# Scattering Angles in Kerr Metrics

Poul H. Damgaard1, Jitze Hoogeveen1, Andres Luna1, and Justin Vines2
1Niels Bohr International Academy
Niels Bohr Institute, University of Copenhagen
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
2Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Muhlenberg 1, Potsdam 14476, Germany

###### Abstract

Scattering angles for probes in Kerr metrics are derived for scattering in the
equatorial plane of the black hole. We use a method that naturally resums all
orders in the spin of the Kerr black hole, thus facilitating comparisons with
scattering-angle computations based on the Post-Minkowskian expansion from
scattering amplitudes or worldline calculations. We extend these results to
spinning black-hole probes up to and including second order in the probe spin
and any order in the Post-Minkowskian expansion, for probe spins aligned with
the Kerr spin. When truncating to third Post-Minkowskian order, our results
agree with those obtained by amplitude and worldline methods.

###### pacs:

04.60.-m, 04.62.+v, 04.80.Cc

††preprint:

##  I Introduction

The gravitational bending of light around the Sun famously provided one of the
earliest observational checks on predictions from Einstein’s general theory of
relativity. Since then, gravitational optics has become one of the common
tools of observational astronomy, in fact now used inversely to infer mass
distributions of massive objects partly obstructing light in the direct line
of sight. The light-like bending angles of general relativity thus have a
central position in modern physics.

Recently, scattering angles of massive objects have attracted renewed
attention from an entirely different direction. For gravitational-wave
predictions of black-hole mergers one needs the effective Hamiltonian that
governs the dynamics of two massive bodies in general relativity. It was
suggested in Ref. Damour:2016gwp  that an improvement of traditional
analytical approaches based on Post-Newtonian expansions could come from the
Post-Minkowskian expansion of the scattering regime. This suggests that modern
amplitude methods of the quantized theory to great advantage may be used to
infer the effective two-body interactions of general relativity Bjerrum-
Bohr:2018xdl ; Cheung:2018wkq ; Cristofoli:2019neg  after properly removing
all non-classical contributions at loop level Kosower:2018adc . In a short
span of time there has been enormous progress in this direction, with results
to third Post-Minkowskian order now fully under control Bern:2019nnu ;
Antonelli:2019ytb ; Parra-Martinez:2020dzs ; Mougiakakos:2021ckm ;
Herrmann:2021tct ; DiVecchia:2020ymx ; Damour:2020tta ; Bjerrum-Bohr:2021vuf ;
Damgaard:2021ipf ; Brandhuber:2021eyq . Even amplitude computations to fourth
Post-Minkowskian order Bern:2021dqo  and, in the probe limit, fifth Post-
Minkowskian order Bjerrum-Bohr:2021wwt  have now been considered. A parallel
track based on effective field theory in the worldline formalism offers
results at similar high orders in the Post-Minkowskian expansion Kalin:2020mvi
; Kalin:2022hph ; Kalin:2020fhe ; Liu:2021zxr ; Dlapa:2021npj ; Mogull:2020sak
; Jakobsen:2021smu ; Jakobsen:2022fcj ; Jakobsen:2022psy ; Saketh:2022wap .
For recent reviews, see, $e.g.$, refs. Bjerrum-Bohr:2022blt ; Kosower:2022yvp
; Buonanno:2022pgc .

Adding classical spin to the Post-Minkowskian expansion leads to interesting
challenges in the amplitude approach due to the traditional barrier at spin-2
in quantum field theory (although recent progress in describing massive higher
spin states has been made in Refs. Bautista:2021wfy ; Chiodaroli:2021eug ;
Cangemi:2022abk ; Ochirov:2022nqz ). Results at the first Post-Minkowskian
order and all orders in the spins were derived by solving Einstein’s field
equations directly Vines:2017hyw . Amplitude-based and worldline approaches
have since made substantial progress towards obtaining Post-Minkowskian
results with spin Guevara:2017csg ; Vines:2018gqi ; Guevara:2018wpp ;
Chung:2018kqs ; Maybee:2019jus ; Chung:2019duq ; Damgaard:2019lfh ;
Aoude:2020onz ; Bern:2020buy ; Haddad:2020tvs ; Kosmopoulos:2021zoq ;
Haddad:2021znf ; Chen:2021qkk ; Bern:2022kto ; Alessio:2022kwv ;
delaCruz:2022nlj ; FebresCordero:2022jts ; Kim:2022iub ; Guevara:2020xjx ;
Arkani-Hamed:2019ymq ; Cristofoli:2021jas ; Siemonsen:2019dsu ;
Bautista:2019tdr ; Adamo:2021rfq ; Menezes:2022tcs ; Riva:2022fru . In order
to have known limits in which to compare amplitude-based results for
scattering angles with those computed directly from general relativity, we
here reconsider the classical problem of scattering in the equatorial plane of
a Kerr black hole. We restrict the spin of the black hole to be parallel with
the orbital angular momentum and the motion is therefore restricted to the
equatorial plane. A single scattering angle can then describe the asymptotic
motion and the situation is rather similar to that of scattering around a
Schwarzschild black hole except for the fact that the scattering angle will
depend on whether the black hole spin is pointing in the same direction as the
orbital angular momentum, or opposite. We will be working with metrics of
signature $(-,+,+,+)$ throughout.

One of the interesting observations of the first-order Post-Minkowskian result
of ref. Vines:2017hyw  was that the spins, to that order in the Post-
Minkowskian expansion, could be provided in an exact (resummed) form. The same
resummed form naturally appears also from amplitude calculations to the same
order in the Post-Minkowskian expansion Guevara:2018wpp  and remnants of such
a structure can be found also at second Post-Minkowskian order, at least up to
fourth order in one of the spins Guevara:2018wpp . It turns out that this
structure of resummed spin is a general feature of the probe limit: if the
probe is spinless, we can show this to any order in the Post-Minkowskian
expansion. Taking the lightlike limit, and expanding in the black hole spin,
we recover the Kerr results for the bending of light Iyer:2009hq . As we shall
detail below, there are several other checks on our results as well.

Finally, an interesting and challenging problem is that of adding spin to the
probe. We shall derive expressions for the Kerr scattering angle for a
spinning probe, with the probe spin aligned with both the orbital angular
momentum and the Kerr spin, valid up to (and including) second order in the
black-hole probe spin. In principle, these calculations can be carried through
to arbitary Post-Minkowskian order and we illustrate that below by providing
analytical expressions up to and including ${\mathcal{O}}(G^{5})$. Truncating
to third Post-Minkowskian order our results agree with those of refs.
Vines:2018gqi ; Jakobsen:2022fcj ; FebresCordero:2022jts . The general
expressions we present here for the probe limit both without and with spin may
be useful for checks on amplitude computations at higher orders in the Post-
Minkowskian expansion.

##  II Warm-up: Scattering in Schwarzschild Metrics

Before we turn to the main subject, it is instructive to describe our method
in a far simpler setting that still retains the important features. We
therefore first consider the computationally easier problem of scattering
around a Schwarzschild black hole. This will highlight the importance of
choosing suitable variables to simplify the calculation.

Consider first a scattering problem in a spherically symmetric effective
potential $V_{\rm eff}(r)$ for which the radial momentum reads

$p_{r}~{}=~{}\sqrt{p_{\infty}^{2}-\frac{L^{2}}{r^{2}}-V_{\rm eff}(r)}~{},$ (1)

where $p_{\infty}$ is the three-momentum at radial infinity and $L$ is the
conserved angular momentum. As is well known from analytical mechanics (say,
fom Hamilton-Jacobi theory), the scattering angle $\chi$ in such a situation
follows from the relation

$\frac{\chi}{2}=-\frac{\partial}{\partial
L}\int_{r_{m}}^{\infty}dr\sqrt{p_{\infty}^{2}-\frac{L^{2}}{r^{2}}-V_{\rm
eff}(r)}-\frac{\pi}{2},$ (2)

where $r_{m}$ is the turning point of the orbit. This is determined by the
condition $p_{r}(r_{m})=0$, i.e. at the (real and positive) point where the
integrand vanishes. One may legitimately move the derivative with respect to
$L$ inside the integral since the boundary term at $r_{m}$ vanishes by
definition. The scattering angle can thus be computed from

$\frac{\chi}{2}=L\int_{r_{m}}^{\infty}\frac{dr}{r^{2}}\frac{1}{\sqrt{p_{\infty}^{2}-\frac{L^{2}}{r^{2}}-V_{\rm
eff}(r)}}-\frac{\pi}{2}~{},$ (3)

which not only appears to depend on $r_{m}$ but even seems to be singular due
to the integrand diverging at the endpoint. As is well known, these subtleties
are only apparent and the whole expression is completely well defined. In
reality, though, except for a very small set of integrable potentials $V_{\rm
eff}(r)$, we wish to find the scattering angle as a perturbative series in the
strength of the potential. A very compact solution to this problem was
recently provided in ref. Bjerrum-Bohr:2019kec , where the scattering angle is
given in terms of a series of finite integrals, with one new integral
appearing for each order in perturbation theory. The final result reads

$\chi=\sum_{k=1}^{\infty}\frac{2b}{k!}\int_{0}^{\infty}du\left(\frac{d}{du^{2}}\right)^{k}\frac{[V_{\rm
eff}(r)]^{k}r^{2(k-1)}}{p_{\infty}^{2k}}~{}.$ (4)

Here $r^{2}=u^{2}+b^{2}$ and the impact parameter $b$ has been introduced in
the usual way by $b=L/p_{\infty}$. Note that all integrals now run along the
full positive line, and they thus become elementary for power-law potentials.
One important example which immediately fits into this framework is that of
scattering in a Schwarzschild metric expressed in isotropic coordinates, and
thus with line element

$ds^{2}=-\left(\frac{1+\frac{GM}{2r}}{1-\frac{GM}{2r}}\right)^{2}dt^{2}+\left(1+\frac{GM}{2r}\right)^{4}\left(dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right).$ (5)

This translates into the effective potential Damgaard:2021rnk

$V_{\rm eff}(r)=m^{2}(\gamma^{2}-1)-m^{2}\left(1+{GM\over
2r}\right)^{4}\left(\gamma^{2}\left(1+{GM\over 2r}\over 1-{GM\over
2r}\right)^{2}-1\right)~{}.$ (6)

Here $\gamma=1/\sqrt{1-v^{2}}$ is the usual Lorentz contraction factor and we
have chosen the scattering to take place in the equatorial plane of
$\theta=\pi/2$. Writing down the Schwarzschild scattering angle to any order
in $G$ is thus straightforward upon expansion of the potential in a power
series and subsequent use of eq. (4).

