# Polarized Image of Equatorial Emission in the Kerr Geometry

Zachary Gelles<EMAIL_ADDRESS>Center for Astrophysics $|$
Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Black Hole
Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Elizabeth Himwich<EMAIL_ADDRESS>Center for the Fundamental Laws of
Nature, Harvard University, Cambridge, MA 02138, USA  Black Hole Initiative at
Harvard University, 20 Garden Street, Cambridge, MA 02138, USA  Michael D.
Johnson<EMAIL_ADDRESS>Center for Astrophysics $|$ Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Black Hole Initiative
at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA  Daniel C.
M. Palumbo<EMAIL_ADDRESS>Center for Astrophysics $|$ Harvard
& Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Black Hole
Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA

###### Abstract

We develop a simple toy model for polarized images of synchrotron emission
from an equatorial source around a Kerr black hole by using a semi-analytic
solution of the null geodesic equation and conservation of the Penrose-Walker
constant. Our model is an extension of Narayan_2020 , which presented results
for a Schwarzschild black hole, including a fully analytic approximation. Our
model includes an arbitrary observer inclination, black hole spin, local
boost, and local magnetic field configuration. We study the geometric effects
of black hole spin on photon parallel transport and isolate these effects from
the complicated combination of relativistic, gravitational, and
electromagnetic processes in the emission region. We find an analytic
approximation, consistent with previous work, for the subleading geometric
effect of spin on observed face-on polarization rotation in the direct image:
$\Delta{\rm EVPA}\sim-2a/r_{\rm s}^{2}$, where $a$ is the black hole spin and
$r_{\rm s}$ is the emission radius. We further show that spin introduces an
order unity effect on face-on subimages: $\Delta{\rm EVPA}\sim\pm
a/\sqrt{27}$. We also use our toy model to analyze polarization “loops”
observed during flares of orbiting hotspots. Our model provides insight into
polarimetric simulations and observations of black holes such as those made by
the EHT and GRAVITY.

##  I Introduction

The first polarized images of the black hole M87*, which reveal a bright ring
of emission with twisting polarization pattern, have recently been released by
the Event Horizon Telescope (EHT) collaboration PaperI ; PaperII ; PaperIII ;
PaperIV ; PaperV ; PaperVI ; PaperVII ; PaperVIII . The polarization structure
in black hole images depends on propagation effects, plasma emission, magnetic
field geometry, and spacetime curvature. Simulations of polarized emission are
an important tool that have been used to study astrophysical and geometric
properties of black hole accretion flows, and as an aid for interpreting
observations Connors:1980 ; Bromley:2001 ; Broderick:2003bg ; Broderick:2004 ;
Broderick:2005 ; Broderick:2006 ; Fish:2009 ; Zamaninasab:2010 ; Penna_2010 ;
Li:2009 ; Shcherbakov:2011 ; Shcherbakov:2012 ; Dexter_2016 ;
Moscibrodzka:2017 ; Gold:2017 ; Marin:2018 ; Jimenez:2018 ; Moscibrodzka:2018
; Palumbo_2020 ; Moscibrodzka:2020 ; Moscibrodzka:2021 . Detailed simulations
that simultaneously incorporate astrophysical and relativistic effects are
physically realistic but are generally computationally expensive. Furthermore,
disentangling astrophysical and relativistic effects in these models can be
challenging.

Toy models offer an efficient pathway to decouple and characterize different
effects on black hole images over a broad range of simplified emission
configurations. An exact description of polarized images from emission around
black holes requires a numerical solution of the geodesic equation, even in
the spherically symmetric Schwarzschild geometry (see, e.g., Chen_1991 ). The
description of polarized images in both Schwarzschild and Kerr is greatly
simplified by conservation of the Penrose-Walker constant Walker_Penrose_1970
. Recently, a simplified toy model of synchrotron emission around a
Schwarzschild black hole was presented in Narayan_2020 , which took advantage
of an additional simplification: an approximation developed by Beloborodov
Beloborodov_2002  with which the polarized image can be computed analytically.
Here, we extend the toy model of Narayan_2020  using tools developed in
Carter_1968 ; Gralla2020a ; Gralla2020b ; Himwich2020  to include the effects
of spin by generalizing to the Kerr geometry, for which a Beloborodov-like
approximation is not available.

Our model consists of an equatorial ring of magnetized fluid orbiting a Kerr
black hole. The images of axisymmetric rings of radiating fluid are described
by analyzing the local frame of an emitting point source in the Kerr geometry.
The semi-analytic description of an unpolarized image from such an emitter has
a long history going back to the 1970s with Cunningham and Bardeen,
Bardeen1972 ; Cunningham1973  and has recently been discussed in specific
contexts such as the near-horizon-extreme-Kerr emission Gralla2018 ; Gates2020
; Gates:2020els . The polarized image has also been treated analytically for
the high-spin case in Gates:2018hub .

A semi-analytic treatment of the geometric effects of spin on the polarized
image of orbiting geodesic rings at arbitrary emission radius and inclination
was performed in the seminal work of Connors, Piran, & Stark Connors:1980 , as
well as the PhD thesis of Eric Agol Agol:1997 . Our model generalizes this
work to semi-analytically compute the polarized image given an arbitrary spin,
emission radius, magnetic field geometry, equatorial fluid velocity, and
observer inclination.

Using conservation of the Penrose-Walker constant Walker_Penrose_1970 , our
model generates predictions for linear polarization angle and relative
polarized intensity, providing insight into how the accretion flow and
spacetime geometry affect the polarized image. In this paper, we provide
illustrative examples of polarized images for a variety of physical
configurations. Our model includes the image of direct emission as well the
corresponding lensed indirect emission, or “subimages.” Though we consider
only emission from a single radius, the image of a disk with finite radial
extent can be modelled by simply summing contributions from individual radii;
this was performed by Agol:1997 , which noted that the sum gave rise to a net
depolarization effect, since adding polarization vectors at different angles
reduces the polarization. An analogous depolarization can arise from summing
contributions of direct and indirect emission, as seen in simulations Jimenez-
Rosales:2021ytz .

For face-on black holes, we find an analytic expression for the effect of spin
on polarization rotation, which is subleading in the direct image ($\Delta{\rm
EVPA}\sim-2a/r_{\rm s}^{2}$) but order unity in subimages ($\Delta{\rm
EVPA}\sim a/\sqrt{27}$), where $a$ is the black hole spin and $r_{\rm s}$ is
the emission radius. The specific effects of frame-dragging on the rotation of
the polarization plane have been studied extensively in the past and have been
interpreted as a gravitational analogue of Faraday rotation Balazs_1958 ;
Pineault_1977 ; Fayos_1982 ; Brodutch_2011 ; Ishihara_1988 ; Nouri_1999 . In
this work, we use our model to re-derive earlier results using simple
techniques that avoid ambiguities about local reference frames. We further
build on prior work by examining the effects of parallel transport on photon
ring subimages.

An important EHT observing target is the black hole Sgr A* at the center of
our own galaxy. The polarization of Sgr A* has significant time variability in
both submillimeter Marrone_2006 ; Johnson:2015 ; Bower:2018  and near-infrared
Eckart:2006 ; Trippe:2007 ; Gravity_2018 ; Gravity_2020  observations, with
particularly rapid variability in near-infrared flares. The flares are likely
born out of various plasma and MHD effects such as magnetic reconnection
Dodds_2010 ; Ripperda_2020 . Simulations are a powerful tool used to
understand these polarized flares and have a long history also going back to
the 1970s, including the work of Connors:1980 , which presents time-dependent
EVPA in the direct emission from an orbiting hotspot. More advanced recent
simulation studies include Broderick:2005 ; Broderick:2006 ; Fish:2009 ;
Zamaninasab:2010 , which attribute certain time-varying features to emission
from a localized orbiting hotspot. These features may be manifested as loops
in the linear Stokes $Q,U$ polarization. Our toy semi-analytic model, which
can isolate geometric effects from the simulated astrophysical processes,
complements these efforts. Our model also isolates the effects of individual
subimages and the ways in which they affect observed polarization patterns.

Using analytic results in the Kerr geometry, we can model the effects of spin
on the image of hotspot emission and generate images of $Q,U$ time variation
in direct and indirect images that reproduce observed polarization loops. Our
model can be directly compared to simulated and observed values for the case
of near-infrared flares, in which the influence of Faraday effects,
absorption, and background emission are insignificant Narayan_2020 .

In particular, using infrared interferometry, the GRAVITY collaboration has
recently reported the first resolved centroid motion and polarization during
flares of Sgr A* Gravity_2018 . Both the centroid and polarization traced
loops over time, which were interpreted using a model of an orbiting
equatorial hotspot Gravity_2018 ; Gravity_2020 . In simulations of motion of a
small Gaussian hotspot, Gravity_2018  (see Appendix D therein) found that the
presence of a single polarization loop in $Q,U$, in which the orbital period
in polarization is the same as the orbital period of the hotspot, is a
signature of magnetic fields perpendicular to the orbital plane (i.e.
vertical). In contrast, for a toroidal field (i.e. equatorial) they found that
the orbital period of polarization is half that of the hotspot, corresponding
to two loops in $Q,U$. The $Q,U$ data for the observed July 28, 2018 flare
were consistent with a single polarization loop observed at low inclination,
as in a poloidal field configuration Gravity_2018 . Follow-up work in
Gravity_2020  considered the July 28 flare in further detail, and compared
observations to simulations of a Gaussian hotspot as well as a simplified non-
relativistic analytic model of a point emitter, finding that a single loop in
$Q,U$ arises only from fields with a nonzero vertical component, with the best
fit to the July 28 flare having a vertical plus azimuthal field. We
substantiate the same claims in Gravity_2018 ; Gravity_2020  about single
(double) loops arising from vertical (equatorial) magnetic fields using
analytic results in the Kerr geometry, providing additional physical insight
into GRAVITY observations and the general observational distinction between
vertical and equatorial fields.

Our work can also be further developed to future extensions that include non-
equatorial emission, and to studies of circular polarization (Stokes $V$)
Bower:2002 ; Huang:2008 ; Shcherbakov:2011 ; Munoz:2012 ; Gold:2017 ;
Ricarte:2021frd ; Moscibrodzka:2021 . These extensions will be useful in
comparisons to general-relativistic magnetohydrodynamic (GRMHD) simulations
and will further our understanding of the extent to which fluid configuration
and magnetic field geometries can be inferred using polarized images.
Comparing our model to ray-traced GRMHD simulations will also provide insight
into whether Faraday effects prevent such inferences.

This paper is organized as follows. Section II describes our simple model for
fluid orbiting a Kerr black hole. Section III reviews the calculation of the
observed appearance of polarized emission around a Kerr black hole. Section IV
presents and discusses sample polarized images produced by our model for a
variety of magnetic fields, observer inclinations, and fluid configurations.
Section V considers the observed EVPA for black holes viewed on-axis,
quantifying the effects of spin on EVPA. Section VI considers the application
of our model to the polarized images of orbiting hotspots and provides
additional details of comparisons with Gravity_2018 ; Gravity_2020 . Section
VII gives a brief summary of our conclusions and directions for future
investigation. We present explicit details of the orbiting fluid model in
Appendix A, details of the semi-analytic solution of the geodesic equation in
Appendix B, a general discussion of image symmetries in Appendix C, and
details of cusp formation in direct emission $Q,U$ loops in Appendix D.

##  II Orbiting Fluid Model

This section introduces our model of accreting fluid around a Kerr black hole
using an orbiting emitter.

###  II.1 Circular Orbiting Emitter in Kerr

The Kerr line element for a black hole of mass $M$ and angular momentum $J=aM$
in Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ is Kerr1963 ;
Chandrasekhar:1985

$ds^{2}=-\frac{\Delta\Sigma}{\Xi}dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+\frac{\Xi\sin^{2}{\theta}}{\Sigma}\left[d\phi-\omega
dt\right]^{2},\\\ $ (1)

where

$\displaystyle\Delta(r)=r^{2}-2Mr+a^{2},\quad\Sigma(r,\theta)=r^{2}+a^{2}\cos^{2}{\theta},$
$\displaystyle\omega=\frac{2aMr}{\Xi},\quad\Xi=(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta.$ (2)

Note that $\omega$ is the angular momentum of the zero-angular-momentum-
observer (ZAMO), which vanishes in the Schwarzschild limit $a\to 0$.

Consider a point source at radius $r_{\rm s}$ on an equatorial $(\theta_{\rm
s}=\frac{\pi}{2})$ circular orbit of zero angular momentum with angular
velocity $\omega_{\rm s}=\omega(r_{\rm s},\theta_{\rm
s}=\frac{\pi}{2})$.111Angular velocity is defined relative to the asymptotic
rest frame by $\frac{d\phi}{dt}=\frac{u^{\phi}}{u^{t}}$, where $u^{\mu}$ is
the four-velocity the emitting source. The tetrad that describes the ZAMO
local rest frame consists of its four-velocity $u^{\mu}=e^{\mu}_{(t)}$
($u^{\mu}u_{\mu}=-1$) and three orthogonal unit spacelike vectors:

$\displaystyle e_{(t)}$ $\displaystyle=\frac{1}{r_{\rm s}}\sqrt{\frac{\Xi_{\rm
s}}{\Delta_{\rm s}}}(\partial_{t}+\omega_{\rm s}\partial_{\phi}),$ (3)
$\displaystyle e_{(r)}$ $\displaystyle=\frac{1}{r_{\rm s}}\sqrt{\Delta_{\rm
s}}\partial_{r},$ $\displaystyle e_{(\phi)}$ $\displaystyle=\frac{r_{\rm
s}}{\sqrt{\Xi_{\rm s}}}\partial_{\phi},$ $\displaystyle e_{(\theta)}$
$\displaystyle=-\frac{1}{r_{\rm s}}\partial_{\theta}.$

Note that the minus sign in the last line implies that the local
$(\hat{\theta})$ and Boyer-Lindquist $\hat{\theta}$ are anti-aligned. Here and
below, the subscript $s$ denotes a quantity evaluated at the source $r_{\rm
s},\theta_{\rm s}=\frac{\pi}{2}$. The local orthonormal frame has the flat
Minkowski metric $\eta^{(a)(b)}$, and the frame components of four-vectors are
given by

$V^{(a)}=\eta^{(a)(b)}e_{(b)}^{\mu}V_{\mu}.$ (4)

The orientation is such that
$(\hat{x},\hat{y},\hat{z})\leftrightarrow\left((\hat{r}),(\hat{\phi}),(\hat{\theta})\right)$.
The tetrad is given explicitly as a matrix in Appendix A.

###  II.2 Boosted Emitter

From the local orthonormal ZAMO frame, consider boosting the emitter in the
$(r),(\phi)$ plane with velocity

$\vec{\beta}=\beta_{v}\left(\cos\chi\,(\hat{r})+\sin\chi\,(\hat{\phi})\right).$
(5)

Vectors are boosted via a Lorentz transformation $\Lambda^{(a)}_{\ \ (b)}$. To
transform from the ZAMO frame to the boosted orthonormal frame, denoted by
primed quantities, take

$\displaystyle V^{\prime(a)}$ $\displaystyle=\Lambda^{(a)}_{\ \ (b)}V^{(b)}$
(6) $\displaystyle=\Lambda^{(a)}_{\ \ (b)}\eta^{(b)(c)}e_{(c)}^{\mu}V_{\mu},$

using equation (4). This defines a new tetrad

$\displaystyle e^{\prime\mu}_{(d)}=\eta_{(d)(a)}\Lambda^{(a)}_{\ \
(b)}\eta^{(b)(c)}e_{(c)}^{\mu}=\Lambda_{(d)}^{\ \ (c)}e_{(c)}^{\mu},$ (7)

with components given by matrix multiplication (recall that for
$\Lambda=\Lambda^{(a)}_{\ \ (b)}$, the inverse matrix
$\Lambda^{-1}=\Lambda_{(a)}^{\ \ (b)}$). For concreteness, the explicit tetrad
components are given in Appendix A. An inverse transformation from the local
frame to the vector in Kerr is given by

$V^{\mu}=e^{\prime\mu}_{(a)}V^{\prime(a)}.$ (8)

The next section uses the local emitter motion to compute its polarized image
seen by a distant observer.

##  III Image Intensity and Polarization

This section describes how to compute the observed location and polarization
of the direct image of an axisymmetric equatorial disk of emitting matter at
radius $r_{\rm s}$ around a Kerr black hole with arbitrary spin $a$, local
magnetic field $\vec{B}$, and observer inclination $\theta_{o}$. We briefly
summarize the important steps in the calculation, details of which are
included in the following subsections:

  1. 1.

Subsections III.1 and III.2: For photons emitted by a source at position
$r_{\rm s},\theta_{\rm s}=\frac{\pi}{2}$, we find the arrival position on the
screen (and corresponding conserved quantities) by solving the geodesic
connecting the source $r_{\rm s},\theta_{\rm s}=\frac{\pi}{2}$ and observer.

