# GYOTO 2.0: a polarized relativistic ray-tracing code

N Aimar1111The three first authors in the list contributed equally to the
article., T Paumard1††footnotemark: , F H Vincent1††footnotemark: , E
Gourgoulhon2,3, and G Perrin1 1 LESIA, Observatoire de Paris, CNRS, Université
Pierre et Marie Curie, Université Paris Diderot, 5 place Jules Janssen, 92190
Meudon, France
2 Laboratoire Univers et Théories, Observatoire de Paris, CNRS, Université
PSL, Université Paris Cité, 5 place Jules Janssen, 92190 Meudon, France
3 Laboratoire de Mathématiques de Bretagne Atlantique, CNRS, Université de
Bretagne Occidentale, 6 avenue Le Gorgeu, 29238 Brest, France
<EMAIL_ADDRESS>
###### Abstract

Polarized general-relativistic radiative transfer in the vicinity of black
holes and other compact objects has become a crucial tool for probing the
properties of relativistic astrophysics plasmas. Instruments like GRAVITY, the
Event Horizon telescope, ALMA, or IXPE make it very timely to develop such
numerical frameworks. In this article, we present the polarized extension of
the public ray-tracing code Gyoto, and offer a python notebook allowing to
easily perform a first realistic computation. The code is very modular and
allows to conveniently add extensions for the specific needs of the user. It
is agnostic about the spacetime and can be used for arbitrary compact objects.
We demonstrate the validity of the code by providing tests, and show in
particular a perfect agreement with the ipole code. Our article also aims at
pedagogically introducing all the relevant formalism in a self-contained
manner.

##  1 Introduction

Generating synthetic images of black hole environments is tricky because of
relativistic effects such as aberration, Doppler beaming, gravitational
redshift and light bending. General-Relativistic Radiative Transfer (GRRT)
computation through ray-tracing method, i.e. integration of photons
trajectories (null geodesics, assuming no impact other than gravitation on the
trajectory), allows to naturally account for all relativistic effects to
generate synthetic observables of such environments that can be compared to
observational data. This technique is key to obtain meaningful constraints on
the parameters describing the emitting accretion flow or the spacetime
geometry. It can be used either on analytically described accretion flows, or
in post-processing of general relativistic magnetohydrodynamical (GRMHD)
simulations. For example, the Numerical Observatory of Violent Accreting
system (NOVAs; Varniere et al., 2018; Mignon-Risse et al., 2021) combines
GRMHD simulations of accretion flows around black holes as computed by the
code GR-AMRVAC (Casse et al., 2017), with the ray-tracing code Gyoto (Vincent
et al., 2011) to produce various synthetic observables.

The recent development of instruments now allows to measure the polarization
of light coming from the extremely close environment of black holes. The Event
Horizon Telescope (EHT) released the millimetric polarized image of M87*
(Event Horizon Telescope Collaboration et al., 2022). GRAVITY (GRAVITY
Collaboration et al., 2018, 2023) in the near infrared, and the Atacama Large
Millimeter/submillimeter Array (ALMA, Wielgus et al., 2022) have detected a
polarized signature of the radiation flares associated with the supermassive
black hole at the center of the Galaxy, Sagittarius A* (Sgr A*). Moreover, the
Imaging X-ray Polarimetry Explorer (IXPE) has obtained important constraints
on the geometry of the accretion flow of an X-ray binary by measuring its
polarized radiation (Krawczynski et al., 2022).

Thus, polarized ray-tracing codes are of particular interest to generate
synthetic observables from accretion models, be they analytic (see e.g.
Broderick et al., 2016; Gralla et al., 2018, 2019; Vincent et al., 2019;
Gralla et al., 2020; GRAVITY Collaboration et al., 2020; Nalewajko et al.,
2020; Dovčiak et al., 2022; Vos et al., 2022; Aimar et al., 2023; Cárdenas-
Avendaño and Lupsasca, 2023), or numeric (see e.g. Chan et al., 2015;
Mościbrodzka et al., 2016; Chael et al., 2018; Davelaar et al., 2018; Chael et
al., 2019; Event Horizon Telescope Collaboration et al., 2019; Anantua et al.,
2020; Dexter et al., 2020; Porth et al., 2021). In particular, polarization
signatures might allow probing the nature of spacetime close to the event
horizon of black hole candidates, making GRRT a crucial tool for constraining
general relativity (GR) in the strong-field regime (Himwich et al., 2020;
Jiménez-Rosales et al., 2021; Vincent et al., 2023).

The need for ray tracing when computing images in GR led to the development of
multiple codes. Many of them were developed for unpolarized light (Noble et
al., 2007; Dexter and Agol, 2009; Dauser et al., 2010; Vincent et al., 2011;
Pu et al., 2016; Chan et al., 2018; Bronzwaer et al., 2018; Younsi et al.,
2020). Some codes also keep track of the electric vector position angle and
polarization degree, assuming a geometrically thin equatorial accretion flow
(Dovčiak et al., 2008; Gelles et al., 2021; Cárdenas-Avendaño et al., 2023),
the latter two being specialized for highly-lensed features by implementing
adaptive ray tracing. Only a handful of codes are able of treating the most
demanding problem of integrating the full polarized radiative transfer:
grtrans (Dexter, 2016), ipole (Mościbrodzka and Gammie, 2018a), Arcmancer
(Pihajoki et al., 2018), Bhoss (Younsi et al., 2020), Raptor (Bronzwaer et
al., 2020), Blacklight (White, 2022), as well as Lemon (Xiao-lin et al., 2021)
which specializes on polarized radiative transfer with scattering. Some of
these polarized GRRT codes were recently compared by Prather et al. (2023).

Gyoto (Vincent et al., 2011) is a backwards ray-tracing code (i.e. integrating
from the observer to the source), operating in any given (analytically or
numerically computed) metric, and solving the radiative transfer equation. It
can also integrate timelike geodesics for computing e.g. stellar orbits
(Grould et al., 2017). The code is publicly available
222https://github.com/gyoto/Gyoto/blob/master/INSTALL.Gyoto.md , built to be
modular so that extensions are easy to integrate and user-friendly with XML
and python interfaces.

The goals of this paper are the following: (i) providing the new version of
Gyoto, with full polarized GRRT included, publicly available at the same
address as the older version, with the addition of a python snippet
333https://github.com/gyoto/Gyoto/blob/master/doc/examples/Gyoto_Polar_example.ipynb
to allow the interested reader to immediately be able to compute a non-trivial
setup; (ii) providing a pedagogical, in-depth presentation of the formalism of
GR polarization, as well as a detailed description of the technical
implementation. In the following discussion, we will focus on polarized
synchrotron radiation, as it is the dominant emission mechanism for our
sources of primary scientific interest (Sgr A* and M87). But the code is able
to compute polarized observables for other emission mechanism as soon as the
electric field is provided in the model. Section 2 presents the formalism of
GR polarized radiative transfer. In section 3, we present various tests that
we made to validate our code. The last section is dedicated to discussion and
conclusion.

##  2 Formalism

We will discuss the problem of polarized GRRT taking the usual point of view
of ray tracing. We thus consider a light ray (mathematically speaking, a null
geodesic) integrated backwards from a distant observer’s screen towards some
source of radiation. The problem can then be divided into three main parts
that will be discussed hereafter:

  * •

the definition of a wave vector at the distant observer’s screen, tangent to
the considered null geodesic, together with a pair of spacelike vectors
forming an orthonormal basis of the observer’s screen;

  * •

the backwards parallel propagation along the considered null geodesic of the
wave vector together with the screen basis, until a source of radiation is
reached;

  * •

the integration of polarized radiative transfer within the source.

Before describing these three steps in detail, we will start by providing
important definitions in the next section.

###  2.1 Geometric optics, light ray, covariant and observer-specific
polarization vectors

We consider a monochromatic plane electromagnetic wave propagating in a given
spacetime. The geometrical optics approximation of Maxwell’s equations under
Lorenz gauge condition allows to describe this wave as follows.

The complex 4-potential 1-form reads

$\mathbf{\hat{A}}=\mathbf{\hat{a}}\,e^{i\Phi},$ (1)

where $\mathbf{\hat{a}}$ is the amplitude 1-form, assumed to vary much slower
than the phase $\Phi$ (this is the basic idea of the geometrical optics
approximation). The hat reminds that we are dealing with complex quantities.
The Faraday electromagnetic 2-form,

$\boldsymbol{\hat{\mathcal{F}}}=\mathbf{d\hat{A}},$ (2)

then reads in components

$\displaystyle\hat{\mathcal{F}}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\nabla_{\alpha}\hat{A}_{\beta}-\nabla_{\beta}\hat{A}_{\alpha}$
$\displaystyle=$ $\displaystyle
e^{i\Phi}\nabla_{\alpha}\hat{a}_{\beta}+\hat{a}_{\beta}\,i\,e^{i\Phi}\,\nabla_{\alpha}\Phi-\left(e^{i\Phi}\nabla_{\beta}\hat{a}_{\alpha}+\hat{a}_{\alpha}\,i\,e^{i\Phi}\,\nabla_{\beta}\Phi\right)$
$\displaystyle\approx$ $\displaystyle
i\left(\hat{a}_{\beta}\,k_{\alpha}-\hat{a}_{\alpha}\,k_{\beta}\right)\,e^{i\Phi},$

where we introduce the wave vector

$\mathbf{k}\equiv\boldsymbol{\nabla}{\Phi},$ (4)

and use the geometric optics approximation to neglect the variations of the
amplitude, i.e. the $\nabla_{\alpha}\hat{a}_{\beta}$ terms.

Plugging this into Maxwell’s equations and assuming Lorenz gauge (that is, the
divergence of $\mathbf{\hat{A}}$ should vanish) leads to the following
results:

  * •

$\mathbf{k}$ is a null vector parallel propagated along itself,

$\mathbf{k}\cdot\mathbf{k}=0,\quad\boldsymbol{\nabla}_{\mathbf{k}}\,\mathbf{k}=\mathbf{0},$
(5)

so that it defines a null geodesic, which we define as a light ray;

  * •

we can introduce the unit spacelike covariant polarization vector

$\mathbf{\hat{f}}\equiv\frac{\mathbf{\hat{a}}}{a},$ (6)

where $\mathbf{\hat{a}}$ is the complex vector corresponding to the amplitude
1-form introduced in Eq. 1 by metric duality (we use the same notation for
both quantities in order to simplify the notation). The scalar quantity $a$ is
the modulus of the complex vector $\mathbf{\hat{a}}$,

$a=\sqrt{\hat{a}_{\mu}\hat{a}^{\mu*}}.$ (7)

Our naming convention specifies that this vector is covariant in order to
differentiate it from the polarization vector as observed by a specific
observer, which will be our quantity of prime interest in the following. The
covariant polarization vector satisfies the two following important
properties: (i) it is perpendicular to the wave vector $\mathbf{k}$ (this is a
consequence of the Lorenz gauge choice), and (ii) it is parallel transported
along $\mathbf{k}$ in vacuum (this is a consequence of Maxwell’s equations):

$\mathbf{\hat{f}}\cdot\mathbf{k}=0,\quad\boldsymbol{\nabla}_{\mathbf{k}}\,\mathbf{\hat{f}}=\mathbf{0}.$
(8)

We can thus reexpress the Faraday tensor in terms of the polarization and wave
vectors as follows

${\hat{\mathcal{F}}_{\alpha\beta}=i\,a\,\left(\hat{f}_{\beta}\,k_{\alpha}-\hat{f}_{\alpha}\,k_{\beta}\right)\,e^{i\Phi}.}$
(9)

From this expression we deduce the important property that it is possible to
add any multiple of the wave vector $\mathbf{k}$ to the polarization vector
$\mathbf{\hat{f}}$ without altering the Faraday tensor. So the polarization
vector can be arbitrarily transformed under

$\mathbf{\hat{f}}\mapsto\mathbf{\hat{f}}+q\mathbf{k}$ (10)

for any scalar field $q$ (note that $q$ is not necessarily a constant, it is
an arbitrary scalar field).

So far, we have only used global quantities that are not defined with respect
to any particular observer. We now want to introduce the electric and magnetic
fields as observed by the distant observer, $\mathcal{O}$, the oscillations of
which define the observed electromagnetic wave. Let us denote by
$\mathbf{u_{0}}$ the 4-velocity of observer $\mathcal{O}$. By definition, the
electric linear form and the magnetic vector as measured by a generic observer
with 4-velocity $\mathbf{u}$ read 444We highlight that the electric and
magnetic fields discussed here are the electromagnetic fields describing the
monochromatic wave that reaches the observer’s screen. They should not be
confused with the electromagnetic fields that might exist at the source
location, for instance the magnetic field of the accretion flow surrounding
the black hole.

