# Charged test-particle scattering and effective one-body metrics with spin

Jitze Hoogeveen1
1Niels Bohr International Academy
Niels Bohr Institute, University of Copenhagen
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark

###### Abstract

Using recently developed techniques, we consider weak-field test-particle
scattering angle calculations in two distinct settings: Charged test-particles
in spacetimes of charged sources and Effective One-Body theory with spin. We
present scattering angle calculations up to $\mathcal{O}(G^{4})$ of charged
particles in the Kerr-Newman metric, including electromagnetic interactions up
to second order in charge. Coulomb scattering is also discussed, and the well-
known Darwin scattering formula is rederived by resummation. An Effective One-
Body metric for a Kerr-Schwarzschild binary is constructed in a post-
Minkowskian framework up to $\mathcal{O}(G^{2})$ and first order in spin.
Facilitated by explicit scattering calculations, our approach is equivalent
with existing literature through gauge-like transformations. Finally, we
investigate if the Newman Janis Algorithm applied to an Effective One-Body
metric of non-spinning binaries represents a binary system with spin.

###### pacs:

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

††preprint:

##  I Introduction

The breakthrough observation of Gravitational Waves Abbott  _et al._ (2016)
opens a new window to the universe, allowing for the first time detailed
testing of General Relativity. Among these initial observations, signals from
binary black holes intricately encode both dynamics of the binary, and single
black hole properties. The prospect of gaining insight into these previously
unprobed areas has in recent years catalysed a great theoretical effort,
developing new analytical and numerical techniques solving the highly non-
linear dynamics of General Relativity. One such area is the investigation of
test-particle trajectories. This historically well-established subject
provides both insight to single black hole properties, and facilitates a
simple setting for developing new calculational tools, some of which have
proven useful even to the full binary problem.

Analytical expressions for geodesics of test-particles have been found for
(off-)equatorial trajectories, expressed with elliptical functions (for a
review, see ref. Lämmerzahl and Hackmann (2016)). Calculations with a non-
spinning Schwarzschild black hole were carried out in Hagihara (1930); Darwin
(1959, 1961) followed by a charged non-spinning Reissner Nordström black hole
Gackstatter (1983); Grunau and Kagramanova (2011). Similar geodesics for a
spinning Kerr black hole Carter (1968); Mino (2003); Čadež  _et al._ (1998);
Fujita and Hikida (2009) including plunging orbits Dyson and van de Meent
(2023), notably introduced the Carter constant Carter (1968), and Mino time
which is essential for integration Mino (2003). A collection of results is
presented e.g. in refs. Sharp (1979); Kraniotis (2005). Kerr-Newman geodesics
are similarly expressible in terms of elliptical functions Hackmann and Xu
(2013), considering even charge on the test-particle.

Recently, test-particle scattering trajectories have seen increased attention.
The associated calculations of the scattering angle are closely related to
binary dynamics. For the purposes of this article, we will restrict ourselves
to planar scattering. Examples of non-planar paths are given in e.g. refs.
Hackmann (2010); Hackmann and Xu (2013). Although planar scattering angles are
easily encapsulated within a Hamilton-Jacobi formalism, difficulties dealing
with integration limits make actual calculations a non-trivial matter. Few
closed form scattering angle calculations are possible (see e.g. ref. Kol  _et
al._ (2022)).

In a weak-field limit, where the scattering angle $\chi$ is expanded in
Newtons gravitational constant $G$, these difficulties may be overcome.
Hadamard regularization is the traditional approach to scattering angle
evaluations in this limit Damour (2016, 2020a). It substitutes the difficult
lower integration limit with something simple and manually removes emergent
divergencies. A different method was recently developed in ref. Damgaard  _et
al._ (2022). Building on work in isotropic metrics Wallace (1973); Bjerrum-
Bohr  _et al._ (2020), this technique provides a simple formula applicable to
very general situations. It considers the scattering angle integral

$\chi/2+\pi/2=\int_{r_{m}}^{\infty}\\!dr\,\frac{d\phi}{dr}.$ (1)

The lower integration limit, $r_{m}$, is the distance of minimum approach,
which may not always be explicitly obtained. This complicates integration.
Ref. Damgaard  _et al._ (2022) showed, by writing the integral in a very
general form and assuming a weak-field limit, that one may explicitly render
the scattering angle a sum of easily calculable integrals independent of
$r_{m}$. Notably, this applies to test-particles in very general metrics (not
necessarily restricting to black holes), for both scalar and spinning
particles. In ref. Damgaard  _et al._ (2022), the formalism was specifically
employed for scattering in the equatorial plane of a Kerr metric for spinning
test-particles up to second order in spin.

Test-particle scattering in a weak-field regime is linked to binary (black
hole) dynamics in numerous ways. The extreme mass ratio limit of a two-body
calculation naturally retrieves the test-particle regime. As such test-
particle scattering in itself provides a useful tool for cross checking two-
body calculations. However beyond this connection, dynamics of test-particles
may encode dynamics of a two-body system through Effective One-Body (EOB)
theory. Crucial for the EOB approach, to be introduced shortly, are the
calculations of full binary Hamiltonians and scattering angles.

Outside the test-particle limit, the work on binary dynamics in GR is rapidly
evolving, due to its connection with Gravitational Wave observations. Precise
knowledge of binary trajectories is crucial for constructing the waveform.
Both numerical and analytical approaches have proven fruitful. On the
analytical side, various classical methods yield the two-body Hamiltonian in
an expanded form; a post-Newtonian (PN) approach expands around weak-field
Newtonian gravity in velocities $v^{2}/c^{2}\ll 1$ and Newton’s gravitational
constant $GM/c^{2}r\ll 1$, whereas a post-Minkowskian (PM) expansion considers
weak-field interactions with arbitrary velocity as perturbations of Minkowski
space in $GM/c^{2}r\ll 1$. This latter regime is equivalent to the weak-field
approach discussed above in the test-particle limit. Partial expressions for
the non-spinning binary black hole Hamiltonian are available up to 6PN, ie.
$\mathcal{O}[(v/c)^{12}]$, in the post-Newtonian expansion Blümlein  _et al._
(2020); Bini  _et al._ (2020a, b); Blümlein  _et al._ (2021); Foffa  _et al._
(2021); Blümlein  _et al._ (2021) and 4PM, ie. $\mathcal{O}(G^{4})$, in the
post-Minkowskian expansion Kälin  _et al._ (2020a, b); Dlapa  _et al._
(2022a). For spinning binaries, results up to 5PN including both spin-orbit
Levi  _et al._ (2021a); Kim  _et al._ (2022a); Mandal  _et al._ (2022) and
spin-spin couplings Levi (2012); Levi and Steinhoff (2014, 2016, 2021); Kim
_et al._ (2022b); Levi  _et al._ (2021b); Cho  _et al._ (2022a); Kim  _et al._
(2022c); Levi and Yin (2022) have been found. For the post-Minkowskian
expansion all-order in spin expressions are available at 1PM Vines (2018),
whereas second order in spin results are available at 2PM and 3PM Bini and
Damour (2017, 2018); Febres Cordero  _et al._ (2023); Jakobsen and Mogull
(2022a). Starting at $\mathcal{O}(G^{4})$, radiative processes contribute to
conservative dynamics. Dealing with these subtleties is still an open problem
Buonanno  _et al._ (2022).

The post-Minkowskian (PM) expansion naturally lends itself to scattering
trajectories. For reviews see e.g. refs. Bjerrum-Bohr  _et al._ (2022a);
Kosower  _et al._ (2022); Buonanno  _et al._ (2022). Importantly, the two-body
Hamiltonian may be recovered from this regime Vines  _et al._ (2019); Jakobsen
and Mogull (2022a); Damour (2016, 2018). Binary scattering angles, computed
order by order in $G$, encode information about the Hamiltonian. These may be
found by a plethora of methods. Early landmark calculations were performed by
Westpfahl Westpfahl (1985). A linearised form of Einstein’s equations have
yielded exact results for aligned spinning binaries at first order in $G$
Vines (2018). Other manifestly classical approaches such as the world-line
formalism Mogull  _et al._ (2021); Jakobsen  _et al._ (2021, 2022a, 2022b);
Jakobsen and Mogull (2022b); Jakobsen  _et al._ (2022c); Saketh and Vines
(2022); Jakobsen and Mogull (2022a); Bastianelli  _et al._ (2022) and
effective field theory Foffa (2014); Kälin and Porto (2020); Kälin  _et al._
(2020a); Kälin and Porto (2020, 2020); Kälin  _et al._ (2020b); Liu  _et al._
(2021); Cho  _et al._ (2022b); Dlapa  _et al._ (2022b, a); Kälin  _et al._
(2022) employ techniques lent from quantum field theory. Using these,
scattering angles with both non-spinning and spinning binaries are available
up to $\mathcal{O}(G^{3})$ and second order in spin. Quantum calculations,
e.g. amplitudes of massive particle scattering mediated by gravitons, provide
further results up to $\mathcal{O}(G^{5})$ Bern  _et al._ (2019); Antonelli
_et al._ (2019); Parra-Martinez  _et al._ (2020); Di Vecchia  _et al._ (2020);
Damour (2020b); Mougiakakos  _et al._ (2021); Herrmann  _et al._ (2021); Di
Vecchia  _et al._ (2021a, b); Bjerrum-Bohr  _et al._ (2021a, b); Damgaard  _et
al._ (2021); Brandhuber  _et al._ (2021); Bern  _et al._ (2022a, b); Bjerrum-
Bohr  _et al._ (2022b) and various orders in spin Guevara (2019); Damgaard
_et al._ (2019); Vines  _et al._ (2019); Guevara  _et al._ (2019a, b);
Bautista and Guevara (2019); Maybee  _et al._ (2019); Chung  _et al._ (2019);
Arkani-Hamed  _et al._ (2020); Siemonsen and Vines (2020); Chung  _et al._
(2020a, b, c); Aoude  _et al._ (2020); Guevara  _et al._ (2021); Cristofoli
_et al._ (2021); Chen  _et al._ (2022a, b); Kim (2022); Haddad and Helset
(2020); Haddad (2022); Aoude  _et al._ (2022a, b); Bern  _et al._ (2021);
Kosmopoulos and Luna (2021); Bern  _et al._ (2022c); de la Cruz (2022); Febres
Cordero  _et al._ (2023); Alessio and Di Vecchia (2022); Adamo  _et al._
(2022); Menezes and Sergola (2022); Riva  _et al._ (2022); Chiodaroli  _et
al._ (2022); Cangemi  _et al._ (2022); Adamo  _et al._ (2023); Bjerrum-Bohr
_et al._ (2023); Kim and Steinhoff (2023). Remarkably, the classical limit of
such amplitude calculations is in correspondence with macroscopic black hole
scattering Cachazo and Guevara (2020); Guevara  _et al._ (2019a, b),
reproducing classical results at least up to 3PM and second order in spin
Guevara  _et al._ (2019a, b); Jakobsen and Mogull (2022a). Especially relevant
in our current treatment is the scattering angle of aligned spinning particles
from amplitudes, up to 2PM and fourth order in spin Guevara  _et al._ (2019a),

$\displaystyle\begin{aligned}
\hfil\displaystyle\begin{split}\chi^{(a_{1},a_{2})}=&\frac{2GE\left(-2(a_{1}+a_{2})v+bv^{2}+b\right)}{v^{2}\left(b^{2}-(a_{1}+a_{2})^{2}\right)}-\pi
G^{2}E\frac{\partial}{\partial
b}[m_{2}f(a_{1},a_{2})+m_{1}f(a_{2},a_{1})]+\mathcal{O}(G^{3})\end{split}\end{aligned}$
(2a) $\displaystyle
f(\sigma,a)=\frac{1}{2a^{2}}\left(-b+\frac{(j+\varkappa-2a)^{5}}{4v\varkappa[(j+\varkappa)^{2}-(2va)^{2}]^{3/2}}\right)+\mathcal{O}(\sigma^{5}),\quad\quad
j=vb+\sigma+a,\quad\varkappa=\sqrt{j^{2}-4va(b-v\sigma)}$ (2b)

evaluated in center of mass coordinates. This matches classical computations
of aligned Kerr black holes at all orders in spin for $\mathcal{O}(G)$, up to
linear order in spin at $\mathcal{O}(G^{2})$, and conjecturally up to
$\mathcal{O}(spin^{4})$. Parameters $(a_{1},a_{2})$ are the binary spins with
$m_{1}$ and $m_{2}$ their masses, $E$ is the total (center of mass) energy of
the system, $v$ is the relative asymptotical (center of mass) velocity between
objects, $b$ is their impact parameter and $\gamma=1/\sqrt{1-v^{2}}$.

As suggested in refs. Damour (2016); Bjerrum-Bohr  _et al._ (2018); Cheung
_et al._ (2018); Cristofoli  _et al._ (2019); Kälin and Porto (2020, 2020);
Cho  _et al._ (2022b), post-Minkowskian angles encode binary dynamics even for
bound orbits. This is crucial for the study of gravitational waves, typically
emitted by inspiraling bound systems. One way to recover bound orbit dynamics
from scattering data is with Effective One-Body (EOB) theory. Originally
formulated in a Post Newtonian expansion of velocities Buonanno and Damour
(1999, 2000); Damour  _et al._ (2000); Damour (2001), in 2016 it was naturally
adopted to a post-Minkowskian, scattering-based approach Damour (2016);
Antonelli  _et al._ (2019). The EOB formalism translates full binary motion
(outside the scattering regime), to an effective test-particle moving in an
EOB metric. This metric may be constructed by matching scattering angles of
the full binary with those of the effective test-particle. This formalism is
thus heavily reliant on scattering angle calculations of test-particles in
complicated (EOB) metrics. The construction of such a metric has three major
benefits, i) calculations of test-particle motion are much easier than
directly solving Einstein’s equations, ii) geodesics of the metric readily
include bound orbits, and iii) the metric in effect resums post-Minkowskian
data, widening the regime of applicability.

EOB formalisms have been constructed up to $\mathcal{O}(G^{2})$ without spin
in refs. Damour (2016, 2018, 2020a). Including spin, current literature goes
up to $\mathcal{O}(G^{2})$ and $\mathcal{O}(spin^{1})$ Bini and Damour (2017,
2018). EOB mappings at higher orders in spin have been considered in ref.
Vines  _et al._ (2019) at $\mathcal{O}(G^{2})$, and an all order in spin
result at $\mathcal{O}(G)$ was published by Justin Vines Vines (2018). For
recent developments in the post-Newtonian approach, see refs. Bini  _et al._
(2020a, b). Ref. Bini and Damour (2012) includes a discussion on non-
conservative contributions. For older results see Buonanno and Damour (1999);
Damour  _et al._ (2000); Damour (2001); Buonanno and Damour (2000) of which a
review is given in Damour (2008). Notably, ref. Damour  _et al._ (2000)
presents a post-geodesic $\mathcal{Q}$ term to the Hamiltonian which is
reintroduced in post-Minkowskian theory in Damour (2018). See also ref. Damour
(2020a) for discussions hereof, especially the rewriting to an effective
potential $W$.

