Radiation in the black hole–plasma system:

propagation in equatorial plane

Vladimír Balek

Department of Theoretical Physics, Comenius University, Bratislava

Barbora Bezděková, Jiří Bičák

Institute of Theoretical Physics, Charles University, Prague

Effect of cold plasma on the form of rays propagating in the equatorial plane
of a rotating black hole is investigated. Two kinds of regions in the
radius–impact parameter plane allowed for the rays are constructed: for
radiation with a given frequency at infinity and for radiation with a given
“telescope frequency” seen by a local observer. The form of allowed regions
for locally nonrotating observers as well as observers falling freely from
infinity is established. The allowed regions contain rays which directly reach
the horizon, or there exists a “neck” connecting the forbidden regions such
that the rays coming from infinity cannot reach the horizon. In case we
considered a set of observers at various radii instead of the neck we find two
different regions – from one the rays reach the horizon and not infinity and
from the other one they reach infinity, but not the horizon. The results are
analyzed by analytical methods and illustrated by figures constructed
numerically.

1\. Introduction

It is widely acknowledged that in 1911 Albert Einstein, in his best known
paper written during his Prague stay, addressed the question of the influence
of gravity on light (“Über den Einfluß der Schwerkraft auf die Ausbreitung des
Lichtes”) and suggested that the bending of rays moving close to the Sun
should be observable. Less is known that Einstein had derived the lens
equation, predicated the double images and the magnifications in the year
1912, as documented in his notebook written during his visit of Berlin from
Prague [1]. As a consequence of many significant developments in both theory
and observations, the effects of gravity on the light propagation have become
subject of numerous studies. For very comprehensive reviews on gravitational
lensing, see, for example, [2, 3, 4, 5].

In the following we do not address directly gravitational lensing, rather, we
turn to more general question about the existence of allowed regions for rays
propagating in the gravitational field of a rotating (Kerr) black hole which
is surrounded by a cold plasma.

Let us first mention works on light rays in the fields of black holes in
vacuum. The pioneering study of the allowed regions for both photons and free
massive particles around a Kerr black hole in the equatorial plane were done
many years ago by De Felice [6]. He determined and analyzed the turning points
in terms of impact parameter, and constructed also diagrams corresponding to
various particle energies and black hole angular momenta. Allowed regions in
lensing problems around a Kerr black hole in vacuum were more recently
discussed, e.g., by Vazquez and Esteban [7] and Gralla and Lupsasca [8]. The
values of impact (“critical”) parameter for which circular light rays exist
were derived by Iyer and Hansen [9]. Their paper also includes the study of
ray deflection angles, proving that there is the light bending asymmetry due
to hole’s rotation.

For related works on the photon escape cones from black hole’s vicinity, see
[10, 11, 12] and a new comprehensive review by Ogasawara and Igata [13].

Only recently the light ray trajectories in Kerr spacetime were studied
assuming also the presence of a medium in the spacetime (see, e.g., [14, 15,
16] and references therein).

In the present paper, we study the allowed regions for rays in cold plasma in
the vicinity of a Kerr black hole. Our basic parameters to characterize the
rays are the impact parameter at infinity and the frequency of radiation. In
case of the frequency we distinguish two physically distinct situations:
either the frequency $\omega$ is fixed at infinity, or we consider a given
“telescope frequency” $\omega_{0}$ which is fixed by the requirement that it
is measured by a given observer located outside the horizon.

Unlike in the study of De Felice [6], we investigate the turning points for
rays around a Kerr black hole surrounded by a refractive and dispersive medium
in geometrical optics approximation by using the elegant Hamiltonian formalism
by Synge [17]. This formalism was employed in developing the radiation
transfer theory in refractive and dispersive media in curved spacetimes by
Bičák and Hadrava [18], for example, and it has been used recently in a number
of studies to describe ray behavior in refractive and dispersive media in the
vicinity of black holes. The focus is usually on gravitational lensing, on
black hole shadow etc. Let us mention just few examples. In the series of
papers by Bisnovatyi-Kogan and Tsupko, e.g., [19, 20, 21, 22], the authors
found the deflection angle formulas in dispersive media around a Schwarzschild
black hole. The form of the shadow of a Kerr black hole surrounded by plasma
was derived by Perlick and Tsupko [14] and further investigated by Chowdhuri
and Bhattacharyya [16]. Fathi et al. [15] used the separability condition for
the Hamilton-Jacobi equation from [14], assumed some simple ansatz for the
radial and longitudinal dependence of plasma frequency, and calculated the
light rays analytically in terms of elliptic integrals. More generally,
separability of the Hamilton-Jacobi equation and shadow in axially symmetric
spacetimes with plasma was recently analyzed in [23].

The dispersion of light rays around a Kerr black hole caused by the presence
of a plasma with a power distributed density was studied by Kimpson et al.
[24]. Sárený and Balek [25] investigated light rays in several models of
plasma in the vicinity of a Kerr black hole.

In our work, we employ the distribution of nonsingular isothermal sphere [26]
for plasma. The distribution of a (non)singular isothermal sphere lies
somewhere between exponential and power-law distributions used as density
models for spiral and elliptical galaxies, respectively [27]. It is frequently
used in the description of gravitational lensing by galaxies or clusters of
galaxies [28]. In such studies, the inhomogeneity parameter in the plasma
distribution is by many orders of magnitude greater than the size of a typical
black hole in the center of galaxy. We will be interested in the effect of
plasma on the propagation of radiation in close proximity of the black hole,
therefore we will suppose the parameter to be of the same order as the radius
of the horizon.

In the following section the Hamiltonian method for rays in a curved spacetime
with a (non-gravitating) medium described by a spacetime dependent and
frequency dependent index of refraction $n$ is introduced. The medium is then
considered to be a cold non-magnetic plasma characterized by the electron
density $N_{e}$ and plasma frequency $\omega_{pl}$. In order to determine
regions to which the light rays can propagate, we have to satisfy conditions
that (i) the Hamilton equations are satisfied, and (ii) the constraint of the
vanishing Hamiltonian is fulfilled. The latter condition guarantees that the
index of refraction remains real along any given ray. The conditions are
specified in the Kerr metric. The frequency of the radiation as seen by a
general observer is expressed, in particular for the observers locally non-
rotating (LNRF) and for observers freely falling from the rest from infinity.

In Section 3 allowed regions for the radiation with the fixed frequency as
measured at infinity are analyzed in detail in analytic terms, and specific
examples are constructed numerically. First the vacuum case is considered,
then plasma is added. As could be anticipated, the forbidden regions are
larger than in vacuum. This is well seen in the diagrams showing the radial
coordinate a ray can reach when having given impact parameter and frequency at
infinity. Frequency can, of course, be negative as seen at infinity if a ray
is close to the horizon and cannot reach infinity. We pay a particular
attention to the formation of “necks” in the radius and impact parameters
plane which connect forbidden regions.

In Section 4 a detailed analysis is devoted to the illustration of the allowed
regions when a ray should reach a given LNRF observer with a given frequency
$\omega_{0}$. In Appendices A–D we give more detailed discussion of the
formation of the neck in both cases when $\omega$ or $\omega_{0}$ are fixed,
we investigate the allowed regions far/near to infinity, and provide the
analysis of the allowed regions when LNRF observer’s position and the impact
parameter are given.

In Section 5 the allowed regions are analyzed for observers freely falling
from the rest at infinity. Again, both analytical and graphical and numerical
results are presented. We give detailed explanations of the results by
combining both analytical considerations and number of figures constructed
numerically. One can concentrate primarily on the introductions to each
section and on the figure captions if one does not wish to follow detailed
“analytical” argumentations.

We use the system of units $c=G=1$ and the signature of the metric $(-+++)$.

2\. Allowed regions for radiation in the medium in the Kerr spacetime

Consider an electromagnetic wave propagating in a refractive and dispersive
medium with refractive index $n$ in flat spacetime. Denote the frequency and
the wave vector of the wave in the rest frame of the medium by
$\omega_{\mbox{\scriptsize m}}$ and $\bf k_{\mbox{\scriptsize m}}$,
respectively. The dispersion relation in this frame, $|{\bf
k}_{\mbox{\scriptsize m}}|=n\omega_{\mbox{\scriptsize m}}$, can be rewritten
in a Lorentz invariant form as

$k^{2}-(n^{2}-1)\omega_{\mbox{\scriptsize
m}}^{2}=0,\quad\omega_{\mbox{\scriptsize m}}=-k\cdot u_{\mbox{\scriptsize
m}},$ (1)

where $k^{\mu}$ is the wave 4-vector and $u_{\mbox{\scriptsize m}}^{\mu}$ is
the 4-velocity of the medium. The equation stays valid, with the replacement
$\eta_{\mu\nu}\to g_{\mu\nu}$, for radiation propagating in an inhomogeneous
medium around a gravitating body in a curved spacetime, provided the
wavelength of the radiation $\lambda$ is much smaller than the typical scale
$l$ on which gravitational field and the properties of the medium are varying
(geometrical optics approximation). In this limit, radiation propagates along
rays, the paths in the spacetime given by the Hamiltonian of the form [17]

$H=-\tfrac{1}{2}\big{[}k^{2}-(n^{2}-1)\omega_{\mbox{\scriptsize
m}}^{2}\big{]}.$ (2)

The Hamiltonian is subject to the constraint $H=0$. Rays can be viewed as
worldlines of light signals, configurations of electromagnetic field localized
in a volume extended over many $\lambda$’s, but still small in comparison with
$l$.

The refractive index of a cold non-magnetic plasma can be expressed in terms
of plasma frequency $\omega_{\mbox{\scriptsize pl}}$, which in turn is given
by electron number density $N_{\mbox{\scriptsize e}}$,

$n^{2}=1-\frac{\omega_{\mbox{\scriptsize pl}}^{2}}{\omega_{\mbox{\scriptsize
m}}^{2}},\quad\omega_{\mbox{\scriptsize
pl}}^{2}=\frac{e^{2}}{\epsilon_{0}m_{\mbox{\scriptsize
e}}}N_{\mbox{\scriptsize e}}.$ (3)

The resulting dispersion relation is

$k^{2}+\omega_{\mbox{\scriptsize pl}}^{2}=0.$ (4)

The Hamiltonian, $H=-\frac{1}{2}(k^{2}+\omega_{\mbox{\scriptsize pl}}^{2})$,
corresponds to a “particle” with variable mass
$m\propto\omega_{\mbox{\scriptsize pl}}$. If we identify momentum $p_{\mu}$
canonically conjugated with spacetime coordinates $x^{\mu}$ with the covariant
components of the wave 4-vector, $p_{\mu}=k_{\mu}$, the 4-velocity of the
light signal will be $\dot{x}^{\mu}=k^{\mu}$, where the dot denotes
differentiation with respect to the parameter of the ray. Note, however, that
the physical momentum differs from $k^{\mu}$; in particular, for a single
photon we have $p_{\mbox{\scriptsize phys}}^{\mu}=\hbar k^{\mu}$ and
$m_{\mbox{\scriptsize phys}}=\hbar\omega_{\mbox{\scriptsize pl}}$.