$\displaystyle\hat{E}_{\alpha}=\hat{\mathcal{F}}_{\alpha\mu}\,u^{\mu},$ (11)
$\displaystyle\hat{B}^{\alpha}=-\frac{1}{2}\epsilon^{\alpha\mu\nu}_{\>\>\>\>\>\>\>\rho}\,\hat{\mathcal{F}}_{\mu\nu}\,u^{\rho}$

where $\boldsymbol{\epsilon}$ is the Levi-Civita tensor. The electric field
vector $\mathbf{\hat{E}_{0}}$ as observed by the distant observer
$\mathcal{O}$ thus reads

$\displaystyle\hat{E}^{\rho}$ $\displaystyle=$ $\displaystyle
g^{\rho\alpha}\,\hat{E}_{\alpha}$ $\displaystyle=$ $\displaystyle
g^{\rho\alpha}\,\hat{\mathcal{F}}_{\alpha\beta}\,u^{\beta}$ $\displaystyle=$
$\displaystyle
i\,a\,e^{i\Phi}\,g^{\rho\alpha}\left(\hat{f}_{\beta}\,k_{\alpha}-\hat{f}_{\alpha}\,k_{\beta}\right)u^{\beta}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\Phi}\,\left(\hat{f}_{\beta}\,k^{\rho}-\hat{f}^{\rho}\,k_{\beta}\right)u^{\beta}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\Phi}\,\left((\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}})\,k^{\rho}+\omega_{0}\hat{f}^{\rho}\right)$

where we drop the lower index $0$ for the components of the various tensors
for simplicity (all of them being evaluated at the distant observer’s
location), and we introduce
$\omega_{0}\equiv-\mathbf{k_{0}}\cdot\mathbf{u_{0}}$, where $\mathbf{k_{0}}$
is the wave vector at the distant observer’s location. This quantity
$\omega_{0}$ is the pulsation of the photon as measured by $\mathcal{O}$. All
vectors with a lower index $0$ are defined at the distant observer’s screen.

Let us decompose the vectors $\mathbf{\hat{f}_{0}}$ and $\mathbf{k_{0}}$ in
parts parallel and orthogonal to the observer’s 4-velocity:

$\displaystyle\mathbf{k_{0}}$ $\displaystyle=$
$\displaystyle\omega_{0}\,\mathbf{u_{0}}+\mathbf{K_{0}},\qquad\qquad\mathbf{K_{0}}\perp\mathbf{u_{0}},$
(13) $\displaystyle\mathbf{\hat{f}_{0}}$ $\displaystyle=$
$\displaystyle-(\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}})\,\mathbf{u_{0}}+\mathbf{\hat{f}_{0}^{\perp}},\quad\mathbf{\hat{f}_{0}^{\perp}}\perp\mathbf{u_{0}}.$

Note that

$\mathbf{\hat{f}_{0}^{\perp}}\cdot\mathbf{\hat{f}_{0}^{\perp}}=1+\left(\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}\right)^{2}$
(14)

so that $\mathbf{\hat{f}_{0}^{\perp}}$ is not a unit vector in general. It is
normalized only if $\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}=0$, in which case
we simply have $\mathbf{\hat{f}_{0}^{\perp}}=\mathbf{\hat{f}_{0}}$. Similary,
$\mathbf{K_{0}}$ is not a unit vector and it is easy to show that

$\mathbf{K_{0}}\cdot\mathbf{K_{0}}=\omega_{0}^{2}.$ (15)

The vector $\mathbf{K_{0}}$ coincides with the incident wave vector as
measured by observer $\mathcal{O}$.

In terms of the vectors normal to $\mathbf{u_{0}}$ we immediately obtain the
final expression of the electric vector as observed by $\mathcal{O}$:

${\mathbf{\hat{E}_{0}}=i\,a\,e^{i\Phi}\left(\omega_{0}\,\mathbf{\hat{f}_{0}^{\perp}}-\frac{\mathbf{K_{0}}\cdot\mathbf{\hat{f}_{0}^{\perp}}}{\omega_{0}}\,\mathbf{K_{0}}\right).}$
(16)

This vector is clearly orthogonal to the direction of propagation
$\mathbf{K_{0}}$, and it is also orthogonal to the 4-velocity
$\mathbf{u_{0}}$, as it should for a vector living in the local rest space of
the observer.

Then, the magnetic field vector $\mathbf{\hat{B}_{0}}$ as observed by
$\mathcal{O}$ reads

$\displaystyle\hat{B}^{\rho}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\epsilon^{\rho\mu\nu}_{\>\>\>\>\>\>\alpha}\hat{\mathcal{F}}_{\mu\nu}u^{\alpha}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}i\,a\,e^{i\Phi}\,\epsilon^{\rho\mu\nu}_{\>\>\>\>\>\>\alpha}\left(\hat{f}_{\nu}k_{\mu}-\hat{f}_{\mu}k_{\nu}\right)u^{\alpha}$
$\displaystyle=$
$\displaystyle-i\,a\,e^{i\Phi}\,\epsilon^{\rho\mu\nu}_{\>\>\>\>\>\>\alpha}\hat{f}_{\nu}k_{\mu}u^{\alpha}$
$\displaystyle=$
$\displaystyle-i\,a\,e^{i\Phi}\,\epsilon^{\rho}_{\>\>\mu\nu\alpha}u^{\alpha}k^{\mu}\hat{f}^{\nu}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\Phi}\,\epsilon^{\>\>\>\>\>\>\>\>\rho}_{\alpha\mu\nu}u^{\alpha}k^{\mu}\hat{f}^{\nu}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\Phi}\,\epsilon^{\>\>\>\>\>\>\>\>\rho}_{\alpha\mu\nu}u^{\alpha}K^{\mu}\hat{f}^{\perp\nu},$

where we have used extensively the antisymmetric nature of the Levi-Civita
tensor. The last expression exactly coincides with the definition of the cross
product in the vector space orthogonal to $\mathbf{u_{0}}$ (which we label by
$\times_{\mathbf{u_{0}}}$), such that finally the magnetic field vector as
measured by $\mathcal{O}$ reads

${\mathbf{\hat{B}_{0}}=i\,a\,e^{i\Phi}\,\mathbf{K_{0}}\times_{\mathbf{u_{0}}}\mathbf{\hat{f}_{0}^{\perp}}.}$
(18)

This vector is also obviously orthogonal to the direction of propagation
$\mathbf{K_{0}}$, and to the electric vector.

We can now define the notion of polarization vector as measured by observer
$\mathcal{O}$:

$\mathbf{\hat{F}_{0}}=\mathbf{K_{0}}\times_{\mathbf{u_{0}}}\mathbf{\hat{B}_{0}},$
(19)

which is by construction normal to the direction of propagation and to the
magnetic field vector. We note that this quantity depends on the observer,
just as the electric and magnetic fields, while the covariant polarization
vector $\mathbf{\hat{f}_{0}}$, defined in Eq. 6, is a covariant quantity. They
obviously differ given that by construction $\mathbf{\hat{F}_{0}}$ is
orthogonal to the observer’s 4-velocity $\mathbf{u_{0}}$, while
$\mathbf{\hat{f}_{0}}$ is defined independently from $\mathbf{u_{0}}$. A
natural question is to investigate the relation between $\mathbf{\hat{F}_{0}}$
and $\mathbf{\hat{f}_{0}^{\perp}}$ that both live in the vector space
orthogonal to $\mathbf{u_{0}}$. These two vectors are completely independent a
priori, because $\mathbf{\hat{F}_{0}}$ is by construction orthogonal to both
$\mathbf{K_{0}}$ and $\mathbf{\hat{B}_{0}}$ (see Eq. 19), while
$\mathbf{\hat{f}_{0}^{\perp}}$ is only orthogonal to $\mathbf{\hat{B}_{0}}$
(see Eq. 18), but not to $\mathbf{K_{0}}$. Indeed, from Eq. 8 and 13, we have

$\displaystyle\mathbf{\hat{f}_{0}}\cdot\mathbf{k_{0}}=0$ (20)
$\displaystyle\Leftrightarrow$
$\displaystyle\left(\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}\right)\,\omega_{0}+\mathbf{\hat{f}_{0}^{\perp}}\cdot\mathbf{K_{0}}=0$

so that only if $\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}=0$ (which has no
reason to hold in general) is $\mathbf{\hat{f}_{0}^{\perp}}$ orthogonal to
$\mathbf{K_{0}}$. In this special case, we saw in Eq. 13 that
$\mathbf{\hat{f}_{0}^{\perp}}=\mathbf{\hat{f}_{0}}$, so that when
$\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}=0$, and only then,
$\mathbf{\hat{f}_{0}}$ is the unit vector along $\mathbf{\hat{F}_{0}}$.
However, we have seen in Eq. 10 that the covariant polarization vector is
defined up to a term proportional to the wavevector, the proportionality
coefficient being a scalar field. We can thus choose to work with a covariant
polarization vector $\mathbf{f^{\prime}}$ such that, at the distant observer’s
location

$\mathbf{\hat{f}^{\prime}_{0}}=\mathbf{\hat{f}_{0}}+\frac{\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}}{\omega_{0}}\,\mathbf{k_{0}}.$
(21)

This vector is such that

$\mathbf{\hat{f}^{\prime}_{0}}\cdot\mathbf{u_{0}}=0,$ (22)

and we saw just above that this implies that $\mathbf{\hat{f}^{\prime}_{0}}$
is then the unit vector along $\mathbf{\hat{F}_{0}}$. Thanks to the degree of
liberty in the definition of the covariant polarization vector expressed by
Eq. 10, we can thus confuse the covariant and non-covariant polarization
vectors at the observer, $\mathbf{\hat{f}_{0}}$ and $\mathbf{\hat{F}_{0}}$.

By virtue of the double vector product law we have

$\displaystyle(\mathbf{K_{0}}\times_{\mathbf{u_{0}}}\mathbf{\hat{B}_{0}})^{\alpha}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\phi}\left[(\mathbf{K_{0}}\cdot\mathbf{\hat{f}_{0}^{\perp}})K^{\alpha}-(\mathbf{K_{0}}\cdot\mathbf{K_{0}})\hat{f}^{\perp\alpha}\right]$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\phi}\,\omega_{0}\,\left[\frac{\mathbf{K_{0}}\cdot\mathbf{\hat{f}_{0}^{\perp}}}{\omega_{0}}K^{\alpha}-\omega_{0}\hat{f}^{\perp\alpha}\right]$

so that finally

${\mathbf{\hat{F}_{0}}=(\mathbf{K_{0}}\times_{\mathbf{u_{0}}}\mathbf{\hat{B}_{0}})=-\omega_{0}\,\mathbf{\hat{E}_{0}}}.$
(24)

We thus conclude that the polarization vector as measured by $\mathcal{O}$
coincides, up to a normalization factor, with the electric field as measured
by $\mathcal{O}$. The various vectors lying in observer $\mathcal{O}$’s local
rest space (that is, the vector space orthogonal to the 4-velocity
$\mathbf{u_{0}}$) are depicted in Fig. 1. We introduce in this figure the
electric vector position angle (EVPA), defined in the local frame of the
distant observer, which is the angle between a reference direction (the local
North of the distant observer) and the polarization vector
$\mathbf{\hat{F}_{0}}$.

Figure 1: Vectors lying in the distant observer $\mathcal{O}$’s local rest
space (the vector space orthogonal to $\mathcal{O}$’s 4-velocity
$\mathbf{u_{0}}$). $\mathbf{K_{0}}$ is the wave vector projected orthogonal to
$\mathbf{u_{0}}$. The vectors $\mathbf{\bar{w}_{0}}$ and
$\mathbf{\bar{n}_{0}}$ are unit vectors pointing towards the local West and
North directions, so that
$(\mathbf{K_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$ forms a direct
orthogonal triad. $\mathbf{\hat{E}_{0}}$ and $\mathbf{\hat{B}_{0}}$ are the
electric and magnetic vectors as measured by $\mathcal{O}$, associated with
the incident light wave, so that
$(\mathbf{K_{0}},\mathbf{\hat{E}_{0}},\mathbf{\hat{B}_{0}})$ is a direct
orthogonal triad. $\mathbf{\hat{F}_{0}}$ is the polarization vector as
measured by $\mathcal{O}$ (defined in Eq. 19), which is antiparallel to
$\mathbf{\hat{E}_{0}}$ as stated by Eq. 24. $\mathbf{\hat{f}_{0}^{\perp}}$ is
the covariant polarization vector (defined in Eq. 6) projected orthogonal to
$\mathbf{u_{0}}$. The plane orthogonal to $\mathbf{K_{0}}$ (observer’s screen
plane) is drawn in green. It contains the electric and magnetic field vectors,
and the polarization vector as measured by $\mathcal{O}$. The plane orthogonal
to $\mathbf{\hat{B}_{0}}$ is drawn in blue-green. It contains the photon’s
wave vector $\mathbf{K_{0}}$, the electric vector $\mathbf{\hat{E}_{0}}$, and
both the covariant and $\mathcal{O}$’s specific polarization vectors. The
observed Electric Vector Position Angle (EVPA0), measured East of North in the
screen’s plane, is shown in dark red.

