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Kerr Black Holes as Elementary Particles

Citation for published version: Arkani-Hamed, N, Huang, Y & O'Connell, D 2020, 'Kerr Black Holes as Elementary Particles', Journal of High Energy Physics, vol. 2020, no. 1, 46. https://doi.org/10.1007/JHEP01(2020)046

Digital Object Identifier (DOI): 10.1007/JHEP01(2020)046

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Kerr Black Holes as Elementary Particles

Nima Arkani-Hamed,1, Yu-tin Huang,2,3, Donal O’Connell,4 1 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA 2 Department of Physics and Astronomy, National Taiwan University, Taipei 10617, Taiwan 3 Physics Division, National Center for Theoretical Sciences, National Tsing-Hua University, No.101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan and 4 Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK

Long ago, Newman and Janis showed that a complex deformation z → z +ia of the Schwarzschild solution produces the Kerr solution. The underlying explanation for this relationship has remained obscure. The complex deformation has an electromagnetic counterpart: by shifting the Coloumb√ potential, we obtain the EM field of a certain rotating charge distribution which we term Kerr. In this note, we identify the origin of this shift as arising from the exponentiation of spin operators for the recently defined “minimally coupled” three-particle amplitudes of spinning particles coupled to gravity, in the large-spin limit. We demonstrate this by studying the impulse imparted to a test particle in the background of the√ heavy spinning particle. We first consider the electromagnetic case, where the impulse due to Kerr is reproduced by a charged spinning particle; the shift of the Coloumb potential is matched to the exponentiated spin-factor appearing in the amplitude. The known impulse due to the Kerr black hole is then trivially derived from the gravitationally coupled spinning particle via the double copy.

INTRODUCTION special three-particle amplitude for massive particles of spin S coupled to gravitons and photons, naturally as- sociated “on-shell” with a notion of “minimal coupling”, The no hair theorem states that black holes are char- given by acterized by only their mass, charge and angular momen- tum, implying that externally the black hole behaves as a q point particle. For a long time this point of view has been h12i2S utilized to derive the spin-independent part of the two- = g(xm)h (1) m2S body classical potential for inspiralling black holes [1–7], S from the scattering amplitudes of gravitationally coupled 1S 2 scalars. (See [8–11] for some recent results, and [12] for √ where h = (1, 2) and g = ( κ , 2e), for positive photons a more comprehensive review.) 2 and gravitons respectively. This coupling was singled out Of course any massive object with spin, viewed from by matching to the (standard, leading) coupling for mass- sufficiently long distances, can be effectively treated as a less spin S particles in the high energy limit. Indeed for point particle. From the perspective of on-shell scatter- low spins, this coupling reproduces all the classical elec- ing amplitudes, the most important first issue is to deter- tric and magnetic moments. mine the three-particle amplitude, coupling the massive We therefore have a three-particle amplitude picked particles to gravitons, and if it is charged, to photons. out as being special purely from the on-shell perspective, A convenient on-shell formalism for describing scattering making the massive particle look as “elementary” as pos- amplitudes for general mass and spin in four dimensions sible to the graviton/photon probe by correctly match- has recently been given in [13]. In particular the formal- ing the high-energy limit. Meanwhile, we also know that ism provides a convenient basis for the cubic couplings the Kerr black hole must make a very special choice for of massive spin-S particles with a graviton or photon. the three-particle amplitude as well. Remarkably, the While for all massless particles of given helicities, three- minimally coupled amplitudes are indeed precisely the particle amplitudes are fixed (up to overall strength) by ones enjoyed by Kerr black holes. Following the work Poincar´esymmetry, for massive particles of spin S cou- of Guevara [14], it was shown in [15] and [16] that the pled to gravitons or photons, there are (2S + 1) different potential for Kerr black holes was indeed reproduced rel- allowed structures, reflecting all the allowed multipole ativistically to all orders in the multipole expansion from moments of the particle. Returning to the Kerr black minimal coupling. hole, this three-particle amplitude coupling to a graviton These results establish the equivalence of the minimal should be completely prescribed, and is clearly expected coupling in eq.