We now wish to generalize the derivation of ref. Bjerrum-Bohr:2019kec  so that
it is amenable to more general metrics. We will follow the standard approach
based on solving for the radial momentum $p_{r}$. However, for general
metrics, and in particular also for the Schwarzschild metric in coordinates
different from isotropic, this will not lead to an expression of the simple
form (1). In order to retain as many as possible of the simplifying features
of the approach followed in ref. Bjerrum-Bohr:2019kec  we will make a suitable
change of variables to a metric which in the limit of no interactions
($G\rightarrow 0$) reduces to the metric of flat Minkowski space in spherical
coordinates. As a consequence, we recover the simple relation

$p_{r}=\sqrt{p_{\infty}^{2}-L^{2}/r^{2}}$ (7)

in that limit. We will refer to metrics with this property as being in normal
form. An example will best illustrate what we mean. To this end, let us
consider the Schwarzschild metric, but now written in standard Schwarzschild
coordinates

$ds^{2}=-\left(1+\frac{r_{s}}{r}\right)^{2}dt^{2}+\left(1-\frac{r_{s}}{r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (8)

where we have defined $r_{s}\equiv 2GM$. For the obvious choice $\theta=\pi/2$
the metric leads to

$\displaystyle
p_{r}^{2}=\frac{(E^{2}-m^{2})r^{3}+m^{2}r^{2}r_{s}-L^{2}(r-r_{s})}{r(r-r_{s})^{2}}.$
(9)

In the free case, the radial momentum eq. (9) reduces to
$p_{r}^{2}=p_{\infty}^{2}-L^{2}/r^{2}$. It is thus possible to separate out
this part and write, equivalently,

$\displaystyle
p_{r}^{2}=p_{\infty}^{2}-\frac{L^{2}}{r^{2}}-r_{s}\left(\frac{m^{2}(r-r_{s})-E^{2}(2r-r_{s})+\frac{L^{2}}{r^{2}}(r-r_{s})}{(r-r_{s})^{2}}\right),$
(10)

where the last term involving the bracket clearly vanishes as $G\rightarrow
0$, and we have simply added and subtracted $L^{2}/r^{2}$. The advantage of
this rewriting is that it makes it natural to interpret the remainder

$U(r,L)~{}\equiv~{}r_{s}\frac{m^{2}(r-r_{s})-E^{2}(2r-r_{s})+\frac{L^{2}}{r^{2}}(r-r_{s})}{(r-r_{s})^{2}}$
(11)

formally as an $L$-dependent potential. The Schwarzschild metric in these
coordinates is therefore already of normal form, and the scattering angle can
thus still be written as

$\frac{\chi}{2}=-\frac{\partial}{\partial
L}\int_{r_{m}}^{\infty}dr\sqrt{p_{\infty}^{2}-\frac{L^{2}}{r^{2}}-U(r,L)}-\frac{\pi}{2},$
(12)

but the derivative will now also act on $U(r,L)$. We hence need to generalize
the derivation of ref. Bjerrum-Bohr:2019kec  to this new situation. Moreover,
we discover that the $L$-derivative of eq. (12), which is so natural from the
canonical formalism, can be disposed of so as to open up for more general
situations including the spin of the probe. Let us jump ahead to the final
result which turns out to be surprisingly simple. In order to introduce it, we
write the scattering angle in the form

$\frac{\chi}{2}=\int_{r_{m}}^{\infty}dr\frac{d\phi}{dr}-\frac{\pi}{2}=-\int_{r_{m}}^{\infty}\frac{h(r)}{p_{\infty}}\left(1-b^{2}/r^{2}-\frac{U(r,b)}{p_{\infty}^{2}}\right)^{-1/2}-\frac{\pi}{2}~{},$
(13)

where

$h(r)~{}\equiv~{}-\frac{d\phi}{dr}p_{r}.$ (14)

This rather trivial rewriting in fact anticipates, in simple cases, a first
order derivative representation of $\chi$ in terms of the radial action as in
eq. (12). Moreover, it allows for greater flexibility regarding the effective
potentials we can handle. In terms of these quantities, the scattering angle
is given by

$\chi=-2\sum_{n=0}^{\infty}\int_{0}^{\infty}du\left(\frac{d}{du^{2}}\right)^{n}h(r)\frac{r^{2n}U(r,b)^{n}}{n!p_{\infty}^{2n+1}}-\pi,\quad\quad
r^{2}=u^{2}+b^{2}~{}.$ (15)

The function $h(r)$ needs to be determined for each specific scattering
situation but it often takes very simple forms. As an example, for the
Schwarzschild metric in isotropic coordinates it is simply
$h(r)=-bp_{\infty}/r^{2}$. The identification of $h(r)$ is useful for both
non-spinning and spinning probes but it is particularly suited for the latter,
where there may be no obvious way in which to relate the integrand of the
scattering angle to a first-order derivative. As detailed in our derivation
below, the formula (44) is valid for any $h(r)$ which is real analytic on the
interval $r\in[r_{m},\infty[$, and falls off as
$\lim_{r\rightarrow\infty}h(r)\sim 1/r^{n}$, with $n\geq 2$. These conditions
are always met for the cases considered in this paper.

###  II.1 A compact formula for the scattering angle in metrics of normal form

Although the final result eq. (44) is surprisingly simple, the steps leading
to it are involved and we display them now with a fair amount of detail. Let
us first introduce some general notation. For any non-trivial metric, $p_{r}$
will depend on $G$ and this dependence carries all of the information about
the scattering angle. We define $T\equiv\left.p_{r}^{2}\right|_{G\rightarrow
0}$ so that we can write

$p_{r}^{2}=T(r)-U(r)$ (16)

where, by construction, $U(r)$ carries all the $G$-dependence. Both $T$ and
$U$ depend on the radial coordinate $r$ as indicated but may also in general
depend on orbital angular momentum $L$ and any other parameters of the metric.
The function $U$ is a close analogue of a classical effective potential
associated with the given metric (for some choice of coordinates). If $U$
carries no $L$-dependence the method used in ref. Bjerrum-Bohr:2019kec  can
straightforwardly be used to derive the scattering angle in perturbation
theory. Here we consider its generalization to the $L$-dependent setting,
focusing on a formulation that will encompass the case of spinning probes.

After having introduced this notation, we now return to the case of a metric
which we assume is already in normal form. As explained above, this means that
$T(r)$ takes the simple form

$T(r)~{}=~{}p_{\infty}^{2}-\frac{L^{2}}{r^{2}}$ (17)

in those coordinates. We recall that we can then write the scattering angle as

$\frac{\chi}{2}=\int_{r_{m}}^{\infty}dr\frac{d\phi}{dr}-\frac{\pi}{2}=-\int_{r_{m}}^{\infty}\frac{h(r)}{p_{\infty}}\left(1-b^{2}/r^{2}-\frac{U(r,b)}{p_{\infty}^{2}}\right)^{-1/2}-\frac{\pi}{2},\quad\quad\frac{d\phi}{dr}=-\frac{h(r)}{p_{r}},$
(18)

with $h(r)$ assumed to obey the analyticity and fall-off requirements listed
above. This will be the starting point for our derivation.