###  2.2 Polarization basis defined at the distant observer’s screen

We take here the typical point of view of a ray-tracing problem where the
initial conditions are fixed at the far-away observer’s screen, and the
integration is performed backwards in time from the screen towards the source.
This allows to save a lot of computing time by integrating only those
geodesics that will approach the source by shooting light rays only within a
small solid angle subtending the source. The aim of this subsection is to
explicitly describe our initial conditions at the observer’s screen, which is
illustrated in Fig. 2. In order to be specific, we will consider a black hole
spacetime, but the discussion is very general and is not restricted to this
particular case.

Figure 2: Initial condition of the polarized ray-tracing problem at the
distant observer’s screen. Left: A black hole (BH) spacetime is represented
for being specific, but the figure is very general and applies to any kind of
spacetime. We consider spherical coordinates and the spacelike orthonormal
basis associated with these coordinates are labeled
$(\mathbf{\bar{e}_{r}},\mathbf{\bar{e}_{\theta}},\mathbf{\bar{e}_{\varphi}})$.
The observer’s local rest space is described by a direct orthonormal triad,
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}},\mathbf{\bar{e}_{3}})$, where
$\mathbf{\bar{e}_{3}}=-\mathbf{\bar{e}_{r}}$, and where the screen’s plane is
contained in $\mathrm{Span}(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}})$. We
consider here that the “upwards” direction of the observer’s screen coincides
with the projection of the black hole’s angular momentum on the screen, i.e.
that $\mathbf{\bar{e}_{2}}=-\mathbf{\bar{e}_{\theta}}$. The unit direction of
photon reception at the observer’s screen is $\mathbf{\bar{K}_{0}}$. The local
polarization basis at the observer’s screen,
$(\mathbf{\bar{K}_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$, is shown,
and corresponds to the central pixel of the screen, that is, to the purely
radial incoming direction of the photon. Right: zoom on the local rest space
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}},\mathbf{\bar{e}_{3}})$ and local
celestial sphere of the observer. For a source of radiation located at S on
the local celestial sphere, the unit direction of incidence is
$\mathbf{\bar{K}_{0}}$ (it is $-\mathbf{\bar{K}_{0}}$ on the figure because
$\mathbf{\bar{e}_{3}}$ points towards the source, while the incidence
direction is of course in the opposite direction). The vector
$\mathbf{\bar{K}_{0}}$ is here not purely along the radial direction
$\mathbf{\bar{e}_{3}}$ (in contrast with the left panel): it thus corresponds
to a pixel that is not located at the center of the screen. The corresponding
equatorial angles labeling the source, $(\alpha,\delta)$, are shown, together
with the corresponding spherical angles on the observer local sky, $(a,b)$.
For typical ray-tracing problems where the observer is far away, we have $a\ll
1$.

The observer’s screen is considered to be a pin-hole camera, with the various
pixels corresponding to different directions on sky. The local rest space of
the observer is spanned by a direct orthonormal triad
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}},\mathbf{\bar{e}_{3}})$. Here and
in the following, a bar on top of a vector denotes a spacelike unit vector.
The vector $\mathbf{\bar{e}_{3}}$ is along the line of sight, normal to the
screen, towards the black hole. If we consider spherical coordinates centered
on the black hole (e.g. Boyer-Lindquist coordinates) and a direct orthonormal
triad
$(\mathbf{\bar{e}_{r}},\mathbf{\bar{e}_{\theta}},\mathbf{\bar{e}_{\varphi}})$
associated with these coordinates, then
$\mathbf{\bar{e}_{3}}=-\mathbf{\bar{e}_{r}}$. The screen’s plane is spanned by
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}})$, and we consider the special
case $\mathbf{\bar{e}_{2}}=-\mathbf{\bar{e}_{\theta}}$, which boils down to
assuming that the projection of the black hole’s angular momentum on the
screen is along $\mathbf{\bar{e}_{2}}$. For a $N\times N$ pixels screen, one
pixel with indices $(i,j)$, with $i,j=1..N$, corresponds to a pair of
equatorial angles (see Fig. 2)

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{f}{N}\left(i-\frac{N+1}{2}\right),$ (25)
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{f}{N}\left(j-\frac{N+1}{2}\right),$

where $f$ is the field of view of the observer. The corresponding spherical
angles (see Fig. 2) on the local sky of the observer are given by standard
spherical trigonometry relations:

$\displaystyle\cos a$ $\displaystyle=$ $\displaystyle\cos\alpha\,\cos\delta,$
(26) $\displaystyle\tan b$ $\displaystyle=$
$\displaystyle\frac{\tan\alpha}{\sin\delta}.$

The local unit direction of photon incidence then reads

$\displaystyle\mathbf{\bar{K}_{0}}$ $\displaystyle=$ $\displaystyle-\sin a\cos
b\,\boldsymbol{\bar{e}_{1}}-\sin a\sin b\,\boldsymbol{\bar{e}_{2}}-\cos
a\,\boldsymbol{\bar{e}_{3}}$ $\displaystyle=$ $\displaystyle\frac{\cos
a}{\sqrt{g_{rr}}}\,\boldsymbol{\partial_{r}}+\frac{\sin a\sin
b}{\sqrt{g_{\theta\theta}}}\,\boldsymbol{\partial_{\theta}}+\frac{\sin a\cos
b}{\sqrt{g_{\varphi\varphi}}}\,\boldsymbol{\partial_{\varphi}},$

where we have used the relations

$\mathbf{\bar{e}_{r}}=\frac{\boldsymbol{\partial_{r}}}{\sqrt{g_{rr}}},\quad\mathbf{\bar{e}_{\theta}}=\frac{\boldsymbol{\partial_{\theta}}}{\sqrt{g_{\theta\theta}}},\quad\mathbf{\bar{e}_{\varphi}}=\frac{\boldsymbol{\partial_{\varphi}}}{\sqrt{g_{\varphi\varphi}}},$
(28)

where $\boldsymbol{\partial_{i}}$ are spherical coordinate basis vectors, and
$g_{ii}=\boldsymbol{\partial_{i}}\cdot\boldsymbol{\partial_{i}}$ are the
corresponding metric coefficients. The null 4-vector tangent to the null
geodesic when incident on the observer’s screen then reads

$\mathbf{k_{0}}=\boldsymbol{\partial_{t}}+\mathbf{\bar{K}_{0}}$ (29)

where we consider that the observer’s 4-velocity is
$\mathbf{u_{0}}=\boldsymbol{\partial_{t}}$ (assuming a static observer, and
that $g_{tt}\to-1$ at the observer’s location) and where we have assumed that
$-\mathbf{k_{0}}\cdot\mathbf{u_{0}}=1$, so that the spacelike vector
$\mathbf{\bar{K}_{0}}$ in the last equation is normalized. This last
assumption means that the photon’s energy as measured by the far-away observer
is unity, which does not change anything to the problem, it simply scales the
energy, and physical values of energies can be easily retrieved when radiative
transfer calculations are performed.

Once the photon arrival direction $\mathbf{\bar{K}_{0}}$ has been defined, it
must be completed by two other vectors to form the local orthonormal
polarization basis
$(\mathbf{\bar{K}_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$. The vector
$\mathbf{\bar{K}_{0}}$ corresponding to the central pixel of the screen
coincides with a purely radial direction of arrival,
$\mathbf{\bar{K}_{0}^{\mathrm{cen}}}=-\mathbf{\bar{e}_{3}}$ (the superscript
’cen’ refers to the central pixel of the screen), see the left panel of Fig.
2. This particular vector can be easily completed by
$\mathbf{\bar{w}_{0}^{\mathrm{cen}}}=-\mathbf{\bar{e}_{1}}$ and
$\mathbf{\bar{n}_{0}^{\mathrm{cen}}}=\mathbf{\bar{e}_{2}}$, see the left panel
of Fig. 2. In the general case of a vector $\mathbf{\bar{K}_{0}}$ defined by
the two spherical angles $(a,b)$ (see the right panel of Fig. 2), A shows that
the observer’s screen polarization basis reads

$\displaystyle\mathbf{\bar{w}_{0}}$ $\displaystyle=$
$\displaystyle\left[-\sin^{2}b\left(1-\cos a\right)-\cos
a\right]\mathbf{\bar{e}_{1}}+\sin b\cos b\left(1-\cos
a\right)\mathbf{\bar{e}_{2}}$ $\displaystyle+\cos b\sin
a\,\mathbf{\bar{e}_{3}},$ $\displaystyle\mathbf{\bar{n}_{0}}$ $\displaystyle=$
$\displaystyle-\sin b\cos b\left(1-\cos
a\right)\mathbf{\bar{e}_{1}}+\left[\cos^{2}b\left(1-\cos a\right)+\cos
a\right]\mathbf{\bar{e}_{2}}$ $\displaystyle-\sin b\sin
a\,\mathbf{\bar{e}_{3}}.$

It is straightforward to check that these vectors are unit vectors, orthogonal
to each other, and to $\mathbf{\bar{K}_{0}}$. Moreover, in a typical ray-
tracing problem where $a\ll 1$, we have as we should
$\mathbf{\bar{w}_{0}}\approx\mathbf{\bar{w}_{0}^{\mathrm{cen}}}=-\mathbf{\bar{e}_{1}}$,
and
$\mathbf{\bar{n}_{0}}\approx\mathbf{\bar{n}_{0}^{\mathrm{cen}}}=\mathbf{\bar{e}_{2}}$.
We note that the plane $(\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$ (where
the polarization angle will be defined) strictly speaking only coincides with
the screen’s plane $(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}})$ for the
central pixel of the screen (with $a=0$). We will neglect this small
difference between the polarization plane and the screen’s plane, which is
perfectly valid as long as $a\ll 1$, that is, as long as the field of view is
sufficiently small.

At this point, we have fully defined our initial condition by specifying the
triad $(\mathbf{k_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$ at the
observer’s screen. The next step is to parallel transport these vectors along
the light ray towards the source.

###  2.3 Relevant frames, parallel transport of polarization basis, EVPA

Let us start by introducing the three relevant frames for describing the
polarized GRRT problem. We will focus on synchrotron radiation, which is our
primary science interest, but most of the discussion is rather general. The
frames of interest are:

  * •

the observer frame, described in detail in the previous section, defined by
the 4-velocity $\mathbf{u_{0}}$,

  * •

the fluid frame, defined by the 4-velocity $\mathbf{u}$ describing the bulk
motion of the emitting fluid (for instance, Keplerian motion around a black
hole),

  * •

the particle frame, which follows the helical motion of the synchrotron-
emitting electron around the magnetic field lines described in the fluid
frame.

This section will mostly deal with the fluid frame, and the link with the
particle frame is further discussed in B.

We consider a light ray, modeled by a null geodesic, joining the far-away
observer to the emitting accretion flow surrounding a black hole. We want to
parallel-transport the null 4-vector $\mathbf{k}$ tangent to the null
geodesic, backwards from the observer towards the emitter. We will also
parallel-transport the local West and North unit spacelike directions,
$\mathbf{\bar{w}}$ and $\mathbf{\bar{n}}$. Note that the index $0$ used for
these three vectors in the previous section meant that they were considered at
the screen position. We now consider their evolution along the ray and drop
the index. So we must integrate the following equations

$\displaystyle\boldsymbol{\nabla_{\mathbf{k}}}\mathbf{k}$ $\displaystyle=$
$\displaystyle\mathbf{0},$ (31)
$\displaystyle\boldsymbol{\nabla_{\mathbf{k}}}\mathbf{\bar{w}}$
$\displaystyle=$ $\displaystyle\mathbf{0},$
$\displaystyle\boldsymbol{\nabla_{\mathbf{k}}}\mathbf{\bar{n}}$
$\displaystyle=$ $\displaystyle\mathbf{0},$

with the initial conditions that
$(\mathbf{k},\mathbf{\bar{w}},\mathbf{\bar{n}})=(\mathbf{k_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$
at the screen. Given that the parallel transport preserves the scalar product
between vectors 555This is an obvious property: let us consider two vectors
$a^{\mu}$ and $b^{\mu}$ parallel-transported along $k^{\mu}$, then
$\nabla_{\mathbf{k}}\left(\mathbf{a}\cdot\mathbf{b}\right)=k^{\mu}\nabla_{\mu}\left(g_{\alpha\beta}a^{\alpha}b^{\beta}\right)=g_{\alpha\beta}\left(a^{\alpha}k^{\mu}\nabla_{\mu}b^{\beta}+b^{\beta}k^{\mu}\nabla_{\mu}a^{\alpha}\right)=0$,
because of the parallel-transport relations $k^{\mu}\nabla_{\mu}a^{\alpha}=0$
and $k^{\mu}\nabla_{\mu}b^{\beta}=0$. We used the fact that the connexion
$\nabla$ is compatible with the metric to get $\nabla_{\mu}g_{\alpha\beta}=0$
and take the metric tensor out of the covariant derivative. , and given that
$(\mathbf{k},\mathbf{\bar{w}},\mathbf{\bar{n}})$ are mutually orthogonal at
the observer, they remain mutually orthogonal when parallel-transported at the
emitter, and $\mathbf{\bar{w}}$ and $\mathbf{\bar{n}}$ remain unit vectors.