Since the Hamiltonian for rays in cold plasma does not contain the frequency
$\omega_{\mbox{\scriptsize m}}=-k\cdot u_{\mbox{\scriptsize m}}$, the
propagation of light signals is not affected by the motion of plasma. Still,
frequency $\omega_{\mbox{\scriptsize m}}$ enters the condition determining
where the radiation can in principle be observed: the rays can penetrate only
regions in which $n$ is real, i.e. the inequality $\omega_{\mbox{\scriptsize
m}}\geq\omega_{\mbox{\scriptsize pl}}$ is satisfied. We would need to add this
condition to the theory if the form of the rays would be given just by
Hamilton equations. However, the theory contains, apart from Hamilton
equations, an additional constraint that the Hamiltonian, of which we know
from the equations only that it is conserved, is to be zero. As a consequence,
the light signals have $\omega_{\mbox{\scriptsize m}}=({\bf
k}_{\mbox{\scriptsize m}}^{2}+\omega_{\mbox{\scriptsize pl}}^{2})^{1/2}$, so
the condition $\omega_{\mbox{\scriptsize m}}\geq\omega_{\mbox{\scriptsize
pl}}$ is fulfilled automatically.

Consider Kerr metric in Boyer-Lindquist coordinates
$(x^{0},x^{1},x^{2},x^{3})=(t,r,\theta,\phi)$ and suppose the space outside
the horizon is filled with plasma with the same symmetries as are those of the
metric. The system has mirror symmetry w.r.t. the equatorial plane
($\theta=\pi/2$), which makes it possible for rays to lie entirely in that
plane. Metric tensor restricted to the equatorial plane is block diagonal,
$g_{\mu\nu}=\displaystyle\bigg{(}\mbox{\hskip-5.69054pt}\left.\begin{array}[]{cc}g_{AB}&0\\\
0&g_{11}\\\ \end{array}\right.\bigg{)}$, where $A,B=0,3$, so
$k^{2}=g^{AB}k_{A}k_{B}+g_{11}\dot{r}^{2}$ and we can rewrite the dispersion
relation (4) as

$\displaystyle\dot{r}^{2}=-g^{11}(g^{AB}k_{A}k_{B}+\omega_{\mbox{\scriptsize
pl}}^{2}).$ (5)

The covariant $t$\- and $\phi$-components of the wave 4-vector can be written
as $k_{0}=-\omega$ and $k_{3}=\omega b$, where $\omega$ and $b$ are frequency
and impact parameter, respectively, measured by distant observers (assuming
the rays can reach them). For the given values of $\omega$ and $b$, the light
signal can reach the given radius $r$ only if the r.h.s. of (5) is non-
negative, the zero value corresponding either to a turning point or to a
circular orbit.

Kerr metric depends on two parameters: the black hole mass $M$ and the Kerr
parameter $a$ (angular momentum per unit mass). If we put $M=1$, denote
$u=r^{-1}$ and introduce functions $f=1-2u$, ${\cal
A}=1+a^{2}u^{2}+2a^{2}u^{3}$ and ${\cal D}=1-2u+a^{2}u^{2}$, the metric in the
equatorial plane becomes

$\displaystyle ds^{2}=-fdt^{2}-4ar^{-1}dtd\phi+{\cal A}r^{2}d\phi^{2}+{\cal
D}^{-1}dr^{2}.$

Note that functions $f$, $\cal A$ and $\cal D$ satisfy the identity $f{\cal
A}+4a^{2}u^{4}={\cal D}$. By inserting for $g^{AB}$, $k_{0}$ and $k_{3}$, we
obtain (denoting by superscript “$\circ$” the vacuum case)

$\displaystyle g^{AB}k_{A}k_{B}=-\omega^{2}\frac{\mathring{F}}{{\cal
D}r^{2}},\quad\mathring{F}={\cal A}r^{2}-4ar^{-1}b-fb^{2}.$

If we plug this into equation (5) and use $g^{11}=\cal D$, we find

$\displaystyle\dot{r}^{2}=\omega^{2}Fr^{-2},\quad\text{where in plasma}\
F=\mathring{F}-{\cal D}r^{2}\frac{\omega_{\mbox{\scriptsize
pl}}^{2}}{\omega^{2}}.$ (6)

Light signals can propagate only in the regions in the $(r,b)$-plane with
$F\geq 0$; they have turning points ($\dot{r}=0$) at $F=0$ and revolve around
the black hole on a circular orbit if, in addition to $F=0$, $F$ also
satisfies $F^{\prime}=0$ (prime denoting differentiation with respect to $r$).

We are interested in regions in the $(r,b)$-plane accessible to rays in the
equatorial plane of Kerr black hole surrounded by plasma with the given
electron number density $N_{\mbox{\scriptsize e}}$. We consider two
situations: (i) the light signal propagating along the ray has a fixed
frequency $\omega$ at infinity, or (ii) it has a fixed “telescope frequency”
$\omega_{0}$ seen by an observer with given 4-velocity $u_{\mbox{\scriptsize
obs}}^{\mu}$ at the given radius $r_{0}$. In the latter case the spectrum of
the light arriving at the observer, be it from distant sources (stars) or from
a source close to the horizon, is supposed to spread across all frequencies
$\omega$, so that among the components of the light there is always one whose
frequency in the observer’s frame is $\omega_{0}$. For plasma, we will employ
the model of a nonsingular isothermal sphere used, for example, in [28, 29],
in which $N_{\mbox{\scriptsize e}}\propto(r^{2}+r_{\mbox{\scriptsize
c}}^{2})^{-1}$. In the problem with fixed “telescope frequency” we will
consider first an observer at rest with respect to a given locally nonrotating
frame (LNRF), and then an observer falling freely from rest at infinity.

The 4-velocity of a particle moving in the equatorial plane is
$u^{\mu}=\Gamma(1,v,0,\Omega)$, where $v$ and $\Omega$ are radial and angular
coordinate velocities, $v=dr/dt$ and $\Omega=d\phi/dt$; $\Gamma=dt/d\tau$ is
the conversion factor between the time of distant observers $t$ and the proper
time $\tau$ of the particle. Let $\Omega$ and $\Gamma$ be the angular velocity
and the Lorentz factor of LNRF, $\Omega=2au^{3}/{\cal A}$, $\Gamma=({\cal
A}/{\cal D})^{1/2}$, and $\hat{v}$ and $\hat{\Gamma}$ be the (coordinate)
radial velocity and the Lorentz factor of a particle falling freely from the
rest at infinity, $\hat{v}=-\ {\cal D}\sqrt{\alpha}/{\cal A}$, where
$\alpha={\cal A}-{\cal D}=2u(1+a^{2}u^{2})$, and $\hat{\Gamma}={\cal A}/{\cal
D}$. (This follows from separated geodesic equations – see [31] for a detailed
discussion, including the integration in this case.) Using this notation, we
have $v_{\mbox{\scriptsize obs}}=0$, $\Omega_{\mbox{\scriptsize
obs}}=\Omega_{0}$ and $\Gamma_{\mbox{\scriptsize obs}}=\Gamma_{0}$ (index 0
refers to the radius $r_{0}$) for a locally nonrotating observer, and
$v_{\mbox{\scriptsize obs}}=\hat{v}_{0}$, $\Omega_{\mbox{\scriptsize
obs}}=\Omega_{0}$, and $\Gamma_{\mbox{\scriptsize obs}}=\hat{\Gamma}_{0}$ for
a freely falling observer.

The frequency seen by an observer with the 4-velocity $u_{\mbox{\scriptsize
obs}}^{\mu}$ is $\omega_{0}=-k\cdot u_{\mbox{\scriptsize
obs}}=\omega\Gamma_{\mbox{\scriptsize obs}}(1-{\cal
D}_{0}^{-1}v_{\mbox{\scriptsize obs}}\eta_{0}-\Omega_{\mbox{\scriptsize
obs}}b)$, where $\eta$ is the rescaled radial velocity of light,
$\eta=\dot{r}_{\mbox{\scriptsize ph}}/\omega=\pm\sqrt{F}u$ (index “ph” stands
for “photon”). For the locally nonrotating observer this yields

$\displaystyle\omega_{0}=\omega\Gamma_{0}(1-\Omega_{0}b),$ (7)

and for the freely falling observer ($\xi={\cal
D}^{-1}\hat{v}=-\sqrt{\alpha}/{\cal A}$)

$\displaystyle\omega_{0}=\omega\hat{\Gamma}_{0}(1-\xi_{0}\eta_{0}-\Omega_{0}b).$
(8)

3\. Radiation with fixed $\omega$

Consider first radiation in vacuum. The boundaries of allowed regions are
given by the solutions to the quadratic equation $\mathring{F}=0$, which can
be expressed in two equivalent forms,

$\displaystyle\mathring{b}_{\pm}=\frac{1}{f}(-2ar^{-1}\pm\sqrt{{\cal
D}}r)=\frac{{\cal A}r^{2}}{2ar^{-1}\pm\sqrt{\cal D}r}.$ (9)

Functions $\mathring{b}_{\pm}$ coincide at the horizon,
$\mathring{b}_{\pm}=b_{\mbox{\scriptsize h}}=\Omega_{\mbox{\scriptsize
h}}^{-1}$ at $r=r_{\mbox{\scriptsize h}}=1+\sqrt{1-a^{2}}$ (the larger root

Fig. 1: Allowed regions for rays in vacuum. Kerr parameter is $a=0.8$ and
radius of the horizon is $r_{\mbox{\scriptsize h}}=1.6$. Shaded areas bounded
by the lines $\mathring{b}_{+}$ and $\mathring{b}_{-}$ (“corners”) are
forbidden, free areas $\mathring{\cal O}_{\mbox{\scriptsize I}}$ and
$\mathring{\cal O}_{\mbox{\scriptsize II}}$ outside them are allowed. The thin
vertical line at $r=2$ is the asymptote of the inner boundary of forbidden
regions.

of ${\cal D}=0$), and approach asymptotes at the angles $\pm 45^{\circ}$ to
the $r$ axis far from the horizon, $\mathring{b}_{\pm}\sim\pm r$ for $r\gg 1$.
From the first expression in (9) we can see that the $\mathring{b}_{-}$ root
is falling to $-\infty$ and rising to $+\infty$ as we approach the radius
$r=2$ (static limit) from the left and right, respectively. From the second
expression in (9), on the other hand, it is seen that the $\mathring{b}_{+}$
root stays finite at $r=2$. As seen in Fig. 1, there are two distinct regions
in the $(r,b)$-diagram forbidden for the rays (regions where
$\mathring{F}<0$): the “upper corner” above $\mathring{b}_{+}$ at $r>2$ and
between $\mathring{b}_{+}$ and $\mathring{b}_{-}$ at $r<2$, and the “lower
corner” under $\mathring{b}_{-}$ at $r>2$. Outside the corners there are two
regions accessible for rays, region $\mathring{\cal O}_{\mbox{\scriptsize I}}$
between the corners at $r>2$ and below the upper corner at $r<2$, and region
$\mathring{\cal O}_{\mbox{\scriptsize II}}$ left to the upper corner at $r<2$.
Photon orbits are located at the lowest point of the upper corner (corotating
orbit) and at the highest point of the lower corner (counter-rotating orbit).

