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Hyperbolic scattering of spinning particles by a Kerr black hole

We investigate the scattering of a spinning test particle by a Kerr black hole within the Mathisson-Papapetrou-Dixon model to linear order in spin. The particle's spin and orbital angular momentum are taken to be aligned with the black hole's spin. Both the particle's mass and spin length are assumed to be small in comparison with the characteristic length scale of the background curvature, in order to avoid backreaction effects. We analytically compute the modifications due to the particle's spin to the scattering angle, the periastron shift, and the condition for capture by the black hole, extending previous results valid for the nonrotating Schwarzschild background. Finally, we discuss how to generalize the present analysis beyond the linear approximation in spin, including spin-squared corrections in the case of a black-hole-like quadrupolar structure for the extended test body.


I. INTRODUCTION
The scenario of a black hole which scatters matter, radiation and even other black holes is especially attractive in view of the possibility to consider these events as sources of gravitational waves, potentially detectable by current detectors such as LIGO and VIRGO [1][2][3] or future detectors. While much is known for "elliptic encounters," or bound motion, namely the spiraling of one body around another up to the merging of the two bodies into a single one, the case of "hyperbolic encounters," or unbound motion, has been poorly investigated in the literature. In fact, up to now the back-reaction on a Kerr background of a small particle moving on an equatorial hyperbolic-like orbit has not yet been analytically computed. The first attempt concerning a spinless particle in a Schwarzschild background has been presented only very recently in Ref. [4]. Our knowledge of the main features of the process mostly relies on the post-Newtonian approximation (at low orders; see, e.g., Refs. [5][6][7] and references therein) or numerical relativity simulations [8][9][10][11]. This lack of information is due to the computational difficulties associated with the problem, like the fact that the hyperbolic gravitational wave spectrum is a continuum, in contrast to the quasi-circular case, where it is mainly monochromatic. Furthermore, apart from the direct relevance to actual hyperbolic motions, the study of unbound motion is also relevant for the analysis of bound motion. In fact, for instance, computing the scattering angle for a hyperbolic encounter can encode gaugeinvariant information characterizing both unbound and bound motion (see, e.g., Ref. [12], where the gaugeinvariant "effective-one-body" function ρ(x) was introduced in the discussion of small eccentricity motion).
A necessary pre-requisite to the study of back-reaction effects due to gravitational wave emission is the complete knowledge of the conservative dynamics of a hyperbolic encounter of a test particle with the (spinning) black hole. When the particle's internal structure is negligible, it moves along a hyperbolic-like geodesic orbit around the black hole. We study here the problem of the scattering of a particle endowed with spin by a Kerr black hole in the framework of the Mathisson-Papapetrou-Dixon (MPD) model [13][14][15], generalizing a previous work in the nonrotating Schwarzschild background spacetime [16]. The motion is non-geodesic due to the presence of a spincurvature coupling force. The two bodies are assumed to have aligned/antialigned spins and the equatorial plane is chosen as the orbital plane. According to the MPD model the extended body is treated as a test body, i.e., backreaction effects on the background metric are neglected, even if it has an internal structure described by its spin. The MPD equations of motion can be solved analytically in terms of elliptic integrals, allowing to explicitly compute the corrections to first-order in spin to the scattering angle, i.e., the most natural gauge-invariant and physical observable associated with the scattering process. We also determine the shift of the periastron position and the modification due to spin to the critical impact parameter for capture by the hole with respect to the well known geodesic case in both ultrarelativistic and non-relativistic regimes. Finally, we show how to extend the present analysis to take into account spinsquared effects as well by suitably modifying the MPD set of equations, qualitatively discussing some features of the second-order-in-spin corrections to the orbital motion in comparison with the linear-in-spin ones, in the simplest case of an extended body endowed with a "black-holelike" quadrupolar structure.
We use geometrical units and assume that Greek indices run from 0 to 3, whereas Latin indices from 1 to 3.