It is useful at this point to note that, in vacuum, the EVPA is a conserved
quantity along a geodesic. Let us demonstrate this result. We have seen that,
at the distant observer’s location, we might confuse the covariant and non-
covariant polarization vectors, $\mathbf{\hat{f}_{0}}$ and
$\mathbf{\hat{F}_{0}}$. Let us consider the point along a photon’s geodesic
corresponding to the exit from the emitting source region, meaning that the
part of the geodesic located in between this point and the distant observer is
in vacuum. We hereafter call this point the exit point. We can make the exact
same reasoning at the exit point as we made at the distant observer’s
location, and conclude that we can confuse the covariant and non-covariant
polarization vectors at the exit point, $\mathbf{\hat{f}_{\mathrm{exit}}}$ and
$\mathbf{\hat{F}_{\mathrm{exit}}}$, where $\mathbf{\hat{F}_{\mathrm{exit}}}$
is the polarization vector as measured by the emitter at the exit point. This
implies more generally that $\mathbf{\hat{f}}$ and $\mathbf{\hat{F}}$ can be
confused at any point along the part of the geodesic located in vacuum. Given
that $\mathbf{\hat{f}}$ and the screen basis
$(\mathbf{\bar{w}},\mathbf{\bar{n}})$ are parallel propagated along the
geodesic in vacuum (see Eqs. 8 and 31), the angle between $\mathbf{\hat{f}}$
and the basis vectors is conserved in vacuum, hence the EVPA is conserved
along the part of the geodesic located in vacuum. This is of course no longer
valid in the source region, where the covariant polarization vector is no
longer parallel propagated (the parallel propagation of $\mathbf{\hat{f}}$ is
a consequence of Maxwell’s equations in vacuum).

We want to project the parallel-transported basis vectors
$(\mathbf{k},\mathbf{\bar{w}},\mathbf{\bar{n}})$ orthogonally to the
4-velocity $\mathbf{u}$ of the emitting fluid, that is, project them in the
local rest space of the fluid. By doing so without further precaution, we
would of course lose the mutual orthogonality between these vectors, which is
not preserved in a projection. Let us consider

$\displaystyle\mathbf{\bar{w}^{\prime}}=\mathbf{\bar{w}}-\frac{\mathbf{\bar{w}}\cdot\mathbf{u}}{\mathbf{k}\cdot\
\mathbf{u}}\,\mathbf{k},$ (32)
$\displaystyle\mathbf{\bar{n}^{\prime}}=\mathbf{\bar{n}}-\frac{\mathbf{\bar{n}}\cdot\mathbf{u}}{\mathbf{k}\cdot\
\mathbf{u}}\,\mathbf{k},$

where the denominator, $\mathbf{k}\cdot\mathbf{u}$, is minus the energy of the
photon as measured in the fluid frame, and as such, non zero, so that these
expressions are well defined. It is easy to check that these two vectors are
spacelike unit vectors, orthogonal to $\mathbf{u}$, to each other, and to

$\mathbf{\bar{K}}=\frac{\mathbf{k}+\left(\mathbf{k}\cdot\mathbf{u}\right)\,\mathbf{u}}{|\mathbf{k}\cdot\mathbf{u}|},$
(33)

the normalized projection of $\mathbf{k}$ orthogonal to $\mathbf{u}$, which
coincides with the unit direction of emission of the photon in the fluid
frame. We thus obtain a well-defined orthonormal direct triad
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$ of
the fluid frame, illustrated in Fig. 3. We note that if we consider a vector
$\mathbf{F}$ in the fluid frame, normal to $\mathbf{\bar{K}}$, then our
definition leads to

$\mathbf{F}\cdot\mathbf{\bar{n}}=\mathbf{F}\cdot\mathbf{\bar{n}^{\prime}},$
(34)

and similarly for $\mathbf{w}$, so that our definition allows to keep
unchanged the angles between such a vector $\mathbf{F}$ and the reference
directions, be they primed or unprimed. This will be important later.

Figure 3: Frames of interest for the polarized ray-tracing problem. The far-
away observer’s rest frame (orthogonal to the observer’s 4-velocity
$\mathbf{u_{0}}$) is shown in green, it is a simplified version of Fig. 1 and
shows the local incidence direction $\mathbf{\bar{K}_{0}}$ (spacelike vector)
of the observed light ray, together with the null 4-vector tangent to the
incident null geodesic $\mathbf{k_{0}}$, the local North and West spacelike
directions $(\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$, such that
$(\mathbf{\bar{K}_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$ is a direct
orthonormal triad, and the polarization vector as measured by the far-away
observer, $\mathbf{\bar{F}_{0}}$. The observed EVPA is labelled EVPA0, lying
between the screen’s North direction and the observed polarization vector
$\mathbf{\bar{F}_{0}}$. Starting from the far-away observer, a null geodesic
is integrated backwards towards the source (red line), until it reaches the
accretion flow (black line) surrounding the black hole (black disk). The fluid
frame (orthogonal to the emitter’s 4-velocity $\mathbf{u}$) is shown in blue.
The null 4-vector tangent to the null geodesic at the emission point is called
$\mathbf{k}$. The magnetic field spacelike 4-vector as measured in the fluid
frame (thus, orthogonal to $\mathbf{u}$) is called $\mathbf{b}$. The
synchrotron-emitting electron’s trajectory is represented by the pale black
helix. The local direction of photon emission, as measured in the fluid frame,
is the spacelike unit vector $\mathbf{\bar{K}}$. The pair of vectors
$(\mathbf{\bar{n}^{\prime}},\mathbf{\bar{w}^{\prime}})$ is related to the pair
$(\mathbf{\bar{n}_{0}},\mathbf{\bar{w}_{0}})$, parallel-propagated along the
null geodesic (see text for details), such that
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$ is a
direct orthonormal triad. The unit polarization vector as measured by the
emitter is called $\mathbf{\bar{F}}$. The radiation field
$\mathbf{E_{\mathrm{rad}}}$ associated with the helical motion of the electron
is shown in pale black, and lies along $\mathbf{\bar{F}}$ for a relativistic
electron (see B for a demonstration). The unit projection of the magnetic
4-vector orthogonal to $\mathbf{\bar{K}}$ is called
$\boldsymbol{\bar{b}_{\perp}}$. Thus
$(\mathbf{\bar{K}},\boldsymbol{\bar{b}_{\perp}},\mathbf{\bar{F}})$ is also a
direct orthonormal triad, rotated with respect to
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$ by
the emission EVPA, labeled EVPAe, lying between the parallel-transported North
direction and the fluid-frame polarization vector $\mathbf{\bar{F}}$.

Let us now consider the magnetic field 4-vector $\mathbf{b}$ of the accretion
flow, as measured in the fluid frame. By construction, this vector lies in the
local rest frame of the fluid, so it is orthogonal to $\mathbf{u}$. We are
also interested in its normalized projection orthogonally to
$\mathbf{\bar{K}}$, which reads

$\boldsymbol{\bar{b}_{\perp}}=\frac{\mathbf{b}-\left(\mathbf{b}\cdot\mathbf{\bar{K}}\right)\mathbf{\bar{K}}}{||\mathbf{b}-\left(\mathbf{b}\cdot\mathbf{\bar{K}}\right)\mathbf{\bar{K}}||},$
(35)

(note that the minus sign in the numerator and denominator of the rhs,
compared to the plus sign in the numerator of the rhs of Eq. 33, comes from
the fact that $\mathbf{\bar{K}}$ is spacelike while $\mathbf{u}$ is timelike),
and in the unit polarization vector as measured in the fluid frame

$\mathbf{\bar{F}}=\frac{\mathbf{\bar{K}}\times\mathbf{b}}{||\mathbf{\bar{K}}\times\mathbf{b}||}=\mathbf{\bar{K}}\times\boldsymbol{\bar{b}_{\perp}}.$
(36)

We thus have constructed a second orthonormal direct triad of the fluid rest
space, $(\mathbf{\bar{K}},\boldsymbol{\bar{b}_{\perp}},\mathbf{\bar{F}})$. We
note that it is not obvious that the vector defined by Eq. 36 coincides with
the emission polarization vector for synchrotron radiation, that is, with the
direction of the radiation electric field emitted by an electron moving around
the $\mathbf{b}$ field lines, given that we have never discussed the emitting
electron motion so far. In B, by relating the particle frame and the fluid
frame, we demonstrate that, provided the emitting electron is relativistic,
this is indeed so.

We have thus at hand two orthonormal triads of the fluid frame, the observer-
related
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$, and
the magnetic-field-related
$(\mathbf{\bar{K}},\boldsymbol{\bar{b}_{\perp}},\mathbf{\bar{F}})$. These two
frames are rotated with respect to each other by the angle

$\chi\equiv(\mathbf{\bar{n}^{\prime}},\mathbf{\bar{F}})=(\mathbf{\bar{n}},\mathbf{\bar{F}})\equiv\mathrm{EVPA_{e}},$
(37)

the EVPA in the fluid frame, where the index e reminds that we are dealing
with an emission EVPA, as compared to the observed EVPA of Fig. 1. Both angles
are illustrated in Fig. 3. We note that the emission EVPA evolves as the light
ray evolves through the emitting fluid; the EVPA is only conserved in vacuum
as demonstrated above. So a sequence of emission EVPAs corresponds to a unique
observed EVPA. Note that the second equality in Eq. 37 is a consequence of Eq.
34. This emission EVPA will be crucial in the polarized radiative transfer
formalism that we introduce in the next section. A practical, code-friendly
expression for the emission EVPA is the following

$\mathrm{EVPA_{e}}=\frac{\pi}{2}-\mathrm{atan2}\left(\mathbf{\bar{b}_{\perp}}\
\cdot\mathbf{\bar{w}^{\prime}},\mathbf{\bar{b}_{\perp}}\cdot\mathbf{\bar{n}^{\prime}}\right).$
(38)

After having discussed the parallel transport of the vectors of interest along
the null geodesic, the last step of the polarized GRRT problem is to integrate
the polarized radiative transfer within the accretion flow surrounding the
compact object.

###  2.4 Polarized radiative transfer

####  2.4.1 Stokes parameters

The most general monochromatic electromagnetic wave has an elliptical
polarization, in the sense that the electric field vector describing the wave
draws an ellipse during its time evolution in the plane normal to the
direction of propagation, see Fig. 4.

Figure 4:  Polarization ellipse of a general monochromatic wave (in blue). The
black axes $(x,y)$ label the frame of interest where we want to formulate the
problem, while the blue axes $(x^{\prime},y^{\prime})$ are along the major and
minor axes of the ellipse and are therefore naturally adapted to the
elliptically polarized wave. The angle between the two bases is called $\chi$
(it would coincide with the notion of EVPA for a linearly polarized wave along
the $x$ axis), and the quantity $\tan\beta$ encodes the ellipse axes ratio.
The dashed black axes $(a,b)$ are tilted by $45^{\circ}$ relative to $(x,y)$
and are useful for defining the U Stokes parameter.

Let us introduce the electric field complex vector of the monochromatic wave

$\mathbf{\hat{E}}=\hat{E}_{x}\mathbf{\bar{e}_{x}}+\hat{E}_{y}\mathbf{\bar{e}_{y}},$
(39)

decomposed in an arbitrary orthonormal basis
$(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$ of the plane orthogonal to the
direction of propagation, where $\hat{E}_{x}$ and $\hat{E}_{y}$ are the
complex components along the axes. The polarization ellipse described by the
vector $\mathbf{\hat{E}}$ can be equivalently described by two sets of
parameters. From a geometrical point of view, it is very natural to provide
the total intensity $|\mathbf{\hat{E}}|^{2}$, together with two angles $\chi$
and $\beta$. The angle $\chi$ lies between the basis
$(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$ and the basis corresponding to
the axes of the ellipse, while $\tan\beta$ encodes the ellipse axes ratio (see
Fig. 4). However, this parametrization is not practical from a physical point
of view given that the two angles are not directly observable. So from a
physical point of view, it is more natural to consider the following set of
four Stokes parameters (Rybicki and Lightman, 1979)

$\displaystyle I$ $\displaystyle=$
$\displaystyle|\hat{E}_{x}|^{2}+|\hat{E}_{y}|^{2},$ (40) $\displaystyle Q$
$\displaystyle=$ $\displaystyle|\hat{E}_{x}|^{2}-|\hat{E}_{y}|^{2}=I\,\cos
2\chi\,\cos 2\beta,$ $\displaystyle U$ $\displaystyle=$
$\displaystyle|\hat{E}_{a}|^{2}-|\hat{E}_{b}|^{2}=I\,\sin 2\chi\,\cos 2\beta,$
$\displaystyle V$ $\displaystyle=$
$\displaystyle|\hat{E}_{r}|^{2}-|\hat{E}_{l}|^{2}=I\,\sin 2\beta,$

where $\hat{E}_{a}$ and $\hat{E}_{b}$ are the complex components of the
electric field in an orthonormal basis
$(\mathbf{\bar{e}_{a}},\mathbf{\bar{e}_{b}})$ rotated by $45^{\circ}$ compared
to $(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$, see the dashed black axes in
Fig. 4. The quantities $\hat{E}_{r}$ and $\hat{E}_{l}$ are the complex
components of the field in an orthonormal complex basis,
$\mathbf{\bar{e}_{l,r}}=\sqrt{2}/2(\mathbf{\bar{e}_{x}}\pm
i\mathbf{\bar{e}_{y}})$. These relations clearly show that the Stokes
parameters are all sums or differences of intensities along specific
directions, and as such are directly observable and well adapted to being
evolved in a radiative transfer problem. Equations 40 show how to construct
the Stokes parameters from the geometrical angular parameters $\chi$, $\beta$
of the polarization ellipse. The reverse expression is easy to find and reads

$\displaystyle\tan 2\chi$ $\displaystyle=$ $\displaystyle\frac{U}{Q},$ (41)
$\displaystyle\sin 2\beta$ $\displaystyle=$ $\displaystyle\frac{V}{I}.$

For a circular polarization, $\beta=\pi/4$, so

$Q=0,\quad U=0,\quad V=I\quad(\mathrm{circular\>polarization}),$ (42)

while for a linear polarization, $\beta=0$, and

$Q=I\cos 2\chi,\quad U=I\sin 2\chi,\quad
V=0\quad(\mathrm{linear\>polarization}),$ (43)

and if the wave is polarized along the $x$ axis of Fig. 4, then $Q=I$ and
$U=0$, while if the wave is polarized at $45^{\circ}$ from the $x$ axis, then
$Q=0$ and $U=I$. So $Q$ and $U$ encode linear polarization along the
directions $\mathbf{\bar{e}_{x}}$ or $\mathbf{\bar{e}_{y}}$ and
$\mathbf{\bar{e}_{a}}$ or $\mathbf{\bar{e}_{b}}$, respectively, and $V$
encodes circular polarization.