Test-particle scattering calculations provide a way to both probe single black
hole properties, and encode full black hole binary dynamics. The method of
ref. Damgaard  _et al._ (2022) provides a novel tool for evaluating angles,
enabling new streamlined analysis of these areas. In this paper, demonstrating
the versatility of the method, we consider scattering in two distinct
settings. First, the formalism of ref. Damgaard  _et al._ (2022) is applied to
scattering of charged test-particles in the Kerr-Newman metric. Applications
to other spacetimes and electric potentials is discussed. Among these is the
treatment of relativistic Coulomb scattering.

Second, the method of ref. Damgaard  _et al._ (2022), given its broad
applicability, is considered with the EOB approach. A post-Minkowskian
framework, such as that from refs. Damour (2016); Damgaard and Vanhove (2021)
is used. Particularly, an EOB metric for a Kerr-Schwarzschild black hole
binary is constructed, up to second order in $G$ and fourth order in spin.
Comparisons with earlier approaches Vines (2018); Damgaard and Vanhove (2021);
Damour (2020a) are made. The formalism is restricted to orbits in the
equatorial plane. As an accompanying study, the Newman-Janis Algorithm (NJA)
is explored in context of the EOB formalism. Does the application of the NJA
algorithm to non-spinning EOB metrics produce an EOB metric with spin? The
success of this approach, based on a Schwarzschild-Schwarzschild binary EOB
metric from ref. Damgaard and Vanhove (2021), will be explicitly checked by
comparing post-Minkowskian scattering angles of the NJA-transformed metric
with those of aligned Kerr black holes from amplitude methods Guevara  _et
al._ (2019a).

Section II introduces the general formalism of ref. Damgaard  _et al._ (2022)
with a view towards Kerr-Newman, but emphasises its general applicability.
Sections III and IV then compute Kerr-Newman scattering angles of scalar and
charged test-particles respectively. Sections V and VI turn to the EOB
formalism based on a post-Minkowskian approach. In section V, a 2PM EOB metric
describing Kerr-Schwarzschild binaries is constructed based on a deformed Kerr
metric. Section VI treats the application of the NJA to the EOB metric from
ref. Damgaard and Vanhove (2021).

Throughout we adopt natural units for the speed of light $c=1$ and the Coulomb
constant $1/(4\pi\epsilon_{0})=1$. Newtons gravitational constant is denoted
$G$, and the mostly plus sign convention ($-+++$) is used.

##  II Scattering angles in post-Minkowskian expansion

We introduce test-particle scattering in this section. Let us first consider a
scalar (uncharged, non-spinning) test-particle. Adding electrodynamic behavior
is covered in section IV. We start by reviewing the work of ref. Damgaard  _et
al._ (2022), establishing a general method of evaluating test-particle
scattering angles. The assumptions listed below are used implicitly throughout
the article.

We concern ourselves only with planar scattering in asymptotically flat
metrics $g_{\mu\nu}=g_{\mu\nu}(r)$ parametrised by polar coordinates
$\\{t,r,\phi\\}$. The test-particle, given mass $m$, follows a geodesic
scattering trajectory $r=\\{\infty\rightarrow r_{m}\rightarrow\infty\\}$ with
angular deflection $\phi=\\{0\rightarrow\pi/2+\chi/2\rightarrow\pi+\chi\\}$.
The incident direction is chosen as $\phi=0$ without loss of generality and
$r_{m}$ denotes the distance of minimum approach. Test-particle impact
parameter is denoted $b$, asymptotical velocity $v$, asymptotical momentum
$p_{\infty}$, energy $E$, and orbital angular momentum $L$. Energy and momenta
$E$, $L$ and $p_{\infty}$ may be expressed in terms of $b$ and $v$ as

$E=\gamma m,\quad p_{\infty}=\gamma mv,\quad
L=bp_{\infty},\quad\text{where}\quad\gamma=1/\sqrt{1-v^{2}}.$ (3)

Hamilton-Jacobi theory readily determines the test-particle trajectory, and
therefore also the scattering angle. For a scalar test-particle, the
Hamiltonian and associated Hamilton-Jacobi equations which normalise canonical
momentum are

$H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu},\quad\quad
p_{\mu}=g_{\mu\nu}\frac{dx^{\nu}}{d\lambda},\quad-m^{2}=g^{\mu\nu}p_{\mu}p_{\nu}\quad\Rightarrow\quad\dot{x}^{\mu}=\frac{\partial
H}{\partial p_{\mu}},$ (4)

with affine parameter $d\lambda=ds/m$, given in terms of line element $ds$,
parameterizing the particle path such that canonical momentum $p_{\mu}$
matches test-particle four-momentum. Dots denote differentiation with respect
to $\lambda$. The equations of motion for $\dot{x}^{\mu}$ may be readily found
from the Hamiltonian. Translational symmetry of $H$ in $t$ and $\phi$ yields
conservation of energy and orbital angular momentum, $p_{t}\equiv-E$ and
$p_{\phi}\equiv L$. The radial component $p_{r}$ may be determined from the
Hamilton-Jacobi equation as a function only of coordinate $r$.

The scattering angle may readily be found from the Hamiltonian. It may
generically be written as

$\chi/2+\pi/2=\int_{r_{m}}^{\infty}\\!dr\,\frac{d\phi}{dr}=-\int_{r_{m}}^{\infty}\\!dr\,\frac{h(r)}{p_{r}},\quad\quad
h(r)\equiv-p_{r}\frac{d\phi}{dr}$ (5)

simply by integrating angular deflection over half a scattering trajectory. A
conventional $\pi/2$ has been added to yield $\chi=0$ for straight-line
motion. The lower integration limit $r_{m}$ may be found for Hamiltonians
quadratic in $p_{r}$ by the requirement

$\left.\dot{r}=0\right|_{r=r_{m}}\quad\Rightarrow\quad p_{r}(r_{m})=0.$ (6)

Following ref. Damgaard  _et al._ (2022), we have rewritten the scattering
angle integral simply by defining the function $h(r)$ in terms of
$\frac{d\phi}{dr}$ and $p_{r}$. However this form is suggestive: Note that the
integral is naturally divergent in the lower integration limit
$p_{r}(r_{m})=0$. Factoring out this divergence, $h(r)$ often takes very
simple non-divergent forms. In fact from eq. (4) one identifies

$h(r)=-\frac{Lg^{\phi\phi}-Eg^{\phi t}}{g^{rr}}$ (7)

for a scalar test-particle in the general metric discussed above. This
identification of $h(r)$ is useful beyond scalar particles, and was also shown
applicable to spinning test-particles in ref. Damgaard  _et al._ (2022).
Below, the same will be shown true also for charged test-particles.

Exact calculation of the scattering angle, often does not yield a closed
expression, instead returning an elliptical integral. However, in a weak-field
expansion in $G$ closed expressions may be found order by order in $G$. In
ref. Damgaard  _et al._ (2022) a general calculation of the scattering angle
in the weak-field regime was provided. It considers a scattering angle written
in the form of eq. (5) with $h(r)$ obeying the requirements

$\text{ $h(r)$ is analytical on $r\in[r_{m},\infty[$ and falls off at least
like $1/r^{2}$ as $r\rightarrow\infty$}.$ (8)

One may readily confirm that eq. (7) indeed obeys these requirements.
Furthermore, the metric is assumed written in what ref. Damgaard  _et al._
(2022) defines as a normal form. A metric is of normal form when it has the
property

$g_{\mu\nu}(r)\rightarrow\text{(Minkowski)}=\begin{pmatrix}-1&0&0\\\ 0&1&0\\\
0&0&r^{2}\end{pmatrix}\quad\quad\text{as $G\rightarrow 0$}$ (9)

in the scattering plane. Metrics may readily be written in normal form by
setting $G\rightarrow 0$, and performing a coordinate transformation to
recover the above structure. For asymptotically flat metrics, $p_{r}$ then
takes the form

$p_{r}^{2}=p_{\infty}^{2}-\frac{J^{2}}{r^{2}}-U(r)$ (10)

with some function $U$ specified by the metric reminiscent of potentials in
classical and isotropic amplitude calculations (see e.g. Bjerrum-Bohr  _et
al._ (2020)). This potential may safely be assumed to drop off to zero as
$r\rightarrow\infty$, and may depend on any metric and test-particle
quantities, e.g. angular momentum, energy etc.

The above requirements are satisfied throughout this article. Under such
requirements, ref. Damgaard  _et al._ (2022) provides the scattering angle in
a summed form which readily yields arbitrary high orders in the weak-field
expansion in $G$,

$\chi+\pi=-2\sum_{n=0}^{\infty}\int_{0}^{\infty}\\!du\,\left(\frac{d}{du^{2}}\right)^{n}h(r)\frac{r^{2n}U(r,b)^{n}}{n!p_{\infty}^{2n+1}},\quad\quad
r^{2}=u^{2}+b^{2}.$ (11)

A very similar formula for isotropic metrics was previously obtained in ref.
Bjerrum-Bohr  _et al._ (2020), and may readily be recovered by restricting
$U(r,b)\rightarrow V_{eff}(r)$ independent of $b$ and
$h(r)=-bp_{\infty}/r^{2}$. In this special case $V_{eff}$ is independent of
impact parameter $b$ or, equivalently, angular momentum $L$.

Restricting to scalar test-particles, eq. (11) obtains another useful form.
Inspired by a derivative form of eq. (5),

$\chi/2+\pi/2=-\frac{d}{dL}\int_{r_{m}}^{\infty}\\!dr\,p_{r},\quad\quad\text{(scalar
test-particle)}$ (12)

the $h(r)$-dependence may be converted to a derivative of the impact
parameter,

$\chi=\sum_{n=0}^{\infty}\int_{0}^{\infty}\\!du\,\left(\frac{d}{du^{2}}\right)^{n}\frac{d}{db}\left[\frac{r^{2n}U(r,b)^{n+1}}{(n+1)!p_{\infty}^{2(n+1)}}\right],\quad\quad
r^{2}=u^{2}+b^{2}\quad\text{(scalar test-particle)},$ (13)

removing the need to specify $h(r)$ for each individual metric.

In the next section, we demonstrate the use of eqs. (11) and (13) by
explicitly calculating the scattering angle in the Kerr-Newman metric of a
scalar test-particle, up to $\mathcal{O}(G^{4})$.

##  III Non-charged test-particles in Kerr-Newman spacetime

The Kerr-Newman metric describes a charged, spinning black hole, its mass
denoted by $M$, charge $Q$ and spin $a$. The $Q\rightarrow 0$ and
$a\rightarrow 0$ limits are Kerr and Reissner-Nordström black holes
respectively. Written in Boyer-Lindquist coordinates, it is rotationally
symmetric only in the $\theta=\pi/2$ plane. In this plane the metric reads

$g_{\mu\nu}=\left(\begin{array}[]{ccc}-\left(1-2GM/r+\frac{r_{Q}^{2}}{r^{2}}\right)&0&-a\left(2GM/r-\frac{r_{Q}^{2}}{r^{2}}\right)\\\
0&\frac{r^{2}}{a^{2}-2GMr+r_{Q}^{2}+r^{2}}&0\\\
-a\left(2GM/r-\frac{r_{Q}^{2}}{r^{2}}\right)&0&a^{2}(1+2GM/r-\frac{r_{Q}^{2}}{r^{2}})+r^{2}\\\
\end{array}\right),$ (14)

which is effectively a Kerr metric with $2GM/r$ replaced by
$2GM/r-r_{Q}^{2}/r^{2}$. Parameter $r_{Q}=GQ^{2}$ encodes the black hole
electric charge.

Notice that eq. (14) is not in normal form, which setting $G\rightarrow 0$
confirms. One recovers a result identical to that of Kerr,

$g_{\mu\nu}\rightarrow\begin{pmatrix}-1&0&0\\\
0&\frac{r^{2}}{r^{2}+a^{2}}&0\\\ 0&0&r^{2}+a^{2}\end{pmatrix}\quad\text{as
$G\rightarrow 0$},$ (15)

which may be brought to normal form by a coordinate transformation
$r^{2}\rightarrow\rho^{2}=r^{2}+a^{2}$. In these coordinates, the potential
$U$ may readily be found from eqs. (4) and (10),

$U(\rho,b)=\frac{G\left(Q^{2}-2M\sqrt{\rho^{2}-a^{2}}\right)\left(2a^{2}b^{2}p_{\infty}^{2}-a^{2}\rho^{2}p_{\infty}^{2}-2abE\rho^{2}p_{\infty}-b^{2}\rho^{2}p_{\infty}^{2}+2\rho^{4}p_{\infty}^{2}+m^{2}\rho^{4}\right)}{\rho^{4}\left(\rho^{2}-a^{2}\right)}+\mathcal{O}(G^{2}),$
(16)

given here up to $\mathcal{O}(G)$ for brevity. Of course, equation (4) readily
yields $U$ to all orders in $G$. With this information, equation (13) straight
forwardly gives the scalar test-particle scattering angle in such a spacetime.
No explicit computation of $h(r)$ is needed. Take the $\mathcal{O}(G^{1})$
calculation as an example. The results at higher orders are listed in table 2.
Extracting only $G^{1}$ terms from eq. (13), one finds

$\begin{split}\chi_{1}=&\int_{0}^{\infty}\\!du\,\frac{d}{db}\left[\frac{U(\rho,b)}{p_{\infty}^{2}}\right],\quad\rho^{2}=u^{2}+b^{2}\\\
=&\frac{1}{p_{\infty}^{2}}\int_{0}^{\infty}\\!du\,\frac{d}{db}\left[\frac{G\left(Q^{2}-2M\sqrt{\rho^{2}-a^{2}}\right)\left(2a^{2}b^{2}p_{\infty}^{2}-a^{2}\rho^{2}p_{\infty}^{2}-2abE\rho^{2}p_{\infty}-b^{2}\rho^{2}p_{\infty}^{2}+2\rho^{4}p_{\infty}^{2}+m^{2}\rho^{4}\right)}{\rho^{4}\left(\rho^{2}-a^{2}\right)}\right]\\\
=&\frac{2GM\left(-2av+bv^{2}+b\right)}{v^{2}\left(b^{2}-a^{2}\right)}+\frac{\pi
GQ^{2}\left(2a^{3}v-a^{2}\left(-v^{2}\sqrt{b^{2}-a^{2}}+2bv^{2}+b\right)+b^{2}v^{2}\left(b-\sqrt{b^{2}-a^{2}}\right)\right)}{2a^{2}v^{2}\left(b^{2}-a^{2}\right)^{3/2}}.\end{split}$
(17)

The $Q\rightarrow 0$ limit is just the Kerr result. At any order
$\mathcal{O}(G^{n})$, $Q$ only appears in even orders no greater than
$Q^{2n}$. The scattering angle $\chi$ may thus be decomposed into orders of
$G$ and $Q$,

$\chi=\sum_{n=1}^{\infty}\sum_{k=0}^{n}\chi^{(k)}_{n},\quad\quad\chi^{(k)}_{n}\sim\pi^{(n+k+1)\text{
mod }2}\frac{Q^{2k}G^{n}M^{n-k}}{v^{2n}(b^{2}-a^{2})^{(3n+k-1)/2}}f_{n,k}(v).$
(18)

These expressions are structurally similar to those found in ref. Damgaard
_et al._ (2022) for Schwarzschild and Kerr metrics. A prefactor $\pi$ appears
in $\chi_{n}^{(k)}$ only when $n+k$ is even, much like the factors of $\pi$
appearing at even order in ref. Damgaard  _et al._ (2022) for Kerr and
Schwarzschild. Indeed, the Kerr-Newman potential $U\sim 1/r^{n+k}$ at order
$\mathcal{O}(G^{n})$ and $\mathcal{O}(Q^{2k})$ resembles in its $r$ dependence
a Kerr potential at order $n+k$ in $G$. Additional similarities may be found.
Polynomials $f_{n,k}(v)$ of velocity $v$ depend on fractional powers of $b$
and $a$ for even $n+k$, and integer powers of $b$ and $a$ for odd $n+k$. This
mimics the identical behavior in Kerr, for $n$ respectively even and odd.
Interestingly, the $v$ and $(a,b)$ dependence of $f_{n,k}$ is partially
factorized at orders $(n,k)=(1,0),(2,1),(3,2),(4,3)$, ie. when $n-k=1$. Here
we observe

$f_{n,k}=[f^{(v)}_{-}f^{(ab)}_{-+}+f^{(v)}_{+}f^{(ab)}_{+-}]_{n,k}$ (19)

where $f^{(v)}$ and $f^{(ab)}$ are polynomials exclusively dependent on either
$v$ or $(a,b)$ respectively. Subscript $+,-$ signs indicate whether $v$, $a$
and $b$ appear in even or odd powers, ie. $f^{(v)}_{+}$ is even in $v$ and
$f^{(ab)}_{+-}$ is even in $a$ but odd in $b$. All polynomials differ in
structure with varying $n,k$. We expect the structure of eq. (19) to continue
to higher orders.

