r → r + , (6.2)which ensures that the in-falling observes sees only the so-called “non-special” fields(= fields that are not singular and not identically equal to zero). Moreover, theseboundary condition guarantees that the energy momentum flux flows strictly intothe black hole [49, 50, 52]. The desired solution can be easily constructed [70, 71], R = const · (1 + x ) − imγ − s x imγ − s F ( − (cid:96) − s, (cid:96) + 1 − s ; 1 + 2 iγm − s ; − x ) . (6.3)Taylor-expanding this solution at spatial infinity x → ∞ (see Appendix A for therelevant analytic continuation formula) we find R = const · x (cid:96) − s r (cid:96)sa (cid:18) x − (cid:96) − Γ( − (cid:96) − (cid:96) − s + 1)Γ( (cid:96) + 2 γim + 1)Γ( − (cid:96) − s )Γ(2 iγm − (cid:96) )Γ(2 (cid:96) + 1) (cid:19) , (6.4)which provides us with the following dimensionless response coefficients k ( s ) (cid:96)m = Γ( − (cid:96) − (cid:96) − s + 1)Γ( (cid:96) + 2 γim + 1)Γ( − (cid:96) − s )Γ(2 iγm − (cid:96) )Γ(2 (cid:96) + 1) (cid:18) r sa r s (cid:19) (cid:96) +1 = ( − s +1 imχ (cid:96) + s )!( (cid:96) − s )!(2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n (1 − χ ) + m χ ) , (6.5)where χ = J/M is black hole’s dimensionless spin, and r s = 2 M is the Schwarzschildradius. This expression recovers the scalar response coefficients for s = 0 (seeEq. (3.54)), the spin-1 (electromagnetic) response coefficients for s = 1 (see Eq. (4.27)),and the spin-2 (gravitational) response coefficients for s = 2 (see Eq. (5.14)). Using the Teukolsky equation, we can also obtain expressions for non-static responsesto all orders in frequency, which we present here for completeness. Let us consider ageneric field ψ of spin weight s that factorizes in the Kerr background as [49, 50] ψ = e − iωt + imφ R ( r ) S ( θ ) . (6.6)– 39 –he functions R and S satisfy the following frequency-dependent equations (cid:34) ( ω ( a + r ) − am ) ( ω ( a + r ) − am + 2 is ( M − r )) a + r ( r − M ) − a ω + 2 amω − A + 4 irsω + (cid:0) a + r ( r − M ) (cid:1) d dr + 2( s + 1)( r − M ) ddr (cid:35) R ( r ) = 0 , θ ddθ (cid:18) sin θ dS ( θ ) dθ (cid:19) + (cid:32) a ω cos θ − m sin θ − aωs cos θ − ms cos θ sin θ − s cot θ + s + A − a ω − amω (cid:33) S ( θ ) = 0 , (6.7)where A denotes angular eigenvalues. For small aω they are given by A = ( (cid:96) − s )( (cid:96) + s + 1) − aω ms (cid:96) ( (cid:96) + 1) + O ( a ω ) . (6.8)This expression provides a rational behind the the analytic continuation (cid:96) → R : theangular eigenvalues are actually always non-integer for non-zero frequencies [60, 70,71]. The purely incoming boundary condition for R at the black hole horizon has thefollowing form in the Boyer-Lindquist coordinates [50, 70] R = const × ( r − r + ) iQ − s , as r → r + , (6.9)where we have introduced Q ≡ γm − M r + r + − r − ω = am − M r + ωr + − r − . (6.10)The solution of the Teukolsky equation at finite frequency that satisfies the purelyincoming boundary condition at the horizon can be obtained in the form of a seriesover hypergeometric functions, see Refs. [72–75], R = e − i (cid:15)x √ − χ x − s − i(cid:15)/ − i ˜ Q (1 + x ) i(cid:15)/ i ˜ Q (cid:32) ∞ (cid:88) n = −∞ a νn Γ(1 − s + 2 iQ )Γ(2 n + 2 ν + 1)Γ( n + ν + 1 + 2 i ˜ Q )Γ( n + ν + 1 − s − i(cid:15) ) × x ν + n F ( − n − ν + 2 i ˜ Q, − n − ν + s + i(cid:15), − n − ν, − x − )+ ∞ (cid:88) n = −∞ a − ν − n Γ(1 − s + 2 iQ )Γ(2 n − ν − n − ν + 2 i ˜ Q )Γ( n − ν − s − i(cid:15) ) × x − ν − n F ( − n + ν + 1 + 2 i ˜ Q, − n + ν + 1 + s + i(cid:15), ν + 2 − n, − x − ) (cid:33) , (6.11)– 40 –here we have used Q = ˜ Q − (cid:15)/ along with (cid:15) ≡ r s ω , ˜ Q = mχ − r s ω (cid:112) − χ , ν = (cid:96) + ∆ (cid:96) , ∆ (cid:96) = (cid:15) (cid:96) + 1 (cid:34) − − s (cid:96) ( (cid:96) + 1) + (( (cid:96) + 1) − s ) (2 (cid:96) + 1)(2 (cid:96) + 2)(2 (cid:96) + 3) − ( (cid:96) − s ) (2 (cid:96) − (cid:96) (2 (cid:96) + 1) (cid:35) + O ( (cid:15) ) , (6.12)The quantity ν is called “renormalized angular momentum.” The coefficients a νn and a − ν − n satisfy certain recursion relations that are given in Refs. [72–75]. In generalthey depend on (cid:15) parametrically and they are suppressed in the low-frequency limit,e.g. a νn = O ( (cid:15) | n | ) for n ≥ − (cid:96) . In what follows we will extract the part of the solutionthat has the desired source and response asymptotics at large distances. We will workat linear order in (cid:15) = r s ω , in which case we will need only the following coefficients, a − ν − = a ν = 1 , a ν − = a − ν − = i(cid:15) ( (cid:96) + s ) ( (cid:96) − i ˜ Q )2 (cid:96) (2 (cid:96) + 1) (cid:112) − χ + O ( (cid:15) ) ,a ν = a − ν − − = i(cid:15) ( (cid:96) − s + 1) ( (cid:96) + 1 + 2 i ˜ Q )2( (cid:96) + 1) (2 (cid:96) + 1) (cid:112) − χ + O ( (cid:15) ) . (6.13)The relevant solution that scales as r ν − s (1 + O ( r − ν − )) at large distances in thesmall-frequency limit is given by: R (cid:12)(cid:12)(cid:12) r →∞ ⊃ const × e − i ωr − χ r ν − s (cid:32) κ ( s ) νm (cid:16) r sa r (cid:17) (cid:96) +1 (cid:33)(cid:34) Γ(2 ν + 1)Γ( ν + 1 + 2 i ˜ Q )Γ( ν + 1 − s − i(cid:15) ) − a ν Γ(2 ν + 3)( − − ν + 2 i ˜ Q )( − − ν + s + i(cid:15) )Γ( ν + 2 + 2 i ˜ Q )Γ( ν + 2 − s − i(cid:15) )( − ν − − a ν ( ν + 1) r + r sa Γ(2 ν + 3)Γ( ν + 2 + 2 i ˜ Q )Γ( ν + 2 − s − i(cid:15) ) (cid:35) + O ( (cid:15) ) , (6.14)where we have used the frequency-dependent response coefficient k ( s ) νm ≡ κ ( s ) νm ( r sa /r s ) (cid:96) +1 , κ ( s ) νm ≡ Γ( ν − s + 1 − i(cid:15) )Γ( ν + 2 i ˜ Q + 1)Γ( − ν − ν + 1)Γ( − ν − s − i(cid:15) )Γ(2 i ˜ Q − ν ) × (cid:32) − a − ν − ( ν + 2 i ˜ Q )(2 ν + 1)( ν − i ˜ Q ) − a ν (2 ν + 1)(1 + ν − i ˜ Q )( ν + 1 + 2 i ˜ Q )+ a ν r + r sa ν + 1) (2 ν + 1)( ν + 1 + 2 i ˜ Q )( ν + 1 − s ) + a − ν − r + r sa ν (2 ν + 1)( ν − i ˜ Q )( ν + s ) (cid:33) + O ( (cid:15) ) . (6.15)– 41 –or ω = 0 the response coefficient that appears in Eq. (6.15) reduces to Eq. (6.5).We will expand now this coefficient to linear order in (cid:15) , while retaining black hole’sspin to all orders. The (cid:15) → limit of Eq. (6.15) is complicated by the presenceof a pole in the gamma functions. The ambiguity associated with this pole can beeliminated if we formally consider ∆ (cid:96) and (cid:15) as independent parameters, and use theexpression Eq. (6.12) only after regularizing the singularity. The presence of the polealso generates a finite logarithmic contribution, see Appendix D. We obtain κ ( s ) , finite νm = (cid:34) iγm sinh(2 πγm ) sinh (cid:26) π r s r + r sa ( ω − m Ω) (cid:27) − r s ω ) γm ln x − maω (2 (cid:96) + 1)2( (cid:96) + 1) (cid:96) (cid:32) (cid:96) + (cid:96) + s + 2 imγ (cid:18) s + s (cid:96) ( (cid:96) + 1) (cid:19) (cid:33) + (2 (cid:96) + 1) γmr + ω (cid:35) × ( − s ( (cid:96) + s )!( (cid:96) − s )!(2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n + 4( γm ) ) + O (( r s ω ) , ( r s ω ) ln x ) , (6.16)where Ω is the black hole’s angular velocity Ω ≡ a/ ( r + a ) = a/ (2 M r + ) , and it isuseful to recall that x = ( r − r + ) /r sa . The first important observation is that if weexpand the response coefficient at leading order in black hole’s spin and frequencyof the external perturbation, we will find that Eq. (6.16) matches the Newtonianexpression (2.10) with the vanishing static Love number λ (cid:96) = 0 , but a non-zerodissipative part, k ( s ) νm = ir s ( ω − m Ω) ( − s ( (cid:96) + s )!( (cid:96) − s )!( (cid:96) !) (2 (cid:96) )!(2 (cid:96) + 1)! + O ( ω Ω , ω , Ω ) . (6.17)The dissipative imaginary response part vanishes for the locking frequency ω = m Ω . Therefore, at leading order in black hole’s spin and frequency of the externalperturbation the Kerr black holes behave like rigidly rotating dissipative spheres.The second important observation is that generically the conservative responsecoefficients O ( m Ω) is not zero. Indeed, at face value, Eq. (6.16) implies that thefollowing time-dependent worldline operator does not vanish (cid:90) dτ E L ˙ E L (cid:48) Λ ( ω Ω) LL (cid:48) , (6.18)where the spin-dependent coefficient Λ ( ω Ω) LL (cid:48) is odd w.r.t. time reversal. The thirdimportant observation is that for non-zero ω the tidal response coefficients exhibitclassical renormalization-group running. This means that only the logarithmic partof the conservative frequency-dependent Love number appearing in Eq. (6.16) isuniversal and independent of the renormalization scheme. This situation can becontrasted with the ω = 0 Love numbers, which do not receive any logarithmiccontributions and hence do not run with distance [6, 76]. Fourth, Eq. (6.11) can This is true in four dimensions. In certain spacetime dimensions the static Love numbers alsoexhibit renormalization-group running [7, 11]. – 42 –e used to extract the Love numbers that depend on the the frequency squared.They will be interesting to compare with recent results on frequency-dependent Lovenumbers given in Ref. [28]. We leave this question for future work.Finally, let us comment on the near-field approximation, which has been recentlyused to compute Love numbers from the Teukolsky equation [26]. We present thiscalculation for a generic spin s perturbation in Appendix D. There we show that thenear-field expansion does not exactly map onto the small frequency expansion. As aresult, the leading order near-field approximation does not fully capture the O ( ω Ω) corrections to the tidal response coefficients. In this work we have computed the static response of Kerr black holes to externalelectromagnetic and scalar perturbations in four dimensions. This complements theanalysis of Refs. [25, 26], which have calculated the response of Kerr black holesto spin-2 (gravitational) perturbations. Our main results are summarized in themaster formula (6.5), which displays the Kerr black hole static response coefficientsfor a perturbing field with generic integer spin s . Importantly, all responses arepurely dissipative, i.e. the Love numbers for spinning black holes identically vanishfor spin-0, spin-1, and spin-2 fields to all orders in black hole spin. We have alsoextended our results to leading frequency-dependent effects, which also include therunning of response coefficients, see Eq. (6.16).We have used the gauge-invariant definition of Love numbers as Wilson coeffi-cients in the point-particle effective field theory (EFT). To that end we have intro-duced local