Let us consider the Stokes parameters $(I,Q,U,V)$ defined in a basis
$(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$, and the parameters
$(I^{\prime},Q^{\prime},U^{\prime},V^{\prime})$ defined in a basis
$(\mathbf{e^{\prime}_{x}},\mathbf{e^{\prime}_{y}})$, rotated by an angle
$\chi$ with respect to $(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$, see Fig.
4. It is easy to show that the $Q$ and $U$ Stokes parameters transform
following

$\left(\begin{array}[]{c}Q\\\
U\end{array}\right)=\left(\begin{array}[]{cc}\cos 2\chi&-\sin 2\chi\\\ \sin
2\chi&\cos 2\chi\end{array}\right)\left(\begin{array}[]{c}Q^{\prime}\\\
U^{\prime}\end{array}\right),$ (44)

while $I$ and $V$ are invariant.

For a monochromatic radiation, there must exist a relation between the four
Stokes parameters, that are equivalent to the set of three parameters
$(I,\chi,\beta)$, and thus cannot be independent. This relation reads

$I^{2}=Q^{2}+U^{2}+V^{2}\quad(\mathrm{monochromatic\>/\>fully\>polarized})$
(45)

and the radiation is then said to be fully polarized. For a superimposition of
waves at different frequencies, the radiation is only partially polarized and
the resulting Stokes parameters verify

$I^{2}\geq Q^{2}+U^{2}+V^{2}\quad(\mathrm{partially\>polarized}).$ (46)

It is then useful to introduce the degree of polarization

$\mathrm{d_{p}}=\frac{\sqrt{Q^{2}+U^{2}+V^{2}}}{I},$ (47)

and the degree of linear polarization

$\mathrm{d_{lp}}=\frac{\sqrt{Q^{2}+U^{2}}}{I}.$ (48)

####  2.4.2 Stokes parameters for synchrotron radiation, conventions

We are primarily interested in polarized synchrotron radiation given that our
main science interest is the millimeter and infrared radiation emitted by
nearby supermassive black hole environments. Let us consider a single electron
following a helical motion around the field lines of a magnetic field
$\mathbf{b}$ as measured in the fluid frame. The emitted synchrotron radiation
is elliptically polarized, with the minor axis of the polarization ellipse
aligned along the direction of the magnetic field projected orthogonally to
the direction of propagation, and major axis along the fluid-frame
polarization vector (Huang et al., 2009). The Stokes parameters are thus
naturally expressed in a basis aligned with the axes of this polarization
ellipse, that is, the $(\mathbf{\bar{F}},-\mathbf{\bar{b}_{\perp}})$ basis
(see the illustration in Fig. 5). We call this basis the synchrotron
polarization basis. This basis is rotated by the emission EVPA with respect to
the observer-related $(\mathbf{\bar{n}^{\prime}},-\mathbf{\bar{w}^{\prime}})$
basis. This last basis is called the parallel-transported polarization basis.
For integrating the radiative transfer in the observer’s frame, we will need
to take care of this rotation between the synchrotron and the parallel-
transported polarization bases. This is described in the next section.

Our sign conventions for the Stokes parameters are illustrated in Fig. 5. It
complies with the convention of the International Astronomical Union
(Transactions of the International Astronomical Union, 1973; Hamaker and
Bregman, 1996, see Fig. 1 of the second reference).

Figure 5: Geometry of the polarized synchrotron problem and Stokes parameter
illustration. All quantities depicted here are defined in the fluid frame. The
magnetic field is $\mathbf{b}$, which lies along the $\mathbf{\bar{e}_{z}}$
unit vector. The direction of emission in this frame is $\mathbf{\bar{K}}$,
which lies along the $\mathbf{\bar{e}_{\textfrak}{c}}$ unit vector. We define
the $\mathbf{\bar{e}_{\textfrak}{a}}$ unit vector as lying along the major
axis of the synchrotron polarization ellipse (shown in dashed pale blue). This
vector is defined up to an unimportant sign convention. The
$\mathbf{\bar{e}_{\textfrak}{b}}$ unit vector is such that
$(\mathbf{\bar{e}_{\textfrak}{a}},\mathbf{\bar{e}_{\textfrak}{b}},\mathbf{\bar{e}_{\textfrak}{c}})$
is a direct orthonormal triad of the fluid frame. The vector
$\mathbf{\bar{e}_{x}}$ is parallel to $\mathbf{\bar{e}_{\textfrak}{a}}$. The
$\mathbf{\bar{e}_{y}}$ unit vector is such that
$(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}},\mathbf{\bar{e}_{z}})$ is a direct
orthonormal triad of the fluid frame. The angle between $\mathbf{b}$ and
$\mathbf{\bar{K}}$ is called $\theta_{B}$. The Stokes parameters Q and U are
defined in the
$(\mathbf{\bar{e}_{\textfrak}{a}}=\mathbf{F},\mathbf{\bar{e}_{\textfrak}{b}}=-\mathbf{\bar{b}_{\perp}})$
basis, that we call the synchrotron polarization basis, illustrated by the
zoom on the right of the figure. This zoom shows the synchrotron polarization
basis (in blue; subscript ’synch’) as well as the parallel-transported
polarization basis (in green; subscript ’$\parallel$trans’),
$(\mathbf{\tilde{e}_{\textfrak}{a}}=\mathbf{\bar{n}},\mathbf{\tilde{e}_{\textfrak}{b}}=-\mathbf{\bar{w}})$.
These two bases are rotated by the emission EVPA angle. The sign conventions
of the Stokes parameters are as shown in this zoom. Note that the orientation
convention used in this figure is the same as that used by e.g. Dexter (2016);
Huang and Shcherbakov (2011), which results in a positive emission coefficient
for Stokes Q. Some authors use an alternative orientation convention, taking
$\mathbf{\bar{e}_{\textfrak}{a}}$ along the minor axis of the polarization
ellipse, see Shcherbakov (2008); Marszewski et al. (2021). This simply leads
to Stokes Q being multiplied by $-1$, and to a negative emission coefficient
for Stokes Q.

####  2.4.3 Transfer equation

Just like in the unpolarized version of Gyoto (Vincent et al., 2011), we will
integrate the radiative transfer equation in the fluid frame, and then
transform the quantities to the observer’s frame. The unpolarized radiative
transfer equation used by Gyoto reads

$\frac{\mathrm{d}I^{\mathrm{em}}_{\nu}}{\mathrm{d}s}=j_{\nu}-\alpha_{\nu}\,I^{\mathrm{em}}_{\nu},$
(49)

where $I^{\mathrm{em}}_{\nu}$ is the specific intensity (the index $\nu$ means
that we are considering an intensity per unit of frequency), $\mathrm{d}s$ is
the element of optical path length, $j_{\nu}$ and $\alpha_{\nu}$ are the
specific emission and absorption coefficients, all these quantities being
measured in the fluid frame (hence the superscript ’em’ for the intensity,
referring to the emitting fluid; we discard it for the other quantities for
simplicity). The intensity in the observer frame (superscript ’obs’) then
follows from

$I^{\mathrm{obs}}_{\nu}=I^{\mathrm{em}}_{\nu}\,\left(\frac{\nu^{\mathrm{obs}}}{\nu^{\mathrm{em}}}\right)^{3},$
(50)

which is a consequence of Liouville’s theorem (Misner et al., 1973).

The polarized radiative transfer equation is naturally written in the
synchrotron polarization basis of the fluid frame. In this basis, the transfer
equation reads

$\frac{\mathrm{d}}{\mathrm{d}s}\left(\begin{array}[]{c}I_{\nu}^{\mathrm{em;synch}}\\\
Q_{\nu}^{\mathrm{em;synch}}\\\ U_{\nu}^{\mathrm{em;synch}}\\\
V_{\nu}^{\mathrm{em;synch}}\end{array}\right)=\left(\begin{array}[]{c}j_{\nu,I}\\\
j_{\nu,Q}\\\ j_{\nu,U}\\\
j_{\nu,V}\end{array}\right)-\left(\begin{array}[]{cccc}\alpha_{\nu,I}&\alpha_{\nu,Q}&\alpha_{\nu,U}&\alpha_{\nu,V}\\\
\alpha_{\nu,Q}&\alpha_{\nu,I}&r_{\nu,V}&-r_{\nu,U}\\\
\alpha_{\nu,U}&-r_{\nu,V}&\alpha_{\nu,I}&r_{\nu,Q}\\\
\alpha_{\nu,V}&r_{\nu,U}&-r_{\nu,Q}&\alpha_{\nu,I}\end{array}\right)\left(\begin{array}[]{c}I_{\nu}^{\mathrm{em;synch}}\\\
Q_{\nu}^{\mathrm{em;synch}}\\\ U_{\nu}^{\mathrm{em;synch}}\\\
V_{\nu}^{\mathrm{em;synch}}\end{array}\right),$ (51)

where the ’em;synch’ label is here to remind that we are dealing with Stokes
parameters expressed in the synchrotron polarization basis of the emitting
fluid frame; moreover, $j_{\nu,X}$ and $\alpha_{\nu,X}$ are emission and
absorption coefficients for the Stokes parameter X, $r_{\nu,Q}$ and
$r_{\nu,U}$ are Faraday conversion parameters, and $r_{\nu,V}$ is the Faraday
rotation parameter. All these transfer coefficients are defined in the
synchrotron polarization basis of the fluid frame (we discard the superscript
for simplicity). We refer to Fig. 5 for the details of the sign conventions.

However, we rather want to integrate this equation in the parallel-transported
polarization basis, $(\mathbf{\bar{n}^{\prime}},-\mathbf{\bar{w}^{\prime}})$,
which is rotated by the emission EVPA with respect to the synchrotron
polarization basis, see Fig. 5. In the parallel-transported polarization basis
of the fluid frame, the transfer equation reads

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\left(\begin{array}[]{c}I_{\nu}^{\mathrm{em;\parallel
trans}}\\\ Q_{\nu}^{\mathrm{em;\parallel trans}}\\\
U_{\nu}^{\mathrm{em;\parallel trans}}\\\ V_{\nu}^{\mathrm{em;\parallel
trans}}\end{array}\right)$ $\displaystyle=$
$\displaystyle\mathbf{R}\left(\chi_{\mathrm{e}}\right)\left(\begin{array}[]{c}j_{\nu,I}\\\
j_{\nu,Q}\\\ j_{\nu,U}\\\ j_{\nu,V}\end{array}\right)$ (69)
$\displaystyle-\mathbf{R}\left(\chi_{\mathrm{e}}\right)\left(\begin{array}[]{cccc}\alpha_{\nu,I}&\alpha_{\nu,Q}&\alpha_{\nu,U}&\alpha_{\nu,V}\\\
\alpha_{\nu,Q}&\alpha_{\nu,I}&r_{\nu,V}&-r_{\nu,U}\\\
\alpha_{\nu,U}&-r_{\nu,V}&\alpha_{\nu,I}&r_{\nu,Q}\\\
\alpha_{\nu,V}&r_{\nu,U}&-r_{\nu,Q}&\alpha_{\nu,I}\end{array}\right)\mathbf{R}\left(-\chi_{\mathrm{e}}\right)\left(\begin{array}[]{c}I_{\nu}^{\mathrm{em;\parallel
trans}}\\\ Q_{\nu}^{\mathrm{em;\parallel trans}}\\\
U_{\nu}^{\mathrm{em;\parallel trans}}\\\ V_{\nu}^{\mathrm{em;\parallel
trans}}\end{array}\right),$

where the superscript ’em;$\parallel$trans’ reminds that we are dealing with
Stokes parameters defined in the parallel-transported polarization basis of
the emitting fluid frame, and

$\mathbf{R}\left(\chi_{\mathrm{e}}\right)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\cos 2\chi_{\mathrm{e}}&-\sin 2\chi_{\mathrm{e}}&0\\\ 0&\sin
2\chi_{\mathrm{e}}&\cos 2\chi_{\mathrm{e}}&0\\\ 0&0&0&1\end{array}\right)$
(70)

is a rotation matrix describing the rotation by the angle
$\chi_{\mathrm{e}}\equiv EVPA_{\mathrm{e}}$, the emission EVPA, between the
synchrotron and the parallel-transported polarization bases. This is the exact
same transformation as that described by Eq. 44.