##  IV Charged test-particles in Kerr-Newman spacetime

Continuing our treatment of the Kerr-Newman metric, we next consider test-
particles with charge $e$ and mass $m$ in a spacetime of a charged source. For
completeness, as before, our initial analysis pertains to a general metric
$g_{\mu\nu}$ as defined in the previous section. Hereafter, the specific case
of the Kerr-Newman metric is treated. A short study of Coulomb scattering will
also be discussed, as results are easily compared to the all-order exact
Coulomb scattering angle Kol  _et al._ (2022) obtained by direct integration
of eq. (5).

Apart from gravitational effects, electromagnetic interactions with coupling
constant $e$ (charge of test-particle) and electromagnetic 4-potential
$A_{\mu}$ need to be accounted for. Charged bodies in curved spacetime may be
treated with Einstein-Maxwell theory (for a review, see e.g. ref. Poisson  _et
al._ (2011), including Westpfahl (1985) for early scattering calculations).
Charged test-particle orbits around black holes are covered in ref. Young
(1976); Johnston and Ruffini (1974); Sharp (1979); Hackmann and Xu (2013) and
fully characterized for Kerr-Newman in ref. Hackmann and Xu (2013). The
treatment presented in this article is restricted to test-particle limits of
$e$ and $m$, neglecting self-force effects Westpfahl (1985); Poisson  _et al._
(2011); Barack and Pound (2019). $A_{\mu}$ is then entirely produced by the
gravitational source.

Subjected to an external potential $A_{\mu}$, the Hamiltonian and associated
equations of motion of a test-particle with charge $e$ and mass $m$ in metric
$g_{\mu\nu}$ are

$H=\frac{1}{2}g^{\mu\nu}(p_{\mu}-eA_{\mu})(p_{\nu}-eA_{\nu}),\quad
p_{\mu}=g_{\mu\nu}\frac{dx^{\nu}}{d\lambda}+eA_{\mu}\quad\Rightarrow\quad\dot{x}^{\mu}=\frac{\partial
H}{\partial p_{\mu}}$ (20)

where $p_{\mu}$ denotes canonical momentum. Affine parameter $d\lambda=ds/m$
is defined in terms of the line element $ds$ as in eq. (4), and we continue to
denote $\dot{x}^{\mu}=\frac{dx^{\mu}}{d\lambda}$. For generality, the
discussion below will not assume any specific form of $A_{\mu}$, save require

$\text{$A_{\mu}\rightarrow 0$ as $r\rightarrow\infty$},\quad\quad
A_{\mu}=(A_{t}(r),0,0,A_{\phi}(r))_{\theta=\pi/2}$ (21)

with $\theta=\pi/2$ the equatorial plane of orbit. The scattering angle may
still be calculated from eq. (5). Equation (24) yields

$\frac{d\phi}{dr}=\frac{(L-eA_{\phi})g^{\phi\phi}+(-E-eA_{t})g^{\phi
t}}{g^{rr}p_{r}},$ (22)

and per identification

$\displaystyle h(r)=-\frac{(L-eA_{\phi})g^{\phi\phi}+(-E-eA_{t})g^{\phi
t}}{g^{rr}}.$ (23)

Asymptotically, $h(r)\sim 1/r^{2}$ as required, provided $A_{\mu}$ obeys eq.
(21). The Hamilton-Jacobi equation from which $p_{r}$ may be determined, is
found by normalizing $\dot{x}^{2}=-m^{2}$ with eq. (20)

$-m^{2}=g^{\mu\nu}(p_{\mu}-eA_{\mu})(p_{\nu}-eA_{\nu}).$ (24)

Having found the radial momentum with $g_{\mu\nu}$ implicitly in normal form,
the corresponding potential $U$ is identified from eq. (10)

$p_{r}^{2}=T-U.$ (25)

Crucially, $T$ is now taken independent of both $G$ and $e$,

$T\equiv\left.p_{r}^{2}\right|_{G=e=0}$ (26)

as we are dealing with two interactions.

We now treat the Kerr-Newman metric, and subsequently relativistic Coulomb
scattering in flat space. Both conform to the requirements set by eq. (11).
The respective electrodynamic potentials read

$A_{\mu}=(-\frac{Q}{r},0,0,\frac{aQ}{r})\quad\text{(Kerr-Newman)},\quad\quad
A_{\mu}=(-\frac{Q}{r},0,0,0)\quad\text{(Coulomb)}.$ (27)

where the Kerr-Newman potential, see ref. Griffiths and Podolsky (2009), has
been evaluated at $\theta=\pi/2$ in Boyer-Lindquist coordinates. The Coulomb
potential may simply be considered the $G=a=0$ limit of the Kerr-Newman
solution.
Let us therefore focus on Kerr-Newman. The normal form of the metric is
recovered by the coordinate transformation of eq. (15), $\rho^{2}=r^{2}+a^{2}$
where $r$ is the Boyer-Lindquist radial coordinate appearing in eq. (14).
Equations (10) and (24) then yield

$\begin{split}U(\rho)=&\frac{Qe\left(-2ab\sqrt{\rho^{2}-a^{2}}p_{\infty}+\rho^{2}\left(2E\sqrt{\rho^{2}-a^{2}}-Qe\right)+a^{2}Qe\right)}{\rho^{4}-a^{2}\rho^{2}}\\\
&\begin{aligned}
+G\frac{\left(Q^{2}-2M\sqrt{\rho^{2}-a^{2}}\right)}{\rho^{6}-a^{2}\rho^{4}}\Big{[}&p_{\infty}\left(p_{\infty}\left(-\rho^{2}\left(a^{2}+b^{2}\right)+2a^{2}b^{2}+2\rho^{4}\right)+4abQe\sqrt{\rho^{2}-a^{2}}-2abE\rho^{2}\right)\\\
&+2Q\rho^{2}e\left(Qe-2E\sqrt{\rho^{2}-a^{2}}\right)-2a^{2}Q^{2}e^{2}+m^{2}\rho^{4}\Big{]}\end{aligned}\\\
&+\mathcal{O}(G^{2}),\end{split}$ (28)

here truncated at $\mathcal{O}(G)$ for simplicity. The scattering angle is now
found readily from eq. (11). It may be expanded simultaneously in $G$ and $e$,
ie. as a collective expansion in gravitational and electromagnetic
interactions,

$\chi=\sum_{n=0}^{\infty}\sum_{j=0}^{\infty}\chi_{n,j},\quad\chi_{n,j}\sim
G^{n}e^{j}.$ (29)

As before, owing to the $Q^{2}$-dependence of the Kerr-Newman metric,
$\chi_{n,j}$ may be decomposed as

$\chi_{n,j}=\sum_{k=0}^{n}\chi_{n,j}^{(k)},\quad\chi_{n,j}^{(k)}\sim Q^{2k},$
(30)

where $k$ denotes the order of $Q^{2}$ coming from the metric.
$\chi_{n,j}^{(k)}$ are tabulated in table 3, B) up to $\mathcal{O}(G^{3})$ and
$\mathcal{O}(e^{3})$. We again find definite structure

$\chi_{n,j}^{(k)}\sim\pi^{(n+j+k+1)\text{mod
}2}\frac{G^{n}e^{j}Q^{2k+j}M^{n-k}}{m^{j}\gamma^{j}v^{2(j+n)}\left(b^{2}-a^{2}\right)^{(3(n+j)+k-1)/2}}\sum_{p=0}^{2(n+j)-1}d_{n,j,k}a^{p}b^{2(n+j)-p}f_{n,j,k}(v),$
(31)

involving only whole powers of $a$ and $b$, contrary to fractional powers
encountered with non-charged test-particles (table 2). This behavior is
structurally equivalent to odd powers of $n+k$ for non-charged test-particles.
$Q$ appears exclusively as a prefactor $Q^{2k+j}$, containing a contribution
from the metric ($Q^{2k}$) and a contribution from the electromagnetic
potential ($Q^{j}$). $f_{n,j,k}(v)$ are polynomials in $v$ and $d_{n,j,k}$ are
numerical constants. As observed with scalar test-particles in eq. (19), some
angles factorise $v$ and $(a,b)$ dependence. Specifically, for
$(n,j,k)=(0,1,0),(1,1,1),(2,1,2)$ and $(3,1,3)$, ie. when $j=1$, the sum in
eq. (31) may be written as two terms

$\sum_{p=0}^{2(n+j)-1}d_{n,j,k}a^{p}b^{2(n+j)-p}f_{n,j,k}(v)=[f^{(v)}_{-}f^{(ab)}_{-+}+f^{(v)}_{+}f^{(ab)}_{+-}]_{n,j,k}$
(32)

where, again, $f^{(v)}$ and $f^{(ab)}$ are polynomials exclusively dependent
on $v$ or $(a,b)$ respectively. Subscript $+,-$ signs indicate whether $v$,
$a$ and $b$ appear as even or odd powers, ie. $f^{(ab)}_{+-}$ is even in $a$
but odd in $b$. The exact structure of polynomials $f$ depends on $n,j$ and
$k$. Furthermore, $f^{(v)}$ and $f^{(ab)}$ share numerical coefficients of $v$
and $(a,b)$ terms. For instance, with $(n,j,k)=(2,1,2)$, the coefficients
$\\{1,5,10\\}$ appear both in $f^{(v)}$ and $f^{(ab)}$. We expect this
behavior to continue to higher orders in $n$ and $k$.

It is similarly a straight forward matter to consider relativistic Coulomb
scattering. The corresponding potential $U$ is given by the $G=a=0$ limit of
eq. (28). It only involves one coupling constant, $e$, in terms of which the
scattering angle is expanded and presented in table 3, A). Order by order
comparisons match the small $Qe/J$ expansion of the well-known Coulomb
scattering angle presented e.g. in ref. Kol  _et al._ (2022),

$\chi+\pi=\frac{J}{\sqrt{J^{2}-Q^{2}e^{2}}}\left(\pi-2\,\arctan\left(\frac{Qe}{v\sqrt{J^{2}-Q^{2}e^{2}}}\right)\right)\quad\quad\text{(Coulomb
scattering)}.$ (33)

##  V Effective One-Body metric for Kerr-Schwarzschild black hole binaries

###  V.1 The EOB formalism and construction of the EOB metric

We now apply the method of ref. Damgaard  _et al._ (2022) to the full binary
problem by means of Effective One-Body (EOB) theory. Dynamics of aligned,
spinning binary black holes are mapped to an effective system consisting of a
test-particle in an EOB metric $g_{\mu\nu}^{(eff)}$. This is achieved by
directly relating equatorial scattering angles of the binary to those of the
effective test-particle. We stress that our approach is not unique - different
EOB metrics may be constructed by similar methods.

The EOB formalism is first presented in the post-Minkowskian regime by
reviewing ref. Damgaard and Vanhove (2021). Consider equatorial scattering of
binary Kerr-Schwarzschild black holes with masses $m_{1}$ and $m_{2}$, and
spins $a_{1}=a$ and $a_{2}=0$. Center of mass coordinates may be used, in
which $E$ denotes the total energy of the system, $v$ is the relative
asymptotical velocity of the binary objects, $p_{\infty}$ is the asymptotical
momentum of a single body, $L$ is the orbital angular momentum of the system,
and $b$ denotes the impact parameter. These quantities are related by

$v=|\mathbf{v}_{1}-\mathbf{v}_{2}|,\quad\quad
p_{\infty}=\frac{(E^{2}-(m_{1}+m_{2})^{2})(E^{2}-(m_{1}-m_{2})^{2})}{4E^{2}}=\frac{m_{1}m_{2}}{E}\gamma
v,\quad\quad L=bp_{\infty},$ (34)

where $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are velocities of the individual
black holes. For convenience we have defined the reduced mass $\mu$, total
mass $M$, and asymptotical Lorentz contraction factor $\gamma$ as

$\mu\equiv\frac{m_{1}m_{2}}{m_{1}+m_{2}},\quad M\equiv
m_{1}+m_{2},\quad\nu\equiv\frac{\mu}{M},\quad\gamma=\frac{1}{\sqrt{1-v^{2}}}.$
(35)

This binary is now described by an effective system consisting of a test-
particle of mass $\mu$ scattering on a metric $g_{\mu\nu}^{(eff)}$. Two-body
quantities are related to effective ones by an EOB map, which we now present.
Label with subscript "eff" quantities of the effective system. From kinematic
considerations of the effective test-particle, one may establish the following
EOB map

$v_{eff}=v,\quad E=M\sqrt{1+2\nu\left(\frac{E_{eff}}{\mu}-1\right)},\quad
p_{eff}=\mu\gamma v=\frac{E}{M}p_{\infty},\quad b_{eff}=b\Rightarrow
L_{eff}=b_{eff}p_{eff}=L\frac{E}{M},$ (36)

where $E_{eff}=\mu\gamma$ is the test-particle energy, and
$v_{eff},p_{eff},L_{eff}$ and $b_{eff}$ are the asymptotical test-particle
velocity, asymptotical momentum, angular momentum and impact parameter
respectively. Furthermore, the effective formalism should have some notion of
spin, call it $a_{eff}$. EOB maps between spin have been discussed in detail
in e.g. refs. Damour (2001); Bini and Damour (2017, 2018); Vines (2018). We
shall here use a simple map, namely