Frequency of radiation seen by an observer with $v_{\mbox{\scriptsize obs}}=0$
is $\omega_{\mbox{\scriptsize obs}}=\omega\Gamma_{\mbox{\scriptsize
obs}}(1-\Omega_{\mbox{\scriptsize obs}}b)$. This frequency is necessarily
$>0$, hence we can determine the sign of $\omega$ at any given point in the
$(r,b)$-plane by analyzing the interval of possible values of the angular
velocity $\Omega_{\mbox{\scriptsize obs}}$. Denote by $\Omega_{\pm}$ angular
velocities of co- and counter-rotating light signals in vacuum, kept on a path
close to a circle with an arbitrary given radius by a system of mirrors
tangential to the circle and placed at equal distances from each other. The
signals have, in the limit when the number of mirrors goes to infinity,
$b=\mathring{b}_{\pm}$, and their 4-velocity squared is $u_{\mbox{\scriptsize
ph}}^{2}=-\omega\dot{t}_{\mbox{\scriptsize
ph}}+\kappa\dot{\phi}_{\mbox{\scriptsize
ph}}\propto-1+b\Omega_{\mbox{\scriptsize ph}}=0$. Thus, $\mathring{b}_{\pm}$
are just inverses to $\Omega_{\pm}$; $\mathring{b}_{\pm}=\Omega_{\pm}^{-1}$.
The value of $\Omega_{\mbox{\scriptsize obs}}$ falls between $\Omega_{-0}$ and
$\Omega_{+0}$, therefore, $b/\mathring{b}_{-0}<\Omega_{\mbox{\scriptsize
obs}}b<b/\mathring{b}_{+0}$ for $b>0$ and $\Omega_{\mbox{\scriptsize
obs}}b<b/\mathring{b}_{-0}$ for $b<0$. As a result, we have different signs of
$\omega$ in different allowed regions: light signals have $\omega>0$ in region
$\mathring{\cal O}_{\mbox{\scriptsize I}}$ and $\omega<0$ in region
$\mathring{\cal O}_{\mbox{\scriptsize II}}$. The radiation with $\omega<0$ is
a special case of a “particle” with a negative energy – a necessary ingredient
of the Penrose process (see, e.g., [30], §33.7).

Now let us add plasma to the picture. Consider radiation with a fixed
frequency $\omega$ at infinity, positive in region $\mathring{\cal
O}_{\mbox{\scriptsize I}}$ and negative in region $\mathring{\cal
O}_{\mbox{\scriptsize II}}$. The function defining the boundaries of the
allowed regions is now shifted, $F=\mathring{F}-{\cal D}r^{2}\zeta^{2}$, where
$\zeta=\omega_{\mbox{\scriptsize pl}}/\omega$. Equation $F=0$ is again
quadratic, and its solutions $b_{\pm}$ can be written in the form of the first
expression in (9) with ${\cal D}\to(1-f\zeta^{2}){\cal D}$ (equation (A.1)).
We can see that functions $b_{\pm}$ have the same asymptotes at $r\gg 1$ as
functions $\mathring{b}_{\pm}$, $b_{\pm}\sim\pm r$, and that the asymptote of
function $b_{-}$, with $b_{-}$ falling to $-\infty$ to the right of it and
rising to $+\infty$ to the left of it, lies at $r=2$ just like that of
function $\mathring{b}_{-}$. However, the similarity between functions
$\mathring{b}_{-}$ and $b_{\pm}$ ends there: $b_{\pm}$, when compared to
$\mathring{b}_{\pm}$, are pulled towards each other or entirely vanish at
$r>2$, and are pushed away from each other at $r<2$. As a result, both
forbidden regions expand, and there may even appear a neck between them, so
that no ray, regardless of its impact parameter, can pass from black hole to
distant observers or vice versa. The formation of the neck is discussed in
Appendix A.

As the forbidden regions expand, allowed regions shrink: instead of regions
$\mathring{\cal O}_{\mbox{\scriptsize I}}$ and $\mathring{\cal
O}_{\mbox{\scriptsize II}}$ in vacuum we have smaller regions ${\cal
O}_{\mbox{\scriptsize I}}$ and ${\cal O}_{\mbox{\scriptsize II}}$ contained
within them. The former region is divided into two parts in the presence of a
neck, ${\cal O}_{{\mbox{\scriptsize I}}A}$ on the lower left of the neck and
${\cal O}_{{\mbox{\scriptsize I}}B}$ on its right. Since ${\cal
O}_{\mbox{\scriptsize I}}$ and ${\cal O}_{\mbox{\scriptsize II}}$ lie inside
$\mathring{\cal O}_{\mbox{\scriptsize I}}$ and $\mathring{\cal
O}_{\mbox{\scriptsize II}}$, we have $\omega>0$ in the former region and
$\omega<0$ in the latter one.

Function $\zeta$ is proportional to $\omega_{\mbox{\scriptsize pl}}$, hence to
$\sqrt{N}_{\mbox{\scriptsize e}}$, so, for our choice of plasma distribution,
to $(r^{2}+r_{\mbox{\scriptsize c}}^{2})^{-1/2}$. To fix the coefficient of
proportionality, we choose some radius $r_{\mbox{\scriptsize ref}}$ close to
$r_{\mbox{\scriptsize h}}$ and some frequency ratio
$q=\omega_{{\mbox{\scriptsize pl}},{\mbox{\scriptsize ref}}}/|\omega|$, and
write $\zeta=\pm q\sqrt{\cal R}$, where ${\cal R}=N_{\mbox{\scriptsize
e}}/N_{{\mbox{\scriptsize e}},{\mbox{\scriptsize ref}}}=(r_{\mbox{\scriptsize
ref}}^{2}+r_{\mbox{\scriptsize c}}^{2})/(r^{2}+r_{\mbox{\scriptsize c}}^{2})$;
the signs $+$ and $-$ refer to region ${\cal O}_{\mbox{\scriptsize I}}$ and
${\cal O}_{\mbox{\scriptsize II}}$, respectively.

Allowed regions for radiation with fixed $\omega$ are shown in Fig. 2. As
expected, forbidden regions in

Fig. 2: Allowed regions for rays in plasma with fixed frequency $\omega$ at
infinity. In addition to $a=0.8$ we have $r_{\mbox{\scriptsize c}}=1$,
$r_{\mbox{\scriptsize ref}}=1.8$ (the reference circle lies halfway between
the horizon and the static limit) and $q=1$, 3.5 in the left and right panel,
respectively. Shaded areas are forbidden, free areas outside them, regions
${\cal O}_{\mbox{\scriptsize I}}$ and ${\cal O}_{\mbox{\scriptsize II}}$ in
the left panel and regions ${\cal O}_{{\mbox{\scriptsize I}}A}$, ${\cal
O}_{{\mbox{\scriptsize I}}B}$ and ${\cal O}_{\mbox{\scriptsize II}}$ in the
right panel, are allowed. For comparison, the boundaries of forbidden regions
in vacuum are shown, too, depicted by lighter lines inside the shaded regions.
The vertical line, located at the static limit just like in vacuum, is the
asymptote of the inner boundary of the forbidden regions, separated from each
other in the left panel and connected by a neck in the right panel. Radiation
has $\omega>0$ in region ${\cal O}_{\mbox{\scriptsize I}}$, divided into
regions ${\cal O}_{{\mbox{\scriptsize I}}A}$ and ${\cal O}_{{\mbox{\scriptsize
I}}B}$ in the right panel, and $\omega<0$ in region ${\cal
O}_{\mbox{\scriptsize II}}$.

plasma are larger as compared with those in vacuum. In the right panel $q$
exceeds $q_{\mbox{\scriptsize crit}}$ defined in Appendix A, therefore
forbidden regions are connected by a neck.

4\. Radiation with fixed $\omega_{0}$: locally nonrotating observers

Consider a LNRF observer at $r=r_{0}$, with the telescope adjusted to the
frequency $\omega_{0}$. Then, we can use equation (7) to express
$\zeta=\omega_{\mbox{\scriptsize pl}}/\omega$ in terms of $\omega_{0}$. In
this way we obtain

$\displaystyle\zeta=\pm q\Gamma_{0}(1-\Omega_{0}b)\sqrt{\cal R},$ (10)

where $q$ is the ratio of the plasma frequency on observer’s orbit to the
absolute value of the telescope frequency, $q=\omega_{{\mbox{\scriptsize
pl}},0}/|\omega_{0}|$, $\cal R$ is the electron density ratio for
$r_{\mbox{\scriptsize ref}}=r_{0}$, ${\cal R}=N_{\mbox{\scriptsize
e}}/N_{{\mbox{\scriptsize e}}0}=(r_{0}^{2}+r_{\mbox{\scriptsize
c}}^{2})/(r^{2}+r_{\mbox{\scriptsize c}}^{2})$, and the sign in front of $q$
equals the relative sign of $\omega$ and $\omega_{0}$. (For the possibility of
$\omega_{0}<0$, see below.) In what follows we assume that $q$ is independent
of $r_{0}$ so that observers set on different orbits have different telescope
frequencies, the less the larger the radius of the orbit; in such a way they
are able to see the effect of plasma on the propagation of radiation even at
$r\gg 1$, where plasma is thin. If plasma is locally nonrotating, we have
$\omega_{{\mbox{\scriptsize m}}0}=\omega_{0}$ and the condition of real-
valuedness of $n$ implies that no ray can reach the observer unless $q<1$.
However, since $n$ is real in the regions where $F\geq 0$ regardless of how
the plasma moves, the constraint on $q$ has to be satisfied in general case as
well (as seen also from equation (D.1), where there appears $1-q^{2}$ under
the square root).

If we insert into $F=\mathring{F}-{\cal D}r^{2}\zeta^{2}$ from (10), we see
that $F$ is quadratic in $b$ just as in the case with fixed $\omega$, when
$\zeta=const$; however, all three coefficients in expression $F=k+2lb+mb^{2}$
now become shifted, not just $k$ (equation (B.1)). This manifests itself in
the behavior of functions $b_{\pm}$ solving equation $F=0$. In particular, the
shift in $m$ implies that the asymptotics of $b_{\pm}$ at $r\gg 1$ is modified
to $b_{\pm}\sim\pm Cr$, where $C<1$, and that the radius $r_{\mbox{\scriptsize
a}}$ of the asymptote of function $b_{-}$ shifts from $r_{\mbox{\scriptsize
a}}=2$ to $r_{\mbox{\scriptsize a}}<2$. In Appendix B we show that $C$
increases from 0 to 1 and $r_{\mbox{\scriptsize a}}$ increases from $r_{h}$ to
2, as the radius of observer’s orbit $r_{0}$ increases from $r_{h}$ to
$\infty$. We also discuss there the case when the function diverging at
$r_{\mbox{\scriptsize a}}$ is $b_{+}$ rather than $b_{-}$.