II. EQUATORIAL PLANE MOTION IN A KERR SPACETIME
The motion of a spinning test particle in any given gravitational background is described by the MPD equations [13][14][15] Here P µ ≡ mu µ (with u · u ≡ u µ u µ = −1) is the total 4-momentum of the body with mass m and unit timelke four-velocity u µ ; S µν is the (antisymmetric) spin tensor; and U µ = dx µ /dτ is the timelike unit 4-velocity vector tangent to the body's "center-of-mass worldline," parametrized by the proper time τ (with parametric equations x µ = x µ (τ )), selected to perform the multipole reduction which is at the basis of the MPD approach. Self-consistency of the model requires that some additional conditions be imposed to the determine reference world line, as Eqs. (1) and (2) represent only 10 equations for the 14 degrees of freedom x µ (τ ), P µ (τ ) and S µν (τ ). Here we shall use the Tulczyjew-Dixon conditions [15,17], which read The spin tensor can then be fully represented by the spatial vector (with respect to u) where we have denoted by η(u) αβγ = η µαβγ u µ the spatial unit volume 3-form (with respect to u) built from the unit volume 4-form η αβγδ = √ −g ǫ αβγδ which orients the spacetime, with ǫ αβγδ (ǫ 0123 = 1) being the Levi-Civita alternating symbol and g the determinant of the metric.
It is also useful to introduce the signed magnitude s of the spin vector which is a constant of motion. Implicit in the MPD model is the condition that the length scale |s|/m naturally associated with the spin should be very small compared to the one associated with the background curvature (say M ), in order to neglect back reaction effects, namely |ŝ| ≡ |s|/mM ≪ 1. This condition leads to a simplified set of linearized evolution equations around the geodesic motion. In fact, the total 4-momentum P of the particle is still aligned with U in this limit, i.e., P µ = mU µ + O(S 2 ), the mass m of the particle remaining constant along the path. Furthermore, Eq.
(2) becomes DS µν /dτ = O(S 2 ), implying that the spin vector is parallel transported along U . Finally, when the background spacetime has Killing vectors, there are conserved quantities along the motion [18], which can be used to simplify the equations of motion. In the case of stationary axisymmetric spacetimes with coordinates adapted to the spacetime symmetries, they are the energy E and the total angular momentum J associated with the timelike and azimuthal Killing vectors ξ = ∂ t and η = ∂ φ , respectively, and read A. Kerr spacetime and ZAMO adapted frame Let us consider the Kerr spacetime, whose metric written in standard Boyer-Lindquist coordinates is given by with ∆ = r 2 − 2M r + a 2 , Σ = r 2 + a 2 cos 2 θ and A = (r 2 + a 2 ) 2 − ∆a 2 sin 2 θ. Here a and M denote the specific angular momentum and the total mass of the spacetime solution, so that the quantityâ = a/M is dimensionless. The inner and outer horizons are located at r ± = M ± √ M 2 − a 2 . Let us introduce the zero angular momentum observer (ZAMO) family of fiducial observers, with 4-velocity where N = (−g tt ) −1/2 = ∆Σ/A and N φ = g tφ /g φφ = −2aM r/A are the lapse and shift functions, respectively. The ZAMO adapted form of the metric then reads ds 2 = −N 2 dt 2 + g rr dr 2 + g θθ dθ 2 + g φφ (dφ+ N φ dt) 2 . (9) A suitable spatial orthonormal frame adapted to ZAMOs is given by The nonzero ZAMO kinematical quantities (i.e., acceleration a(n) = ∇ n n and expansion vector θφ(n) α = θ(n) α β eφ β ) all belong to the r-θ 2-plane of the tangent space [19][20][21][22], i.e., On the equatorial plane (θ = π/2) the only nonvanishing components are given by B. Spinning particle motion on the equatorial plane Let us consider a spinning particle moving on the equatorial plane of a Kerr spacetime with the spin vector aligned with the spacetime rotation axis, i.e., Its 4-velocity U decomposed with respect to the ZAMOs is where and γ(U, n) = 1/ 1 − ||ν(U, n)|| 2 is the associated Lorentz factor. Hereafter we will use the abbreviated notation γ(U, n) ≡ γ and ν(U, n)â ≡ νâ. The equations of motion (1) then imply where denotes the Lie relative curvature [20,21] evaluated at θ = π/2, defined by k (lie) = −∂r ln √ g φφ . The frame components of the spin force with respect to ZAMOs are given by where are the nontrivial components of the electric and magnetic parts of the Riemann tensor with respect to ZAMOs, andŝ ≡ s/mM is the dimensionless signed spin magnitude. The remaining nonvanishing component Ft (spin) follows from the condition F (spin) · U = 0. Finally, Eq. (16) must be coupled with the decomposition (14) of providing the evolution of t, r and φ.