$a_{eff}=a.$ (37)

Last, a map between scattering angles is required. Denote by $\chi$ and
$\chi_{eff}$ the full two-body and effective test-particle scattering angles.
The most natural mapping between these, as discussed in ref. Damour (2016), is
simply

$\chi_{eff}=\chi.$ (38)

Both angles are treated perturbatively in $G$,
$\chi=\sum_{n=0}^{\infty}\chi_{n}G^{n}$. The calculation of $\chi_{eff}$
depends on $g_{\mu\nu}^{(eff)}$. With the full two-body system amplitude
calculations give the individual components $\chi_{n}$. As stated before, ref.
Guevara  _et al._ (2019a) provides the scattering angle of a Kerr-
Schwarzschild binary with spin parameter $a$ and respective masses $m_{1}$ and
$m_{2}$,

$\begin{split}\chi^{(a,0)}=&\frac{2GE\left(-2av+bv^{2}+b\right)}{v^{2}\left(b^{2}-a^{2}\right)}\\\
&\begin{split}+\frac{\pi
EG^{2}}{32b^{6}v^{4}}\big{[}&5a^{4}\left(m_{1}\left(35v^{4}+180v^{2}+24\right)+24\left(4m_{2}v^{2}+m_{2}\right)\right)-96a^{3}bv\left(m_{1}\left(5v^{2}+4\right)+m_{2}\left(2v^{2}+3\right)\right)\\\
&+6a^{2}b^{2}\left(m_{1}\left(15v^{4}+72v^{2}+8\right)+4m_{2}\left(2v^{4}+11v^{2}+2\right)\right)-16ab^{3}v\left(3v^{2}+2\right)(4m_{1}+3m_{2})\\\
&+24b^{4}v^{2}\left(v^{2}+4\right)(m_{1}+m_{2})\big{]}+\mathcal{O}(a^{5})\\\
\end{split}\\\ &+\mathcal{O}(G^{3}).\end{split}$ (39)

The spin configuration is indicated by $(a,0)$. As mentioned, the connection
between eq. (39) and Kerr black hole scattering has only been confirmed at
$\mathcal{O}(G)$ for all orders in $a$, and at $\mathcal{O}(G^{2})$ only up to
$\mathcal{O}(a)$ Guevara  _et al._ (2019a). We restrict to these orders in the
text, and provide a conjectural Kerr binary EOB metric matching the full
$\mathcal{O}(a^{4})$ result in table 1.

The EOB metric $g_{\mu\nu}^{(eff)}$ is constructed in such a way that eq. (38)
is satisfied. An ansatz is provided, the parameters of which are constrained
by eq. (38). As scattering is planar by construction, a rotationally
symmetric, asymptotically flat ansatz is natural. We further demand that the
metric reduces to Kerr in the test-particle limit $m_{2}\rightarrow 0$.

One may thus naturally search for EOB metrics among generalizations of Kerr.
It is instructive to review in short the results obtained by ref. Damgaard and
Vanhove (2021). Here binary Schwarzschild black holes were considered. The EOB
metric had a generalized Schwarzschild form

$ds^{2}=-\left(\frac{1-\alpha(r)}{1+\alpha(r)}\right)^{2}dt^{2}+\left(1+\alpha(r)\right)^{4}(dr^{2}+r^{2}d\Omega^{2}),\quad\quad\alpha(r)=\sum_{n=1}^{\infty}\alpha_{n}\frac{G^{n}}{r^{n}},$
(40)

which simply replaces $GM/(2r)\rightarrow\alpha(r)$ in the Schwarzschild
spacetime written in isotropic coordinates. Adopting an isotropic calculation
renders the scattering angle integrand, $d\phi/dr$ in a form comparable
directly to amplitude calculations of refs. Bjerrum-Bohr  _et al._ (2020);
Damgaard and Vanhove (2021). This allows specification of $\alpha_{n}$
directly from comparing scattering angle integrands, as opposed to merely the
scattering angles. One finds, comparing up to 2PM,

$\alpha_{1}=\frac{1}{2}E,\quad\alpha_{2}=-\frac{3(5\gamma^{2}-1)}{8(2\gamma^{2}-1)}\left(1-\frac{M}{E}\right)E^{2}.$
(41)

The Schwarzschild metric in isotropic coordinates, meaning $\alpha\rightarrow
GM/(2r)$, is recovered in the test-particle limit.

Now turn to Kerr-Schwarzschild binaries. We provide an EOB metric ansatz
written in non-isotropic coordinates, meaning $g_{rr}\neq r^{2}g_{\phi\phi}$.
Our method differs from that of ref. Damgaard and Vanhove (2021) by explicitly
computing the effective scattering angle, instead of comparing integrands.
Comparing our result with eq. (39) determines the EOB metric parameters. We
choose an ansatz constructed from equatorial Kerr in Boyer-Lindquist
coordinates by replacing $GM/r\rightarrow\kappa(r)$,

$g_{\mu\nu}^{(eff)}=\left(\begin{array}[]{ccc}-\left(1-\kappa(r)\right)&0&-a\kappa(r)\\\
0&\frac{r^{2}}{a^{2}+r^{2}\left(1-\kappa(r)\right)}&0\\\
-a\kappa(r)&0&a^{2}\left(1+\kappa(r)\right)+r^{2}\\\
\end{array}\right),\quad\kappa(r)=\sum_{n=1}^{\infty}\kappa_{n}\frac{G^{n}}{r^{n}}$
(42)

parametrized with coordinates $\\{t,r,\phi\\}$. This choice of EOB metric is
entirely arbitrary. A different ansatz could be equally viable, producing a
different final result. The metric above resums orders in $a$, in a structure
similar to Kerr(-Newman) metrics. Note however that $\kappa$ may also depend
on $a$. Below we present $\kappa$ to $\mathcal{O}(a^{4})$ as eq. (39)
naturally restricts hereto.

Equation (13) readily yields the test-particle scattering angle. However, one
should be careful about the parameter-dependence of $\kappa$, which may
possibly depend on all test-particle quantities, namely energy $E_{eff}$,
impact parameter $b_{eff}$, and angular momentum $L_{eff}$, including also
asymptotical velocity $v$ and asymptotical momentum $p_{\infty}$. As an
example, eq. (40) is dependent on $E$. However impact parameter, or
equivalently, angular momentum dependence influences the application of
$\frac{d}{db}$ in eq. (13). The result of calculating the post-Minkowskian
scattering angle by eq. (13) (see footnote 111Note a subtlety with applying
eq. (13). If $\kappa$ depends on $b$, $h(r)$ is no longer given by the usual
scalar structure of eq. (7). However, as explained in ref. Damgaard  _et al._
(2022), the scattering angle integral takes the form $d\phi/dr=dp_{r}/dL$
regardless. $L$ is angular momentum and $p_{r}$ is canonical radial momentum.
Consequently eq. (13) still applies), with arbitrary b-dependent $\kappa$, is
therefore an expression with first-order derivatives of $\kappa_{n}$ in $b$.
In our case, it is sufficient to assume $\kappa_{1}$ is independent of $b$,
which will yield consistent solutions. Other
$\kappa_{n}\rightarrow\kappa_{n}(b)$ will remain unspecified functions of $b$.
Imposing the EOB map of eqs. (36)-(38), the effective test-particle scattering
angle of the EOB metric from eq. (42) becomes

$\begin{split}\chi_{eff}=&\frac{G\kappa_{1}\left(-2av+bv^{2}+b\right)}{v^{2}\left(b^{2}-a^{2}\right)}\\\
&\begin{split}+\frac{\pi
G^{2}}{128b^{6}v^{4}}&\Big{(}4v^{2}\big{[}\kappa_{2}(b)\left(5a^{4}\left(5v^{2}+6\right)-48a^{3}bv+6a^{2}b^{2}\left(3v^{2}+4\right)-32ab^{3}v+8b^{4}\left(v^{2}+2\right)\right)\\\
&\hskip
17.22217pt-b\left(a^{4}\left(5v^{2}+6\right)-12a^{3}bv+2a^{2}b^{2}\left(3v^{2}+4\right)-16ab^{3}v+8b^{4}\left(v^{2}+2\right)\right)\frac{d\kappa_{2}(b)}{db}\big{]}\\\
&+\kappa_{1}^{2}\big{[}5a^{4}\left(35v^{4}+180v^{2}+24\right)-96a^{3}bv\left(5v^{2}+4\right)+6a^{2}b^{2}\left(15v^{4}+72v^{2}+8\right)\\\
&\hskip
17.22217pt-64ab^{3}v\left(3v^{2}+2\right)+24b^{4}v^{2}\left(v^{2}+4\right)\big{]}\Big{)}+\mathcal{O}(a^{5})\end{split}\\\
&+\mathcal{O}(G^{3}),\end{split}$ (43)

presented here up to $\mathcal{O}(G^{2})$ and truncated to
$\mathcal{O}(a^{4})$ to facilitate direct comparison with eq. (39). Of course,
the angle could be evaluated to any order in $G$ and $a$. At the current
precision, equating eq. (39) with eq. (43), $\kappa_{n}$ may be determined up
to $\mathcal{O}(a^{4})$. This produces a first order differential equation.
The solutions will therefore naturally involve a $b$-independent integration
constant $\mathcal{C}$. For brevity we will only present the $\mathcal{O}(a)$
result in the text, leaving the complete $\mathcal{O}(a^{4})$ result to table
1. One finds

$\begin{split}\kappa_{1}=&2E,\\\
\kappa_{2}=&\left(\frac{3\left(v^{2}+4\right)E(m_{1}+m_{2}-E)}{v^{2}+2}+\frac{b\,\mathcal{C}}{v^{2}+2}\right)\\\
&+a\left(\frac{E\left[-2(m_{1}-E)\left(3v^{4}+4v^{2}+8\right)-3m_{2}\left(v^{4}+4\right)\right]}{bv\left(v^{2}+2\right)^{2}}+\frac{2\mathcal{C}v}{(v^{2}+2)^{2}}\right)+\mathcal{O}(a^{2}).\end{split}$
(44)

The integration constant $\mathcal{C}$ may be set to $0$ by requiring
$g_{\mu\nu}^{(eff)}\rightarrow\text{Kerr}$ in the test-particle limit of
$m_{2}$. Contrary, the limit $m_{1}\rightarrow 0$ describes a spinning test-
particle in Schwarzschild. Here the EOB metric does not reduce to
Schwarzschild, as it also encodes spin of the probe.

We stress the simplicity of the EOB construction presented here. An EOB metric
is readily found from an ansatz and EOB map, by directly matching scattering
angles of the full two-body system with those of the effective test-particle.
Non-metric, post-geodesic Finsler-type contributions of e.g. refs. Damour
(2020a); Bini and Damour (2017) are not needed. Similar observations were made
without spin in ref. Damgaard and Vanhove (2021). We leave to future work the
extension to higher Post Minkowskian orders, by the inclusion of
$\kappa_{n>2}$ terms. We emphasize that the EOB metric found above is by no
means unique. Other solutions, based on a different ansatz, may exist.

$n$ |  $[\kappa_{2}$ at $\mathcal{O}(a^{n})$] / $\frac{a^{n}E}{b^{n}v^{n}(v^{2}+2)^{n+1}}$
---|---
$0$ |  $3\left(v^{2}+4\right)(m_{1}+m_{2}-E)$
$1$ |  $-2(m_{1}-E)\left(3v^{4}+4v^{2}+8\right)-3m_{2}\left(v^{4}+4\right)$
$2$ |  $\frac{1}{4}(m_{1}-E)\left(6v^{8}+18v^{6}+148v^{4}+96v^{2}+32\right)+\frac{1}{4}m_{2}\left(-v^{8}-14v^{6}+72v^{4}+16v^{2}+32\right)$
$3$ |  $-\frac{v^{2}}{4}(m_{1}-E)\left(12v^{8}+144v^{6}+176v^{4}+352v^{2}+192\right)+\frac{v^{2}}{4}m_{2}\left(v^{8}-34v^{6}+60v^{4}-184v^{2}-128\right)$
$4$ |  $\frac{v^{2}}{16}(m_{1}-E)\left(22v^{12}+230v^{10}+576v^{8}+3240v^{6}+3456v^{4}+1984v^{2}+512\right)\newline +\frac{v^{2}}{16}m_{2}\left(-27v^{12}-156v^{10}-484v^{8}+1264v^{6}+864v^{4}+832v^{2}+512\right)$
Table 1: Parameter $\kappa_{2}$ from the EOB metric of eq. (42) computed up to
$\mathcal{O}(a^{4})$ by matching the scattering angle of the amplitude
calculations from Guevara  _et al._ (2019a), provided in eq. (39) above.
Equation (39) only matches black hole scattering at $\mathcal{O}(G^{2})$ for
$\mathcal{O}(a)$, with higher orders in $a$ matched only by conjecture. Each
row, labelled by $n$ contains the $\mathcal{O}(a^{n})$ contribution to
$\kappa_{2}$. Expressions are given in terms of full two-body quantities.
These solutions follow from equating eq. (43) with eq. (39) and inserting the
1PM result $\kappa_{1}=2E$ from eq. (70). The arbitrary b-independent
integration constant $\mathcal{C}$ associated with these solutions is set to
$\mathcal{C}=0$.

###  V.2 Comparison with earlier approaches without spin

Our EOB metric is first compared to previous approaches without spin. Letting
$a=0$ in the previous section, below our treatment is shown equivalent with
that of refs. Damgaard and Vanhove (2021); Damour (2020a). Particularly, the
above results are related to ref. Damour (2020a) by a gauge transformation of
the post-geodesic $\mathcal{Q}$ term and a coordinate-shift of the scattering
angle integral to incorporate differing EOB maps. Ref. Damour (2020a) presents
both Schwarzschild-like and isotropic EOB metrics. We will compare exclusively
with the Schwarzschild type.

Ref. Damour (2020a) uses an EOB map differing from eq. (36) and (38) purely by
equating angular momenta and not impact parameters

$\mathllap{L=L_{eff}\quad}\Rightarrow\mathrlap{\quad
b=b_{eff}\frac{E}{M}.\qquad\qquad\quad\text{(EOB map of ref.
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Damour2020_claqua}{\@@citephrase{(}}{\@@citephrase{)}}})}}$
(45)

We have expressly indicated above that these are relations used in ref. Damour
(2020a) and continue to denote $L$, $L_{eff}$ and $b$ as defined by eq. (36)
in everything following below. This EOB map can readily be converted to eq.
(36) as both maps employ $\chi=\chi_{eff}$. Equation (13) therefore implies

$\frac{d}{dL}\int_{R_{m}}^{\infty}\\!dR\,p_{R}=\frac{d}{dL_{eff}}\int_{r_{m}}^{\infty}\\!dr\,p_{r}=\frac{M}{E}\frac{d}{dL}\int_{r_{m}}^{\infty}\\!dr\,p_{r},$
(46)

where the integral in $R$ is that of ref. Damour (2020a), and the integral in
$r$ that of the current paper. $p_{R}$ and $p_{r}$ are the corresponding
canonical radial momenta. This equality allows the natural identification,
applicable when both formalisms use the same ansatz (Schwarzschild-like)
metric,

$R=\frac{M}{E}r,\quad\quad p_{R}=p_{r},$ (47)

amounting to a coordinate-shift of the scattering angle integral. This is
exactly the identification made by ref. Damgaard and Vanhove (2021), there
interpreted as a canonical transformation between effective test-particle
momentum of ref. Damour (2020a) and center of mass momentum of the full two-
body system. Particularly, eq. (54) and (51) in ref. Damgaard and Vanhove
(2021) is exactly eq. (47) above.

