# Polarization Patterns of Non-Circular Hotspots around Kerr Black Holes: A
Preliminary Study

Bin Chen1,2,3 E-mail<EMAIL_ADDRESS>Yehui Hou2 E-mail:
<EMAIL_ADDRESS>Yu Song2 E-mail<EMAIL_ADDRESS>Zhenyu Zhang2
E-mail<EMAIL_ADDRESS>
###### Abstract

The multi-wavelength polarized light signals from supermassive black holes
have sparked a range of studies on polarized images of accretion disks and
hotspots. However, the polarization patterns within the innermost stable
circular orbit (ISCO) region remain to be explored. In this study, we focus on
two specific types of orbits, namely the plunging geodesics inward from the
ISCO and homoclinic geodesics, to uncover the polarization features associated
with non-circular motion in a Kerr spacetime. For an on-axis observer, we
specifically develop an approximate function to describe gravitational lensing
along the azimuthal direction, and establish a simplified synchrotron emission
model. Based on these, we analyze the polarized patterns of hotspots
accumulated over time and their Stokes parameters. Moreover, we explore the
polarized image of the plunging region within a thin accretion disk.

1 Department of Physics, School of Physical Science and Technology, Ningbo
University, Ningbo, Zhejiang 315211, China

2 School of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P.R.
China

3 Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

##  1 Introduction

Optical observations of supermassive black holes have yielded significant
results in recent years. Whether it be the millimeter-band images of
Sagittarius A* (Sgr A*) and M87* captured by the Event Horizon Telescope (EHT)
collaboration [1, 2] or the astrometry of near-infrared (NIR) flares from Sgr
A* reported by the GRAVITY collaboration [3], both instances prompt a deeper
exploration of black hole physics. The findings from EHT and GRAVITY encompass
the information of polarization, offering more insights into the surrounding
astronomical environments near the black holes [4, 5, 6]. The observed
polarization angle, stemming from synchrotron radiation, encodes details about
the plasma properties and magnetic field structures at the horizon-scale [7,
8, 9], aiding in the investigation of strong field physics [10, 11, 12, 13,
14].

A simple but efficient way to analyze the NIR flares near Sgr A* is to
consider these flares as time-varying images of a hotspot. This hotspot can be
envisioned as a small, high-temperature, and neutral plasmoid, with its
internal electrons emitting thermal synchrotron radiation [15, 16, 17, 18, 19,
20, 21]. One plausible mechanism for generating such a plasmoid is magnetic
reconnection, a process in which magnetic field lines are severely curved and
undergo reconnection to dissipate their energy [22, 23, 24, 25, 26]. The
energy release in this process drives the plasma into high-energy plasmoids
that are expelled from the reconnection region. The radiation characteristics
of these plasmoids render them as compelling candidates for the flares
observed from Sgr A* [26].

In theoretical researches, it has been proposed a ring model, as outlined in
[27] that the emissions originate from a circular ring encircling a
Schwarzschild black hole. Viewing the ring as the accumulation of an orbiting
hotspot over time, this model offers a simplified depiction of orbiting
hotspots [28], where the locally defined magnetic field can be modulated to
generate various optical appearances. Furthermore, by superimposing rings of
different radii, the ring model can serve as a simplified representation of
the accretion disk, facilitating the analysis of polarized images captured by
the EHT [27]. The ring model was then expanded to encompass more general
spherically symmetric spacetimes [29] and the rotating Kerr spacetime [30].
Using this analytic ring model, it has been shown in [31] that the twisty
morphology in linear polarization is driven by the magnetic field structure in
the emitting region, with the Doppler boosting and the lensing being
subdominant factors. In [32], a more realistic accretion model near a
Schwarzschild black hole was developed analytically. Additionally,
investigations have delved into the impact of various types of spacetime on
the polarization pattern within this ring model [33, 34, 35, 36].

There exist several unexplored avenues in the current research on the
polarization images. One of the most challenging tasks is to explore the
horizon-scale magnetic field, which is intricately linked to the background
magnetofluid [7]. Traditionally, the strength and the orientation of the
magnetic field have been treated as phenomenological parameters in studies.
Moreover, while the ring model adeptly captures the key characteristics of an
orbiting hotspot or accretion disk, it is not well-suited for the domain
within the Innermost Stable Circular Orbit (ISCO). The inner-ISCO region of an
accretion flow, referred to as the plunging region, characterized by
streamlines that can be roughly approximated by a cluster of plunging
geodesics, could potentially make significant contributions to multi-
wavelength observations. These observations could encompass the inner shadow
observations by the EHT [37] and the X-ray spectra of stellar mass black holes
[38, 39, 40]. For non-circular hotspots, particularly those perturbed from
unstable orbits, the astrometry and the polarization patterns may diverge
significantly from those of circular hotspots. This divergence could present
new avenues for theoretical exploration into the flares observed from Sgr A*
and potentially unlock novel insights into their nature.

Recent studies have demonstrated that the trajectory functions of timelike
geodesics perturbed from unstable circular orbits or the ISCO can be elegantly
expressed using elementary functions [41, 42]. These findings have enabled the
generation of trajectory images through purely analytical ways. In this work,
we extend the methodology employed in studying circular orbit imaging and
polarizations [27, 30] to encompass non-circular orbits within a Kerr
spacetime. This extension utilize the trajectory integrals outlined in [41,
42]. In our study, we position the distant observer along the axis of rotation
of the black hole for simplicity, thereby deriving a set of analytical
expressions for gravitational lensing. Our focus lies on the hotspots moving
along two distinct types of trajectories: plunging geodesics and homoclinic
orbits [41]. We analyze the polarized images accumulated over time for these
hotspots, and carefully examine the variations in total intensity, the
polarization vector along the trajectory, and the polarization Stokes
parameters under diverse magnetic fields. We identify a range of polarization
characteristics exhibited along different trajectories. Furthermore, we apply
the aforementioned methodology to explore the polarized image of the plunging
region within an accretion disk. We explore the distributions of total
intensity and polarization vector across the disk plane.

The remaining parts of the paper are organized as follows. In Sec. 2, we
review the timelike geodesic equations in the equatorial plane of the Kerr
spacetime, and discuss the characteristics of plunging and homoclinic orbits.
In Sec. 3, we discuss the light propagations, the imaging screen for the on-
axis observer, gravitational lensing formulas, and synchrotron radiation at
the light source. In Sec. 4, we showcase the polarization patterns of
different trajectories and discuss their distinctive features. We conclude our
study in Sec. 4. We will work in the unit with $G=c=1$.