finite-size operators in the EFT and have extracted the relevant Wilsoncoefficients by matching the EFT and full GR calculations. We have also shownthat the EFT allows one to clearly separate dissipative and conservative responses.The key ingredient of our matching procedure is the analytic continuation of relevantstatic response solutions to non-integer values of the orbital multipole number (cid:96) . Wehave explicatively shown that this procedure allows one to extract the response co-efficients from full general relativity solutions in a coordinate-independent fashion.Moreover, we have interpreted this procedure in the EFT context and have shownthat it helps to separate non-linear gravity corrections to perturbing sources (i.e.source-graviton EFT diagrams) from corrections generated by the induced multipolemoments. Curiously, we have found that the subleading source corrections exactlycancel the response part in the advanced Kerr coordinates. It will be interesting tounderstand the origin of this cancellation in the future.We have demonstrated that spinning black holes are very similar to the staticones from the EFT point of view: both of them can be described with a single point-particle term in the worldline action in the static limit. Our analysis suggests thatthe case of spinning black holes may be useful to understand the vanishing of tidal– 43 –ove numbers in four dimensions and a possible EFT naturalness problem related tothat. In particular, we have shown that this problem may be addressed at the levelof the massless scalar field. This is a very simplistic model, yet it captures manyqualitative details of Love number calculations relevant for both static and spinningblack holes. This suggests that the scalar field toy model may play an importantrole in elucidating the nature of vanishing of local finite-size EFT operators for blackholes.Our analysis can be extended in multiple ways. First, one can compute Lovenumbers and the relevant Wilson coefficients of the point-particle EFT for spinningblack holes in spacetime dimensions greater than four. The properties of higherdimensional spinning black holes are known to differ significantly from their fourdimensional counterparts (see e.g. [77, 78]), and hence we can expect interesting con-sequences for response coefficients there. Second, one can study the relationship be-tween dissipative spin-0 and spin-1 response coefficients that we have computed andthe phenomenon of black hole torques along the lines of [25, 27, 54]. Third, it wouldbe interesting to compute the Love coefficients for charged spinning Kerr–Newmanblack holes [79]. Fourth, one can carry out a systematic analysis of the frequency-dependent Love numbers for spinning black holes. Eventually, it will be important tounderstand if there is an extra symmetry of the Schwarzschild and Kerr spacetimeswhich makes the conservative static black hole response vanish in four dimensions.We leave these research directions for future work. Note added.
While this paper was being prepared, Refs. [26–28] appeared. Thesepapers have some overlap with our work in the interpretation of dissipative responsecoefficients. In particular, we have independently obtained that the response co-efficients presented as “Love numbers” in Ref. [25] actually correspond to purelynon-conservative effects.
Acknowledgments
This work is supported in part by the NSF award PHY-1915219 and by the BSF grant 2018068. MI is partially supported by the SimonsFoundation’s Origins of the Universe program.
A Useful Mathematical Relations
A.1 Spherical HarmonicsScalar Spherical Harmonics.
We use the following definition for the (scalar)spherical harmonics Y (cid:96)m ( θ, φ ) = ( − (cid:96) + | m | + m (cid:96) (cid:96) ! (cid:20) (cid:96) + 14 π ( (cid:96) − | m | )!( (cid:96) + | m | )! (cid:21) / e imφ (sin θ ) | m | (cid:18) dd cos θ (cid:19) (cid:96) + | m | (sin θ ) (cid:96) , (A.1)– 44 –alid for (cid:96) ≥ , − (cid:96) < m < (cid:96) . These harmonics obey the following relations ∆ S Y (cid:96)m = − (cid:96) ( (cid:96) + 1) Y (cid:96)m , Y ∗ (cid:96)m ( n ) = ( − m Y (cid:96) ( − m ) ( n ) , (cid:73) S d Ω Y (cid:96)m Y ∗ (cid:96) (cid:48) m (cid:48) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) , (A.2)where ∆ S is the two-sphere Laplacian. Spin-Weighted Spherical Harmonics.
One can introduce the following spin s -raising and spin s -lowering operators, ð ≡ − (cid:18) ∂ θ + i sin θ ∂ φ − s cos θ sin θ (cid:19) , ¯ ð ≡ − (cid:18) ∂ θ − i sin θ ∂ φ + s cos θ sin θ (cid:19) . (A.3)Applying these operators on the the (scalar) spherical harmonics Y (cid:96)m ≡ Y (cid:96)m one candefine the spin-weighted spherical harmonics for (cid:96) ≥ | s | , ð ( s Y (cid:96)m ) = + (cid:112) ( (cid:96) − s )( (cid:96) + s + 1) s +1 Y (cid:96)m , ¯ ð ( s Y (cid:96)m ) = − (cid:112) ( (cid:96) + s )( (cid:96) − s + 1) s − Y (cid:96)m . (A.4)These harmonics obey the following relations s Y ∗ (cid:96)m ( n ) = ( − m + s − s Y (cid:96) ( − m ) ( n ) , (cid:73) S d Ω s Y (cid:96)ms Y ∗ (cid:96) (cid:48) m (cid:48) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) . (A.5) Transverse Vector Spherical Harmonics.
