Hyperbolic scattering of spinning particles by a Kerr black hole

# Hyperbolic scattering of spinning particles by a Kerr black hole

## Abstract

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.

Kerr black hole; spinning bodies; scattering process.
04.20.Cv

## 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 gauge-invariant information characterizing both unbound and bound motion (see, e.g., Ref. (12), where the gauge-invariant “effective-one-body” function 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 spin-curvature 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 spin-squared 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-hole-like” quadrupolar structure.

We use geometrical units and assume that Greek indices run from to , whereas Latin indices from to .

## 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)

 DPμdτ = −12RμναβUνSαβ≡Fμ(spin), (1) DSμνdτ = 2P[μUν]. (2)

Here (with ) is the total 4-momentum of the body with mass and unit timelke four-velocity ; is the (antisymmetric) spin tensor; and is the timelike unit 4-velocity vector tangent to the body’s “center-of-mass worldline,” parametrized by the proper time (with parametric equations ), 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 , and . Here we shall use the Tulczyjew-Dixon conditions (15); (17), which read

 Sμνuν=0. (3)

The spin tensor can then be fully represented by the spatial vector (with respect to )

 S(u)α=12η(u)αβγSβγ,Sμν=η(u)μναS(u)α, (4)

where we have denoted by the spatial unit volume 3-form (with respect to ) built from the unit volume 4-form which orients the spacetime, with () being the Levi-Civita alternating symbol and the determinant of the metric.

It is also useful to introduce the signed magnitude of the spin vector

 s2=S(u)βS(u)β=12SμνSμν, (5)

which is a constant of motion. Implicit in the MPD model is the condition that the length scale naturally associated with the spin should be very small compared to the one associated with the background curvature (say ), in order to neglect back reaction effects, namely . This condition leads to a simplified set of linearized evolution equations around the geodesic motion. In fact, the total 4-momentum of the particle is still aligned with in this limit, i.e., , the mass of the particle remaining constant along the path. Furthermore, Eq. (2) becomes , implying that the spin vector is parallel transported along .

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 and the total angular momentum associated with the timelike and azimuthal Killing vectors and , respectively, and read

 E=−Pt+12Sαβξα;β,J=Pϕ−12Sαβηα;β. (6)

### ii.1 Kerr spacetime and ZAMO adapted frame

Let us consider the Kerr spacetime, whose metric written in standard Boyer-Lindquist coordinates is given by

 ds2 = −(1−2MrΣ)dt2−4aMrΣsin2θdtdϕ+ΣΔdr2 (7) +Σdθ2+AΣsin2θdϕ2,

with , and . Here and denote the specific angular momentum and the total mass of the spacetime solution, so that the quantity is dimensionless. The inner and outer horizons are located at .

Let us introduce the zero angular momentum observer (ZAMO) family of fiducial observers, with 4-velocity

 n=N−1(∂t−Nϕ∂ϕ), (8)

where and are the lapse and shift functions, respectively. The ZAMO adapted form of the metric then reads

 ds2=−N2dt2+grrdr2+gθθdθ2+gϕϕ(dϕ+Nϕdt)2. (9)

A suitable spatial orthonormal frame adapted to ZAMOs is given by

 e^t = n,e^r=1√grr∂r≡∂^r, e^θ = 1√gθθ∂θ≡∂^θ,e^ϕ=1√gϕϕ∂ϕ≡∂^ϕ. (10)

The nonzero ZAMO kinematical quantities (i.e., acceleration and expansion vector ) all belong to the - 2-plane of the tangent space (19); (20); (21); (22), i.e.,

 a(n) = a(n)^re^r+a(n)^θe^θ ≡ ∂^r(lnN)e^r+∂^θ(lnN)e^θ, θ^ϕ(n) = θ^ϕ(n)^re^r+θ^ϕ(n)^θe^θ (11) ≡ −√gϕϕ2N(∂^rNϕe^r+∂^θNϕe^θ).

On the equatorial plane () the only nonvanishing components are given by

 a(n)^r = Mr2√Δ(r2+a2)2−4a2Mrr3+a2r+2a2M, θ^ϕ(n)^r = −aM(3r2+a2)r2(r3+a2r+2a2M). (12)

### ii.2 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.,

 S(U)=−se^θ. (13)

Its 4-velocity decomposed with respect to the ZAMOs is

 U=γ(U,n)[n+ν(U,n)], (14)

where

 ν(U,n)≡ν(U,n)^re^r+ν(U,n)^ϕe^ϕ, (15)

and is the associated Lorentz factor. Hereafter we will use the abbreviated notation and .