Now, using the condition $p_{r}(r_{m})=0$, and following the derivation of
ref. Bjerrum-Bohr:2019kec , we find it convenient to isolate

$\frac{b^{2}}{r^{2}}=\frac{r_{m}^{2}}{r^{2}}-\frac{r_{m}^{2}}{r^{2}}\frac{U(r_{m},b)}{p_{\infty}^{2}},$
(19)

and insert this into eq. (18). This gives

$\chi/2=-\int_{r_{m}}^{\infty}\\!dr\,\frac{h(r)}{p_{\infty}}\left(1-\frac{r_{m}^{2}}{r^{2}}-W(r,b)\right)^{-1/2}-\pi/2,$
(20)

where

$W(r,b)\equiv\frac{1}{p_{\infty}^{2}}\left(U(r,b)-\frac{r_{m}^{2}}{r^{2}}U(r_{m},b)\right).$
(21)

Changing integration variable to $u$ through $r^{2}=u^{2}+r_{m}^{2}$ (where
$u\geq 0$), we get

$\frac{\chi}{2}=-\int_{0}^{\infty}\\!du\,\frac{h(r)}{p_{\infty}}\left(1-\frac{r^{2}}{u^{2}}W(r,b)\right)^{-1/2}-\frac{\pi}{2},$
(22)

where $r$ just stands for $r=\sqrt{u^{2}+r_{m}^{2}}$. Use of the binomial
expansion

$(1+x)^{-1/2}=1+\sum_{n=0}^{\infty}\begin{pmatrix}-1/2\\\
n+1\end{pmatrix}x^{n+1}=1+\sum_{n=0}^{\infty}\frac{(-1)^{n+1}(2n+1)!!}{2^{n+1}(n+1)!}x^{n+1},$
(23)

yields the following expression for the angle

$\frac{\chi}{2}=F_{0}(r_{m})-\sum_{n=0}^{\infty}\frac{(2n+1)!!}{2^{n+1}(n+1)!}\int_{0}^{\infty}\\!du\,\frac{1}{u^{2(n+1)}}\left(\frac{h(r)}{p_{\infty}}r^{2(n+1)}W(r,b)^{n+1}\right)-\frac{\pi}{2},$
(24)

where we have defined the function

$F_{0}(r_{m})\equiv-\frac{1}{p_{\infty}}\int_{0}^{\infty}du~{}h(r)\quad\quad
r^{2}=u^{2}+r_{m}^{2}~{}.$ (25)

Although this integral is often elementary (such as for the Schwarzschild
metric in isotropic coordinates), we do not need to evaluate it explicitly.
This will become clear below. In fact, this function, being dependent on
$r_{m}$ must disappear in the end since the scattering angle should not depend
on $r_{m}$. The remaining terms above can be rewritten by means of the
integration-by-parts identity Wallace:1973iu ,

$\int_{0}^{\infty}\frac{du}{u^{2(n+1)}}f(u)=\frac{1}{(2n+1)!!}\int_{0}^{\infty}du\left(\frac{1}{u}\frac{d}{du}\right)^{n+1}f(u)=\frac{2^{n+1}}{(2n+1)!!}\int_{0}^{\infty}du\left(\frac{d}{du^{2}}\right)^{n+1}f(u),$
(26)

valid for any $\mathcal{C}^{\infty}$ function $f$ for which $f(u)/u^{2n+1}$
vanishes at zero and infinity. On account of our assumptions about $h(r)$, eq.
(26) may be applied to eq. (24) to obtain

$\displaystyle\frac{\chi}{2}$ $\displaystyle=$ $\displaystyle
F_{0}(r_{m})-\sum_{n=0}^{\infty}\Delta_{n}-\frac{\pi}{2},$ (27)

where we have defined

$\Delta_{n}~{}\equiv~{}\frac{1}{(n+1)!}\int_{0}^{\infty}\\!du\left(\frac{d}{du^{2}}\right)^{n+1}\left[\frac{h(r)}{p_{\infty}}r^{2(n+1)}W(r,b)^{n+1}\right].$
(28)

Furthermore, writing

$W(r,b)^{n+1}=\frac{U(r,b)^{n+1}}{p_{\infty}^{2(n+1)}}(1-x)^{n+1}\quad\text{
with }\quad x\equiv-\frac{r_{m}^{2}}{r^{2}}\frac{U(r_{m},b)}{U(r,b)},$ (29)

we can again Taylor expand, this time in powers of $x$, to get

$\Delta_{n}=\int_{0}^{\infty}du\left(\frac{d}{du^{2}}\right)^{n+1}\,\,\sum_{k=0}^{n+1}\frac{1}{(n-k+1)!k!}\frac{h(r)}{p_{\infty}}\frac{r^{2(n+1)}U(r,b)^{n-k+1}}{p_{\infty}^{2(n+1)}}\left[-\frac{r_{m}^{2}}{r^{2}}U(r_{m},b)\right]^{k},$
(30)

and once again we can use eq. (19) to substitute the $U(r_{m},b)$ in the
square brackets. This results in

$\Delta_{n}=\sum_{k=0}^{n+1}\frac{(b^{2}-r_{m}^{2})^{k}}{k!}\int_{0}^{\infty}du\left(\frac{d}{du^{2}}\right)^{n+1}\,\,\frac{h(r)}{p_{\infty}}\frac{r^{2(n-k+1)}U(r,b)^{n-k+1}}{(n-k+1)!\,p_{\infty}^{2(n-k+1)}}.$
(31)

Note that the only explicit $r_{m}$-dependence in the integrand occurs through
$r=\sqrt{u^{2}+r_{m}^{2}}$. Since $r$ is symmetric in $r_{m}^{2}$ and $u^{2}$,
we can exchange derivatives in $u^{2}$ for derivatives in $r_{m}^{2}$, and
consider the identity

$\left(\frac{d}{du^{2}}\right)^{n+1}=\left(\frac{d}{dr_{m}^{2}}\right)^{k}\left(\frac{d}{du^{2}}\right)^{n-k+1}~{}.$
(32)

Applying this to the sum in eq. (31), we find

$\Delta_{n}=\sum_{k=0}^{n+1}\Delta_{n,k}(r_{m})~{},$ (33)

where we have defined

$\Delta_{n,k}\equiv\frac{(b^{2}-r_{m}^{2})^{k}}{k!}\left(\frac{d}{dr_{m}^{2}}\right)^{k}\int_{0}^{\infty}\\!du\,\left(\frac{d}{du^{2}}\right)^{n-k+1}\,\frac{h(r)}{p_{\infty}}\frac{r^{2(n-k+1)}U(r,b)^{n-k+1}}{(n-k+1)!\,p_{\infty}^{2(n-k+1)}}~{}.$
(34)

We observe that the term with $k=n+1$ is $U$-independent. Crucially, as we
shall demonstrate next, this fact will make the apparent $r_{m}$-dependence
disappear, cancelling the $r_{m}$-dependent piece $F_{0}(r_{m})$. We start by
evaluating the $k=n+1$ and $F_{0}(r_{m})$ terms together, and introduce (the
reason for the factor 1/2 on the left hand side will become clear shortly),

$\frac{1}{2}\zeta_{-1}\equiv
F_{0}(r_{m})-\sum_{n=0}^{\infty}\Delta_{n,n+1}(r_{m}).$ (35)

Consider now the Taylor expansion of $F_{0}(r_{m})$ around $r_{m}=b$. This
reads

$F_{0}(b)=-\sum_{n=0}^{\infty}\frac{(b^{2}-r_{m}^{2})^{n}}{n!}\left(\frac{d}{dr_{m}^{2}}\right)^{n}\int_{0}^{\infty}\\!du\,\,\frac{h(r)}{p_{\infty}}.$
(36)

Furthermore, we note that the sum $\sum_{n=0}^{\infty}\Delta_{n,n+1}$ can be
rewritten as

$\begin{split}\sum_{n=0}^{\infty}\Delta_{n,n+1}&=\sum_{n=0}^{\infty}\frac{(b^{2}-r_{m}^{2})^{n+1}}{(n+1)!}\left(\frac{d}{dr_{m}^{2}}\right)^{n+1}\int_{0}^{\infty}\\!du\,\,\frac{h(r)}{p_{\infty}},\\\
&=-\int_{0}^{\infty}du\frac{h(r)}{p_{\infty}}+\sum_{n=0}^{\infty}\frac{(b^{2}-r_{m}^{2})^{n}}{n!}\left(\frac{d}{dr_{m}^{2}}\right)^{n}\int_{0}^{\infty}\\!du\,\,\frac{h(r)}{p_{\infty}},\end{split}$
(37)

where in the second line we have added and subtracted $F_{0}(r_{m})$. Making
use of eq. (36) and the definition of $F_{0}(r_{m})$ we find

$\sum_{n=0}^{\infty}\Delta_{n,n+1}=F_{0}(r_{m})-F_{0}(b)~{}.$ (38)

Inserting this into eq. (35) results in $\frac{1}{2}\zeta_{-1}=F_{0}(b)$. The
$r_{m}$-dependence has explicitly disappeared from this term. It follows from
the above that the scattering angle can be written in the form

$\chi-\zeta_{-1}+\pi=-2\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{(b^{2}-r_{m}^{2})^{k}}{k!}\left(\frac{d}{dr_{m}^{2}}\right)^{k}\int_{0}^{\infty}du\left(\frac{d}{du^{2}}\right)^{n-k+1}\,\,\frac{h(r)}{p_{\infty}}\frac{r^{2(n-k+1)}U(r,b)^{n-k+1}}{(n-k+1)!\,p_{\infty}^{2(n-k+1)}}$
(39)

To simplify our notation, we now define

$\zeta_{n}(x)\equiv-2\int_{0}^{\infty}du\left(\frac{d}{du^{2}}\right)^{n+1}\frac{h(r)}{p_{\infty}}\frac{r^{2(n+1)}U(r,b)^{n+1}}{(n+1)!p_{\infty}^{2(n+1)}},\quad\quad
r^{2}=u^{2}+x^{2}$ (40)

so that

$\displaystyle\chi-\zeta_{-1}+\pi$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{(b^{2}-r_{m}^{2})^{k}}{k!}\left(\frac{d}{dr_{m}^{2}}\right)^{k}\zeta_{n-k}(r_{m})$
(41) $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{\ell=0}^{\infty}\frac{(b^{2}-r_{m}^{2})^{\ell}}{\ell!}\left(\frac{d}{dr_{m}^{2}}\right)^{\ell}\zeta_{n}(r_{m})$
(42)

which we recognize as the Taylor expansion of $\zeta_{n}(b)$ around the point
$r_{m}$. In this way, the turning point $r_{m}$ has explicitly disappeared
from all terms of the scattering angle, as it should. No regularization of the
involved integrals and no use of ad hoc rules has been needed. The final
formula for the scattering angle thus becomes

$\chi+\pi=\sum_{n=0}^{\infty}\zeta_{n}(b)+\zeta_{-1},$ (43)

where $\zeta_{n}(b)$ is given in eq. (40) evaluated, as we see, at
$r^{2}=u^{2}+b^{2}$. The choice of notation for $\zeta_{-1}$ is now clear, as
this is precisely $\zeta_{n}$ from eq. (40) evaluated at $x=b$ and $n=-1$.
Thus we can write eq. (43) as

$\chi=-2\sum_{n=0}^{\infty}\int_{0}^{\infty}du\left(\frac{d}{du^{2}}\right)^{n}h(r)\frac{r^{2n}U(r,b)^{n}}{n!p_{\infty}^{2n+1}}-\pi,\quad\quad
r^{2}=u^{2}+b^{2}$ (44)

This is a very general result, valid for any well-behaved $h(r)$ as stipulated
in precise terms above. As we shall see next, it will apply to both scalar and
spinning test bodies up to cubic order in the spin of the test particle $S$
and we see no obstacle towards it being applicable to any order in the spin of
the probes. The essential ingredient is that the spin of the test body is
considered in perturbation theory. Note that when interactions are turned off,
the function $h(r)$ reduces to $h(r)=-bp_{\infty}/r^{2}$ and the $n=0$ term in
the sum above thus produces zero scattering angle up to terms that vanish when
interactions are set to zero.