We show below that the coordinate transformation of eq. (47) yields a
transformed momentum $p_{R}$ related to that found in ref. Damour (2020a) by a
gauge-transformation in $\mathcal{Q}$. The formalisms are thus equivalent. To
see this, consider the specific forms of $p_{r}$ and $p_{R}$. One finds

$p_{r}=p_{eff}^{2}-\frac{L_{eff}^{2}}{r^{2}}-U(r)=p_{eff}^{2}-\frac{L^{2}}{R^{2}}-U(r),$
(48a) with
$\begin{split}U(r)&=\frac{2GM}{R^{3}}\left(L^{2}+\frac{\mu^{2}R^{2}\left(v^{2}+1\right)}{v^{2}-1}\right)\\\
&+\frac{G^{2}M^{2}\left(L^{2}\left(v^{2}-1\right)\left(3M\left(v^{2}+4\right)+\left(v^{2}-4\right)E\right)+\mu^{2}R^{2}\left(3M\left(v^{4}+5v^{2}+4\right)+\left(v^{4}+v^{2}+4\right)E\right)\right)}{R^{4}\left(v^{2}-1\right)\left(v^{2}+2\right)E}\\\
&+\mathcal{O}(G^{3}),\end{split}$ (48b)

by inserting eq. (42) in eq. (4), subsequently rewriting in terms of
$R=\frac{M}{E}r$ and $L_{eff}=\frac{E}{M}L$.

Canonical momentum $p_{R}$ is calculated very similarly in ref. Damour
(2020a), from a modified Hamilton-Jacobi equation,

$g_{D}^{\mu\nu}p_{\mu}p_{\nu}=-\mu^{2}-\mathcal{Q},$ (49)

with post-geodesic correction $\mathcal{Q}$. Subscript D indicates quantities
from ref. Damour (2020a). It was shown in refs. Damour (2018, 2016) that, i)
the EOB metric $g^{D}_{\mu\nu}$ can be chosen to be a Schwarzschild metric
with mass $M=m_{1}+m_{2}$, ii) $\mathcal{Q}$ starts at $G^{2}$, and iii) the
scattering angle is invariant under certain gauge-like transformations of
$\mathcal{Q}$. Furthermore, it was shown a suitable gauge could be chosen,
such that $\mathcal{Q}$ only depends on $R$ and quantities relating to energy.
We therefore write

$\mathcal{Q}=\sum_{n=2}^{\infty}\mathcal{Q}_{n}(R)G^{n}$ (50)

without loss of generality. Truncating at $\mathcal{O}(G^{2})$, one finds

$p_{R}=p_{eff}^{2}-\frac{L^{2}}{R^{2}}-U(R),$ (51a)
$U(R)=\frac{2GM\left(L^{2}+\frac{\mu^{2}R^{2}\left(v^{2}+1\right)}{v^{2}-1}\right)}{R^{3}}+G^{2}\mathcal{Q}_{2}+\frac{G^{2}M^{2}\left(4L^{2}+R^{2}\left(\frac{4\mu^{2}\left(v^{2}+2\right)}{v^{2}-1}\right)\right)}{R^{4}}+\mathcal{O}(G^{3}),$
(51b)

where $U(R)$ denotes the corresponding potential, suitably identified as
$g_{\mu\nu}^{D}$ is already in normal form. Note in particular that, in the
potential, $\mathcal{Q}_{2}$ term appears isolated from other metric-dependent
terms. This becomes important in a moment.

$\mathcal{Q}$ may be found from our EOB formalism by imposing eq. (47) and
inserting eqs. (48) and (51). One finds

$U(r)=U(R)$ (52)

to all orders in $G$. This equality is trivial up to 1PM as both EOB metrics
are Schwarzschild-like. $\mathcal{Q}$ therefore starts at $\mathcal{O}(G^{2})$
as expected. At 2PM the equality determines the lowest order coefficient of
$\mathcal{Q}$

$\mathcal{Q}=\mathcal{Q}^{M}_{2}G^{2}+\mathcal{O}(G^{3})=\frac{3\left(v^{2}+4\right)(M-E)\left(L^{2}\left(v^{2}-1\right)+\mu^{2}R^{2}\left(v^{2}+1\right)\right)}{R^{2}\left(v^{2}-1\right)\left(v^{2}+2\right)E}\frac{G^{2}M^{2}}{R^{2}}+\mathcal{O}(G^{3}),$
(53)

corresponding to translating the EOB metric of eq. (70) to a Finsler-type
post-geodesic form directly comparable with ref. Damour (2018, 2016, 2020a).
The obtained value of $\mathcal{Q}_{2}$ is dependent on angular momentum, and
thus definitely not in the gauge used in ref. Damour (2020a). Results for
$\mathcal{Q}$ from refs. Damour (2018, 2016, 2020a) instead yield,

$\mathcal{Q}^{D}_{2}=\frac{3\mu^{2}\left(v^{2}+4\right)(M-E)}{2\left(v^{2}-1\right)E},$
(54)

which is different from eq. (53) by

$\Delta\mathcal{Q}\equiv\mathcal{Q}_{2}^{D}-\mathcal{Q}^{M}_{2}=-\frac{3M^{2}\left(v^{2}+4\right)(M-E)\left(2L^{2}\left(v^{2}-1\right)+\mu^{2}R^{2}v^{2}\right)}{2R^{4}\left(v^{2}-1\right)\left(v^{2}+2\right)E}.$
(55)

However, $\mathcal{Q}_{2}^{M}$ may be related to $\mathcal{Q}_{2}^{D}$ by a
gauge-transformation. After all, the scattering angle calculated with each is
the same. Gauge-transformations may be introduced by considering, as in ref.
Damour (2018), the scattering angle integral eq. (13). Plug in eq. (51) with
unspecified $\mathcal{Q}_{2}$. Denoting by $\chi_{\mathcal{Q}}$ contributions
to the scattering angle that come from $\mathcal{Q}$, one finds up to
$\mathcal{O}(G^{2})$

$\chi_{\mathcal{Q}}=\frac{1}{p_{eff}}\frac{d}{dL}\int_{0}^{\infty}\\!du\,\mathcal{Q}_{2}(R)G^{2}+\mathcal{O}(G^{3}),\quad\quad
R^{2}=u^{2}+L^{2}/p_{eff}^{2}=u^{2}+b^{2}\frac{M^{2}}{E^{2}},$ (56)

rewriting eq. (13) in terms of $R$, imposing in effect the EOB map of ref.
Damour (2016). Consider how the full scattering angle integral changes with
$\mathcal{Q}^{M}_{2}$ compared to $\mathcal{Q}_{2}^{D}$, keeping the
Schwarzschild metric $g_{\mu\nu}^{D}$. By construction, the scattering angles
calculated in either case are equal. One therefore concludes

$\frac{d}{dL}\int_{0}^{\infty}\\!du\,\mathcal{Q}^{D}_{2}(R)=\frac{d}{dL}\int_{0}^{\infty}\\!du\,\mathcal{Q}^{M}_{2}(R),\quad\quad
R^{2}=u^{2}+b^{2}\frac{M^{2}}{E^{2}}.$ (57)

Subtracting the LHS from the RHS yields

$\frac{d}{dL}\int_{0}^{\infty}\\!du\,\Delta\mathcal{Q}=0,$ (58)

as confirmed by an explicit calculation. One may thus interpret
$\mathcal{Q}^{M}_{2}$ as being related to $\mathcal{Q}_{2}^{D}$ by a gauge-
like transformation which keeps the scattering angle invariant,

$\mathcal{Q}_{2}^{D}=\mathcal{Q}^{M}_{2}+\frac{d}{du}G(R),\quad
R^{2}=u^{2}+b^{2}\frac{M^{2}}{E^{2}},$ (59)

with contributions from $G(R)$ vanishing in the integration limits of eq.
(56). A very similar result was found in ref. Damour (2018), based on
analogous considerations. Incidentally, an identical relation holds when
imposing the EOB map of eq. (36), with the replacements $b\rightarrow
b\frac{E}{M}$, $R\rightarrow r$ and $u$ continuing to denote the integration
parameter in eq. (13). For reference, $G(R)$ inferred from eq. (55) is

$G(R)=\frac{3G^{2}\mu^{2}M^{2}uv^{2}\left(v^{2}+4\right)(M-E)}{2\left(v^{2}-1\right)\left(v^{2}+2\right)E\left(b^{2}\frac{M^{2}}{E^{2}}+u^{2}\right)},\quad\quad
u^{2}=R^{2}-b^{2}\frac{M^{2}}{E^{2}}.$ (60)

The gauge-transformation presented above is explicitly restricted to
$\mathcal{O}(G)$. Similar arguments may be made at higher orders in $G$ from a
more complicated gauge-relation derived from multiple terms of eq. (13).

Concluding, we may interpret our EOB metric without spin as a gauge-specific
embedding of a Schwarzschild metric with post-geodesic $\mathcal{Q}$
contribution.

We next compare our results with ref. Damgaard and Vanhove (2021), which is
related to ref. Damour (2020a) simply by the coordinate-shift of eq. (47). By
means of the previous analysis, our result is therefore related to ref.
Damgaard and Vanhove (2021) purely by the gauge-transformation in
$\mathcal{Q}$. Apparent from isotropic and Schwarzschild EOB constructions of
ref. Damour (2020a), this transformation cannot be interpreted as a simple
coordinate-shift between Schwarzschild and isotropic metrics, eqs. (42) and
(40). Such behavior is to be expected since the EOB metric is by no means
unique - multiple distinct metrics, ie. not related by coordinate
transformations, may encode full binary dynamics. To see this, try bringing
the isotropic EOB metric of ref. Damgaard and Vanhove (2021) to a
Schwarzschild form which satisfies $g_{\phi\phi}=\rho^{2}$ and
$g_{tt}=-1/g_{\rho\rho}$. Denoting the transformed coordinates
$\\{t,\rho,\phi\\}$, and the transformed metric $\tilde{g}_{\mu\nu}$, one does
not recover eq. (42) with $a=0$, nor even Schwarzschild structure. Instead

$\tilde{g}_{\mu\nu}=\left(\begin{array}[]{ccc}-\left(1-\mathcal{A}\right)&0&0\\\
0&\frac{1}{1-\widetilde{\mathcal{A}}}&0\\\ 0&0&\rho^{2}\\\
\end{array}\right),$ (61a) with
$\mathcal{A}=\frac{4\alpha_{1}G}{\rho}+\frac{4\alpha_{2}G^{2}}{\rho^{2}}+\mathcal{O}(G^{3}),\quad\quad\widetilde{\mathcal{A}}=\frac{4\alpha_{1}G}{\rho}+\frac{8\alpha_{2}G^{2}}{\rho^{2}}+\mathcal{O}(G^{3}).$
(61b) $\mathcal{A}$ and $\widetilde{\mathcal{A}}$ are unequal starting at
$\mathcal{O}(G^{2})$, breaking the Schwarzschild-like characteristics.
$\alpha_{1}$ and $\alpha_{2}$ are given in eq. (41). We have used
$r=\rho-2\alpha_{1}G-\left(\alpha_{1}^{2}+2\alpha_{2}\right)G^{2}/\rho+\mathcal{O}(G^{3})$
(61c)

and neglected $\mathcal{O}(G^{3})$ terms in the metric. This is warranted as
$\alpha(r)$ is only specified up to $\mathcal{O}(G^{2})$ anyway. A resummation
in $G$ is presented merely to highlight deviations from the Schwarzschild
form. As a consequence, the gauge-transformation in $\mathcal{Q}$ of eq. (59),
does not correspond to a coordinate-transformation of the EOB metric.

###  V.3 Comparison with earlier approaches with spin

We now turn to consistency checks with earlier approaches including spin.
Observations are similar to those without spin. At 1PM, our EOB metric is
compared with that of Justin Vines ref. Vines (2018), section III,b. In ref.
Vines (2018) the EOB map of ref. Damour (2020a) is used, introducing a Kerr-
like EOB metric with mass $M=m_{1}+m_{2}$ and spin $\tilde{a}=\frac{M}{E}a$.
By an analysis identical to that without spin, one may propose the connection
of eq. (46) yielding eq. (47)

$R=\frac{M}{E}r,\quad\quad p_{R}(R,\tilde{a},L)=p_{r}(r,a,L_{eff}).$ (62)

This identification is indeed correct, as an explicit calculations of $p_{r}$
and $p_{R}$ shows. We remind the reader that $L$ and $L_{eff}$ are defined in
eq. (36). The simplicity of this coordinate connection is purely due to the
Kerr-like structure of both metrics. Our result is thus consistent with ref.
Vines (2018).

##  VI Spinning Binary EOB metrics and Newman-Janis Algorithm

Finally, we consider the Newman-Janis Algorithm (NJA), a remarkable procedure
for introducing spin to a non-spinning, so-called seed metric. Although
presented first as an ad-hoc observation by Newman and Janis in 1965 Newman
and Janis (1965), its uniqueness has subsequently been investigated Drake and
Szekeres (2000). Beyond useful only for its original application in obtaining
the Kerr metric from a Schwarzschild spacetime, it successfully produces also
the Kerr-Newman metric from a Reissner Nordström seed.

As such it is interesting to explore the application of the NJA to EOB
metrics. Namely, does one recover an EOB metric for aligned spinning Kerr
black holes by applying the NJA to the EOB metric of two Schwarzschild black
holes? We consider only equatorial, aligned spin scattering throughout.
Adopting the EOB formalism above with a scalar effective test-particle of mass

$\mu=\frac{m_{1}m_{2}}{M},$ (63)

and EOB maps given by eqs. (36) and (38), the requirement for such a metric is
that it reproduces the aligned spinning binary scattering angles, eq. (2),
order by order in $G$. We will concern ourselves with 1PM and 2PM scattering
angles below.

We choose to construct our NJA metric from the non-spinning binary
Schwarzschild EOB metric of eqs. (40) and (41). We employ the NJA in the form
discussed by refs. Newman and Janis (1965); Drake and Szekeres (2000), and
restrict ourselves purely to the equatorial plane ($\theta=\pi/2$). Only the
final result is presented in the main text, and specifics of the procedure are
included in appendix A. A few remarks are worth noting. First, notice that eq.
(40) is symmetric in binary masses $m_{1}$ and $m_{2}$. The NJA-transformed
metric will naturally preserve this symmetry, and we therefore expect the
metric to describe some equal-in-spin binary, for which the scattering angle
of eq. (2) is symmetric in masses. Furthermore, note we are only interested in
scattering angles up to $\mathcal{O}(G^{2})$, and therefore we take as the
seed metric the $\mathcal{O}(G^{2})$ accurate Schwarzschild-form of eq. (40)
presented in eq. (61a). In the application of the NJA, the transformed tetrads
defining the transformed metric are similarly truncated to
$\mathcal{O}(G^{2})$ (see step 4 in appendix A).