Forbidden regions in the $(r,b)$-plane may be connected by a neck just as in
the case with fixed $\omega$. The neck is formed in an interval of $r$ where
$\Delta=l^{2}-km<0$ (the discriminant of equation $F=0$ is negative). As shown
in Appendix C, the neck is there for $r_{0}\sim r_{\mbox{\scriptsize h}}$ as
well as for $r_{0}\gg 1$, but it disappears in some interval
$\rho_{0A}<r_{0}<\rho_{0B}$. The band connecting forbidden regions shrinks to
a point at some radius $r=\rho_{A}$, as $r_{0}$ approaches the value
$\rho_{0A}$ from the left, and at some radius $r=\rho_{B}>\rho_{A}$, as
$r_{0}$ approaches the value $\rho_{0B}$ from the right. The values of the
radii $(\rho_{0A},\rho_{A})$ and $(\rho_{0B},\rho_{B})$ are discussed in
Appendix C.

With $\omega_{0}$ fixed, we can either restrict to the rays that actually
reach the observer, or to consider also “ghost rays” that bounce on their way
back to the observer, but have such frequency $\omega$ and impact parameter
$b$ that their frequency measured by the observer, provided they reach them,
would have just the desired value. In the latter case we obtain allowed
regions ${\cal O}_{\mbox{\scriptsize I}}$ and ${\cal O}_{\mbox{\scriptsize
II}}$ similar in form to those in the problem with fixed $\omega$. The
frequency $\omega_{0}$, however, must be taken with minus sign for some “ghost
rays”, depending on the value of $b$ when compared to the limit impact
parameter $b_{\mbox{\scriptsize lim}}=\Omega_{0}^{-1}$. By (7),
$\omega_{0}\propto\omega(1-b/b_{\mbox{\scriptsize lim}})$, and since
$\omega>0$ in ${\cal O}_{\mbox{\scriptsize I}}$ and $\omega<0$ in ${\cal
O}_{\mbox{\scriptsize II}}$, we need to put $\omega_{0}<0$ in the upper part
of ${\cal O}_{\mbox{\scriptsize I}}$, where $b>b_{\mbox{\scriptsize lim}}$, as
well as in the lower part of ${\cal O}_{\mbox{\scriptsize II}}$, where
$b<b_{\mbox{\scriptsize lim}}$.

Allowed regions for radiation with fixed $\omega_{0}$ are shown in Fig. 3.

Fig. 3: Allowed regions for rays in plasma with fixed frequency $\omega_{0}$
seen by a LNRF observer. In addition to $a=0.8$ and $r_{\mbox{\scriptsize
c}}=1$ we have $q=2/3$ and $r_{0}=$ 1.8, 1.65 and 7.25 in the left panel, left
part of the right panel and right part of the right panel. Shaded areas are
forbidden regions in an extended theory with “ghost rays”, thin lines inside
them indicate where the forbidden regions lie in vacuum, solid vertical line
is the asymptote of the inner boundary of forbidden regions and dotted
horizontal line represents $b_{\mbox{\scriptsize lim}}$, the value of $b$
dividing regions with “ghost rays” in the upper part of the diagrams into two
parts with different sign of $\omega_{0}$: in order that $\omega$ has the
right sign, $\omega_{0}$ needs to be $<0$ above $b_{\mbox{\scriptsize lim}}$
for $r>2$ and under $b_{\mbox{\scriptsize lim}}$ for $r<2$ and
$b>b_{\mbox{\scriptsize h}}$. Position of the observer is marked by a bullet
on the $r$ axis, dashed lines depict rays at the moment of arriving at the
observer and areas within the thick lines, regions $\cal F$, ${\cal
G}_{{\mbox{\scriptsize I}}+}$ and ${\cal G}_{{\mbox{\scriptsize I}}-}$ in the
left panel and ${\cal G}_{{\mbox{\scriptsize I}}A}$, ${\cal
G}_{\mbox{\scriptsize II}}$ and ${\cal G}_{{\mbox{\scriptsize I}}B}$ in the
right panel, are allowed regions for rays reaching the observer. Region ${\cal
G}_{\mbox{\scriptsize II}}$ in the left panel and line $b_{\mbox{\scriptsize
lim}}$ in the right part of the right panel did not make it into the diagram
since they lie too high, the former at $b>25.7$ and the latter at $b=276.5$.

The areas bounded by thick lines are allowed regions for rays that eventually
reach the observer, either directly or after bouncing back. The remaining free
areas outside the shaded corners, or shaded strip in the case with a neck, are
allowed regions for “ghost rays” that never arrive at the observer.

In the left panel, $r_{0}$ is in the interval $(\rho_{0A},\rho_{0B})$,
therefore the corners are separated from each other and rays can propagate
freely from infinity to the horizon and back within free band $\cal F$.
Outside $\cal F$ there are allowed regions bounded from one side: if $r_{0}$
is less than the radius at which $b_{+}$ has minimum, as in the figure, rays
can reach the observer from the regions ${\cal G}_{{\mbox{\scriptsize I}}+}$,
${\cal G}_{{\mbox{\scriptsize I}}-}$ adjacent to $\cal F$ and (not seen in the
figure) region ${\cal G}_{\mbox{\scriptsize II}}$ separated from $\cal F$, all
bounded from outside. In the right panel, $r_{0}$ lies outside the interval
$(\rho_{0A},\rho_{0B})$, hence the corners merge and rays accessible to the
observer are restricted to regions ${\cal G}_{{\mbox{\scriptsize I}}A}$ and
${\cal G}_{\mbox{\scriptsize II}}$ bounded from outside for $r_{0}<\rho_{0A}$
(left part of the panel) and region ${\cal G}_{{\mbox{\scriptsize I}}B}$
bounded from inside for $r_{0}>\rho_{0B}$ (right part of the panel). In the
case with $r_{0}<\rho_{0A}$, we have $q<q_{\mbox{\scriptsize rev}}$, with
$q_{\mbox{\scriptsize rev}}$ defined in Appendix B; hence, the asymptote at
$r=r_{\mbox{\scriptsize a}}$ is approached by function $b_{+}$.

Rays accessible to the LNRF observers, who are orbiting the black hole at
different radii $r_{0}$, occupy regions $\tilde{\cal O}_{\mbox{\scriptsize
I}}$ and $\tilde{\cal O}_{\mbox{\scriptsize II}}$ in the $(r_{0},b)$-plane,
similar in form to regions ${\cal O}_{\mbox{\scriptsize I}}$ and ${\cal
O}_{\mbox{\scriptsize II}}$ in the problem with fixed $\omega$ and
$q<q_{\mbox{\scriptsize crit}}$ (left panel of Fig. 2), with the boundaries
$B_{\pm}$ given by equation $F(r=r_{0})=0$. As shown in Appendix D, functions
$B_{\pm}$ are given by the second expression in (9), with $r$ replaced by
$r_{0}$ and with the function under the square root rescaled as ${\cal
D}_{0}\to{\cal D}_{0}/(1-q^{2})$ (equation (D.1)).

In the allowed regions in $(r_{0},b)$-plane there are two rays with the
telescope frequency passing through each point, one directed towards the black
hole (with $\dot{r}_{{\mbox{\scriptsize ph}},0}<0$) and one directed away from
it (with $\dot{r}_{{\mbox{\scriptsize ph}},0}>0$). Let us single out two
distinct parts of the allowed regions by the behavior of rays extended
backwards in time, region ${\cal O}_{\mbox{\scriptsize h}}$ in which at least
one of the rays starts at the horizon and region ${\cal O}_{\mbox{\scriptsize
i}}$ in which at least one of the rays starts at infinity (in the sense of
limit) – see Fig. 4. Denote, furthermore, by ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ the intersection of ${\cal O}_{\mbox{\scriptsize
h}}$ and ${\cal O}_{\mbox{\scriptsize i}}$ and by ${\cal O}_{\mbox{\scriptsize
h}}^{\prime}$ and ${\cal O}_{\mbox{\scriptsize i}}^{\prime}$ the parts of
${\cal O}_{\mbox{\scriptsize h}}$ and ${\cal O}_{\mbox{\scriptsize i}}$
outside of ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$. Region
${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ consists of rays
from the free bands in the $(r,b)$-plane, like band $\cal F$ in the left panel
of Fig. 3, while regions ${\cal O}_{\mbox{\scriptsize h}}^{\prime}$ and ${\cal
O}_{\mbox{\scriptsize i}}^{\prime}$ are composed of rays from allowed regions
bounded from one side, like regions ${\cal G}_{{\mbox{\scriptsize I}}+}$ and
${\cal G}_{{\mbox{\scriptsize I}}-}$ in the left panel of Fig. 3 and regions
${\cal G}_{{\mbox{\scriptsize I}}A}$, ${\cal G}_{\mbox{\scriptsize II}}$ and
${\cal G}_{{\mbox{\scriptsize I}}B}$ in the right panel of Fig. 3. If we
denote by ${\cal B}_{+}$ the upper limit of $\cal F$ (minimum of $b_{+}$) and
by ${\cal B}_{-}$ the lower limit of $\cal F$ (maximum of $b_{-}$ for
$r>r_{\mbox{\scriptsize a}}$), we can define ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ as a region in the $(r_{0},b)$-plane bounded from
above by the line ${\cal B}_{+}$ and from below by the line ${\cal B}_{-}$.
Note that the boundaries actually do not belong to ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$, since for rays with
$b={\cal B}_{\pm}$ the access to the observer is free from one side only; when
arriving from the other side, the rays do not get past the peak of the
forbidden corner standing in their way. Since the radii $R_{\pm}$, where
$b_{+}$ has minimum and $b_{-}$ has maximum, depend continuously on the radius
$r_{0}$, and since $R_{\pm}>r_{0}$ at $r_{0}=\rho_{0A}$ and $R_{\pm}<r_{0}$ at
$r_{0}=\rho_{0B}$, $r_{0}$ eventually passes through $R_{+}$ as well as
$R_{-}$ as it rises from $\rho_{0A}$ to $\rho_{0B}$. As a result, lines ${\cal
B}_{\pm}$ touch lines $B_{\pm}$ in some points $P_{\pm}$, dividing the
boundary of region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$
into two segments where regions ${\cal O}_{\mbox{\scriptsize h}}^{\prime}$ and
${\cal O}_{\mbox{\scriptsize i}}^{\prime}$ are attached to region ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$, ${\cal
O}_{\mbox{\scriptsize h}}^{\prime}$ from the left and ${\cal
O}_{\mbox{\scriptsize i}}^{\prime}$ from the right. The position of points
$P_{\pm}$ as well as the behavior of functions $B_{\pm}$, ${\cal B}_{\pm}$ in
their vicinity are discussed in Appendix D.

