# Polarized image of equatorial emission in horizonless spacetimes I:
traversable wormholes

Valentin Delijski1111E-mail<EMAIL_ADDRESS>Galin
Gyulchev1222E-mail<EMAIL_ADDRESS>Petya Nedkova1333E-mail:
<EMAIL_ADDRESS>
Stoytcho Yazadjiev1,2444E-mail<EMAIL_ADDRESS>

1 Faculty of Physics, Sofia University,
5 James Bourchier Boulevard, Sofia 1164, Bulgaria
2 Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences, Acad. G. Bonchev 8,
Sofia 1113, Bulgaria

###### Abstract

We study the linear polarization from the accretion disk around a class of
static traversable wormholes. Applying the simplified model of a magnetized
fluid ring orbiting in the equatorial plane, we search for characteristic
signatures, which could distinguish wormhole from black hole spacetimes by
their polarization properties. For the purpose we analyse the direct polarized
images for different inclination angles, the strongly lensed indirect images,
and the polarized radiation which reaches the asymptotic observer through the
wormhole throat, and compare to the Schwarzschild black hole. For small
inclination angles the two types of compact objects lead to a very similar
polarization pattern of the direct images. More significant distinctions are
observed for the strongly lensed indirect images, where the polarization
intensity in the wormhole spacetimes can grow up to an order of magnitude
compared to the Schwarzschild black hole. Detecting radiation from the region
across the wormhole throat leads to the formation of an additional structure
of ring images with distinct polarization properties. The twist of the
polarization vector around the ring is less pronounced, thus modifying the
polarization pattern, and the polarization intensity can increase with an
order of magnitude compared to the radiation from our universe. Thus, while it
could be difficult to distinguish wormhole spacetimes by their direct
polarized images, the strongly lensed images and the polarization of the
radiation through the wormhole throat provide characteristic signatures which
can serve as probes for horizonless objects.

##  1 Introduction

The Event Horizon Telescope opened the possibility to observe directly the
supermassive black holes in the galactic centers in the radio band [1]-[7].
However, a major challenge is to interpret the radio images by answering
questions about the nature of the compact objects and the gravitational theory
describing them, and as well as about the specifications of the accretion
flux. These issues are hard to be determined since the information about the
gravitational field is coupled to the magneto-hydrodynamics of the accretion
process in a non-linear way depending on many parameters and physical
ambiguities.

The situation is more complicated since a large variety of compact objects are
characterized by qualitatively very similar shadows. They can deviate only
minimally from the Kerr black hole and the deviation frequently falls within
the limits of the experimental error. This class of self-gravitating systems
includes black holes in the modified theories of gravity [8]-[14], but also
more exotic objects like wormholes and naked singularities [15]-[19]. In
addition, it is theoretically possible that two different compact objects
possess exactly the same shadow in the strict sense, proving that the shadow
boundary cannot determine the compact object uniquely [20].

In order to constrain further the spacetime geometry, we should consider
additional properties of the images. We can analyse, for example, the apparent
structure of the accretion disk and the observable intensity of its emission.
However, the investigation of the relativistic images of some simplified
models of accretion disks demonstrates that the problem still contains a lot
of degeneracy. The accretion disks around compact objects of very different
physical nature closely resemble each other and we obtain again a large sample
of Kerr black hole mimickers [21]-[27].

Another approach to break the degeneracy and differentiate more effectively
between different compact objects by means of their radio images is to
consider the polarization of the electromagnetic radiation from the accretion
disk. The polarization characterizes the structure of the magnetic field in
the strong-gravity region probing in particular the interaction of the local
magnetic field with the spacetime geometry. In this way it brings
supplementary information in addition to the observable intensity of the
accretion disk.

Recently, the Event Horizon Telescope Collaboration published a survey of the
observable polarization from the radio source M87*, revealing a linearly
polarized emission on event-horizon scales, which is most probably produced by
synchrotron radiation [28]-[29]. The observed images are interpreted by
comparing with the polarization structure which would follow theoretically
from different scenarios of the accretion disk physics by means of performing
general relativistic magnetohydrodynamic (GRMHD) simulations. The analysis
strongly suggests that the magnetically arrested disk models (MAD), which are
characterized by dynamically important magnetic fields in the emission region,
are most relevant for describing the vicinity of M87*.

In addition, a simple analytical toy model was developed and compared with the
observational data [30]. The polarized emission is represented as the
synchrotron radiation, which originates from a thin ring of gas while orbiting
in the gravitational field of the Schwarzschild black hole assuming that the
local magnetic field is constant. The superposition of the radiation produced
by several copies of such fluid rings builds up the emission from the
innermost region of the accretion disk. The model is characterized only by a
few parameters which include the radius of the ring and its tilt with respect
to the observer, the velocity of the fluid in the local rest frame and the
magnitude and the direction of the magnetic field. However, despite its
simplicity it leads to similar predictions as the fully non-linear MAD models,
and manages to reproduce the basic features of the observable polarization.

The model of a synchrotron emitting ring was generalized to the Kerr black
hole in the context of the EHT observations [31] and extended to the modified
theories of gravity [32]. The polarization structure for rotating black holes
with near-extremal angular momenta was considered recently in [33], while [34]
investigated the universal properties of the strongly lensed polarimetric
images in the vicinity of the photon ring. We should note, however, that semi-
analytical studies of the linear polarization and the lensed images of an
equatorial circular emitter in black hole spacetimes date back already from
the 70s and have resulted in a range of classical works [35]-[40] .

Our aim in this paper is to explore the polarization signatures in the
spacetime of horizonless compact objects. On the one hand, we search for
qualitatively new features of the polarized images which could distinguish
exotic compact objects observationally. On the other hand, we probe how
sensitive is the observable polarization to the spacetime geometry and how
effectively we can use its structure in order to determine the physical nature
of the compact objects at the galactic centres. For the purpose we apply the
analytical model of a synchrotron emitting ring of gas in the gravitational
field of a class of static traversable wormholes and investigate whether the
non-black hole nature of spacetime would leave imprints on the properties of
the polarization. Since we would like to isolate effects which can be
attributed primarily to the absence of an event horizon, we select geometries
possessing similar structure of the circular geodesics as the Schwarzschild
black hole. In particular, the wormhole solutions contain a single photon
ring, which is located at a similar radial distance as for the Schwarzschild
black hole. Such relation applies also for the marginally bound and the
marginally stable particle orbits, which determine the characteristic
distances for the accretion process.

The paper is organized as follows. In the next section we present the static
traversable wormholes which we consider and some of the characteristics of the
geodesic motion, which are relevant for our studies. In section 3 we describe
the physical model of the linear polarization, which results from synchrotron
radiation propagating in curved spacetime and the computational procedure for
obtaining its observable image. We show that in static spherically symmetric
spacetime the procedure is completely algebraic due to the presence of Killing
and Killing-Yano tensors. In section 4 we present our simulated images for the
linear polarization in wormhole geometry and discuss their properties in
comparison to the Schwarzschild black hole. We consider direct images at
different inclination angles, as well as the strongly lensed indirect images.
In the last section we summarize our results.

##  2 Static traversable wormholes

In their seminal paper [41] Morris and Thorne suggested the general geometry
which describes a static traversable wormhole

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

The metric functions depend only on the radial coordinate $r$, and we assume
that the redshift function $N(r)$ is finite and non-zero in all the coordinate
range. Then, the metric describes a completely regular spacetime and we can
infer the properties of this geometry by considering the embedding of the
constant $t$ and $\theta$ cross-sections in the three-dimensional Euclidean
space. The embedding describes a tunnel connecting two regions in the
spacetime with a throat located at the constant radius $b(r)=r_{0}$ where the
metric function $g_{rr}$ becomes divergent. It further possesses the
characteristic flaring out shape of the wormhole if we require that the
condition $db/dr<1$ is satisfied at the throat. In addition, the two ends are
asymptotically flat provided that the metric functions behave as

$\displaystyle N$ $\displaystyle=$ $\displaystyle
1-\frac{M}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right),\quad~{}~{}~{}\frac{b}{r}=\mathcal{O}\left(\frac{1}{r}\right),$
(2)

at the spacetime infinity, where $M$ is the ADM mass of the solution.

In this work we consider a particular class of wormhole solutions with metric
functions given by

$\displaystyle
N=\exp\left(-\frac{r_{0}}{r}-\alpha\frac{r^{2}_{0}}{r^{2}}\right),\quad~{}~{}~{}b=r_{0},$
(3)

where $r_{0}$ and $\alpha$ are positive constants. The parameter $r_{0}$
determines the ADM mass of the wormhole, while the parameter $\alpha$ controls
the redshift of the solution. For convenience we can further represent the
metric in a dimensionless form by making the conformal transformation and
rescaling by the wormhole mass

$\displaystyle dS^{2}=r_{0}^{-2}ds^{2},\quad~{}~{}~{}t\rightarrow
r_{0}t,\quad~{}~{}~{}r\rightarrow r_{0}r.$

In this way we obtain a wormhole solution with a unit mass and a throat
located at $r=1$.