  2. 2.

Subsection III.3: The photon conserved quantities give its initial momentum at
the source. Using the initial momentum and magnetic field, we calculate the
polarization in the emitter frame and the Penrose-Walker constant at the
source.

  3. 3.

Subsection III.4: Using the conserved Penrose-Walker constant and the photon’s
arrival position, we calculate its polarization on the observer screen, taking
redshift and path length effects into account.

Subsection III.5 then reviews simplifying aspects of the calculation in the
special case of an on-axis observer.

###  III.1 Light Propagation in Kerr

We review photon propagation and polarization in Kerr following Himwich2020 .
From the geodesic equation, a photon’s (energy-rescaled) four-momentum is
given in terms of its position and conserved quantities ($\lambda$,$\eta$)
corresponding to the energy-rescaled angular momentum parallel to the axis of
symmetry and Carter integral, respectively:

$p_{\mu}dx^{\mu}=-dt\pm_{r}\frac{\sqrt{\mathcal{R}(r)}}{\Delta(r)}dr\pm_{\theta}\sqrt{\Theta(\theta)}d\theta+\lambda
d\phi,$ (9)

given in terms of the radial and angular potentials

$\displaystyle\mathcal{R}(r)$
$\displaystyle=(r^{2}+a^{2}-a\lambda)^{2}-\Delta\left[\eta+(a-\lambda)^{2}\right],$
(10) $\displaystyle\Theta(\theta)$
$\displaystyle=\eta+a^{2}\cos^{2}{\theta}-\lambda^{2}\cot^{2}{\theta}.$

The photon trajectory is determined by its initial position, conserved
quantities $(\lambda,\eta)$, and signs $\pm_{r},\pm_{\theta}$ of its initial
motion. The photon’s linear polarization $f^{\mu}$ is parallel transported
along its trajectory,

$f^{\mu}p_{\mu}=0,\ \ \ p^{\mu}\nabla_{\mu}f^{\nu}=0.$ (11)

From the photon momentum and polarization, one can construct the Penrose-
Walker constant $\kappa$, a complex constant conserved along the photon
trajectory Walker_Penrose_1970 ; Chandrasekhar:1985 :

$\displaystyle\kappa$
$\displaystyle=\kappa_{1}+i\kappa_{2}=(\mathcal{A}-i\mathcal{B})(r-ia\cos\theta),$
(12) $\displaystyle\mathcal{A}$
$\displaystyle=(p^{t}f^{r}-p^{r}f^{t})+a\sin^{2}\theta(p^{r}f^{\phi}-p^{\phi}f^{r}),$
$\displaystyle\mathcal{B}$
$\displaystyle=\left[(r^{2}+a^{2})(p^{\phi}f^{\theta}-p^{\theta}f^{\phi})-a(p^{t}f^{\theta}-p^{\theta}f^{t})\right]\sin\theta.$

Given the Penrose-Walker constant, one can solve for the polarization
$f^{\mu}$ at any point along the photon path.

###  III.2 Screen Coordinates & Conserved Quantities

For a photon with conserved quantities ($\lambda$,$\eta$), its arrival
position is given by the screen coordinates Cunningham1973

$\alpha=-\frac{\lambda}{\sin\theta_{o}},\ \ \
\beta=\pm_{o}\sqrt{\Theta(\theta)},$ (13)

where $\theta_{o}$ is the observer’s polar inclination from the spin axis and
$\pm_{o}$ is the sign of $p^{\theta}$ at the observer.

Conversely, the photon’s arrival position on the screen determines its
corresponding conserved quantities:

$\displaystyle\lambda$ $\displaystyle=-\alpha\sin\theta_{o},$ (14)
$\displaystyle\eta$
$\displaystyle=(\alpha^{2}-a^{2})\cos^{2}\theta_{o}+\beta^{2}.$

For time-averaged, axisymmetric images of an equatorial disk, the relevant
photon motion is in $(r,\theta)$. Photon trajectories from an initial position
$(r_{\rm s},\theta_{\rm s}=\frac{\pi}{2})$ to a final position
$(r_{o}\rightarrow\infty,\theta_{o})$ are given by the null geodesic equation
Carter_1968 :

$I_{r}=\fint_{r_{\rm
s}}^{r_{o}}\frac{dr}{\pm_{r}\sqrt{\mathcal{R}(r)}}=\fint_{\theta_{\rm
s}}^{\theta_{o}}\frac{d\theta}{\pm_{r}\sqrt{\Theta(\theta)}}=G_{\theta},$ (15)

where the slash denotes a monotonic path integral with the signs
$\pm_{r},\pm_{\theta}$ changing at radial and angular turning points,
respectively. These have closed-form solutions in terms of elliptic integrals,
which have been described in a variety of formalisms by many authors,
including Cunningham_1975redshift ; Rauch:1994 ; Agol:1997 ; Kapec2020 . We
follow the conventions of Gralla2020b  (see references therein and App. B.1
for definitions). Following (81) of Gralla2020a , for a trajectory with $m$
turning points in $\theta$ and $\theta_{\rm s}=\frac{\pi}{2}$ the geodesic
equation becomes222 When $a=0$ exactly (for which (15) is not well-defined)
the geodesic equation reduces to (see e.g. B1 of Gates2020 ):  $\displaystyle
I_{r}$
$\displaystyle=G_{\theta}=\frac{1}{\sqrt{\eta+\lambda^{2}}}\left[m\pi-{\rm
sign}(\beta)\arcsin\left(\sqrt{1+\frac{\lambda^{2}}{\eta}}\cos\theta_{o}\right)\right].$
(16)

$I_{r}=G_{\theta}^{m}=\frac{1}{\sqrt{-u_{-}a^{2}}}\left(2mK\left(\frac{u_{+}}{u_{-}}\right)-\text{sign}(\beta)F_{o}\right),$
(17)

where

$F_{o}=F\left(\arcsin\frac{\cos{\theta_{o}}}{\sqrt{u_{+}}}\Big{|}\frac{u_{+}}{u_{-}}\right),$
(18)

and

$u_{\pm}=\Delta_{\theta}\pm\sqrt{\Delta_{\theta}^{2}+\frac{\eta}{a^{2}}},\ \ \
\Delta_{\theta}=\frac{1}{2}\left(1-\frac{\eta+\lambda^{2}}{a^{2}}\right).$
(19)

Given $m$ and $r_{\rm s}$, (17) defines a relationship between $\lambda$ and
$\eta$ and therefore between image $\alpha$ and $\beta$ via (13). 333For
$\theta_{o}>\pi/2$, (82) of Gralla2020a  generalizes to
$\overline{m}=m-H(\beta\cos\theta_{o})$. If $\theta_{o}<\pi/2$, the observer
is above the midplane, so geodesics with odd $m$ arrive on the top half of the
image, while geodesics with even $m$ arrive on the bottom. For
$\theta_{o}>\pi/2$, the observer is below the midplane, so the parity of $m$
switches. See Fig. 7 of Gralla2020a .

The geodesic equation (17) can be inverted (see e.g. (30) of Gralla2020a ) to
find $r_{\rm s}(I_{r}=G_{\theta}^{m})$, which gives

$r_{\rm
s}(I_{r})=\frac{r_{4}r_{31}-r_{3}r_{41}\text{sn}^{2}\left(\frac{1}{2}\sqrt{r_{31}r_{42}}I_{r}-\mathcal{F}_{o}\big{|}k\right)}{r_{31}-r_{41}\text{sn}^{2}\left(\frac{1}{2}\sqrt{r_{31}r_{42}}I_{r}-\mathcal{F}_{o}\big{|}k\right)},$
(20)

where

$\mathcal{F}_{o}=F\left(\arcsin{\sqrt{\frac{r_{31}}{r_{41}}}}\Big{|}k\right),$
(21)

and

$k=\frac{r_{32}r_{41}}{r_{31}r_{42}},\ \ \ r_{ij}=r_{i}-r_{j},$ (22)

with the roots $\left\\{r_{i}\right\\}$ of $\mathcal{R}(r)$ (given below in
App. B.2 as well as App. A of Gralla2020a ). Because (17) defines a relation
for $r_{\rm s}(G_{\theta}^{m})$, and $G_{\theta}^{m}$ can be written in terms
of $(\lambda,\eta)$ or equivalently $(\alpha,\beta)$, (20) defines an equation
$r_{\rm s}(\alpha,\beta)$, which can be solved numerically to find the allowed
curves of $(\alpha,\beta)$ that describe a source at radius $r_{\rm s}$.

In practice, (20) is computed using specified values of
$a,\sin{\theta_{o}},r_{\rm s},\varphi\equiv\arctan(\beta/\alpha)$ and a test
value(s) of $b\equiv\sqrt{\alpha^{2}+\beta^{2}}$ to solve for the
corresponding impact parameter using FindRoot in Mathematica 12 or
scipy.optimize in python 3.444Notebooks available upon request to the
corresponding author. Results of this calculation are displayed for a variety
of $a$ and $\theta_{o}$ in Fig. 6 of Gralla2020a .

After computing the set of allowed screen positions $(\alpha,\beta)$ for a
photon emitted at a given equatorial radius $r_{\rm s}$, (14) is used to
determine the corresponding conserved quantities $(\lambda,\eta)$, which
specify the photon momentum at the source and can be used to determine its
polarization, as described in the next section.

###  III.3 Polarization in the Local Frame

Having determined $(\lambda,\eta)$ for $r_{\rm s},a,\theta_{o}$ as described
in the previous section, the photon (energy-rescaled) momentum at the source
is given by (9):

$\displaystyle p_{t}$ $\displaystyle=-1,$ (23) $\displaystyle p_{r}$
$\displaystyle=\pm_{r},\frac{\sqrt{\mathcal{R}(r_{\rm s})}}{\Delta_{\rm s}},$
$\displaystyle p_{\phi}$ $\displaystyle=\lambda,$ $\displaystyle p_{\theta}$
$\displaystyle=\pm_{s}\sqrt{\eta},$

where the sign of $p^{\theta}$ at the source is
$\pm_{s}=(-1)^{m}\pm_{o}$.555For $\theta_{o}<\frac{\pi}{2}$, $\pm_{s}=-1$ for
both $m=0$ on the bottom of the image (where $\pm_{o}=-1$) and $m=1$ on the
top (where $\pm_{o}=1$).

The sign $\pm_{r}$ depends on $\\{\lambda,\eta,r_{\rm s},a,\theta_{o},m\\}$
and must be computed semi-analytically as described in Appendix B.3. With the
sign of $p^{r}$ determined, we compute the components of the four-momenta at
the source as

$\displaystyle p^{t}$ $\displaystyle=\frac{1}{r_{\rm
s}^{2}}\left(-a(a-\lambda)+\frac{(r_{\rm s}^{2}+a^{2})(r_{\rm
s}^{2}+a^{2}-a\lambda)}{\Delta_{\rm s}}\right),$ (24) $\displaystyle p^{r}$
$\displaystyle=\pm_{r}\frac{1}{r_{\rm s}^{2}}\sqrt{\mathcal{R}(r_{\rm s})},$
$\displaystyle p^{\phi}$ $\displaystyle=\frac{1}{r_{\rm
s}^{2}}\left(-(a-\lambda)+\frac{a}{\Delta_{\rm s}}\left(r_{\rm
s}^{2}+a^{2}-a\lambda\right)\right),$ $\displaystyle p^{\theta}$
$\displaystyle=\pm_{s}\frac{\sqrt{\eta}}{r_{\rm s}^{2}}.$

To compute the local photon polarization at the source, $p^{\mu}$ is
transformed to the local frame of the emitter via (6). In the local frame
$f^{(t)}=0$, and the spatial components
$\vec{f}=\left(f^{(r)},f^{(\phi)},f^{(\theta)}\right)$ are given by a cross
product of the local three-momentum
$\vec{p}=\left(p^{(r)},p^{(\phi)},p^{(\theta)}\right)$ with the local magnetic
field $\vec{B}=\left(B^{(r)},B^{(\phi)},B^{(\theta)}\right)$:

$\vec{f}=\frac{\vec{p}\times\vec{B}}{|\vec{p}|},$ (25)

as expected for synchrotron radiation Rybicki_1979 ; Narayan_2020 . Note that
the axes $(\hat{r}),(\hat{\phi}),$ and $(\hat{\theta})$ are orthogonal in the
fluid frame with $(\hat{r})\times(\hat{\phi})=(\hat{\theta})$, so the cross
product has its standard form in $\mathbb{R}^{3}$. For later reference, we
record explicitly:

$\displaystyle f^{(r)}$ $\displaystyle\propto
p^{(\phi)}B^{(\theta)}-p^{(\theta)}B^{(\phi)},$ (26) $\displaystyle
f^{(\phi)}$ $\displaystyle\propto-p^{(r)}B^{(\theta)}+p^{(\theta)}B^{(r)},$
$\displaystyle f^{(\theta)}$ $\displaystyle\propto
p^{(r)}B^{(\phi)}-p^{(\phi)}B^{(r)}.$

Following Narayan_2020 , the polarization is normalized to satisfy

$f^{\mu}f_{\mu}=\sin^{2}\zeta|\vec{B}|^{2},$ (27)

where $\zeta$ is the angle between $\vec{p}$ and the magnetic field $\vec{B}$:

$\sin\zeta=\frac{|\vec{p}\times\vec{B}|}{|\vec{p}||\vec{B}|}.$ (28)

Given $f^{(a)}$ in the local frame, $f^{\mu}$ in Boyer-Lindquist coordinates
is computed via (8). Then, the Penrose-Walker constant (12) for the trajectory
is computed from $f^{\mu}$ and $p^{\mu}$ at the source $r_{\rm s},\theta_{\rm
s}=\frac{\pi}{2}$ (see (24)), which simplifies to

$\displaystyle\kappa$ $\displaystyle=\kappa_{1}+i\kappa_{2}=r_{\rm
s}\left(\mathcal{A}-i\mathcal{B}\right),$ (29) $\displaystyle\mathcal{A}$
$\displaystyle=(p^{t}f^{r}-p^{r}f^{t})+a(p^{r}f^{\phi}-p^{\phi}f^{r}),$
$\displaystyle\mathcal{B}$ $\displaystyle=\left[(r_{\rm
s}^{2}+a^{2})(p^{\phi}f^{\theta}-p^{\theta}f^{\phi})-a(p^{t}f^{\theta}-p^{\theta}f^{t})\right],$

giving the simple relation

$\left(\kappa_{1},\kappa_{2}\right)=r_{\rm
s}\left(\mathcal{A},-\mathcal{B}\right)$ (30)

for equatorial sources.666An analogous calculation to analytically compute
geodesics and parallel transport $f$ is implemented in the ray-tracing code
grtrans Dexter_2016 . Our model differs from this code because we ray-trace
_forwards_ from a fixed emission radius, as opposed to backwards from a fixed
impact parameter. Furthermore, we use the Legendre elliptic formalism
presented by Gralla2020a  as opposed to the Carlson elliptic formalism
presented by Dexter_2009 , and we formulate the fluid-frame tetrad entirely in
terms of Lorentz boosts as opposed to using the results of a Gram-Schmidt
orthogonalization.

###  III.4 Observed Polarization and Redshift

Given the photon arrival position $(\alpha,\beta)$ and its Penrose-Walker
constant $\kappa$, computed as described in the previous subsections, the
observed polarization (direction of electric field transverse to photon
momentum) is computed from the components of $f^{\mu}$ at large radius
projected along the $\hat{\alpha}$ and $\hat{\beta}$ directions on the
observer screen Himwich2020 :

$\displaystyle\left(f^{\alpha},f^{\beta}\right)$
$\displaystyle=\frac{1}{\mu^{2}+\beta^{2}}\left(\beta\kappa_{2}-\mu\kappa_{1},\beta\kappa_{1}+\mu\kappa_{2}\right),$
(31) $\displaystyle\mu$ $\displaystyle=-(\alpha+a\sin\theta_{o}).$

Here, ($f^{\alpha}$,$f^{\beta})$ is a two-vector on the asymptotic observer
screen (see Appendix A of Himwich2020  for details).