The transverse vector analog of scalarspherical harmonics are defined as follows (cid:126)Y
T(cid:96)m ≡ − (cid:112) (cid:96) ( (cid:96) + 1) (cid:126)r × (cid:126) ∇ Y (cid:96)m , Y Ti(cid:96)m = − √ det g ε ijk (cid:112) (cid:96) ( (cid:96) + 1) x j ∇ k Y (cid:96)m , (A.6)where g is the 3d metric, ε ijk is the three-dimensional Levi-Civita symbol ( ε = 1 )and ∇ k is the corresponding covariant derivative. Note that there is a sign differencebetween our definition and the one adopted in Ref. [42]. We also stress that ε ijk denotes the fully-antisymmetric symbol , whereas (cid:15) ijk stands for the anti-symmetric tensor , (cid:15) ijk = √ g ε ijk , (cid:15) ijk = ε ijk √ g . (A.7)The transverse spin-1 spherical harmonics are related to the spin-weighted sphericalharmonics through Y Ti(cid:96)m = i √ − Y (cid:96)m m i + +1 Y (cid:96)m m ∗ i ) . (A.8)Since (cid:126)Y T(cid:96)m are orthogonal to the radial direction n i , it is also convenient to use theirprojections onto S , known as the Regge-Wheeler (RW) vector spherical harmon-ics [64, 80] Y RW a (cid:96)m ≡ (cid:112) (cid:96) ( (cid:96) + 1) √ g ε ab g bc ∇ c Y (cid:96)m , (A.9)– 45 –here a = ( θ, φ ) , g ab is the metric tensor on S , g ≡ det g , ∇ a is the covariantderivative on S , and we have introduced the flat-space 2-dimensional Levi-Civitasymbol ε θφ = − ε φθ = 1 , ε θθ = ε φφ = 0 . (A.10)In our conventions the RW and the vector harmonics defined in Eq. (A.6) coincide inthe orthonormal spherical coordinates basis. The 2d transverse spherical harmonicssatisfy [7, 80]: ∆ S Y RW a(cid:96)m = − ( (cid:96) ( (cid:96) + 1) − Y RW a(cid:96)m , (cid:73) S d Ω g ab Y RW a(cid:96)m Y RW ∗ b(cid:96) (cid:48) m (cid:48) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) . (A.11) Symmetric Trace-Free Tensors.
Finally, instead of the spherical harmonics itmay be conveneient to use the basis of the symmetric trace-free tensors of rank (cid:96) (“STF- (cid:96) tensors”) [42]. These tensors generate an irreducible representation of SO (3) and hence there exists a one-to-one mapping between them and the sphericalharmoncis. This mapping is realized via Y (cid:96)m = Y ∗ L(cid:96)m n (cid:104) L (cid:105) , or n (cid:104) L (cid:105) = 4 π(cid:96) !(2 (cid:96) + 1)!! (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m Y L(cid:96)m , (A.12)where the constant STF tensors Y L(cid:96)m satisfy Y L(cid:96)m = (2 (cid:96) + 1)!!4 π(cid:96) ! (cid:73) S d Ω n (cid:104) L (cid:105) Y ∗ (cid:96)m , Y L(cid:96) ( − m ) = ( − m Y ∗ L(cid:96)m . (A.13)Since Y L(cid:96)m tensors form a basis for the ( (cid:96) + 1 ) dimensional vector space of the STFtensors on S , any STF tensor can be expanded over them as F L = (cid:96) (cid:88) m = − (cid:96) Y ∗ L(cid:96)m F (cid:96)m , F (cid:96)m = 4 π(cid:96) !(2 (cid:96) + 1)!! Y L(cid:96)m F L . (A.14)All in all, any scalar function on S can be represented as F ( θ, φ ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) F (cid:96)m Y (cid:96)m = ∞ (cid:88) (cid:96) =0 F L n (cid:104) L (cid:105) . (A.15)Some other important identities are ε jpq Y ∗ (cid:96)mp ( L − Y (cid:96)mq ( L − = − im (2 (cid:96) + 1)!!4 π(cid:96) ! (cid:96) ˆ ξ j ,ε jpq Y ∗ (cid:96)mp ( L − Y (cid:96) ( m +1) q ( L − = − i (2 (cid:96) + 1)!!4 π(cid:96) !2 (cid:96) [2( (cid:96) − m )( (cid:96) + m + 1)] / ˆ ξ − j ,ε jpq Y ∗ (cid:96)mp ( L − Y (cid:96) ( m − q ( L − = i (2 (cid:96) + 1)!!4 π(cid:96) !2 (cid:96) [2( (cid:96) + m )( (cid:96) − m + 1)] / ˆ ξ +1 j ,ε jpq Y ∗ (cid:96)mp ( L − Y (cid:96) ( m + µ ) q ( L − = 0 if µ (cid:54) = 0 or ± , (A.16)where ˆ ξ j = δ j , ˆ ξ ± j = ∓ ( δ j ± iδ j ) / √ . – 46 – .2 Gamma Function The Euler Gamma funciton is defined via Γ( x + 1) = Γ( x ) x . (A.17)We use several important relations in the main text | Γ(1 + (cid:96) + bi ) | = πb sinh( πb ) (cid:96) (cid:89) n =1 ( n + b ) for (cid:96) ∈ N , (A.18)as well as Γ( z ∗ ) = Γ ∗ ( z ) , and Euler’s reflection formula, Γ( z )Γ(1 − z ) = π sin( πz ) . (A.19)We also need the Taylor expansions of the Gamma function around its poles thatcorrespond to natural values of the orbital number (cid:96) . To obtain them, we shift theargument of the Gamma function as (cid:96) → (cid:96) + ε , ε (cid:28) , which yields (cid:96) + 1)Γ( − (cid:96) ) = − ε , − (cid:96) ) = ( − (cid:96) +1 ( (cid:96) !) ε . (A.20)These expressions lead to the following relations Γ( − (cid:96) − − (cid:96) ) = ( − (cid:96) +1 (cid:96) !2(2 (cid:96) + 1)! , Γ( − (cid:96) − − (cid:96) −
2) = ( − (cid:96) +1 ( (cid:96) + 2)!2(2 (cid:96) + 1)! , Γ( − (cid:96) − − (cid:96) −
1) = ( − (cid:96) ( (cid:96) + 1)!2(2 (cid:96) + 1)! , Γ( − (cid:96) )Γ( − (cid:96) ) = ( − (cid:96) (cid:96) !2(2 (cid:96) )! . (A.21) A.3 Gauss Hypergeometric Function