The described spacetime possesses a photon sphere located at the maximum of
the effective potential for the null geodesics

$V^{ph}_{eff}=L^{2}\frac{N(r)}{r^{2}},$ (4)

where $L$ is the specific angular momentum of the photon, i.e. satisfying
$V^{ph}_{eff}=0$, $dV^{ph}_{eff}/dr=0$ and $dV^{ph}_{eff}/dr<0$. We can
further obtain the position of the marginally stable timelike orbit,
corresponding to the inflection point of the effective potential for the
timelike geodesics

$V_{eff}=N(r)\left(1+\frac{L^{2}}{r^{2}}\right),$ (5)

as a solution to the equations $V_{eff}=0$, $dV_{eff}/dr=0$ and
$d^{2}V_{eff}/dr^{2}=0$. The locations of the photon sphere and the marginable
stable orbit can be given explicitly in the form

$\displaystyle r_{\text{ph}}=\frac{M}{2}\left(1+\sqrt{1+8\alpha}\right),$
$\displaystyle
r_{\text{ms}}=2M\left[\sqrt{\frac{4}{9}\left(6\alpha+1\right)}\cosh{\left(\frac{1}{3}\operatorname{arcosh}{\frac{1+9\alpha+\frac{27}{2}\alpha^{2}}{(6\alpha+1)^{\frac{3}{2}}}}\right)}+\frac{1}{3}\right].$
(6)

On the other hand, the marginally bound timelike orbit is determined by the
condition

$\displaystyle E=\frac{N^{2}(r)}{\sqrt{N^{2}(r)-rN(r)N^{{}^{\prime}}(r)}}=1,$
(7)

where $E$ is the specific energy of the orbiting particle.

The behavior of the characteristic circular orbits for the wormhole solution
(3) is summarized in Fig. 1. We observe that in the range $\alpha\in[2,3]$
their locations approach the corresponding values for the Schwarzschild
solution. For example, for $\alpha=2.2$ the marginally stable orbit is located
at $r=6M$, for $\alpha=2.4$ the marginally bound orbit corresponds to $r=4M$,
while for $\alpha=3$ we obtain the photon sphere at the radial distance
$r=3M$.

Figure 1: Location of the photon sphere $r_{ph}$, the marginally bound orbit
$r_{mb}$ and marginally stable orbit $r_{ms}$ for traversable wormholes with
different redshift parameter $\alpha$.

##  3 Polarized image of an equatorial emitting ring in static spherically
symmetric spacetime

We consider a ring of magnetized fluid orbiting in the equatorial plane around
a static spherically symmetric compact object. Due to the interaction with the
local magnetic field the fluid emits linearly polarized synchrotron radiation.
In this section we will discuss the apparent polarization of the
electromagnetic radiation generated in this setting as seen by an asymptotic
observer and describe the general procedure for obtaining its observable
pattern and intensity.

Let us consider a static spherically symmetric metric in the general form

$\displaystyle
ds^{2}=-e^{2\nu(r)}dt^{2}+e^{2\lambda(r)}dr^{2}+R^{2}(r)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (8)

For our purpose it is convenient to express the polarization in the fluid rest
frame by defining a local orthonormal tetrad $\\{e^{\mu}_{(a)}\\}$ aligned
with the rotating fluid element at a given radius $r$ and azimuthal angle
$\phi$. The natural tetrad adapted to the spacetime symmetries is given by

$\displaystyle
e_{(t)}=e^{-\nu(r)}\partial_{t},\,\,\,\,\,e_{(r)}=e^{-\lambda(r)}\partial_{r},\,\,\,\,\,e_{(\theta)}=\frac{1}{R(r)}\partial_{\theta},\,\,\,\,\,e_{(\phi)}=\frac{1}{R(r)\sin\theta}\partial_{\phi}.$
(9)

We further allow for the fluid to move with velocity $\vec{\beta}$ in the
$(r)$-$(\phi)$ plane of the local frame, which we parameterize as

$\vec{\beta}=\beta\left(\cos\chi\,(r)+\sin\chi\,(\phi)\right).$ (10)

Defining a local rest frame attached to the boosted emitter

$\displaystyle\hat{e}^{\,\mu}_{(a)}=\Lambda^{\hskip
8.5359pt(b)}_{(a)}e^{\mu}_{(b)},$ (11)

vectors are transformed by the Lorenz transformation $\Lambda^{(a)}_{\hskip
8.5359pt(b)}$

$\hat{V}^{(a)}=\Lambda^{(a)}_{\ \ (b)}V^{(b)},$ (12)

where the matrix $\Lambda^{(a)}_{\hskip 8.5359pt(b)}$ is given in explicit
form as

$\displaystyle\Lambda=\begin{pmatrix}\gamma&-\beta\gamma\cos\chi&-\beta\gamma\sin\chi&0\\\
-\beta\gamma\cos\chi&(\gamma-1)\cos^{2}\chi+1&(\gamma-1)\sin\chi\cos\chi&0\\\
-\beta\gamma\sin\chi&(\gamma-1)\sin\chi\cos\chi&(\gamma-1)\sin^{2}\chi+1&0\\\
0&0&0&1\\\ \end{pmatrix},$ (13) $\displaystyle\gamma=(1-\beta^{2})^{-1/2}$
(14)

Given a vector $\hat{V}^{(a)}$ in the local rest frame, we can recover its
components in the spacetime coordinates $V^{\mu}$ by the inverse
transformation

$V^{\mu}=\hat{e}^{\,\mu}_{(a)}\hat{V}^{(a)}.$ (15)

In the local rest frame we introduce the magnetic field
$\vec{B}=\left(\hat{B}^{r},\hat{B}^{\phi},\hat{B}^{\theta}\right)$ and denote
the local three momentum by
$\vec{p}=\left(\hat{p}^{r},\hat{p}^{\phi},\hat{p}^{\theta}\right)$. Under the
action of the magnetic field the fluid emits synchrotron radiation with
polarization

$\vec{f}=\frac{\vec{p}\times\vec{B}}{|\vec{p}|},$ (16)

where we have $\hat{f}^{(t)}=0$. For convenience we further normalize the
polarization vector to satisfy

$\hat{f}^{a}\hat{f}_{a}=\sin^{2}\zeta|\vec{B}|^{2},$ (17)

denoting the angle between $\vec{p}$ and the magnetic field $\vec{B}$ by
$\zeta$

$\sin\zeta=\frac{|\vec{p}\times\vec{B}|}{|\vec{p}||\vec{B}|}.$ (18)

The components of the polarization 4-vector $f^{\mu}$ in the spacetime
coordinates are recovered in the standard way by performing the transformation
(15).

Having defined the polarization at the point of emission we should calculate
how it transforms while the electromagnetic radiation propagates through the
curved spacetime in order to reach the asymptotic observer. According to the
geometric optics approximation the right rays propagate on null geodesics,
while the polarization $4$-vector is parallel transported along the geodesic,
and it is normal to the $4$-momentum $p^{\mu}$, i.e.

$\displaystyle p^{\mu}\nabla_{\mu}p_{\nu}=0$ $\displaystyle
p^{\mu}\nabla_{\mu}f_{\nu}=0,\;\;\;p^{\mu}f_{\mu}=0.$ (19)

In general, in order to obtain the polarization vector at the observer’s
position we have to solve the described differential equations with the
corresponding boundary conditions. However, in static spherically symmetric
spacetime the procedure reduces to algebraic manipulations due to the high
degree of symmetry. The spacetime possesses four Killing vectors which make
the geodesic motion completely integrable. In addition, we have the hidden
symmetries in the form of irreducable Killing and Killing-Yano tensors which
don’t generate isometries but lead to conserved constants of motion. Using
their explicit form we can derive an algebraic expression for the polarization
$4$-vector at any point of the spacetime.