The photon conserved energy $E=1$ is the energy measured by stationary
observers at infinity. The energy measured in the rest frame of the emitting
source is

$E_{s}=p^{(t)}=-p_{\mu}u^{\mu},$ (32)

and the ratio of these two energies is the redshift,

$g=\frac{E}{E_{s}}=\frac{1}{p^{(t)}}.$ (33)

As discussed in Narayan_2020 , the intensity of synchrotron radiation emitted
along $\vec{p}$ depends on $\sin\zeta$. The emitted intensity varies with
frequency $\nu$ as $I_{\nu}\sim\nu^{-\alpha_{\nu}}$, with angular dependence
$\left(\sin\zeta\right)^{1+\alpha_{\nu}}$. In this paper we take
$\alpha_{\nu}\sim 1$, which is consistent with M87* observed at 230 GHz
Narayan_2020  and with the range of values $\alpha_{\nu}\sim 0.5-1.5$ that
describe very bright flares ($\alpha_{\nu}\sim 0.5$) to the average lower-flux
density state ($\alpha_{\nu}\sim 1.5$) of Sgr A* (see Witzel:2018kzq  and
references therein). The specific intensity is Doppler-boosted by a factor of
$g^{3+\alpha_{\nu}}$ when it reaches the observer. Note that in this paper,
“intensity” (“flux”) always refers to specific intensity (flux). For an
optically and geometrically thin disk, which we assume throughout Section IV,
intensity grows linearly with the geodesic path length $l_{p}$ through the
emitting material Narayan_2020 :

$\displaystyle\l_{p}$ $\displaystyle=\frac{p^{(t)}_{s}}{p^{(z)}_{s}}H,$ (34)

with $H$ the height of the disk, taken to be a constant. Since the intensity
is proportional to the square of the electric field (i.e. the polarization
vector), the observed components are

$\left(f^{\alpha}_{\text{obs}},f^{\beta}_{\text{obs}}\right)\propto\sqrt{l_{p}}g^{2}\left(f^{\alpha},f^{\beta}\right).$
(35)

Note that following Himwich2020 , we measure the Electric Vector Position
Angle (EVPA) counter-clockwise from $+\hat{\beta}$:

$\displaystyle\text{EVPA}\equiv\arctan\left({-\frac{f^{\alpha}_{\rm
obs}}{f^{\beta}_{\rm obs}}}\right).$ (36)

Our model assumes pure synchrotron radiation with a constant polarization
fraction.

###  III.5 Face-On Coordinates and EVPA

This subsection reviews the special case of an on-axis observer. For a black
hole viewed face-on $(\theta_{o}=0^{\circ})$, the coordinates $(\alpha,\beta)$
on the screen degenerate because the $\hat{\beta}$ axis becomes a point.
Still, the radial screen coordinate

$b=\sqrt{\alpha^{2}+\beta^{2}}=\sqrt{\eta+\lambda^{2}+a^{2}\cos^{2}\theta_{0}}\rightarrow\sqrt{\eta+a^{2}}$
(37)

remains well-defined. There is a single spherical photon orbit that crosses
the spin axis, corresponding to photons with $\lambda=0$. Taking
$(b,\varphi=\arctan\frac{\beta}{\alpha})$ in the limit $\lambda\to 0$
$(\theta_{o}\to 0)$ defines Cartesian coordinates
$(b\cos\varphi,b\sin\varphi)$ that smoothly interpolate to the usual
definition for observers with $\theta_{o}>0$. The polarization components are

$(f^{\alpha},f^{\beta})\propto\left(b\sin\varphi\kappa_{2}+b\cos\varphi\kappa_{1},b\sin\varphi\kappa_{1}-b\cos\varphi\kappa_{2}\right),$
(38)

and the EVPA is given by

$\text{EVPA}=\arctan\left(-\frac{\kappa_{2}\sin\varphi+\kappa_{1}\cos\varphi}{\kappa_{1}\sin\varphi-\kappa_{2}\cos\varphi}\right).$
(39)

The direct image of an axisymmetric source for a face-on observer is entirely
described by $m=0$ with $\pm_{s}=-1$, and the geodesic equation for all spins
is approximated well by the simple relation of “just adding 1,” $b\approx
r_{\rm s}+1$ ($M=1$) Gralla2020a . Including subleading terms (see Eq. 10 of
Gates2020 ),

$\displaystyle r_{\rm s}$
$\displaystyle=b-1+\frac{1-a^{2}}{2b}+\frac{3(5\pi-16)}{4b^{2}}+\mathcal{O}(1/b^{3}),$
(40)

which may be inverted to yield

$\displaystyle b$ $\displaystyle=r_{\rm s}+1+\frac{a^{2}-1}{2r_{\rm
s}}+\frac{50-2a^{2}-15\pi}{4r_{\rm s}^{2}}+\mathcal{O}(1/r_{\rm s}^{3}).$ (41)

We will make use of this analytic expansion to quantify the effects of spin on
observed EVPA in Section V.

##  IV Polarization Visualizations

Figure 1: Polarized intensity tick plots for three idealized magnetic field
configurations: from left to right, radial $B=B^{(r)}$, azimuthal
$B=B^{(\phi)}$, and vertical $B=B^{(\theta)}$, in the case the direct image
seen by an on-axis observer $\theta_{o}=0$. The fluid is modelled by an
unboosted ZAMO emitter ($\beta_{v}=0$ in (5)). Each plot shows two spins
($a=0$ and $a=-0.99$ in red and blue, respectively), as well as two emission
radii ($r_{\rm s}=3$ and $r_{\rm s}=6$, corresponding to the inner and outer
rings, respectively).  Figure 2: Polarized intensity tick plots for the
indirect image corresponding to the direct image shown in Fig. 1 for
$\theta_{o}=0$ and $\beta_{v}=0$, now displayed for the single emission radius
$r_{\rm s}=6$, and with critical impact parameters shown as dashed lines.

In this section, we use our model to compute the polarized image of equatorial
emitting sources at a variety of different spins, magnetic fields, and
inclination angles. Using (35), we compute the polarized intensity of various
configurations and present the resulting images as “tick plots” as in
Narayan_2020 . To draw comparisons to M87*, we consider $a<0$,
$\theta_{o}<\frac{\pi}{2}$, corresponding to the clockwise fluid motion on the
sky seen around M87*.777Note that this terminology differs from that used in
EHT Paper V PaperV , in which $a<0$ corresponds to retrograde accretion flow
(counterclockwise on the sky). For us, $a<0$ corresponds to a prograde
accretion flow (clockwise on the sky) with the spin axis pointed away from the
observer. In each tick plot we display $a=-0.99$ to represent the extremal
Kerr limit, as exact expressions for $|a|=1$ and $\theta_{o}=\frac{\pi}{2}$
require additional care (see e.g. Gates2020 ). The relationship between
positive and negative spin is discussed in detail in Appendix C.

#### On-axis Observers

Three sample tick plots for on-axis observers of purely radial, azimuthal, and
vertical magnetic fields are presented respectively in the left, middle, and
right panels of Figure 1. We emphasize that these purely radial, azimuthal,
and vertical magnetic field geometries are idealized and are specified
pointwise. They have been chosen to probe the breadth of potential magnetic
field geometries that could be present in realistic horizon-scale black hole
accretion flows or subsections of accretion flows. In reality, physical
magnetic fields vary throughout the accretion region and contain nonzero
radial, azimuthal, and vertical components. Each tick has orientation aligned
with the EVPA of the arriving photon and length proportional to intensity
(square of the electric field). In each panel, the inner and outer circles
correspond to emission radii of $r_{\rm s}=3$ and $r_{\rm s}=6$, respectively,
and the boost parameter from (5) is taken to be $\vec{\beta}_{v}=0$. As
expected from the axisymmetric model, all three panels of Figure 1 are
radially symmetric. The ticks in the rightmost panel, which displays a
vertical magnetic field, are significantly shorter than those in the cases of
radial and azimuthal fields; for purely vertical fields $\vec{B}=B^{(\theta)}$
in (28), $|\sin\zeta|\propto\vec{p}\times\vec{B}\ll 1$ and is only nonzero due
to relativistic effects Gravity_2020 .

In addition to the direct image, our model includes general subimages, also
referred to as indirect images, which are indexed by number of radial turning
points $m$. For large $m$ in the face-on case, the impact parameter $b$ will
approach the critical impact parameter $b_{c}$, up to exponentially suppressed
corrections Gralla2020a ; Johnson_2020 . The definition of critical parameters
and the critical curve corresponding to bound photon orbits are presented in
Appendix B.4. For illustration, tick plots for the first subimage
corresponding to the direct image in Figure 1 are shown in Figure 2, now
displayed for clarity for just a single emission radius $r_{\rm s}=6$. The
critical impact parameters $b_{c}$ are also shown as dashed lines. Note that
all of the tick plots display relative polarized intensity from (35). The
polarized flux from subimages is suppressed by a demagnification factor, which
is described in Appendix B.6. When direct and indirect images are summed, this
can cause depolarization in the total image Jimenez-Rosales:2021ytz .

The low-spin and high-spin signatures in Figure 1 showcase relatively similar
EVPAs. The effect of spin becomes slightly more visible at smaller emission
radii due to the stronger lensing effects of the higher curvature nearer to
the black hole. Relative to Figure 1, the photon arrival positions in Figure 2
have all been pushed closer to the critical curve, and the distinction between
the low-spin and high-spin EVPAs is consequently more pronounced in the
indirect image than in the direct image. We quantify these effects for the
face-on case in Section V.

Figure 3: Polarized intensity tick plots for direct image with
$\theta_{o}=50^{\circ}$ and boost parameter $\beta_{v}=0.3$, shown for various
equatorial field configurations $\vec{B}=(B^{(r)},B^{(\phi)},B^{(\theta)})$
and fluid velocities $\chi$ (see (5)). The fluid configurations are taken,
from left to right, to match those in the top left, top right, and bottom left
of Figure 5 of Narayan_2020 , with boost direction anti-parallel to the
magnetic field. Each plot shows two spins ($a=0,-0.99$ in red and blue,
respectively) and two emission radii ($r_{\rm s}=3,6$, the inner and outer
rings, respectively).  Figure 4: Polarized intensity tick plots for the
indirect images corresponding to the direct images shown in Fig. 3 for
$\theta_{o}=50^{\circ}$ and $\beta_{v}=0.3$, displayed for the single emission
radius $r_{\rm s}=6$ with critical impact parameters shown as dashed lines.

#### Inclined Observers

The effects of spin on the observed polarization pattern become more
pronounced with increasing observer inclination $\theta_{o}$, as one may
intuitively expect from the appearance of $a\sin\theta_{o}$ in (31). An
illustrative example is shown in Figure 3, which displays tick plots for the
direct image of various equatorial field configurations
$\vec{B}=(B^{(r)},B^{(\phi)},B^{(\theta)})$ and boost directions $\chi$ (see
(5)) observed at a moderately high inclination $(\theta_{o}=50^{\circ})$ and
boost parameter ($\beta_{v}=0.3$). Again, the inner and outer circles
correspond to emission radii of $r_{\rm s}=3$ and $r_{\rm s}=6$, respectively.
The fluid rotation and boost configurations are taken, from left to right, to
match those in the top left, top right, and bottom left of Figure 5 of
Narayan_2020 . The boost direction is chosen so that it is parallel to the
magnetic field. As in the face-on case, the effect of spin becomes stronger at
smaller emission radii. Due to the moderately high observer inclination, in
all three panels, the bottom half of the image (corresponding to emission from
the near half of the midplane) is pushed towards the $\alpha$ axis and has
significantly lower intensity.

Note that for the cases of purely radial and toroidal fields (left two panels)
the polarized intensity at $r_{\rm s}=6$ is greater than at $r_{\rm s}=3$,
while the reverse holds for vertical fields (right panel). This is due to the
stronger lensing effects at smaller radii, which increase the relative local
$p^{(r)}$ and therefore decrease the pitch angle (28) for
$B=B^{(r)},B^{(\phi)}$ and increase it for $B=B^{(\theta)}$ (see (26)).

In Figure 4, we also show the corresponding tick plots for the first subimage
($m=1$ for bottom half of image and $m=2$ for top half of image), now
displayed for clarity for only a single emission radius $r_{\rm s}=6$. As in
Figure 2, we overlay the critical curves for the low-spin and high-spin
geometries as dashed lines. The high-spin critical curve is substantially
displaced from the image origin, significantly affecting the photon arrival
position. The corresponding polarization is altered both by the change in
arrival position and the explicit dependence of polarization on spin in (31).
Because the photon arrival positions are closer to the critical curve, the
subimage shows a much higher degree of symmetry across the $\hat{\alpha}$ axis
than the direct image. Note that compared with the direct image, the intensity
in the indirect image is no longer diminished for vertical fields because
photons in the indirect image experience stronger lensing effects and their
emission angle (28) relative to the vertical magnetic field no longer must be
small.

##  V EVPA of On-Axis Observers

This section presents an approximate analytic expression for the effect of
spin on EVPA in face-on images, and demonstrates the accuracy of this
approximation using the toy model.

###  V.1 Direct Image

Figure 5: Face-on EVPA as a function of emission radius (in units of $M$) at
$\varphi=0$ on the image for purely radial $B=B^{(r)}$, toroidal
$B=B^{(\phi)}$, and vertical $B=B^{(\theta)}$ magnetic fields. The numerically
computed values are shown for $a=0$ and $a=-0.99$ in blue and green,
respectively, and the approximations for zero-spin (Beloborodov) and high-spin
in (44) are shown as dashed yellow and purple lines, respectively.

As described in Section III.5, the direct image seen by an on-axis observer
corresponds to source momenta with $\lambda=0$ and is parametrized by the
impact parameter $b=\sqrt{\eta+a^{2}}$. To leading order, $b$ is related to
the source radius $r_{\rm s}$ by “just adding 1” with subleading corrections
given in (41). Computing $(\kappa_{1},\kappa_{2})$ as described in Section
III.3 with $(\lambda=0,\eta=\sqrt{b^{2}-a^{2}})$ yields an expression for the
EVPA as a function of $b$ and $r_{\rm s}$, which may be series expanded in
$r_{\rm s}$ using (41). Applying this expansion to (39) with $\beta_{v}=0$, we
obtain

$\displaystyle\text{EVPA}=\eta_{e}+\varphi+\frac{B^{(\theta)}\sin\eta_{e}}{B_{\rm
eq}r}-\frac{2a}{r^{2}}$ $\displaystyle\ \
+\frac{\sin\eta_{e}\left(16\cos\eta_{e}(B_{\rm
eq}^{2}{+}2(B^{(\theta)})^{2}){+}3B_{\rm
eq}B^{(\theta)}(5\pi{-}16)\right)}{8B_{\rm eq}^{2}r^{2}}$ $\displaystyle\ \
+\mathcal{O}\left(\frac{1}{r_{\rm s}^{3}}\right)$ (42)

with $\vec{B}_{\rm eq}\equiv B_{\rm
eq}(\cos\eta_{e}(\hat{r})+\sin\eta_{e}(\hat{\phi}))$ (as in Narayan_2020 ).
The leading order correction to EVPA due to spin is $-2a/r^{2}$, which is
independent of the magnetic field and hence a purely geometric effect. The
effect of spin is suppressed by two orders of magnitude compared to the
leading order term, explaining the similarity between the low-spin and high-
spin EVPAs seen in Figure 1.

As discussed in detail in Narayan_2020 , the Schwarzschild impact parameter is
extremely well-approximated by the Beloborodov relation Beloborodov_2002 ,
which for a face-on black hole is (compare $b^{2}=r_{\rm s}^{2}+2r_{\rm
s}+a^{2}+\mathcal{O}(r_{\rm s}^{-1})$ using (41)):

$\displaystyle b_{\rm Belo}=\sqrt{r_{\rm s}(2+r_{\rm s})}.$ (43)

Defining $\text{EVPA}_{\rm Belo}\equiv\text{EVPA}(b_{\rm Belo})$, this
implies888We also remark that we have not yet found an accurate series
expansion akin to (44) for arbitrary inclinations.

EVPA $\displaystyle\approx\text{EVPA}_{\rm belo}-\frac{2a}{r_{\rm s}^{2}},$
(44)

which holds in the limit where $B_{\rm eq}\to 0$, even though (V.1) will not
be well-defined. One can also show that this relation holds for nonzero
$\beta_{v}$, indicating that the primary effect of frame-dragging on
polarization does not depend on the orbital speed of the accreting material.
Note that the minus sign in (44) implies that increasing $a$ will cause the
EVPA to rotate in the opposite direction of the spin.

The $-2a/r_{\rm s}^{2}$ correction matches the calculations performed by
Pineault_1977 ; Fayos_1982 ; Brodutch_2011  and first termed “gravitational
Faraday rotation” by Ishihara_1988 .999 The setup of our problem is slightly
different than that of Pineault_1977 ; Fayos_1982 ; Brodutch_2011 . While we
fix $\varphi$, they effectively fix $\phi$. The results that appear within the
calculation, however, still match. If one instead fixes $\phi$, the arrival
coordinate $\varphi$ will shift by $\sim 2a/r_{\rm s}^{2}$ and the net
polarization rotation will be $\mathcal{O}(r_{\rm s}^{-3})$ Pineault_1977 ;
Fayos_1982 . Weak-field expansions performed by Ishihara_1988 ; Nouri_1999
show a leading order correction of $\mathcal{O}(r_{\rm min}^{-3})$, with
$r_{\rm min}$ the largest root of the radial potential. Past work derived this
result by analyzing local reference frames/directions at every point on the
geodesic in question. Because such reference frames are not unique, the
ambiguity historically led to disagreements about the effects of spin on
parallel transport Fayos_1982 . By combining conservation of $\kappa$ with the
“just add one” approximation, we entirely overstep this issue, providing a
clean approximation of the gravitational Faraday effect that is recast in
terms of only the emitter and observer frames.