Allowed regions for rays seen by LNRF observers at various distances from the
black hole are shown in

Fig. 4: Allowed regions in the $(r_{0},b)$ plane for radiation seen by LNRF
observers. Parameters $a$, $r_{\mbox{\scriptsize c}}$ and $q$ are the same as
in Fig. 3. Shaded areas are forbidden for rays seen by the observer at the
given radius $r_{0}$, thin lines inside them are boundaries of the forbidden
regions in vacuum (they look the same in the $(r_{0},b)$ plane as in the
$(r,b)$ plane, since the form of rays in vacuum does not depend on $\omega$),
solid vertical line is the asymptote of the inner boundary of forbidden
regions, region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ in
the central part of the diagrams consists of rays that can reach observers
from the horizon as well as from infinity, regions ${\cal
O}_{{\mbox{\scriptsize h}}\pm}^{\prime}$ in the left panel and region ${\cal
O}_{\mbox{\scriptsize i}}^{\prime}$ in the right panel are composed of rays
that can reach observers from one side only, either from the horizon (regions
${\cal O}_{{\mbox{\scriptsize h}}\pm}^{\prime}$) or and from infinity (region
${\cal O}_{\mbox{\scriptsize i}}^{\prime}$), and points $P_{\pm}$ divide the
boundary of ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ into
segments adjacent to these regions. Rays that can come to the local observers
from the horizon form the region ${\cal O}_{\mbox{\scriptsize h}}$ to the left
of the thick line in the left panel, and rays that can arrive at the local
observers from infinity comprise the region ${\cal O}_{\mbox{\scriptsize i}}$
to the right of the thick line in the right panel.

Fig. 4. Region ${\cal O}_{\mbox{\scriptsize h}}$, consisting of ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ in the center of
$\tilde{\cal O}_{\mbox{\scriptsize I}}$, region ${\cal O}_{{\mbox{\scriptsize
h}}-}^{\prime}$ adjacent to it and region ${\cal O}_{{\mbox{\scriptsize
h}}+}^{\prime}=\tilde{\cal O}_{\mbox{\scriptsize II}}$ disconnected from both
of them, is shown in the left panel. Region ${\cal O}_{\mbox{\scriptsize i}}$,
composed of regions ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$
and ${\cal O}_{\mbox{\scriptsize i}}^{\prime}$, is displayed in the right
panel. The Kerr parameter is less than critical, $a<a_{\mbox{\scriptsize
crit}}$, where $a_{\mbox{\scriptsize crit}}$ is defined in Appendix D,
therefore the point $P_{+}$ is shifted to the left of the static limit. The
boundary of region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$
is not horizontal at points $P_{\pm}$ as it may appear from the figure;
instead, as discussed in Appendix D, it is slightly inclined downwards at
$P_{+}$ and upwards at $P_{-}$.

5\. Radiation with fixed $\omega_{0}$: freely falling observers

As seen from equation (8), the ratio $\zeta=\omega_{\mbox{\scriptsize
pl}}/\omega$ for an observer falling freely from rest at infinity can be
written as

$\displaystyle\zeta=q\hat{\Gamma}_{0}(1-\xi_{0}\eta_{0}-\Omega_{0}b)\sqrt{\cal
R}.$ (11)

The expression is applicable only to rays that can reach the observer,
otherwise the parameter $\eta_{0}=\pm\sqrt{F_{0}}u_{0}$ is purely imaginary.
Thus, the frequency $\omega_{0}$ is necessarily positive; unlike in the case
of locally nonrotating observer, there exist no ,,ghost rays“ with
$\omega_{0}=-|\omega_{0}|$. As a result, parameter $q$ can be defined just as
$q=\omega_{{\mbox{\scriptsize pl}},0}/\omega_{0}$ and no $\pm$ is needed in
equation (11).

Function ${\cal F}=Fu^{2}$ satisfies the identity ${\cal F}_{0}=\eta_{0}^{2}$,
which can be written as a quadratic equation for $\eta_{0}$:
$K_{0}+2L_{0}\eta_{0}+M_{0}\eta_{0}^{2}=0$. We can define allowed regions
$\tilde{\cal O}_{\mbox{\scriptsize I}}$ and $\tilde{\cal O}_{\mbox{\scriptsize
II}}$ in the $(r_{0},b)$-plane by the condition ${\cal F}_{0}\geq 0$ just as
for the LNRF observers, however, now this inequality is not saturated at the
boundaries of $\tilde{\cal O}_{\mbox{\scriptsize I}}$ and $\tilde{\cal
O}_{\mbox{\scriptsize II}}$. Therefore, if we want to determine the shape of
these regions, we need to start from the requirement that equation ${\cal
F}_{0}=\eta_{0}^{2}$ has real solution, i.e., from the inequality
$\delta_{0}=L_{0}^{2}-K_{0}M_{0}\geq 0$. As shown in Appendix E, this leads to
the boundaries $\tilde{B}_{\pm}$ which are given by the same expression as the
boundaries $B_{\pm}$ in the problem with LNRF observers, just with $q$
replaced by $\tilde{q}_{\mbox{\scriptsize
NR}}=q\Gamma_{0}/\big{[}1+q^{2}(\Gamma_{0}^{2}-1)\big{]}{}^{1/2}$. If we
choose $q$ as a function of $r_{0}$ so that $\tilde{q}_{\mbox{\scriptsize
NR}}=const$, regions $\tilde{\cal O}_{\mbox{\scriptsize I}}$ and $\tilde{\cal
O}_{\mbox{\scriptsize II}}$ will have the same shape for freely falling
observers as for locally nonrotating ones.

We have again, as in the problem with LNRF observers, two rays at each point
of regions $\tilde{\cal O}_{\mbox{\scriptsize I}}$ and $\tilde{\cal
O}_{\mbox{\scriptsize II}}$; however, the rays now have rescaled radial
velocities $\eta_{0}^{(\pm)}=(-L_{0}\pm\sqrt{\delta_{0}})/M_{0}$ which do not
differ just by sign. They can even have the same sign: $\eta_{0}^{(-)}$ is
negative in the whole region $\tilde{\cal O}_{\mbox{\scriptsize I}}$,
including its boundaries, but $\eta_{0}^{(+)}$ is positive in a smaller
region, passing through zero at lines $B_{\pm}$ where $K_{0}=0$. In region
$\tilde{\cal O}_{\mbox{\scriptsize II}}$, on the other hand, $\eta_{0}^{(+)}$
is positive everywhere and $\eta_{0}^{(-)}$ is negative outside the line
$B_{-}$. However, $\dot{r}_{{\mbox{\scriptsize ph}},0}$ has sign different
from $\eta_{0}$ in region $\tilde{\cal O}_{\mbox{\scriptsize II}}$, since it
equals $\omega\eta_{0}$ and $\omega<0$ there. Let us call rays with velocities
$\eta_{0}^{(+)}$ in region $\tilde{\cal O}_{\mbox{\scriptsize I}}$ and
$\eta_{0}^{(-)}$ in region $\tilde{\cal O}_{\mbox{\scriptsize II}}$ the “rays
of class I”, and rays with the other velocities the “rays of class II”. As
suggested by the discussion of the signs of $\eta_{0}^{(\pm)}$, the rays of
class II, as well the rays of class I, lying between the lines
$\tilde{B}_{\pm}$ and $B_{\pm}$, are directed towards the horizon, while the
remaining rays of class I are directed away from it.

Should we define regions ${\cal O}_{\mbox{\scriptsize h}}$ and ${\cal
O}_{\mbox{\scriptsize i}}$ as before, we would have diagrams containing the
rays of both classes I and II, with region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ divided into three zones, each with rays of
different behavior. We will consider the two classes separately, extending the
definition of regions ${\cal O}_{\mbox{\scriptsize h}}$ and ${\cal
O}_{\mbox{\scriptsize i}}$ by using rays prolonged in both directions in time:
the ray will be supposed to belong to region ${\cal O}_{\mbox{\scriptsize h}}$
if it has, when maximally extended, either starting or ending point on the
horizon, and to region ${\cal O}_{\mbox{\scriptsize i}}$, respectively, if it
has either starting or ending point at infinity.

Allowed regions for rays seen by freely falling observers at various distances
from black hole are depicted in Fig. 5. In the left panel we show the behavior
of rays of class I, in the right panel – of class II. Regions ${\cal
O}_{\mbox{\scriptsize h}}$ and ${\cal O}_{\mbox{\scriptsize i}}$ are displayed
both in the same diagram, marked by continuous and intermittent thick line,
respectively. In Appendix E we explain how we determined those regions without
having an explicit expression for the boundaries of forbidden regions in the
$(r,b)$-plane. If we return to the original definition of regions ${\cal
O}_{\mbox{\scriptsize h}}$ and ${\cal O}_{\mbox{\scriptsize i}}$, the whole
region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ in the right
panel, as well as the part of region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ lying between the lines $\tilde{B}_{\pm}$ and
$B_{\pm}$ in the left panel, will pass to region ${\cal O}_{\mbox{\scriptsize
i}}^{\prime}$, while the rest of region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ in the left panel will become a part of region
${\cal O}_{\mbox{\scriptsize h}}^{\prime}$.

Since the rays with limit impact parameters $\tilde{B}_{\pm}$ have nonzero
rescaled radial velocities $\tilde{\eta}_{0}=-L_{0}/M_{0}$, they do not bounce
back as they arrive at the observer; instead, rays coming from infinity bounce
back after they pass by the observer and rays coming from the horizon bounce
back before that (as both have radial velocity
$\tilde{\dot{r}}_{{\mbox{\scriptsize ph}},0}=\omega\tilde{\eta}_{0}<0$). In
addition, there are also rays which, rather than bouncing back, proceed
directly from infinity to the horizon. These rays form arcs $P_{A+}P_{B+}$ and
$P_{A-}P_{B-}$, marked by the black segments of thick lines in Fig. 5.
Similarly as the points $P_{+}$ and $P_{-}$ in case of LNRF observers, arcs
$P_{A+}P_{B+}$ and $P_{A-}P_{B-}$ divide the boundary of region ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ into segments where it
borders on regions ${\cal O}_{\mbox{\scriptsize h}}^{\prime}$ and ${\cal
O}_{\mbox{\scriptsize i}}^{\prime}$.