Having derived this general expression for the scattering angle, it is of
interest to see the form it takes when the scattering angle can be expressed
in terms of an $L$-derivative of the radial action as in eq. (12). We will
then also see how it relates to the formula derived in ref. Bjerrum-
Bohr:2019kec  for the special case where $U(r,b)$ does not depend on $b$.
Carrying out the derivative in eq. (12), we see that

$\frac{d\phi}{dr}=-\frac{\partial}{\partial
L}p_{r}=\frac{1}{p_{r}}\left(\frac{bp_{\infty}}{r^{2}}+\frac{1}{2p_{\infty}}\frac{\partial}{\partial
b}U(r,b)\right),$ (45)

and thus

$h(r)=-\left(\frac{bp_{\infty}}{r^{2}}+\frac{1}{2p_{\infty}}\frac{\partial}{\partial
b}U(r,b)\right)~{}.$ (46)

Let us define

$\Upsilon(r,b)\equiv\frac{\partial}{\partial b}U(r,b)~{}.$ (47)

Inserting $h(r)$ from eq. (46) into (44), we get

$\begin{split}\chi&=2\sum_{n=0}^{\infty}\int_{0}^{\infty}\\!du\,\left(\frac{d}{du^{2}}\right)^{n}\left[\frac{bp_{\infty}}{r^{2}}+\frac{1}{2p_{\infty}}\Upsilon(r,b)\right]\frac{r^{2n}U(r,b)^{n}}{n!p_{\infty}^{2(n+1)-1}}-\pi\\\
&=2b\sum_{n=0}^{\infty}\int_{0}^{\infty}\\!du\,\left(\frac{d}{du^{2}}\right)^{n+1}\frac{r^{2n}U(r,b)^{n+1}}{(n+1)!p_{\infty}^{2(n+1)}}+\sum_{n=0}^{\infty}\int_{0}^{\infty}\\!du\,\left(\frac{d}{du^{2}}\right)^{n}\Upsilon(r,b)\frac{r^{2n}U(r,b)^{n}}{n!p_{\infty}^{2(n+1)}}\end{split}$
(48)

where we recall the notation $r^{2}=u^{2}+b^{2}$. Note that the $n=0$ term
from the free part has cancelled the explicit $\pi$, and we have relabelled
the remaining terms accordingly. This is a valid and compact form for the
scattering angle but we can simplify it further by use of the identity (which
is valid for $r^{2}=u^{2}+b^{2}$),

$\frac{d}{db}\left[r^{2n}U(r,b)^{n+1}\right]=(n+1)r^{2n}\Upsilon(r,b)U(r,b)^{n}+2b\frac{d}{du^{2}}\left[r^{2n}U(r,b)^{n+1}\right]~{}.$
(49)

Substituting this into eq. (48) we obtain the compact expression

$\chi=\sum_{n=0}^{\infty}\int_{0}^{\infty}\\!du\,\left(\frac{d}{du^{2}}\right)^{n}\frac{d}{db}\left[\frac{r^{2n}U(r,b)^{n+1}}{(n+1)!p_{\infty}^{2(n+1)}}\right],\quad\quad
r^{2}=u^{2}+b^{2}$ (50)

In the special case of a $b$-independent $U(r,b)$ this is seen to reduce to
the formula (4). Remarkably, whether based on the general formula (44) or on
the special case (50), we have found that these final results for the
scattering angle are almost as simple as those given in ref. Bjerrum-
Bohr:2019kec  even though $U(r,b)$, which effectively acts as a potential,
here can depend on angular momentum $L$ (or $b=L/p_{\infty}$). The more
general representation (44) is valid even when we cannot obviously write the
scattering angle as an $L$-derivative of the radial action.

If we substitute the specific form of $U(r,L)$ for Schwarzschild coordinates
as in eq. (11) we do indeed recover the correct Schwarzschild scattering
angles from eq. (44). To illustrate this point, we perform the Taylor
expansion of the $U(r,b)$ in eq. (11) to second order in $G$. This results in

$U(r,b)=U_{1}(r,b)+U_{2}(r,b)+\mathcal{O}(G^{3})=-\frac{(2E^{2}-m^{2})r^{2}-L^{2}}{r^{3}}r_{s}-\frac{(3E^{2}-m^{2})r^{2}-L^{2}}{r^{4}}r_{s}^{2}+\mathcal{O}(G^{3})~{}.$
(51)

To leading order in $G$ only the single term $U_{1}(r,b)$ contributes to the
scattering angle and we thus get

$\chi_{1}=\int_{0}^{\infty}du\frac{d}{db}\frac{1}{p_{\infty}^{2}}U_{1}(r,b)=\frac{2GM(2E^{2}-m^{2})}{bp_{\infty}^{2}}=\frac{2(2\gamma^{2}-1)GMm^{2}}{bp_{\infty}^{2}},$
(52)

where $E^{2}=\gamma^{2}m^{2}$ and $\gamma=1/\sqrt{1-v^{2}}$ is the Lorentz
factor of the test-particle with velocity $v$ at infinity. This is the well-
known leading order result. At second order in $G$ there are two
contributions: the $U_{1}$ term from $n=1$ in the sum (50) and the $U_{2}$
term from $n=0$ part. The second-order contribution $\chi_{2}$ to the
scattering angle is thus

$\displaystyle\chi_{2}=\int_{0}^{\infty}du\frac{\partial}{\partial
b}\left(\frac{d}{du^{2}}\right)\frac{r^{2}U^{2}_{1}(r,b)}{2p_{\infty}^{4}}+\int_{0}^{\infty}du\frac{\partial}{\partial
b}\frac{U_{2}(r,b)}{p_{\infty}^{2}}$
$\displaystyle=r_{s}^{2}\left(\frac{\pi(6E^{2}-2m^{2}-p_{\infty}^{2})}{4p_{\infty}^{2}b^{2}}-\frac{\pi(8E^{2}-4m^{2}-3p_{\infty}^{2})}{16b^{2}p_{\infty}^{2}}\right)$
$\displaystyle=4G^{2}M^{2}m^{2}\pi\left(\frac{5\gamma^{2}-1}{4p_{\infty}^{2}b^{2}}-\frac{5\gamma^{2}-1}{16p_{\infty}^{2}b^{2}}\right)$
$\displaystyle=\frac{3G^{2}M^{2}m^{2}\pi(5\gamma^{2}-1)}{4p_{\infty}^{2}b^{2}}$
(53)

which is the known answer. In Table 1 below we list the contributions up to
and including tenth order in $G$ computed straightforwardly in this manner,
but expressed in terms of velocity $v$ rather than $\gamma$ for the sake of
compactness.