In Figure 5, we test the accuracy of $\eqref{eq:faceonapprox}$ by plotting the
EVPA as a function of emission radius (scaled in units of $M$) at $\varphi=0$,
on the image for purely radial $B=B^{(r)}$, toroidal $B=B^{(\phi)}$, and
vertical $B=B^{(\theta)}$ magnetic fields. By axisymmetry, the EVPA at an
arbitrary angle $\varphi=\vartheta$ is offset by exactly $\vartheta$ from the
EVPA at $\varphi=0$. The numerically computed values are shown for $a=0$ and
$a=-0.99$ in blue and green, respectively, and the approximations for $a=0$
(Beloborodov) and $a=1$ in (44) are shown as dashed yellow and purple lines,
respectively. The approximation (44) is demonstrated to hold reasonably well
in all cases for $r\gtrsim 6M$. In the case of radial and toroidal fields with
$\eta_{e}=0$ and $\eta_{e}=\pi$, respectively, the EVPA will asymptote to
$\eta_{e}+\varphi$ (up to additive factors of $\pi$). For the purely vertical
field, the EVPA asymptotes to $\varphi$.

Note that the Schwarzschild EVPA is constant over emission radius due to
spherical symmetry of the Schwarzschild solution, which implies that fixed
screen angle $\varphi$ corresponds to fixed emission angle $\phi$ (see (49)
below). In this case, the EVPA at fixed $\varphi$ remains unchanged as a
function of $r_{\rm s}$ for magnetic fields that are aligned with the local
frame axes.

Since the effect of spin is to rotate all polarization ticks by an equal
amount $(-2a/r_{\rm s}^{2})$, the geometric imprint of spin on polarization is
indistinguishable from that of electromagnetic Faraday rotation by a coherent
external screen at a fixed observing frequency. Therefore, single-frequency
observations at the resolution of the EHT will likely be unable to determine
black hole spin from the geometric rotation alone. However, because the
geometric rotation is achromatic while Faraday effects are chromatic, it could
be isolated by observations at multiple frequencies. In addition, if the
magnetic field and spin of the black hole can be determined via other methods,
deviations from the uniform geometric rotation could provide information about
internal Faraday rotation, which causes differential relative rotation across
the image.

While the geometric effect of spin on photon trajectories is subleading, spin
can strongly influence the accretion dynamics and emissivity profile of the
plasma surrounding the black hole. For instance, the location of the innermost
stable circular orbit (ISCO) depends strongly on spin and in some cases may
control where the accretion disk truncates, altering the resultant appearance
of the black hole. This principle has helped guide numerous measurements of
black hole spin using both X-ray reflection spectroscopy and continuum fitting
Remillard:2006 ; Reynolds_2013x ; Mcclintock_2011 ; Brenneman_2013 ).
Comparison of the simple geometric approximation (44) with observed data and
GRMHD simulations could provide additional ways to distinguish geometric
effects of spin from these astrophysical effects of spin. These phenomena,
which cannot be disentangled in observed data, are also difficult to
disentangle in GRMHD simulations. The GRMHD simulations are evolved from
initial conditions in a fully general relativistic framework, allowing the
effects of spin simultaneously change the geometry, the distribution of
matter, and the magnetic field configurations therein PaperV .

It is instructive to compare the results of (44) to the work of Palumbo_2020 ,
which analyzed the EHT GRMHD library by examining rotational modes of images
with low-inclination $(17^{\circ})$ and found that the phase of the second
rotational Fourier mode (which describes rotational symmetry and is denoted
$\beta_{2}$ Palumbo_2020 ) shifted by over $40^{\circ}$ from $a=0$ to
$a=0.94$. By contrast, (44) suggests that for an emission radius of e.g.
$r_{\rm s}=6$ (the Schwarzschild ISCO), $\beta_{2}$ should only rotate by
$4a/r_{\rm s}^{2}\sim 6^{\circ}$. The spin-dependent effects present in the
images analyzed by Palumbo_2020  are therefore likely a result of the spin-
dependent accretion dynamics and not of the changing underlying geometry in
which the geodesics propagate.

###  V.2 Subimages

The preceding face-on EVPA approximation may be generalized to subimages using
the fact that for large $m$, the impact parameter $b$ approaches the critical
impact parameter $b_{c}$ up to exponentially suppressed corrections
Gralla2020a ; Johnson_2020 . Performing the series expansion of (39) in
$r_{\rm s}^{-1}$ with the replacement $b\to b_{c}$, one finds that the frame
dragging produces an $\mathcal{O}(1)$ effect on the subimage EVPA:

$\displaystyle\text{EVPA}_{m}\approx\text{EVPA}_{m,{\rm
Sch}}+(-1)^{m+1}\sin^{-1}(a/b_{c})+\mathcal{O}\left(\frac{1}{r_{\rm
s}}\right),$ (45)

where $\text{EVPA}_{m,{\rm Sch}}$ is the Schwarzschild face-on subimage
($m>0$) EVPA. The critical impact parameter $b_{c}$ has a closed form
expression in terms of $a,M$ (given by e.g. Eq. (67) of Gralla2020a ). Note
that $\sin^{-1}(a/b_{c})$ is approximated by $a/\sqrt{27}$ to within $8\%$ for
all spins $-1\leq a\leq 1$, and $\sin^{-1}(a/b_{c})|_{a=1}\approx 12^{\circ}$.

Like the approximation (V.1) for the direct image, (45) holds for any magnetic
field configuration and boost parameter. The correction $\sin^{-1}(a/b_{c})$
is analogous to the “gravitational Faraday rotation” in the indirect image,
thus extending the work of Balazs_1958 ; Pineault_1977 ; Fayos_1982 ;
Ishihara_1988 ; Nouri_1999 ; Brodutch_2011  to the strongly lensed regime.

Unlike in the direct case, the effects of spin on the indirect image remain
important even at large emission radii since the trajectory of a photon in the
indirect image includes a radial turning point, which implies that the photon
must move towards the black hole at some point after its emission into regions
of higher curvature in which geometrical effects are important.

We demonstrate this effect in Figure 6, which displays subimage EVPA as a
function of emission radius (scaled in units of $M$) at $\varphi=0$. Again, by
axisymmetry, the EVPA at an arbitrary angle $\varphi=\vartheta$ is offset by
exactly $\vartheta$ from the EVPA at $\varphi=0$. For maximal spin $a=-0.99$
the numerically computed $m=1,2,3$ subimage EVPAs are shown in red, green, and
blue, respectively. The Schwarzschild EVPA, which does not depend on $m$ at
fixed $\varphi$ for magnetic fields aligned with the local frame axes (see
discussion of Figure 5 in the previous subsection), is shown in black, and the
approximation (45) of $\text{EVPA}_{\rm Sch}\pm\sin^{-1}(a/b_{c})$ is shown
with a dashed black line. In Figure 6, the higher order subimages near a high-
spin black hole oscillate in $m$ about the Schwarzschild EVPA before
asymptoting to $\text{EVPA}_{\rm Sch}\pm\sin^{-1}(a/b_{c})$ at large $r_{\rm
s}$ and $m$.101010Note that we at small radii, the EVPA of the $a=-0.99$,
$m=1,2,3$ subimages shown in Figure 6 is closer to the Schwarzschild value
than the corresponding $a=-0.99$ direct $m=0$ image shown in Figure 5. We have
yet to find an intuitive explanation for this.

Figure 6: Face-on EVPA at $\varphi=0$ as a function of emission radius (in
units of $M$) for the first three sub-images. The columns are displayed for
purely radial $(B=B^{(r)},\eta_{e}=0)$, toroidal
$(B=B^{(\phi)},\eta_{e}=\pi)$, and vertical $(B=B^{(\theta)})$ magnetic
fields. For $a=-0.99$ the numerically computed $m=1,2,3$ subimage EVPAs are
shown in red, green, and blue, respectively. The Schwarzschild EVPA is shown
in black and the approximation $\text{EVPA}_{\rm Sch}\pm\sin^{-1}(a/b_{c})$ is
shown with a dashed line.

##  VI Orbiting Hotspots

An important application of our model is the computation of the polarization
of an orbiting “hotspot.” As discussed in the introduction, such hotspots are
frequently invoked to explain the observed flares in near-infrared and sub-
millimeter frequencies near Sgr A*. These flares can be tracked on short time-
scales.

###  VI.1 Setup

In our framework, the hotspot can be modelled as a point-source emitter
orbiting on a circular, prograde, equatorial geodesic. Our simple model of
point-source emission neglects any effects associated with the internal
structure of the hotspot, such as shearing and cooling. Simulations of
hotspots that include shearing and other plasma effects can be found in, e.g.,
Jeter_2020 ; Tiede_2020 .

Measured from the ZAMO (which is not on a geodesic), the geodesic emitter has
apparent boost (see (5) and e.g. Eq. (13) of Gates2020 ):

$\displaystyle\beta_{v}$
$\displaystyle=\frac{a^{2}-2|a|\sqrt{r}+r^{2}}{\sqrt{a^{2}+r(r-2)}\left(|a|+r^{3/2}\right)},\
\ \ \chi=-\pi/2,$ (46)

where we explicitly include the absolute value to emphasize that under
$a\to-a$, the boost remains in the same direction relative to the ZAMO. To
track hotspot polarization, it is useful to work with the Stokes parameters
$Q$ and $U$, defined following Himwich2020  by

$\displaystyle Q$ $\displaystyle=(f^{\beta})^{2}-(f^{\alpha})^{2},\qquad
U=-2f^{\alpha}f^{\beta}.$ (47)

Note that our model of pure synchrotron radiation has a polarization fraction
of $1$, i.e. $\sqrt{Q^{2}+U^{2}}=I$.

Tracking the motion of the hotspot in the $Q,U$ plane gives rise to
polarization loops that trace the screen polarization pattern over time. The
phase in $Q,U$ space tracks the EVPA via the relation

${\rm EVPA}=\frac{1}{2}\arctan{\left(\frac{U}{Q}\right)},$ (48)

following (36), and the magnitude in $Q,U$ space tracks the polarized flux of
the hotspot image. To convert intensity to flux, we multiply by a
magnification factor that relates a differential area element on the
observer’s screen to a differential area element on the emitter’s screen
(described in detail in Appendix B.6).

Figures 7 and 8 illustrate $Q,U$ loops produced by our semianalytic model for
the intensity of direct emission of a hotspot in a purely vertical magnetic
field and a purely equatorial field, respectively. Each panel of these figures
shows spins of $a=0,-0.99$ in red and blue, respectively, as well as the three
inclinations of $\theta_{o}=20^{\circ},45^{\circ},70^{\circ}$ in rows from top
to bottom, and three emission radii of $r_{\rm s}=6M,8M,10M$ in columns from
left to right. Note that radii $r_{\rm s}\geq 6$ are outside (or at) the ISCO
and admit stable orbits for all spins. The top left panel of each figure also
indicates the location of four azimuthal emission coordinates $\phi$, spaced
by $90^{\circ}$ on the Schwarzschild loop, computed using the fact that
Schwarzschild geodesics lie in a plane and taking $\phi=\varphi=0$ aligned on
the sky Narayan_2020 ,

$\displaystyle\tan(\varphi-n\pi)=\tan\phi\cos\theta_{o},$ (49)

where $n$ denotes subimage number (recall that $n=0$ has $m=1$ for $\beta>0$
and $m=0$ for $\beta<0$; see Footnote 5).

Figure 7: Polarized flux $Q$ vs. $U$ for equatorial magnetic fields. Plots are
shown for spins of $a=0,-0.99$ in red and blue, respectively, as well as three
inclinations of $\theta_{o}=20^{\circ},45^{\circ},70^{\circ}$ in rows from top
to bottom, and three emission radii of $r_{\rm s}=6M,8M,10M$ in columns from
left to right. The top left panel uses black dots to indicate four azimuthal
emission coordinates $\phi$ on the Schwarzschild loop, spaced by $90^{\circ}$.
Black crosshairs indicate the origin of each plot. Figure 8: Polarized flux
$Q$ vs. $U$ for vertical magnetic field. Plots are shown for spins of
$a=0,-0.99$ in red and blue, respectively, as well as three inclinations of
$\theta_{o}=20^{\circ},45^{\circ},70^{\circ}$ in rows from top to bottom, and
three emission radii of $r_{\rm s}=6M,8M,10M$ in columns from left to right.
The top left panel uses black dots to indicate four azimuthal emission
coordinates $\phi$ on the Schwarzschild loop, spaced by $90^{\circ}$. Black
crosshairs indicate the origin of each plot.

As discussed in Sections IV and V, the differences between polarization
emitted from an orbiter of fixed boost for high and low spin black holes is
generally small at $r_{\rm s}\geq 6$, with the largest differences appearing
at high inclination. For the geodesic orbiter (46) considered in this section,
the boost parameter $\beta_{v}$ depends strongly on $a$, which is the primary
source of differences seen in the $Q,U$ loops for low and high spins in
Figures 7 and 8.

The $Q,U$ loops arising from equatorial and vertical magnetic field
configurations have different topological structure. In particular, the
equatorial field configuration gives rise to $Q,U$ plots with two loops
enclosing the origin at all spins, emission radii, and inclination angles,
while two loops are only present for low observer inclinations in the case of
vertical fields. We explore the variations in $Q,U$ loop topology in more
detail in the following section.

The $Q,U$ loop topology produced by our model, shown in Figures 7 and 8, is
qualitatively consistent with the results of polarization loop studies for Sgr
A* by GRAVITY Gravity_2018 ; Gravity_2020 . In particular, we find that purely
equatorial fields always give rise to two origin-enclosing $Q,U$ loops, while
fields with a nonzero vertical component can give rise to a single
loop.111111Results of Gravity_2018 ; Gravity_2020  are displayed in intensity-
normalized polarization fraction $Q/I,U/I$ and therefore cannot be
quantitatively compared to our $Q,U$ (we assume a polarization fraction of
$1$). However, normalization does not affect the observed phase in $Q,U$
space, so it is still meaningful to compare topological features of our loops.
Note that the double loops in the ray-traced and observed figures presented in
Gravity_2018 ; Gravity_2020  do not encircle the origin. A shift in observed
or simulated $Q,U$ loops relative to the results of our simple geometric model
of a point emitter could be due to the inclusion of indirect images, as we
illustrate in Section VI.3. A shift could also arise as a result of Faraday
conversion of Stokes $Q,U$ to circularly polarized Stokes $V$ and could also
arise for hotspots with physical extent large enough for
$\vec{p}\times\vec{B}$ to vary significantly over the emitting region. Any
stable polarized background will also add an offset (possibly
$\phi$-dependent) in $Q,U$.

The lack of strong dependence on spin in Figures 7 and 8 parallels prior
studies of both simulations and data. The work of Broderick:2006  ray-traced
hot-spots around Sgr A* in both NIR and radio frequencies and found minimal
spin-dependence of the hotspot light curve for fixed orbital radius.
Similarly, Gravity_2020  fit data for the July 22 Sgr A* flare using three
values of $a$ and saw minimal discrepancies.121212While the polarization
pattern of a point-like emitter orbiting on a fixed radius depends negligibly
on the spin of the black hole, unpolarized observations of physical hotspots
do provide an important observational target for the interferometric
experiments that can constrain spin very well Tiede_2020 .

###  VI.2 $Q,U$ Loop Topology

#### Equatorial Fields

For the equatorial fields $B^{(\theta)}=0$ shown in Figure 7, the $Q,U$ loops
in the direct emission all wrap twice around the origin, corresponding to
$\psi_{\rm net}=2\pi$, where $\psi_{\rm net}$ is the net rotation of the EVPA
over the course of one hotspot orbit. To explain this feature, we first note
that for $\theta_{o}=0$, axisymmetry dictates that the $Q,U$ diagram will
consist of two identical loops stacked on top of each other, both of which
encircle the origin. Note that axisymmetry also implies that the EVPA will
rotate in the same direction as the hotspot orbit. As $\theta_{o}$ increases,
the two loops will become distinct. Suppose for contradiction that one loop
were to disappear; by continuity, one of its points must pass through the
origin $(Q=U=0)$. But this is impossible for $B^{(\theta)}=0$ since
$p^{(z)}\neq 0$ implies that locally
$\sqrt{Q^{2}+U^{2}}\propto|\vec{B}\times\vec{p}|\neq 0$, which in the absence
of Faraday conversion is directly proportional to the observed value.