The static spherically symmetric spacetime admits a second order Killing-Yano
tensor $Y_{\mu\nu}$ with the following non-zero components in the standard
coordinate basis

$\displaystyle Y_{\theta\phi}=-Y_{\phi\theta}=R^{3}(r)\sin\theta.$ (20)

In addition we have a conformal Killing-Yano tensor ${\tilde{Y}}_{\mu\nu}$
arizing as the Hodge dual of $Y_{\mu\nu}$, i.e.
${\tilde{Y}}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}Y^{\alpha\beta}$.
Using $Y_{\mu\nu}$ and ${\tilde{Y}}_{\mu\nu}$ we can construct two constants
of motion $\kappa_{1}$ and $\kappa_{2}$ which are conserved along the photon
trajectory and related to the photon’s 4-momentum $p^{\mu}$ and polarization
vector $f^{\mu}$ in the following way

$\displaystyle\kappa_{1}=\frac{1}{2}{\tilde{Y}}_{\mu\nu}p^{\mu}f^{\nu},\;\;\;\kappa_{2}=Y_{\mu\nu}p^{\mu}f^{\nu}.$
(21)

In explicit form we have

$\displaystyle\kappa_{1}=R(r)e^{\nu(r)+\lambda(r)}(p^{t}f^{r}-p^{r}f^{t}),$
$\displaystyle\kappa_{2}=R^{3}(r)\sin\theta(p^{\theta}f^{\phi}-p^{\phi}f^{\theta}).$
(22)

These expressions allow us to determine the observable polarization entirely
in terms of the boundary data on the geodesic. Let us consider a light ray
emitted at spacetime coordinates $x^{\mu}_{s}$ with linear polarization
$f^{\mu}_{s}$ which reaches a distant observer located at $x^{\mu}_{obs}$ with
polarization $f^{\mu}_{obs}$. In order to obtain the observable polarization
we should define a local reference frame at the observer’s position. We
introduce the natural orthonornal tetrad given by (9) and denote by $p^{(a)}$
the projection of the $4$-momentum in the local basis. The projection is
completely determined by two angles $\alpha$ and $\beta$ defined as [42]

$\displaystyle\sin\alpha=\frac{p^{(\theta)}}{p^{(t)}},\quad\,\,\,\tan\beta=\frac{p^{(\phi)}}{p^{(r)}},$
(23)

which we can choose as celestial coordinates in the observer’s sky. Since
$\alpha$ and $\beta$ scale inversely with the radial coordinate at large
distances, it is convenient to rescale them defining the celestial coordinates
$x=r\alpha$ and $y=r\beta$. For an asymptotic observer we can take the limit
$r\rightarrow\infty$ and the celestial coordinates reduce to

$\displaystyle x$ $\displaystyle=$
$\displaystyle-R(r)p^{(\phi)}=-\frac{p_{\phi}}{\sin\theta_{0}},$
$\displaystyle y$ $\displaystyle=$ $\displaystyle
R(r)p^{(\theta)}=p_{\theta},$ (24)

where $\theta_{0}$ is the inclination angle of the observer.

The constants of motion $\kappa_{1}$ and $\kappa_{2}$ can be expressed by
means of the projection of the $4$-momentum and the polarization vector in the
local reference frame of the observer

$\displaystyle\kappa_{1}=R(r)(p^{(t)}f^{(r)}-p^{(r)}f^{(t)}),$
$\displaystyle\kappa_{2}=R(r)(p^{(\theta)}f^{(\phi)}-p^{(\phi)}f^{(\theta)}).$
(25)

Using the orthogonality condition $p^{\mu}f_{\mu}=0$ we can solve (3) for
$f^{(\theta)}$ and $f^{(\phi)}$ and obtain

$\displaystyle
f^{(\theta)}=\frac{p^{(\theta)}}{(p^{(\theta)})^{2}+(p^{(\phi)})^{2}}\left[-\frac{\kappa_{1}}{R}\frac{p^{(r)}}{p^{(t)}}+\left((p^{(t)})^{2}-(p^{(r)})^{2}\right)f^{(t)}\right]-\frac{p^{(\phi)}}{(p^{(\theta)})^{2}+(p^{(\phi)})^{2}}\frac{\kappa_{2}}{R},$
(26) $\displaystyle
f^{(\phi)}=\frac{p^{(\phi)}}{(p^{(\theta)})^{2}+(p^{(\phi)})^{2}}\left[-\frac{\kappa_{1}}{R}\frac{p^{(r)}}{p^{(t)}}+\left((p^{(t)})^{2}-(p^{(r)})^{2}\right)f^{(t)}\right]+\frac{p^{(\theta)}}{(p^{(\theta)})^{2}+(p^{(\phi)})^{2}}\frac{\kappa_{2}}{R}.$

For an asymptotic observer we can simplify these relations taking advantage of
a further constant of motion associated with the hidden symmetries in static
spherically symmetric spacetimes. The square of the Killing-Yano tensor
$Y_{\mu\nu}$ gives rise to the Killing tensor
$K_{\mu\nu}=Y_{\mu\alpha}Y_{\nu}^{\;\;\alpha}$, which generates the conserve
quantity $C=K_{\mu\nu}p^{\mu}p^{\nu}$ along the photon trajectories. In
explicit form we have

$\displaystyle
C=K_{\mu\nu}p^{\mu}p^{\nu}=R^{4}(r)\left[(p^{\theta})^{2}+\sin^{2}\theta(p^{\phi})^{2}\right]=R^{2}(r)\left[(p^{(\theta)})^{2}+(p^{(\phi)})^{2}\right],$
(27)

Using this integral of motion and the property of the null geodesics
$p_{\mu}p^{\mu}=0$ we can show that in the limit $r\rightarrow\infty$ the
terms containing the polarization component $f^{(t)}$ vanish. Then, we take
into account the definition of the celestial coordinates (3) and obtain the
following expressions for the components of the polarization vector in the
coordinate system on the asymptotic observer’s screen

$\displaystyle f^{x}=\frac{x\kappa_{1}+y\kappa_{2}}{x^{2}+y^{2}},$
$\displaystyle f^{y}=\frac{y\kappa_{1}-x\kappa_{2}}{x^{2}+y^{2}}.$ (28)

. We can further normalize the polarization vector measured by the observer to
unity as

$\displaystyle
f^{x}_{obs}=\frac{x\kappa_{1}+y\kappa_{2}}{\sqrt{(\kappa_{1}^{2}+\kappa_{2}^{2})(x^{2}+y^{2})}},$
$\displaystyle
f^{y}_{obs}=\frac{y\kappa_{1}-x\kappa_{2}}{\sqrt{(\kappa_{1}^{2}+\kappa_{2}^{2})(x^{2}+y^{2})}}.$
(29)

In this way we obtain expressions for the observable polarization at infinity
which depend only on the constants of motion $\kappa_{1}$ and $\kappa_{2}$ and
the projection of the photon trajectory on the observer screen $(x,y)$. The
conserved quantities $\kappa_{1}$ and $\kappa_{2}$ can be determined by the
initial data for the polarization, i.e. by the parameters of the physical
process which gives rise to it in the accreting fluid. We consider the
components of the polarization vector in the fluid local rest frame
$\hat{f}^{(a)}$ following from the synchrotron radiation (16). Then, we
project them in the local tetrad attached to the orbiting fluid at certain
radius $r_{s}$ (9) and substituting in (3) we obtain expressions for
$\kappa_{1}$ and $\kappa_{2}$ only in terms of the initial data in the fluid
rest frame

$\displaystyle\kappa_{1}=\gamma
R(r_{s})\left[{\hat{p}^{(t)}}{\hat{f}^{(r)}}+\beta\left({\hat{p}^{(r)}}{\hat{f}^{(\theta)}}-{\hat{f}^{(r)}}{\hat{p}^{(\theta)}}\right)\right],$
$\displaystyle\kappa_{2}=\gamma
R(r_{s})\left[\left({\hat{p}^{(\theta)}}{\hat{f}^{(\phi)}}-{\hat{f}^{(\theta)}}{\hat{p}^{(\phi)}}\right)-\beta{\hat{p}^{(t)}}{\hat{f}^{(\phi)}}\right].$
(30)

As we defined previously in our model, $\beta$ is the physical velocity of the
orbiting emitter and $\gamma=1/\sqrt{1-\beta^{2}}$.