Solving Eq. 69 is a well-known problem that we briefly discuss in C. The
corresponding Stokes parameters in the observer frame then follow from

$X^{\mathrm{obs}}_{\nu}=X^{\mathrm{em;\parallel
trans}}_{\nu}\,\left(\frac{\nu^{\mathrm{obs}}}{\nu^{\mathrm{em}}}\right)^{3},$
(71)

similarly as in Eq. 50, where $X$ is either of the Stokes parameters.

####  2.4.4 Polarized synchrotron coefficients

We now need to express the synchrotron coefficients in the synchrotron
polarization basis. In this basis, the transfer coefficients for the Stokes
parameter U, i.e. $j_{\nu,U}$, $\alpha_{\nu,U}$ and $r_{\nu,U}$ are zero by
definition. However, the computation of emission, absorption and rotation
synchrotron coefficients for the others Stokes parameters, from an arbitrary
distribution of electrons could be quite heavy. Indeed, even for isotropic
distributions, the computation of the emission coefficients require a double
integral (Rybicki and Lightman, 1979) and the others are even more complex
using the susceptibility tensor (see Appendix B of Marszewski et al., 2021).
Fortunately, Marszewski et al. (2021); Dexter (2016); Huang and Shcherbakov
(2011) derived fitted formulae for the emissivities, absorptivities and
rotativities for well-defined isotropic distributions of electrons : Thermal
(Maxwell-Jüttner), Power Law or Kappa (thermal core with a power law tail). We
choose to implement the formulae of Marszewski et al. (2021) in Gyoto to
compute the synchrotron coefficients as they are the only one who provides
formulae for a kappa distribution. These formulae are valid for a specific
range of parameters.

For a thermal distribution, parametrized by the dimensionless temperature
$\Theta_{e}=k_{B}T/m_{e}c^{2}$, the fits are accurate for $3<\Theta_{e}<40$
and for $\nu/\nu_{c}\gg 1$ with $\nu_{c}=eB/(2\pi m_{e}c)$ the cyclotron
frequencies (Marszewski et al., 2021).

For a Power Law distribution, parametrized by a minimum and maximum Lorentz
factor, $\gamma_{\mathrm{min}}$ and $\gamma_{\mathrm{max}}$ respectively, and
by a power law index $p$, the fits are accurate for
$\gamma_{\mathrm{min}}<10^{2}$, $1.5<p<6.5$ and, as before, for
$\nu/\nu_{c}\gg 1$ (Marszewski et al., 2021).

The Kappa distribution is characterized by two parameters $w$ (equivalent to
the dimensionless temperature) and $\kappa=p+1$. Contrary to the other
distributions where the fits are continuous functions of the parameters, the
fits for rotation coefficients for the Kappa distribution have been done for
four specific values of $\kappa=(3.5,4.0,4.5,5.0)$ and are not defined for any
other value. The fits are valid while $3<w<40$, $\nu/\nu_{c}\gg 1$ and
$X_{\kappa}\gg 10^{-1}$ where $X_{\kappa}=\nu/\nu_{c}(w\kappa)^{2}\sin\theta$
with $\theta$ the angle between the magnetic field vector and the photon
tangent vector (Marszewski et al., 2021).

For some tests in section 3, we will use the formulae from Dexter (2016),
especially for the comparison with the ray-tracing code ipole which use the
formulae of Dexter (2016) (as the code grtrans). The order of the maximum
relative error between all the fits in Marszewski et al. (2021) and the true
values is of 30%. For the typical parameters of the accretion flow of Sgr A*,
the difference between the formulae of Marszewski et al. (2021) and Dexter
(2016) is lower or equal to $10\%$.

##  3 Tests

###  3.1 Test of the parallel transport

The first test that we have to make is to check that the observer polarization
basis’ vectors, i.e. $\mathbf{\bar{n}}$ and $\mathbf{\bar{w}}$, are well
parallel transported along the null geodesics. The parallel transport equation
given by Eq. 31 is fully general and agnostic about the particular spacetime
considered. However, in the special case of the Kerr spacetime, it is well
known that the special algebraic kind of the spacetime allows the existence of
the Walker-Penrose constant (Walker and Penrose, 1970), defined as follows. If
$\mathbf{k}$ is the tangent vector to a null geodesic, and if $\mathbf{f}$ is
a vector orthogonal to $\mathbf{k}$ and parallel-transported along
$\mathbf{k}$, then the following complex quantity

$\displaystyle K_{1}-iK_{2}=$
$\displaystyle(r-ia\cos\theta)\bigg{[}(k^{t}f^{r}-k^{r}f^{t})+a\sin^{2}\theta(k^{r}f^{\varphi}-k^{\varphi}f^{r})\bigg{.}$
$\displaystyle\bigg{.}-i\sin\theta\left\\{(r^{2}+a^{2})(k^{\varphi}f^{\theta}-k^{\theta}f^{\varphi})-a(k^{t}f^{\theta}-k^{\theta}f^{t})\right\\}\bigg{]},$

here expressed in Boyer-Lindquist coordinates, is conserved along the null
geodesic. Thus, knowing the evolution of $\mathbf{k}$ along the null geodesic,
the vectors $\mathbf{\bar{n}}$ and $\mathbf{\bar{w}}$ (that obviously fulfill
the conditions on $\mathbf{f}$ in Eq. 3.1) can be immediately obtained without
further computation by using this constant. This Kerr-specific result is only
used for testing purposes in Gyoto. The code is agnostic about the spacetime
and does not use this property.

We thus check the conservation of $K_{1}$ and $K_{2}$ for specific geodesics
and obtain a conservation to within $10^{-5}$ for default integration
parameters of Gyoto. This can be improved by setting a lower tolerance value
for the integration steps, but at the cost of a longer calculation time.

###  3.2 EVPA calculation test

Parallel transport having been tested, the observer polarization basis is well
defined in the rest frame of the emitter
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$,
through Eq. (32). As said in section 2.3, the natural basis to express the
synchrotron coefficients is
$(\mathbf{\bar{K}},\boldsymbol{\bar{b}_{\perp}},\mathbf{\bar{F}})$ with
$\mathbf{\bar{K}}$ a common vector between the two bases. Thus, to express the
radiative transfer in the observer-related basis, rather than in the
synchrotron basis, we just need to apply the rotation matrix defined in Eq.
(70). The angle between these two bases corresponds to the emission EVPA (see
section 2.3).

We define a simple setup to check the computation of this crucial angle. We
consider a Page-Thorne disk (geometrically thin, optically thick; Page and
Thorne, 1974) in a Minkowski metric (to avoid GR effects), seen face-on. We
consider two magnetic field configurations $\mathbf{b}$, toroidal and radial.
Expressed in Boyer-Lindquist coordinates, the two magnetic field
configurations read, in the rest frame of the emitter

$\mathrm{\textbf{Toroidal:}}\>b^{\alpha}=\left\\{\begin{array}[]{l}b^{t}=\sqrt{\frac{g_{\varphi\varphi}}{g_{tt}}\frac{\Omega^{2}}{g_{tt}+g_{\varphi\varphi}\Omega^{2}}},\\\
b^{r}=0,\\\ b^{\theta}=0,\\\
b^{\varphi}=\sqrt{\frac{g_{tt}}{g_{\varphi\varphi}}\frac{1}{g_{tt}+g_{\varphi\varphi}\Omega^{2}}},\end{array}\right.$
(73)

where $\Omega=u^{\varphi}/u^{t}$ and $\mathbf{u}$ is the 4-velocity of the
emitting fluid assumed to be Keplerian, and

$\mathrm{\textbf{Radial: }}\>b^{\alpha}=\left\\{\begin{array}[]{l}b^{t}=0,\\\
b^{r}=\sqrt{\frac{1}{g_{rr}}},\\\ b^{\theta}=0,\\\
b^{\varphi}=0.\end{array}\right.$ (74)

In the toroidal case, for all azimuthal angles of the disk, the wave-vector
$\mathbf{\bar{K}}$ is made of two components, one almost vertical (face-on
view), and one azimuthal component resulting from special-relativistic
aberration (Vincent et al., 2023). The magnetic field $\mathbf{b}$ is in the
toroidal direction. Thus, the resulting polarization vector
$\mathbf{\bar{F}}=\mathbf{\bar{K}}\times\boldsymbol{\bar{b}_{\perp}}$ is in
the radial direction. Similarly, for the radial magnetic field, the resulting
polarization vector is in the toroidal direction.

We consider an arbitrary $I_{\nu}=1$ emission for unpolarized intensity. Gyoto
computes images for the four Stokes parameters from which we can compute the
orientation of the polarization vector, i.e. the EVPA. As we are only
interested by the computation of the EVPA for the moment, i.e. without
radiative transfer, we assume a fully linearly polarized radiation by taking
$Q_{\nu}=I_{\nu}\,\cos\left(2*EVPA\right)$ and
$U_{\nu}=I_{\nu}\,\sin\left(2*EVPA\right)$. This means that we do not take
into account any absorption nor Faraday rotation.

Figure 6: Image of total (unpolarized) intensity of a Page-Thorne disk with
Keplerian orbit in a Minkowski space-time seen face-on for two magnetic field
configurations (toroidal left, radial right). The inner radius is at
$r=3\,r_{g}$. The green lines represent the orientation of polarization
vectors.

We show in the left panel of Fig. 6 in background the unpolarized images of
the setup described above and the polarization vector with the green ticks.
The position angle of these vectors is the observed EVPA. As expected, the
polarization vectors are radial for a toroidal magnetic field and toroidal for
radial magnetic field.

We now want to make a similar test in curved spacetime using a Kerr metric. To
validate the computation of EVPA in Gyoto, we compute the polarization from a
geometrically thin ring as in Gelles et al. (2021) taking as previously
$I_{\nu}=1$, $Q_{\nu}=I_{\nu}\,\cos\left(2*EVPA\right)$ and
$U_{\nu}=I_{\nu}\,\sin\left(2*EVPA\right)$. Gelles et al. (2021) consider a
synchrotron emission that we discard here, our only interest being in testing
the EVPA. Here, we are only interested in the orientation of the polarization
vector, that is, in the EVPA, at $r_{1}=3\,r_{g}$ and $r_{2}=6\,r_{g}$ (the
inner and outter radius of the ring). Figure 7 shows the tick plots for three
magnetic field configurations : radial, toroidal and vertical with the same
setup as in Fig. 1 of Gelles et al. (2021), the fluid being assumed to be
comoving with the Zero Angular Momentum Observer frame. The vertical
configuration implemented in Gyoto reads

$\mathrm{\textbf{Vertical}}\>b^{\alpha}=\left\\{\begin{array}[]{l}b^{t}=0,\\\
b^{r}=\frac{\cos{\theta}}{\sqrt{g_{rr}}},\\\
b^{\theta}=\frac{\sin{\theta}}{\sqrt{g_{\theta\theta}}},\\\
b^{\varphi}=0.\end{array}\right.$ (75)

The results of Gyoto shown in Fig. 7 are in perfect agreement with the ones in
the Fig. 1 in Gelles et al. (2021). We note that Gelles et al. (2021) scale
the length of the ticks by the observed synchrotron intensity, while our ticks
are all of unit length: we are only interested in the EVPA. This confirms that
the calculation of the EVPA works correctly and we can now test the radiative
transfer part and compare the results with another ray-tracing code.

Figure 7: Polarized tick plots for three idealized magnetic field
configurations: radial (left), toroidal (middle), and vertical (right) from a
geometrically thin ring seen by an almost face-on observer $i=0.1^{\circ}$.
The fluid is comoving with the Zero Angular Momentum Observer frame. Each plot
shows two spins ($a=0$ and $a=-0.99$ in red and blue, respectively) as two
emission radii ($r_{1}=3\,r_{g}$ and $r_{2}=6\,r_{g}$, corresponding to the
inner and outer rings, respectively).