Thus, two origin-enclosing $Q,U$ loops are always present in our geometric
model with axisymmetric equatorial field configurations. In Figure 7, the
inner loop dramatically shrinks at high inclinations due to Doppler deboosting
and small pitch angles (see (28)), but always encircles the origin. With two
nested loops present in the image, the direction of EVPA rotation matches the
direction of motion of the hotspot on the sky.

#### Vertical Fields

In nonzero vertical fields $B^{(\theta)}\neq 0$, $Q,U$ loops can deform more
dramatically. In particular, as the observer inclination $\theta_{o}$
increases, the inner loop may contract to a point and then vanish altogether,
as demonstrated in each column of Figure 8 and also noted in Gravity_2020 .

In images with two $Q,U$ loops, $\psi_{\rm net}=2\pi$, and the direction of
EVPA rotation never changes over the hotspot orbit. In images with one $Q,U$
loop, the EVPA rotation direction instead changes twice over the hotspot orbit
before the EVPA returns to its initial value.

At moderate inclinations, the $Q,U$ curve in Figure 8 at
$\theta_{o}=45^{\circ}$ with $r=10M$ displays a lemniscate (figure-eight)
shape that crosses over itself due to strong Doppler effects, but has the same
EVPA rotation behavior as a single loop. In general, these lemniscates can
form and then pull all the way through themselves as $\theta_{o}$ increases,
producing the patterns seen in Figure 8 at $\theta_{o}=70^{\circ}$.

For illustration, in Figure 9, we plot the Schwarzschild EVPA as a function of
orbital azimuth (again using (49)) for a hotspot in a purely vertical field at
two observer inclinations, $\theta_{o}=38^{\circ},42^{\circ}$. The EVPA
continuously rotates in the same direction at $\theta_{o}=38^{\circ}$,
corresponding to two $Q,U$ loops, while the EVPA changes rotation direction
twice at $\theta_{o}=42^{\circ}$, corresponding to one loop.

Figure 9: Schwarzschild EVPA as a function of azimuth $\phi$ for a hotspot
orbiting in a purely vertical field. Two inclinations
$\theta_{o}=38^{\circ},42^{\circ}$ are shown in blue and red, respectively.

When considering only the direct image in our model, the inner $Q,U$ loop
always contracts to the origin as $\theta_{o}$ increases, as it must for
$\psi_{\rm net}=2\pi$ to change to $\psi_{\rm net}<180^{\circ}$; the single
loop with a point at the origin corresponds to the boundary case of $\psi_{\rm
net}=180^{\circ}$. We remark that the point to which the inner $Q,U$ loop
contracts always appears as a cusp in $Q,U$ space, where the curvature of the
$Q,U$ curve diverges. Though the cusp is smoothed out in physical observations
with finite resolution, it is instructive to consider its geometric
interpretation, included in Appendix D.

When the inner loop of a double-loop configuration is shifted off the origin,
it no longer has $\psi_{\rm net}=2\pi$. But as long as the outer loop
encircles the origin, then $\psi_{\rm net}\geq 180^{\circ}$. In Gravity_2018 ,
the observed single $Q,U$ loop for the July 28, 2018 flare had $\psi_{\rm
net}\geq 180^{\circ}$, implying that it encircles the $Q,U$ origin. For the
range of inferred radii, their simulations required a certain range of
observer inclination $\theta_{o}\sim 15{-}30^{\circ}$ for a loop enclosing the
origin to form (see Gravity_2018  Appendix D). This is consistent with our
analysis: for $Q,U$ loops of direct images in vertical fields, we find loops
enclosing the origin only at low-to-moderate inclinations. There is a single
origin-enclosing (and for us, origin-intersecting) loop for which the
polarization and hotspot orbital periods are equal, only present at a (radius-
dependent) critical inclination. Physical effects, discussed in the previous
and following subsections, that shift the loops would only function to modify
these ranges.

We note additionally that for fields with zero azimuthal component (such as
those that best fit the July 28 flare in Gravity_2020 ), if $\psi_{\rm
net}\leq 180^{\circ}$, then increasing $\theta_{o}$ will never cause
$\psi_{\rm net}>180^{\circ}$. To prove this, note that for a fixed $a$,
$r_{\rm s}$, and non-azimuthal $\vec{B}$ field, there is only one pair of
conserved quantities $(\eta,\lambda)$ that satisfy $Q=U=0$ in the direct
image, so there is only one inclination angle that satisfies $Q=U=0$ (see
Appendix D for further justification). Hence loops of the direct image can
pass through the origin at most once; if $\psi_{\rm net}\leq 180^{\circ}$,
increasing $\theta_{o}$ cannot cause $\psi_{\rm net}>180^{\circ}$ again.

###  VI.3 Including Subimages

Relative to the direct image, subimages appear both time-delayed and
demagnified due to the longer path length of the corresponding photon
trajectories. These effects are reviewed in Appendix B.5 and B.6,
respectively, and must be taken into account to compute the contribution of
indirect images to the polarized flux of an orbiting hotspot.

A sample polarized flux loop including the direct and first indirect image is
displayed in Figure 10 for a hotspot orbiting a Schwarzschild black hole with
$r_{\rm s}=6,\theta_{o}=11^{\circ}$, and $\vec{B}=\frac{1}{\sqrt{6}}(1,1,2)$.
As discussed in the previous subsection, the secondary loops in the direct
emission are origin-enclosing. Figure 10 illustrates that the time-delayed and
demagnified $Q,U$ contribution from subimages can generally displace the
secondary loop from encircling the origin.

When the direct and indirect images are summed, this can cause depolarization;
in this case, the depolarization will be strongest when the subimage has the
largest relative flux Broderick:2006 . Future investigations may reveal
whether depolarization in the total image could be a used as a robust
signature of photon subrings, and could be an interesting future application
of our model and traditional ray-tracing codes.

Figure 10: Polarized $Q,U$ flux loop for an orbiting hotspot at $r_{\rm
s}=6,\,a=0,\,\theta_{o}=11^{\circ},\,\vec{B}=\frac{1}{\sqrt{6}}(1,1,2)$,
including the direct image (solid line) and the sum of the direct plus first
indirect image (dashed line).

##  VII Summary

In this paper, we developed a semi-analytic toy model for polarized equatorial
synchrotron emission in the Kerr geometry. We computed the polarized images of
axisymmetric fluid orbiting in various magnetic field configurations. Our
simple model allowed us to isolate the individual effects of spin, magnetic
field, and observer inclination on polarization signatures of the direct image
and photon ring.

In the face-on case, we found that an analytic approximation to photon
trajectories yields a series expansion for the face-on EVPA, revealing a
$\mathcal{O}(1/r_{\rm s}^{2})$ correction from spin to the EVPA of the direct
image but a $\mathcal{O}(1)$ correction to the EVPA of the indirect image.
These calculations confirm results from previous studies of “gravitational
Faraday rotation” Balazs_1958 ; Pineault_1977 ; Fayos_1982 ; Brodutch_2011 ;
Ishihara_1988 ; Nouri_1999  and extend prior work to the strongly lensed
regime.

Our results bear direct relevance to observations and simulations of the
supermassive black hole M87*, which is believed to be viewed at nearly face-on
inclination (Walker_2018, ). Our model suggests that differences between
images of low-spin and high-spin simulations of M87* arise primarily from
differences in accretion dynamics, as opposed to the frame-dragging of null
geodesics. In the future, we hope to extend our calculations of analytic
approximations to inclined observers as well.

As an additional application of our toy model, we computed the polarized image
of orbiting hotspots and provided an analytical framework for interpreting the
topology of $Q,U$ loops observed in Sgr A* flares. In particular, we explained
how the magnetic field direction and observer inclination individually control
the number and shape of $Q,U$ loops present, consistent with the
interpretation provided by the GRAVITY collaboration Gravity_2018 ;
Gravity_2020 .

While our toy models provide insight through simplicity, more complicated
astrophysical effects are necessary to fully describe observed polarization
patterns. For example, a realistic accretion model that includes turbulence
cannot be modelled by axisymmetric rings of matter orbiting in a constant
magnetic field, and Faraday effects can scramble the observed polarization
pattern. Our work does not attempt to provide a complete, physically realistic
model; many highly-developed ray-tracing codes already exist for this purpose
(such as Dexter_2016 ; Moscibrodzka:2018 ; Younsi_2020 ). Instead, our simple
toy model provides intuition for the geometric effects of the Kerr spacetime
on photon propagation and its imprint on polarized images.

We have disentangled the roles of astrophysics and geometry, paving the way
for a more complete understanding of polarimetric observations and how they
are affected by a multitude of competing factors. Simple toy models have
already played a key role in interpreting polarized black hole images (e.g.,
PaperVIII ; Narayan_2020 ), and we hope that our model and its future
developments – such as extensions to include non-equatorial emission and
circular polarization – will continue to provide new insights into
polarimetric observations and simulations.

## Acknowledgements

We thank Avery Broderick, Dominic Chang, Andrew Chael, Delilah Gates, Ramesh
Narayan, Angelo Ricarte, Paul Tiede, and Maciek Wielgus for useful
discussions. We also thank Ziri Younsi for helpful comments on our manuscript.
This work was supported in part by the Black Hole Initiative, which is funded
by grants from the John Templeton Foundation and the Gordon and Betty Moore
Foundation to Harvard University. We thank the National Science Foundation
(AST-1716536, AST-1935980) and the Gordon and Betty Moore Foundation
(GBMF-5278) for financial support of this work.

##  Appendix A Details in Orbiting Fluid Model

In this appendix, we include for concreteness the explicit expressions for the
ZAMO tetrad and boost matrices described in Section II. The ZAMO tetrad
appearing in (4) can be arranged into a matrix

$\left(e_{\rm{Z}}\right)^{\mu}_{(a)}=\begin{pmatrix}\frac{1}{r_{\rm
s}}\sqrt{\frac{\Xi_{\rm s}}{\Delta_{\rm s}}}&0&\frac{\omega_{\rm s}}{r_{\rm
s}}\sqrt{\frac{\Xi_{\rm s}}{\Delta_{\rm s}}}&0\\\ 0&\frac{\sqrt{\Delta_{\rm
s}}}{r_{\rm s}}&0&0\\\ 0&0&\frac{r_{\rm s}}{\sqrt{\Xi_{\rm s}}}&0\\\
0&0&0&-\frac{1}{r_{\rm s}},\\\ \end{pmatrix}$ (50)

so that

$\displaystyle\left(\begin{array}[]{c}V_{(t)}\\\ V_{(r)}\\\ V_{(\phi)}\\\
V_{(\theta)}\\\ \end{array}\right)$
$\displaystyle=e_{Z}\,\left(\begin{array}[]{c}V_{t}\\\ V_{r}\\\ V_{\phi}\\\
V_{\theta}\\\ \end{array}\right).$ (59)

The Lorentz transformation $\Lambda^{(a)}_{\ \ (b)}$ corresponding to (5) is
given by the matrix

$\small\Lambda=\begin{pmatrix}\gamma&-\beta\gamma\cos\chi&-\beta\gamma\sin\chi&0\\\
-\beta\gamma\cos\chi&(\gamma-1)\cos^{2}\chi+1&(\gamma-1)\sin\chi\cos\chi&0\\\
-\beta\gamma\sin\chi&(\gamma-1)\sin\chi\cos\chi&(\gamma-1)\sin^{2}\chi+1&0\\\
0&0&0&1\\\ \end{pmatrix}.\normalsize$ (60)

The explicit components of the tetrad (7) relating the boosted local frame to
Kerr are:

$\displaystyle e^{\prime}_{(t)}$ $\displaystyle=\frac{\gamma}{r_{\rm
s}}\sqrt{\frac{\Xi_{\rm s}}{\Delta_{\rm
s}}}\partial_{t}+\frac{\beta\gamma\cos\chi}{r_{\rm s}}\sqrt{\Delta_{\rm
s}}\partial_{r}$ (61) $\displaystyle+\left(\frac{\gamma\omega_{\rm s}}{r_{\rm
s}}\sqrt{\frac{\Xi_{\rm s}}{\Delta_{\rm s}}}+\frac{r_{\rm
s}\beta\gamma\sin\chi}{\sqrt{\Xi_{\rm s}}}\right)\partial_{\phi},$
$\displaystyle e^{\prime}_{(r)}$
$\displaystyle=\frac{\beta\gamma\cos\chi}{r_{\rm s}}\sqrt{\frac{\Xi_{\rm
s}}{\Delta_{\rm s}}}\partial_{t}+\frac{1+(\gamma-1)\cos^{2}\chi}{r_{\rm
s}}\sqrt{\Delta_{\rm s}}\partial_{r}$
$\displaystyle+\left(\frac{\beta\gamma\omega_{\rm s}\cos\chi}{r_{\rm
s}}\sqrt{\frac{\Xi_{\rm s}}{\Delta_{\rm s}}}+\frac{r_{\rm
s}(\gamma-1)\cos\chi\sin\chi}{\sqrt{\Xi_{\rm s}}}\right)\partial_{\phi},$
$\displaystyle e^{\prime}_{(\phi)}$
$\displaystyle=\frac{\beta\gamma\sin\chi}{r_{\rm s}}\sqrt{\frac{\Xi_{\rm
s}}{\Delta_{\rm s}}}\partial_{t}+\frac{(\gamma-1)\cos\chi\sin\chi}{r_{\rm
s}}\sqrt{\Delta_{\rm s}}\partial_{r}$
$\displaystyle+\left(\frac{\beta\gamma\omega_{\rm s}\sin\chi}{r_{\rm
s}}\sqrt{\frac{\Xi_{\rm s}}{\Delta_{\rm s}}}+\frac{r_{\rm
s}((\gamma-1)\sin^{2}\chi+1)}{\sqrt{\Xi_{\rm s}}}\right)\partial_{\phi},$
$\displaystyle e^{\prime}_{(\theta)}$ $\displaystyle=-\frac{1}{r_{\rm
s}}\partial_{\theta}.$

##  Appendix B Details of Semi-Analytic Calculation

###  B.1 Definition of Special Functions

The special functions in Section III.2 are defined:

$\displaystyle F(x|k)$
$\displaystyle=\int_{0}^{x}\frac{d\theta}{\sqrt{1-k\sin^{2}\theta}}=\int_{0}^{\sin{x}}\frac{dt}{\sqrt{(1-t^{2})(1-kt^{2})}},$
(62) $\displaystyle K(k)$ $\displaystyle=F\left(\frac{\pi}{2}\big{|}k\right),$
$\displaystyle\text{sn}(u|k)$ $\displaystyle=\sin\left(F^{-1}(x|k)\right),$

compatible with Mathematica 12 Kapec2020 . Elliptic functions are computed in
python using the package mpmath mpmath .

###  B.2 Radial Roots

See Appendix A of Gralla2020a . We introduce

$\displaystyle\mathcal{A}$ $\displaystyle=a^{2}-q^{2}-\lambda^{2},$ (63)
$\displaystyle\mathcal{B}$
$\displaystyle=2M\left[q^{2}+(\lambda-a)^{2}\right],$
$\displaystyle\mathcal{C}$ $\displaystyle=-a^{2}q^{2},$
$\displaystyle\mathcal{P}$
$\displaystyle=-\frac{\mathcal{A}^{2}}{12}-\mathcal{C},$
$\displaystyle\mathcal{Q}$
$\displaystyle=-\frac{A}{3}\left[\left(\frac{\mathcal{A}}{6}\right)^{2}-\mathcal{C}\right]-\frac{\mathcal{B}^{2}}{8},$

and

$\displaystyle z$
$\displaystyle=\sqrt{\frac{-2(3^{1/3}\mathcal{P})+2^{1/3}\mathcal{H}^{2/3}}{2(6^{2/3}\mathcal{H}^{1/3})}-\frac{\mathcal{A}}{6}}>0,$
(64) $\displaystyle\mathcal{H}$
$\displaystyle=-\mathcal{Q}+\sqrt{12\mathcal{P}^{3}+81\mathcal{Q}^{2}}.$

In terms of these definitions, the four roots are

$\displaystyle r_{1}$
$\displaystyle=-z-\sqrt{-\frac{\mathcal{A}}{2}-z^{2}+\frac{\mathcal{B}}{4z}},$
(65) $\displaystyle r_{2}$
$\displaystyle=-z+\sqrt{-\frac{\mathcal{A}}{2}-z^{2}+\frac{\mathcal{B}}{4z}},$
$\displaystyle r_{3}$
$\displaystyle=z-\sqrt{-\frac{\mathcal{A}}{2}-z^{2}-\frac{\mathcal{B}}{4z}},$
$\displaystyle r_{4}$
$\displaystyle=z+\sqrt{-\frac{\mathcal{A}}{2}-z^{2}-\frac{\mathcal{B}}{4z}}.$

Note that we have altered the root of the resolvent cubic from Gralla2020b  so
that all cube roots can be taken as principal (i.e. we take
$(-1)^{1/3}=e^{i\pi/3}$ as opposed to $-1$). This convention is default in
Mathematica 12 and numpy.