For rays of class I, allowed region $\tilde{\cal O}_{\mbox{\scriptsize I}}$
between line $\tilde{B}_{+}$ and the lower line $\tilde{B}_{-}$ contains a
smaller region $\tilde{\cal O}_{\mbox{\scriptsize I}}^{\prime}$ between line
$B_{+}$ and the lower line $B_{-}$, where rays do bounce

Fig. 5: Allowed regions in the $(r_{0},b)$ plane for radiation seen by freely
falling observers from rest at infinity. Parameter $a$ is the same as in Fig.
1–4, parameter $r_{\mbox{\scriptsize c}}$ is the same as in Fig. 2–4 and
parameter $\tilde{q}_{\mbox{\scriptsize NR}}$ coincides with $q$ used in Fig.
4. Allowed regions for rays of class I (with radial velocity $\eta_{0}^{(+)}$)
and of class II (with radial velocity $\eta_{0}^{(-)}$) are shown in the left
and right panel, respectively. The meaning of shaded areas, thin lines inside
them and solid vertical line is the same as in Fig. 4, a new feature are only
lines $B_{\pm}$ (dashed lines in the left panel), where velocity
$\eta_{0}^{(+)}$ changes its sign. Thick lines mark, as in Fig. 4, regions
${\cal O}_{\mbox{\scriptsize h}}$ and ${\cal O}_{\mbox{\scriptsize i}}$; both
are now drawn in the same diagram, boundary of ${\cal O}_{\mbox{\scriptsize
h}}$ as continuous (“put on the top”) and boundary of ${\cal
O}_{\mbox{\scriptsize i}}$ as intermittent (“lying under it”). For rays of
class I, region ${\cal O}_{\mbox{\scriptsize h}}$ contains, in addition to
region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$, regions
${\cal O}_{{\mbox{\scriptsize h}}\pm}^{\prime}$ to the left of ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$, and region ${\cal
O}_{\mbox{\scriptsize i}}$ contains region ${\cal O}_{\mbox{\scriptsize
i}}^{\prime}$ to the right of ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$, just as in Fig. 4; for rays of class II, region
${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ extends all the way
up to the horizon and region ${\cal O}_{{\mbox{\scriptsize h}}-}^{\prime}$ is
divided into regions ${\cal O}_{{\mbox{\scriptsize h}}-+}^{\prime}$ and ${\cal
O}_{{\mbox{\scriptsize h}}--}^{\prime}$, adjacent to ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ from above and from below,
respectively. Points $P_{\pm}$, where region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ has touched forbidden regions in the problem with
LNRF observers, are stretched into finite segments, the same for rays of both
classes (arcs $P_{A+}P_{B+}$ and $P_{A-}P_{B-}$ marked by thick black line).
For rays of class I, broken arcs $P_{A+}P_{A-}$ and $P_{B+}P_{B-}$, forming
together the boundary of ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize
i}}}$, merge at the points $Q_{\pm}$ lying on the lines $B_{\pm}$.

back when arriving at the observer. (As explained in Appendix E, in order that
the latter region extends to the horizon as in Fig. 5,
$\tilde{q}_{\mbox{\scriptsize NR}}$ needs to be $<1/\sqrt{2}$.) At lines
$B_{\pm}$, region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$
shrinks to a point. Thus, it is composed of two triangles and a central part,
attached to the boundaries of region $\tilde{\cal O}_{\mbox{\scriptsize
I}}^{\prime}$ at some points $Q_{\pm}$, just as in the problem with LNRF
observers region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$
touched the boundaries of region $\tilde{\cal O}_{\mbox{\scriptsize I}}$ at
points $P_{\pm}$. For rays of class II, region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ is considerably wider than for rays of class I;
it also does not stop at a finite distance from the horizon, but extends all
the way up. The shape of allowed regions for rays of both classes I and II is
discussed in more detail in Appendix E.

6\. Conclusions

Using the Hamiltonian approach for tracing rays in curved spacetime possibly
filled with dispersive and refractive medium we studied their behavior in the
Kerr metric. We restricted to equatorial plane and focused on comparing the
form of the regions accessible to rays in vacuum case with the case when the
black hole is surrounded by cold plasma. We assumed that plasma has
distribution of an isothermal nonsingular sphere with inhomogeneity parameter
comparable with the size of the black hole and with the density large enough
to affect the propagation of radiation substantially.

In the presence of plasma, forbidden regions in the radius–impact parameter
plane, the $(r,b)$-plane, become larger than in vacuum. They can even develop
a neck preventing rays from passing from infinity to the horizon and back. To
understand these effects, note that the plasma is acting on a ray repulsively
if its density decreases with radius. Thus, rays are less bent in plasma than
in vacuum, and since extremal values of $b$ at given $r$ correspond to rays
tangent to the circle with radius $r$, plasma necessarily makes these values
less in absolute value in case the rays are coming from infinity. (If such a
ray is straightened, it needs to enter gravitational field closer to the
radial line parallel to it, in order to get to the same distance from the
source.) The neck connecting forbidden regions is formed when the repulsive
effect is large enough to cause signals coming from infinity to bounce back,
even if they have impact parameter $b$ close to zero. As for signals coming
from the horizon, it might seem that since they are dragged away from the
black hole by plasma, no matter how small the repulsive effect is near the
horizon, they necessarily escape from the black hole. However, in the
expression for radial acceleration of a light signal there appears, in
addition to a centrifugal acceleration proportional to $b^{2}$, the term (with
plus sign) proportional to $\dot{r}^{2}$. Since that term is less in the
presence of plasma than in vacuum, the effect of gravity is enhanced by the
presence of plasma, and if it is dense enough, signals coming from the horizon
bounce back.

We considered also observers located at various radii $r_{0}$, equipped with
telescopes with a given frequency $\omega_{0}$, and constructed allowed
regions for rays with that frequency. LNRF observers were supposed to have
frequency $\omega_{0}$ proportional to the local plasma frequency, while for
freely falling observers we added an extra factor rising to infinity as the
observer’s Lorentzian factor $\Gamma_{0}$, as the horizon is approached.
Frequency of light signals with the extremal values of $b$ blue-shifts by a
factor of order $\Gamma_{0}$ as we pass from the LNRF to the freely falling
frame, hence it needs to be finite in the LNRF, just like the telescope
frequency with LNRF observers. The frequency is blue-shifted despite the fact
that the observer is moving away from the source, because of the transversal
Doppler effect. However, rays coming from infinity can reach the horizon only
if their impact parameter is separated by a finite gap from the maximum $b$ in
the region with positive frequency at infinity $\omega$; and with such $b$,
rays with the velocity directed to the black hole are redshifted rather than
blueshifted by the transition from the LNRF to the observer’s frame. Their
redshift is quite substantial, of order $\Gamma_{0}^{-1}$, so that their
frequency in the LNRF is of the same order of magnitude as the Lorentzian
factor $\hat{\Gamma}_{0}=\Gamma_{0}^{2}$ of a freely falling observer.

In the diagrams with multiple observers we constructed regions in which at
least one of the two rays passing through each point of the diagram started at
the horizon (region ${\cal O}_{\mbox{\scriptsize h}}$) or at infinity (region
${\cal O}_{\mbox{\scriptsize i}}$). The setting with freely falling observers
is more general than that with LNRF observers, since the rays seen by them are
independent of each other. For LNRF observers, the rays are related by a
simultaneous change of sign of coordinates $t$ and $\phi$; thus, they are
mirror images of each other with respect to the radial line drawn through the
point of observation, while their velocities are mirror images of each other
with respect to the tangential line drawn through that point. As a result, the
starting point of each ray, obtained by extending it maximally backwards in
time, coincides with the ending point of the other ray, obtained by extending
it maximally forwards in time, provided it passed by the observer without
being detected. For freely falling observers the two rays are not mirror
images of each other, since the radial motion of the observer breaks the
symmetry with respect to the tangential line. Thus, we have two classes of
rays that are not related by symmetry, one containing a subclass of rays
directed away from the black hole and the other consisting entirely of rays
directed towards the black hole. By extending the rays in both directions in
time we obtain a separate diagram for each class. The diagrams look quite
different from the one constructed with LNRF observers: the intersection of
regions ${\cal O}_{\mbox{\scriptsize h}}$ and ${\cal O}_{\mbox{\scriptsize
i}}$ is a narrow strip with ragged boundary, located close to the horizon, for
rays of the first class, and a wide band with smooth boundary, reaching far
from the horizon on one side and adjacent to the horizon on the other side,
for rays of the second class.

Regions ${\cal O}_{\mbox{\scriptsize h}}$ and ${\cal O}_{\mbox{\scriptsize
i}}$ arise in the diagrams with multiple observers because of the presence of
plasma; they reflect the fact that if the plasma is sufficiently dense, there
may appear necks in the diagrams with one observer. No such regions exist for
rays in vacuum, whose form does not depend on frequency. Thus, the shape of
regions ${\cal O}_{\mbox{\scriptsize h}}$ and ${\cal O}_{\mbox{\scriptsize
i}}$ may be regarded as a visualization of the interplay between strong
gravitational field and plasma as an example of a refractive and dispersive
medium, as they act on radiation propagating through them.

Declarations

Acknowledgements. V. B. was supported by the grant VEGA 1/0719/23, B. B.
acknowledges the support of the Charles University Grant Agency under the
Contract No. 317421. B. B. and J. B. were supported by the Czech Grant Agency
under the Contract No. 21/112685.

Competing interests. The authors have no relevant financial or non-financial
interests to disclose.

A. Formation of neck in the case with fixed $\omega$

For radiation with fixed frequency $\omega$, let us find the condition for the
formation of a neck between forbidden regions. Functions $b_{\pm}$ in the case
with fixed $\omega$ are

$b_{\pm}=\frac{1}{f}(-2au\pm\sqrt{\Delta}),\quad\Delta=(1-q^{2}f{\cal R}){\cal
D}r^{2}.$ (A.1)

As mentioned in Section 3, this is just the first expression in (9) with $\cal
D$ multiplied by $1-f\zeta^{2}$ (since $\zeta^{2}=q^{2}\cal R$). Function
$\hat{\cal R}=f{\cal R}$ appearing in the definition of $\Delta$ can be
written as

$\hat{\cal R}\propto(1-2u)u^{2}/(1+r_{\mbox{\scriptsize c}}^{2}u^{2}).$

The function is positive for $0<u<1/2$, it vanishes at both $u=0$ and $u=1/2$,
and its derivative, $d\hat{\cal R}/du\propto 1-3u-r_{\mbox{\scriptsize
c}}^{2}u^{3}$, decreases monotonically with $u$, crossing zero at some
$u_{C}<1/2$. Thus, $\hat{\cal R}$ has the maximum at $r_{C}=u_{C}^{-1}>2$.
Denote $q_{\mbox{\scriptsize crit}}=\hat{\cal R}_{C}^{-1/2}$. For
$q=q_{\mbox{\scriptsize crit}}$, the curves $b_{+}$ and $b_{-}$ merge at
$r_{C}$; in fact, they develop spikes directed against each other, whose tips
touch at $r_{C}$. The reason is that in the vicinity of $r=r_{C}$ the function
$\Delta$ is proportional to $\delta^{2}$, where $\delta=r-r_{C}$, so that
functions $b_{\pm}$ assume the form $b_{\pm}=A+B\delta\pm C|\delta|$, and as a
result, have spike-like minimum and maximum at $\delta=0$. For
$q>q_{\mbox{\scriptsize crit}}$, $\Delta$ becomes negative in some interval
$(r_{A},r_{B})$ containing $r_{C}$, the larger the greater the value of $q$;
therefore, the two regions inaccessible for rays become connected by a neck
that extends from $r_{A}$ to $r_{B}$. For the parameters used in Fig. 2 we
have $r_{C}=3.104$, which yields $\hat{\cal R}_{C}=0.1418$ and
$q_{\mbox{\scriptsize crit}}=2.656$. Note that the interval $(r_{A},r_{B})$
lies entirely above $r=2$ for any $q>q_{\mbox{\scriptsize crit}}$, since
$\Delta=(1+|f|\zeta^{2}){\cal D}r^{2}>0$ for $r<2$. For $q=3.5$ (the value
used in the right panel of Fig. 2), the neck extends from $r_{A}=2.268$ to
$r_{B}=5.727$.