The equations of motion (1) then imply

 dν^rdτ = −γ{(ν^ϕ)2k(lie) + (1−(ν^r)2)[a(n)^r+2θ^ϕ(n)^rν^ϕ]} + 1mγ[(1−(ν^r)2)F^r(spin)−ν^rν^ϕF^ϕ(spin)], dν^ϕdτ = γν^rν^ϕ[k(lie)+a(n)^r+2θ^ϕ(n)^rν^ϕ] + 1mγ[−ν^rν^ϕF^r(spin)+(1−(ν^ϕ)2)F^ϕ(spin)],

where

 k(lie)=−(r3−a2M)√Δr2(r3+a2r+2a2M) (17)

denotes the Lie relative curvature (20); (21) evaluated at , defined by . The frame components of the spin force with respect to ZAMOs are given by

 F^r(spin) = mM^sγ2[(E^θ^θ−E^r^r)ν^ϕ+(1+(ν^ϕ)2)H^r^θ], F^ϕ(spin) = −mM^sγ2[(E^θ^θ−E^ϕ^ϕ)ν^r+ν^rν^ϕH^r^θ], (18)

where

 E^r^r = −M(2r4+5r2a2−2a2Mr+3a4)r4(r3+a2r+2a2M), E^θ^θ = −E^ϕ^ϕ−E^r^r, E^ϕ^ϕ = Mr3, H^r^θ = −3Ma(r2+a2)√Δr4(r3+a2r+2a2M), (19)

are the nontrivial components of the electric and magnetic parts of the Riemann tensor with respect to ZAMOs, and is the dimensionless signed spin magnitude. The remaining nonvanishing component follows from the condition . Finally, Eq. (II.2) must be coupled with the decomposition (14) of the 4-velocity , i.e.,

 dtdτ = γN, drdτ = γν^r√grr, dϕdτ = γ√gϕϕ(ν^ϕ−√gϕϕNϕN), (20)

providing the evolution of , and .

In order to solve the full set of equations (II.2) and (II.2) we take advantage of the existence of the conserved quantities (6), which become

 ^E = Nγ{1−√gϕϕNϕNν^ϕ +M^s[a(n)^rν^ϕ+θ^ϕ(n)^r +√gϕϕNϕN(k(lie)+θ^ϕ(n)^rν^ϕ)]}, ^J = √gϕϕMγ[ν^ϕ−M^s(k(lie)+θ^ϕ(n)^rν^ϕ)], (21)

where and are dimensionless. Eq. (II.2) thus provide two algebraic relations for the frame components and of the linear velocity, which once inserted in Eq. (II.2) finally yield

 ΔM2dtdτ = ^E(r2M2+^a2)+Mr[(2^a+3^s)^E−(2^a+^s)^J]−M3r3^a2^s(^J−^a^E), (drdτ)2 = ^E2−1+2Mr+M2r2[^a2(^E2−1)−^J(^J−2^E^s)] +2M3r3(^J−^a^E)(^J−^a^E−3^E^s)+2M5r5^a^s(^J−^a^E)2, ΔMdϕdτ = ^J−^E^s−2Mr(^J−^a^E−^E^s)−M3r3^a^s(^J−^a^E), (22)

to first order in spin.

Let us introduce a conical-like representation of the radial variable, i.e.,

 r=Mp1+ecosχ, (23)

where both the semi-latus rectum and the eccentricity are dimensionless parameters (23). Bound orbits have and and oscillate between a minimum radius (periastron, ) and a maximum radius (apastron, )

 r(per)=Mp1+e,r(apo)=Mp1−e, (24)

corresponding to the extremal points of the radial motion, i.e.,

 drdτ∣∣r=r(per)=0=drdτ∣∣r=r(apo). (25)

Unbound (hyperbolic-like) orbits, instead, have eccentricity and energy parameter . In particular, a typical scattering orbit starts far from the hole at radial infinity, reaches a minimum approach distance , and then comes back to radial infinity, corresponding to a scattering angle , (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 express and in terms of as follows

 ^E=^E0+^s^E^s,^J=^J0+^s^J^s, (26)

to first order in spin. The geodesic values are given by

 ^E20=1p[(1−e2)2^x20p2+p−(1−e2)], (27)

where

 ^x20=−N∓√N2−4CF2F, (28)

with , and dimensionless coefficients given by (24)

 F = (1−3+e2p)2−4^a2(1−e2)2p3, −N2 = (p−3−e2)+^a2(1+1+3e2p), C = (^a2−p)2, (29)

and

 N2−4CF=16^a2p3{[p2−2p+^a2(1+e2)]2−4e2(p−^a2)2]}. (30)

The upper (lower) sign corresponds to prograde (retrograde) motion. The geodesic angular momentum is simply related to and by . Formulas valid for retrograde orbits can be obtained from those for prograde orbits by and , under which .