In order to solve the full set of equations (16) and (20) we take advantage of the existence of the conserved quantities (6), which becomê whereÊ ≡ E/m andĴ ≡ J/mM are dimensionless. Eq. (21) thus provide two algebraic relations for the frame components νr and νφ of the linear velocity, which once inserted in Eq. (20) finally yield to first order in spin. Let us introduce a conical-like representation of the radial variable, i.e., where both the semi-latus rectum p and the eccentricity e ≥ 0 are dimensionless parameters [23]. Bound orbits have 0 ≤ e < 1 and 0 <Ê < 1 and oscillate between a minimum radius r (per) (periastron, χ = 0) and a maximum radius r (apo) (apastron, χ = π) corresponding to the extremal points of the radial motion, i.e., Unbound (hyperbolic-like) orbits, instead, have eccentricity e > 1 and energy parameterÊ > 1. In particular, a typical scattering orbit starts far from the hole at radial infinity, reaches a minimum approach distance r (per) , and then comes back to radial infinity, corresponding to a scattering angle χ ∈ [−χ (max) , χ (max) ], χ (max) = arccos(−1/e) (see Eq. (23)). In this case the apoastron does not exist anymore, as it corresponds to a negative value of the radial variable. However, the parametrization (24) can still be adopted. The conditions (25) allow one to expressÊ andĴ in terms of (p, e) as followŝ to first order in spin. The geodesic values are given bŷ with F , N and C dimensionless coefficients given by [24] and The upper (lower) sign corresponds to prograde (retrograde) motion. The geodesic angular momentum is simply related toÊ 0 andx 0 byx 0 =Ĵ 0 −âÊ 0 . Formulas valid for retrograde orbits can be obtained from those for prograde orbits byâ → −â andĴ 0 → −Ĵ 0 , under whicĥ The first-order corrections to the energy and angular momentum are given bŷ

III. HYPERBOLIC-LIKE MOTION
After converting the radial equation into an equation for χ through Eq. (23), the azimuthal equation finally becomes to first order in spin, with u p = 1/p. The solution of this equation can be obtained analytically in terms of elliptic functions as where with φ(0) = 0 and Here K(k) and F (ϕ, k) and E(k) and E(ϕ, k) are the complete and incomplete elliptic integrals of the first kind and of the second kind, respectively, defined by whereas are the incomplete and complete elliptic integrals of the third kind, respectively [25].
Unbound orbits which are not captured by the black hole start at an infinite radius at the azimuthal angle φ = φ(−χ (max) ), the radius decreases to its periastron value at φ = 0 and then returns back to infinite value at φ = φ(χ (max) ), undergoing a total increment of ∆φ = φ(χ (max) ) − φ(−χ (max) ). This scattering process is symmetric with respect to the minimum approach (χ = 0) as in the case of a spinless particle, i.e., φ(−χ (max) ) = −φ(χ (max) ), so that ∆φ = 2φ(χ (max) ), and the deflection angle from the original direction of the orbit is δ(u p , e,ŝ) = δ 0 (u p , e) +ŝδŝ(u p , e) , with δ 0 (u p , e) = 2φ 0 (χ (max) ) − π and δŝ(u p , e) = 2φŝ(χ (max) ). Figure 1 shows a typical hyperbolic-like orbit of a spinning particle with spin aligned along the positive z-axis and in the opposite direction compared with the corresponding geodesic orbit of a spinless particle. The orbital parameters are chosen as p = 20 and e = 2, implying that the distance of minimum approach is r (per) ≈ 6.66667M . The trajectory of the spinning particle thus depends on the same parameters p and e as in the spinless case. Once these parameters have been fixed, orbits with different values ofŝ have the same closest approach distance, but different values of energy and angular momentum. On the contrary, setting a pair of values of (Ê,Ĵ) leads to a shift of the periastron due to spin, as shown below. To first order in the rotation parameterâ Eq. (32) becomes where also terms of the orderâŝ have been neglected. Integration then gives The terms φ 0 and φŝ are given in Eq. (24) of Ref. [16], whereas the term linear inâ is with ν = 4eu p /(1 − 2u p + 2eu p ).

