High-energy hyperbolic scattering by neutron stars and black holes

# High-energy hyperbolic scattering by neutron stars and black holes

Donato Bini    Andrea Geralico Istituto per le Applicazioni del Calcolo “M. Picone,” CNR, I-00185 Rome, Italy
July 13, 2019
###### Abstract

We investigate the hyperbolic scattering of test particles, spinning test particles and particles with spin-induced quadrupolar structure by a Kerr black hole in the ultrarelativistic regime. We also study how the features of the scattering process modify if the source of the background gravitational field is endowed with a nonzero mass quadrupole moment as described by the (approximate) Hartle-Thorne solution. We compute the scattering angle either in closed analytical form, when possible, or as a power series of the (dimensionless) inverse impact parameter. It is a function of the parameters characterizing the source (intrinsic angular momentum and mass quadrupole moment) as well as the scattered body (spin and polarizability constant). Measuring the scattering angle thus provides useful information to determine the nature of the two components of the binary system undergoing high-energy scattering processes.

High energy scattering; spinning particles; Kerr black hole; Hartle-Thorne solution
04.20.Cv

## I Introduction

The gravitational scattering of a two-body system consisting of two black holes or a black hole and a neutron star has recently received much attention Glampedakis:2002ya (); Shibata:2008rq (); Sperhake:2008ga (); Sperhake:2009jz (); Sperhake:2012me (), in connection with the possibility to detect the associated emission of gravitational wave (GW) signals by the advanced phase of currently operating GW Earth-based interferometers as well as (more likely) by future, forthcoming satellite missions involving space-based interferometers (see, e.g., Refs. ligo (); virgo (); Abbott:2016blz (); lisa ()).

In spite of the fact that the occurrence of hyperbolic encounters, in general, is expected to be as probable as that of coalescing phenomena, the analytical treatment of either case is not equally developed in the literature. For example, in the case of coalescence several analytical and semi-analytical methods are available, in addition to the full numerical relativity (NR) simulations: Post-Newtonian (PN), Post-Minkowskian (PM), effective-one-body (EOB), perturbation theory and gravitational self-force (GSF), etc. Buonanno:1998gg (); Buonanno:2000ef (); Damour:2000we (); Damour:2001tu (); Schafer:2009dq (); Blanchet:2013haa (); Barack:2009ux (); Poisson:2011nh (); Detweiler:2008ft (); Barack:2011ed (). The same methods can be applied, in principle, also to scattering processes, but with additional complications. For instance, the spectrum of the emitted waves in the latter case spreads all over the infinite range of all possible frequencies, whereas in the case of coalescence it is peaked around a single frequency (that of the inspiralling, quasi-circular motion) up to the very end of the process, i.e., the merger phase. This fact mainly explains why there are no analytical results coming from perturbation theory yet.

In general relativity, the scattering problem is well established and fully solved when one body has mass much smaller than the other so that backreaction effects can also be neglected, i.e., in the test-field approximation. In this regime the problem reduces to study the hyperbolic-like motion of a pointlike massive particle in a given background spacetime, starting at radial infinity along a certain asymptotic direction, reaching a minimum approach distance to the gravitational source, and then coming back to radial infinity being deflected from the original direction of an angle which is the main observable of the process. For instance, if the source of the gravitational field is a black hole, the scattering angle can be expressed in terms of Elliptic integrals and is a function of the energy and angular momentum of the particle (a compendium can be found in the textbook of Chandrasekhar Chandrasekhar:1985kt ()). A renewed interest in this problem can be found in the recent literature, with studies involving PN and PM expansions of the scattering angle Damour:2014afa (); Damour:2016gwp (); Hopper:2017qus (); Bini:2017xzy (); Bini:2017wfr (); Vines:2017hyw ().