$n$ |  $\chi_{n}/\frac{G^{n}M^{n}}{b^{n}v^{2n}}$
---|---
1 |  $2\left(v^{2}+1\right)$
2 |  $(3\pi/4)v^{2}\left(v^{2}+4\right)$
3 |  $(2/3)\left(5v^{6}+45v^{4}+15v^{2}-1\right)$
4 |  $(105\pi/64)v^{4}\left(v^{4}+16v^{2}+16\right)$
5 |  $(2/5)\left(21v^{10}+525v^{8}+1050v^{6}+210v^{4}-15v^{2}+1\right)$
6 |  $(1155\pi/256)v^{6}\left(v^{6}+36v^{4}+120v^{2}+64\right)$
7 |  $(2/35)\left(429v^{14}+21021v^{12}+105105v^{10}+105105v^{8}+15015v^{6}-1001v^{4}+91v^{2}-5\right)$
8 |  $(45045\pi/16384)v^{8}\left(5v^{8}+320v^{6}+2240v^{4}+3584v^{2}+1280\right)$
9 |  $(2/63)(2431v^{18}+196911v^{16}+1837836v^{14}+4288284v^{12}$
$+2756754v^{10}+306306v^{8}-18564v^{6}+1836v^{4}-153v^{2}+7)$
10 |  $(2909907\pi/65536)v^{10}\left(v^{10}+100v^{8}+1200v^{6}+3840v^{4}+3840v^{2}+1024\right)$
Table 1: Scattering angle of a non-spinning test particle in a Schwarzschild
background up to 10th order in $G$.
$\chi_{n}$ is the $G^{n}$ contribution to the full scattering angle $\chi=\sum_{n}\chi_{n}$. $n$ |  $\chi_{n}/\frac{G^{n}M^{n}}{v^{2n}(b^{2}-a^{2})^{(3n-1)/2}}$
---|---
1 |  $2\big{[}-2av+b(1+v^{2})\big{]}$
2 |  $(\pi/2a^{2})\Big{[}(b^{2}-a^{2})^{5/2}v^{4}+(a-bv)^{3}\big{[}-4a^{2}v+3ab+b^{2}v\big{]}\Big{]}$
3 |  $(2/3)\big{[}2a^{5}v\left(3v^{4}-10v^{2}-9\right)-3a^{4}b\left(v^{6}-15v^{4}-45v^{2}-5\right)-4a^{3}b^{2}v\left(15v^{4}+70v^{2}+27\right)$
$+2a^{2}b^{3}\left(11v^{6}+135v^{4}+105v^{2}+5\right)-18ab^{4}v\left(5v^{4}+10v^{2}+1\right)+b^{5}\left(5v^{6}+45v^{4}+15v^{2}-1\right)\big{]}$
4 |  $(3\pi/16a^{4})\Big{[}2(b^{2}-a^{2})^{11/2}v^{8}+(a-bv)^{5}\big{[}-8a^{6}v\left(14v^{2}+5\right)+5a^{5}b\left(72v^{2}+7\right)$
$+a^{4}b^{2}v\left(16v^{2}-305\right)-5a^{3}b^{3}\left(11v^{2}-14\right)-a^{2}b^{4}v\left(11v^{2}-30\right)+10ab^{5}v^{2}+2b^{6}v^{3}\big{]}\Big{]}$
5 |  $(2/15)\big{[}-2a^{9}v\left(15v^{8}-60v^{6}+378v^{4}+900v^{2}+175\right)$
$+15a^{8}b\left(v^{10}-15v^{8}+210v^{6}+1050v^{4}+525v^{2}+21\right)+8a^{7}b^{2}v\left(45v^{8}-780v^{6}-6426v^{4}-6300v^{2}-875\right)$
$-140a^{6}b^{3}\left(v^{10}-45v^{8}-630v^{6}-1050v^{4}-315v^{2}-9\right)-84a^{5}b^{4}v\left(45v^{8}+1020v^{6}+2814v^{4}+1500v^{2}+125\right)$
$+14a^{4}b^{5}\left(67v^{10}+3375v^{8}+15750v^{6}+14070v^{4}+2295v^{2}+27\right)-1400a^{3}b^{6}v\left(9v^{8}+84v^{6}+126v^{4}+36v^{2}+1\right)$
$+36a^{2}b^{7}\left(29v^{10}+875v^{8}+2450v^{6}+1190v^{4}+65v^{2}-1\right)-50ab^{8}v\left(63v^{8}+420v^{6}+378v^{4}+36v^{2}-1\right)$
$+3b^{9}\left(21v^{10}+525v^{8}+1050v^{6}+210v^{4}-15v^{2}+1\right)\big{]}$
6 |  $(5\pi/128a^{6})\Big{[}8(b^{2}-a^{2})^{17/2}v^{12}+(a-bv)^{7}\big{[}-4a^{10}v\left(1584v^{4}+1540v^{2}+189\right)$
$+7a^{9}b\left(5720v^{4}+2808v^{2}+99\right)-a^{8}b^{2}v\left(2200v^{4}+85232v^{2}+17829\right)+14a^{7}b^{3}\left(260v^{4}+5391v^{2}+330\right)$
$-2a^{6}b^{4}v\left(334v^{4}-1491v^{2}+14070\right)+21a^{5}b^{5}\left(85v^{4}-272v^{2}+176\right)+a^{4}b^{6}v\left(255v^{4}-1904v^{2}+1680\right)$
$-28a^{3}b^{7}v^{2}\left(17v^{2}-24\right)-4a^{2}b^{8}v^{3}\left(17v^{2}-56\right)+56ab^{9}v^{4}+8b^{10}v^{5}\big{]}\Big{]}$
Table 2: Scattering angle of a non-spinning test particle in a Kerr background
up to 6th order in $G$.
$\chi_{n}$ is the $G^{n}$ contribution to the full scattering angle
$\chi=\sum_{n}\chi_{n}$.

##  III Scattering in Kerr Metrics

We next consider applying the formula we found in the previous section to the
scattering of a small non-spinning probe around a Kerr black hole. A standard
choice for the metric is Boyer-Lindquist coordinates $(t,r,\phi)$, for which,
when restricted to the equatorial $\theta=\pi/2$ plane, the metric reads

$g_{\mu\nu}=\begin{pmatrix}-\left(1-\frac{r_{s}}{r}\right)&0&-\frac{r_{s}a}{r}\\\
0&\frac{r^{2}}{r^{2}-r_{s}r+a^{2}}&0\\\
-\frac{r_{s}a}{r}&0&\frac{(r+r_{s})a^{2}+r^{3}}{r}\end{pmatrix}.$ (54)

Letting a test body orbit in this $\theta=\pi/2$ plane, it will have its
orbital angular momentum $L$ conserved, and it will therefore remain in that
plane. This allows for a well-defined scattering angle and it will also allow
us to rewrite the Kerr metric in normal form. We find the radial momentum
$p_{r}$ from the Hamilton-Jacobi equation

$p_{r}^{2}=\frac{r\left(p_{\infty}^{2}r^{3}+m^{3}r^{2}r_{s}+(a^{2}p_{\infty}^{2}-L^{2})r+r_{s}(Ea-L)^{2}\right)}{(a^{2}+r^{2}-r_{s}r)^{2}},$
(55)

and we can write it in the form $p_{r}^{2}=T(r,L,a)-U(r,L,a)$, with

$T(r,L,a)\equiv\frac{r^{2}}{r^{2}+a^{2}}\left(p_{\infty}^{2}-\frac{L^{2}}{r^{2}+a^{2}}\right),$
(56)

which indeed is independent of $G$, and

$\displaystyle U(r,L,a)$ $\displaystyle\equiv$
$\displaystyle-\frac{\left[(2E^{2}-m^{2})r^{6}+(m^{2}-E^{2})r^{5}r_{s}+((4E^{2}-m^{2})a^{4}-4ELa^{3})r^{2}\right]rr_{s}}{(a^{2}+r^{2}-r_{s}r)^{2}(a^{2}+r^{2})^{2}}$
(57) $\displaystyle-$
$\displaystyle\frac{\left[((5E^{2}-2m^{2})a^{2}-2ELa-L^{2})r^{4}-r_{s}((E^{2}-m^{2})a^{2}-L^{2})r^{3}+a^{4}(Ea-L)^{2}\right]rr_{s}}{(a^{2}+r^{2}-r_{s}r)^{2}(a^{2}+r^{2})^{2}},$
(58)

which carries all $G$-dependence. Although well separated into $T$ and $U$
pieces, we notice that $T$ is not of the free kind shown in eq. (17). Thus,
Boyer-Lindquist coordinates are not of normal form and we need to choose
different coordinates in order for our formalism to be applicable. As noted in
the previous section, the needed change of integration variables in the radial
action can equivalently be viewed as a coordinate transformation away from
Boyer-Lindquist coordinates, thus leading to a different metric.

Indeed, in the $G\to 0$ limit the Boyer-Lindquist metric eq. (54) takes the
form

$g_{\mu\nu}=\begin{pmatrix}-1&0&0\\\ 0&\frac{r^{2}}{r^{2}+a^{2}}&0\\\
0&0&a^{2}+r^{2}\end{pmatrix}$ (59)

which does not correspond to flat Minkowski space in ordinary polar
coordinates. Since the Kerr metric is diagonal and only depends on the radial
coordinate $r$ in this limit, we can find a coordinate change
$r\rightarrow\rho(r)$ which allows us to recover the free structure of $T$.
This change is given simply by

$\rho^{2}=r^{2}+a^{2}~{}.$ (60)

For this new radial coordinate $\rho$ the Kerr metric takes the form

$\tilde{g}_{\mu\nu}=\begin{pmatrix}-1&0&0\\\ 0&1&0\\\
0&0&\rho^{2}\end{pmatrix}$ (61)

in the $G\to 0$ limit, corresponding to a metric in normal form. The
transformation eq. (60), also automatically produces the needed

$T(\rho,L,a)=p_{\infty}^{2}-\frac{L^{2}}{r^{2}}~{}.$ (62)

Note that the free part of the Kerr metric becomes independent of the black-
hole spin $a$ in these coordinates. The radial momentum $p_{r}$ transforms
like

$p_{\rho}=\frac{dr}{d\rho}p_{r}$ (63)

under this coordinate change, and so we obtain the new effective potential

$\tilde{U}(\rho,b,a)=\left(\frac{dr}{d\rho}\right)^{2}U(r,b,a).$ (64)

We may thus write the formula for the scattering angle (50) in terms of $\rho$
as

$\chi=\sum_{n=0}^{\infty}\int_{0}^{\infty}du\frac{\partial}{\partial
b}\left(\frac{d}{du^{2}}\right)^{n}\frac{\rho^{2n}\tilde{U}(\rho,b)^{n+1}}{(n+1)!p_{\infty}^{2(n+1)}},\quad\quad\rho^{2}=u^{2}+b^{2}.$
(65)

As a first quick check, we compute the scattering angle $\chi_{1}$ to leading
order in $G$ using eq. (65). We find

$\chi_{1}=\frac{2GM\left(\gamma^{2}(2b-2av)-b\right)}{\gamma^{2}v^{2}\left(b^{2}-a^{2}\right)}$
(66)

where $v$ is the asymptotic velocity of the test-particle. This agrees with
the scattering angle computed in ref. Vines:2017hyw  when restricted to the
test-body limit. We emphasize that as in ref. Vines:2017hyw  our result gives
the scattering angle to all orders in the spin of the black hole $a$. It is
easy to verify that in the light-like limit $v\to 1,\gamma\to\infty$, the
result above reproduces the terms of the expansion provided in ref.
Iyer:2009hq .