B. Allowed regions far from the horizon and close to it in the case with fixed
$\omega_{0}$

For radiation with fixed frequency $\omega_{0}$ seen by a LNRF observer,
function $F$ determining allowed regions has the form $F=k+2lb+mb^{2}$, where
coefficients $k$, $l$, $m$ can be expressed in terms of function
$Q=q\Gamma_{0}\sqrt{\cal DR}r$ as

$k={\cal A}r^{2}-Q^{2},\quad l=-2au+Q^{2}\Omega_{0},\quad
m=-f-Q^{2}\Omega_{0}^{2}.$ (B.1)

Solutions to equation $F=0$ are $b_{\pm}=$ $-(l\pm\sqrt{\Delta}$ $)/m$, where
$\Delta=l^{2}-km$.

For $r\gg 1$, $Q$ approaches the value $Q_{\infty}=q\Gamma_{0}r_{0+}$, where
$r_{+}=(r^{2}+r_{\mbox{\scriptsize c}}^{2})^{1/2}$; hence, $F$ can be written
as $F\doteq
r^{2}+2Q_{\infty}^{2}\Omega_{0}b-(1+Q_{\infty}^{2}\Omega_{0}^{2})b^{2}$ and
the solutions to the equation $F=0$ in the leading order in $r$ are
$b_{\pm}=\pm Cr$, where $C=(1+Q_{\infty}^{2}\Omega_{0}^{2})^{-1/2}$. Since
$C<1$, the wedge-like band between the forbidden regions narrows down if we
fix $\omega_{0}$ instead of $\omega$.

In order to determine the behavior of constant $C$ as a function of $r_{0}$,
let us express constant $\hat{q}=Q_{\infty}\Omega_{0}$ appearing in the
definition of $C$ as

$\hat{q}\propto r_{0+}u_{0}^{3}/\sqrt{{\cal A}_{0}{\cal
D}_{0}}=r_{0+}/(\sqrt{r_{0}A_{0}D_{0}}),$

where $A={\cal A}r^{3}=r^{3}+a^{2}r+2a^{2}$ and $D={\cal
D}r^{2}=r^{2}-2r+a^{2}$. The expression for $\hat{q}$ can be written as a
product of functions $r_{+}/r$, $\sqrt{r/A}$ and $1/\sqrt{D}$, evaluated at
$r=r_{0}$; and since function $r_{+}/r$ obviously decreases with $r$ for all
$r>0$, while functions $A/r$ and $D$ can be shown to increase with $r$ for
$r>r_{\mbox{\scriptsize h}}$, constant $\hat{q}$ decreases and constant
$C=(1+\hat{q}^{2})^{-1/2}$ increases as $r_{0}$ goes from
$r_{\mbox{\scriptsize h}}$ to $\infty$. The former constant decreases from
$\infty$ to 0 for $r_{0}$ increasing from $r_{\mbox{\scriptsize h}}$ to
$\infty$, hence, the latter constant increases from 0 to 1. For the parameters
used in Fig. 3 we have $C=$ 0.681 (left panel) and 0.9997 (right part of the
right panel).

Consider now the radius $r_{\mbox{\scriptsize a}}$ at which either $b_{-}$ or
$b_{+}$ has an asymptote. The value of $r_{\mbox{\scriptsize a}}$ is given by
the equation $m_{\mbox{\scriptsize a}}=-f_{\mbox{\scriptsize
a}}-Q_{\mbox{\scriptsize a}}^{2}\Omega_{0}^{2}=0$, i.e., $f_{\mbox{\scriptsize
a}}=-\hat{q}^{2}D_{\mbox{\scriptsize a}}/r_{{\mbox{\scriptsize a}}+}^{2}$
(since $Q=Q_{\infty}\sqrt{D}/r_{+}$). Function $f$ is negative only for $r<2$
so that the asymptote lies necessarily there; and since it is located at $r=2$
in case with fixed $\omega$, the inner borderline of forbidden regions, or
region in case with a neck, shifts towards $r_{\mbox{\scriptsize h}}$ as we
fix $\omega_{0}$ instead of $\omega$.

Function $-f$ falls from a positive value to 0 as $r$ rises from
$r_{\mbox{\scriptsize h}}$ to 2, while function $D/r_{+}^{2}$ can be shown to
rise from 0 to a positive value. Thus, the asymptote is the closer to the
static limit the smaller the value of $\hat{q}$; and since $\hat{q}$ falls
from $\infty$ to 0 as $r_{0}$ rises from $r_{\mbox{\scriptsize h}}$ to
$\infty$, the asymptote shifts with increasing $r_{0}$ from
$r_{\mbox{\scriptsize h}}$ to 2. For the parameters used in Fig. 3 we have
$r_{\mbox{\scriptsize a}}=$ 1.836 (left panel) and 1.691 (the left part of
right panel).

For radiation with fixed $\omega$ we had $r_{\mbox{\scriptsize a}}=2$ and the
divergent function was $b_{-}$, just as in case without plasma. Now we have
$r_{\mbox{\scriptsize a}}<2$ and the divergent function may be $b_{+}$, too:
it is $b_{-}$ if $l_{\mbox{\scriptsize a}}<0$ and $b_{+}$ if
$l_{\mbox{\scriptsize a}}>0$. In the latter case, forbidden regions are
necessarily connected with a neck and the way their boundary approaches the
asymptote at $r=r_{\mbox{\scriptsize a}}$ is reversed as compared with the
former case: function $b_{+}$ is falling to $-\infty$ to the left of the
asymptote and rising to $+\infty$ to the right. Let us write
$l_{\mbox{\scriptsize a}}=-2au_{\mbox{\scriptsize a}}-f_{\mbox{\scriptsize
a}}/\Omega_{0}\propto\lambda_{\mbox{\scriptsize a}}-1$, where
$\lambda=-f/(2a\Omega_{0}u)=A_{0}(2-r)/(4a^{2})$. As $r$ rises from
$r_{\mbox{\scriptsize h}}$ to 2, $\lambda$ falls linearly from
$\lambda_{\mbox{\scriptsize h}}=A_{0}r_{{\mbox{\scriptsize h}}-}/(4a^{2})$ to
0, where $r_{{\mbox{\scriptsize h}}-}$ is the radius of the inner horizon,
$r_{{\mbox{\scriptsize h}}-}=1-\sqrt{1-a^{2}}$ (the smaller root of ${\cal
D}=0$). It can be easily proven that $\lambda_{\mbox{\scriptsize h}}>1$, so
that $\lambda>1$ below some radius $r_{1}>r_{\mbox{\scriptsize h}}$;
explicitly, $r_{1}=r_{\mbox{\scriptsize h}}+r_{{\mbox{\scriptsize
h}}-}(1-1/\lambda_{\mbox{\scriptsize h}})$. For the parameters used in the
left part of the right panel of Fig. 3 we have
$r_{1}=1.625<r_{\mbox{\scriptsize a}}=1.691$; thus, the function diverging at
$r_{\mbox{\scriptsize a}}$ is $b_{-}$. Note that in order that the diverging
function is $b_{+}$, the parameter $q$ needs to satisfy
$q>q_{\mbox{\scriptsize
rev}}=(\Gamma_{0}\Omega_{0})^{-1}\sqrt{-f_{1}/(D_{1}{\cal R}_{1})}$ (“rev”
stands for “reversed”). In our case $q_{\mbox{\scriptsize rev}}=1.421$, so
that $q<q_{\mbox{\scriptsize rev}}$.

C. Formation of neck in case with fixed $\omega_{0}$

In the case radiation has fixed frequency $\omega_{0}$ with respect to a LNRF
observer, the discriminant of equation $F=0$ is

$\Delta=(1-q^{2}\Gamma_{0}^{2}f_{+}{\cal R}){\cal D}r^{2},\quad
f_{+}=f+4ar^{-1}\Omega_{0}-{\cal A}r^{2}\Omega_{0}^{2}.$ (C.1)

If forbidden regions in the $(r,b)$-plane are connected by a neck, the radii
$r_{A}$ and $r_{B}$, where the inner edge of the neck is farthest from the
horizon and the outer edge is closest to it, are given by the equation $X=0$,
where $X=1-q^{2}\Gamma_{0}^{2}f_{+}{\cal R}$ is the extra factor by which
$\Delta$ differs from $\mathring{\Delta}={\cal D}r^{2}$ (see equation (C.1)).
While in case with fixed $\omega$ the neck forms only for large enough $q$,
now it can exist for arbitrary $q$, provided $r_{0}$ lies outside a certain
interval $(\rho_{0A},\rho_{0B})$. To see this, consider asymptotics of $X$ for
$r_{0}\sim r_{\mbox{\scriptsize h}}$ and $r_{0}\gg 1$.

Close to the horizon, at $\epsilon=r-r_{\mbox{\scriptsize h}}\ll 1$, we have
${\cal D}\doteq 2\sqrt{1-a^{2}}u_{\mbox{\scriptsize h}}^{2}\epsilon\equiv
u_{\mbox{\scriptsize h}}^{2}\hat{\epsilon}$, and the value of function $\cal
A$ at the horizon is ${\cal A}_{\mbox{\scriptsize h}}=4u_{\mbox{\scriptsize
h}}^{2}$. Thus, if the observer is shifted w.r.t. the horizon by
$\epsilon_{0}\ll 1$, their $\Gamma$-factor is $\Gamma_{0}\doteq({\cal
A}_{\mbox{\scriptsize h}}/{\cal D}_{0})^{1/2}\doteq
2\hat{\epsilon}_{0}^{-1/2}$; furthermore, for the angular velocity of the
observer we have $\Omega_{0}\doteq\Omega_{\mbox{\scriptsize
h}}=\frac{1}{2}au_{\mbox{\scriptsize h}}$. As seen from these expressions,
function $X$ reduces close to the horizon to $X\doteq
1-4q^{2}\hat{\epsilon}_{0}^{-1}f_{+}$, where
$rf_{+}=r-2+4a\Omega_{0}(1-\frac{1}{4}a^{-1}A\Omega_{0})\doteq
r-2+2a^{2}u_{\mbox{\scriptsize
h}}(1-\frac{1}{8}u_{h}A)\doteq[1-\frac{1}{4}a^{2}(3+a^{2}u_{\mbox{\scriptsize
h}}^{2})]\epsilon\equiv C\epsilon$. (We have used the fact that if $r_{0}$ as
well as $r$ are $\sim r_{\mbox{\scriptsize h}}$, then ${\cal R}\doteq 1$.) As
a result, for $\epsilon_{0}\ll 1$ there exists a neck in the $(r,b)$-plane
with the inner edge bent away from the horizon up to
$\epsilon_{A}=\frac{1}{4}q^{-2}(r_{\mbox{\scriptsize
h}}/C)\hat{\epsilon}_{0}$. If we denote $\alpha=\sqrt{1-a^{2}}$, we have
$\hat{\epsilon}_{0}=2\alpha\epsilon_{0}$ and $4r_{\mbox{\scriptsize
h}}^{2}C=(4-3a^{2})r_{\mbox{\scriptsize h}}^{2}-a^{4}=2\alpha
r_{\mbox{\scriptsize h}}^{3}$ (as seen by writing $r_{\mbox{\scriptsize h}}$
as $1+\alpha$); hence, $\epsilon_{A}=q^{-2}\epsilon_{0}$. Note that from the
expansion of $F$ up to the first order in $\epsilon$ it follows that the
asymptote of $b_{-}$ is shifted w.r.t. the horizon by the same value,
$\epsilon_{a}=q^{-2}\epsilon_{0}$. Thus, if we want to check that
$\epsilon_{A}>\epsilon_{a}$ (the point where $b_{-}$ merges with $b_{+}$ lies
beyond the asymptote of either $b_{-}$ or $b_{+}$), we need to proceed to the
next-to-leading order in $\epsilon_{0}$.