##  2 Trajectory of the hotspot

We begin with a brief review of the trajectory of the hotspot on the
equatorial plane in a Kerr spacetime. In the Boyer-Lindquist coordinates, the
line element of the spacetime takes the form

$\displaystyle\mathrm{d}s^{2}=$
$\displaystyle-\mathopen{}\mathclose{{}\left(1-\frac{2Mr}{\Sigma}}\right)\mathrm{d}t^{2}+\frac{\Sigma}{(r-r_{+})(r-r_{-})}\mathrm{d}r^{2}+\Sigma\mathrm{d}\theta^{2}-\frac{4Mra}{\Sigma}\mathrm{sin}^{2}\theta\mathrm{d}t\mathrm{d}\phi$
$\displaystyle+\sin^{2}{\theta}\mathopen{}\mathclose{{}\left(r^{2}+a^{2}+\frac{2Mra^{2}}{\Sigma}\mathrm{sin}^{2}\theta}\right)\mathrm{d}\phi^{2}\,,$
(2.1)

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$, $M$ and $a$ are the ADM mass and
spin parameters, respectively. The event horizons reside at $r_{\pm}=M\pm
M\sqrt{1-a^{2}}$. In subsequent discussions we focus on the case $a>0$.

###  2.1 Timelike circular orbits

The four-velocity vectors are represented as
$U^{\mu}=\mathrm{d}x^{\mu}/\mathrm{d}\tau$ with $\tau$ the proper time. The
Killing symmetries along $t$ and $\phi$ directions yield conserved energy
density and angular momentum density, symbolized by $E\equiv-U_{t}$ and
$L\equiv U_{\phi}$, respectively. The geodesic equation in the equatorial
plane reads

$\displaystyle(U^{r})^{2}+\frac{2M}{r}-\frac{L^{2}-a^{2}(E^{2}-1)}{r^{2}}+\frac{2M(L-aE)^{2}}{r^{3}}=E^{2}-1\,.$
(2.2)

For our purposes, we rewrite the above equation as $(U^{r})^{2}+V_{\rm
eff}(r)=0$, where $V_{\rm eff}(r)$ functions as the effective potential,

$\displaystyle V_{\rm
eff}(r)=(E^{2}-1)\mathopen{}\mathclose{{}\left(\frac{r_{1}}{r}-1}\right)\mathopen{}\mathclose{{}\left(\frac{r_{2}}{r}-1}\right)\mathopen{}\mathclose{{}\left(\frac{r_{3}}{r}-1}\right)\,.$
(2.3)

Here $r_{1}$, $r_{2}$ and $r_{3}$ represent the roots of $V_{\rm eff}(r)$.
Typically, these roots are complex, suggesting that there may be zero to three
physical roots outside the black hole, depending on the values of $E,L$ as
well as the spin. The potential simplifies in the presence of a real double
root, denoted as $r_{1}=r_{2}=r_{c}$. In this case, one has $V_{\rm
eff}(r_{c})=V_{\rm eff}^{\prime}(r_{c})=0$, indicating that the potential
permits a circular orbit at $r=r_{c}$. The expressions for the conserved
quantities of a circular orbit are provided by [43]

$\displaystyle
L_{c}=\pm\frac{(Mr_{c})^{1/2}}{\mathcal{D}}\mathopen{}\mathclose{{}\left(1+\frac{a^{2}}{r_{c}^{2}}\mp\frac{2aM^{1/2}}{r_{c}^{3/2}}}\right)\,,\quad
E_{c}=\frac{1}{\mathcal{D}}\mathopen{}\mathclose{{}\left(1-\frac{2M}{r_{c}}\pm\frac{aM^{1/2}}{r_{c}^{3/2}}}\right)\,,$
(2.4)

where

$\displaystyle\mathcal{D}^{2}=1-\frac{3M}{r_{c}}\pm\frac{2aM^{1/2}}{r_{c}^{3/2}}\,,$
(2.5)

and the “$\pm$” indicates the prograde/retrograde orbits. Comparing Eq. (2.2)
and Eq. (2.3), we get the expression of $r_{3}$ for the case with double root,

$\displaystyle
r_{3}=\frac{2M}{r^{2}_{c}}\frac{(L_{c}-aE_{c})^{2}}{1-E_{c}^{2}}\,.$ (2.6)

Hence, for a potential that possesses a double root at $r=r_{c}$, the
conserved quantities and the third root are dictated by the value of $r_{c}$.
The stability of circular orbits is determined by the sign of
$V^{\prime\prime}_{\rm eff}(r_{c})$. Marginal stability occurs when
$V^{\prime\prime}_{\rm eff}(r_{c})=0$, which yields the radius of the
innermost-stable-circular orbit (ISCO), denoted by $r_{I}$. The circular
orbits outside the ISCO are stable and converge towards their Newtonian
behavior at substantial radii. For $r_{c}<r_{I}$, the circular orbit becomes
unstable under perturbations. The radius of the ISCO is given by

$\displaystyle r_{I}/M=3+Z_{2}\mp\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}\,,$
$\displaystyle
Z_{1}=1+(1-a_{\star}^{2})^{1/3}\mathopen{}\mathclose{{}\left[(1+a_{\star})^{1/3}+(1-a_{\star})^{1/3}}\right]\,,\quad
Z_{2}=\sqrt{3a_{\star}^{2}+Z_{1}^{2}}\,,$ (2.7)

where $a_{\star}=a/M$ is the dimensionless spin parameter. In addition to the
stability of circular orbits, another characteristic pertains to whether the
orbits are bound or unbound. The notion of the bound orbits can be understood
from Eq. (2.6): the root $r_{3}$ exists only when $E_{c}<1$. This leads to a
class of pseudo-circular orbits where a particle, once perturbed outward from
$r=r_{c}$, travels out to $r_{3}$ and subsequently returns to $r_{c}$. For
$E_{c}\geq 1$, $r_{3}$ ceases to exist, and a point particle at the circular
orbit $r=r_{c}$ will become unbound and escape to infinity under
perturbations. The marginal case is the innermost-bound-circular orbit (IBCO),
the radius of which is determined by setting $E_{c}=1$ in Eq. (2.4), resulting
in

$\displaystyle r_{ib}/M=\mathopen{}\mathclose{{}\left(\,1+\sqrt{1\mp
a_{\star}}\,}\right)^{2}\,.$ (2.8)

For $r_{ib}<r_{c}<r_{I}$, the circular orbits have $E_{c}<1$ and remain bound,
though they are unstable. The orbits inside the IBCO has $E_{c}>1$ and are
both unstable and unbound.