A. Periastron shift
A different (equivalent) parametrization of the orbit can be adopted in terms of energyÊ and angular mo-mentumĴ instead of p and e. In this case the periastron distance depends onŝ, and can be determined from the turning points for radial motion.
In the case of a spinless particle the radial and azimuthal equations can be conveniently written in the following factorized form in terms of the dimensionless inverse radial variable u = M/r du dτ which can be combined to yield Here u 1 (0) < u 2 (0) < u 3 (0) are the ordered roots of the equation For hyperbolic orbits we have u 1 (0) < 0 < u ≤ u 2 (0) < u 3 (0) , with u 2 (0) corresponding to the closest approach distance, i.e., r (per) = M/u 2 (0) [23].
The orbital equation for a spinning particle can be cast in a similar form as Eq. (43), i.e., to first order inŝ, where 1) . The corrections to the geodesic values are The periastron shift is then given by to first order in spin. Let us turn to the orbital equation (46), which yields with where the ± sign should be chosen properly during the whole scattering process (with u 1 < 0 < u ≤ u 2 < u 3 ), depending on the choice of initial conditions. For instance, by choosing φ(u 2 ) = 0 at periastron, integration between u 2 and u gives to first order inŝ, where Π(α, 0, m) = F (α, m) and Π(0, m) = K(m), and (53) The total change in φ for the complete scattering process is then given by 2φ(0) determined by Eq. (52) with α = α(0) = −u 1 /(u 2 − u 1 ), so that the deflection angle is δ(Ê,Ĵ,ŝ) = 2φ(0) − π.
A typical orbit using this parametrization is shown in Fig. 2. The two branches (upper +, lower −) join at the periastron on the horizontal axis.

B. Capture by the hole
The condition for capture by the black hole is u 2 (Ê,Ĵ,ŝ) ≥ u 3 (Ê,Ĵ,ŝ), i.e., the cubic at the right hand side of Eq. (46) has a double root at the critical point u 2 = u 3 [26,27]. Solving this condition forĴ then yields the critical value of the dimensionless total angular mo-mentumĴ crit for capture, with associated dimensionless The case of a spinless particle moving on a hyperboliclike geodesic orbit in the equatorial plane is well known in both ultrarelativistic (Ê ≫ 1) and non-relativistic (β 2 ≡ E 2 − 1 ≪ 1) regimes (see, e.g., Ref. [28], Sec. 3.4.5, and references therein). For example, the critical impact parameter for corotating orbits andâ > 0 in the limit E → ∞ is given by (see Eq. For a spinning particle we find b crit (0) u.r. =b (0) 1 + 1 where with w = 3(b 2 (0) −â 2 ). To linear order in the rotational parameterâ (also neglecting termsâŝ) the previous expression becomeŝ In the non-relativistic limit the critical impact parameter for corotating geodesic orbits andâ > 0 is given by (see Eq. (3.4.33) of Ref. [28]) For a spinning particle we find which in the limit of small rotation parameter reduces tô

IV. INCLUDING QUADRATIC-IN-SPIN CORRECTIONS
In the previous section we have seen how (linear) corrections due to spin affect the geodesic scattering angle. According to the positive/negative sign of the spin the scattering angle is larger/smaller than the corresponding geodesic value. It is interesting to consider quadratic-inspin corrections too, not sensitive to the sign of the spin. This requires generalizing the above discussion to include quadrupolar terms in the MPD equations. The presence of a completely general quadrupolar modification to the MPD equations and subsequent derivation of the scattering angle is beyond the scope of this work. We will limit here our analysis to the simpler case of a "black-holelike" extended body only, i.e., to the case of a body with special spin-squared quadrupolar structure, as specified below.