In order to examine the polarization structure we use two observables - the
polarization intensity and the direction of the polarization vector. The
polarization intensity is proportional to the norm of the observable
polarization vector but we need to take into account additional factors
determined by the phenomenology of the synchrotron emission [30]. The
intensity of the synchrotron radiation depends on the angle $\sin\zeta$
between the direction of the magnetic field and the 4-velocity in the fluid
rest frame, which is modulated by the frequency of the emitted photons
$\alpha_{\nu}$ to a power law $(\sin\zeta)^{1+\alpha_{\nu}}$. Since this is
the emission per unit volume, it should be further multiplied by the geodesic
path $l_{p}$ length in the emitting region. For an optically and geometrically
thin disk, which we assume in our model, the geodesic path length can be
expressed as

$\displaystyle l_{p}=\frac{\hat{p}^{(t)}}{\hat{p}^{(\theta)}}\,H,$ (31)

by means of the 4-momentum in the fluid rest frame and the disk height $H$.
While propagating in the spacetime in order to reach the observer, the
intensity is further Doppler-boosted by a factor $\delta^{3+\alpha}$, where
$g$ is the gravitational redshift

$g=\frac{E_{obs}}{E_{s}}=\frac{1}{\hat{p}^{(t)}},$ (32)

given by ratio of the photon’s energy measured at the observer’s frame
$E_{obs}$ and the emitter’s frame $E_{s}$. Thus, assuming that the
polarization vector in the observer’s frame is normalized to unity, the
observed linear polarization intensity is given by the expression

$\displaystyle|P|=\delta^{3+\alpha}\,l_{p}\,(\sin\zeta)^{1+\alpha_{\nu}},$
(33)

up to a proportionality constant. Models of M87* are consistent with the power
$\alpha_{\nu}=1$, therefore in our studies we adopt this value.

Since the intensity is proportional to the norm of the polarization vector we
can define the components of the observed electric field as

$\displaystyle E^{x}_{obs}$ $\displaystyle=$
$\displaystyle\delta^{2}\,l_{p}^{1/2}\,\sin\zeta\,f^{x}_{obs},$ $\displaystyle
E^{y}_{obs}$ $\displaystyle=$
$\displaystyle\delta^{2}\,l_{p}^{1/2}\,\sin\zeta\,f^{y}_{obs},$
$\displaystyle|P|$ $\displaystyle=$
$\displaystyle(E^{x}_{obs})^{2}+(E^{y}_{obs})^{2},$ (34)

where $f^{x}_{obs}$ and $f^{y}_{obs}$ are given by eq. (3).

The direction of the polarization vector in the observer’s frame is measured
by the Electric Vector Position Angle (EVPA) defined as

$\displaystyle EVPA=\arctan\left({-\frac{f^{x}_{obs}}{f^{y}_{obs}}}\right).$
(35)

In our simulations we estimate the EVPA counter-clockwise with respect to the
positive semi-axis $x>0$ on the observer’s sky.

##  4 Linear polarization in static wormhole spacetime

In this section we apply the model of a magnetized fluid ring emitting
synchrotron radiation to obtain the observable polarization in static wormhole
spacetime. Using a ray-tracing procedure we integrate numerically the null
geodesic equations for trajectories originating from a particular equatorial
circular orbit with radius $r_{s}$. As a result, each trajectory projects on
the observer’s sky with celestial coordinates $x$ and $y$. The emission model
in the fluid rest frame determines the constants of motion $\kappa_{1}$ and
$\kappa_{2}$ on the photon trajectory in terms of the components of the
polarization vector and the 4-velocity in the local frame as expressed by eq.
(3). Then, the apparent polarization at each point of the observer’s sky with
coordinates $(x,y)$ is completely defined by the values of integrals of motion
$\kappa_{1}$ and $\kappa_{2}$ according to eq. (3). We further calculate the
polarization intensity and direction EVPA, which represent the experimentally
observable quantities. In our images the polarization intensity is
proportional to the length of the polarization vector while EVPA is determined
by its tilt with respect to the positive direction of the $x$-axis.

The images of the circular orbits can be classified into direct and indirect
ones according to the type of the photon trajectories which lead to their
formation. Direct images arise from photon trajectories which are
characterized by the variation of the azimuthal angle in the range
$\phi\in[0,\pi)$. On the other hand, for the indirect images the azimuthal
angle varies in the interval $\phi\in[0,(k+1)\pi]$, where the positive integer
$k$ determines the order of the image. This implies that the photon
trajectories which form indirect images of order $k$ perform $k$ half-loops
around the compact object before reaching the observer. In our study we will
consider both direct images and indirect images of order $k=1$.

In our simplified model of a magnetized fluid ring orbiting around the central
compact object the observable polarization depends on a range of physical
parameters. On the one hand we have the characteristics of the accretion flux,
which include the direction of the magnetic field and the fluid velocity. On
the other hand the spacetime geometry affects the images by means of the
parameters of the metric, which describes the compact object, the inclination
angle of the observer, and the emission radius. Since we would like to probe
the hypothesis that an exotic compact object may lead to a similar
polarization distribution as observed for M87*, we give preference to physical
parameters which reproduce qualitatively the EHT polarimetric observations.
For this reason we consider small inclination angles close to the estimated
inclination angle $\theta=17^{\circ}$ for the galaxy M87*. The fluid is
assumed to move inwards which is the natural direction of the accretion flow,
and it rotates in a clockwise direction as observed for M87*. We further pick
emission orbits located in the region of strong gravitational field in the
vicinity of the ISCO. In particular, we consider the ISCO for the
Schwarzschild solution and the orbit with radius $r=4.5M$ which corresponds to
the apparent size of the emission ring in the EHT observations of M87*.

We further put constraints on the local magnetic field by taking advantage of
the insights provided by previous studies [30]. Modelling the polarization of
M87* by means of an equatorial synchrotron emission in the Schwarzschild
background, it was demonstrated that vertical magnetic field leads to a very
different polarization pattern than the experimentally detected. The asymmetry
of the observed polarization intensity along the ring is controlled by the
physical effects of Doppler beaming and abberation. For vertical magnetic
field and in-falling fluid, which rotates in the clockwise direction the two
effects compete and favor the increasing of the polarization intensity at the
opposite sides of the image. The effect of the abberation dominates and causes
that the largest intensity is distributed in the left or down part of the
images depending on the direction of the fluid velocity. Similar pattern is
observed if we consider only the influence of the strong gravitational field
on the vertically magnetized fluid assuming that the fluid is at rest. The
gravitational lensing leads to maximum observable intensity of the polarized
flux concentrated at the bottom of the ring.

This pattern contradicts the observations by the EHT since the image of M87*
is characterized by a strong linearly polarized flux on its right-hand side
which gradually shades down towards the north and south poles. Such a
distribution of the polarization intensity can be achieved by restricting the
magnetic field to the equatorial plane. If the magnetic field contains only
equatorial components, the Doppler beaming and the abberation lead to the same
type of flux asymmmetry and enhance each other. As a result, the polarization
intensity concentrates on the right-hand side of the image resembling the
observed pattern.

Purely radial or azimuthal magnetic fields are not suitable for modeling M87*
since they don’t reproduce the twist of the polarization vector. They lead to
polarization in purely tangential or radial direction, respectively, while the
observed image is characterized by a twisting polarization pattern where the
polarization vector rotates up to $70^{\circ}$ around the ring, measured with
respect to the radial direction. Thus, it was estimated in [30] that assuming
that the spacetime geometry is described by the Schwarzschild solution the
emission from a magnetized fluid ring reproduces the observed polarization
pattern most accurately if the magnetic field is purely equatorial with radial
component dominating over the azimuthal one. Probing several directions of the
magnetic field it was suggested that a magnetic field aligned as
$B=[B_{r},B_{\phi},B_{\theta}]=[0.87,0.5,0]$ describes best the observed
polarization of M87*.

We can consider that the assumption of a purely equatorial magnetic field
already determines the direction of the fluid velocity. In the inner region of
the disk it is most probable that the magnetic field follows the flow of the
fluid velocity, i.e. the angles $\eta$ and $\xi$ are related as $\eta=\xi$ or
$\eta=\pi+\xi$. The two choices correspond to parallel magnetic and velocity
fields with coinciding or opposite directions as the ambiguity does not
influence the polarization pattern.

In the following we investigate how the polarization properties of the fluid
ring model are influenced by the spacetime geometry by performing a range of
simulations in static wormhole spacetime for various inclination angles and
magnetic field directions. We consider directly lensed images as well as
indirect images of order $k=1$, which form by photon trajectories making one
half-turn around the compact object before reaching the observer.

###  4.1 Direct images

As a first step we study the polarization in vertical magnetic field. In
particular, we are interested in examining whether vertical magnetic field in
wormhole spacetime leads to a polarization picture which is inconsistent with
the observations of M87* similar to the conclusion for the Schwarzschild black
hole. The effect of abberation which causes increased polarization intensity
in the left-hand side of the image is kinematic one and it will appear in any
static spacetime. We should investigate however the influence of the
gravitational lensing in the region of strong gravitational field and ensure
that it does not have the opposite impact, so that it compensates the
abberation and reverses the asymmetry in the observable polarization
intensity.

For the purpose we simulate the observable polarization produced by a fluid
ring at rest with purely vertical magnetic field. In order to probe the strong
gravitational interaction we choose two values of the ring radius given by
$r=6M$ and $r=4.5M$. We vary the redshift function of the wormhole geometry in
the limits $\alpha\in[0,3]$ and compare the apparent linear polarization with
the emission in the spacetime of the Schwarzschild black hole under the same
conditions.