###  2.2 Inspiraling orbits and homoclinic orbits

For an unstable circular orbit, the hotspot will deviate from the circle under
small perturbations. Consider a hotspot with $E_{c}=E(r_{c}),L_{c}=L(r_{c})$,
which is perturbed off the orbit $r=r_{c}$. If $r_{I}\geq r_{c}>r_{ib}$, the
equations of motion are

$\displaystyle\frac{\mathrm{d}r}{\mathrm{d}\tau}=\pm_{r}\mathopen{}\mathclose{{}\left(\frac{r_{c}}{r}-1}\right)\sqrt{\frac{2M(L_{c}-aE_{c})^{2}}{r_{c}^{2}\,r}+E_{c}^{2}-1}\,,\quad\frac{\mathrm{d}\phi}{\mathrm{d}\tau}=\frac{2MaE_{c}+L_{c}(r-2M)}{r(r-r_{+})(r-r_{-})}\,,$
(2.9)

from which we may obtain the trajectory of the hotspot parameterized by $r$,

$\displaystyle\sqrt{1-E_{c}^{2}}\,\phi=\int\frac{r^{3/2}\mathopen{}\mathclose{{}\left[2MaE_{c}/r+L_{c}(1-2M/r)}\right]}{(r-r_{+})(r-r_{-})(r-r_{c})\sqrt{r_{3}-r}}\,\mathrm{d}r\,.$
(2.10)

It is evident that there are poles at $r_{\pm}$ and $r_{c}$ in the integral,
suggesting that the hotspot warps infinitely near the poles and approaches the
radius in the asymptotic future or past. The poles at $r_{\pm}$ correspond to
the frame-dragging effect of the Kerr black hole, while the pole at $r_{c}$
reveals the nature of the unstable circular orbit. In the regime $|r-r_{c}|\ll
r_{c}$, if the hotspot moves from $r_{1}$ to $r_{2}$ within a time interval
$\Delta\tau=\tau_{2}-\tau_{1}$, one finds

$\displaystyle\frac{r_{2}-r_{c}}{r_{1}-r_{c}}\approx\,e^{\pm\gamma(\tau_{2}-\tau_{1})}\,,\quad\gamma=\frac{1}{r_{c}}\sqrt{\frac{2M}{r_{c}^{3}}(L_{c}-aE_{c})^{2}+E_{c}^{2}-1}\,,$
(2.11)

where the “$\pm$” denotes whether the motion is outward/inward relative to
$r=r_{c}$. The parameter $\gamma$ is the Lyapunov exponent encoding the
asymptotic behavior near $r=r_{c}$ [41]. At the ISCO, we have $V_{\rm
eff}(r_{I})=V^{\prime}_{\rm eff}(r_{I})=V^{\prime\prime}_{\rm eff}(r_{I})=0$,
resulting in a triple root, $r_{c}=r_{3}=r_{I}$. Under small perturbations,
the hotspot at the ISCO will plunge into the horizon, whose radial velocity
takes a simple form [43]

$\displaystyle
U_{I}^{r}=-\sqrt{\frac{2M}{3r_{I}}}\mathopen{}\mathclose{{}\left(\frac{r_{I}}{r}-1}\right)^{3/2}.$
(2.12)

In this case, Eq. (2.10) can be integrated to yield the trajectory function
parameterized by $r$ [44]

$\displaystyle\phi(r)=f_{+}(r)+f_{-}(r)+f_{I}(r)\,,$ $\displaystyle
f_{\pm}(r)=\pm\mathopen{}\mathclose{{}\left[\frac{2L_{c}r_{\pm}^{3/2}+4Mr_{\pm}^{1/2}(aE_{c}-L_{c})}{(r_{+}-r_{-})(r_{c}-r_{\pm})\sqrt{r_{3}-r_{\pm}}}}\right]\frac{1}{\sqrt{1-E_{c}^{2}}}\tanh^{-1}\mathopen{}\mathclose{{}\left(\sqrt{\frac{r_{3}/r-1}{r_{3}/r_{\pm}-1}}}\right)\,,$
$\displaystyle
f_{I}(r)=\sqrt{\frac{6r_{I}}{M}}\mathopen{}\mathclose{{}\left(\frac{2M(L_{c}-aE_{c})-r_{I}L_{c}}{r_{I}^{2}-2Mr_{I}+a^{2}}}\right)\sqrt{\frac{r}{r_{I}-r}}\,.$
(2.13)

The trajectories with different initial conditions can be related to Eq. (2.2)
through the rotations along the $\phi$ direction. From Eq. (2.2), we see that
$f_{\pm}(r)$ governs the asymptotic behavior near $r_{\pm}$ respectively,
while $f_{I}(r)$ characterizes the asymptotic behavior near the ISCO, which
differs from the exponential behavior near $r_{c}$ given by Eq. (2.11). The
left panel of Fig. 1 showcases an inspiral orbit, from which it is clear that
the trajectory near $r_{I}$ exhibits significantly denser loops.

Figure 1: Left: The red circle represents the prograde ISCO, $r=r_{I}$, and
the blue curve represents the inspiral orbit from the ISCO. Right: The red
circle represents the prograde, bound circular orbit $r_{c}=r_{ib}+0.6M\,$,
and the blue curve represents the homoclinic orbit outward from $r_{c}$. The
gray dot denotes the radius of $r_{3}$. The black hole spin is set to be
$a=0.2M$.

For a circular orbit inside the ISCO, i.e., $r_{ib}<r_{c}<r_{I}$, the orbit is
unstable but remains bound. There emerge two types of trajectories under the
perturbations at $r=r_{c}$: the orbits spiral outwards from $r_{c}$ towards
$r_{3}$, or the orbits spiral inwards from $r_{c}$ and then plunge into the
outer horizon. The latter case is qualitatively similar to the inspiral orbit
from the ISCO. For the orbits in the former case, the trajectory function
parameterized by $r$ takes the form of

$\displaystyle\phi(r)=f_{+}(r)+f_{-}(r)+f_{h}(r)\,,$ $\displaystyle
f_{h}(r)=-\mathopen{}\mathclose{{}\left[\frac{2L_{c}r_{c}^{3/2}+4Mr_{c}^{1/2}(aE_{c}-L_{c})}{(r_{c}-r_{+})(r_{c}-r_{-})\sqrt{r_{3}-r_{c}}}}\right]\frac{1}{\sqrt{1-E_{c}^{2}}}\tanh^{-1}\mathopen{}\mathclose{{}\left(\sqrt{\frac{r_{3}/r-1}{r_{3}/r_{c}-1}}}\right)\,,$
(2.14)

where $f_{\pm}(r)$ is the same as in Eq. (2.2). These orbits are referred to
as homoclinic orbits [41]. Physically, a homoclinic orbit corresponds to a
hotspot forming at an unstable but bound circular orbit, and is subsequently
perturbed away from the circular orbit. The right panel in Fig. 1 depicts a
homoclinic orbit with parameters $a=0.2,r_{c}=r_{ib}+0.6M$, and $r_{3}\approx
11.96$. In this scenario, the hotspot is perturbed outwards from $r=r_{c}$,
circles around it multiple times, and eventually reaches the “aphelion” at
$r=r_{3}$. Subsequently, the hotspot retreats from the aphelion back towards
$r=r_{c}$.

##  3 Imaging the hotspot

As our interest lies in the optical appearance of the hotspot, we need to
compute the light propagations from the hotspot to a distant observer. In this
section, we initially revisit the basic equations of light propagation in Kerr
spacetime. Then, we discuss the map between the screen coordinates of the
observer and the hotspot’s position, and provide a new fitting function of the
map, which proves crucial for an analytical examination of the hotspot’s
images. Subsequently, we introduce the setting for synchrotron radiation of
the hotspot under external magnetic field.