###  B.3 Sign $\pm_{r}$ of Initial Radial Motion

The sign $\pm_{r}$ in (9) depends on $\\{\lambda,\eta,r_{\rm
s},a,\theta_{o},m\\}$ and must be computed semi-analytically. To do so, one
must first check whether a given geodesic contains a radial turning point
(equivalently, whether the geodesic terminates inside or outside the critical
curve Gralla2020a ). Geodesics that do not contain a turning point (and thus
fall inside the critical curve) must have $\pm_{r}=+1$ to reach infinity. For
geodesics that do contain a turning point, one must check whether a ray shot
backwards encounters the turning point before or after the desired emission
coordinates. For geodesics with a turning point, the radial integral $I_{r}$
has antiderivative given by (A9) of Gralla2020b :

$\displaystyle\int\frac{dr}{\sqrt{\mathcal{R}(r)}}$
$\displaystyle=\frac{2}{\sqrt{r_{31}r_{42}}}F\left(\arcsin{\sqrt{\frac{r-r_{4}}{r-r_{3}}\frac{r_{31}}{r_{41}}}}\bigg{|}k\right).$
(66)

The turning point is located at $r_{4}$, at which point the antiderivative
vanishes. Following Gralla2020a , the radial integral $I_{r}^{\rm turn}$ for
the portion of the geodesic connecting the turning point to the observer is
simply the limit of Eq. (66) with $r\to\infty$, as we assume the observer is
located at infinity,

$\displaystyle I_{r}^{\rm turn}$
$\displaystyle=\frac{2}{\sqrt{r_{31}r_{42}}}F\left(\arcsin{\sqrt{\frac{r_{31}}{r_{41}}}}\bigg{|}k\right).$
(67)

Since $I_{r}=G_{\theta}$ is strictly increasing along the geodesic Gralla2020a
, then rays with $G_{\theta}^{m}<I_{r}^{\rm turn}$ will not encounter the
turning point, and rays with $G_{\theta}^{m}>I_{r}^{\rm turn}$ will encounter
the turning point. We therefore have

$\displaystyle\pm_{r}$ $\displaystyle=\begin{cases}+1&b<b_{c}\\\
\text{sign}(I_{r}^{\rm turn}-G^{m}_{\theta})&b>b_{c}.\end{cases}$ (68)

As $m$ grows, the boundary between rays with $\pm_{r}=1$ and $\pm_{r}=-1$
approaches the critical curve; rays with $m\gg 1$ that terminate outside the
critical curve must be emitted with $\pm_{r}=-1$ so that they can asymptote to
a spherical photon orbit before escaping to infinity. Thus
$\pm_{r}\to\text{sign}(b-b_{c})$ as $m\to\infty$.

###  B.4 Critical Parameters and Critical Curve

We follow Gralla2020a , to which we refer readers for more details and
discussion. Tildes denote “critical” parameters, evaluated at radii
$\tilde{r}$ at which the Kerr geometry admits bound photon orbits, lying
within $\tilde{r}_{-}\leq\tilde{r}\leq\tilde{r}_{+}$, with

$\tilde{r}_{\pm}=2M\left[1+\cos\left(\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right)\right].$
(69)

Photons on orbits with radius $\tilde{r}$ have critical parameters

$\displaystyle\tilde{\lambda}$
$\displaystyle=a+\frac{\tilde{r}}{a}\left[\tilde{r}-\frac{2\tilde{\Delta}}{\tilde{r}-M}\right],$
(70) $\displaystyle\tilde{\eta}$
$\displaystyle=\frac{\tilde{r}^{3}}{a^{2}}\left[\frac{4M\tilde{\Delta}}{(\tilde{r}-M)^{2}}-\tilde{r}\right].$

On the observer screen, these parameters define a closed “critical curve”
parametrized by $\tilde{r}$. Note that each radius $\tilde{r}$ defines two
points on the critical curve corresponding to the two possible values
$\pm_{o}$ (two possible values $\pm\beta$).

###  B.5 Schwarzschild Subimage Time Delay

Due to longer path lengths between the source and observer, subimages of an
emitting source appear at a time delay relative to the direct image. The
expression for time elapsed between source and observer follows from the
geodesic equation (see e.g. (7c) of Gralla2020a ), and in the Schwarzschild
$(a\to 0)$ case is given by

$\displaystyle\Delta t_{n}=t_{o,n}-t_{s,n}$ $\displaystyle=\fint
dr\,\frac{r^{3}}{(r-2)\sqrt{\mathcal{R}(r)}},$ (71)

where $t_{s,m}$ and $t_{o,m}$ denote the times of source emission and
observation, respectively, for the $n$-th subimage. To incorporate time delay
between the direct and indirect image in the hotspot model, we suppose that
the hotspot is located at $\phi=0$ when $t_{s}=0$. In the $a=0$ case, we may
use (49) to obtain

$\displaystyle\tan(\varphi-n\pi)=\tan[\omega_{\rm s}\,(t_{o,n}-\Delta
t_{n})]\cos\theta_{o},$ (72)

where $\omega_{\rm s}=-r_{\rm s}^{-3/2}$ is the angular velocity of the
equatorial geodesic in Schwarzschild Gralla2018  (corresponding to (46) with
$a=0$), and we have added the minus sign to be consistent with clockwise
motion on the sky ($\chi=-\pi/2$). Using (71-72), we numerically compute
$t_{o,n}(\varphi)$, and interpolate the inverse function $\varphi(t_{o,n})$ to
find $Q(\varphi(t_{o,n})),U(\varphi(t_{o,n}))$ as a function of $t_{o,n}$. We
sum the Stokes parameters of distinct subimages arriving at the same $t_{o}$
to obtain the total observed intensity or flux.

###  B.6 Magnification and Flux

To compute the image flux, we must consider a bundle of null geodesics with
infinitesimal but nonzero area emanating from the emitting source. The total
flux from such a bundle of null rays is the integral of the observer intensity
over the projection of the area of the bundle of rays on the observer’s
screen. In practice, this involves computing the Jacobian relating
differential area elements between the emitter and observer screen, which is
outlined in e.g. Cunningham1973 ; Gralla2018 . The area element of the solid
angle corresponding to the bundle of rays of the observer screen at radius
$r_{o}$ in Bardeen coordinates is $d\alpha d\beta/r_{o}^{2}$, so the Stokes
linearly polarized flux is the area integral

$\oiint\frac{d\alpha\
d\beta}{r_{o}^{2}}\left(Q,U\right)=\frac{\left(Q,U\right)}{r_{o}^{2}}\mathcal{A},\
\ \ \ \mathcal{A}=\oiint d\alpha\ d\beta,$ (73)

where $\mathcal{A}$ is the area of the image on the observer screen, and $Q,U$
are defined using (35) and (47), including the redshift but without including
the path length, as the hotspot is an isotropic circular emitter rather than a
thin disk. In (73) we take $Q,U$ (along with $r_{o}$) to be constant over the
image of an infinitesimal bundle of rays. We compute the area $\mathcal{A}$
directly following Gralla2018 , defining the local “source screen” $(t)=(r)=0$
in the frame of the orbiting geodesic emitter (corresponding to boost (46)).
Note that the factor of $r_{o}^{2}$ drops out of the ratio of direct and
indirect image flux.

##  Appendix C Image Symmetries

This appendix records changes in the polarized image under various parameter
sign flips. Such transformations correspond to symmetries of geodesics, which
only in certain fluid and magnetic field configurations correspond to
symmetries of the polarized image.

Figure 11: Sample images produced under various transformations of the “original image” in the leftmost panel, aligned with the corresponding symmetry transformation in Table 1. The original image has $r_{\rm s}=6,\,a=-0.5,\,\theta_{o}=30^{\circ},\,\vec{B}=\frac{1}{\sqrt{3}}(1,1,1)$, and boost parameters corresponding to a prograde geodesic (46). |  $a,\alpha$ $\downarrow$ $-a,-\alpha$ |  $a,\alpha,\chi$ $\downarrow$ $-a,-\alpha,-\chi$ |  $a,\alpha,\chi,B^{(\phi)}$ $\downarrow$ $-a,-\alpha,-\chi,-B^{(\phi)}$ |  $\theta_{o},\beta$ $\downarrow$ $\pi-\theta_{o},-\beta$ |  $\theta_{o},\beta,B^{(\theta)}$ $\downarrow$ $\pi-\theta_{o},-\beta,-B^{(\theta)}$
---|---|---|---|---|---
Arrival Position |  ✓ |  ✓ |  ✓ |  ✓ |  ✓
Doppler Boost |  ✗ |  ✓ |  ✓ |  ✓ |  ✓
$I,Q,U,V$ |  ✗ |  ✗ |  ✓ |  ✗ |  ✓
Table 1: Effects of various parameter transformations on geodesic arrival
position, Doppler boost factor, and the Stokes parameters $I,Q,U,V$ in our
model of equatorial emission. A check ✓ ( ✗ ) indicates that a quantity is (is
not, generically) invariant under the given transformation.

#### Observer Inclination $\theta_{o}\to\pi-\theta_{o}$

Consider a photon arriving at position $(\alpha,\beta)$ for an observer at
inclination $\theta_{o}<\frac{\pi}{2}$ relative to the positive black hole
spin axis. The observer at location $\pi-\theta_{o}$ will observe photons with
an opposite sign momentum along the spin axis, $\pm_{o}\to\mp_{o}$, and the
arrival position in $\beta$ will correspondingly flip: $\beta\to-\beta$. By
inspection of (14) the geodesic conserved quantities are invariant.131313The
direct image observed at inclination $\pi-\theta_{o}$ will thus have $m=0$ for
photons arriving on the top half of the image and $m=1$ for photons arriving
on the bottom half of the image. Note that Eq. 82 of Gralla2020a  generalizes
to $\overline{m}=m-H(\beta\cos\theta_{o})$, meaning the direct image with
$\overline{m}=0$ is unchanged by the transformation
$\theta_{o}\to\pi-\theta_{o},\beta\to-\beta$.

For the polarized image with a non-vertical field ($B^{(\theta)}=0$), the sign
flip $\pm_{o}$ will cause $f^{(r)}$ and $f^{(\phi)}$ to flip sign by
inspection of (26). In this case, $\kappa_{1}$ flips sign and the polarization
corresponding to $(f^{\alpha},f^{\beta})$ at $(\alpha,\beta)$ for an observer
at $\theta_{o}<\frac{\pi}{2}$ will appear as $(-f^{\alpha},f^{\beta})$ at
position $(\alpha,-\beta)$ for an observer at $\pi-\theta_{o}$. For fields
that contain nonzero $B^{(\theta)}$, the same result is produced under the
additional transformation $B^{(\theta)}\to-B^{(\theta)}$.

Furthermore, an observer at $\pi-\theta_{o}$ with flipped values of the
equatorial magnetic field $\vec{B}_{\rm eq}\to-\vec{B}_{\rm eq}$ corresponds
to a flip $f^{\theta}\to-f^{\theta}$, which complex conjugates
$\kappa\to\bar{\kappa}$. With this flip in fields, the polarization
corresponding to $(f^{\alpha},f^{\beta})$ at $(\alpha,\beta)$ for an observer
at $\theta_{o}<\frac{\pi}{2}$ will appear as $(f^{\alpha},-f^{\beta})$ at
position $(\alpha,-\beta)$ for an observer at $\pi-\theta_{o}$.

Since the polarization on the screen is defined up to an overall sign, in all
of the cases above we find that the polarized image will be reflected across
the $\alpha$ axis. Note also that at $\pi-\theta_{o}$, the apparent direction
of clockwise/counterclockwise fluid motion flips.

Finally, we note that $\theta_{o}\to-\theta_{o}$ corresponds to a rotation in
$\phi_{o}$, which will not change the image position of axisymmetric
configurations or the polarization pattern of axisymmetric magnetic field
configurations.

#### Spin Direction $a\to-a$

Consider again a photon arriving at position $(\alpha,\beta)$ for an observer
at inclination $\theta_{o}<\frac{\pi}{2}$ relative to the positive black hole
spin axis, which is defined as $\hat{z}$. If the spin flips sign, its angular
momentum changes to be aligned with $-\hat{z}$, so the definition of
$\hat{\beta}$ changes orientation and the observer will be at an angle
$\pi-\theta_{o}$ relative to the spin vector. In this new coordinate system,
rotation in $\phi$ changes from counterclockwise to clockwise on the screen,
and the image that appeared at $\alpha\hat{\alpha}+\beta\hat{\beta}$ in the
old coordinates will appear at $(-\alpha)\hat{\alpha}+(-\beta)(-\hat{\beta})$
in the new coordinates , i.e., with $\alpha$ flipped relative to the old image
position. By inspection of (14), $\lambda$ flips sign and $\eta$ is invariant;
thus $p^{\phi}$ flips sign.

For the polarized image of _unboosted_ ZAMO emission, if $B^{(\phi)}=0$, the
sign flip in $p^{\phi}$ will cause $f^{(r)}$ and $f^{(\theta)}$ to flip sign
by inspection of (26). In this case, $\kappa_{1}$ flips sign and $\mu$ also
flips sign, so the polarization $(f^{\alpha},-f^{\beta})$ at $(\alpha,\beta)$
for an observer of spin $+a$ aligned with $+\hat{z}$ will appear as the
polarization $(f^{\alpha},f^{\beta})$ to an observer with spin aligned $-a$ at
position $(-\alpha,\beta)$ in the original coordinates. For fields with
$B^{(\phi)}\neq 0$, the same result is produced under the additional
transformation $B^{(\phi)}\to-B^{(\phi)}$.

Furthermore, the transformation with unboosted emission
$a,B^{(r)},B^{(\theta)}\to-a,-B^{(r)},-B^{(\theta)}$ corresponds to a flip
$f^{\phi}\to-f^{\phi}$, which complex conjugates $\kappa\to\overline{\kappa}$.
Hence, under this transformation, the polarization corresponding to
$(f^{\alpha},f^{\beta})$ at screen coordinates $(\alpha,\beta)$ and spin $+a$
aligned with $+\hat{z}$ will appear as polarization $(-f^{\alpha},f^{\beta})$
to an observer with spin aligned $-a$ at position $(-\alpha,\beta)$.

Since the EVPA is defined up to a sign in $\pm(f^{\alpha},f^{\beta})$, for all
the $a\to-a$ cases above, the polarized image will be reflected across the
$\beta$ axis. Note also that with $a\to-a$, the direction of ZAMO motion
changes direction and will appear clockwise on the observer screen, with the
boost $\chi$ now in the opposite direction of the ZAMO motion. The
transformation $a,\chi\to-a,-\chi$ leaves the fluid frame tetrad
$e_{(a)}^{\,\mu}$ invariant, so the above transformations for $a\to-a$ apply
to boosted emission as well if the azimuthal boost direction flips
$\chi\to-\chi$.

#### Combined Symmetries

Since the polarization symmetries for $\theta_{o}\to\pi-\theta_{o}$ reflect
the image across the $\alpha$ axis while the polarization symmetries for
$a,\chi\to-a,-\chi$ flip the image across the $\beta$ axis, the two sets of
transformations will produce images that are rotated by $\pi$ relative to each
other.

We note additionally that $Q$ and $U$ are invariant under a symmetry
transformation if and only if the arrival positions of the geodesics are
unchanged and the pitch angle (28) is unchanged, which is equivalent to
preserving $I$. Hence $I,Q,U$ all obey the same set of symmetry relations.

The effect of a variety of the symmetry transformations on arrival position,
boost factor, and polarization presented in this section are listed in Table
1. A sample image with $r_{\rm
s}=6,\,a=-0.5,\,\theta_{o}=30^{\circ},\,\vec{B}=\frac{1}{\sqrt{3}}(1,1,1)$,
and boost parameters corresponding to a prograde geodesic (46) is explicitly
transformed to showcase these symmetries in Figure 11.

#### Circular Polarization

Our model can naturally be extended to include circular polarization, which is
encoded in Stokes $V$ and is invariant on geodesics in the absence of Faraday
effects. At the source, $V$ is a complicated function of $\vec{B}$ such that
the sign of the magnetic field components enters only into the sign of $V$ via
the relation ${\rm sign}(V)={\rm sign}(\vec{p}\cdot\vec{B})$ Ricarte:2021frd .
With this relation, we can derive the effect of the same symmetry
transformations, which only affect $V$ through sign$(V)$ (assuming the
circular polarization is entirely intrinsic). As previously stated, the
transformation $a,\alpha,\chi\to-a,-\alpha,-\chi$ flips the sign of
$p^{(\phi)}$, so taking
$a,\alpha,\chi,B^{(\phi)}\to-a,-\alpha,-\chi,-B^{(\phi)}$ leaves
$\vec{p}\cdot\vec{B}$ and hence sign$(V)$ and $V$ unchanged. Similarly, the
transformation $\theta_{o},\beta\to\pi-\theta_{o},-\beta$ flips the sign of
$p^{(\theta)}$, so taking
$\theta_{o},\beta,B^{(\theta)}\to\pi-\theta_{o},-\beta,-B^{(\theta)}$ also
leaves sign$(V)$ and $V$ unchanged.