The classic hypergeometric equation has the following form x (1 − x ) y (cid:48)(cid:48) + ( c − (1 + a + b ) x ) y (cid:48) − aby = 0 . (A.22)If c (cid:54) = 0 , − , − , ... , this equation has the following solution in terms of the Gausshypergeometric function y = F ( a, b, c, x ) ≡ ∞ (cid:88) n =0 Γ( n + a )Γ( n + b )Γ( a )Γ( b ) Γ( c )Γ( n + c ) x n n ! . (A.23)The other indepedent solution of Eq. (A.22) is given by y = x − c F ( b − c + 1 , a − c + 1 , − c, x ) . (A.24)This solution is singular at z = 0 . – 47 –f c = − n , where n = 0 , , , ... , the (regular at x = 0 ) solution to Eq. (A.22)takes the following form y = x n F ( a + n + 1 , b + n + 1 , n + 2 , x ) . (A.25)If a = − n , n = 0 , , , ... and c = − m , m = n, n + 1 , n + 2 , ... the hypergeometricseries truncates at order m . If a + b − c < , the hypergeometric series converges at | x | = 1 . Otherwise it generically converges for | x | < (unless it is a polynomial).The hypergeometric function (A.23) can be analytically continued at x = ∞ via F ( a, b, c, x ) = Γ( c )Γ( b − a )Γ( b )Γ( c − a ) ( − x ) − a F ( a, a + 1 − c, a + 1 − b, x − )+ Γ( c )Γ( a − b )Γ( a )Γ( c − b ) ( − x ) − b F ( b, b + 1 − c, b + 1 − a, x − ) , (A.26)and around x = 1 via F ( a, b, c, x ) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) F ( a, b, a + b + 1 − c, − x )+ Γ( c )Γ( − c + a + b )Γ( a )Γ( b ) (1 − x ) c − a − b F ( c − a, c − b, c + 1 − a − b, − x ) . (A.27) B Calculation of Maxwell-Newman-Penrose Scalars
In this Appendix we compute the stationary electromagnetic field around the Kerrblack hole for a source located at spatial infinity. This is the calculation relevant forthe extraction of the response coefficients. In the Newman-Penrose formalism, theelectromagnetic tensor F µν is represented in terms of 3 complex scalar functions, Φ = F µν l µ m ν , Φ = 12 F µν ( l µ n ν + m ∗ µ m ν ) , Φ = F µν m ∗ µ n ν , (B.1)where l µ , n µ , m µ are the NP null tetrades and m ∗ is the complex conjugate of m µ .In what follows we will use the Boyer-Lindquist coordinates, in which the Kinner-sley tetrades are given by Eq. (4.11). Instead of the usual scalars Φ , Φ , Φ , it isconvenient to work in terms of the rescaled scalars, ˜Φ = Φ , ˜Φ = ( r − ia cos θ ) ( r + − r − ) Φ , ˜Φ = ( r − ia cos θ ) ( r + − r − ) Φ . (B.2)– 48 –he stationary (i.e. ω = 0 ) vacuum Maxwell equations take the following form interms of the NP quantities [49, 50]: √ r sa (cid:16) ∂ r + a ∆ ∂ φ (cid:17) ˜Φ − ( r − ia cos θ ) (cid:18) ∂ θ + cot θ − i sin θ ∂ φ (cid:19) ˜Φ + ia sin θ ˜Φ = 0 , (B.3a) √ r sa (cid:18) ∂ θ + i sin θ ∂ φ (cid:19) ˜Φ + ( r − ia cos θ ) (cid:16) ∂ r − a ∆ ∂ φ (cid:17) ∆ ˜Φ − ∆ ˜Φ = 0 , (B.3b) √ (cid:18) ∂ θ − i sin θ ∂ φ (cid:19) ˜Φ − ( r − ia cos θ ) (cid:16) ∂ r + a ∆ ∂ φ (cid:17) ˜Φ + ˜Φ = 0 , (B.3c) √ (cid:16) ∂ r − a ∆ ∂ φ (cid:17) ˜Φ + ( r − ia cos θ ) (cid:18) ∂ θ + cot θ + i sin θ ∂ φ (cid:19) ˜Φ ∆ − ia sin θ ∆ ˜Φ = 0 . (B.3d)Teukolsky has shown that ˜Φ and ˜Φ factorize in the Kerr background as ˜Φ = (cid:88) (cid:96)m a (cid:96)m R (2) (cid:96)m ( r ) − Y (cid:96)m ( θ, φ ) , ˜Φ = (cid:88) (cid:96)m a (cid:96)m R (0) (cid:96)m ( r ) +1 Y (cid:96)m ( θ, φ ) , (B.4)where the radial harmonic R (2) (cid:96)m satisfies Eq. (4.16). Applying the operator ∂ θ − i∂ φ / sin θ to Eq. (B.3a) and the operator ∂ r + a∂ φ / ∆ to Eq. (B.3c), we find Eq. (4.21), which means we have obtained both ˜Φ and ˜Φ .It is important to express the radial function R (0) (cid:96)m as a second derivative over R (2) (cid:96)m because we will have to integrate over it to get ˜Φ .The calculation of ˜Φ is more intricate as it does not factorize in θ and x . Theaxial symmetry suggests the following ansatz ˜Φ = ∞ (cid:88) m = −∞ (cid:18) x x (cid:19) − iγm e imφ ˜Φ m ( x, θ ) . (B.5)Plugging this into Eq. (B.3a) and integrating over x (which is related to the radialcoordinate r ) we obtain (4.22) plus an integration constant, which corresponds toblack hole’s charge. Since we consider the neutral black holes, we put this constantto zero. C Spin-1 Magnetic Love Numbers
Due to the presence of magnetic-electric duality in four dimensions, we have antici-pated that the electric and magnetic response coefficients would coincide in the Kerr– 49 –ackground. In this Appendix we explicitly show it. To that end, we extract themagnetic response from the Maxwell-Newman-Penrose scalar Φ and match them tothe Wilson coefficients of the magnetic field worldline EFT.Our first goal is to extract the Newtonian response coefficients from the Kerrsolution using Eq. (4.8). We will match one particular component, F θφ . To proceed,we need to simplify the commutator ∇ [ a Y RW b ] (cid:96)m . A crucial observation is that ∇ [ θ Y RW φ ] (cid:96)m = 12 (cid:112) (cid:96) ( (cid:96) + 1) (cid:104) ∇ θ (cid:16)(cid:112) det g ε φθ g θθ ∇ θ (cid:17) − ∇ φ (cid:16)(cid:112) det g ε θφ g φφ ∇ φ (cid:17)(cid:105) Y (cid:96)m = 12 (cid:112) (cid:96) ( (cid:96) + 1) (cid:104) − (cid:112) det g ∆ S (cid:105) Y (cid:96)m = sin