###  3.1 Light propagations

It is well-established that in a Kerr spacetime, a photon has three conserved
quantities along its trajectory, namely the energy, angular momentum, and the
Carter constant, expressed as

$\displaystyle\mathcal{E}=-p_{t},\quad\mathcal{L}=p_{\phi},\quad\mathcal{Q}=p^{2}_{\theta}-\cos^{2}\theta(a^{2}p_{t}^{2}-p_{\phi}^{2}\csc^{2}\theta).$
(3.1)

These quantities originate from the Killing symmetries along $t,\phi$
directions and a high-rank symmetry [45]. The magnitude of $\mathcal{E}$ can
be absorbed into the scaling of the affine parameter and does not affect null
geodesic trajectories. Therefore, we work with energy-rescaled quantities,
defined as

$\displaystyle\lambda=\frac{\mathcal{L}}{\mathcal{E}}\,,\quad\eta=\frac{\mathcal{Q}}{\mathcal{E}^{2}}\,,$
(3.2)

also referred to as the impact parameters. A photon with an affinely
parameterized trajectory has the energy-rescaled four-momentum

$\displaystyle
p_{\mu}\mathrm{d}x^{\mu}=-\mathrm{d}t\pm_{r}\frac{\sqrt{\mathcal{R}(r)}}{(r-r_{+})(r-r_{-})}\mathrm{d}r\pm_{\theta}\sqrt{\Theta(\theta)}\mathrm{d}\theta+\lambda\mathrm{d}\phi\,,$
(3.3)

given in terms of the radial and angular potentials

$\displaystyle\mathcal{R}(r)=(r^{2}+a^{2}-a\lambda)^{2}-(r-r_{+})(r-r_{-})[\eta+(\lambda-a)^{2}]\,,$
$\displaystyle\Theta(\theta)=\eta+a^{2}\cos^{2}\theta-\lambda^{2}\cot^{2}\theta.$
(3.4)

The symbols $\pm_{r}$ and $\pm_{\theta}$ indicate the sign of $p_{r}$ and
$p_{\theta}$, respectively. The variations of the impact parameters change the
shape of the potentials and determine the behaviors of null geodesics. For our
purpose, we restrict to the case of $\eta>0$, corresponding to the light rays
that can reach the equatorial plane.

As the hotspot emits synchrotron radiation under an external magnetic field,
we need to consider the transport of the polarization vector along a light
ray. The unit-norm spacelike polarized vector, denoted by $f^{\mu}$, is
orthogonal to the photon’s four-momentum and parallel transported along the
trajectory,

$\displaystyle f\cdot p=0\,,\quad f\cdot f=1\,,\quad p\cdot\nabla
f^{\nu}=0\,,$ (3.5)

where we have omitted the indices when performing the inner product. With the
help of the Newman-Penrose null tetrad $\\{l,n,m,\bar{m}\\}$, one can
construct a conserved quantity for the polarization vector along null
geodesics [46], typically written as

$\displaystyle\kappa\equiv\kappa_{1}+i\kappa_{2}=(A-iB)(r-ia\cos\theta)\,,$
(3.6) $\displaystyle
A=(p^{t}f^{r}-p^{r}f^{t})+a\sin^{2}{\theta}(p^{r}f^{\phi}-p^{\phi}f^{r})\,,$
$\displaystyle
B=[(r^{2}+a^{2})(p^{\phi}f^{\theta}-p^{\theta}f^{\phi})-a(p^{t}f^{\theta}-p^{\theta}f^{t})]\sin{\theta}\,,$
(3.7)

where $\kappa_{1},\kappa_{2}$ represent the real and imagine part of $\kappa$,
respectively. The conserved quantity is often referred to as the Penrose-
Walker constant.

###  3.2 Coordinates on the screen

The observations indicate a small inclination angles for the supermassive
black holes [1, 6, 8]. In this study, for simplicity, we restrict to an on-
axis observer, $\theta_{o}=0^{\circ}$. From the angular potential, it is
evident that only the photons with zero angular momentum, $\lambda=0$, can
approach the spin axis. When a photon reaches the screen of the on-axis
observer at the distance, its position of arrival is given by the polar
coordinates [30]

$\displaystyle\rho=\frac{\sqrt{\eta+a^{2}}}{r_{o}}\,,\quad\varphi=\phi_{o}\,,$
(3.8)

where $\rho$ represents the angular radius of the photon on the screen,
$r_{o}$ and $\phi_{o}$ are the radius and azimuthal angle of the arriving
photon, respectively. In what follows we take a rescaling

$\displaystyle\rho\rightarrow\frac{r_{o}}{M}\rho$ (3.9)

to simplify the calculation. The polar coordinates encodes the four-momentum
of the photon and allows us to determine the route of the arriving photon. Our
study focuses on a hotspot at the equatorial plane that serves as the sole
light source. For the photons reaching $(\rho,\varphi)$, the emission point
$(r_{s},\theta_{s},\phi_{s})$ is determined by combining the expression of the
photon’s trajectory with the constrain $\theta_{s}=\pi/2$, yielding a
relationship between the hotspot’s position and the arriving photon’s polar
coordinates, the latter of which provides the hotspot’s image point on the
observer’s screen.

Owing to the strong gravitational lensing effect, a null geodesic can loop
around the black hole multiple times, potentially resulting in more than one
image point associated with a single equatorial plane position, known as the
lensed images of the hotspot [20]. However, our interest lies solely in the
direct image of the hotspot, which is produced by the photons that do not loop
around the black hole and directly escape to the distant observer. For this
direct image, the relationship between $(r_{s},\phi_{s})$ and $(\rho,\varphi)$
can be well approximated by the analytical expansions presented in [30],

$\displaystyle\frac{r_{s}}{M}=\rho-1+\frac{1-a^{2}}{2\rho}+\frac{3(5\pi-16)}{4\rho^{2}}+\mathcal{O}(1/\rho^{3})\,,$
(3.10)
$\displaystyle\rho=\frac{r_{s}}{M}+1+\frac{a^{2}-M^{2}}{2Mr_{s}}+\frac{50M^{2}-2a^{2}-15M^{2}\pi}{4r_{s}^{2}}+\mathcal{O}(M^{3}/r_{s}^{3})\,.$
(3.11)

When deriving the above equations, the limit $r_{s}\gg M$ has been used.
Despite this, Eq. (3.10) and Eq. (3.11) work well for nearly all values of
$r_{s}>r_{+}$.