Let us recall that the MPD equations for bodies with general quadrupolar structure read [29][30][31][32] where J µναβ denotes the relativistic quadrupole tensor, which has all the algebraic symmetries of a Riemann tensor. When specialized to the "spin-squared" case it reads corresponding to an extended body with a "black-holelike quadrupole" tensor 2 . As before, it is useful to trade S µν for the spin vector S(u) µ obtained by the spatial duality operation of Eq. (4). The relationship between U and P = mu then follows from differentiating along U the spin supplementary conditions and using Eqs. (62). For the "black-hole-like" extended body a direct calculation shows that P and U remain aligned, i.e., u µ = U µ +O(S 3 ) (see, e.g., Eq. 4.21 of Ref. [33] with C Q = 1). However, the mass m = ||P || is no longer a conserved quantity, and is given by where E(U ) νβ = R µναβ U µ U α is the electric part of the Riemann tensor with respect to U and m 0 denotes the (constant) "bare" mass of the body. For equatorial motion with aligned spin (see Eq. (13)) the spin evolution equations are satisfied with constant signed spin magnitude s, and the orbital motion is fully determined by the conservation of energy and total angular momentum still written in the form (6). The orbital equation then reads du dφ 2 = V(u;Ê,L,ŝ) + O(ŝ 3 ) = V 0 (u;Ê,L) +ŝV 1 (u;Ê,L) with 2 A more general expression uses a polarizability parameter C Q as a proportionality constant in the rhs of Eq. (63), related to the shape of the test body, and can be found for example in [31]; see also Ref. [33], Eqs. (2.6) and (2.7) where C Q = 1 to compare with the present case.
(70) It turns out that theŝ 2 corrections are more involved with respect to the linear ones and the discussion best proceeds by working numerically, or by using approximation schemes such as the post-Newtonian and post-Minkowskian ones. Fig. 2 can be redone then to include theŝ 2 modifications to the orbit; however, with the same choice of parameters as in Fig. 2 the differences cannot be visually appreciated. It is interesting to study the behavior of the various separated contributions φ n (u;Ê,Ĵ) (n = 0, 1, 2) as shown in Fig. 3. At the minimum approach distance the geodesic term φ 0 vanishes (as the full quantity φ(u)), while both the linear and quadratic corrections diverge, one positively (φ 2 ) and the other negatively (φ 1 ), leading to compensating effects.  Fig. 2 (corresponding to the lower branch there), namelyâ = 0.5,Ê ≈ 1.08411 and J ≈ 5.28307. The spin dependent term are both indefinitely growing as the minimum approach distance is reached, but with opposite behavior. With these values of parameters one can compute the minimum approach distance and the result is u2 ≈ 0.15 + 0.0467ŝ + 0.0124ŝ 2 .

V. CONCLUDING REMARKS
We have studied here the hyperbolic scattering of a two body system consisting of a rotating (Kerr) black hole and a spinning test body described according to the Mathisson-Papapetrou-Dixon model. The motion has been assumed to be equatorial, i.e., confined on the reflection-symmetry plane of the Kerr spacetime, with the spin of the body aligned/anti-aligned with that of the hole, implying meaningful simplifications and allowing for a handy treatment. Backreaction effects on the background metric are neglected because of the smallness condition of the characteristic spin length scale with respect to the background curvature length scale implicit in the MPD model.
Taking advantage of the existence of conserved quan-tities related to Killing symmetries, we have been able to analytically solve the "linearized in spin" equations of motion in terms of elliptic integrals. We have computed the scattering angle to linear order in spin by using two equivalent parametrizations of the orbit: in terms of either eccentricity and (inverse) semi-latus rectum, or total energy and angular momentum (which are gaugeinvariant variables). In the former case orbits with different values of the spin magnitude have the same closest approach distance, but different values of energy and angular momentum. On the contrary, a given pair of values of energy and angular momentum (the same for both geodesic and accelerated orbits) leads to a shift of the periastron due to spin. In this case the geodesic value of the scattering angle is modified in such a way that it increases/decreases according to the spin orientation being aligned/anti-aligned with the black hole rotation axis, respectively. Finally, we have shown how to generalize the present analysis to the case of extended bodies with quadrupolar structure, even if in the simplified situation of spinsquared quadrupole interactions for a "black-hole-like" body. The presence of a quadrupolar force term in the equations of motion complicates matters, so that obtaining a closed form expression for the orbit is not as simple as in the linear-in-spin case. We leave to a forthcoming paper the discussion of more general quadrupolar situations, which plays a role in the current analysis of binary dynamics.