The first-order corrections to the energy and angular momentum are given by

 ^E^s = −^x02p5(1−e2)2[(3+e2)^x02+p2], ^J^s = ^E0−^x02p2^E0^J0(3+e2)−2^x0p3^a(1−e2). (31)

## Iii Hyperbolic-like motion

After converting the radial equation into an equation for through Eq. (23), the azimuthal equation finally becomes

 dϕdχ = u1/2p^J0−2up^x0(1+ecosχ)[1+u2p^x20(e2−2ecosχ−3)]1/2[1−2up(1+ecosχ)+^a2u2p(1+ecosχ)2] (32) −^su5/2p^x0(^x0^E0+^a)ecosχ+3[1+u2p^x20(e2−2ecosχ−3)]3/2,

to first order in spin, with . The solution of this equation can be obtained analytically in terms of elliptic functions as

 ϕ(χ)=ϕ0(χ)+^sϕ^s(χ), (33)

where

 ϕ0(χ) = κ2^a2e2u2p√eup^x20(b+−b−){[^J0−2up^x0(1+eb+)][Π(π2−χ2,k+,κ)−Π(k+,κ)] −[^J0−2up^x0(1+eb−)][Π(π2−χ2,k−,κ)−Π(k−,κ)]}, ϕ^s(χ) = κ(^E0^x0+^a)2√eup^x20{K(κ)−F(π2−χ2,κ) (34) −C+6^x20C−2e^x20[E(κ)−E(π2−χ2,κ)+κ√e^x0sinχ(A−2e^x20cosχ)1/2]},

with and

 κ2 = 4eu2p^x201+(e−1)(e+3)^x20, k± = 21+b±, b± = 1−^a2up±√1−^a2^a2eup, u2pC = 1+u2p^x20(e2−3). (35)

Here and and and are the complete and incomplete elliptic integrals of the first kind and of the second kind, respectively, defined by

 F(φ,k) = ∫φ0dz√1−k2sin2z,K(k)=F(π/2,k), E(φ,k) = ∫φ0√1−k2sin2zdz,E(k)=E(π/2,k),

whereas

 Π(ϕ,n,k) = ∫φ0dz(1−nsin2z)√1−k2sin2z, Π(n,k) = Π(π/2,n,k), (37)

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 , the radius decreases to its periastron value at and then returns back to infinite value at , undergoing a total increment of . This scattering process is symmetric with respect to the minimum approach () as in the case of a spinless particle, i.e., , so that , and the deflection angle from the original direction of the orbit is

 δ(up,e,^s)=δ0(up,e)+^sδ^s(up,e), (38)

with and .

Figure 1 shows a typical hyperbolic-like orbit of a spinning particle with spin aligned along the positive -axis and in the opposite direction compared with the corresponding geodesic orbit of a spinless particle. The orbital parameters are chosen as and , implying that the distance of minimum approach is . The trajectory of the spinning particle thus depends on the same parameters and as in the spinless case. Once these parameters have been fixed, orbits with different values of have the same closest approach distance, but different values of energy and angular momentum. On the contrary, setting a pair of values of (, ) leads to a shift of the periastron due to spin, as shown below.

To first order in the rotation parameter Eq. (32) becomes

 dϕdχ = 1√1−6up−2eupcosχ−4^a^E0up^J01(1−6up−2eupcosχ)3/2(1−2up−2eupcosχ) (39) − ^s^E0up^J0ecosχ+3(1−6up−2eupcosχ)3/2 ≡ ddχϕ0+ddχϕ^a+ddχϕ^s,

where also terms of the order have been neglected. Integration then gives

 ϕ(χ)=ϕ0(χ)+ϕ^a(χ)+ϕ^s(χ). (40)

The terms and are given in Eq. (24) of Ref. (16), whereas the term linear in is

 ϕ^a(χ) = ^a^E0^J0κ√eup[E[(π−χ)/2,κ]−E(κ)1−6up−2eup−Π[(π−χ)/2,ν,κ]−F[(π−χ)/2,κ]1−2up+2eup − κ√eup1−6up−2eupsinχ√1−6up−2eupcosχ],

with .

### iii.1 Periastron shift

A different (equivalent) parametrization of the orbit can be adopted in terms of energy and angular momentum instead of and . In this case the periastron distance depends on , 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

 (dudτ)2 = 2^x2M2u4(u−u1(0))(u−u2(0))(u−u3(0)), dϕdτ = 2^xM^a2u2u4(0)−u(u−u+)(u−u−), (42)

which can be combined to yield

 (dudϕ)2 = ^a42(u−u+)2(u−u−)2(u4(0)−u)2× (43) (u−u1(0))(u−u2(0))(u−u3(0)).

Here are the ordered roots of the equation

 u3−(^x2+2^a^E^x+^a2)u22^x2+u^x2+^E2−12^x2=0, (44)

with , whereas

 u±=Mr±,u4(0)=^J2^x. (45)

For hyperbolic orbits we have , with corresponding to the closest approach distance, i.e., (23).

The orbital equation for a spinning particle can be cast in a similar form as Eq. (43), i.e.,

 (dudϕ)2 = A^a42(1−^sBu)(u−u+)2