For an axisymmetric light source, such as a time-averaged accretion disk, Eq.
(3.10) and Eq. (3.11) are sufficient to determine the optical appearance. This
is because the emission profile does not depend on $\phi_{s}$, and the image
dose not depend on $\varphi$. However, when the hotspot serves as a point
light source, a significant distortion along the $\phi$ direction occurs
between the hotspot and its image on the screen, i.e., $|\varphi-\phi_{s}|$
could be large. This distortion originates from the frame-dragging effect on
the light trajectory. Consequently, it is necessary to determine the map
$\varphi(r_{s},\phi_{s})$ for hotspot imaging. For a photon emitted at
$(r_{s},\phi_{s})$, the angle of arrival takes

$\displaystyle\varphi=\phi_{s}+\int_{r_{s}}^{\infty}\frac{2Mardr}{(r-r_{+})(r-r_{-})\sqrt{\mathcal{R}}}\,,\quad\mathcal{R}=(r^{2}+a^{2})^{2}-(r-r_{+})(r-r_{-})(\eta+a^{2})\,,$
(3.12)

where $\eta$ is related to $r_{s}$ through Eq. (3.8) and Eq. (3.11), for the
direct image. To make a preliminary discussion, we take the limit $r_{s}\gg
r_{+}$ in Eq. (3.12), and the radial potential can be expanded to $R\approx
r^{4}-r_{s}^{2}r^{2}+{\mathcal{O}}\mathopen{}\mathclose{{}\left(r_{s}r^{2}}\right)+{\mathcal{O}}\mathopen{}\mathclose{{}\left(r_{s}^{2}r}\right)$.
This expansion is valid if $r$ is greater than $r_{s}$, a condition easily met
for the light rays associated with the direct image. If we consider only the
first two terms in the expansion, the integral simplifies to

$\displaystyle\varphi-\phi_{s}\approx\int_{r_{s}}^{\infty}\frac{2Madr}{r^{2}\sqrt{r^{2}-r_{s}^{2}}}=\frac{2Ma}{r_{s}^{2}}\,.$
(3.13)

This expression works well for a large $r_{s}$. However, when $r\rightarrow
r_{+}$, the integral in Eq. (3.12) diverges for a nonzero spin, a phenomenon
attributed to the inevitable dragging effect, and Eq. (3.13) is not applicable
in this region. To derive an approximate equation for $r_{+}\leq r<\infty$,
one has to take the pole at $r=r_{+}$ into account. The purely analytical
expansion for Eq. (3.12) is complicated, and in this work we resort to
numerical schemes for function fitting. Based on the asymptotic behavior
described by Eq. (3.13) and the behavior near $r=r_{+}$, we develop a
piecewise function to approximate Eq. (3.12), which takes the form

$\displaystyle\varphi-\phi_{s}\approx\mathopen{}\mathclose{{}\left\\{\begin{aligned}
&\,\,\frac{2Ma}{r_{s}^{2}}\bigg{(}1-\frac{\tilde{\beta}}{(\frac{r_{s}}{r_{+}}-1)^{\tilde{\alpha}}}\ln{(\frac{r_{s}}{r_{+}}-1)}\bigg{)}\,,\quad\,\,r_{s}\textless
r_{I}\,,\\\
&\,\,\frac{2Ma}{r_{s}^{2}}\bigg{(}1+\frac{\tilde{\gamma}}{(\frac{r_{s}}{r_{+}}-1)}\bigg{)}\,,\quad\,\,r_{s}\geq
r_{I}\,,\\\ \end{aligned}}\right.$ (3.14)

where $r_{I}$ is the prograde ISCO, and
$\tilde{\alpha},\tilde{\beta},\tilde{\gamma}$ are three functions of
$a_{\star}=a/M$,

$\displaystyle\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\tilde{\alpha}&=0.035(1-a_{\star})+\frac{0.0059}{(1-a_{\star})^{0.4577}}+0.1163\,,\\\
\tilde{\beta}&=0.2093\arctan{(a_{\star}^{12})}+0.3467\,,\\\
\tilde{\gamma}&=0.07815\arctan{(a_{\star}^{3})}+0.0983\,.\end{aligned}}\right.$
(3.15)

Fortunately, the approximation function Eq. (3.14) works good for a large
parameter range of $a_{\star}$. In Fig. 2 we present the comparision between
the approximation function Eq. (3.14) and the precise result of Eq. (3.12),
under a small spin $a_{\star}=0.1$ and a high spin $a_{\star}=0.9$. The
piecewise function is slightly discontinuous at the ISCO, where the largest
error occurs. However, the error is rather small and can be neglected in
subsequent discussions. By using Eq. (3.14), we can easily obtain the hotspot
image for the on-axis observer, without the need to deal with lengthy elliptic
integrals.

Figure 2: Comparsion between Eq. (3.12) and Eq. (3.14).

###  3.3 Synchrotron radiation of the hotspot

In order to obtain the polarized image at the screen, we need to specify the
synchrotron radiation of the hotspot. The polarized intensity emitted from the
hotspot is determined by its magnetic field, $B_{\mu}=-\star
F_{\mu\nu}U^{\nu}$, with $\star F_{\mu\nu}$ being the Hodge dual of the
electromagnetic tensor, and $U^{\mu}$ being the four-velocity of the hotspot.
In the hotspot’s comoving frame, we introduce the spatial four-momentum of the
radiated photons, as well as the magnetic field perpendicular to this
momentum,

$\displaystyle p_{\perp}^{\mu}=p^{\mu}+(U\cdot p)U^{\mu}\,,\quad
B^{\mu}_{\perp}=B^{\mu}-\frac{B\cdot p}{(U\cdot p)^{2}}\,p_{\perp}^{\mu}\,.$
(3.16)

From the above expressions we find $U\cdot p_{\perp}=U\cdot
B_{\perp}=p_{\perp}\cdot B_{\perp}=0$. For thermal synchrotron emission, the
polarization vector is perpendicular to the photon momentum and the magnetic
field [47]. Thus, we can express the polarization vector as

$\displaystyle
f^{\mu}=-\frac{\epsilon^{\mu\nu\alpha\beta}U_{\nu}p_{\alpha}(B_{\perp})_{\beta}}{(U\cdot
p)\sqrt{B_{\perp}^{2}}}\,.$ (3.17)

Besides, the emitted total intensity concentrates on the direction
perpendicular to the magnetic field in the hotspot’s comoving frame. Following
[27], we set the total intensity to be ${\mathcal{I}}=\xi B_{\perp}^{2}$, with
the overall coefficient $\xi$ determined by the details of the hotspot. In
what follows we simply set $\xi=1$.

Strictly speaking, the structure of the magnetic field within the hotspot
depends on the specific mechanism that generates the hotspot. In the present
study, we assume that the hotspot is embedded within a strong background
magnetic field, the structure of which remains unaffected by the motion of the
hotspot. We consider two types of magnetic fields, with vertical and radial
configurations, to study the imprint of magnetic field structure on the
polarization images. First, we employ the source-free Papapetrou-Wald solution
[48] as the vertical field, in which the gauge potential reads

$A^{\mu}\partial_{\mu}=a\mathcal{B}\,\partial_{t}+\frac{\mathcal{B}}{2}\,\partial_{\phi}\,,$
(3.18)

where the constant $\mathcal{B}$ indicates the field strength observed by a
distant static observer. Such vertical configuration implies a dynamical
important magnetic field near the equatorial plane, which has been suggested
by some observations [49, 6].