(1) and Kerr black holes in the context of to be “special” in some way, so the natural question is: classical observables, but why did this happen? In this what three-particle amplitude is dictated by the no-hair note we would like to give a more fundamental under- theorem? standing of why minimally coupled higher-spin particles From a completely different motivation, [13] defined a at large spin correspond to Kerr black holes. We will do 2 this by relating minimal coupling to some classic features COMPLEXIFYING SCHWARZSCHILD AND THE of the Kerr solution. DOUBLE COPY Not long after Kerr wrote down the solution for spin- ning black holes [17], Newman and Janis observed that An early example of the utility of complexified space- one can “rederive” the Kerr metric by complexifying time was the derivation of the Kerr metric from a com- the Schwarzschild solution in null polar coordinates and plex coordinate transformation of the Schwarzschild met- performing a shift [18]. The construction was later ex- ric [18]. We will make use of the metric in Kerr-Schild tended to a derivation of the Kerr-Newman solution from form: Reissner-Nordstrom [19]. For other solutions derived in a 0 gµν = g + kµkν φ , (3) similar fashion, see [20]. The methods of amplitudes al- µν 0 low us to understand the origin of the complex shift. We where gµν is the flat Minkowski metric, and the vector kµ will demonstrate that the shift is a consequence of the is null with respect to both gµν and g0,µν . In particular, spin effects generated when one goes from a minimally the Schwarzschild solution takes the form coupled scalar to a spinning particle. In particular start- r ∂ ∂ ing with a spin-S particle and taking the classical limit, Schwarzschild : φ (r) = 0 , kµ∂ = − (4) Sch r µ ∂t ∂r S → ∞, ~ → 0 while keeping ~S fixed, the minimal cou- pling exponentiates [15]. This exponent can be identified where r0 = 2GM. For the Kerr solution, one instead has as q·s , where sµ is the Pauli-Lubanski pseudovector, qµ m r r ∂ ∂ the massless momentum and m the mass. When applied 0 µ Kerr : φKerr(r) = 2 2 2 , k ∂µ = − . to the computation of classical observables, such as the r + a cos θ ∂t ∂r (5) change in momentum a probe experiences in a gravita- Unlike the Schwarzschild case, for Kerr (r, θ) are not the tional or electromagnetic field, this exponentiation pre- usual polar coordinates but are defined by: cisely induces the relevant shift, after Fourier transform- ing to position space. In other words, the exponenti- x = (r sin φ+a cos φ) sin θ, y=(a sin φ−r cos φ) sin θ ation incurred going from minimally-coupled scalars to z = r cos θ . (6) spinning particles, is the momentum space image of the complex shift that relates the Schwarzschild to the Kerr In particular, in the Kerr case r is the solution to the solution. This sharpens the equivalence between black equation holes and particles. x2 + y2 z2 This connection also provides an on-shell realization + = 1 . (7) r2 + a2 r2 of the double copy relation for classical solutions. In an earlier work by one of the authors [21], it was shown It is remarkable that φKerr can be obtained from φSch that stationary Kerr-Schild metrics admit a double copy by a complex shift, which is as simple as z → z + ia. construction. In particular the double and single copy To see how this connects the Schwarzschild to the Kerr solutions take the form: solution, note that the quantity r2 = x2 + y2 + z2 shifts to x2 + y2 + z2 − a2 + 2iaz = r2 − a2 cos2 θ + 2iar cos θ = (r+ia cos θ)2, where now r is the solution to equation (7). 