In Fig. 2 we see that the polarization pattern for the wormhole spacetime
resembles closely the distribution in the Schwarzschild case. The polarization
intensity is smaller than for the Schwarzschild black hole and further
decreases when we increase the redshift parameter $\alpha$. However, it varies
in the same manner along the ring reaching its maximum values at the bottom
part of the image and declining towards the top. Thus, we can conclude that if
we describe the spacetime by a wormhole geometry the emission of a fluid ring
in a vertical magnetic field would still leads to contradiction with the
observed polarization of M87* similar to the Schwarzschild black hole. For
this reason we focus in our analysis on purely equatorial magnetic fields.

Figure 2: Polarization in vertical magnetic field for wormholes with different
redshift parameter $\alpha$. Each color represents the observable polarization
of the orbits located at $r=6M$ (outer ring) and $r=4.5M$ (inner ring) for a
particular wormhole solution with $\alpha\in[0,3]$. The polarization for the
Schwarzschild black hole is given by a black dotted line as a reference. The
inclination angle is $\theta=20^{\circ}$.

In order to get intuition about the variation of the observable polarization
due to the different spacetime geometry we plot the polarized image of the
rings located at $r=6M$ and $r=4.5M$ for a continuous distribution of the
redshift parameter $\alpha$ and several representative configurations of
equatorial magnetic field.

Figure 3: Polarization in equatorial magnetic field for wormholes with
different redshift parameter $\alpha$. Each color represents the observable
polarization of the orbits located at $r=6M$ (outer ring) and $r=4.5M$ (inner
ring) for a particular wormhole solution with $\alpha\in[0,3]$. The
polarization for the Schwarzschild black hole is given by a black dotted line
as a reference. The inclination angle is $\theta=20^{\circ}$.

We observe that the polarization in the wormhole spacetime obeys a
qualitatively similar distribution as for the Schwarzschild black hole. The
polarization intensity increases with respect to the Schwarzschild case for
small $\alpha$ and decreases when the redschift parameter grows. However, its
relative distribution around the ring follows the same pattern as for the
Schwarzschild black hole. In Fig. 3 we see that the maximum intensity of the
polarized flux is again concentrated at the right-hand side of the image and
decays in a similar fashion towards the north and south poles. The direction
of the polarization vector deviates very slightly from the Schwarzschild
solution for all the values of the azimuthal coordinate and the redshift
parameter. As a result it reproduces a similar twist around the ring.

From these arguments we can conclude that the direct gravitational lensing
around wormholes can lead to a similar polarization picture as for black holes
if we consider small inclination angles. Thus, the direct equatorial emission
in wormhole spacetimes can reproduce the polarization data for M87* in a
comparable way as the corresponding model in the Schwarzschild spacetime.

In order to assess quantitatively the variation in the polarization picture
for the two types of compact objects we analyse the deviation of the
polarization intensity and its direction. Since we would like to investigate
whether we can distinguish wormholes and black holes observationally by means
of polarization measurements, we compare the features of the linear
polarization for the same apparent radius rather than for the same emitting
one. Due to the different degree of focusing resulting from the gravitational
lensing, the polarized flux which we consider originates from different
emission radii in the two spacetimes. However, if we restrict ourselves to
direct images, the deviation in the emitting radius is not substantial (see
Appendix A).

In Fig. 4 we present the analysis of the relative polarization properties of
the Schwarzschild black hole and the class of static traversable wormholes
with different redshift parameter $\alpha$. For each $\alpha$ we consider the
polarization at the observable location of the ISCO for the Schwarzschild
black hole. In this way we can explore how the polarization properties at a
given point of the celestial sphere will be modified if we vary the spacetime
geometry. The inclination angle is $\theta=20^{\circ}$ and the magnetic field
is purely equatorial.

Figure 4: Linear polarization for static wormholes at the inclination angle
$\theta=20^{\circ}$. We analyze the polarization intensity I and direction
EVPA as a function of the redshift parameter $\alpha$, as well as their
deviation from the Schwarzschild black hole $\Delta$I and $\Delta$EVPA (see
main text). The two critical values of $\alpha$, which lead to a minimal
deviation $\Delta$I and $\Delta$EVPA are presented in red solid and red dashed
lines, respectively.

For each image we plot the variation of the polarization intensity and its
direction around the ring as a function of the azimuthal coordinate. We
further present their deviation from the Schwarzschild solution defining the
relative intensity $\Delta\text{I}=\text{I}_{\text{WH}}-\text{I}_{\text{Sch}}$
and the relative direction angle $\Delta\text{EVPA}=\text{
EVPA}_{\text{WH}}-\text{EVPA}_{\text{Sch}}$ at each point of the image.
Examining these quantities we can reach the following conclusions. For each
$\alpha$ the variation of the intensity and the direction angle as a function
of the azimuthal angle possesses a similar profile as for the Schwarzschild
solution. The deviation between the wormhole and the black hole polarization
characteristics is not substantial for all the values of $\alpha\in[0,3]$.
Furthermore, we notice that for small values of the redshift parameter the
polarization intensity in the wormhole spacetime is larger than that for the
Schwarzschild black hole at each point of the image while for larger $\alpha$
the opposite inequality applies. This implies that there exists some critical
value of the redshift parameter $\alpha^{(1)}_{crit}$, for which the deviation
in the polarization intensity for the two types of geometries is minimal.

The behavior of the polarization direction angle possesses a similar feature.
For small values of $\alpha$ the deviation $\Delta$ EVPA is positive in some
region at the right-hand side of the image and negative for the other
azimuthal angles while for large redshift parameter we observe the opposite
trend. Thus, we can conclude that there exists a second critical value
$\alpha^{(2)}_{crit}$, for which the direction of the polarization vector for
the wormhole solution deviates minimally from the Schwarzschild black hole.

(a)

(b)

Figure 5: Maximum deviation of the polarization properties of the wormhole
from the Schwarzschild black hole as a function of the redshift parameter
$\alpha$. We plot the maximum deviation of the intensity max $\Delta$I (a),
and the polarization direction max $\Delta$EVPA (b) which is reached in the
images presented in Fig. 4 for each value of $\alpha$. Negative values imply
that the corresponding quantity is larger for the Schwarzschild solution than
for the wormhole.

In Fig. 5 we examine in more details the deviation of the intensity $\Delta$I
and the polarization angle $\Delta$EVPA in the wormhole geometry with respect
to the Schwarzschild solution. Since these quantities vary depending on the
azimuthal angle around the ring we estimate the maximum deviation of each of
them, which can be reached in the image for a fixed value of the redshift
parameter and plot it as a function of $\alpha$. In this way we can measure
the upper limit of the possible discrepancy in the polarization properties of
the two geometries for any redshift parameter. From this analysis we can
deduce the two critical values of $\alpha$, which lead to minimal deviation in
the polarization intensity and direction, respectively, and estimate its
amount.

The results are presented in Table 1 for the three configurations of
equatorial magnetic field, which we consider. Thus, for $B=[0.87,0.5,0]$ we
obtain that the wormhole solution with $\alpha^{(1)}_{crit}=1.64$ possesses
polarization intensity whose distribution resembles most closely the
Schwarzschild black hole deviating less than $3.3\,\%$ at any point of the
image. For the same value of the redshift parameter the polarization direction
deviates less than $1.1\,\%$. On the other hand, the wormhole solution for
$\alpha^{(2)}_{crit}=2.86$ resembles most closely the Schwarzschild black hole
in the polarization twist around the ring with deviation of the direction
vector less than $0.04\,\%$. For the same value of $\alpha$ the polarization
direction deviates less than $23.4\,\%$ at any point of the image. The values
for the other magnetic field configurations are interpreted in a similar way.

Magnetic field |  Minimization criteria |  $\frac{\text{max}\,\Delta\text{I}}{\text{I}_{\text{Sch}}}\,\,[\%]$ |  $\frac{\text{max}\,\Delta\text{EVPA}}{\text{EVPA}_{\text{Sch}}}\,\,[\%]$ | $\phi\,[rad]$ | $\alpha_{crit}$
---|---|---|---|---|---
B = [0.5, 0.87, 0] |  $\Delta$I $\Delta$EVPA |  3.8 23.0 |  2.2 0.3 |  $0.48\pi$ $0.73\pi$ |  1.39 2.85
B= [0.71, 0.71, 0] |  $\Delta$I $\Delta$EVPA |  3.6 23.1 |  1.8 0.07 |  $0.53\pi$ $1.32\pi$ |  1.54 2.85
B= [0.87, 0.5, 0] |  $\Delta$I $\Delta$EVPA |  3.3 23.4 |  1.1 0.04 |  $0.53\pi$ $0.32\pi$ |  1.64 2.86
Table 1: Deviation of the wormhole polarization from the Schwarzschild black
hole for the critical values of the redshift parameter $\alpha$, for which
$\Delta$I or $\Delta$EVPA are minimal. In each case we give the maximum
relative deviations $\text{max}\,\Delta\text{I}/\text{I}_{\text{Sch}}$ and
$\text{max}\,\Delta\text{EVPA}/\text{EVPA}_{\text{Sch}}$ with respect to the
Schwarzschild solution, which are reached in the polarized images and the
corresponding azimuthal angle.