###  3.3 Comparison with ipole

To check that all parts of our code work correctly, we compare the results of
Gyoto with the ones of another polarized ray-tracing code : ipole
(Mościbrodzka and Gammie, 2018b). We will focus on polarized observables of a
thick disk around a Schwarzschild black hole. We take power law profiles of
the physical quantities following Vos et al. (2022). To compute the emission,
absorption and rotation coefficients for the radiative transfer, we assume a
thermal distribution of electrons (as in Vos et al., 2022), and, for this
comparison only, we use the fitting formula of Dexter (2016) as used in Ipole
(we remind that Gyoto implements the formulae from Marszewski et al., 2021).

We compared the three magnetic field configurations (toroidal, radial and
vertical) described in Vos et al. (2022) at two inclinations, close to face-on
with $i=20$° and close to edge-on with $i=80$°. We define, as in Prather et
al. (2023), the normalized mean squared error (NMSE) as

$NMSE(A,B)=\frac{\sum|A_{j}-B_{j}|^{2}}{\sum|A_{j}|^{2}}$ (76)

where $A_{j}$ and $B_{j}$ are the intensities of a particular Stokes parameter
in two images at pixel $j$. The results of Gyoto are in perfect agreement with
Ipole with a NMSE $<10^{-4}$ for all configurations and Stokes parameters
except for Stoke U in the radial cases at high inclination for which the NMSE
is around $10^{-3}$. This can be compared to the worst NMSE of $\sim 0.01$
obtained in the code comparison made in Prather et al. (2023) showing the
perfect agreement between Gyoto and Ipole.

We also performed pixel-to-pixel comparisons, not restricting our comparison
to integrated quantities like the NMSE. Fig. 8 illustrates this for 128x128
pixels images of the four Stokes parameters computed by Gyoto and their
relative difference with Ipole, in a field of view of $40\,r_{g}$, for the
three magnetic field configurations described above, at low inclination
$i=20^{\circ}$. The relative error map is very close to zero for the vast
majority of pixels, with typical values $\lesssim 0.1\%$. Higher values are
only reached on specific tracks, that correspond to the zeroes of the
corresponding Stokes parameters (as is clear by comparing with the panels
showing the maps of the corresponding Stokes parameters). It is thus not
surprising to get higher residuals there, and it does not affect the radiative
transfer, given that the Stokes parameters are anyway close to zero in these
regions. Besides these tracks corresponding to the zeroes of the Stokes
parameters, the interior of the “shadow” region (i.e. geodesics that
asymptotically approach the horizon when backward ray traced) lead to higher
error. This is due to the stop condition of the geodesic integration that
differs in the two codes. Given that this part of the image is anyway strongly
redshifted and leads to a very low flux, this has no impact on the field-of-
view integrated comparison of the NMSE.

Figure 8: The first, third and fifth lines show images from left to right of
the four Stokes parameters (I, Q, U, V) generated by Gyoto and $Q^{2}+U^{2}$
as the last column for the three magnetic field configuration at low
inclination ($i=20$°). Their relative error maps with ipole images are shown
in the second, fourth and sixth lines. The relative errors scale is linear
between $\pm 5\%$.

##  4 Conclusion

This article presents the polarized version of the ray-tracing code Gyoto.
After reviewing the formalism for polarized GRRT, our main aim is to explain
the details of our implementation, and to provide tests of our code. In
particular, we have shown that in the framework of the GRRT computation of a
geometrically thick, optically thin accretion flow, we find results in perfect
agreement with the ipole code.

Polarized GRRT is of fundamental importance for interpreting current and
future observed data, in particular that of GRAVITY, the EHT, the polarized
loops of ALMA associated with Sgr A* flares, or the data of IXPE. Properly
interpreting these data is key to better understanding the properties of
plasmas in the extreme environments of black holes, and might offer new
interesting probes of strong-field gravity.

The polarized Gyoto code is public, actively maintained, and in constant
development for offering ever more diversed setups for relativistic
astrophysics. The recent polarized version of the code is accompanied by a
python notebook, available at
https://github.com/gyoto/Gyoto/blob/master/doc/examples/Gyoto_Polar_example.ipynb,
offering a quick and user-friendly first example of the new environment.

##  Appendix A Observer’s screen polarization basis

Figure 9: Local observer’s basis
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}},\mathbf{\bar{e}_{3}})$, in green,
as represented in Fig. 2. The polarization basis corresponding to the central
pixel of the screen, that is, to a purely radial direction of incidence, is
$(\mathbf{\bar{K}_{0}^{\mathrm{cen}}},\mathbf{\bar{w}_{0}^{\mathrm{cen}}},\mathbf{\bar{n}_{0}^{\mathrm{cen}}})$,
and is aligned with the oberver’s local basis vectors. When considering a non-
central pixel, the direction of photon incidence is no longer purely radial.
The corresponding vector $\mathbf{\bar{K}_{0}}$ is obtained from
$\mathbf{\bar{K}_{0}^{\mathrm{cen}}}$ by a rotation of angle $a$ around the
vector $\mathbf{v}$ (in red) normal to the red plane that contains
$\mathbf{\bar{e}_{3}}$ and makes an angle $b$ with $\mathbf{\bar{e}_{1}}$. The
same applies for $\mathbf{\bar{w}_{0}}$ and $\mathbf{\bar{n}_{0}}$.

A general vector $\mathbf{\bar{K}_{0}}$, defined by the two spherical angles
$(a,b)$, see the right panel of Fig. 2, is obtained by rotating the vector
$\mathbf{\bar{K}_{0}^{\mathrm{cen}}}$ by an angle $a$ in a plane containing
$\mathbf{\bar{e}_{3}}$ and making an angle $b$ with the $\mathbf{\bar{e}_{1}}$
vector (see the red plane in Fig. 9). This corresponds to a rotation of angle
$a$ around the vector $\mathbf{v}=-\sin b\,\mathbf{\bar{e}_{1}}+\cos
b\,\mathbf{\bar{e}_{2}}$. The corresponding rotation matrix reads

$R(a,b)=\left(\begin{array}[]{cccc}\sin^{2}b\left(1-\cos a\right)+\cos a&-\sin
b\cos b\left(1-\cos a\right)&\cos b\sin a\\\ -\sin b\cos b\left(1-\cos
a\right)&\cos^{2}b\left(1-\cos a\right)+\cos a&\sin b\cos a\\\ -\cos b\sin
a&-\sin b\sin a&\cos a\end{array}\right).$ (77)

The general vector $\mathbf{\bar{K}_{0}}$ then reads
$\mathbf{\bar{K}_{0}}=R(a,b)\,\mathbf{\bar{K}_{0}^{\mathrm{cen}}}$, and
similarly for the two other polarization vectors. It is then easy to express
the general local unit West and North directions that complete the local
incoming photon direction given by Eq. 2.2. They read

$\displaystyle\mathbf{\bar{w}_{0}}$ $\displaystyle=$
$\displaystyle\left[-\sin^{2}b\left(1-\cos a\right)-\cos
a\right]\mathbf{\bar{e}_{1}}+\sin b\cos b\left(1-\cos
a\right)\mathbf{\bar{e}_{2}}$ $\displaystyle+\cos b\sin
a\,\mathbf{\bar{e}_{3}},$ $\displaystyle\mathbf{\bar{n}_{0}}$ $\displaystyle=$
$\displaystyle-\sin b\cos b\left(1-\cos
a\right)\mathbf{\bar{e}_{1}}+\left[\cos^{2}b\left(1-\cos a\right)+\cos
a\right]\mathbf{\bar{e}_{2}}$ $\displaystyle-\sin b\sin
a\,\mathbf{\bar{e}_{3}}.$

##  Appendix B Electron gyration and polarization vector direction

Our discussion of the polarization vector in section 2.3 took place in the
fluid frame. Here, we need to discuss a third natural frame (after the
observer frame and the fluid frame) naturally associated with the GRRT
problem, namely the particle frame, the frame comoving with an individual
electron swirling around the magnetic field lines and emitting synchrotron
radiation as a consequence of this accelerated motion (see Fig. 3). The
description of this infinitesimal 666Infinitesimal as compared to the natural
scale of our problem coinciding with the gravitational radius. motion is the
topic of this appendix. Our goal is to express the radiation electric field of
an individual electron, as measured in the fluid frame. The direction of this
vector should coincide with that of the polarization vector, which was
introduced in Eq. 36 without any reference to the electron’s motion, nor to
its radiation field.

Let us consider the standard picture of synchrotron emission by a relativistic
electron illustrated in Fig. 10, using the same notation and following the
derivation of Westfold (1959). In the fluid frame, we consider a direct
orthonormal triad,
$(\boldsymbol{\mathcal{I}},\boldsymbol{\mathcal{J}},\boldsymbol{\mathcal{K}})$,
such that $\boldsymbol{\mathcal{I}}$ is antiparallel to the acceleration
vector at some initial time $t=0$ (coinciding with the proper time of the
fluid frame), $\boldsymbol{\mathcal{K}}$ is along the ambient magnetic field
$\mathbf{b}$ (measured in the fluid frame), and $\boldsymbol{\mathcal{J}}$
completes the triad. We call $\boldsymbol{\beta}$ the velocity 3-vector of the
electron in the fluid frame and $\boldsymbol{\dot{\beta}}$ the corresponding
acceleration. A synchrotron wave is emitted by the accelerated electron in the
unit direction $\mathbf{\bar{K}}$ (measured in the fluid frame). The emission
angle $\theta_{B}$ and the pitch angle $\alpha$ are illustrated in Fig. 10.

The velocity and acceleration read

$\displaystyle\boldsymbol{\beta}$ $\displaystyle=$
$\displaystyle\beta\sin\alpha\left(-\sin\omega_{B}t\,\boldsymbol{\mathcal{I}}+\cos\omega_{B}t\,\boldsymbol{\mathcal{J}}\right)+\beta\cos\alpha\,\boldsymbol{\mathcal{K}},$
(79) $\displaystyle\boldsymbol{\dot{\beta}}$ $\displaystyle=$
$\displaystyle-\omega_{B}\beta\sin\alpha\left(\cos\omega_{B}t\,\boldsymbol{\mathcal{I}}+\sin\omega_{B}t\,\boldsymbol{\mathcal{J}}\right),$

where $\beta$ is the velocity of the electron in units of the speed of light,
and $\omega_{B}$ is the cyclotron gyrofrequency. It is easy to check that at
$t=0$, the projection of the velocity vector orthogonal to $\mathbf{b}$ is
along $\boldsymbol{\mathcal{J}}$, and the acceleration vector is along
$-\boldsymbol{\mathcal{I}}$, which is the setup illustrated in Fig. 10.

The radiation field of the moving charge in the fluid frame satisfies

$\mathbf{E_{\mathrm{rad}}}\propto\mathbf{\bar{K}}\times\left[\left(\mathbf{\bar{K}}-\boldsymbol{\beta}\right)\times\boldsymbol{\dot{\beta}}\right].$
(80)

Let us consider the unit direction of emission of a synchrotron photon,
written in full generality as

$\mathbf{\bar{K}}=a\,\boldsymbol{\mathcal{I}}+b\,\boldsymbol{\mathcal{J}}+c\,\boldsymbol{\mathcal{K}},$
(81)

where $(a,b,c)$ are arbitrary real numbers such that $\mathbf{\bar{K}}$ is a
unit vector. It is easy to show that

$\mathbf{E_{\mathrm{rad}}}\cdot\boldsymbol{\mathcal{K}}\propto\beta\cos\alpha-c.$
(82)

The lhs quantity represents the projection of the radiating electric field
along the ambient magnetic field direction. It is well known that the beaming
effect leads to the radiation being confined within a narrow cone around the
pitch angle of the relativistically moving electron (see Fig. 6.5 of Rybicki
and Lightman, 1979). This exactly means that

$\beta\cos\alpha-c\approx 0,$ (83)

such that the radiating electric field is orthogonal to the ambient magnetic
field. It follows that

$\mathbf{E_{\mathrm{rad}}}\propto\mathbf{\bar{K}}\times\mathbf{b},$ (84)

so that the rhs can be used to define the direction of the synchrotron
polarization vector, as done in Eq. 36.

Figure 10: Geometry of the synchrotron-emitting electron motion in the fluid
frame. $\boldsymbol{\beta}$ is the velocity 3-vector, $\mathbf{\bar{K}}$ is
the direction of emission, $\mathbf{b}$ is the magnetic vector, all these
quantities defined in the fluid frame. The emission angle between the magnetic
field direction and the direction of emission is called $\theta_{B}$ while
$\alpha$ is the pitch angle between the magnetic field direction and the
velocity of the electron.