0 µa a µ gµν = gµν + kµkν φ(r),A = c k φ(r) , (2) In short, the replacement z → z + ia is equivalent to the replacement r → r + ia cos θ. The action on φ(r) is where φ(r) is the universal part for the gravity/gauge the-   r0 1 1 ory solution and kµ a null (r, θ)-dependent vector. Pass- φSch(r)|r→r+ia cos θ = + 2 r r¯ r→r+ia cos θ ing from the φ(r) for Schwarzschild to Kerr, one simply r r takes a complex shift. On the other hand, as discussed = 0 = φ (r) . (8) r2 + a2 cos2 θ Kerr previously, the difference between the three-point ampli- tude for gravitational and electromagnetic minimal cou- Indeed it is straightforward to show that the Riemann pling is simply the squaring of the x factor, whilst the tensors of the two solution are related via this complex spin-dependent part is untouched. The later corresponds shift [18]. We are unaware of any classical understanding to the shifted φ(r), while the squaring can be identified of why this remarkably simple procedure should work; with the squaring√ of kν . To illustrate this, we compute however, we will see that it follows directly from the na- the impulse for Kerr and match it to that from the ture of observables computed from on-shell amplitudes. minimally coupled charged spinning particle. One then The Kerr-Schild form of the metric is particularly obtains the gravitational counterpart by squaring all x- convenient for revealing double copy relations between factors, which simply translate to a factor of two in ra- classical solutions of the Yang-Mills and Einstein equa- pidity. Remarkably this simple factor of two converts the tions. It was previously shown [21] that for every sta- electromagnetic impulse to the gravitational version. tionary Kerr-Schild solution to the Einstein equations, 3 i.e. ∂0φ = ∂0kµ = 0, one finds a solution to the Yang- impulse on the particle is computed via: Mills equation with Z ∞ µ µν ∆p1 = e1Re dτ F (x1)u1ν . (11) Aµa = cakµφ(r) . (9) −∞ This impulse can be computed perturbatively by iterat- r0 ing the Lorentz force. At leading order, the trajectory of For example consider the Schwarzschild case φ(r) = r . a Using the replacement r0 → gcaT , one finds the static particle 1 is simply a straight line, which is also the all- Coulomb potential after a suitable gauge transforma- order trajectory of the source—which we take to be par- tion. On the other hand, beginning with the Coulomb ticle 2. Thus for both particles we have xi(τ) = bi + uiτ, charge but performing a complex coordinate shift, one where ui is the proper√ velocity; while b1 is real and b2 is finds the electromagnetic field of a rotating disc with ra- complex, reflecting the Kerr nature of particle 2. With- dius a [21]. This Yang-Mills solution is the “square√ root” out loss of generality we set b1 = 0, and b2 = −b−ia so of the√ Kerr solution, and therefore we call it Kerr. In that b1−b2 = b+ia. It is convenient to work in Fourier fact, Kerr was discussed by Newman and Janis [18] space, with the field strength due to the source written as a complex deformation of Coulomb, and also more re- as cently by Lynden-Bell [22]. It correspond to the EM field Z F µν (x ) = dˆ4q¯F µν (¯q) e−iq¯·x1 . (12) of Kerr-Newman where both M and S are sent to zero 2 1 2 while holding a fixed. In the following we will compute Our notation is thatq ¯ is a wavenumber (momentum the impulse probe particles incur in this background and transfer q with a scaled out), dˆq¯ ≡ dq/¯ (2π) and relate the results to Kerr. ~ δˆ(x) = (2π)δ(x). One then has: Z µ ˆ4 µν ˆ √ ∆p = e1 Re d q¯F (¯q)u1ν δ(¯q · u1) FROM Kerr TO SPINNING PARTICLES 1 2 Z   ˆ4 µν i µναβ ˆ =e1 Re d q¯ F2 (¯q)+  F2αβ(¯q) u1ν δ(¯q · u1). We first study the√ equivalence between the electromag- 2 netic field of the Kerr solution with the minimally cou- (13) pled spinning particle, in the infinite spin limit, by com- We need an expression for the field strength F in Fourier puting the impulse induced on a charged particle. In the space. Using the Maxwell equation process we will identify the Kerr parameter a with s , m Z where s and m are the absolute value of the spin-vector 2 µ µ 4 ∂ A2 (x) = e2 dτu2 δ (x − x2(τ)) , (14) and mass of the particle, respectively. √ it’s easy to see that, to all orders for static Kerr ,

√ µν −iq¯·(b+ia) ˆ 1 µ ν ν µ Impulse from Kerr F (¯q) = ie2 e δ(¯q · u2) (¯q u − q¯ u ). (15) 2 q¯2 2 2 With this information, we obtain our final expression for Performing the complex shift z → z + ia on the the impulse in momentum space: Coloumb electric field Ec, we obtain a complex quan- tity, E → E. The interpretation is simple: Re E is the Z e−iq¯·(b+ia) c √ ∆pµ =e e Re dˆ4q¯δˆ(¯q · u )δˆ(¯q · u ) (iq¯µu · u electric field of Kerr, while Im E is the magnetic field. 1 1 2 1 2 q¯2 1 2 Covariantly, the complex shift induces a complex field µναβ
+ q¯ν u1αu2β . (16) strength F µν . The Lorentz force on a particle with mass µ µ Note that the presence of the Levi-Civita tensor is a re- m, momentum p and√ proper velocity u moving under the influence of the Kerr fields is flection of the complexification of the field strength. We will now reproduce the above result from the scattering dpµ amplitude involving minimally coupled spinning parti- = e Re F µν u , (10) dτ ν cles.