These symmetries are also listed in Table 1. The other transformations listed
in Table 1 flip the sign of only certain components of the dot product
$\vec{p}\cdot\vec{B}$, so the effect on sign$(V)$ cannot be generically
determined.

#### Universal Subimage Symmetries

Our result (45) is also consistent with the subimage polarization symmetry
relations derived in Himwich2020 . In particular, suppose that
$\vec{B}=B^{(\theta)}$ so that equatorial sources have
$f_{m}^{\theta}=-f_{m+1}^{\theta}$. Together with
$p_{m}^{\theta}=-p_{m+1}^{\theta}$, this implies that $\kappa$ is complex
conjugated across subrings (as shown in Himwich2020 ). Then Eqs. (29), (38),
and (45) imply

$\displaystyle\frac{f^{\alpha}_{m}{+}f^{\alpha}_{m+1}}{f^{\beta}_{m}{+}f^{\beta}_{m+1}}$
$\displaystyle\propto\frac{\cos{({\rm EVPA}_{\rm
sch})}\frac{a}{b_{c}}}{\sin{({\rm EVPA}_{\rm sch})}\frac{a}{b_{c}}}=\cot({\rm
EVPA}_{\rm sch}),$
$\displaystyle\frac{f^{\beta}_{m}{-}f^{\beta}_{m+1}}{f^{\alpha}_{m}{-}f^{\alpha}_{m+1}}$
$\displaystyle\propto\frac{\pm(-1)^{m}\cos{({\rm EVPA}_{\rm
sch})}\sqrt{1{-}\frac{a^{2}}{b_{c}^{2}}}}{\mp(-1)^{m}\sin{({\rm EVPA}_{\rm
sch})}\sqrt{1{-}\frac{a^{2}}{b_{c}^{2}}}},$ $\displaystyle\qquad=-\cot({\rm
EVPA}_{\rm sch}),$ (74)

Noting that

$\cot({\rm EVPA}_{\rm sch})=\cot\varphi,$ (76)

the result agrees with Eqs. (36) and (37) of Himwich2020 . In re-deriving
these relations, we have used the additional fact that the screen polarization
vector $(f^{\alpha},f^{\beta})$ flips sign across subrings for the specific
case of $a=0$, $B=B^{(\theta)}$.

##  Appendix D $Q,U$ Cusp Formation in Vertical Fields

Figure 12: Polarization at $\theta_{\rm cusp}$ for a hotspot orbiting in a
magnetic field $\vec{B}=\frac{1}{\sqrt{2}}(0,1.0,1.0)$ around a Schwarzschild
black hole. Left panel: $\theta_{\rm cusp}$ as a function of emission radius
$r_{\rm s}$, with dashed lines intersecting at $r_{\rm s}=8$, $\theta_{\rm
cusp}=32.95^{\circ}$. Center panel: polarization $Q,U$ loop at $r_{\rm s}=8$,
$\theta_{\rm cusp}=32.95^{\circ}$. Right panel: EVPA as a function of orbital
azimuth for observer inclinations of $\theta_{o}=\theta_{\rm cusp}\pm
2^{\circ}$ in green and blue, respectively.

This appendix discusses an interesting feature of the $Q,U$ loop topology
(Section VI.2) for vertical fields displayed in Figure 8: as $\theta_{o}$
increases, the point to which the inner loop of the direct image contracts
always appears as a cusp in $Q,U$ space, where the curvature of the $Q,U$
curve diverges. The cusp will not appear in physical observations with finite
resolution, but it is still instructive to examine the geometric origin of
$Q,U$ loop cusp formation.

To gain intuition for this phenomenon, note as in Section VI.2 that for two
$Q,U$ loops, one has $\psi_{\rm net}=2\pi$ with constant rotational direction,
while for one $Q,U$ loop, $\psi_{\rm net}\neq 2\pi$; in the latter case, the
EVPA rotation direction changes twice over the orbit before the EVPA returns
to its initial value. As the observer inclination increases, a cusp forms at
$\theta_{\rm cusp}$ at the boundary between these two types of EVPA behavior.

Figure 9 plots the Schwarzschild EVPA as a function of orbital azimuth for a
hotspot in a purely vertical field at two observer inclinations
$\theta_{o}=42^{\circ},38^{\circ}$ slightly above and below $\theta_{\rm
cusp}$. As notes in Section VI.2, the EVPA continuously rotates in the same
direction at $\theta_{o}=38^{\circ}$, corresponding to two $Q,U$ loops, while
the EVPA changes rotation direction twice at $\theta_{o}=42^{\circ}$,
corresponding to one $Q,U$ loop. At $\theta_{\rm cusp}$, which lies between
these two values, the EVPA changes discontinuously, i.e. its derivative
becomes singular, as Figure 9 suggests. In the case of Figure 9, $\theta_{\rm
cusp}=40.7^{\circ}$, which can be calculated explicitly as described below.

As noted above, when an origin-enclosing loop contracts, by continuity one of
its points must pass through the origin $(Q=U=0)$. For vertical fields, we
observe that the cusp formed when a loop contracts is always located at the
origin, where the EVPA is not well-defined (i.e. rotates infinitely quickly).
The $Q,U$ cusp and the $Q,U$ origin both reflect a singularity in EVPA
rotation, and we intuitively expect that they coincide, i.e. that there is
always a cusp in the $Q,U$ loop if $Q=U=0$.141414We are able to prove this
explicitly in the specific case of a purely vertical field. We expect that we
should be able to extend the proof to any field configuration with nonzero
field, and that the converse also holds. We leave this to future work.

Thus, given $\vec{B}$, $a$, and $r_{\rm s}$, one can solve for the “cusp
angle” $\theta_{\rm cusp}$ simply by solving $Q=U=0$ for $\theta_{o}$. In
general, this entails solving three nonlinear equations: $Q(\alpha_{\rm
cusp},\beta_{\rm cusp},\theta_{\rm cusp})=0$, $U(\alpha_{\rm cusp},\beta_{\rm
cusp},\theta_{\rm cusp})=0$, and the geodesic equation. However, the
calculation for purely vertical fields can be done analytically, and we
present it here. Observe from (29) that $\kappa_{1}=\kappa_{2}=0$ for a purely
vertical field only if $p^{r}\propto\mathcal{R}(r_{\rm s})=0$ and $f^{(r)}=0$.
Expanding the fluid-frame components (26), the latter condition is equivalent
to $p^{(\phi)}=0$. Solving $p^{(\phi)}(r_{\rm s},\lambda_{\rm cusp},\eta_{\rm
cusp})=0$ (with $\pm_{s}=-1$ for the direct image) and $\mathcal{R}(r_{\rm
s},\lambda_{\rm cusp},\eta_{\rm cusp})=0$ gives the conserved quantities
$\lambda_{\rm cusp}$ and $\eta_{\rm cusp}$ at which the cusp will form.
$\theta_{\rm cusp}$ is then found from the constraint that $\lambda_{\rm
cusp}$ and $\eta_{\rm cusp}$ must be a solution to the geodesic equation for
$r_{\rm s}$ and $\theta_{\rm s}=\frac{\pi}{2}$. Following Gralla2020a  (see
(73) therein) the geodesic equation can be inverted to find:

$\theta_{\rm cusp}=\arccos\left[\pm\sqrt{u_{+}}{\rm
sn}\left(\sqrt{-u_{-}a^{2}}I_{r}^{\rm
turn}\big{|}\frac{u_{+}}{u_{-}}\right)\right],$ (77)

where we have used $I_{r}^{\rm turn}$ (67) because by construction $p^{r}_{\rm
s}(\lambda_{\rm cusp},\eta_{\rm cusp})=0$, so the ray must begin at a radial
turning point. To compare with (73) of Gralla2020a , note that $\eta>0$,
$I_{r}^{\rm total}$ therein is $2I_{r}^{\rm turn}$, and use (17) for
$m=0$.151515For the exact case of $a=0$, (77) reduces to
$\displaystyle\theta_{\rm cusp}$
$\displaystyle=\arccos\left[\frac{\pm\sin\left(I_{r}^{\rm
turn}\sqrt{\lambda_{\rm cusp}^{2}+\eta_{\rm
cusp}}\right)}{\sqrt{1+\frac{\lambda_{\rm cusp}^{2}}{\eta_{\rm
cusp}}}}\right].$

We introduce the explicit $\pm$ in (77) to allow for $\theta_{o}>\pi/2$. Note
that the $\pm$ indicates that for purely vertical fields, one loop in $Q,U$
space will exist for all observer inclinations that satisfy $\theta_{\rm
cusp}<\theta_{o}<\pi-\theta_{\rm cusp}$. This follows additionally from
Appendix C, wherein we show that the polarization pattern for a purely
vertical field will be reflected across the $\alpha$ axis upon taking
$\theta_{o}\to\pi-\theta_{o}$.

More generally, fields with both vertical and equatorial components require a
numerical solution for $\lambda_{\rm cusp},\eta_{\rm cusp}$, which can be
substituted into (73) of Gralla2020a  to solve for $\theta_{\rm cusp}$ (now
without the explicit $\pm$).

Figure 12 illustrates some features of the direct image polarization
corresponding to $\theta_{\rm cusp}$ for a sample magnetic field
$\vec{B}=(0,1,1)$ around a Schwarzschild black hole, for which $\theta_{\rm
cusp}$ must be computed numerically. The left panel displays $\theta_{\rm
cusp}$ as a function of emission radius $r_{\rm s}$, with dashed lines
intersecting at $r_{\rm s}=8$, $\theta_{\rm cusp}=32.95^{\circ}$. The central
panel displays the $Q,U$ loop at $r_{\rm s}=8$, $\theta_{\rm
cusp}=32.95^{\circ}$, in which a cusp forms at the origin. The right panel
displays the EVPA as a function of orbital azimuth for observer inclinations
of $\theta_{o}=\theta_{\rm cusp}\pm 2^{\circ}$ in green and blue,
respectively. As in Figure 9, the EVPA rotates $2\pi$ for
$\theta_{o}<\theta_{\rm cusp}$ and less than $2\pi$ for
$\theta_{o}>\theta_{\rm cusp}$.

Finally, note that as the inclination continues to increase beyond
$\theta_{\rm cusp}$, the lemniscate (figure-eight) that develops in the $Q,U$
curves and pulls through itself in Figure 8 at very large inclinations does
not represent a singularity in EVPA.

As remarked in Section VI, the location of the cusp can be shifted off the
origin by including subimages (as suggested by Figure 10) and by Faraday
conversion of Stokes $Q,U$ to circularly polarized Stokes $V$. It will also be
shifted for hotspots with physical extent large enough for
$\vec{p}\times\vec{B}$ to vary significantly over the emitting region.