θ (cid:112) (cid:96) ( (cid:96) + 1) Y (cid:96)m . (C.1)Thus, we have F θφ = (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) sin θY (cid:96)m β (cid:96)m r (cid:96) (cid:34) − (cid:96) + 1 (cid:96) ˜ k (1) (cid:96)m (cid:18) rr s (cid:19) − (cid:96) − (cid:35) . (C.2)To extract the magnetic Love numbers, we need to compare this expression with ourformula for F θφ that we have obtained by solving for Φ . We have F θφ (cid:12)(cid:12)(cid:12) r →∞ = 2 Im Φ r sin θ = 2 r sin θ √ r sa r Im (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m Y (cid:96)m ( (cid:96) ( (cid:96) + 1)) / (cid:20) x ddx ( y (cid:96)m ) − y (cid:96)m (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =( r − r + ) /r sa = sin θ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) β (cid:96)m Y (cid:96)m F θφ(cid:96)m ( r ) , (C.3)where in the last line we have used Eq. (4.24) and we have also introduced the newfunction F θφ(cid:96)m = r (cid:96) +1 sa x (cid:96) +1 · (cid:20) − x − (cid:96) − (cid:96) + 1 (cid:96) Γ( − (cid:96) − (cid:96) + 2)Γ( (cid:96) + 2 imγ + 1)Γ(1 − (cid:96) )Γ(2 (cid:96) + 1)Γ(2 imγ − (cid:96) ) (cid:21) . (C.4)Note that the sum (cid:80) (cid:96)m β (cid:96)m Y (cid:96)m ( xy (cid:48) (cid:96)m − y (cid:96)m ) is real.Matching this with Eq. (C.2) we obtain that the magnetic and electric responsecoefficients coincide in the Kerr background ˜ k (1) (cid:96)m = k (1) (cid:96)m . (C.5) It can be shown that in general ∇ [ a √ g ε b ] c ∇ c = −√ g ε ab ∇ . – 50 – atching to the EFT can be done using the EFT electromagnetic action Eq. (4.1).In analogy with the electric field we introduce an external background magnetic fieldsource as ¯ A j = r (cid:96) (cid:112) (cid:96) ( (cid:96) + 1) (cid:96) (cid:88) m = − (cid:96) ¯ β (cid:96)m Y Tj(cid:96)m = − √ g ε jik x i ∇ k (cid:96) ( (cid:96) + 1) (cid:96) (cid:88) m = − (cid:96) r (cid:96) ¯ β (cid:96)m Y (cid:96)m = − √ g ε jik x i ∇ k (cid:96) ( (cid:96) + 1) ¯ β i ...i (cid:96) x i ...i (cid:96) , (C.6)where ¯ β i ...i (cid:96) is the STF tensor. By construction, our source ¯ A j is manifestly harmonic ∇ ¯ A j = 0 and transverse ∇ j ¯ A j = 0 and hence it satisfies the spatial part of theMaxwell equations ∇ µ F µj = 0 . Expanding the Maxwell action to quadratic order in A j and solving perturbatively its equation of motion with the coupling to the sourceincluded, we obtain A j = (cid:96) (cid:88) m = − (cid:96) ¯ β (cid:96)m r (cid:96) Y Tj(cid:96)m − √ g ε ji (cid:48) k (cid:48) x i (cid:48) ∇ k (cid:48) (cid:96) ( (cid:96) + 1) ˜ λ (1) i ...i (cid:96) i (cid:48) ...i (cid:48) (cid:96) ¯ β i ...i (cid:96) n (cid:104) i (cid:48) ...i (cid:48) (cid:96) (cid:105) ( − (cid:96) (cid:96) − π / Γ(1 / − (cid:96) ) r − (cid:96) − . (C.7)Now we are in position to match the angular component of the magnetic tensor F θφ .As a first step we compute the response part of the Maxwell tensor F response ab fromEq. (C.7) and use that √ g ε [ bjk x j ∇ k ∇ a ] = √ g ε ab r ∇ S in the spherical coordinatebasis, which yields F response ab = − √ g ε ab r ∇ S (cid:96) ( (cid:96) + 1) ˜ λ (1) i ...i (cid:96) i (cid:48) ...i (cid:48) (cid:96) ¯ β i ...i (cid:96) n (cid:104) i (cid:48) ...i (cid:48) (cid:96) (cid:105) ( − (cid:96) (cid:96) − π / Γ(1 / − (cid:96) ) r − (cid:96) − . (C.8)This can be compared with Eq. (C.3), which we rewrite for a single orbital harmonic (cid:96) as follows: F θφ = − √ g ε θφ r ∇ S (cid:96) ( (cid:96) + 1) (cid:96) (cid:88) m = − (cid:96) β (cid:96)m Y (cid:96)m F θφ(cid:96)m ( r ) /r . (C.9)Now we can rewrite the expression above using the Thorne STF tensors and arriveat the anticipated result ˜ λ (1) i (cid:48) ...i (cid:48) (cid:96) i ...i (cid:96) = r (cid:96) +1 s B (cid:96) π(cid:96) !(2 (cid:96) + 1)!! − ( (cid:96) + 1) (cid:96) (cid:96) (cid:88) m = − (cid:96) ˜ k (1) (cid:96)m Y ∗ i ...i (cid:96) (cid:96)m Y i (cid:48) ...i (cid:48) (cid:96) (cid:96)m , (C.10)where B (cid:96) a constant is given in Eq. (3.31). This expression coincides with the electricresponse coefficient tensor given in Eq. (4.32) up to a factor ( (cid:96) + 1) /(cid:96) . D Comment on the Near-Field Approximation
In this appendix we discuss the validity of the solution of the frequency-dependentradial Teukolsky equation (6.7) in the near-field approximation. We start with theKlein-Gordon equation in the Schwarzschild background.– 51 – .1 Scalar Field Example