We have studied in previous works how the features of the scattering process modify if the scattered particle is no more pointlike, but is endowed with an internal structure given by its spin Bini:2017ldh (); Bini:2017pee (). The motion is no longer geodesic, but accelerated due to the spin-curvature coupling, according to the the Mathisson-Papapetrou-Dixon (MPD) model Mathisson:1937zz (); Papapetrou:1951pa (); tulc59 (); Dixon:1970zza (); ehlers77 (). The aim of the present paper is to compute the scattering angle in the high-energy limit by adding more structure to both the scattered body and the source of the gravitational field. To this end, we will consider extended bodies with spin-induced quadrupole moment, using available results for quadrupolar particle motion in both Schwarzschild and Kerr spacetimes Bini:2013nw (); Bini:2013uwa (); Bini:2015zya (). We will provide corrections to the scattering angle with respect to the spinless case up to the quadratic order in spin, also depending on the “shape” of the body, covering both cases of “black hole-like” and “neutron star-like” objects. Furthermore, we will investigate the companion situation of a structureless particle moving along a hyperbolic-like geodesic orbit in the spacetime of a (slowly) rotating (slightly) deformed source endowed with a mass quadrupole moment, as described by the (approximate) Hartle-Thorne solution Hartle:1968si (). We will provide corrections to the scattering angle with respect to the Kerr case which are linear in the quadrupole parameter of the source.

We follow the notation and conventions of Ref. Misner:1974qy (). The signature of the metric is and Greek indices run from 0 to 3, whereas Latin ones from 1 to 3.

## Ii Hyperbolic-like equatorial motion in a Kerr spacetime

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

 ds2 = −(1−2MrΣ)dt2−4aMrΣsin2θdtdϕ+ΣΔdr2 (1) +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 .

The motion in the equatorial plane is governed by the geodesic equations (see, e.g., Ref. Bini:2016ovy ())

 ΔM2dtdτ = (^E−M2r2^a^x)(r2M2+^a2)+Δr2^a^x, (drdτ)2 = (^E−M2r2^a^x)2−Δr2(1+M2r2^x2), ΔMdϕdτ = (1−2Mr)^x+^a^E, (2)

where and are the conserved energy and azimuthal angular momentum per unit mass of the particle, respectively, and (so that is dimensionless).

The radial and azimuthal equations can be conveniently written in the following factorized form in terms of the dimensionless inverse radial variable

 M2(dudτ)2 = 2^x2u4(u−u1)(u−u2)(u−u3), Mdϕdτ = 2^x^a2u2u4−u(u−u+)(u−u−), (3)

which can be combined to yield

 (dudϕ)2 = ^a42(u−u+)2(u−u−)2(u4−u)2× (4) (u−u1)(u−u2)(u−u3).

Here are the ordered roots of the equation

 u3−(^x2+2^a^E^x+^a2)u22^x2+u^x2+^E2−12^x2=0, (5)

while

 u±=Mr±,u4=^L2M^x=12(1+^a^E^x). (6)

For hyperbolic orbits we have , with corresponding to the closest approach distance Chandrasekhar:1985kt ().

We are interested in the ultrarelativistic limit () for fixed values of the impact parameter (so that is dimensionless). In this limit the orbital equation (4) becomes

 dϕdu = ±√2^a2u4−u(u−u+)(u−u−)× (7) 1√(u−u1)(u2−u)(u3−u)+O(^E−2),

where the sign should be chosen properly during the whole scattering process (with ), depending on the choice of initial conditions. Here are the ordered roots of the equation

 2(1−^aα)2u3−(1−^a2α2)u2+α2=0, (8)

with

 α=1^b,^b=^LM√^E2−1, (9)

while

 u±=Mr±,u4=12(1−^aα). (10)

They are given by

 u1 = ¯u[1−2cos(2Θ3)], u2 = ¯u[1+2cos(2Θ+π3)], u3 = ¯u[1−2cos(2(Θ+π)3)], (11)

where

 ¯u=161+^aα1−^aα,cosΘ=3√3α√1−^a2α2(1+^aα)2, (12)

and we have assumed for simplicity. The condition gives the critical value of corresponding to capture by the black hole. For instance, the critical impact parameter for corotating orbits in the limit is given by (see, e.g., Ref. Frolov:1998wf ())

 ^bcrit=^a+8cos3(13arccos(−^a)), (13)

so that .