##  Appendix C Solving the polarized radiative transfer equation

Let us start with the unpolarized radiative transfer equation in the emitter
frame

$\frac{\mathrm{d}I}{\mathrm{d}s}=-\alpha I+j$ (85)

with obvious notations. The formal solution reads

$I(\tau)=\int_{s_{0}}^{s}\mathrm{exp}\left(-(\tau-\tau^{\prime})\right)S(\tau^{\prime})\mathrm{d}\tau^{\prime}$
(86)

where $S=j/\alpha$ is the source function, $\mathrm{d}\tau=\alpha\,\delta s$
is the optical depth, $s$ is the proper length in the emitter frame and
$s_{0}$ is some initial value of $s$ where the intensity is assumed to be
zero. This equation can equivalently be written

$I(s)=\int_{s_{0}}^{s}\mathrm{exp}\left(-\alpha(s-s^{\prime})\right)j(s^{\prime})\mathrm{d}s^{\prime}$
(87)

where $\alpha$ is assumed constant over the integration interval. Considering
a small range $\delta s=s-s^{\prime}$ between some previous location
$s^{\prime}$ and some current location $s$, over which interval $j$ and
$\alpha$ can be considered constant in a realistic problem, the increment of
intensity reads

$\delta I(s)=j(s)\,\delta s\,\mathrm{exp}\left(-\alpha(s)\,\delta s\right).$
(88)

Let us now come back to the polarized version of the radiative transfer
equation (Eq. 69) which reads

$\frac{\mathrm{d}\boldsymbol{\mathcal{I}}}{\mathrm{d}s}=-\boldsymbol{\mathcal{K}}\boldsymbol{\mathcal{I}}+\boldsymbol{\mathcal{J}}$
(89)

where $\boldsymbol{\mathcal{I}}$ is the vector of Stokes parameters and

$\boldsymbol{\mathcal{K}}=\mathbf{R}\left(\chi\right)\left(\begin{array}[]{cccc}\alpha_{I}&\alpha_{Q}&\alpha_{U}&\alpha_{V}\\\
\alpha_{Q}&\alpha_{I}&r_{V}&-r_{U}\\\ \alpha_{U}&-r_{V}&\alpha_{I}&r_{Q}\\\
\alpha_{V}&r_{U}&-r_{Q}&\alpha_{I}\end{array}\right)\mathbf{R}\left(-\chi\right);\qquad\boldsymbol{\mathcal{J}}=\mathbf{R}\left(\chi\right)\left(\begin{array}[]{c}j_{I}\\\
j_{Q}\\\ j_{U}\\\ j_{V}\end{array}\right)$ (90)

with $\mathbf{R}\left(\chi\right)$ being the rotation matrix given in Eq. 70.
Its formal solution is the direct generalization of Eq. 86, provided that
$\boldsymbol{\mathcal{K}}$ is constant with $s$:

$\boldsymbol{\mathcal{I}}(s)=\int_{s_{0}}^{s}\mathrm{exp}\left(-\boldsymbol{\mathcal{K}}(s-s^{\prime})\right)\boldsymbol{\mathcal{J}}(s^{\prime})\mathrm{d}s^{\prime}.$
(91)

Let us introduce the matrix

$\mathbf{O}(s,s^{\prime})=\mathrm{exp}\left(-\boldsymbol{\mathcal{K}}(s-s^{\prime})\right).$
(92)

Over a small interval of proper length $\delta s=s-s^{\prime}$, over which the
absorption matrix and emission vector can be considered constant, the
elementary increase of Stokes parameters is

$\delta\boldsymbol{\mathcal{I}}(s)=\mathbf{O}(\delta
s)\boldsymbol{\mathcal{J}}(s)\delta s$ (93)

which is the direct generalization of Eq. 88. This is the equation used in
Gyoto to update the Stokes parameters along the light ray.

We still need to compute the exponential of matrix appearing in Eq. 92. Landi
Degl’Innocenti and Landi Degl’Innocenti (1985) have given an expression for
this matrix. It reads

$\displaystyle\mathbf{O}(\delta s)=\mathrm{exp}\left(-\alpha_{I}\delta
s\right)$ $\displaystyle\left\\{\left[\mathrm{cosh}\left(\Lambda_{1}\delta
s\right)+\cos\left(\Lambda_{2}\delta s\right)\right]\mathbf{M_{1}}/2\right.$
$\displaystyle\left.-\sin\left(\Lambda_{2}\delta
s\right)\mathbf{M_{2}}\right.$
$\displaystyle\left.-\mathrm{sinh}\left(\Lambda_{1}\delta
s\right)\mathbf{M_{3}}\right.$
$\displaystyle\left.+\left[\mathrm{cosh}\left(\Lambda_{1}\delta
s\right)-\cos\left(\Lambda_{2}\delta s\right)\right]\mathbf{M_{4}}/2\right\\}$

with

$\displaystyle\mathbf{M_{1}}$ $\displaystyle=$ $\displaystyle\mathbf{1},$ (95)
$\displaystyle\mathbf{M_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Theta}\left(\begin{array}[]{cccc}0&\Lambda_{2}\tilde{\alpha}_{Q}-\sigma\Lambda_{1}\tilde{r}_{Q}&\Lambda_{2}\tilde{\alpha}_{U}-\sigma\Lambda_{1}\tilde{r}_{U}&\Lambda_{2}\tilde{\alpha}_{V}-\sigma\Lambda_{1}\tilde{r}_{V}\\\
\Lambda_{2}\tilde{\alpha}_{Q}-\sigma\Lambda_{1}\tilde{r}_{Q}&0&\sigma\Lambda_{1}\tilde{\alpha}_{V}+\Lambda_{2}\tilde{r}_{V}&-\sigma\Lambda_{1}\tilde{\alpha}_{U}-\Lambda_{2}\tilde{r}_{U}\\\
\Lambda_{2}\tilde{\alpha}_{U}-\sigma\Lambda_{1}\tilde{r}_{U}&-\sigma\Lambda_{1}\tilde{\alpha}_{V}-\Lambda_{2}\tilde{r}_{V}&0&\sigma\Lambda_{1}\tilde{\alpha}_{Q}+\Lambda_{2}\tilde{r}_{Q}\\\
\Lambda_{2}\tilde{\alpha}_{V}-\sigma\Lambda_{1}\tilde{r}_{V}&\sigma\Lambda_{1}\tilde{\alpha}_{U}+\Lambda_{2}\tilde{r}_{U}&-\sigma\Lambda_{1}\tilde{\alpha}_{Q}-\Lambda_{2}\tilde{r}_{Q}&0\end{array}\right),$
(100) $\displaystyle\mathbf{M_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Theta}\left(\begin{array}[]{cccc}0&\Lambda_{1}\tilde{\alpha}_{Q}+\sigma\Lambda_{2}\tilde{r}_{Q}&\Lambda_{1}\tilde{\alpha}_{U}+\sigma\Lambda_{2}\tilde{r}_{U}&\Lambda_{1}\tilde{\alpha}_{V}+\sigma\Lambda_{2}\tilde{r}_{V}\\\
\Lambda_{1}\tilde{\alpha}_{Q}+\sigma\Lambda_{2}\tilde{r}_{Q}&0&-\sigma\Lambda_{2}\tilde{\alpha}_{V}+\Lambda_{1}\tilde{r}_{V}&\sigma\Lambda_{2}\tilde{\alpha}_{U}-\Lambda_{1}\tilde{r}_{U}\\\
\Lambda_{1}\tilde{\alpha}_{U}+\sigma\Lambda_{2}\tilde{r}_{U}&\sigma\Lambda_{2}\tilde{\alpha}_{V}-\Lambda_{1}\tilde{r}_{V}&0&-\sigma\Lambda_{2}\tilde{\alpha}_{Q}+\Lambda_{1}\tilde{r}_{Q}\\\
\Lambda_{1}\tilde{\alpha}_{V}+\sigma\Lambda_{2}\tilde{r}_{V}&-\sigma\Lambda_{2}\tilde{\alpha}_{U}+\Lambda_{1}\tilde{r}_{U}&\sigma\Lambda_{2}\tilde{\alpha}_{Q}-\Lambda_{1}\tilde{r}_{Q}&0\end{array}\right),$
(105) $\displaystyle\mathbf{M_{4}}$ $\displaystyle=$
$\displaystyle\frac{2}{\Theta}$ (110) $\displaystyle\times$
$\displaystyle\left(\begin{array}[]{cccc}(\tilde{\alpha}^{2}+\tilde{r}^{2})/2&\tilde{\alpha}_{V}\tilde{r}_{U}-\tilde{\alpha}_{U}\tilde{r}_{V}&\tilde{\alpha}_{Q}\tilde{r}_{V}-\tilde{\alpha}_{V}\tilde{r}_{Q}&\tilde{\alpha}_{U}\tilde{r}_{Q}-\tilde{\alpha}_{Q}\tilde{r}_{U}\\\
\tilde{\alpha}_{U}\tilde{r}_{V}-\tilde{\alpha}_{V}\tilde{r}_{U}&\tilde{\alpha}_{Q}^{2}+\tilde{r}_{Q}^{2}-(\tilde{\alpha}^{2}+\tilde{r}^{2})/2&\tilde{\alpha}_{Q}\tilde{\alpha}_{U}+\tilde{r}_{Q}\tilde{r}_{U}&\tilde{\alpha}_{V}\tilde{\alpha}_{Q}+\tilde{r}_{V}\tilde{r}_{Q}\\\
\tilde{\alpha}_{V}\tilde{r}_{Q}-\tilde{\alpha}_{Q}\tilde{r}_{V}&\tilde{\alpha}_{Q}\tilde{\alpha}_{U}+\tilde{r}_{Q}\tilde{r}_{U}&\tilde{\alpha}_{U}^{2}+\tilde{r}_{U}^{2}-(\tilde{\alpha}^{2}+\tilde{r}^{2})/2&\tilde{\alpha}_{U}\tilde{\alpha}_{V}+\tilde{r}_{U}\tilde{r}_{V}\\\
\tilde{\alpha}_{Q}\tilde{r}_{U}-\tilde{\alpha}_{U}\tilde{r}_{Q}&\tilde{\alpha}_{V}\tilde{\alpha}_{Q}+\tilde{r}_{V}\tilde{r}_{Q}&\tilde{\alpha}_{U}\tilde{\alpha}_{V}+\tilde{r}_{U}\tilde{r}_{V}&\tilde{\alpha}_{V}^{2}+\tilde{r}_{V}^{2}-(\tilde{\alpha}^{2}+\tilde{r}^{2})/2\end{array}\right)$

where

$\displaystyle\tilde{\alpha}^{2}$ $\displaystyle=$
$\displaystyle\tilde{\alpha}_{Q}^{2}+\tilde{\alpha}_{U}^{2}+\tilde{\alpha}_{V}^{2},$
(111) $\displaystyle\tilde{r}^{2}$ $\displaystyle=$
$\displaystyle\tilde{r}_{Q}^{2}+\tilde{r}_{U}^{2}+\tilde{r}_{V}^{2},$
$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\sqrt{\frac{1}{4}(\tilde{\alpha}^{2}-\tilde{r}^{2})^{2}+(\tilde{\alpha}_{Q}\tilde{r}_{Q}+\tilde{\alpha}_{U}\tilde{r}_{U}+\tilde{\alpha}_{V}\tilde{r}_{V})^{2}}+\frac{1}{2}(\tilde{\alpha}^{2}-\tilde{r}^{2})},$
$\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\sqrt{\frac{1}{4}(\tilde{\alpha}^{2}-\tilde{r}^{2})^{2}+(\tilde{\alpha}_{Q}\tilde{r}_{Q}+\tilde{\alpha}_{U}\tilde{r}_{U}+\tilde{\alpha}_{V}\tilde{r}_{V})^{2}}-\frac{1}{2}(\tilde{\alpha}^{2}-\tilde{r}^{2})},$
$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle\Lambda_{1}^{2}+\Lambda_{2}^{2},$ $\displaystyle\sigma$
$\displaystyle=$
$\displaystyle\mathrm{sign}\left(\tilde{\alpha}_{Q}\tilde{r}_{Q}+\tilde{\alpha}_{U}\tilde{r}_{U}+\tilde{\alpha}_{V}\tilde{r}_{V}\right),$

and where the tilde quantities $\tilde{\alpha}_{X}$, $\tilde{r}_{X}$ take into
account the basis rotation by an angle $\chi$ and read

$\left(\begin{array}[]{c}\tilde{\alpha}_{Q}\\\
\tilde{\alpha}_{U}\end{array}\right)=\left(\begin{array}[]{cc}\cos 2\chi&-\sin
2\chi\\\ \sin 2\chi&\cos
2\chi\end{array}\right)\left(\begin{array}[]{c}\alpha_{Q}\\\
\alpha_{U}\end{array}\right),$ (112)

and similarly for $\tilde{r}_{Q}$ and $\tilde{r}_{U}$, while
$\tilde{\alpha}_{I,V}$ and $\tilde{r}_{V}$ are the same as their counterparts
without a tilde, given that $I$ and $V$ are not affected by the rotation.

## Acknowledgements

It is a pleasure for the authors to acknowledge continuous discussions with
Maciek Wielgus on the topic of the paper. The authors gratefully acknowledge
fruitful discussions with B. Cerutti and B. Crinquand, as well as very useful
exchanges with J. Vos for the code comparison of section 3.3. E. G.
acknowledges funding by l’Agence Nationale de la Recherche, Project StronG
ANR-22-CE31-0015.

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