For the raidal configuration, we consider a magnetic field sourced by a
background accretion flow infalling from the infinity [50], with four-velocity
$\bar{U}_{\mu}dx^{\mu}=-dt-\sqrt{2r(r^{2}+a^{2})}(r-r_{+})^{-1}(r-r_{-})^{-1}dr$.
By considering the ideal-MHD condition, the electromagnetic tensor near the
equatorial plane takes

$\displaystyle
F_{\theta\phi}=\Psi\,,~{}~{}~{}~{}F_{r\phi}=\Psi\,\frac{\bar{U}^{\phi}}{\bar{U}^{r}}\,.$
(3.19)

Here, $\Psi$ denotes the overall strength and keeps constant along the
streamline. Within the accretion flow the magnetic field lines align with the
streamlines, which appear as a radial pattern at a distance from the black
hole, but near the horizon $r_{+}$, they become highly curled due to the
frame-dragging effect.

##  4 Polarization patterns

The polarization state of the observed photons is encoded in the Stokes
parameters [4, 5, 6]. Following the convention in [4], we denote the Stokes
parameters as
$\\{{\mathcal{I}}_{o},{\mathcal{Q}}_{o},{\mathcal{U}}_{o},\mathcal{V}_{o}\\}$,
where ${\mathcal{I}}_{o}$ is the observed total intensity, ${\mathcal{Q}}_{o}$
and ${\mathcal{U}}_{o}$ describe the linear polarization components along two
perpendicular axes, and $\mathcal{V}_{o}$ represents the circular polarization
[47]. In the case of pure linear polarization, the Stokes parameters read

$\displaystyle{\mathcal{P}}_{o}={\mathcal{Q}}_{o}+i\,{\mathcal{U}}_{o}={\mathcal{I}}_{o}\,e^{2i\chi_{o}}\,,\quad\mathcal{V}_{o}=0\,,$
(4.1)

where ${\mathcal{I}}_{o}$ is the observed total intensity, related to the
emitted intensity through ${\mathcal{I}}_{o}=g^{4}{\mathcal{I}}$ with $g$ the
redshift factor from the hotspot to the observer [51]. The electric vector
polarization angle (EVPA), denoted by $\chi_{o}$, is obtained by projecting
the polarization vector on the screen’s tetrad. The result takes [30]

$\displaystyle\chi_{o}=\arctan{\frac{\kappa_{1}}{\kappa_{2}}}+\varphi=\chi_{net}+\varphi\,,$
(4.2)

where we have introduced the “net EVPA”, $\chi_{net}=\chi_{o}-\varphi\,$, to
denote the angle between the polarization vector and the line from the origin
of the screen to the image point $(\rho,\varphi)$. The values of the EVPA and
net EVPA are set to be $\chi_{o},\chi_{net}\in[\,0,\pi)$, for the reason that
only the direction of the linear polarization is meaningful. It becomes
evident that it is $\chi_{net}$ that encodes the pattern of the polarization.
For instence, for an axisymmetric light source, $\chi_{net}=0$ reflects a
radial pattern, with the polarization vectors all pointing to the $\rho$
direction; $\chi_{net}=\pi/2$ indicates a toroidal pattern, where the
polarization vectors all point to the $\varphi$ direction. Moreover, the case
of $\pi/2>\chi_{net}>0$ corresponds to the “counterclockwise” pattern, that
is, relative to the radial pattern, the polarization vector is combed in the
counterclockwise direction. On the contrary, $\pi>\chi_{net}>\pi/2$
corresponds to the “clockwise” pattern, with the polarization vector combed
clockwise.

###  4.1 Results for inspiraling hotspots

Figure 3: From left to right column, we present the trajectories, the
polarization images, and the Q-U curve of the hotspot plunging from the
prograde ISCO, under the vertical magnetic field Eq. (3.18). The constant
$\mathcal{B}$ within the Papapetrou-Wald potential is set to be 1, without
loss of generality. From top to bottom row, the black hole spin is set to be
$a=0.1M,0.5M,0.9M$. For each case, the gray area denotes the region enclosed
by $r_{+}$; the color bar represents the value of the total intensity; the
polarization vector is represented by the red line segments on the screen; the
Stokes parameters $Q,U$ are normalized by the maximal total intensity
$I_{max}$ along the trajectory. Figure 4: From left to right column, we
present the trajectories, the polarization images, and the Q-U curve of the
hotspot plunging from the prograde ISCO, under the radial magnetic field Eq.
(3.19), where the constant $\Psi$ is set to be 1, without loss of generality.
From top to bottom row, the black hole spin is set to be $a=0.1M,0.5M,0.9M$.
For each case, the gray area denotes the region enclosed by $r_{+}$; the color
bar represents the value of the total intensity; the polarization vector is
represented by the red line segments on the screen; the Stokes parameters
$Q,U$ are normalized by the maximal total intensity $I_{max}$ along the
trajectory.

In Fig. 3 we depict the trajectory and the polarization image of the
inspiraling hotspot under the vertical magnetic field Eq. (3.18), for various
black hole spins. When depicting the images, we employ the Cartesian
coordinates on the screen,
$\alpha=\rho\cos{\varphi}\,,\beta=\rho\sin{\varphi}$. Due to the gravitational
redshift, the total intensity exhibits a decreasing trend as the hotspot
approaches the horizon. In the right colume of Fig. 3, the Stokes parameters
along the hotspot’s trajectory is displayed on the Q-U plane, which we call
the Q-U curve. Similar to the trajectory, the Q-U curve exhibits a right-
handed spiral shape111The chirality mentioned in this study is defined
opposite to the direction of the observer’s line of sight.. From their
definitions,
${\mathcal{Q}}_{o}+i\,{\mathcal{U}}_{o}={\mathcal{I}}_{o}\,e^{2i\chi_{net}+2i\varphi}$,
it is evident that the phase variation of $Q,U$ is mainly determined by the
hotspot’s motion in the $\phi$ direction.

Under the radial magnetic field Eq. (3.19), the polarization of the
inspiraling hotspot significantly differs, as depicted in the middle column of
Fig. 4. In particular, for the high-spin case, the polarization vector is
almost parallel to the trajectory, which does not appear in the case of a
vertical magnetic field. From the right column of Fig. 4, we see that the Q-U
curve first spirals outward and then back towards the origin for
$a=0.1M,0.5M$, indicating a non-monotonic behavior of the total intensity
along the trajectory.

Figure 5: The observed total intensity ($\mathcal{I}_{o}$) and net EVPA
($\chi_{net}$) along a prograde, inspiraling trajectory, under various black
hole spin. The top and bottom row represent the case of vertical and radial
magnetic field, respectively. For each case, the dashed line represents the
projected prograde ISCO on the observer’s screen. To make a comparison, the
region outside the ISCO is filled with luminous hotspots moving along prograde
circular orbits. Since the constants for the two types of magnetic field
($\mathcal{B}$ and $\Psi$) only affect the overall magnitude of
$\mathcal{I}_{o}$, we have normalized them to 1.