Although the integrals are slightly more involved than those of Schwarzschild
scattering, the final results for massive-probe scattering are relatively
simple. In table 2 we list results up to and including sixth order in $G$
(this table can readily be extended based on our general formula). Again,
expanding in powers of $a$ and taking the massless limit this reproduces the
well-known light-bending formulas for Kerr metrics.

We note that the resulting scattering angle contribution $\chi_{n}$ to any
order in $G$ displays some simple patterns. First, the scattering angle
naturally has emerged in a form that re-sums all orders in $a$. Second, to
order $n$ one may identify the prefactor

$c_{n}\equiv\frac{G^{n}M^{n}}{(b^{2}-a^{2})^{(3n-1)/2}v^{2n}}.$ (67)

which accounts for the all-order-in-spin behaviour. We also note that even
orders in powers of $G$ are relatively simpler, and one can easily identify
more structural patterns in them. To be concrete, we observe that after
factorising the term in eq. (67), the angle takes the form

$\chi_{n}/c_{n}=(d_{1,n}\pi/a^{2n})\Big{[}d_{2,n}v^{2n}(b^{2}-a^{2})^{(3n-1)/2}+(a-bv)^{n+1}\sum_{\ell=0}^{2n-2}a^{\ell}b^{2n-2-\ell}f_{n,\ell}(v)\Big{]}$
(68)

where $d_{i,n}$ are numerical constants, and $f_{n,k}(v)$ is a polynomial in
$v$ of degree $n-1$ for even $\ell$ or $n-2$ for odd $\ell$.

$(n,k)$ |  $\chi_{n,k}/\frac{G^{n}M^{n}(S/m)^{k}}{v^{2n}(b^{2}-a^{2})^{(3n+2k-1)/2}}$
---|---
$(1,1)$ |  $-4(av-b)(a-bv)$
$(1,2)$ |  $-4a^{3}v+6a^{2}b\left(v^{2}+1\right)-12ab^{2}v+2b^{3}\left(v^{2}+1\right)$
$(2,1)$ |  $(3\pi/2)(av-b)(a-bv)\big{[}a^{2}\left(-2v^{2}-3\right)+10abv-b^{2}\left(3v^{2}+2\right)\big{]}$
$(2,2)$ |  $(3\pi/4)\big{[}-10a^{5}v\left(v^{2}+1\right)+a^{4}b\left(12v^{4}+71v^{2}+12\right)-90a^{3}b^{2}v\left(v^{2}+1\right)$
$+a^{2}b^{3}\left(21v^{4}+128v^{2}+21\right)-40ab^{4}v\left(v^{2}+1\right)+b^{5}\left(2v^{4}+11v^{2}+2\right)\big{]}$
$(3,1)$ |  $8(av-b)(a-bv)\big{[}a^{4}\left(-v^{4}-10v^{2}-5\right)+8a^{3}bv\left(3v^{2}+5\right)$
$-2a^{2}b^{2}\left(5v^{4}+38v^{2}+5\right)+8ab^{3}v\left(5v^{2}+3\right)-b^{4}\left(5v^{4}+10v^{2}+1\right)\big{]}$
$(3,2)$ |  $4\big{[}-2a^{7}v\left(7v^{4}+30v^{2}+11\right)+5a^{6}b\left(3v^{6}+55v^{4}+65v^{2}+5\right)$
$-6a^{5}b^{2}v\left(51v^{4}+190v^{2}+63\right)+5a^{4}b^{3}\left(17v^{6}+265v^{4}+275v^{2}+19\right)$
$-10a^{3}b^{4}v\left(53v^{4}+170v^{2}+49\right)+3a^{2}b^{5}\left(19v^{6}+255v^{4}+225v^{2}+13\right)$
$-10ab^{6}v\left(11v^{4}+30v^{2}+7\right)+b^{7}\left(3v^{6}+35v^{4}+25v^{2}+1\right)\big{]}$
$(4,1)$ |  $(105\pi/16)(av-b)(a-bv)^{3}\big{[}a^{4}\left(-8v^{4}-20v^{2}-5\right)+12a^{3}bv\left(6v^{2}+5\right)$
$-2a^{2}b^{2}\left(10v^{4}+79v^{2}+10\right)+12ab^{3}v\left(5v^{2}+6\right)-b^{4}\left(5v^{4}+20v^{2}+8\right)\big{]}$
$(4,2)$ |  $(15\pi/32)(a-bv)\big{[}-2a^{8}v\left(24v^{6}+320v^{4}+485v^{2}+95\right)$
$+7a^{7}b\left(248v^{6}+1100v^{4}+635v^{2}+30\right)-a^{6}b^{2}v\left(760v^{6}+15808v^{4}+25345v^{2}+4980\right)$
$+105a^{5}b^{3}\left(92v^{6}+466v^{4}+276v^{2}+13\right)-15a^{4}b^{4}v\left(106v^{6}+2272v^{4}+3860v^{2}+769\right)$
$+21a^{3}b^{5}\left(405v^{6}+2050v^{4}+1258v^{2}+60\right)-a^{2}b^{6}v\left(585v^{6}+12140v^{4}+20270v^{2}+4196\right)$
$+7ab^{7}\left(160v^{6}+775v^{4}+460v^{2}+24\right)-5b^{8}v\left(4v^{6}+79v^{4}+124v^{2}+24\right)\big{]}$
$(5,1)$ |  $4(av-b)(a-bv)\big{[}a^{8}\left(v^{8}-36v^{6}-378v^{4}-420v^{2}-63\right)$
$+64a^{7}bv\left(v^{6}+27v^{4}+63v^{2}+21\right)-4a^{6}b^{2}\left(9v^{8}+668v^{6}+3222v^{4}+2268v^{2}+105\right)$
$+64a^{5}b^{3}v\left(27v^{6}+289v^{4}+405v^{2}+63\right)-2a^{4}b^{4}\left(189v^{8}+6444v^{6}+18094v^{4}+6444v^{2}+189\right)$
$+64a^{3}b^{5}v\left(63v^{6}+405v^{4}+289v^{2}+27\right)-4a^{2}b^{6}\left(105v^{8}+2268v^{6}+3222v^{4}+668v^{2}+9\right)$
$+64ab^{7}v\left(21v^{6}+63v^{4}+27v^{2}+1\right)-b^{8}\left(63v^{8}+420v^{6}+378v^{4}+36v^{2}-1\right)\big{]}$
$(5,2)$ |  $-2\big{[}2a^{11}v\left(11v^{8}+500v^{6}+2114v^{4}+1652v^{2}+203\right)$
$-7a^{10}b\left(3v^{10}+467v^{8}+4214v^{6}+6734v^{4}+2087v^{2}+63\right)$
$+2a^{9}b^{2}v\left(1665v^{8}+35036v^{6}+110726v^{4}+73052v^{2}+8001\right)$
$-21a^{8}b^{3}\left(51v^{10}+3491v^{8}+22358v^{6}+28910v^{4}+7703v^{2}+207\right)$
$+12a^{7}b^{4}v\left(2845v^{8}+41836v^{6}+103726v^{4}+56812v^{2}+5341\right)$
$-42a^{6}b^{5}\left(133v^{10}+6517v^{8}+32410v^{6}+33922v^{4}+7489v^{2}+169\right)$
$+84a^{5}b^{6}v\left(829v^{8}+9580v^{6}+19054v^{4}+8428v^{2}+637\right)$
$-6a^{4}b^{7}\left(1029v^{10}+40789v^{8}+164122v^{6}+137410v^{4}+23617v^{2}+393\right)$
$+42a^{3}b^{8}v\left(795v^{8}+7604v^{6}+12194v^{4}+4148v^{2}+219\right)$
$-a^{2}b^{9}\left(1449v^{10}+49049v^{8}+162722v^{6}+105770v^{4}+12437v^{2}+93\right)$
$+14ab^{10}v\left(203v^{8}+1652v^{6}+2114v^{4}+500v^{2}+11\right)$
$-b^{11}\left(35v^{10}+1043v^{8}+2870v^{6}+1358v^{4}+71v^{2}-1\right)\big{]}$
$(6,1)$ |  $(3465\pi/128)(av-b)(a-bv)^{5}\big{[}a^{6}\left(-32v^{6}-112v^{4}-70v^{2}-7\right)$
$+26a^{5}bv\left(16v^{4}+28v^{2}+7\right)-a^{4}b^{2}\left(112v^{6}+1796v^{4}+1337v^{2}+70\right)$
$+52a^{3}b^{3}v\left(14v^{4}+57v^{2}+14\right)-a^{2}b^{4}\left(70v^{6}+1337v^{4}+1796v^{2}+112\right)$
$+26ab^{5}v\left(7v^{4}+28v^{2}+16\right)-b^{6}\left(7v^{6}+70v^{4}+112v^{2}+32\right)\big{]}$
Table 3: Scattering angle of a spinning probe in a Kerr background up to 5th
order in $G$.
$\chi_{n,k}$ is the $G^{n}S^{k}$ contribution to the full scattering angle
$\chi=\sum_{n,k}\chi_{n,k}$.