The NJA transformed metric of eq. (61a), following the original procedure of
refs. Newman and Janis (1965); Drake and Szekeres (2000), becomes

$g_{\mu\nu}^{\text{(NJA)}}=\left(\begin{array}[]{cccc}-\frac{r^{2}\left(r(r-4\alpha_{1}G)-8\alpha_{2}G^{2}\right)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}&0&-\frac{ar^{2}(4\alpha_{1}Gr+6\alpha_{2}G^{2})}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}\\\
0&\frac{r^{2}}{a^{2}+r(r-4\alpha_{1}G)-8\alpha_{2}G^{2}}&0\\\
-\frac{ar^{2}(4\alpha_{1}Gr+6\alpha_{2}G^{2})}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}&0&r^{2}\left(\frac{a^{2}\left(r(4\alpha_{1}G+r)+4\alpha_{2}G^{2}\right)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}+1\right)\\\
\end{array}\right),$ (64)

which has Boyer-Lidquist structure reminiscent of a Kerr-Newman metric.
Specifically, eq. (64) recovers the equatorial Kerr-Newman metric by setting
$\alpha_{1}=\frac{M}{2}-\frac{Q^{2}}{4r}$ and $\alpha_{2}=0$, for which eq.
(61a) becomes the Reissner-Nordström metric.

We now discuss EOB interpretation of such a metric, based on scattering data.
The equatorial scattering angle of a scalar (effective) test-particle in
$g_{\mu\nu}^{\text{(NJA)}}$ is readily calculated by eq. (13),

$\begin{split}\chi^{\text{NJA}}_{\text{EOB}}=&\frac{2GE\left(-2av+bv^{2}+b\right)}{v^{2}\left(b^{2}-a^{2}\right)}\\\
&+\frac{3\pi G^{2}M\left(v^{2}+4\right)E}{4b^{2}v^{2}}-\frac{a\left(\pi
G^{2}E\left(9M\left(v^{2}+4\right)v^{2}+\left(15v^{4}+4v^{2}+16\right)E\right)\right)}{4\left(b^{3}v^{3}\left(v^{2}+1\right)\right)}+\mathcal{O}(a^{2})\\\
&+\mathcal{O}(G^{3}),\end{split}$ (65)

after coordinate transformation $r^{2}\rightarrow\rho^{2}=r^{2}+a^{2}$
retrieving the normal form. $M=m_{1}+m_{2}$ denotes the total mass of the
binary. To obtain EOB interpretation, this should reproduce a full binary
scattering angle. We compare explicitly with the aligned Kerr binary from ref.
Guevara  _et al._ (2019a), adopting the EOB map of eqs. (36) and (38). The 2PM
angle above has been truncated at $\mathcal{O}(a)$ to make such comparisons.
Further analysis could equally be carried out at higher orders in $a$, however
eq. (2) is not yet confirmed to represent scattering of black holes at these
orders. Consider all configurations $(a_{1},a_{2})$ of eq. (2) conceivably
described by the NJA. The 1PM scattering angle matches exactly that of eq. (2)
with configuration $a_{1}+a_{2}=a$. Spins $a_{1}$ and $a_{2}$ are further
restricted by going to 2PM. Equation (65) is symmetric in masses $m_{1}$ and
$m_{2}$. This symmetry is only recovered in equation (2) if $a_{1}=a_{2}=a/2$.
The corresponding full binary scattering angle from eq. (2) is

$\begin{split}\chi^{(a/2,a/2)}=&\frac{2GE\left(-2av+bv^{2}+b\right)}{v^{2}\left(b^{2}-a^{2}\right)}\\\
&+\frac{3\pi G^{2}M\left(v^{2}+4\right)E}{4b^{2}v^{2}}-\frac{7a\left(\pi
G^{2}M\left(3v^{2}+2\right)E\right)}{4b^{3}v^{3}}+\mathcal{O}(a^{2})\\\
&+\mathcal{O}(G^{3}),\end{split}$ (66)

which is the only binary Kerr scattering angle conceivably matched by eq.
(65). However comparing eq. (65) and eq. (66), they do not match up. In the
present setting, based on the non-spinning EOB metric of eq. (40) and eq.
(41), the NJA thus fails to produce an EOB metric for aligned spinning
binaries.

Although the present NJA interpretation fails, we have only considered scalar
test-particles. One could equally consider scattering of a spinning test-
particle on the metric of eq. (64). Whether this combination has EOB
interpretation, and produces correct two-body scattering angles, is not
pursued here.

##  VII Conclusion

Using the method of ref. Damgaard  _et al._ (2022), eq. (11), scattering
angles of systems involving both electromagnetic and gravitational
interactions have been calculated. As a specific case-study, post-Minkowskian
scattering angles in the Kerr-Newman metric have been computed for charged
test-particles (table 2 and table 3 B). Purely electromagnetic situations are
also treatable - the well known Coulomb scattering angle is readily obtained
in resummed form in the weak field limit (table 3 A). Charged test-particle
scattering angles in Kerr-Newman show definite structure. At order
$\mathcal{O}(G^{n})$ and $\mathcal{O}(e^{j})$,

$\chi=\sum_{n=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{n}\chi_{n,j}^{(k)},\quad\quad\chi_{n}^{(k)}\sim\pi^{(n+j+k+1)\text{mod
}2}\frac{G^{n}e^{j}Q^{2k+j}M^{n-k}}{m^{j}\gamma^{j}v^{2(j+n)}\left(b^{2}-a^{2}\right)^{(3(n+j)+k-1)/2}}f_{n,j,k}(v),$
(67)

where $f_{n,j,k}(v)$ is a polynomial in $v$. For $j=0$ and odd (even) $n$,
$f_{n,j,k}(v)$ contains integer (fractional) orders of $a$ and $b$. When
$j\neq 0$ only integer orders of $a$ and $b$ are present, with the structure

$f_{n,j,k}(v)\sim\sum_{p=0}^{2(n+j)}a^{p}b^{2(n+j)-p}\tilde{f}_{n,j,k}(v),$
(68)

where $\tilde{f}_{n,j,k}$ is some polynomial in $v$, independent of $a$ and
$b$.

The flexibility of the scattering angle formula of ref. Damgaard  _et al._
(2022) is aptly suited for EOB formalisms with planar orbits. An EOB metric
describing full binary motion may be constructed by explicitly matching
scattering angles and adopting the EOB map of ref. Damgaard and Vanhove (2021)
(eq. (36) of the present paper). An ansatz,

$\begin{split}g_{\mu\nu}^{(eff)}=\left(\begin{array}[]{ccc}-\left(1-\kappa(r)\right)&0&-a\kappa(r)\\\
0&\frac{r^{2}}{a^{2}+r^{2}\left(1-\kappa(r)\right)}&0\\\
-a\kappa(r)&0&a^{2}\left(1+\kappa(r)\right)+r^{2}\\\
\end{array}\right),\quad\kappa(r)=&\sum_{n=1}^{\infty}\kappa_{n}\frac{G^{n}}{r^{n}},\end{split}$
(69)

for the EOB metric may be constructed from the Kerr metric in Boyer-Lindquist
coordinates by replacing $2GM/r\rightarrow\kappa(r)$. The result is a resummed
metric in spin parameter $a$ and Newtons gravitational constant $G$.
$\kappa_{n}$ may be determined by matching the resulting test-particle
scattering angle with that of a Kerr-Schwarzschild binary Guevara  _et al._
(2019a), order by order in $G$ and $a$. Our current treatment extends to 2PM
and $\mathcal{O}(a)$, yielding

$\begin{split}\kappa_{1}=&2E,\\\
\kappa_{2}=&\left(\frac{3\left(v^{2}+4\right)E(m_{1}+m_{2}-E)}{v^{2}+2}\right)\\\
&+a\left(-\frac{E\left(6m_{1}v^{4}+8m_{1}v^{2}+16m_{1}+3m_{2}v^{4}+12m_{2}-6v^{4}E-8v^{2}E-16E\right)}{bv\left(v^{2}+2\right)^{2}}\right)+\mathcal{O}(a^{2})\end{split}$
(70)

by requiring that the metric reduces to Kerr in the $m_{2}\rightarrow 0$
limit. By identical comparisons with higher orders of $a$ in ref. Guevara  _et
al._ (2019a), $\kappa_{2}$ is presented up to $\mathcal{O}(a^{4})$ in table 1.
We see no obstruction in continuing the treatment to further orders in $G$ and
$a$. In the non-spinning limit, our EOB formalism is related to previous
approaches of refs. Damour (2018, 2020a); Damgaard and Vanhove (2021) through
a gauge-transformation of the post-geodesic Finsler-type $\mathcal{Q}$ term.
Including spin, at 1PM, our formalism is equivalent to that of ref. Vines
(2018) after a coordinate-reparametrisation identical to the connection of
ref. Damgaard and Vanhove (2021) with ref. Damour (2020a). Furthermore, for
spinning binaries, a subtlety with $b$ dependence of the metric was noted.
Corresponding to angular momentum dependence, the EOB metric may readily
depend on it, however requires extra care when evaluating test-particle
scattering angles. In particular with the current ansatz metric, the
scattering angle becomes a function of linear derivatives of $\kappa_{n}$ with
respect to $b$.

Supplementary to our EOB analysis, the Newman-Janis Algorithm was explored in
its application to the non-spinning EOB metric of ref. Damgaard and Vanhove
(2021), see eq. (40). The complexification technique and coordinate-
transformations involved are those originally introduced by Newman and Janis
Newman and Janis (1965), and the EOB map of eqs. (36) and (38) is assumed.
Applying the NJA to eq. (40), the result is

$g_{\mu\nu}^{\text{(NJA)}}=\left(\begin{array}[]{cccc}-\frac{r^{2}\left(r(r-4\alpha_{1}G)-8\alpha_{2}G^{2}\right)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}&0&-\frac{ar^{2}(4\alpha_{1}Gr+6\alpha_{2}G^{2})}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}\\\
0&\frac{r^{2}}{a^{2}+r(r-4\alpha_{1}G)-8\alpha_{2}G^{2}}&0\\\
-\frac{ar^{2}(4\alpha_{1}Gr+6\alpha_{2}G^{2})}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}&0&r^{2}\left(\frac{a^{2}\left(r(4\alpha_{1}G+r)+4\alpha_{2}G^{2}\right)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}+1\right)\\\
\end{array}\right),$ (71)

derived with $\mathcal{O}(G^{2})$-accurate manipulations. Scalar test-particle
scattering angles in this metric are computed, and compared to different
combinations of aligned spin $(a_{1},a_{2})$ of the full binary result Guevara
_et al._ (2019a). No spin map is a priori assumed. At 1PM we find that angles
indeed match, due to natural Kerr-like structure in both cases. At 2PM,
however, no spin configurations are possible for which the angles are equal.
It is therefore, within the confines of the current construction, not possible
to interpret the NJA transformed metric above as an EOB metric of aligned
binary Kerr black holes. This analysis is based on a scalar effective test-
particle. Alternatively, the same analysis including spin on the test-particle
might conceivably have EOB interpretation. This is left to future work.

Acknowledgements: I thank Poul Henrik Damgaard and Andres Luna for helpful
discussions. This work was supported in part by the Independent Research Fund
Denmark, Grant No. 0135-00089B.