µν µν µνρσ where F = F + i Fρσ/2. In electrodynamics, the field strength is gauge invari- Impulse from x ant and observable. However this fact already fails for Yang-Mills theories, and therefore it is desirable to un- The impulse for scalar particles was computed from derstand these classical solutions from a different, more amplitudes in [23] via: gauge invariant point of view. To that end we consider 1 Z the impulse, that is the total change of momentum, from ∆pµ = dˆ4q¯ δˆ(¯q · u )δˆ(¯q · u )e−iq¯·b× 1 4m m 1 2 past infinity to future infinity, of a light particle (particle 1 2 √ µ 3 0 0 1) moving in the (very heavy) Kerr background. The iq¯ ~ M4 (1, 2 → 1 , 2 ) |q¯2→0 (17) 4

2 S indices will be contracted with appropriate wave func- 1 tions. We now take the limit S → ∞ and ~ → 0 with S~ fixed. The amplitudes become q √ 2S √ ±1  q¯ · s  ±1 ±q¯·a h = ±1 : lim ie2 2m x I± = ie2 2mx e S→∞ 2Sm (24)

s where the quantity a = m parameterises the spin, but S has dimensions of length. 2 1 Now let’s consider the classical limit of the four point FIG. 1: The exchange of a photon between a spin-S and a amplitude between a charged particle of spin S, with S → scalar particle. ∞, and a scalar particle:   0 0 e1e2m1m2 x110 −q¯·a x220 q¯·a 2 M4 (1, 2→1 , 2 ) |q →0 = 2 2 e + e 2¯q x220 x110 where M4 correspond to a four point amplitude exchang- (25) ing gravitons or a photons with momentum transfer q. As Note that it is given by two terms with different helicity we will see using this prescription√ we indeed reproduce configurations. We will see that these terms have a cru- the correct impulse for the Kerr electromagnetic field cial role, allowing us to understand the emergence of the in eq.(16), by the scattering of a scalar particle 1 with the real-part operation in the impulse. The x ratios are little minimally coupled spin-S particle 2, illustrated in fig.1. group invariant, and can be shown to be given by: When dressed with the external polarization tensors, x 0 x 0 the three-point, minimally-coupled amplitude is given 11 = ew, 22 = e−w , (26) as [13] x220 x110 where w is the rapidity. Thus we have, √ h220i2S √ 1 [220]2S h = +1 : 2ie2x , h = −1 : 2ie2 . m2S−1 x m2S−1 0 0 e1e2m1m2 w −q¯·a −w q¯·a
M (1, 2→1 , 2 ) | 2 = 2 e e +e e (18) 4 q →0 3q¯2 0 ~ Since q is small, the spinor |2 i is only a small boost of (27) the spinor |2i. We may therefore write 1 We now proceed to compute the impulse by inserting this |20i = |2i + ω (σµσ¯ν − σν σ¯µ)|2i , (19) four-point amplitude into the general expression eq.(17): 8 µν Z µ e1e2 q¯ where the boost parameters ωµν are small. It is easy to µ ˆ4 ˆ ˆ −iq¯·b ∆p1 = i d q¯ δ(¯q · u1)δ(¯q · u2)e compute these boost parameters because 2 q¯2 X eα(w−q¯·a) . (28) 0µ µ µ ν 1 α=± p2 = (δν + ω ν )p2 =⇒ ωµν = − 2 (p2µqν − p2ν qµ) , m2 (20) To proceed, it’s helpful to rewrite the impulse as 2 taking account of the on-shell relation 2p2 · q = q ' 0. Z µ We therefore learn that µ e1e2 q¯ ∆p = i d4q¯ δ(¯q · u )δ(¯q · u )e−iq¯·b 1 2 1 2 q¯2 0 1 |2 i = |2i + /q|2]. (21) (cosh w+ sinh w)e−q¯·Πa+(cosh w− sinh w)eq¯·a