### ii.1 Scattering angle

The solution to Eq. (7) can be expressed in terms of elliptic integrals as

 ϕ(u)=2√2^a2(u+−u−)√u3−u1(A+(u)−A−(u)), (14)

where

 A±(u) = u4−u±u1−u±(Π(ψ,β±,m)−Π(β±,m)), ψ = ψ(u;u1,u2)=√u−u1u2−u1, β± = u2−u1u±−u1=1[ψ(u±;u1,u2)]2, m = √u2−u1u3−u1=1ψ(u3;u1,u2), (15)

having assumed at periastron. Here

 Π(φ,n,k) = ∫φ0dz(1−nsin2z)√1−k2sin2z, Π(n,k) = Π(π/2,n,k), (16)

are the incomplete and complete elliptic integrals of the third kind, respectively Gradshteyn (). The total change in for the complete scattering process is then given by determined by Eq. (14) with , so that the deflection angle is . Its behavior as a function of is shown in Fig. 1 for selected values of .

For small values of , i.e., for large values of the impact parameter, we find

 δ(^a,α) = 4α+(154π−4^a)α2+(1283−10π^a+4^a2)α3+(346564π−192^a+28516π^a2−4^a3)α4 (17) +(35845−6932π^a+512^a2−27π^a3+4^a4)α5 +(255255256π−179203^a+7969564π^a2−32003^a3+119532π^a4−4^a5)α6 +(983047−32818532π^a+27136^a2−133654π^a3+1920^a4−1954π^a5+4^a6)α7 +(33463930516384π−172032^a+1156305152048π^a2−89600^a3+76611151024π^a4−3136^a5+15645256π^a6−4^a7)α8 +O(α9).

In the Schwarzschild limit we have and and , so that

 ϕ(u)=2√2√u3−u1[K(m)−F(ψ,m)], (18)

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

 F(φ,k)=∫φ0dz√1−k2sin2z,K(k)=F(π/2,k). (19)

The roots , and are given by Eqs. (II) with and , whence

 u2−u1 = 1√3cos(23arccos(3√3α)+π6), u3−u1 = 1√3cos(23arccos(3√3α)−π6). (20)

The scattering angle (17) then becomes

 δ(0,α) = 4α+154πα2+1283α3+346564πα4 (21) + 35845α5+255255256πα6+983047α7 + 33463930516384πα8+O(α9),

and can be rewritten as

 δ(0,α)=δπ/(0,α)+πδπ(0,α), (22)

with

 δπ/(0,α) = 4α+1283α3+35845α5+983047α7 +O(α9), δπ(0,α) = 154α2+346564α4+255255256α6 (23) +33463930516384α8+O(α9).

In the Kerr case, instead, one recognizes that the /non- structure is related to even/odd powers of , i.e., terms with even values of have a .

### ii.2 Effects induced by the multipolar structure of the moving body

The hyperbolike-like equatorial motion of a particle endowed with spin-induced quadrupolar structure has been investigated in Ref. Bini:2017pee () according to the Mathisson-Papapetrou-Dixon (MPD) model Mathisson:1937zz (); Papapetrou:1951pa (); tulc59 (); Dixon:1970zza (); ehlers77 (). In the case of aligned spin the signed magnitude of the spin vector is a constant of motion and the orbital equation reads

 (dudϕ)2 = V(u;^E,^L,^s)+O(^s3), (24)

with

 V(u;^E,^L,^s) = ^Δ2^w2{(^E2−1)(1+^a2u2)−^L2u2+(u+^x2u3)(2+3^Δu2^s2) (25) + ^Δu3^w[−(^E^x+^a)[2^x^s+(2^E+3^a^x2u3^w)^s2]+^x3u3^s2 + (CQ−1)[(1−3^x2u2)^x(1−2u)+(1−9^x2u2)^E^a−6^x(^Δ−^E2)]]},

where

 ^Δ=1−2u+^a2u2,^w=^L−2^xu. (26)