$(n,k)$ |  $\chi^{(k)}_{n}/\frac{G^{n}Q^{2k}M^{n-k}}{v^{2n}(b^{2}-a^{2})^{(3n+k-1)/2}}$
---|---
(1,0) |  $2\left(-2av+bv^{2}+b\right)$
(1,1) |  $\pi/(2a^{2})\left(2a^{3}v-a^{2}\left(-v^{2}\sqrt{b^{2}-a^{2}}+2bv^{2}+b\right)+b^{2}v^{2}\left(b-\sqrt{b^{2}-a^{2}}\right)\right)$
(2,0) |  $\pi/(2a^{2})\big{[}-4a^{5}v-4a^{3}b^{2}v\left(3v^{2}+2\right)+2a^{2}b^{2}v^{2}\left(b\left(2v^{2}+3\right)-v^{2}\sqrt{b^{2}-a^{2}}\right)\newline +b^{4}v^{4}\left(\sqrt{b^{2}-a^{2}}-b\right)+a^{4}\left(v^{4}\sqrt{b^{2}-a^{2}}+3b\left(4v^{2}+1\right)\right)\big{]}$
(2,1) |  $(8a^{3}+24ab^{2})(1+v^{2})v+(-6a^{2}b-2b^{3})(1+6v^{2}+v^{4})$
(2,2) |  $-(3\pi)/(16a^{4})\big{[}4a^{7}\left(2v^{3}+v\right)+4a^{5}b^{2}v\left(3v^{2}+4\right)+2b^{6}v^{4}\left(b-\sqrt{b^{2}-a^{2}}\right)+a^{2}b^{4}v^{4}\left(6\sqrt{b^{2}-a^{2}}-7b\right)\newline -a^{6}\left(b\left(8\left(v^{2}+3\right)v^{2}+3\right)-2v^{4}\sqrt{b^{2}-a^{2}}\right)+2a^{4}b^{2}\left(b\left(4v^{4}-3v^{2}-1\right)-3v^{4}\sqrt{b^{2}-a^{2}}\right)\big{]}$
(3,0) |  $\frac{1}{3}\big{[}4a^{5}v\left(3v^{4}-10v^{2}-9\right)+6a^{4}b\left(-v^{6}+15v^{4}+45v^{2}+5\right)-8a^{3}b^{2}v\left(15v^{4}+70v^{2}+27\right)\newline +4a^{2}b^{3}\left(11v^{6}+135v^{4}+105v^{2}+5\right)-36ab^{4}v\left(5\left(v^{2}+2\right)v^{2}+1\right)+2b^{5}\left(5v^{2}\left(v^{4}+9v^{2}+3\right)-1\right)\big{]}$
(3,1) |  $(3\pi)/(8a^{4})\big{[}2a^{9}v\left(20v^{2}+9\right)+12a^{7}b^{2}v\left(10v^{4}+35v^{2}+12\right)+6a^{5}b^{4}v\left(15v^{4}+40v^{2}+8\right)+2b^{8}v^{6}\left(b-\sqrt{b^{2}-a^{2}}\right)\newline +a^{2}b^{6}v^{6}\left(8\sqrt{b^{2}-a^{2}}-9b\right)-a^{8}\left(2v^{6}\sqrt{b^{2}-a^{2}}+15b\left(8v^{4}+12v^{2}+1\right)\right)\newline -a^{6}b^{2}\left(5b\left(8v^{6}+72v^{4}+63v^{2}+4\right)-8v^{6}\sqrt{b^{2}-a^{2}}\right)+3a^{4}b^{4}v^{2}\left(b\left(4v^{4}-15v^{2}-10\right)-4v^{4}\sqrt{b^{2}-a^{2}}\right)\big{]}$
(3,2) |  $(-4a^{5}-40a^{3}b^{2}-20ab^{4})(3+10v^{2}+3v^{4})v+(10a^{4}b+20a^{2}b^{3}+2b^{5})(1+15v^{2}+15v^{4}+v^{6})$
(3,3) |  $(5\pi)/(128a^{6})\big{[}2a^{11}v\left(24v^{4}+40v^{2}+9\right)+4a^{9}b^{2}v\left(60v^{4}+205v^{2}+54\right)+18a^{7}b^{4}v\left(5v^{4}+20v^{2}+8\right)\newline +8b^{10}v^{6}\left(b-\sqrt{b^{2}-a^{2}}\right)+4a^{2}b^{8}v^{6}\left(10\sqrt{b^{2}-a^{2}}-11b\right)-a^{10}\left(3b\left(16v^{6}+120v^{4}+90v^{2}+5\right)-8v^{6}\sqrt{b^{2}-a^{2}}\right)\newline +5a^{8}b^{2}\left(b\left(8v^{6}-108v^{4}-123v^{2}-8\right)-8v^{6}\sqrt{b^{2}-a^{2}}\right)-a^{6}b^{4}\left(b\left(118v^{6}+45v^{4}+60v^{2}+8\right)-80v^{6}\sqrt{b^{2}-a^{2}}\right)\newline +a^{4}b^{6}v^{6}\left(99b-80\sqrt{b^{2}-a^{2}}\right)\big{]}$
(4,0) |  $(3\pi)/(16a^{4})\big{[}-8a^{11}v\left(14v^{2}+5\right)-8a^{9}b^{2}v\left(140v^{4}+273v^{2}+60\right)-8a^{7}b^{4}v\left(70v^{6}+455v^{4}+392v^{2}+40\right)\newline -56a^{5}b^{6}v^{3}\left(5\left(v^{2}+4\right)v^{2}+8\right)+2b^{10}v^{8}\left(\sqrt{b^{2}-a^{2}}-b\right)+a^{2}b^{8}v^{8}\left(11b-10\sqrt{b^{2}-a^{2}}\right)\newline +a^{10}\left(35b\left(16\left(v^{4}+v^{2}\right)+1\right)-2v^{8}\sqrt{b^{2}-a^{2}}\right)+10a^{8}b^{2}\left(v^{8}\sqrt{b^{2}-a^{2}}+7b\left(16v^{6}+56v^{4}+26v^{2}+1\right)\right)\newline +2a^{6}b^{4}v^{2}\left(7b\left(8v^{6}+120v^{4}+195v^{2}+40\right)-10v^{6}\sqrt{b^{2}-a^{2}}\right)+4a^{4}b^{6}v^{4}\left(5v^{4}\sqrt{b^{2}-a^{2}}+b\left(-4v^{4}+35v^{2}+35\right)\right)\big{]}$
(4,1) |  $\frac{4}{3}\big{[}8a^{7}v\left(-v^{6}+7v^{4}+21v^{2}+5\right)+5a^{6}b\left(v^{8}-28v^{6}-210v^{4}-140v^{2}-7\right)+24a^{5}b^{2}v\left(7\left(v^{4}+13v^{2}+19\right)v^{2}+25\right)\newline -5a^{4}b^{3}\left(13v^{8}+420v^{6}+1190v^{4}+532v^{2}+21\right)+40a^{3}b^{4}v\left(7\left(3v^{4}+19v^{2}+17\right)v^{2}+15\right)\newline -3a^{2}b^{5}\left(31v^{8}+700v^{6}+1330v^{4}+364v^{2}+7\right)+40ab^{6}v\left(7\left(v^{4}+5v^{2}+3\right)v^{2}+1\right)\newline +b^{7}\left(1-7v^{2}\left(v^{6}+20v^{4}+30v^{2}+4\right)\right)\big{]}$
(4,2) |  $(15\pi)/(128a^{6})\big{[}-8a^{13}v\left(56v^{4}+84v^{2}+15\right)-8a^{11}b^{2}v\left(7\left(24v^{4}+200v^{2}+237\right)v^{2}+270\right)\newline -8a^{9}b^{4}v\left(7\left(60v^{4}+395v^{2}+372\right)v^{2}+360\right)-24a^{7}b^{6}v\left(7\left(5\left(v^{2}+6\right)v^{2}+24\right)v^{2}+16\right)\newline +8b^{12}v^{8}\left(\sqrt{b^{2}-a^{2}}-b\right)+4a^{2}b^{10}v^{8}\left(13b-12\sqrt{b^{2}-a^{2}}\right)+a^{12}\left(8v^{8}\sqrt{b^{2}-a^{2}}+21b\left(8\left(8v^{4}+30v^{2}+15\right)v^{2}+5\right)\right)\newline +4a^{10}b^{2}\left(7b\left(16v^{8}+360v^{6}+930v^{4}+395v^{2}+15\right)-12v^{8}\sqrt{b^{2}-a^{2}}\right)\newline +2a^{8}b^{4}\left(60v^{8}\sqrt{b^{2}-a^{2}}+7b\left(8v^{8}+540v^{6}+1185v^{4}+400v^{2}+12\right)\right)\newline +4a^{6}b^{6}v^{2}\left(b\left(58v^{6}+105v^{4}+210v^{2}+56\right)-40v^{6}\sqrt{b^{2}-a^{2}}\right)+a^{4}b^{8}v^{8}\left(120\sqrt{b^{2}-a^{2}}-143b\right)\big{]}$
(4,3) |  $(16a^{7}+336a^{5}b^{2}+560a^{3}b^{4}+112ab^{6})(1+7v^{2}+7v^{4}+v^{6})v\newline +(-14a^{6}b-70a^{4}b^{3}-42a^{2}b^{5}-2b^{7})(1+28v^{2}+70v^{4}+28v^{6}+v^{8})$
(4,4) |  $(35\pi)/(2048a^{8})\big{[}-8a^{15}v\left(2v^{2}+5\right)\left(8\left(v^{4}+v^{2}\right)+1\right)-24a^{13}b^{2}v\left(7\left(8v^{4}+52v^{2}+45\right)v^{2}+40\right)\newline -120a^{11}b^{4}v\left(7\left(2v^{4}+15v^{2}+16\right)v^{2}+16\right)-8a^{9}b^{6}v\left(7\left(5\left(v^{2}+8\right)v^{2}+48\right)v^{2}+64\right)+16b^{14}v^{8}\left(\sqrt{b^{2}-a^{2}}-b\right)\newline +8a^{2}b^{12}v^{8}\left(15b-14\sqrt{b^{2}-a^{2}}\right)+a^{14}\left(b\left(32\left(4v^{6}+56v^{4}+105v^{2}+35\right)v^{2}+35\right)-16v^{8}\sqrt{b^{2}-a^{2}}\right)\newline +14a^{12}b^{2}\left(8v^{8}\sqrt{b^{2}-a^{2}}+15b\left(32v^{6}+80v^{4}+30v^{2}+1\right)\right)\newline +42a^{10}b^{4}\left(b\left(16v^{8}+80v^{6}+225v^{4}+104v^{2}+4\right)-8v^{8}\sqrt{b^{2}-a^{2}}\right)\newline +4a^{8}b^{6}\left(140v^{8}\sqrt{b^{2}-a^{2}}+b\left(-200v^{8}+35v^{6}+105v^{4}+56v^{2}+4\right)\right)+5a^{6}b^{8}v^{8}\left(143b-112\sqrt{b^{2}-a^{2}}\right)\newline +6a^{4}b^{10}v^{8}\left(56\sqrt{b^{2}-a^{2}}-65b\right)\big{]}$
Table 2: Scattering angle of a scalar test-particle in the Kerr-Newman metric,
orbiting in the equatorial plane. $\chi^{(k)}_{n}$ is the $\mathcal{O}(G^{n})$
contribution to the full scattering angle, of which the $Q^{2k}$-proportional
part is taken. $k$ ranges from $0$ to $n$ owing to the $GQ^{2}$ charge-
dependence of the metric, eq. (14). A) Coulomb scattering
---
$\chi=-\frac{2Q\sqrt{1-v^{2}}e}{bmv^{2}}-\frac{\pi
Q^{2}\left(v^{2}-1\right)e^{2}}{2\left(b^{2}m^{2}v^{2}\right)}-\frac{2Q^{3}\left(1-v^{2}\right)^{3/2}\left(3v^{2}-1\right)e^{3}}{3b^{3}m^{3}v^{6}}+\cdots=\frac{J}{\sqrt{J^{2}-Q^{2}e^{2}}}\left(\pi-2\,arctan\left(\frac{Qe}{v\sqrt{J^{2}-Q^{2}e^{2}}}\right)\right)-\pi$
B) Kerr-Newman equatorial scattering of charged test-particle
$(n,j,k)$ |  $\chi^{(k)}_{n,j}/\frac{G^{n}e^{j}Q^{2k+j}M^{n-k}}{m^{j}v^{2(j+n)}\left(b^{2}-a^{2}\right)^{(3(n+j)+k-1)/2}}$
(0,1,0) |  $2\sqrt{1-v^{2}}(av-b)$
(0,2,0) |  $\frac{1}{2}\pi\left(v^{2}-1\right)(a-bv)\left(2a^{2}v-3ab+b^{2}v\right)$
(0,3,0) |  $\frac{2}{3}\left(1-v^{2}\right)^{3/2}\big{[}a^{5}v\left(v^{2}+9\right)-3a^{4}b\left(9v^{2}+5\right)+2a^{3}b^{2}v\left(7v^{2}+27\right)-2a^{2}b^{3}\left(21v^{2}+5\right)+9ab^{4}v\left(v^{2}+1\right)+b^{5}\left(1-3v^{2}\right)\big{]}$
(1,1,0) |  $3\pi\sqrt{1-v^{2}}(av-b)(a-bv)^{2}$
(1,1,1) |  $2\left(v^{2}+3\right)\left(-a^{3}v-3ab^{2}v\right)+2\left(2v^{2}+1\right)\left(3a^{2}b+b^{3}\right)$
(1,2,0) |  $2\left(v^{2}-1\right)\big{[}4a^{5}v\left(v^{2}+3\right)-3a^{4}b\left(v^{4}+18v^{2}+5\right)+8a^{3}b^{2}v\left(7v^{2}+9\right)-2a^{2}b^{3}\left(9v^{4}+42v^{2}+5\right)\newline +12ab^{4}v\left(3v^{2}+1\right)+b^{5}\left(1-3v^{2}\left(v^{2}+2\right)\right)\big{]}$
(1,2,1) |  $-\frac{3}{8}\pi\left(v^{2}-1\right)(a-bv)\left[4a^{4}v\left(2v^{2}+3\right)-15a^{3}\left(4bv^{2}+b\right)+3a^{2}b^{2}v\left(8v^{2}+27\right)-5ab^{3}\left(9v^{2}+4\right)+3b^{4}v\left(v^{2}+4\right)\right]$
(1,3,0) |  $\frac{15}{4}\pi\left(1-v^{2}\right)^{3/2}(a-bv)^{2}\big{[}a^{5}v\left(4v^{2}+5\right)-a^{4}b\left(30v^{2}+7\right)+a^{3}b^{2}v\left(13v^{2}+46\right)\newline -a^{2}b^{3}\left(31v^{2}+14\right)+4ab^{4}v\left(v^{2}+3\right)-2b^{5}v^{2}\big{]}$
(1,3,1) |  $-\frac{4}{3}\left(1-v^{2}\right)^{3/2}\big{[}a^{7}v\left(v^{4}+30v^{2}+25\right)-5a^{6}b\left(15v^{4}+50v^{2}+7\right)+3a^{5}b^{2}v\left(13v^{4}+190v^{2}+125\right)\newline -5a^{4}b^{3}\left(85v^{4}+190v^{2}+21\right)+5a^{3}b^{4}v\left(19v^{4}+170v^{2}+75\right)-3a^{2}b^{5}\left(95v^{4}+130v^{2}+7\right)+25ab^{6}v\left(v^{4}+6v^{2}+1\right)\newline +b^{7}\left(1-5v^{2}\left(3v^{2}+2\right)\right)\big{]}$
(2,1,0) |  $-2\sqrt{1-v^{2}}\big{[}a^{5}v\left(v^{4}-10v^{2}-15\right)+15a^{4}b\left(v^{4}+6v^{2}+1\right)-10a^{3}b^{2}v\left(v^{4}+14v^{2}+9\right)+10a^{2}b^{3}\left(9v^{4}+14v^{2}+1\right)\newline -15ab^{4}v\left(v^{4}+6v^{2}+1\right)+b^{5}\left(15v^{4}+10v^{2}-1\right)\big{]}$
(2,1,1) |  $-\frac{15}{4}\pi\sqrt{1-v^{2}}(av-b)(a-bv)^{2}\left(a^{2}\left(4v^{2}+3\right)-14abv+b^{2}\left(3v^{2}+4\right)\right)$
(2,1,2) |  $2v\left(v^{4}+10v^{2}+5\right)\sqrt{1-v^{2}}\left(a^{5}+10a^{3}b^{2}+5ab^{4}\right)+2\left(5v^{4}+10v^{2}+1\right)\sqrt{1-v^{2}}\left(-5a^{4}b-10a^{2}b^{3}-b^{5}\right)$
(2,2,0) |  $\frac{45}{8}\pi\left(v^{2}-1\right)(a-bv)^{3}\left[a^{4}\left(8v^{3}+6v\right)-7a^{3}\left(6bv^{2}+b\right)+3a^{2}b^{2}v\left(4v^{2}+17\right)-7ab^{3}\left(3v^{2}+2\right)+b^{4}v\left(v^{2}+6\right)\right]$
(2,2,1) |  $4(v-1)(v+1)\big{[}-6a^{7}v\left(v^{4}+10v^{2}+5\right)+5a^{6}b\left(v^{6}+45v^{4}+75v^{2}+7\right)-6a^{5}b^{2}v\left(39v^{4}+190v^{2}+75\right)\newline +15a^{4}b^{3}\left(5v^{6}+85v^{4}+95v^{2}+7\right)-10a^{3}b^{4}v\left(57v^{4}+170v^{2}+45\right)+3a^{2}b^{5}\left(5\left(5v^{4}+57v^{2}+39\right)v^{2}+7\right)\newline -30ab^{6}v\left(5\left(v^{2}+2\right)v^{2}+1\right)+b^{7}\left(5v^{2}\left(v^{4}+9v^{2}+3\right)-1\right)\big{]}$
(2,2,2) |  $\frac{45}{128}\pi\left(v^{2}-1\right)(a-bv)\big{[}2a^{6}v\left(8\left(v^{2}+5\right)v^{2}+15\right)-35a^{5}b\left(8v^{4}+12v^{2}+1\right)+5a^{4}b^{2}v\left(24v^{4}+232v^{2}+101\right)\newline -70a^{3}b^{3}\left(v^{2}+2\right)\left(10v^{2}+1\right)+10a^{2}b^{4}v\left(9v^{4}+101v^{2}+58\right)-7ab^{5}\left(25v^{4}+60v^{2}+8\right)+5b^{6}v\left(v^{4}+12v^{2}+8\right)\big{]}$
(3,1,0) |  $\frac{105}{4}\pi\sqrt{1-v^{2}}(av-b)(a-bv)^{4}\left[a^{2}\left(2v^{2}+1\right)-6abv+b^{2}\left(v^{2}+2\right)\right]$
(3,1,1) |  $4\sqrt{1-v^{2}}\big{[}a^{7}v\left(v^{6}-21v^{4}-105v^{2}-35\right)+35a^{6}b\left(v^{2}+1\right)\left(v^{4}+14v^{2}+1\right)-21a^{5}b^{2}v\left(v^{6}+39v^{4}+95v^{2}+25\right)\newline +35a^{4}b^{3}\left(15v^{6}+85v^{4}+57v^{2}+3\right)-35a^{3}b^{4}v\left(3v^{6}+57v^{4}+85v^{2}+15\right)+21a^{2}b^{5}\left(25v^{6}+95v^{4}+39v^{2}+1\right)\newline -35ab^{6}v\left(v^{2}+1\right)\left(v^{4}+14v^{2}+1\right)+b^{7}\left(7v^{2}\left(5\left(v^{2}+3\right)v^{2}+3\right)-1\right)\big{]}$
(3,1,2) |  $\frac{315}{64}\pi\sqrt{1-v^{2}}(av-b)(a-bv)^{2}\big{[}a^{4}\left(8v^{4}+20v^{2}+5\right)-12a^{3}bv\left(6v^{2}+5\right)+2a^{2}b^{2}\left(10v^{4}+79v^{2}+10\right)\newline -12ab^{3}v\left(5v^{2}+6\right)+b^{4}\left(5\left(v^{2}+4\right)v^{2}+8\right)\big{]}$
(3,1,3) |  $2v\left(v^{6}+21v^{4}+35v^{2}+7\right)\sqrt{1-v^{2}}\left(-a^{7}-21a^{5}b^{2}-35a^{3}b^{4}-7ab^{6}\right)\newline +2\left(7\left(v^{4}+5v^{2}+3\right)v^{2}+1\right)\sqrt{1-v^{2}}\left(7a^{6}b+35a^{4}b^{3}+21a^{2}b^{5}+b^{7}\right)$
Table 3: Weak-field scattering angle of a charged test-particle; A) off
another stationary charge $Q$ (Coulomb scattering), and B) in the equatorial
plane of the Kerr-Newman metric. $\chi^{(k)}_{n,j}$ is the scattering angle at
$\mathcal{O}(G^{n})$ and $\mathcal{O}(e^{j})$. Results are decomposed into
$Q^{2k}$-proportional pieces with $0\leq k\leq n$. Purely gravitational
contributions to the scattering angle, $\chi_{n,0}^{(k)}$, may be found in
table 2 and are not displayed here.