The differential equation defining the radial mode function of the scalar field in theSchwarzschild background takes the following form [60, 70] x (1 + x ) R (cid:48)(cid:48) + (2 x + 1) R (cid:48) + (cid:20) ( r s ω ) (1 + x ) x (1 + x ) − (cid:96) ( (cid:96) + 1) (cid:21) R = 0 . (D.1)The near field approximation amounts to replacing [60, 70, 71] ( r s ω ) (1 + x ) x (1 + x ) → ( r s ω ) x (1 + x ) . (D.2)The corrections to the r.h.s. term are small as long as ( r s ω ) x (cid:28) ( (cid:96) + 1) . (D.3)The near-zone approximation can be systematically formulated by an introductionof a formal expansion parameter α such that (1 + x ) → (1 + αx ) = 1 + O ( αx ) . (D.4)The final expressions have to be evaluated at α = 1 . Then the corrections beyondEq. (D.2) can be systematically computed order-by-order in α . However, we notethat this does not correspond to a low-frequency expansion with the small parameter r s ω (cid:28) . Indeed, sufficiently far from the horizon, i.e. for x = O (1) , we have ( r s ω ) ∼ ( r s ω ) x , (D.5)and hence keeping the term ( r s ω ) while neglecting the term ( r s ω ) x in Eq. (D.2)is not justified by the smallness of r s ω .Now let us write Eq. (D.1) in the zeroth order near zone approximation ( αx (cid:28) ), x (1 + x ) R (cid:48)(cid:48) + (2 x + 1) R (cid:48) + (cid:20) ( r s ω ) x (1 + x ) − (cid:96) ( (cid:96) + 1) (cid:21) R = 0 . (D.6)The solution consistent with the purely incoming boundary condition at the horizonis given by R = const · (cid:18) x x (cid:19) iωr s F ( (cid:96) + 1 , − (cid:96), ir s ω, − x ) . (D.7)If we now analytically continue this solution for x (cid:29) , we will obtain, R = const · (cid:18) x x (cid:19) iωr s (cid:32) Γ(1 + 2 ir s ω )Γ(2 (cid:96) + 1)Γ( (cid:96) + 1)Γ(1 + (cid:96) + 2 ir s ω ) x (cid:96) · F (cid:0) − (cid:96), − (cid:96) − ir s ω, − (cid:96), − x − (cid:1) + Γ(1 + 2 ir s ω )Γ( − (cid:96) − − (cid:96) )Γ( − (cid:96) + 2 ir s ω ) x − (cid:96) − · F (cid:0) (cid:96) + 1 , (cid:96) + 1 − ir s ω, (cid:96) + 2 , − x − (cid:1) (cid:33) −−−→ x →∞ r (cid:96)s x (cid:96) (cid:16) k (0) NF (cid:96) x − (cid:96) − (cid:17) , (D.8)– 52 –here the coefficient k (0) NF (cid:96) might be interpreted as a frequency-dependent responsecoefficient, k (0) NF (cid:96) ≡ Γ( − (cid:96) − (cid:96) + 1)Γ(1 + (cid:96) + 2 ir s ω )Γ(2 (cid:96) + 1)Γ( − (cid:96) )Γ( − (cid:96) + 2 ir s ω ) . (D.9)Indeed, for ω = 0 this expression reproduces the scalar Love number for the Schwarzschildblack hole. However, strictly speaking, we cannot use Eq. (D.9) for the Love numbermatching because there may be other frequency-dependent contributions that havebeen omitted in the near zone approximation. This will be shown shortly when wecompare Eq. (D.9) with the accurate solution to the Teukolsky equation. D.2 Teukolsky Equation in the Near-Field Approximation
Now we compute the solution of the frequency-dependent radial Teukolsky equa-tion (6.7) for the mode function R in the potential region (near-field zone), charac-terized by x ( r + − r − ) (cid:28) ( (cid:96) + 1) /ω . (D.10)for a rotating black hole and perturbation of a generic spin s . We additionallyexpand over the small parameter ωM ∼ ωr s (cid:28) . Using that ωa ≤ ωM (cid:28) , wecan approximate A = ( (cid:96) − s )( (cid:96) + s + 1) and hence the radial differential equation canbe written in the following simple form [70, 71], (cid:34) s + s − (cid:96) − (cid:96) + Q + isQ (2 x + 1) x ( x + 1) + ( s + 1)(2 x + 1) ddx + x ( x + 1) d dx (cid:35) R ( x ) = 0 . (D.11)This is the same equation as (6.1), but with γm replaced by Q . Hence, the requiredsolution is given by Eq. (6.3) with γm → Q . If we now formally analytically continuethis solution for x > , we can obtain the following expression for response coefficients k ( s ) NF (cid:96)m = Γ( − (cid:96) − (cid:96) − s + 1)Γ( (cid:96) + 2 iQ + 1)Γ( − (cid:96) − s )Γ(2 iQ − (cid:96) )Γ(2 (cid:96) + 1) (cid:18) r + − r − r s (cid:19) (cid:96) +1 = ( − s +1 i mχ − r + ω ) ( (cid:96) + s )!( (cid:96) − s )!(2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 (cid:2) n (1 − χ ) + ( mχ − r + ω ) (cid:3) = ( − s +1 ir + ( m Ω − ω ) ( (cid:96) + s )!( (cid:96) − s )!(2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 (cid:2) n (1 − χ ) + 4 r ( m Ω − ω ) (cid:3) . (D.12)where we used black hole’s angular velocity Ω ≡ a/ ( r + a ) = a/ (2 M r + ) . We cansee that this expression is not invariant under time reversal transformations ω → − ω , m → − m , which implies that the near-field response is purely dissipative. However,this result is uncertain up to other frequency-dependent corrections. To estimate– 53 –hese corrections, let us use the perturbed angular eigenvalues (6.8) instead of theusual ones. We have: ˜ ν = (cid:96) + ∆ (cid:96) = (cid:96) + aω ms (cid:96) ( (cid:96) + 1)(2 (cid:96) + 1) + ... (D.13)Now we can easily find a solution to Eq. (D.11) with (cid:96) replaced by ν . It is given by R (cid:12)(cid:12)(cid:12) x →∞ = const × x ˜ ν (cid:18) x − ν − Γ( − ν − ν − s + 1)Γ( ν + 2 iQ + 1)Γ( − ˜ ν − s )Γ(2 iQ − ˜ ν )Γ(2˜ ν + 1) (cid:19) . (D.14)Therefore, the relevant response coefficient reads k ( s ) NF (cid:96)m = Γ( − ν − ν − s + 1)Γ(˜ ν + 2 iQ + 1)Γ( − ˜ ν − s )Γ(2 iQ − ˜ ν )Γ(2˜ ν + 1) (cid:18) r sa r s (cid:19) (cid:96) +1 (1 − (cid:96) ln x ) , (D.15)where the logarithm comes from the Taylor expansion of x − ν − , x − ν − = x − (cid:96) − (1 − (cid:96) ln x ) . After some simplifications we obtain k ( s ) NF (cid:96)m =( − s +1 sin(2 iQπ − ∆ (cid:96)π ) 2 iQ sin(2 iQπ ) ( (cid:96) − s )!( (cid:96) + s )!2(2 (cid:96) )!(2 (cid:96) + 1)! (cid:96) (cid:89) n =1 ( n + 4 Q ) × ( r sa /r s ) (cid:96) +1 (1 − (cid:96) ln x ) + O (∆ (cid:96) ) . (D.16)For ∆ (cid:96) = 0 the response coefficients reduce to Eq. (D.12). However, we can see thatthe near field approximation misses O ( ωa ) and O ( ωa ln x ) corrections. This can beconfirmed by an explicit comparison with a solution obtained in a small-frequencyexpansion of the Teukolsky equation [72–75]. D.3 Comparison with the Low-Frequency Solution