The detailed behaviors of the total intensity and the net EVPA are present in
Fig. 5. The region outside the ISCO is set to be filled with luminous hotspots
moving along prograde circular geodesics, which have been well studied in the
literature. It is easy to see the distinction in $I_{o},\chi_{net}$ outside
and inside the ISCO. Note that in all four subplots, the connections at the
ISCO are continuous. Under the vertical magnetic field, the net EVPA increases
as the hotspot nears the horizon. For a higher black hole spin, $\chi_{net}$
displays a larger value at the prograde ISCO, but a smaller value at the
horizon. However, the growth rate of $\chi_{net}$ is largely insensitive to
the black hole spin, and in all cases $\chi_{net}$ exhibit a counterclockwise
pattern, as shown in the top-right panel of Fig. 5. In the case of radial
magnetic fields, the total intensity initially increases as the hotspot
departs from the prograde ISCO, and then drops towards zero as the hotspot
approaches the horizon. This behavior can be attributed to the combined
effects of gravitational redshift and the radial magnetic field. As the black
hole continuously focuses the streamlines and magnetic field lines of the
magneofluids, the magnetic field strength at smaller radii intensifies. When
the hotspot spirals inward from the ISCO, the amplification of the magnetic
field predominates in controlling its synchrotron radiation, which results in
an increase in total intensity. However, as the hotspot gets closer to the
black hole, the overpowering gravitational redshift reduces the radiations. In
terms of the polarization pattern, the net EVPA maintains a fixed value of
$\chi_{net}=\pi/2$ far from the black hole. This is due to the polarization
vector being almost perpendicular to the radial magnetic field lines when
$\rho\gg 1$. As $\rho$ decreases, two scenarios emerge, as illustrated in the
bottom-right panel of Fig. 5. For a higher spin of $a=0.9M$, $\chi_{net}$
decreases monotonically as the hotspot approaches the horizon, indicating a
counterclockwise pattern. Conversely, for a lower spin of $a=0.1M$ or $0.5M$,
$\chi_{net}$ initially increases before rapidly decreasing, but $\chi_{net}$
always remains greater than $\pi/2$, indicating a clockwise pattern.

###  4.2 Results for hotspots from retrograde ISCO

Figure 6: The polarization image, the Q-U curve, and the net EVPA of the
hotspot plunging from the retrograde ISCO, under the vertical magnetic field
Eq. (3.18). The constant $\mathcal{B}$ within the Papapetrou-Wald potential is
set to be 1. In the left two panels, the black hole spin is set to be
$a=0.9M$. In the middle panel, we provide an enlarged view near the origin,
where the scale of the coordinates should be multiplied by $10^{-10}$. In the
right panel, the dashed line represents the projected retrograde ISCO on the
observer’s screen, outside which $\chi_{net}$ is produced by the hotspots
moving along retrograde circular orbits.

Fig. 6 depicts the polarization pattern of a hotspot inspiraling from the
retrograde ISCO, located at a greater distance from the black hole. Generally,
under the vertical magnetic field, the net EVPA exhibits a clockwise pattern,
$\pi>\chi_{net}>\pi/2$, as depicted in the right panel of Fig. 6. As the
hotspot approaches the black hole, it reaches a turning point where the
rotation switches to prograde due to the frame-dragging effect, as
demonstrated in the left panel of Fig. 6. Near this turning point, the net
EVPA reaches $\pi/2$ and then transitions to a counterclockwise pattern with
$\pi/2>\chi_{net}>0$. Interestingly, the Q-U curve in this case follows a
left-handed spiral shape, contrary to the case that the hotspot spirals inward
from the prograde ISCO. This behavior is influenced by the concentration of
total intensity near the retrograde ISCO, which shapes the Q-U curve. In the
magnified view in the lower right corner of the Q-U plane, a right-handed
shape only emerges very close to the origin, corresponding to the polarization
near the horizon.

###  4.3 Results for homoclinic hotspots

Figure 7: The polarization images, the Q-U curves, and the net EVPAs of the
hotspot traveling along a homoclinic orbit, under the vertical magnetic field
(top row) and radial magnetic field (bottom row). In the left two columns, we
set $a=0.9M,r_{c}=r_{ib}+0.15M$. For each case, the projected $r_{c}$ is
denoted by the blue dashed circle on the screen; the blue segment of the Q-U
curve corresponds to the portion of the homoclinic orbit moving outward from
$r_{c}$ to $r_{3}$, while the red segment represents the portion of the orbit
falling from $r_{3}$ to $r_{c}$. The right column presents the variation of
the net EVPA of the hotspots falling from $r_{3}$ (the green point) to
$r_{c}$, under different spins. For the ease of presentation, the radii of
$r_{c}$ are chosen as $r_{c}=r_{ib}+1.2M/0.8M/0.15M$ for $a=0.1M/0.5M/0.9M$.

Fig. 7 illustrates the polarization patterns of a hotspot along a homoclinic
orbit, where the hotspot is perturbed to move outward from an unstable
circular orbit. The net EVPA is greater under the radial magnetic field
compared to the vertical magnetic field, a characteristic also observed in the
case of an inspiraling hotspot. However, the Q-U curve exhibits a distinctive
multi-loop structure, as illustrated in the middle column of Fig. 7. In the
previous studies, the Q-U loop has been considered to be a significant
characteristic in the hotspot astrometry, attributed to the nonzero
observation angle [13]. We now highlight that even when being observed face-
on, such a loop structure can arise from a homoclinic orbit. It is crucial to
note that one-loop structure in the Q-U curves is a general feature of
elliptical-like orbits, depending solely on the variation of total intensity
along the orbit and the completion of a full period in $\varphi$. However,
only the homoclinic orbit can generate a multi-loop structure on the Q-U
plane, where the hotspot wraps around $r_{c}$ an infinite number of times.
Furthermore, the black hole spin introduces a slight breaking of mirror
symmetry in the homoclinic orbit [41]. This asymmetry is amplified in the
corresponding Q-U curve, as can be seen in the right column of Fig. 7.

###  4.4 Results for accretion flows within the plunging region

In the intra-ISCO area of a thin accretion disk, the fluid inevitably adopts a
plunging trajectory towards the black hole’s horizon, forming a distinct
plunging region. This region holds significant astrophysical implications [52,
39, 42]. Notably, it plays a crucial role in explaining the observational
spectrum of accretion disks in the $X$-ray band [38, 40]. If the plunging
region is present near a supermassive black hole like M87* or SgrA*, it can
also yield unique optical signatures in the radio band, potentially detectable
by the EHT. Within this plunging region, the fluid dynamics are primarily
governed by the gravitational pull, causing the streamlines to closely follow
timelike geodesics. The analytical framework outlined in Sec.2 and Sec. 3 can
be directly applied to study the plunging region. For example, superimposing
the trjectories of Eq. (2.2) with varying initial azimuthal angles can depict
a stable, axisymmetric profile of an intra-ISCO accretion disk.