## Appendix

###  VII.1 The Newman-Janis Algorithm and its application to equation (61a)

For completeness, we first present the Newman-Janis algorithm with a review of
ref. Drake and Szekeres (2000), before discussing its specific application to
eq. (61a). The Newman-Janis Algorithm is a five-step procedure, proceeding as
follows;

  1. 1.

Consider a given spherically symmetric seed metric to which spin should be
endowed. Write it in advanced null coordinates, generically

$ds^{2}=-e^{2\Phi(r)}du^{2}-2e^{\Phi(r)+\lambda(r)}dudr+r^{2}d\Omega^{2},$
(72)

specifying functions $\Phi(r)$ and $\lambda(r)$ on a case-to-case basis.

  2. 2.

Invert the metric and express it in terms of a null tetrad of vectors
$\\{l^{\mu},n^{\mu},m^{\mu}\\}$,

$g^{\mu\nu}=-l^{\mu}n^{\nu}-n^{\mu}l^{\nu}+m^{\mu}\bar{m}^{\nu}+\bar{m}^{\mu}m^{\nu},$
(73)

where $m^{\mu}$ can be complex valued, and $\bar{m}^{\mu}$ is its complex
conjugate, such that the metric is real-valued. Furthermore $l,m,n$ obey

$l_{\mu}l^{\mu}=m_{\mu}m^{\mu}=n_{\mu}n^{\mu}=0,\quad
l_{\mu}n^{\mu}=-m_{\mu}\bar{m}^{\mu}=1,\quad l_{\mu}m^{\mu}=n_{\mu}m^{\mu}=0.$
(74)

One may abbreviate notation by introducing
$Z_{a}^{\mu}=(l^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu})$. The metric of eq. (72)
has

$\displaystyle l^{\mu}$ $\displaystyle=\delta^{\mu}_{1},$ (75a) $\displaystyle
n^{\mu}$
$\displaystyle=e^{-\lambda(r)-\Phi(r)}\delta_{0}^{\mu}-\frac{1}{2}e^{-2\lambda(r)}\delta_{1}^{\mu},$
(75b) $\displaystyle m^{\mu}$
$\displaystyle=\frac{1}{\sqrt{2}r}(\delta_{2}^{\mu}+\frac{i}{\sin\theta}\delta_{3}^{\mu}).$
(75c)

  3. 3.

Complexify coordinates by letting $x^{\rho}\rightarrow
x^{\prime\rho}=x^{\rho}+iy^{\rho}\in\mathbb{C}$ and $Z^{\mu}_{a}\rightarrow
Z^{\prime\mu}_{a}$. The transformation only requires
$g^{\prime}_{\mu\nu}\in\mathbb{R}$ and $Z^{\prime}=Z$ when
$x^{\prime}=\bar{x^{\prime}}$, a bar denoting complex conjugation. Multiple
complexification transformations obey this requirement, and as such the
Newman-Janis Algorithm is ambiguous. The transformation adopted here is that
originally introduced by Newman and Janis,

$\frac{1}{r}\rightarrow\frac{1}{2}(\frac{1}{r^{\prime}}+\frac{1}{\bar{r}^{\prime}}),\quad\quad\frac{1}{r^{2}}\rightarrow\frac{1}{r^{\prime}\bar{r}^{\prime}}.$
(76)

Note in particular that $1/r^{2}$ terms transform differently from $1/r$
terms. Justification of this fact was given in ref. Drake and Szekeres (2000).
However, as will be demonstrated, this ambiguity vanishes in the equatorial
plane.

  4. 4.

Perform a complex coordinate transformation
$x^{\prime\mu}=x^{\mu}+iy^{\mu}(x)$ whereby $Z^{\prime\mu}_{a}$ transforms as

$Z^{\mu}_{a}=\frac{\partial x^{\mu}}{\partial
x^{\prime\nu}}Z^{\prime\nu}_{a}.$ (77)

The new coordinates $x^{\mu}\in\\{u,r,\theta,\phi\\}$ are real-valued. To
avoid clutter, names of coordinates have been reused from step 1. The
particular transformation used by Newman and Janis is

$x^{\mu}=x^{\prime\mu}-ia\cos\theta(\delta_{0}^{\mu}-\delta_{1}^{\mu}),$ (78)

which introduces the spin parameter $a$. This transformation will be adopted
here as well. In the equatorial plane $x^{\mu}=x^{\prime\mu}$, and step 3
becomes unambiguous with $1/r$ and $1/r^{2}$ transforming in the same way.

  5. 5.

Finally, it is assumed that one can convert the metric to a Kerr-like
structure by a coordinate transformation in $u$ and $\phi$ of the form
$u=t+F(r)$, $\phi=\psi+G(r)$. The purpose is to remove all off-diagonal terms
except $g_{t\phi}$.

We now apply the NJA to the seed metric of eqs. (40) and (41). $\alpha_{1}$
and $\alpha_{2}$ are kept symbolic, their values only reinstated when analysis
is complete.

Null coordinates may readily be identified from eq. (61a), yielding the line
element (step 1 of the NJA)

$ds^{2}=-(1-\mathcal{A})du^{2}-2\sqrt{\frac{1-\mathcal{A}}{1-\widetilde{\mathcal{A}}}}\,\,dud\rho+\rho^{2}d\Omega^{2},$
(79a) where
$\mathcal{A}=\frac{4\alpha_{1}G}{\rho}+\frac{4\alpha_{2}G^{2}}{\rho^{2}}+\mathcal{O}(G^{3})\quad\text{and}\quad\widetilde{\mathcal{A}}=\frac{4\alpha_{1}G}{\rho}+\frac{8\alpha_{2}G^{2}}{\rho^{2}}+\mathcal{O}(G^{3}).$
(79b)

As mentioned in ref. Drake and Szekeres (2000), the same analysis with a
Reissner-Nordström metric yields instead

$\mathcal{A}=\widetilde{\mathcal{A}}=2GM/\rho-
GQ^{2}/\rho^{2},\quad\quad\text{(Reissner Nordstr\"{o}m metric)}$ (80)

simplifying the $dud\rho$ component. This difference follows from the
observation of eq. (61a); eq. (40) cannot be brought to exact Schwarzschild
form, meaning $g_{\rho\rho}\neq-1/g_{tt}$. Were this the case, the NJA-
transformed metric would simply have Kerr-Newman structure.

The line-element of eq. (79a) is written in terms of null-tetrads (step 2 of
the NJA), which may easily by identified by comparing eq. (79a) with eq. (72)

$e^{\Phi(\rho)}=(1-\mathcal{A})^{1/2},\quad
e^{\lambda(\rho)}=(1-\widetilde{\mathcal{A}})^{-1/2},$ (81a)
$l^{\mu}=\delta_{1}^{\mu},\quad\quad
n^{\mu}=\sqrt{\frac{1-\widetilde{\mathcal{A}}}{1-\mathcal{A}}}\delta_{0}^{\mu}-\frac{1}{2}(1-\widetilde{\mathcal{A}})\delta_{1}^{\mu},\quad\quad
m^{\mu}=\frac{1}{\sqrt{2}\rho}(\delta_{2}^{\mu}+\frac{i}{\sin\theta}\delta_{3}^{\mu}),$
(81b)

and coordinates are complexified (step 3 of the NJA) by replacing

$1/\rho\rightarrow\frac{1}{2}(1/r^{\prime}+1/\bar{r}^{\prime})\text{ and
}1/\rho^{2}\rightarrow 1/(r^{\prime}\bar{r}^{\prime}).$ (82)

A coordinate transformation
$x^{\mu}=x^{\prime\mu}-ia\cos\theta(\delta_{0}^{\mu}-\delta_{1}^{\mu})$ is now
performed (step 4 of the NJA). Reuse notation by denoting
$x^{\mu}=\\{u,r,\theta,\phi\\}$. The tetrads transform according to eq. (77).
In particular, expressions are reduced by expanding $\tilde{Z}^{\mu}_{a}$ to
$\mathcal{O}(G^{2})$. The transformed metric in the equatorial
($\theta=\pi/2$) plane reads

$\displaystyle\tilde{g}_{\mu\nu}^{\text{(NJA)}}$
$\displaystyle=-\tilde{l}_{\mu}\tilde{n}_{\nu}-\tilde{n}_{\mu}\tilde{l}_{\nu}+\tilde{m}_{\mu}\bar{\tilde{m}}_{\nu}+\bar{\tilde{m}}_{\mu}\tilde{m}_{\nu}$
$\displaystyle=\left(\begin{array}[]{cccc}-(1-\mathcal{A})&-\frac{\sqrt{1-\mathcal{A}}}{\sqrt{1-\mathcal{\widetilde{A}}}}&-a\left(\frac{\sqrt{1-\mathcal{A}}}{\sqrt{1-\mathcal{\widetilde{A}}}}+\mathcal{A}-1\right)\\\
-\frac{\sqrt{1-\mathcal{A}}}{\sqrt{1-\mathcal{\widetilde{A}}}}&0&\frac{a\sqrt{1-\mathcal{A}}}{\sqrt{1-\mathcal{\widetilde{A}}}}\\\
-a\left(\frac{\sqrt{1-\mathcal{A}}}{\sqrt{1-\mathcal{\widetilde{A}}}}+\mathcal{A}-1\right)&\frac{a\sqrt{1-\mathcal{A}}}{\sqrt{1-\mathcal{\widetilde{A}}}}&r^{2}-a^{2}\left(-\frac{2\sqrt{1-\mathcal{A}}}{\sqrt{1-\mathcal{\widetilde{A}}}}-\mathcal{A}+1\right)\\\
\end{array}\right)$ (86)
$\displaystyle=\left(\begin{array}[]{cccc}-\frac{r^{2}\left(r(r-4\alpha_{1}G)-8\alpha_{2}G^{2}\right)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}&-\frac{r^{2}}{r^{2}-2\alpha_{2}G^{2}}&-\frac{aGr^{2}(4\alpha_{1}r+6\alpha_{2}G)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}\\\
-\frac{r^{2}}{r^{2}-2\alpha_{2}G^{2}}&0&-\frac{ar^{2}}{2\alpha_{2}G^{2}-r^{2}}\\\
-\frac{aGr^{2}(4\alpha_{1}r+6\alpha_{2}G)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}&-\frac{ar^{2}}{2\alpha_{2}G^{2}-r^{2}}&-r^{2}\left(-\frac{a^{2}\left(r(4\alpha_{1}G+r)+4\alpha_{2}G^{2}\right)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}-1\right)\\\
\end{array}\right),$ (90)

where the first expression uses all-order expressions of $\tilde{Z}^{\mu}_{a}$
and the second uses their expansion to 2PM.

Last (step 5 of the NJA), $g_{tr}$ terms may be removed by the coordinate
transformation $u\rightarrow t$ and $\phi\rightarrow\psi$

$\begin{gathered}dt=du+D_{01}(r)dr,\quad\quad d\psi=d\phi+D_{31}(r)dr,\\\
D_{01}=\frac{a^{2}-2\alpha_{2}G^{2}+r^{2}}{a^{2}-4\alpha_{1}Gr-8\alpha_{2}G^{2}+r^{2}},\quad\quad
D_{31}=\frac{a}{a^{2}-4\alpha_{1}Gr-8\alpha_{2}G^{2}+r^{2}}\end{gathered}$
(91)

producing the final result

$g_{\mu\nu}^{\text{(NJA)}}=\left(\begin{array}[]{cccc}-\frac{r^{2}\left(r(r-4\alpha_{1}G)-8\alpha_{2}G^{2}\right)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}&0&-\frac{ar^{2}(4\alpha_{1}Gr+6\alpha_{2}G^{2})}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}\\\
0&\frac{r^{2}}{a^{2}+r(r-4\alpha_{1}G)-8\alpha_{2}G^{2}}&0\\\
-\frac{ar^{2}(4\alpha_{1}Gr+6\alpha_{2}G^{2})}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}&0&r^{2}\left(\frac{a^{2}\left(r(4\alpha_{1}G+r)+4\alpha_{2}G^{2}\right)}{\left(r^{2}-2\alpha_{2}G^{2}\right)^{2}}+1\right)\\\
\end{array}\right),$ (92)

which is the NJA metric from the seed of eq. (61a), presented in eq. (64) of
the main text.

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