##  IV Adding Spin to the Probe

It is well known that it is possible to consider a probe limit in which the
mass is negligible but the (rescaled) spin of the probe is finite. In this
section we extend the scattering angle calculation to the case of a spinning
probe in a Kerr metric. We will be able to derive results up to and including
second order in the probe spin.

The description of the motion of extended bodies in general relativity is a
complicated problem, and one usually needs to resort to the use of
approximation schemes. For example, one may utilize a multipolar approximation
method originally devised by Tulczyjew Tulczyjew:1959 , to explicitly work out
the equations of motion. This method was applied by Steinhoff and Puetzfeld in
ref. Steinhoff:2009tk , using a multipole approximation up to the quadrupolar
order, i.e. keeping the covariant monopole $t^{\mu\nu}$, dipole
$t^{\mu\nu\alpha}$ and quadrupole $t^{\mu\nu\alpha\beta}$ moments. They write
the stress-energy tensor $T^{\mu\nu}$ in the manifestly covariant form

$\sqrt{-g}T^{\mu\nu}=\int
d\tau\bigg{[}t^{\mu\nu}\delta_{(4)}-\nabla_{\alpha}(t^{\mu\nu\alpha}\delta_{(4)})+\frac{1}{2!}\nabla_{\alpha\beta}(t^{\mu\nu\alpha\beta}\delta_{(4)})\bigg{]}\,.$
(69)

Here $\tau$ is the proper time of the worldline $x^{\rho}(\tau)$, and
$\delta_{(4)}\equiv\delta(y^{\rho}-x^{\rho}(\tau))$. The dynamics of the
multipolar test body follow from demanding that the stress-energy tensor (69)
is covariantly conserved

$\nabla_{\nu}{T^{\mu\nu}}=0\,.$ (70)

This is sometimes referred to as Mathisson’s variational equations of
mechanics Mathisson:1937 ; Mathisson:2010  and imposes certain conditions on
the multipole moments. Building on Tulczyjew’s method Tulczyjew:1959 , eq.
(70) is explicitly evaluated in ref. Steinhoff:2009tk  where it was found that
the multipole tensors $t^{\mu\nu\dots}$ can be expressed in terms of a vector
$p^{\mu}$, an antisymmetric tensor $S^{\mu\nu}$, and Dixon’s reduced moment
$J^{\mu\nu\alpha\beta}$, which has the same symmetries as the Riemann tensor.
In terms of them, the stress-energy tensor becomes Steinhoff:2009tk

$\sqrt{-g}T^{\mu\nu}=\\!\int\\!d\tau\bigg{[}\dot{x}^{(\mu}p^{\nu)}\delta_{(4)}+\frac{1}{3}R_{\alpha\beta\rho}{}^{(\mu}J^{\nu)\rho\beta\alpha}\delta_{(4)}+\nabla_{\alpha}\left(\dot{x}^{(\mu}S^{\nu)\alpha}\delta_{(4)}\right)-\frac{2}{3}\nabla_{\alpha}\nabla_{\beta}\left(J^{\mu\alpha\beta\nu}\delta_{(4)}\right)\\!\bigg{]}\\!,$
(71)

where $\dot{x}^{\mu}$ is the tangent to the worldline, $R_{\mu\nu\rho\sigma}$
is the Riemann tensor (defined via
$2\nabla_{[\mu}\nabla_{\nu]}w_{\rho}=R_{\mu\nu\rho}{}^{\sigma}w_{\sigma}$),
where the (square) brackets denote (anti-) symmetrization of enclosed indices
(e.g. $A_{(\mu\nu)}=\frac{1}{2}(A_{\mu\nu}+A_{\nu\mu})$). The vector $p^{\mu}$
and tensor $S^{\mu\nu}$ are then identified as the linear momentum vector and
spin tensor of the object (which now play the role of monopole and dipole
moment). The motion of a multipolar test body (or probe) in a generic curved
background spacetime is described by the two equations governing the evolution
of its momentum and spin along the worldline, which are also obtained by
evaluating eq. (70) in ref. Steinhoff:2009tk . Through the quadrupolar order
in the multipole expansion, they read

$\displaystyle\frac{Dp_{\mu}}{d\tau}\,+\,\frac{1}{2}R_{\mu\nu\rho\sigma}\dot{x}^{\nu}S^{\rho\sigma}$
$\displaystyle=-\frac{1}{6}\nabla_{\mu}R_{\kappa\lambda\rho\sigma}J^{\kappa\lambda\rho\sigma},$
(72a) $\displaystyle\frac{DS^{\mu\nu}}{d\tau}-2p^{[\mu}\dot{x}^{\nu]}$
$\displaystyle=\frac{4}{3}R^{[\mu}{}_{\lambda\rho\sigma}J^{\nu]\lambda\rho\sigma}.$
(72b)

These are the Mathisson-Papapetrou-Dixon equations Mathisson:1937 ;
Mathisson:2010 ; Dixon:1974 . They may also be derived from (70) without the
assumption of a distributional $T^{\mu\nu}$ (see _e.g._ ref. Harte:2014wya ),
or alternatively from an effective action (see refs. Marsat:2014xea ;
Vines:2016unv  for recent examples). For a specific quadrupole tensor given as
a function of $p^{\mu}$ and $S^{\mu\nu}$, a closed set of evolution equations
is completed by the imposition of a “spin supplementary condition”. We will
here employ the Tulczyjew-Dixon choice Tulczyjew:1959 ; Dixon:1974 ,

$p_{\mu}S^{\mu\nu}=0,$ (73)

which, together with (72), determines the worldline tangent $\dot{x}^{\mu}$ in
terms of the other quantities. Given a Killing vector field $\xi^{\mu}$ of the
background space-time, and regardless of the choice of the spin supplementary
condition, an important property of the equations (72) is that the quantity

$\mathcal{P}_{\xi}=\xi^{\mu}p_{\mu}+\frac{1}{2}S^{\mu\nu}\nabla_{\mu}\xi_{\nu}$
(74)

is conserved along the worldline, $i.e.$, $D\mathcal{P}_{\xi}/d\tau=0$. This
holds to all orders in the multipole expansion Harte:2014wya . The system of
equations (72)–(73) is explicitly invariant under reparametrizations of the
worldline, but for simplicity we will here adopt the condition
$\dot{x}^{2}\equiv\dot{x}_{\mu}\dot{x}^{\mu}=-1$, making $\tau$ the proper
time.

A form of the quadrupole tensor $J$ appropriate to describe a spin-induced
quadrupole, quadratic in the spin, and assuming eq. (73) is given by

$J^{\mu\nu\rho\sigma}=\frac{-3}{(-p^{2})^{3/2}}p^{[\mu}S^{\nu]}p^{[\rho}S^{\sigma]},$
(75)

for the case of a black-hole probe Steinhoff:2011sya . We will restrict
ourselves to such probes here but stress that probes with internal and finite-
size structure can be treated in this formalism as well. Here, $S^{\mu}$ is
the Pauli-Lubanski spin vector,

$S^{\mu}=-\frac{1}{2}\epsilon^{\mu}{}_{\nu\rho\sigma}\frac{p^{\nu}}{\sqrt{-p^{2}}}S^{\rho\sigma}\qquad\Leftrightarrow\qquad
S^{\mu\nu}=\epsilon^{\mu\nu}{}_{\rho\sigma}\frac{p^{\rho}}{\sqrt{-p^{2}}}S^{\sigma},$
(76)

with $p_{\mu}S^{\mu}=0$. It has invariant magnitude

$S^{2}\equiv S_{\mu}S^{\mu}=\frac{1}{2}S_{\mu\nu}S^{\mu\nu}.$ (77)

We next solve for the worldline tangent $\dot{x}^{\mu}$ by covariantly
differentiating (73) with respect to $\tau$ and inserting equations (72). For
our case of a black-hole probe with its associated spin-induced quadrupole
(75), and working perturbatively in the test body’s spin $S$, one finds after
a remarkable cancellation the simple relation

$\dot{x}^{\mu}=\frac{p^{\mu}}{\sqrt{-p^{2}}}+\mathcal{O}(S^{3}),$ (78)

as noted in ref. Bini:2015zya . That is, the tangent is still proportional to
the momentum through this order, for a black hole. Finally, one can verify
from (72) with (75)–(76) that the quantity

$m^{2}\equiv
p^{2}+R_{\mu\nu\rho\sigma}\frac{p^{\mu}p^{\rho}}{-p^{2}}S^{\nu}S^{\rho}+\mathcal{O}(S^{3})$
(79)

is conserved to the order shown. Taking the flat space limit, we identify $m$
with the mass of the scattered probe. All of this holds in a general curved
background.