## References

  * (1) R. Narayan, D. C. M. Palumbo, M. D. Johnson, Z. Gelles, E. Himwich, D. O. Chang, A. Ricarte, J. Dexter, C. F. Gammie, A. A. Chael, and The Event Horizon Telescope Collaboration, “The Polarized Image of a Synchrotron-emitting Ring of Gas Orbiting a Black Hole,” The Astrophysical Journal 912 no. 1, (May, 2021) 35, arXiv:2105.01804 [astro-ph.HE].
  * (2) Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Baloković, J. Barrett, and et al., “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” The Astrophysical Journal Letters 875 (Apr., 2019a) L1, arXiv:1906.11238.
  * (3) Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Baloković, J. Barrett, and et al., “First M87 Event Horizon Telescope Results. II. Array and Instrumentation,” The Astrophysical Journal Letters 875 (Apr., 2019b) L2, arXiv:1906.11239 [astro-ph.IM].
  * (4) Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Baloković, J. Barrett, and et al., “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration,” The Astrophysical Journal Letters 875 (Apr., 2019c) L3.
  * (5) Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Baloković, J. Barrett, and et al., “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole,” The Astrophysical Journal Letters 875 (Apr., 2019d) L4.
  * (6) Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Baloković, J. Barrett, and et al., “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring,” The Astrophysical Journal Letters 875 (Apr., 2019e) L5, arXiv:1906.11242.
  * (7) Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Baloković, J. Barrett, and et al., “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole,” The Astrophysical Journal Letters 875 (Apr., 2019f) L6, arXiv:1906.11243.
  * (8) The Event Horizon Telescope Collaboration, “First M87 Event Horizon Telescope Results. VII. Polarization of the Ring,” arXiv e-prints (May, 2021) arXiv:2105.01169, arXiv:2105.01169 [astro-ph.HE].
  * (9) The Event Horizon Collaboration, “First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon,” arXiv e-prints (May, 2021) arXiv:2105.01173, arXiv:2105.01173 [astro-ph.HE].
  * (10) P. A. Connors, T. Piran, and R. F. Stark, “Polarization features of X-ray radiation emitted near black holes.,” The Astrophysical Journal 235 (Jan., 1980) 224–244.
  * (11) B. C. Bromley, F. Melia, and S. Liu, “Polarimetric Imaging of the Massive Black Hole at the Galactic Center,” The Astrophysical Journal Letters 555 no. 2, (July, 2001) L83–L86, arXiv:astro-ph/0106180 [astro-ph].
  * (12) A. Broderick and R. Blandford, “Covariant magnetoionic theory I: Ray propagation,” Mon. Not. Roy. Astron. Soc. 342 (2003) 1280, arXiv:astro-ph/0302190.
  * (13) A. Broderick and R. Blandford, “Covariant magnetoionic theory - II. Radiative transfer,” Monthly Notices of the Royal Astronomical Society 349 no. 3, (Apr., 2004) 994–1008, arXiv:astro-ph/0311360 [astro-ph].
  * (14) A. E. Broderick and A. Loeb, “Imaging bright-spots in the accretion flow near the black hole horizon of Sgr A*,” Monthly Notices of the Royal Astronomical Society 363 no. 2, (Oct., 2005) 353–362, arXiv:astro-ph/0506433 [astro-ph].
  * (15) A. E. Broderick and A. Loeb, “Imaging optically-thin hotspots near the black hole horizon of Sgr A* at radio and near-infrared wavelengths,” Monthly Notices of the Royal Astronomical Society 367 no. 3, (Apr., 2006) 905–916, arXiv:astro-ph/0509237 [astro-ph].
  * (16) V. L. Fish, S. S. Doeleman, A. E. Broderick, A. Loeb, and A. E. E. Rogers, “Detecting Changing Polarization Structures in Sagittarius A* with High Frequency VLBI,” The Astrophysical Journal 706 no. 2, (Dec., 2009) 1353–1363, arXiv:0910.3893 [astro-ph.GA].
  * (17) M. Zamaninasab, A. Eckart, G. Witzel, M. Dovciak, V. Karas, R. Schödel, R. Gießübel, M. Bremer, M. García-Marín, D. Kunneriath, K. Mužić, S. Nishiyama, N. Sabha, C. Straubmeier, and A. Zensus, “Near infrared flares of Sagittarius A*. Importance of near infrared polarimetry,” Astronomy & Astrophysics 510 (Jan., 2010) A3, arXiv:0911.4659 [astro-ph.GA].
  * (18) R. F. Penna, J. C. McKinney, R. Narayan, A. Tchekhovskoy, R. Shafee, and J. E. McClintock, “Simulations of magnetized discs around black holes: effects of black hole spin, disc thickness and magnetic field geometry,” Monthly Notices of the Royal Astronomical Society 408 no. 2, (Oct., 2010) 752–782, arXiv:1003.0966 [astro-ph.HE].
  * (19) L.-X. Li, R. Narayan, and J. E. McClintock, “Inferring the Inclination of a Black Hole Accretion Disk from Observations of its Polarized Continuum Radiation,” The Astrophysical Journal 691 no. 1, (Jan., 2009) 847–865, arXiv:0809.0866 [astro-ph].
  * (20) R. V. Shcherbakov and L. Huang, “General relativistic polarized radiative transfer: building a dynamics–observations interface,” Monthly Notices of the Royal Astronomical Society 410 no. 2, (12, 2010) 1052–1063.
  * (21) R. V. Shcherbakov, R. F. Penna, and J. C. McKinney, “Sagittarius A* Accretion Flow and Black Hole Parameters from General Relativistic Dynamical and Polarized Radiative Modeling,” The Astrophysical Journal 755 no. 2, (Aug., 2012) 133, arXiv:1007.4832 [astro-ph.HE].
  * (22) J. Dexter, “A public code for general relativistic, polarised radiative transfer around spinning black holes,” Monthly Notices of the Royal Astronomical Society 462 no. 1, (Jun, 2016) 115–136. http://dx.doi.org/10.1093/mnras/stw1526.
  * (23) M. Mościbrodzka, J. Dexter, J. Davelaar, and H. Falcke, “Faraday rotation in GRMHD simulations of the jet launching zone of M87,” Monthly Notices of the Royal Astronomical Society 468 no. 2, (Jun, 2017) 2214–2221, arXiv:1703.02390 [astro-ph.HE].
  * (24) R. Gold, J. C. McKinney, M. D. Johnson, and S. S. Doeleman, “Probing the Magnetic Field Structure in Sgr A* on Black Hole Horizon Scales with Polarized Radiative Transfer Simulations,” The Astrophysical Journal 837 no. 2, (Mar., 2017) 180, arXiv:1601.05550 [astro-ph.HE].
  * (25) F. Marin, M. Dovčiak, F. Muleri, F. F. Kislat, and H. S. Krawczynski, “Predicting the X-ray polarization of type 2 Seyfert galaxies,” Monthly Notices of the Royal Astronomical Society 473 no. 1, (Jan., 2018) 1286–1316, arXiv:1709.03304 [astro-ph.HE].
  * (26) A. Jiménez-Rosales and J. Dexter, “The impact of Faraday effects on polarized black hole images of Sagittarius A*,” Monthly Notices of the Royal Astronomical Society 478 no. 2, (Aug, 2018) 1875–1883, arXiv:1805.02652 [astro-ph.HE].
  * (27) M. Mościbrodzka and C. F. Gammie, “IPOLE - semi-analytic scheme for relativistic polarized radiative transport,” Monthly Notices of the Royal Astronomical Society 475 no. 1, (Mar., 2018) 43–54, arXiv:1712.03057 [astro-ph.HE].
  * (28) D. C. M. Palumbo, G. N. Wong, and B. S. Prather, “Discriminating accretion states via rotational symmetry in simulated polarimetric images of m87,” The Astrophysical Journal 894 no. 2, (May, 2020) 156. http://dx.doi.org/10.3847/1538-4357/ab86ac.
  * (29) M. Mościbrodzka, “General relativistic polarized radiative transfer with inverse-Compton scatterings,” Monthly Notices of the Royal Astronomical Society 491 no. 4, (Feb., 2020) 4807–4815, arXiv:1910.02413 [astro-ph.HE].
  * (30) M. Moscibrodzka, A. Janiuk, and M. De Laurentis, “Unraveling circular polarimetric images of magnetically arrested accretion flows near event horizon of a black hole,” arXiv e-prints (Feb., 2021) arXiv:2103.00267, arXiv:2103.00267 [astro-ph.HE].
  * (31) K. Chen and D. M. Eardley, “Polarization Properties of Emission Lines from Relativistic Accretion Disks,” The Astrophysical Journal 382 (Nov., 1991) 125.
  * (32) M. Walker and R. Penrose, “On quadratic first integrals of the geodesic equations for type { 22} spacetimes,” Communications in Mathematical Physics 18 no. 4, (Dec., 1970) 265–274.
  * (33) A. M. Beloborodov, “Gravitational Bending of Light Near Compact Objects,” The Astrophysical Journal Letters 566 no. 2, (Feb., 2002) L85–L88, arXiv:astro-ph/0201117 [astro-ph].
  * (34) B. Carter, “Global structure of the kerr family of gravitational fields,” Phys. Rev. 174 (Oct, 1968) 1559–1571. https://link.aps.org/doi/10.1103/PhysRev.174.1559.
  * (35) S. E. Gralla and A. Lupsasca, “Lensing by Kerr black holes,” Physical Review D 101 no. 4, (Feb., 2020) 044031, arXiv:1910.12873 [gr-qc].
  * (36) S. E. Gralla and A. Lupsasca, “Null geodesics of the Kerr exterior,” Physical Review D 101 no. 4, (Feb., 2020) 044032, arXiv:1910.12881 [gr-qc].
  * (37) E. Himwich, M. D. Johnson, A. Lupsasca, and A. Strominger, “Universal polarimetric signatures of the black hole photon ring,” Physical Review D 101 no. 8, (Apr., 2020) 084020, arXiv:2001.08750 [gr-qc].
  * (38) J. M. Bardeen, W. H. Press, and S. A. Teukolsky, “Rotating Black Holes: Locally Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation,” The Astrophysical Journal 178 (Dec., 1972) 347–370.
  * (39) C. T. Cunningham and J. M. Bardeen, “The Optical Appearance of a Star Orbiting an Extreme Kerr Black Hole,” The Astrophysical Journal 183 (July, 1973) 237–264.
  * (40) S. E. Gralla, A. Lupsasca, and A. Strominger, “Observational signature of high spin at the Event Horizon Telescope,” Monthly Notices of the Royal Astronomical Society 475 no. 3, (Apr., 2018) 3829–3853, arXiv:1710.11112 [astro-ph.HE].
  * (41) D. E. A. Gates, S. Hadar, and A. Lupsasca, “Maximum Observable Blueshift from Circular Equatorial Kerr Orbiters,” arXiv e-prints (Sept., 2020) arXiv:2009.03310, arXiv:2009.03310 [gr-qc].
  * (42) D. E. A. Gates, S. Hadar, and A. Lupsasca, “Photon Emission from Circular Equatorial Kerr Orbiters,” arXiv:2010.07330 [gr-qc].
  * (43) A. Lupsasca, D. Kapec, Y. Shi, D. E. A. Gates, and A. Strominger, “Polarization whorls from M87* at the event horizon telescope,” Proc. Roy. Soc. Lond. A 476 no. 2237, (2020) 20190618, arXiv:1809.09092 [hep-th].
  * (44) E. Agol, The Effects of Magnetic Fields, Absorption, and Relativity on the Polarization of Accretion Disks around Supermassive Black Holes.  PhD thesis, University of California, Santa Barbara, Jan., 1997.
  * (45) A. Jiménez-Rosales et al., “Relative depolarization of the black hole photon ring in GRMHD models of Sgr A* and M87*,” Mon. Not. Roy. Astron. Soc. 503 no. 3, (2021) 4563–4575, arXiv:2103.06292 [astro-ph.HE].
  * (46) N. L. Balazs, “Effect of a Gravitational Field, Due to a Rotating Body, on the Plane of Polarization of an Electromagnetic Wave,” Physical Review 110 no. 1, (Apr., 1958) 236–239.
  * (47) S. Pineault and R. C. Roeder, “Applications of Geometrical Optics to the Kerr Metric. Analytical Results,” The Astrophysical Journal 212 (Mar., 1977) 541–549.
  * (48) F. Fayos and J. Llosa, “Gravitational effects on the polarization plane,” General Relativity and Gravitation 14 no. 10, (Oct., 1982) 865–877.
  * (49) A. Brodutch, T. F. Demarie, and D. R. Terno, “Photon polarization and geometric phase in general relativity,” Phys. Rev. D 84 no. 10, (Nov., 2011) 104043, arXiv:1108.0973 [gr-qc].
  * (50) H. Ishihara, M. Takahashi, and A. Tomimatsu, “Gravitational Faraday rotation induced by a Kerr black hole,” Phys. Rev. D 38 no. 2, (July, 1988) 472–477.
  * (51) M. Nouri-Zonoz, “Gravitoelectromagnetic approach to the gravitational Faraday rotation in stationary spacetimes,” Phys. Rev. D 60 no. 2, (July, 1999) 024013, arXiv:gr-qc/9901011 [gr-qc].
  * (52) D. P. Marrone, J. M. Moran, J.-H. Zhao, and R. Rao, “The submillimeter polarization of sgr a*,” Journal of Physics: Conference Series 54 (Dec, 2006) 354–362. http://dx.doi.org/10.1088/1742-6596/54/1/056.
  * (53) M. D. Johnson, V. L. Fish, S. S. Doeleman, D. P. Marrone, R. L. Plambeck, J. F. C. Wardle, K. Akiyama, K. Asada, C. Beaudoin, L. Blackburn, R. Blundell, G. C. Bower, C. Brinkerink, A. E. Broderick, R. Cappallo, A. A. Chael, G. B. Crew, J. Dexter, M. Dexter, R. Freund, P. Friberg, R. Gold, M. A. Gurwell, P. T. P. Ho, M. Honma, M. Inoue, M. Kosowsky, T. P. Krichbaum, J. Lamb, A. Loeb, R.-S. Lu, D. MacMahon, J. C. McKinney, J. M. Moran, R. Narayan, R. A. Primiani, D. Psaltis, A. E. E. Rogers, K. Rosenfeld, J. SooHoo, R. P. J. Tilanus, M. Titus, L. Vertatschitsch, J. Weintroub, M. Wright, K. H. Young, J. A. Zensus, and L. M. Ziurys, “Resolved magnetic-field structure and variability near the event horizon of Sagittarius A*,” Science 350 no. 6265, (Dec., 2015) 1242–1245, arXiv:1512.01220 [astro-ph.HE].
  * (54) G. C. Bower, A. Broderick, J. Dexter, S. Doeleman, H. Falcke, V. Fish, M. D. Johnson, D. P. Marrone, J. M. Moran, M. Moscibrodzka, A. Peck, R. L. Plambeck, and R. Rao, “ALMA Polarimetry of Sgr A*: Probing the Accretion Flow from the Event Horizon to the Bondi Radius,” The Astrophysical Journal 868 no. 2, (Dec., 2018) 101, arXiv:1810.07317 [astro-ph.HE].
  * (55) A. Eckart, R. Schödel, L. Meyer, S. Trippe, T. Ott, and R. Genzel, “Polarimetry of near-infrared flares from Sagittarius A*,” Astronomy & Astrophysics 455 (Aug., 2006) 1–10, astro-ph/0610103.
  * (56) S. Trippe, T. Paumard, T. Ott, S. Gillessen, F. Eisenhauer, F. Martins, and R. Genzel, “A polarized infrared flare from Sagittarius A* and the signatures of orbiting plasma hotspots,” Monthly Notices of the Royal Astronomical Society 375 (Mar., 2007) 764–772, astro-ph/0611737.
  * (57) R. Abuter, A. Amorim, M. Bauböck, J. P. Berger, H. Bonnet, W. Brandner, Y. Clénet, V. Coudé du Foresto, P. T. de Zeeuw, and et al., “Detection of orbital motions near the last stable circular orbit of the massive black hole sgra*,” Astronomy & Astrophysics 618 (Oct, 2018) L10. http://dx.doi.org/10.1051/0004-6361/201834294.
  * (58) A. Jiménez-Rosales, J. Dexter, F. Widmann, M. Bauböck, R. Abuter, A. Amorim, J. P. Berger, H. Bonnet, W. Brandner, and et al., “Dynamically important magnetic fields near the event horizon of sgr a*,” Astronomy & Astrophysics 643 (Nov, 2020) A56. http://dx.doi.org/10.1051/0004-6361/202038283.
  * (59) K. Dodds-Eden, P. Sharma, E. Quataert, R. Genzel, S. Gillessen, F. Eisenhauer, and D. Porquet, “Time-Dependent Models of Flares from Sagittarius A*,” The Astrophysical Journal 725 no. 1, (Dec., 2010) 450–465, arXiv:1005.0389 [astro-ph.GA].
  * (60) B. Ripperda, F. Bacchini, and A. A. Philippov, “Magnetic reconnection and hot spot formation in black hole accretion disks,” The Astrophysical Journal 900 no. 2, (Sep, 2020) 100. http://dx.doi.org/10.3847/1538-4357/ababab.
  * (61) G. C. Bower, H. Falcke, R. J. Sault, and D. C. Backer, “The Spectrum and Variability of Circular Polarization in Sagittarius A* from 1.4 to 15 GHz,” The Astrophysical Journal 571 no. 2, (June, 2002) 843–855, arXiv:astro-ph/0202138 [astro-ph].
  * (62) L. Huang, S. Liu, Z.-Q. Shen, M. J. Cai, H. Li, and C. L. Fryer, “Linearly and Circularly Polarized Emission in Sagittarius A*,” The Astrophysical Journal Letters 676 no. 2, (Apr., 2008) L119, arXiv:0802.3561 [astro-ph].
  * (63) D. J. Muñoz, D. P. Marrone, J. M. Moran, and R. Rao, “The Circular Polarization of Sagittarius A* at Submillimeter Wavelengths,” The Astrophysical Journal 745 no. 2, (Feb., 2012) 115, arXiv:1105.0427 [astro-ph.HE].
  * (64) A. Ricarte, R. Qiu, and R. Narayan, “Black Hole Magnetic Fields and Their Imprint on Circular Polarization Images,” arXiv:2104.11301 [astro-ph.HE].
  * (65) R. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics,” Physical Review Letters - PHYS REV LETT 11 (09, 1963) 237–238.
  * (66) S. Chandrasekhar, The mathematical theory of black holes.  1985\.
  * (67) C. T. Cunningham, “The effects of redshifts and focusing on the spectrum of an accretion disk around a Kerr black hole.,” The Astrophysical Journal 202 (Dec., 1975) 788–802.
  * (68) K. P. Rauch and R. D. Blandford, “Optical Caustics in a Kerr Spacetime and the Origin of Rapid X-Ray Variability in Active Galactic Nuclei,” The Astrophysical Journal 421 (Jan., 1994) 46.
  * (69) D. Kapec and A. Lupsasca, “Particle motion near high-spin black holes,” Classical and Quantum Gravity 37 no. 1, (Jan., 2020) 015006, arXiv:1905.11406 [hep-th].
  * (70) G. B. Rybicki and A. P. Lightman, Radiative processes in astrophysics.  1979\.
  * (71) J. Dexter and E. Agol, “A Fast New Public Code for Computing Photon Orbits in a Kerr Spacetime,” The Astrophysical Journal 696 no. 2, (May, 2009) 1616–1629, arXiv:0903.0620 [astro-ph.HE].
  * (72) G. Witzel et al., “Variability Timescale and Spectral Index of Sgr A* in the Near Infrared: Approximate Bayesian Computation Analysis of the Variability of the Closest Supermassive Black Hole,” Astrophys. J. 863 no. 1, (2018) 15, arXiv:1806.00479 [astro-ph.HE].
  * (73) M. D. Johnson, A. Lupsasca, A. Strominger, G. N. Wong, S. Hadar, D. Kapec, R. Narayan, A. Chael, C. F. Gammie, P. Galison, and et al., “Universal interferometric signatures of a black hole’s photon ring,” Science Advances 6 no. 12, (Mar, 2020) eaaz1310. http://dx.doi.org/10.1126/sciadv.aaz1310.
  * (74) R. A. Remillard and J. E. McClintock, “X-Ray Properties of Black-Hole Binaries,” Annual Review of Astronomy & Astrophysics 44 no. 1, (Sept., 2006) 49–92, arXiv:astro-ph/0606352 [astro-ph].
  * (75) C. S. Reynolds, “Measuring Black Hole Spin Using X-Ray Reflection Spectroscopy,” Space Science Reviews 183 no. 1-4, (Sept., 2014) 277–294, arXiv:1302.3260 [astro-ph.HE].
  * (76) J. E. McClintock, R. Narayan, S. W. Davis, L. Gou, A. Kulkarni, J. A. Orosz, R. F. Penna, R. A. Remillard, and J. F. Steiner, “Measuring the spins of accreting black holes,” Classical and Quantum Gravity 28 no. 11, (June, 2011) 114009, arXiv:1101.0811 [astro-ph.HE].
  * (77) L. Brenneman, Measuring the Angular Momentum of Supermassive Black Holes.  2013\.
  * (78) B. Jeter, A. E. Broderick, and R. Gold, “Differentiating disc and black hole-driven jets with eht images of variability in m87,” Monthly Notices of the Royal Astronomical Society 493 no. 4, (Mar, 2020) 5606–5616. http://dx.doi.org/10.1093/mnras/staa679.
  * (79) P. Tiede, H.-Y. Pu, A. E. Broderick, R. Gold, M. Karami, and J. A. Preciado-López, “Spacetime Tomography Using the Event Horizon Telescope,” The Astrophysical Journal 892 no. 2, (Apr., 2020) 132, arXiv:2002.05735 [astro-ph.HE].
  * (80) R. C. Walker, P. E. Hardee, F. B. Davies, C. Ly, and W. Junor, “The Structure and Dynamics of the Subparsec Jet in M87 Based on 50 VLBA Observations over 17 Years at 43 GHz,” The Astrophysical Journal 855 no. 2, (Mar., 2018) 128, arXiv:1802.06166 [astro-ph.HE].
  * (81) Z. Younsi, O. Porth, Y. Mizuno, C. M. Fromm, and H. Olivares, “Modelling the polarised emission from black holes on event horizon-scales,” in Perseus in Sicily: From Black Hole to Cluster Outskirts, K. Asada, E. de Gouveia Dal Pino, M. Giroletti, H. Nagai, and R. Nemmen, eds., vol. 342, pp. 9–12.  Jan., 2020.  arXiv:1907.09196 [astro-ph.HE].
  * (82) F. Johansson et al., mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.18), December, 2013\.  http://mpmath.org/.