The dimensionless energy and angular momentum and as well as the test-body’s orbital angular momentum at infinity (and also ) are now defined by using the conserved bare mass , namely

 ^E = Em0,^L=^J−^E^s, ^L = Lm0M,^s=sm0M. (27)

Note that Eq. (25) has been generalized by including the polarizability parameter related to the shape of the body (see, e.g., Refs. Laarakkers:1997hb (); Steinhoff:2012rw (); Hinderer:2013uwa (); Bini:2015zya ()) with respect to that of Ref. Bini:2017pee (), where only the simplest case corresponding to a “black-hole-like” extended body was considered. In particular in Ref. Laarakkers:1997hb () the (fitted) values of for different equations of state of a rotating neutron star are explicitly shown (see Table VII there).

In the ultrarelativistic limit and for small values of , defined in terms of ,

 α≡αJ=√^E2−1^J, (28)

the deflection angle turns out to be

 δ(^s,^a,α) = δK(^a,α)+(CQ−1)^s2δ^s2(^a,α) (29) +^E−2δ(−2)(^s,^a,α)+O(^E−4),

with

 δ^s2(^a,α) = 4α3+(13516π−12^a)α4+(7685−45π^a+24^a2)α5+(1732564π−1152^a+427532π^a2−40^a3)α6 (30) +(4608−103954π^a+4608^a2−12154π^a3+60^a4)α7 +(160810652048π−53760^a+1673595128π^a2−13440^a3+75285128π^a4−84^a5)α8 +O(α9).

Therefore, in the high-energy limit there is no contribution linear in spin to the scattering angle. Corrections to the spinless (geodesic) value start at the order , provided that , i.e., for extended bodies which are not “black-hole-like.” This circumstance reflects the peculiarity of the MPD model for these objects, as already pointed out in Ref. Bini:2015zya () for what concerns the alignment of the generalized momentum and the unit tangent vector to the world line representative of the body (see Eq. (4.21) there).

Corrections linear in spin appear at the order . In fact, we find the following expression for

 δ(−2)(^s,^a,α)=δ(−2)K(^a,α)+^sδ(−2)^s(^a,α)+(CQ−1)^s2δ(−2)^s2(^a,α), (31)

with

 δ(−2)K(^a,α) = 2α+(3π−2^a)α2+(48−9π^a+2^a2)α3+(3154π−240^a+332π^a2−2^a3)α4 +(1280−22054π^a+672^a2−512π^a3+2^a4)α5 +(13513564π−11520^a+166958π^a2−1440^a3+2858π^a4−2^a5)α6 +(1720325−148648564π^a+55040^a2−463058π^a3+2640^a4−3758π^a5+2^a6)α7 +(14549535256π−22364165^a+17162145128π^a2−188160^a3+42430532π^a4−4368^a5+94516π^a6−2^a7)α8 +O(α9), δ(−2)^s(^a,α) = 2α2+(3π−4^a)α3+(48−272π^a+6^a2)α4+(3154π−320^a+33π^a2−8^a3)α5 +(1280−1102516π^a+1120^a2−2554π^a3+10^a4)α6 +(13513564π−13824^a+5008516π^a2−2880^a3+8558π^a4−12^a5)α7 +(1720325−3468465128π^a+77056^a2−32413532π^a3+6160^a4−262516π^a5+14^a6)α8 +O(α9), δ(−2)^s2(^a,α) = 2α3+(9π−6^a)α4+(224−932π^a+12^a2)α5+(787516π−1696^a+10958π^a2−20^a3)α6 (32) +(9984−387458π^a+6816^a2−24758π^a3+30^a4)α7 +O(α9).

Again, terms quadratic in spin vanish for .