2m2 (29) Thus, we have, Note that on the support of δ(¯q · u1)δ(¯q · u2), the Gram 1 0 1 1 determinant constraint takes the form h22 i = I + 2 ~h2|/q¯|2] = I + q¯ · s , (22) m2 2m2 2Sm2 2 2 2 2 (u1, u2, a, q¯) = − sinh w(a · q¯) + O(¯q ). (30) where sµ is the Pauli-Lubanski pseudovector associated with a spin S particle: We may neglect any terms of orderq ¯2, because in the Fourier integral such terms lead to a delta function in im- 1 sµ = S h2|σµ|2] . (23) pact parameter space. Furthermore, using a “Schouten” m ~ 2 identity we have The operators and sµ are now operators acting on the I ν α β little group space of particle 2. In the end, all little group q¯µ(u1, u2, a, q¯) = (a · q¯)µναβq¯ u1 u2 . (31) 5

2 where u1 · q¯, u2 · q¯,q ¯ are all set to zero. Thus we can To compute the Fourier integrals, note that q · b = q · Πb identify: on the support of the integral, where Π is the projec- tor onto the space orthogonal to u and u . With this 1 1 2 ν α β replacement, the Fourier integral is straightforward to sinh w q¯µ = i(u1, u2, a, q¯) q¯µ = iµναβq¯ u1 u2 (32) a · q¯ compute1, and the result is With this result, the impulse is then:  µ µναβ  µ 2m1m2G cosh 2w b⊥+i2 cosh w u1αu2βb⊥ν Z ∆p1 = − Re 2 , µ e1e2 i sinh w b ∆p = d4q¯ δ(¯q · u )δ(¯q · u ) ⊥ 1 2 1 2 q¯2 (38) h µ µναβ −iq¯·(b−ia) where b⊥ = Π(b + ia), in agreement with eq.(34). (¯q cosh w+i q¯ν u1αu2β)e DISCUSSIONS AND CONCLUSIONS µ µναβ −iq¯·(b+ia)i +(¯q cosh w−i q¯ν u1αu2β)e Z i In this paper we demonstrated that the exponentiation = e e Re d4q¯ δ(¯q · u )δ(¯q · u ) 1 2 1 2 q¯2 induced in taking the large-spin limit of minimally cou- h i pled spinning particles, precisely maps to the Newman- (¯qµ cosh w−iµναβq¯ u u )e−iq¯·(b+ia) . (33) ν 1α 2β Janis complex shift relating the Schwarzschild and Kerr solutions in position space. As one can see we have recovered eq.(16): importantly, we identify the shift in Kerr solution explicitly with the Note that these are very general features, applying to s a wide range of observables (eg the total change in spin exponentiation of m for spinning particles in the large spin limit! Evidently the shift b → b + ia arises because of a particle during a scattering event [27]) and in a wide of the exponential structure of minimally coupled am- range of theories, including Einstein gravity. Moreover, plitudes, and the Fourier factor eiq¯·b in expressions for while we have described the situation in detail at lowest observables in terms of amplitudes. order for the impulse, one can compute the impulse to all orders using scattering amplitudes. The only three point vertex available√ for particles moving in the static Impulse for Kerr black hole background field is the Kerr amplitude of equation (24). Thus the replacement must hold to all orders, as well as The leading order impulse for a spinning black hole for its gravitational counterpart. was derived to all orders in spin by Vines [24]. In impact There is still much to learn by studying classical grav- parameter space it takes the form: ity from the point of view of on-shell methods. We need to learn more