The systematic treatment of the Teukolsky equation in the low-frequency limit [72–75] gives a solution which is somewhat different from the near-field expression, c.f.Eq (D.12) and Eq. (6.16). Importantly, the relevant response coefficients are notpurely imaginary in this case. To see this, let us expand Eq. (6.14) to linear order in (cid:15) , while keeping all powers of ˜ Q . Because of the presence of the simple pole at (cid:15) = 0 ,is important that we also expand the renormalized angular momentum ν = (cid:96) + ∆ (cid:96) ,where (cid:96) is an integer number satisfying (cid:96) ≥ | s | , and ∆ (cid:96) = O ( (cid:15) ) . We have Γ( − ν − ν + 1) = 12∆ (cid:96) (2 (cid:96) + 1)!(2 (cid:96) )! + O ( (cid:15) ) , Γ( ν + 1 − i(cid:15) − s )Γ( − ν − s − i(cid:15) ) = ( − (cid:96) + s +1 ( i(cid:15) + ∆ (cid:96) )( (cid:96) − s )!( (cid:96) + s )! + O ( (cid:15) ) , Γ( ν + 2 i ˜ Q + 1)Γ( − ν + 2 i ˜ Q ) = ( − (cid:96) sin(2 i ˜ Qπ − ∆ (cid:96)π ) 2 i ˜ Q sin(2 i ˜ Qπ ) (cid:96) (cid:89) n =1 ( n + 4 ˜ Q ) + O ( (cid:15) ) , (D.17)– 54 –his gives κ ( s ) νm = (cid:34) − Q(cid:15) ∆ (cid:96) + 2 i ˜ Q sinh(2 ˜ Qπ ) (cid:16) sinh(2 π ˜ Q ) − (cid:15)π cosh(2 π ˜ Q ) (cid:17) − ∆ (cid:96) (2 π ˜ Q ) cosh(2 π ˜ Q )sinh(2 π ˜ Q ) (cid:35) × ( − s +1 ( (cid:96) − s )!( (cid:96) + s )!2(2 (cid:96) + 1)!(2 (cid:96) )! (cid:32) (cid:96) (cid:89) n =1 ( n + 4 ˜ Q ) (cid:33) . (D.18)We see that our response coefficient has a pole at ∆ (cid:96) = 0 . When we match theEFT result to the GR calculation, we use only finite parts, and hence this singularcontribution can be ignored. However, it is important to note that this term alsogenerates a finite logarithmic contribution, κ ( s ) νm x − ν − = κ ( s ) νm x − (cid:96) − (1 − (cid:96) ln x + O ( (cid:15) ))= x − (cid:96) − (cid:34) Q(cid:15) ln x + 2 i ˜ Q sinh(2 Qπ ) (cid:16) sinh(2 π ˜ Q ) − (cid:15)π cosh(2 π ˜ Q ) (cid:17) (cid:35) × ( − s +1 ( (cid:96) − s )!( (cid:96) + s )!2(2 (cid:96) + 1)!(2 (cid:96) )! (cid:32) (cid:96) (cid:89) n =1 ( n + 4 ˜ Q ) (cid:33) + O ( (cid:15) ) . (D.20)Using that sinh(2 π ˜ Q ) − (cid:15)π cosh(2 π ˜ Q ) = sinh(2 π ˜ Q − (cid:15)π ) + O ( (cid:15) ) , (D.21)we obtain the first correction in Eq. (6.16). We see that this expression coincidentlymatches the near-zone result Eq. (D.12) at linear order in ω and zeroth order in ω Ω . However, the near-field approximation does not correctly capture O ( (cid:15) , (cid:15) Ω) corrections and their logarithmic running.Extracting the other contributions from Eq. (6.15) is straightforward. Collectingeverything together we arrive at Eq. (6.16). It is also worth stressing that the full GR solution is regular, the pole in Eq. (D.18) in fact getscanceled by a similar singularity that is contained in the source series. To see this, we have to getback to the original solution (6.14) and regularize the hypergeometric function that is attached tothe source solution ∝ x ν as follows F ( − ν + 2 i ˜ Q, − ν + s + i(cid:15), − ν, − x − )= ( − x ) − (cid:96) − Γ( (cid:96) + 1 + 2 i ˜ Q )Γ( (cid:96) + s + 1 + i(cid:15) )Γ(2 (cid:96) + 2)Γ( − (cid:96) + 2 i ˜ Q )Γ( − (cid:96) + s + i(cid:15) ) − (cid:96) )!2∆ (cid:96) × F ( (cid:96) + 1 + 2 i ˜ Q, (cid:96) + s + 1 + i(cid:15), (cid:96) + 2 , − x − ) + O ( (cid:15) ) , (D.19)which exactly cancels the divergence that we have encountered in the term x − ν − κ ( s ) νm , seeEq. (D.18). – 55 – eferences [1] A. E. H. Love, The Yielding of the Earth to Disturbing Forces , Proceedings of theRoyal Society of London Series A (1909) 73.[2] E. E. Flanagan and T. Hinderer, Constraining neutron star tidal Love numbers withgravitational wave detectors , Phys. Rev. D (2008) 021502 [ ].[3] V. Cardoso, E. Franzin, A. Maselli, P. Pani and G. Raposo, Testing strong-fieldgravity with tidal Love numbers , Phys. Rev. D (2017) 084014 [ ].[4] W. D. Goldberger and I. Z. Rothstein, An Effective field theory of gravity forextended objects , Phys. Rev. D (2006) 104029 [ hep-th/0409156 ].[5] R. A. Porto, The effective field theorist’s approach to gravitational dynamics , Phys.Rept. (2016) 1 [ ].[6] R. A. Porto,
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