Figure 8: The observed total intensity and the net EVPA for a plunging-region
disk comprised of entirely zero-conductivity plasma. We set the radial
velocity at the ISCO crossing as $\delta{U}=10^{-3}$. For each case, the
dashed line denotes the location of the inner shadow calculated using Eq.
(3.10).

In a typical accretion disk, the magnetic field is intricately linked with the
fluid dynamics. In the case of a plasma with zero conductivity, the
accompanying magnetic field is governed by the ideal-MHD condition [50],
resulting in

$\displaystyle
B^{t}=-\frac{\Psi}{\sqrt{-g}}\bigg{(}U_{r}+\frac{U^{\phi}}{U^{r}}L\bigg{)}\,,\quad
B^{r}=-\frac{\Psi}{\sqrt{-g}}E\,,\quad
B^{\phi}=-\frac{\Psi}{\sqrt{-g}}\frac{U^{\phi}}{U^{r}}E\,.$ (4.3)

Here $g$ is the determinant of the metric. Note that $B^{r},B^{\phi}$ diverge
at the ISCO where $U^{r}=0$. This divergence should be addressed by
introducing a non-zero radial velocity when crossing the ISCO, as suggested by
[42], given by $U^{r}=U^{r}_{I}-\delta U$, where $\delta U\ll 1$ is a small
correction based on Eq. (2.2). In the zero-conductivity fluid, the magnetic
field lines align with the streamlines, leading to a negligible vertical
component near the equatorial plane. However, real accretion flows may exhibit
some conductivity near the equatorial plane, resulting in a non-zero vertical
component there. For simplicity, the Papapetrou-Wald potential given by Eq.
(3.18) can be utilized to represent a typical vertical magnetic field within
the plunging region. Consequently, the results depicted in the top row of Fig.
5 can be interpreted as the polarization pattern of the plunging region with a
vertical magnetic field. The suppression of central intensity outlines the
horizon on the screen, commonly referred to as the inner shadow of the
accretion disk [37].

Under the accompanying magnetic field of Eq. (4.3), the optical appearance of
the disk is illustrated in Fig. 8, where we have chosen $\delta{U}=10^{-3}$ as
a conservative estimation. The behavior of $\mathcal{I}_{o}$ as the radius
varies reveals a significant reduction in the observed intensity not so close
to the horizon. For instance, in the case of $a=0.9M$, $\mathcal{I}_{o}$ at
$r=(r_{+}+r_{I})/2$ diminishes to nearly $1/100$ of the value at $r=r_{I}$.
Thus, with such a magnetic field configuration, the plunging region appears
much darker, resulting in a larger central dark region on the screen compared
to the inner shadow produced by the vertical field. Regarding the polarization
pattern, as the radius decreases from the ISCO, $\chi_{net}$ exhibits a
monotonically increasing trend. For higher black hole spins, $\chi_{net}$
demonstrates a larger value at the ISCO but a smaller value at the horizon,
akin to the behavior under the vertical magnetic field. However, for the cases
that $a=0.1M,0.5M$, $\chi_{net}$ surpasses $\pi/2$ as the black hole is
approached, transitioning from a counterclockwise to a clockwise pattern. In
contrast, when $a=0.9M$, the net EVPA maintains a counterclockwise pattern
throughout the plunging region. Conversely, as depicted in the top-right panel
of Fig. 5, $\chi_{net}$ is less than $\pi/2$ across all scenarios under the
vertical magnetic field, indicating that the net EVPA consistently displays a
counterclockwise pattern.

##  5 Summary

In this study, we studied the polarized images of luminous hotspots in close
proximity to the horizon of a Kerr black hole. We focused on the thermal
synchrotron emission from the hotspots moving along the plunging orbits and
the homoclinic orbits [44, 41], while considering two distinct magnetic field
configurations: the vertical and radial fields [48, 50]. For an on-axis
observer, we introduced a new approximate function to facilitate the
computation of the light bending, thereby simplifying the calculation of the
imaging process. To elucidate the polarization patterns, we conducted the
calculations involving the Stokes parameters and derived the total intensity
and the polarization vector along a hotspot’s trajectory. The main results of
this study are listed below:

  * •

During the inspiral of a hotspot from the prograde ISCO, our study revealed
that when subject to a vertical magnetic field, the total intensity of the
hotspot decreases monotonically as it approaches the horizon, while the net
EVPA continually increases. In contrast, with a radial magnetic field, both
the total intensity and net EVPA do not follow monotonic patterns. The Stokes
parameters demonstrate a right-handed spiral shape on the Q-U plane, with the
phase determined by the hotspot’s motion in the $\phi$ direction.

  * •

When a hotspot plunges from the retrograde ISCO, the projected trajectory on
the observer’s screen can be divided into two parts: one segment inspiraling
retrogradely, and the other inspiraling progradely. The retrograde segment
appears significantly brighter, with a net EVPA greater than $\pi/2$,
contributing to nearly the entire Q-U curve. In contrast, the prograde segment
is notably dim, to the extent that it is scarcely discernible in the Stokes
parameters, as demonstrated in Fig. 6. The net EVPA of the prograde segment is
less than $\pi/2$.

  * •

When a hotspot travels along a homoclinic orbit, the resulting Q-U curve
showcases a novel multi-loop pattern, as depicted in Fig. 7. It is noteworthy
that while a Q-U one loop is a common characteristic of elliptical-like
orbits, it is exclusively the homoclinic orbit that can produce a multi-loop
structure on the Q-U plane. This phenomenon arises from the hotspot
continuously encircling an unstable circular orbit.

Moreover, by considering the plunging region of an accretion flow approximated
by plunging geodesics, we extended our study to the polarized images of the
intra-ISCO region of a thin disk. Our study provided, for the first time, the
details of the polarizations of a plunging region disk under different
magnetic field configurations. Under the vertical magnetic field or the
accompanying magnetic field given by Eq. (4.3), the optical appearances of the
disk are totally different, as illustrated in Fig. 5 and Fig. 8. Upon
comparing the total intensities between the two scenarios, we observed that
the plunging region with the accompanying magnetic field appears notably
darker than when under the vertical magnetic field. Furthermore, the behaviors
of the polarization vector in these two cases demonstrate marked differences.

We end this work by outlining several limitations and future perspectives of
this study. Firstly, for the purpose of an analytical study, we considered the
hotspots traveling along the plunging and the homoclinic geodesics with
integrable trajectory functions. It is essential to consider the hotspots with
other types of motion, such as the bound orbits and the deflecting orbits
[53], or the hotspots not confined to the equatorial plane. Secondly, although
the inclination angles for observing M87* and Sgr A* are small, it is still
necessary to consider the effects of a nonzero inclination, especially how it
distorts the geometric shape of the Q-U curve [13]. Lastly, we neglected the
multi-level images of the hotspots or the inner-ISCO disk due to strong
gravitational lensing in this study, which manifest as highly distorted curves
closely following the critical curve of the Kerr black hole. The lensed images
could contribute both to the total flux and the observed Stokes parameters.

## Acknowledgments

The work is partly supported by NSFC Grant No. 12275004.

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