Examining the particular form of the functions max $\Delta$I($\alpha$) and max
$\Delta$EVPA($\alpha$) we see that the wormhole solution with the minimal
deviation $\Delta I$ can be considered with a good precision to possesses the
most similar polarization structure to the Schwarzschild black hole. Thus, we
can assume that for $\alpha=\alpha^{(1)}_{crit}$ the polarization of the
wormhole resembles most closely the Schwarzschild solution taking into account
both its intensity and twist.

In Fig. 5 we obtain also an estimate for the maximal deviation from the
polarization properties of the Schwarzschild black hole which is possible for
our class of wormholes. We see that the wormhole solutions are most clearly
distinguished for small values of $\alpha$ as the discrepancy from the
Schwarzschild solution depends on the magnetic field. For example, in the
magnetic field configuration $B=[0.87,0.5,0]$ we get for $\alpha=0$ that the
maximal deviation in the intensity is
$\text{max}\,\Delta\text{I}/\text{I}_{\text{Sch}}=43\%$ and the maximal
deviation in the polarization angle is
$\text{max}\,\Delta\text{EVPA}/\text{EVPA}_{\text{Sch}}=2.3\%$ , while for
$B=[0.5,0.87,0]$ we get the values
$\text{max}\,\Delta\text{I}/\text{I}_{\text{Sch}}=28\%$ and
$\text{max}\,\Delta\text{EVPA}/\text{EVPA}_{\text{Sch}}=3.8\%$.

The minimal deviation from the Schwarzschild solution, which is possible for
every magnetic field configuration, depends insignificantly in magnitude on
the magnetic field direction but it corresponds to a different value of the
redshift parameter $\alpha$.

The deviation from the Schwarzschild solution for a fixed value of $\alpha$
depends on the magnetic field configuration. For $\alpha<\alpha^{(1)}_{crit}$
$\Delta I$ increases when the radial component $B_{r}$ grows, while for
$\alpha>\alpha^{(1)}_{crit}$ we observer the opposite correlation. The
deviation of the polarization angle increases when $B_{r}$ declines for all
the values of the redshift parameter which we consider.

We reach similar conclusions if we vary the radius of the emitting ring in the
region of strong interaction. In Appendix B we study the polarization observed
at location of the image of the orbit with radius $r=4.5M$ for the
Schwarzschild black hole. The polarization pattern reproduces qualitatively
the characteristics which we described in the case of $r=6M$. The minimal
deviation from the Schwarzschild solution which can be achieved for this
emission radius is slightly larger than in the previous case reaching
approximately $6\%$ for the intensity and $3\%$ for the polarization
direction.

---

---

Figure 6: Polarization in equatorial magnetic field for wormholes with
different redshift parameter $\alpha$. Each color represents the polarization
for a particular wormhole solution with $\alpha\in[0,3]$ at the observable
location of the ISCO for the Schwarzschild solution. The polarization for the
Schwarzschild black hole is given by a black dotted line as a reference. The
inclination angle is $\theta=70^{\circ}$.

As a next step we consider the influence of the inclination angle on the
polarization properties. We obtain the polarized images in the same physical
set-up as in the previous examples, however we assume that the disk is tilted
at $\theta=70^{\circ}$ with respect to the observer. In Fig. 6 we plot the
polarization at the observable location of the ISCO for the Schwarzschild
solution for a continuous distribution of the redshift parameter $\alpha$. In
this way we explore the qualitative features of the polarization pattern for
wormhole solutions in comparison to the Schwarzschild black hole. In Fig. 7 we
further analyse quantitatively the deviation between the polarization
properties of the two types of compact objects.

Figure 7: Linear polarization for static wormholes at the inclination angle
$\theta=70^{\circ}$. We analyze the polarization intensity I and direction
EVPA as a function of the redshift parameter $\alpha$, as well as their
deviation from the Schwarzschild black hole $\Delta$I and $\Delta$EVPA (see
main text). The two critical values of $\alpha$, which lead to a minimal
deviation $\Delta$I and $\Delta$EVPA are presented in red solid and dashed
lines, respectively.

(a)

(b)

Figure 8: Maximum deviation of the polarization properties of the wormhole
from the Schwarzschild black hole as a function of the redshift parameter
$\alpha$. We plot the maximum deviation of the intensity max $\Delta$I (a),
and the polarization direction max $\Delta$EVPA (b) which is reached in the
images presented in Fig. 7 for each value of $\alpha$. Negative values imply
that the corresponding quantity is larger for the Schwarzschild solution than
for the wormhole.

We see that higher inclination angles lead to more pronounced differences in
the polarization properties of the wormhole and black hole geometries. The
images possess some common qualitative features as for small inclinations. The
polarization pattern is similar to the Schwarzschild black hole with highest
intensity at the right-hand side of the image and similar twist around the
ring. For small redshift parameter the polarization intensity for the wormhole
is higher than for the Schwarzschild solution, while for larger values of
$\alpha$ we observe the opposite relation. The variation of $\Delta$EVPA
around the image is more complicated than for small inclination angles.
However, we still observe that for smaller $\alpha$ $\Delta$EVPA is positive
in some region in the right-hand side of the image and negative for the other
azimuthal angles while for larger $\alpha$ the opposite inequality applies in
each of the regions. Thus, there exist a couple of values of the redshift
parameter $\alpha^{(1)}_{crit}$ and $\alpha^{(2)}_{crit}$ for which the
polarization intensity or the twist pattern deviate minimally from the
Schwarzschild solution, respectively.

In Fig. 8 we access the deviation from the Schwarzschild black hole for each
$\alpha$ by plotting the maximum deviation in the intensity and the EVPA which
is reached in the image for the corresponding value of the redshift parameter.
Thus, in the case $B=[0.87,0.5,0]$ we obtain that for
$\alpha^{(1)}_{crit}=1.82$ the wormhole polarization intensity resembles most
that of the Schwarzschild solution with deviation less than $6.3\%$ , while
for $\alpha^{(2)}_{crit}=2.75$ the polarization twist is most similar with
deviation in $\Delta$EVPA less than $0.9\%$. The analysis of the minimal
deviation of $\Delta I$ and $\Delta$EVPA for the different equatorial magnetic
field configurations is presented in Table 2.

Magnetic field |  Minimization criteria |  $\frac{\text{max}\,\Delta\text{I}}{\text{I}_{\text{Sch}}}\,\,[\%]$ |  $\frac{\text{max}\,\Delta\text{EVPA}}{\text{EVPA}_{\text{Sch}}}\,\,[\%]$ | $\phi\,[rad]$ | $\alpha_{crit}$
---|---|---|---|---|---
B = [0.5, 0.87, 0] |  $\Delta$I $\Delta$EVPA |  12.2 21.5 |  11.2 1.6 |  $0.06\pi$ $1.04\pi$ |  1.33 2.70
B= [0.71, 0.71, 0] |  $\Delta$I $\Delta$EVPA |  7.3 21.3 |  6.3 1.3 |  $0.06\pi$ $1.08\pi$ |  1.63 2.72
B= [0.87, 0.5, 0] |  $\Delta$I $\Delta$EVPA |  6.3 21.5 |  3.6 0.9 |  $1.98\pi$ $1.14\pi$ |  1.82 2.75
Table 2: Deviation of the wormhole polarization from the Schwarzschild black hole for the critical values of the redshift parameter $\alpha$, for which $\Delta$I or $\Delta$EVPA are minimal. In each case we give the maximum relative deviations $\text{max}\,\Delta\text{I}/\text{I}_{\text{Sch}}$ and $\text{max}\,\Delta\text{EVPA}/\text{EVPA}_{\text{Sch}}$ with respect to the Schwarzschild solution, which are reached in the polarized images, and the corresponding azimuthal angle.  Magnetic field |  Inclination angle |  $\frac{\text{max}\,\Delta\text{I}}{\text{I}_{\text{Sch}}}\,\,$ |  $\frac{\text{max}\,\Delta\text{EVPA}}{\text{EVPA}_{\text{Sch}}}\,\,$
---|---|---|---
B = [0.5, 0.87, 0] |  $20^{\circ}$ $70^{\circ}$ |  0.28 0.82 |  0.038 0.25
B= [0.87, 0.5, 0] |  $20^{\circ}$ $70^{\circ}$ |  0.43 1.28 |  0.023 0.11
Table 3: Maximal deviation of the wormhole polarization properties from the
Schwarzschild black hole for our class of solutions. The wormhole geometries
are most clearly distinguished for the redshift parameter $\alpha=0$.