about amplitudes for particles with large µ ρ σ ∆p1 = −2Gm1m2 Re [(cosh 2w ηµν +2i cosh w µνρσu1u2 ) spins in order to understand the dynamics of Kerr black (b + iΠa)ν  hole scattering (not just probe scattering) in more de- (34) tail. Moreover, the interplay between the double copy, sinh w(b + iΠa)2 massive particles, and Einstein gravity needs to be ex- √ This result follows straightforwardly from our Kerr dis- plored in more detail [25], especially in light of recent difficulties [26]. cussion, by simply√ “squaring” the x-factors in eq.(25), and replacing 2e → −κ/2. The result is just a factor It will be interesting to explore the correspondence to of two for the rapidity factor in eq.(28) other solutions where either complex shifting or double copy relations hold. This includes the shifting relation Z q¯µ between Kerr-Newman and Reissner-Nordstrom, as well ∆pµ = −i2πGm m d4q¯ δ(¯q · u )δ(¯q · u )e−iq¯·b 1 1 2 1 2 q¯2 as the double copy relation between dyons and the Taub- (cosh 2w+ sinh 2w)eq¯·a+(cosh 2w− sinh 2w)e−q¯·a
NUT solution [28]. We leave this for future work. (35) Finally, in this note we have focused on understanding the three-point couplings of Kerr black holes in the sim- Again, using the identities for sinh w we derived previ- plest and most physical way, involving the scattering of ously, we can rewrite probe particles in asymptotically Minkowski spacetime. We did not directly consider the on-shell three-particle ν α β q¯µ sinh 2w = 2¯qµ cosh w sinh w = i2 cosh w µναβq¯ u1 u2 scattering. Indeed, it is a standard (and important) fact (36) of basic kinematics that the 3-particle amplitude is never and thus “on-shell” in asymptotically Minkowski space. Instead Z ie−iq¯·(b+ia) ∆pµ = −4πGm m Re d4q¯ δ(¯q · u )δ(¯q · u ) 1 1 2 1 2 q¯2

µ ν α β 1 (¯q cosh 2w+i2 cosh w µναβq¯ u1 u2 ) . (37) For example, see page 33 of [23] 6 the three-particle amplitude makes sense for general com- doi:10.1103/PhysRevLett.121.251101 [arXiv:1808.02489 plex momenta, and also for real momenta, not in (3, 1) [hep-th]]. but in (2, 2) signature. Clearly, the complexification asso- [9] Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon ciated with the Kerr solution is begging for a formulation and M. Zeng, Phys. Rev. Lett. 122, no. 20, 201603 (2019) doi:10.1103/PhysRevLett.122.201603 [arXiv:1901.04424 in (2, 2) signature, where an even more direct computa- [hep-th]]. tion of the three-particle amplitude should be possible. [10] S. Foffa, R. A. Porto, I. Rothstein and R. Sturani, Given the important role of (2, 2) signature physics in arXiv:1903.05118 [gr-qc]. many other aspects of four-dimensional scattering ampli- [11] A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard tudes, this may well be more generally a fruitful avenue and P. Vanhove, arXiv:1906.01579 [hep-th]. of exploration for future work. [12] B. Maybee, D. O’Connell and J. Vines, arXiv:1906.09260 [hep-th]. [13] N. Arkani-Hamed, T. C. Huang and Y. t. Huang, ACKNOWLEDGEMENTS arXiv:1709.04891 [hep-th]. [14] A. Guevara, JHEP 1904, 033 (2019) doi:10.1007/JHEP04(2019)033 [arXiv:1706.02314 [hep- We thank Andr´esLuna, Lionel Mason, Ricardo Mon- th]]. teiro and Justin Vines for useful discussions. NAH is sup- [15] A. Guevara, A. Ochirov and J. Vines, arXiv:1812.06895 ported by DOE grant de-sc0009988, YTH is supported [hep-th]. by MoST Grant No. 106-2628-M-002-012-MY3, while [16] M. Z. Chung, Y. T. Huang, J. W. Kim and S. Lee, DOC is supported by the STFC grant “Particle Theory JHEP 1904, 156 (2019) doi:10.1007/JHEP04(2019)156 [arXiv:1812.08752 [hep-th]]. at the Higgs Centre”. YTH and DOC would like to thank [17] R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963). Simons Foundation for its support for the “Amplitudes doi:10.1103/PhysRevLett.11.237 meets cosmology” workshop, during which this work was [18] E. T. Newman and A. I. Janis, J. Math. Phys. 6, 915 done. (1965). doi:10.1063/1.1704350 [19] T. Adamo and E. T. Newman, Scholarpedia 9, 31791 (2014) doi:10.4249/scholarpedia.31791 [arXiv:1410.6626 [gr-qc]]. [1] J. F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994) [20] A. Y. Burinskii, Sov. Phys. JETP 39, 193 (1974). doi:10.1103/PhysRevLett.72.2996 [gr-qc/9310024]. [21] R. Monteiro, D. O’Connell and C. D. White, [2] J. F. Donoghue, Phys. Rev. D 50, 3874 (1994) JHEP 1412, 056 (2014) doi:10.1007/JHEP12(2014)056 doi:10.1103/PhysRevD.50.3874 [gr-qc/9405057]. [arXiv:1410.0239 [hep-th]]. [3] J. F. Donoghue and T. Torma, Phys. Rev. D 54, 4963 [22] D. Lynden-Bell, astro-ph/0207064. (1996) doi:10.1103/PhysRevD.54.4963 [hep-th/9602121]. [23] D. A. Kosower, B. Maybee and D. O’Connell, [4] J. F. Donoghue, B. R. Holstein, B. Garbrecht and JHEP 1902, 137 (2019) doi:10.1007/JHEP02(2019)137 T. Konstandin, Phys. Lett. B 529, 132 (2002) Erra- [arXiv:1811.10950 [hep-th]]. tum: [Phys. Lett. B 612, 311 (2005)] doi:10.1016/S0370- [24] J. Vines, Class. Quant. Grav. 35, no. 8, 084002 2693(02)01246-7, 10.1016/j.physletb.2005.03.018 [hep- (2018) doi:10.1088/1361-6382/aaa3a8 [arXiv:1709.06016 th/0112237]. [gr-qc]]. [5] N. E. J. Bjerrum-Bohr, J. F. Donoghue and [25] H. Johansson and A. Ochirov. “Double copy for massive B. R. Holstein, Phys. Rev. D 68, 084005 (2003) quantum particles with spin,” in progress. Erratum: [Phys. Rev. D 71, 069904 (2005)] [26] J. Plefka, C. Shi, J. Steinhoff and T. Wang, doi:10.1103/PhysRevD.68.084005, 10.1103/Phys- arXiv:1906.05875 [hep-th]. RevD.71.069904 [hep-th/0211071]. [27] A. Guevara, A. Ochirov and J. Vines. “General Spin De- [6] N. E. J. Bjerrum-Bohr, J. F. Donoghue and pendence of Black-Hole Scattering from Amplitudes at B. R. Holstein, Phys. Rev. D 67, 084033 (2003) First Post-Minkowskian Order,” in progress. Erratum: [Phys. Rev. D 71, 069903 (2005)] [28] A. Luna, R. Monteiro, D. O’Connell and doi:10.1103/PhysRevD.71.069903, 10.1103/Phys- C. D. White, Phys. Lett. B 750, 272 (2015) RevD.67.084033 [hep-th/0211072]. doi:10.1016/j.physletb.2015.09.021 [arXiv:1507.01869 [7] B. R. Holstein and J. F. Donoghue, Phys. Rev. Lett. 93, [hep-th]]. 201602 (2004) doi:10.1103/PhysRevLett.93.201602 [hep- th/0405239]. [8] C. Cheung, I. Z. Rothstein and M. P. Solon, Phys. Rev. Lett. 121, no. 25, 251101 (2018)