We now restrict ourselves to the background of a Kerr spacetime outside a
black hole of mass $M$ and spin $Ma$ in Boyer-Lindquist coordinates
$x^{\mu}=(t,r,\theta,\phi)$. We again consider the motion in the equatorial
plane $\theta=\pi/2$, and with the probe spin aligned (or anti-aligned) with
the symmetry axis, $S^{\mu}=-Se_{\theta}{}^{\mu}$ where $e_{\theta}{}^{\mu}$
is the unit vector in the $\theta$ direction. We take the constant scalar $S$
to carry a sign: positive when the probe spin is aligned with the Kerr spin,
and negative when anti-aligned. Note that the motion will remain in the
equatorial plane only when the spin is aligned, and the spin will remain
aligned only when the motion is in the equatorial plane. In this case, the
evolution equation (72b) for $S^{\mu\nu}$ is automatically satisfied, and the
content of evolution equation (72a) for $p^{\mu}$ is equivalent to the
conservation equations for three constants of motion: the invariant mass $m$
of (79) and the two constants (74) from the two Killing vectors of the Kerr
background. The timelike Killing vector $t^{\mu}\equiv(\partial_{t})^{\mu}$
gives the conserved energy $E\equiv\mathcal{P}_{t}$, and the axial Killing
vector $\phi^{\mu}\equiv(\partial_{\phi})^{\mu}$ gives in this aligned-
spin/equatorial case the total angular momentum $J\equiv\mathcal{P}_{\phi}$,

$\displaystyle E$ $\displaystyle=-p_{a}t^{a}-\frac{1}{2}S^{ab}\nabla_{a}t_{b}$
$\displaystyle\qquad\quad J$
$\displaystyle=p_{a}\phi^{a}+\frac{1}{2}S^{ab}\nabla_{a}\phi_{b}$ (80)
$\displaystyle=-p_{t}+\frac{GMS}{r^{3}\sqrt{-p^{2}}}(p_{\phi}+ap_{t}),$
$\displaystyle=p_{\phi}+\frac{S}{\sqrt{-p^{2}}}\Big{[}{-}p_{t}+\frac{GMa}{r^{3}}(p_{\phi}+ap_{t})\Big{]},$
(81)

where the second line has evaluated in terms of the momentum components
$p_{\mu}=(p_{t},p_{r},p_{\theta}=0,p_{\phi})$ in Boyer-Lindquist coordinates
with $\theta=0,p_{\theta}=0$ and with the spin tensor as specified above.
Similarly evaluating (79) yields

$m^{2}=-p^{2}+\frac{GMS^{2}}{r^{3}}\bigg{[}1+3\frac{(p_{\phi}+ap_{t})^{2}}{r^{2}(-p^{2})}\bigg{]}+\mathcal{O}(S^{3}),$
(82)

with

$-p^{2}=-g^{\mu\nu}p_{\mu}p_{\nu}=\frac{[(r^{2}+a^{2})p_{t}+ap_{\phi}]^{2}}{r^{2}\Delta}-\frac{(p_{\phi}+ap_{t})^{2}}{r^{2}}-\frac{\Delta}{r^{2}}p_{r}^{2},$
(83)

and $\Delta\equiv r^{2}+a^{2}-2GMr$. Now the system (80)–(83) can be solved,
working perturbatively in $S$, for the momentum components $p_{t}$, $p_{r}$
and $p_{\phi}$ as functions of only the Boyer-Lindquist radial coordinate $r$
and the constants $M$, $a$, $E$, $J$ and $m$.

From the relation (78) for the tangent vector, with
$p^{\mu}=g^{\mu\nu}p_{\nu}$, evaluating the $r$ and $\phi$ components yields

$\displaystyle\dot{\phi}$
$\displaystyle=\frac{1}{\Delta\sqrt{-p^{2}}}\Big{[}p_{\phi}-\frac{2GM}{r}(p_{\phi}+ap_{t})\Big{]}+\mathcal{O}(S^{3}),\phantom{\Bigg{|}}$
(84) $\displaystyle\dot{r}$
$\displaystyle=\frac{\Delta\,p_{r}}{r^{2}\sqrt{-p^{2}}}+\mathcal{O}(S^{3}),$
(85)

which can each be expressed as functions of $r$ and the constants of motion
from the results above. Then the scattering angle $\chi$ can be computed from

$\chi=2\int_{r_{m}}^{\infty}dr\,\frac{\dot{\phi}}{\dot{r}}-\pi\,.$ (86)

From this expression we can immediately make contact with our general formula
(18). Note that $d\phi/dr$ up to and including $\mathcal{O}(S^{2})$ has the
correct form to readily identify $h(r)$. First,

$\frac{d\phi}{dr}=\frac{r(lr-2G\kappa
M)}{p_{r}\left(a^{2}+r(r-2GM)\right)^{2}}-\frac{aG\kappa
MS}{mrp_{r}\left(a^{2}+r(r-2GM)\right)^{2}}+\frac{G\kappa
MS^{2}(r-2GM)}{mr^{2}p_{r}\left(a^{2}+r(r-2GM)\right)^{2}}+\mathcal{O}(S^{3})$
(87)

where for simplicity we have introduced $\gamma=E/m$ and $l\equiv L/m$, where
$L\equiv J-\gamma S$ is the orbital angular momentum (cf. eq. (81) as
$r\to\infty$), and $\kappa\equiv l-\gamma a$. The radial momentum $p_{r}$ is
the positive root of eq. (83),

$p_{r}=\sqrt{\frac{r^{2}}{\Delta}\left[\frac{[(r^{2}+a^{2})p_{t}+ap_{\phi}]^{2}}{r^{2}\Delta}-\frac{(p_{\phi}+ap_{t})^{2}}{r^{2}}+p^{2}\right]}\,.$
(88)

This identifies

$h(r)=-\frac{r(lr-2G\kappa M)}{\left(a^{2}+r(r-2GM)\right)^{2}}+\frac{aG\kappa
MS}{mr\left(a^{2}+r(r-2GM)\right)^{2}}-\frac{G\kappa
MS^{2}(r-2GM)}{m^{2}r^{2}\left(a^{2}+r(r-2GM)\right)^{2}}+\mathcal{O}(S^{3})~{}.$
(89)

We can readily use the expression above in normal coordinate systems by means
of the transformation (60). Furthermore, $h(r)$ obeys our requirements of
being real analytic on the interval $r\in[r_{m},\infty[$ and with falls-off as
$\lim_{r\rightarrow\infty}h(r)\sim 1/r^{n}$, with $n\geq 2$. Therefore, eq.
(44) can be used. Note that to each order in $S$, the radial momentum $p_{r}$
must be expanded correspondingly. An equivalent form of this integrand was
first derived (by the same methods) in ref. Bini:2017pee .

Results up to sixth order in $G$ and up to second order in probe spin $S$ are
given in Table III. We note that the pattern of resummation in the Kerr black
hole spin is generalized to an overall prefactor of

$c_{n,k}\equiv\frac{G^{n}M^{n}(S/m)^{k}}{v^{2n}(b^{2}-a^{2})^{(3n+2k-1)/2}}$
(90)

to first ($k=1$) and second ($k=2$) order in the probe spin. It is tempting to
conjecture that this pattern will hold to higher orders ($k>2$) in the probe
spin. Furthermore, we observe that the remainders after factorising eq. (90)
again shows remarkable structures to linear order in the spin of the probe
$S$, i.e. for $k=1$

$\displaystyle\chi_{n,1}/c_{n,1}$
$\displaystyle=(av-b)(a-bv)\sum_{\ell=0}^{2n-2}a^{\ell}b^{2n-2-\ell}f_{n,1,\ell}(v)\qquad\textrm{for
odd $n$},$ (91) $\displaystyle\chi_{n,1}/c_{n,1}$
$\displaystyle=(av-b)(a-bv)^{n-1}\sum_{\ell=0}^{n}a^{\ell}b^{n-\ell}f_{n,1,\ell}(v)\qquad\
\textrm{for even $n$},$ (92)

where $f_{n,1,\ell}(v)$ are polynomials in $v$ of order $n$ for even $n$ and
order $2n-2$ for odd $n$. We have not found any discernible structure for the
results at quadratic order in $S$.

##  V Conclusion

We have derived a simple formula for the scattering angle of massless probes
in external black hole metrics. Building on the compact formula presented in
ref. Bjerrum-Bohr:2019kec , we have found a scattering angle expression that
straightforwardly handles metrics in any choice of coordinates belonging to a
class we have denoted as normal. In such coordinates the metric enjoys the
property of reducing to flat Minkowski metric in polar coordinates when one
takes the limit $G\to 0$. To illustrate, we have derived the scattering angles
of massive and massless probes in the metric of a Schwarzschild black hole in
Schwarzschild coordinates. The final scattering angle formula is manifestly
free of any dependence on the turning point $r_{m}$ of the orbit without any
need of regularization or prescription.

While of interest in itself, the existence of such a compact formula for the
scattering angle becomes more important in the case of scattering in the
equatorial plane of Kerr black hole metrics. Choosing standard Boyer-Lindquist
coordinates, one notices that the Kerr metric is not in normal form in those
coordinates. We show that a simple transformation of the radial coordinate
brings the Kerr metric to normal form and we are then able to rather
effortlessly calculate the scattering angle in this Kerr metric to any desired
order in $G$. Interestingly, we find that the resulting expressions all re-sum
the dependence on the black hole spin $a$ to all orders, for any fixed order
in $G$. Finally, we have extended these scattering angle calculations to the
case of spinning black-hole probes in the aligned (or anti-aligned) case of
spins in the equatorial plane of the Kerr metric. Our results display
regularities up to second order in the probe spin that may lead to a better
understanding of all-order results in the case of scattering of spinning black
holes. We expect the resulting expressions to be useful for the community
presently computing scattering angles from gravitational scattering amplitudes
in the Post-Minkowskian expansion.

###### Acknowledgements.

Acknowledgements: The work of P.H.D. and A.L. was supported in part by
Independent Research Fund Denmark, grant number 0135-00089B. This research was
also supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958. AL is supported partially by the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 847523 INTERACTIONS. We thank the Kavli Institute of theoretical
Physics for hospitality during the program "High-Precision Gravitational
Waves" where part of this work was done.

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