We see that the minimal discrepancy in the wormhole and black hole
polarization increases when the inclination angle grows. The wormholes differ
most from the Schwarzschild black hole for $\alpha=0$ as the deviation depends
on the magnetic field. When the radial component $B_{r}$ increases the
difference in the intensity $\Delta I$ grows while $\Delta$EVPA decreases. For
equatorial magnetic fields the maximal discrepancy is significantly larger
compared to the results for small inclination angles. For the magnetic field
configuration $B=[0.87,0.5,0]$ we have approximately 2 times difference in the
polarization intensity and $11\%$ deviation in EVPA. For $B=[0.5,0.87,0]$ the
maximal deviation for $\alpha=0$ is
$\text{max}\,\Delta\text{I}/\text{I}_{\text{Sch}}=82\%$ and
$\text{max}\,\Delta\text{EVPA}/\text{EVPA}_{\text{Sch}}=25\%$. We summarize
the data for the maximal discrepancy in the polarization properties with
respect to the Schwarzschild solution for small and large inclination angles
in Table 3.

###  4.2 Indirect images

In this section we explore the polarization resulting from strongly lensed
photon trajectories, which perform $k$ half-loops around the compact object
before reaching the observer. In particular, we consider the images of order
$k=1$. While the direct photon trajectories exist for a wide range of impact
parameters, the values of the impact parameters for the strongly lensed null
geodesics are very limited as the interval decreases when the order $k$ grows.
As a result, the images of order $k=1$ are restricted to a significantly more
compact regions on the observer’s sky than the direct images.

Due to this property there exists no overlap between the indirect images of
the thin disk around the Schwarzschild black hole and the wormholes with
certain values of the redshift parameter $\alpha$. In Fig. 9 we present the
extend of the thin disk image of order $k=1$ in the impact parameter space for
wormholes with various value of $\alpha$ and for the Schwarzschild black hole
at the inclination angle $\theta=20^{\circ}$. In particular, in each case we
plot the range of the impact parameter $\xi$ which represents the effective
radius of each point of the image on the observer’s sky555For an asymptotic
observer the impact parameter $\xi$ is expressed by means of the celestial
coordinates as $\xi=\sqrt{x^{2}+y^{2}}$.. The lower boundary for each $\alpha$
represents the image of the ISCO for the corresponding solution, while the
upper boundary corresponds to the orbit at $r=50M$. Orbits with larger radii
lead to images infinitesimally close to the upper boundary.

Figure 9: Range of the impact parameter for the indirect image of the thin
disk for wormholes with different redshift parameter $\alpha$ and the
Schwarzschild black hole (black dashed line). The lower boundary represents
the image of the ISCO, while the upper limit corresponds to the circular orbit
at $r=50M$. The inclination angle is $\theta=20^{\circ}$. Figure 10:
Polarization of the indirect images of order $k=1$ for wormholes with
different redshift parameter $\alpha$. Each color represents the observable
polarization of the orbit located at $r=6M$ for a particular wormhole solution
with $\alpha\in[0,3]$. The polarization for the Schwarzschild black hole is
given by a black dotted line as a reference. The inclination angle is
$\theta=20^{\circ}$.

We see that the indirect images of the thin disk around wormholes intersect
with the thin disk image around the Schwarzschild black hole only for a very
narrow range of the redshift parameter close to the value $\alpha=1.8$. Thus,
outside this range we cannot compare the polarization which is observed at a
given point of the celestial sphere as a result of the gravitational lensing
in a wormhole or black hole spacetime. However, these wormhole solutions can
be distinguished by the different apparent radius of the indirect disk image
with respect to the Schwarzschild black hole.

The observable polarization for such wormholes is presented in Fig. 10 for a
continuous distribution of the redshift parameter. For each value of
$\alpha\in[0,3]$ we illustrate the polarized image of the circular orbit
located at $r=6M$. We see that the polarization pattern resembles the
Schwarzschild black hole. However, the observable radius of the orbit can
deviate considerably.

Figure 11:  Polarization of the indirect images of order $k=1$ for wormholes
at the inclination angle $\theta=20^{\circ}$. We analyze the polarization
intensity I and direction EVPA as a function of the redshift parameter
$\alpha$, as well as their deviation from the Schwarzschild black hole
$\Delta$I and $\Delta$EVPA (see main text).

In order to access quantitative effects, in Fig. 11 we study the observable
polarization for wormholes which possess certain overlap of their indirect
disk images with the Schwarzschild solution. Thus, we can compare the
polarization properties at a given point of the celestial sphere for the two
types of solutions. As in our previous analysis, we present the polarization
which is observed at the location of the lensed image of the ISCO for the
Schwarzschild black hole. This is possile for a limited range of the redshift
parameter $\alpha\in[1.78,1.96]$. We see that the strong gravitational lensing
leads to much more pronounced deviation from the polarization properties of
the Schwarzschild black hole. The polarization intensity can deviate
considerably getting more than $8$ times larger for the magnetic field
direction $B=[0.5,0.87,0]$ and $\alpha=1.78$. The maximum deviation in the
intensity in the image decreases when the radial component of the magnetic
field increases. For example, for the magnetic field $B=[0.87,0.5,0]$ and
$\alpha=1.78$ it becomes approximately $3.5$ times larger than for the
Schwarzschild black hole. The deviation in the polarization direction
$\Delta$EVPA possesses the contrary behavior. It is largest for predominantly
radial magnetic fields and declines when the azimuthal component of the
magnetic field increases. The maximum deviation in the polarization intensity
and direction $\Delta$EVPA is presented in Table 4 for the two boundary value
of $\alpha$.

Magnetic field |  $\frac{\text{max}\,\Delta\text{I}}{\text{I}_{\text{Sch}}}$ |  $\frac{\text{max}\,\Delta\text{EVPA}}{\text{EVPA}_{\text{Sch}}}$ | $\alpha$
---|---|---|---
B = [0.5, 0.87, 0] |  7.5 0.93 |  0.53 0.25 |  1.78 1.96
B= [0.71, 0.71, 0] |  5.1 0.58 |  1.8 0.64 |  1.78 1.96
B= [0.87, 0.5, 0] |  2.5 0.18 |  6.5 38 |  1.78 1.96
Table 4: Deviation of the wormhole polarization from the Schwarzschild black
hole for the indirect images of order $k=1$ and inclination angle
$\theta=20^{\circ}$. In each case we give the maximum relative deviations
$\text{max}\,\Delta\text{I}/\text{I}_{\text{Sch}}$ and
$\text{max}\,\Delta\text{EVPA}/\text{EVPA}_{\text{Sch}}$ with respect to the
Schwarzschild solution, which are reached in the polarized image.

We further observe that the intensity around the ring changes its distribution
with respect to the Schwarzschild black hole. When $\alpha$ decreases the
maximum of the observable intensity moves upwards at the right-hand side of
the image. Thus, for $\alpha=1.96$ it is located approximately at the same
azimuthal angle as for the Schwarzschild solution while for $\alpha=1.78$ it
already reaches the top of the image. This qualitative effect combined with
the considerable deviation in the intensity magnitude at predominantly
azimuthal magnetic field can serve as an observational signature for
distinguishing wormhole spacetimes from the Schwarzschild black hole.

###  4.3 Double-disk wormholes

In this section we consider the possibility that the wormholes possess
accretion disks on both sides of their throats and signals from the region
beyond the throat are able to reach our universe in a reasonable time and get
detected by an asymptotic observer. We can further assume the matter accretes
in similar physical conditions on both sides of the throat and emits linearly
polarized radiation. In this case, the synchrotron radiation from the region
beyond the throat will have impact on the polarized image of the disk on the
observer’s sky in our universe.

---
$a)$ $b)$
Figure 12: Polarized images for wormholes with accretion disks on both sides
of the throat. In b) we represent the observable polarization for the orbit at
$r=6M$ in our universe, while a) is the image of the orbit located
symmetrically at the same radial distance on the other side of the throat.
Each color represents a particular wormhole solution with $\alpha\in[0,3]$.
The polarization intensity is represented with a different scale factor in the
two images. Thus, a line element in a) is in fact 4 times larger than a line
element in b). The inclination angle is $\theta=20^{\circ}$.