Let us recall that using the conserved energy and angular momentum associated with the Killing vectors and of the Kerr spacetime evaluated at one finds the relation (see Ref. Bini:2017pee (), footnote 1 on p. 8)

 ^J=^L+^E^s. (33)

Let and be defined as

 ^bL=^L√^E2−1,^bJ=^J√^E2−1. (34)

Inserting these expressions in Eq. (33) leads to

 √^E2−1^bJ=√^E2−1^bL+^E^s, (35)

or

 ^bJ=^bL+^E√^E2−1^s, (36)

which in the limit implies simply a shift by in the two definitions, namely

 ^bJ=^bL+^s. (37)

Let us define then

 αL=1^bL,αJ=1^bJ, (38)

so that

 αL=αJ1−αJ^s≈αJ+α2J^s+α3J^s2+O(^s3). (39)

Consistently, Eq. (24) implies in the limit . It is clear that our results (29) and (30) could have been formulated by using instead of . In terms of the spin corrections to the scattering angle would then appear already at leading order, and not at order . In fact, in this case we find

 δ(^s,^a,αL) = δK(^a,αL)+^sδ^s(^a,αL) (40) +^s2δ^s2(^a,αL)+O(^E−2),

with

 δ^s(^a,αL) = −4α2L+(8^a−152π)α3L +(−128+30π^a−12^a2)α4L +(−346516π+768^a−2854π^a2+16^a3)α5L +O(α6L), δ^s2(^a,αL) = 4CQα3L+[454π−12^a (41) +(13516π−12^a)(CQ−1)]α4L +[256−60π^a+24^a2 +(7685−45π^a+24^a2)(CQ−1)]α5L +O(α6L),

which agree with Eq. (5.23) of Ref. Barrabes:2003ey () (see also Ref. Barrabes:2004gn ()), where a completely different method were actually used.

### ii.3 Two-body scattering angle in PM theory

In a recent work Damour has shown how to translate the ultra high-energy quantum scattering results of Amati, Ciafaloni and Veneziano Amati:1987wq (); Amati:1987uf () into a classical relativistic gravitational two-body scattering angle in PM perturbation theory Damour:2017zjx (). The scattering process between the two (nonrotating) bodies is equivalently described in terms of the scattering of a massless particle in a modified Schwarzschild metric of the form

 ds2=−ftdt2+f−1rdr2+r2(dθ2+sin2θdϕ2), (42)

with

 ft(u) = (1−2u)(1+152u2−18u3+184516u4+…), fr(u) = (1−2u), (43)

where is a dimensionless “inverse radial” variable. The scattering equation in this case reduces to

 dϕdu=±√N(u)D(u), (44)

where

 N(u) = 1+152u2−18u3+184516u4, D(u) = α2−u2(1−2u+152u2−33u3 (45) +242116u4−18458u5),

and the scattering angle results in

 δ(α)=4α+563α3+O(α5), (46)

with vanishing both the coefficients at and , as shown in Ref. Damour:2017zjx (). Note that to integrate Eq. (44) we have used the prescription given there (see Eqs. (7.23)-(7.24)).

A general modified Schwarzschild metric has the form (42), with functions

 ft(u) = (1−2u)(1+c(2)tu3+c(3)tu3+c(4)tu4+…), fr(u) = (1−2u)(1+c(2)ru2+c(3)ru3+c(4)ru4+…).

This includes spherically symmetric solutions from extended theories of gravity (see e.g., Refs. Maeda:2006hj (); Sotiriou:2013qea (); Mukherjee:2017fqz ()), the functions and being parametrized by a number of coefficients generically associated with scalar charges. The scattering angle in the high energy limit turns out to be

 δ(c(i)t,c(i)r,α) = 4α+(154−14c(2)r−12c(2)t)πα2 (48) + (1283−8c(2)t−83c(2)r−23c(3)r − 2c(3)t)α3+(346564+98(c(2)t)2 + 964(c(2)r)2−1058c(2)t−10532c(2)r + 38c(2)rc(2)t−1516c(3)r−154c(3)t − 316c(4)r−34c(4)t)πα4+O(α5).

Note that the absence in of even powers of is compatible with the choice

 c(2)t = 152−12c(2)r, c(4)t = 40516−5c(3)t−258c(2)r+516(c