We explore the contribution of the second accretion disk by considering the
polarized image of a magnetized fluid ring located in the region beyond the
wormhole throat. For the purpose we extend the wormhole metric (1) across the
throat by introducing a global radial coordinate $l$ defined as

$\displaystyle dl=\pm\frac{dr}{\sqrt{1-\frac{r_{0}}{r}}},$ (36)

or explicitly

$\displaystyle
l=\pm\left[\sqrt{\frac{r}{r_{0}}\left(\frac{r}{r_{0}}-1\right)}+\ln\left(\sqrt{\frac{r}{r_{0}}}+\sqrt{\frac{r}{r_{0}}-1}\right)\right].$
(37)

The coordinate $l$ takes the range $-\infty<l<+\infty$ as the limits
$r\rightarrow-\infty$ and $r\rightarrow+\infty$ correspond to the two
asymptotic ends, while the wormhole throat is located at $l=0$. In this way
the metric becomes regular at the wormhole throat

$ds^{2}=-N^{2}(l)dt^{2}+dl^{2}+r^{2}(l)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (38)

and describes smoothly the transition between the two asymptotically flat
regions for positive and negative values of $l$.

In Fig. 12 we present the polarized image of the circular orbit at $r=6M$ in
our universe and the image of the orbit located symmetrically at the same
radial distance with respect to the throat, however in the other
asymptotically flat region. We see that the radiation which originates from
the region beyond the throat possesses rather distinct polarization
properties. The polarization pattern changes since we observe a less
pronounced twist of the polarization direction around the ring. The
polarization is almost radial in the left-hand side of the image and slightly
deviates from the radial direction at the right-hand side. The deviation from
the radial distribution depends on the redshift parameter as it decreases for
small $\alpha$. The polarization intensity increases substantially compared to
the radiation from the asymptotically flat region for positive $l$. Depending
on the value of $\alpha$ it grows approximately from 3.5 to 6.5 times in
magnitude. Thus, the radiation from the asymptotically flat region across the
throat will contribute substantially to the observed polarized image and
provide a characteristic signature for the detection of wormhole geometry. We
will discuss these effects in more detail for different magnetic field
configurations in a further work.

##  5 Conclusion

In this work we studied the linear polarization of the emission from the
accretion disk around a class of static traversable wormholes. Our aim is to
investigate the influence of the spacetime metric on the observable polarized
images and explore the possibility for detecting exotic compact objects with
polarization experiments. Using the simplified model of a magnetized fluid
ring which orbits in the equatorial plane and emits synchrotron radiation, we
simulate the observable polarization in wormhole geometries for a range of
physical parameters and compare with the Schwarzschild black hole. In order to
be able to reproduce the observed polarization of M87*, we focus mainly on
equatorial magnetic fields in our analysis.

We consider three types of polarized images with different properties.
Initially, we study the weakly lensed direct images of orbits in the region of
the strong gravitational interaction for small and large inclination angles.
In this case the polarization pattern is qualitatively very similar for
wormhole and black hole spacetimes. For our sample of physical parameters the
deviation in the polarization intensity between the two types of compact
objects is less than $43\%$ for small inclination angles, while the
polarization direction EVPA deviates less than $3.8\%$. Moreover, by fine-
tuning the wormhole geometry it is possible to obtain horizonless spacetimes
which closely mimic the Schwarzschild black hole in its polarization
properties. Thus, we can construct wormholes which deviate from the
Schwarzschild solution less than $4\%$ in their polarization intensity and
less than $3\%$ in EVPA for a range of equatorial magnetic fields. Increasing
the inclination angle the distinction between the two types of compact objects
grows. For observer located at the inclination angle $\theta=70^{\circ}$ we
estimate that the polarization intensity can increase approximately 2 times
with respect to the Schwarzschild solution, while the deviation of the
polarization direction is within $25\%$.

Next, we consider the strongly lensed indirect images which lead to more
pronounced phenomenological effects. We demonstrate that for images of the
order $k=1$ the deviation in the polarization properties for small inclination
angles can grow with an order in magnitude. The polarization intensity in
wormhole spacetimes can become more than eight times larger than for the
Schwarzschild black hole, while the deviation in EVPA can reach $50\%$. In
addition, the maximum of the polarization intensity shifts upwards towards the
top of the image compared to the Schwarzschild black hole. This qualitative
effect combined with the considerable deviation in the intensity magnitude
could serve as an observational signature for distinguishing wormhole
spacetimes.

Finally, we assume that the wormhole possesses accretion disks on both sides
of its throat and we can detect radiation from the asymptotically flat region
across the throat. We see that the polarized radiation which reaches the
observer through the throat can become more than six times more intensive
compared to the radiation in our universe and it forms an additional structure
of ring-like images at smaller observational radii. The twist of the
polarization direction around these rings is less pronounced and the
polarization pattern changes. Thus, the properties of the polarized images of
the radiation across the throat are significantly different, and can be
considered as a characteristic signature for detecting horizonless objects.

Based on the analysis of our class of wormhole geometries we conclude that at
small inclination angles it could be difficult to distinguish wormhole from
black hole spacetimes by their direct polarized images. Strongly lensed
indirect images provide more reliable probes of the underlying spacetime, as
well as characteristic effects such as the detection of the polarized
radiation from the region across the wormhole throat.

##  6 Appendix

### A. Lensing properties of the circular orbits

Figure 13: Effective radius of the lensed image of the equatorial circular
orbits for direct (left) and indirect (right) images.

We investigate the apparent radius of the equatorial circular orbits at the
observer’s sky as a function of the wormhole redshift parameter $\alpha$ and
the inclination angle $\theta$ in comparison to the Schwarzschild black hole.
Since the radial position of each point of the lensed image depends on the
azimuthal angle we define the effective radius

$\displaystyle\bar{r}_{image}=\frac{1}{4}[(x_{max}-x_{min})|_{y=0}+(y_{max}-y_{min})|_{x=0}],$
(39)

where we consider the maximal deviations in the image along the $x$\- and
$y$-axes. For small inclination angles the images are close to circular and
eq. (39) gives a reasonable approximation for their radii. For larger
inclination angles the images are considerably squashed in the vertical
direction. Still, we can consider the quantity $r_{image}$ as a measure for
comparing the lensing properties of the wormhole solutions and the
Schwarzschild black hole. In Fig. 13 we see that the circular orbits for the
two types of compact objects are visualized at similar radial locations at the
observer’s sky for all the inclination angles and redshift parameters. We
should note however that we consider here only the direct images. The strong
gravitational lensing causes the apparent location of the indirect images to
depend substantially on the redshift parameter (see Fig. 9 in section 4.2).

### B. Direct polarized images: examples

Here we give further examples of the observable polarization for wormholes in
comparison to the Schwarzschild black hole. In particular we explore the
influence of the emission radius on the polarization pattern by considering
magnetized fluid rings which are located deeper in the gravitational field of
the compact objects. In Figs. 14 \- 17 we visualize the wormhole polarization
at the observable location of the orbit with radius $r=4.5M$ for the
Schwarzschild black hole and analyse its properties following the discussion
in section 4.1. We consider purely equatorial magnetic fields and two cases of
different inclination angles.

Figure 14: Linear polarization for static wormholes at the inclination angle
$\theta=20^{\circ}$. We analyze the polarization intensity I and direction
EVPA as a function of the redshift parameter $\alpha$, as well as their
deviation from the Schwarzschild black hole $\Delta$I and $\Delta$EVPA. The
two critical value of $\alpha$, which lead to minimal deviation $\Delta$I and
$\Delta$EVPA are presented in red solid and red dashed lines, respectively.

(a)

(b)

Figure 15: Maximum deviation of the polarization properties of the wormhole
from the Schwarzschild black hole as a function of the redshift parameter
$\alpha$ for the inclination angle $\theta=20^{\circ}$. We plot the maximum
deviation of the intensity max $\Delta$I (a), and the polarization direction
max $\Delta$EVPA (b), which is reached in the images presented in Fig. 14 for
each value of $\alpha$. Negative values imply that the corresponding quantity
is larger for the Schwarzschild solution than for the wormhole.

Figure 16: Linear polarization for static wormholes at the inclination angle
$\theta=70^{\circ}$. We follow the same conventions as in Fig. 14.

(a)

(b)

Figure 17: Maximum deviation of the polarization properties of the wormhole
from the Schwarzschild black hole as a function of the redshift parameter
$\alpha$ for the inclination angle $\theta=70^{\circ}$. We follow the same
conventions as in Fig. 15.

## Acknowledgments

We gratefully acknowledge support by the Bulgarian NSF Grant KP-06-H38/2.

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