In the opposite limit, when $r_{0}\gg 1$, we have $\Gamma_{0}\doteq 1$ and
$\Omega_{0}=O(u_{0}^{3})$; if, moreover, $r\sim r_{0}$, we have also
$f_{+}\doteq 1$ and ${\cal R}\doteq r_{0}^{2}u^{2}$. Thus, the asymptotics of
function $X$ far from the horizon is $X\doteq 1-q^{2}r_{0}^{2}u^{2}$; and,
consequently, there exists a neck in the $(r,b)$-plane with the outer edge
bent towards the horizon down to $r_{B}=qr_{0}$.

The values of $r_{0}$ for which the neck shrinks to a point, along with the
values of $r$ where the point is located, are given by equations
$X=X^{\prime}=0$. The first equation is just $q^{2}\Gamma_{0}^{2}f_{+}{\cal
R}=1$, while the second equation can be written as the cubic equation for
$u=r^{-1}$,

$(f_{+}{\cal R})^{\prime}\propto(1-a\Omega_{0})^{2}u(3+r_{\mbox{\scriptsize
c}}^{2}u^{2})-(r_{\mbox{\scriptsize c}}^{2}-a^{2})\Omega_{0}^{2}-1=0.$

If we express $u$ as function of $\Omega_{0}$ by Cardano’s formula and solve
equation $X=0$ with this $u$ inserted into it numerically, we obtain two pairs
of values $(r_{0},r)$. For the parameters used in Fig. 4 these values are
$(\rho_{0A},\rho_{A})=(1.726,2.176)$ and $(\rho_{0B},\rho_{B})=(6.769,3.080)$.
As seen from our analysis of the limit cases $r_{0}\sim r_{\mbox{\scriptsize
h}}$ and $r_{0}\gg 1$, forbidden regions develop a neck for $r_{0}<\rho_{0A}$
as well as for $r_{0}>\rho_{0B}$; hence, for $r_{0}$ from the interval
$(\rho_{0A},\rho_{0B})$ they are necessarily separated by a free band.
Moreover, the fact that the interval $(\rho_{A},\rho_{B})$ lies entirely
inside the interval $(\rho_{0A},\rho_{0B})$ suggests that the inequalities
$\epsilon_{A}=q^{-2}\epsilon_{0}>\epsilon_{0}$ and $r_{B}=qr_{0}<r_{0}$,
obtained in the limits $r_{0}\sim r_{\mbox{\scriptsize h}}$ and $r_{0}\gg 1$,
stay valid all the way from $r_{0}=r_{\mbox{\scriptsize h}}$ up to
$r_{0}=\rho_{0A}$ and from $r_{0}=\infty$ down to $r_{0}=\rho_{0B}$. This is
easily verified numerically, because equation $X=0$ is cubic in $r$, therefore
we can once again use Cardano’s formula to express its solutions, radii
$r_{A}$ and $r_{B}$, analytically as functions of $r_{0}$.

D. Constructing allowed regions in the $(r_{0},b)$-plane

Let us find functions $B_{\pm}=b_{\pm}(r_{0})$ for a LNRF observer. Define
${\cal F}=Fu^{2}=\mathring{\cal F}-{\cal D}\zeta^{2}$, where $\mathring{\cal
F}=\mathring{F}u^{2}={\cal A}-4au^{3}b-fu^{2}b^{2}$ and $\zeta$ is given by
equation (10). If we rewrite $\mathring{\cal F}$ as $\mathring{\cal F}={\cal
A}(1-\Omega b)^{2}-{\cal D}u^{2}b^{2}/{\cal A}$, we obtain $\mathring{\cal
F}_{0}={\cal A}_{0}\kappa_{0}^{2}-{\cal D}_{0}u_{0}^{2}b^{2}/{\cal A}_{0}$,
where $\kappa_{0}=1-\Omega_{0}b$. [The expression for $\mathring{\cal F}$
which we have started from follows immediately from the identity $f{\cal
A}+4a^{2}u^{4}={\cal D}$, but it can also be obtained by rewriting the 2D
metric as $ds^{2}_{2}=-{\cal D}dt^{2}/{\cal A}+{\cal
A}r^{2}d\tilde{\phi}^{2}$, where $d\tilde{\phi}=d\phi-\Omega dt$, and by
regarding the formula $\mathring{\cal F}=-\omega^{-2}{\cal
D}\tilde{g}^{AB}\tilde{k}_{A}\tilde{k}_{B}$ with
$\tilde{k}_{A}=\omega(-1+\Omega b,b)$.] Since
$\zeta_{0}^{2}=q^{2}\Gamma_{0}^{2}\kappa_{0}^{2}=q^{2}{\cal
A}_{0}\kappa_{0}^{2}/{\cal D}_{0}$, we find ${\cal F}_{0}=(1-q^{2}){\cal
A}_{0}\kappa_{0}^{2}-{\cal D}_{0}u_{0}^{2}b^{2}/{\cal
A}_{0}\propto\mathring{\cal F}_{0}({\cal D}_{0}\to{\cal D}_{0}/(1-q^{2}))$,
and by putting this expression equal to zero we obtain

$B_{\pm}=\frac{{\cal A}_{0}r_{0}^{2}}{2ar_{0}^{-1}\pm\sqrt{{\cal
D}_{0}/(1-q^{2})}r_{0}}.$ (D.1)

Functions $B_{\pm}$ have the asymptotics $B_{\pm}=$ $\pm\sqrt{1-q^{2}}r_{0}$
for $r_{0}\gg 1$. Furthermore, the asymptotics of ${\cal F}_{0}$ for
$b\to\pm\infty$ is ${\cal F}_{0}=\big{[}(1-q^{2}){\cal
A}_{0}\Omega_{0}^{2}-{\cal D}_{0}u_{0}^{2}/{\cal
A}_{0}\big{]}b^{2}\propto\big{[}1-q^{2}-{\cal
D}_{0}r_{0}^{4}/(4a^{2})\big{]}b^{2}$; hence, the radius
$r_{0{\mbox{\scriptsize a}}}$ at which function $B_{-}$ has an asymptote is
given by $\sqrt{{\cal D}_{0{\mbox{\scriptsize a}}}}r_{0{\mbox{\scriptsize
a}}}^{2}/(2a)=\sqrt{1-q^{2}}$. For the parameters used in Fig. 4 we have
$r_{0{\mbox{\scriptsize a}}}=1.874$.

For any given $r_{0}$ from the interval $(\rho_{0A},\rho_{0B})$ we can find
the impact parameters ${\cal B}_{\pm}$ at the lowest point of the upper
forbidden corner and at the highest point of the lower forbidden corner in the
$(r,b)$-plane, and by repeating this procedure for a sequence of nearby
$r_{0}$’s covering the interval $(\rho_{0A},\rho_{0B})$ we can construct the
lines ${\cal B}_{\pm}$ in the $(r_{0},b)$-plane. The line ${\cal B}_{-}$ lies
under the line ${\cal B}_{+}$ inside the interval $(\rho_{0A},\rho_{0B})$ and
merges with it at the end points of the interval; therefore, the two lines
form a loop in the $(r_{0},b)$-plane. In region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ inside the loop, rays can reach observers from
both sides, from the horizon as well as from infinity, while at the boundary
of ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ (the line
$\partial{\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ composed of
${\cal B}_{+}$ and ${\cal B}_{-}$) the rays arrive at the observers from one
side only, either from the horizon or from infinity. Let us discuss the
position of the points $P_{\pm}$ separating the two parts of the
$\partial{\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ with
opposite directions of the incoming light, and the behavior of the lines
$B_{\pm}$ and ${\cal B}_{\pm}$ in their neighborhood.

As $r_{0}$ rises from $\rho_{0A}$ to $\rho_{0B}$, the radii $R_{+}$ and
$R_{-}$, at which function $b_{+}$ has minimum (radius $R_{+}$) and function
$b_{-}$ has maximum (radius $R_{-}$), change from $\rho_{A}$ to $\rho_{B}$:
the radius $R_{+}$ first rises and then falls, while the radius $R_{-}$ first
falls and then rises. Since $\rho_{A}>\rho_{0A}$ and $\rho_{B}<\rho_{0B}$,
both radii necessarily equal $r_{0}$ at some point, the radius $R_{+}$ at
$r_{0}=R_{0+}$ and the radius $R_{-}$ at $r_{0}=R_{0-}$. By the definition of
points $P_{\pm}$, the radii $R_{0\pm}$ are the values of the radius $r_{0}$ at
which these points are located in the $(r_{0},b)$-plane. Obviously, functions
$B_{+}(r_{0})$ and ${\cal B}_{+}(r_{0})$ coincide at $r_{0}=R_{0+}$, and
functions $B_{-}(r_{0})$ and ${\cal B}_{-}(r_{0})$ coincide at $r_{0}=R_{0-}$.
The radii $r=R_{\pm}$ are given by equation $\partial_{r}b_{\pm}(r_{0},r)=0$
or, equivalently, by equation
$\partial_{r}F(r_{0},r,b)\big{|}_{b=b_{\pm}(r_{0},r)}=0$. Thus, for radii
$r_{0}=R_{0\pm}$ we have equation
$\partial_{r}F(r_{0},r_{0},b)\big{|}_{b=B_{\pm}(r_{0})}=0$. To put this into a
compact form, introduce new variables $x=fb+2au$ and $y=1-\Omega_{0}b$ and
write functions $F$ and $\partial_{r}F$ as $fF=D-x^{2}-fQ^{2}y^{2}$ and
$\partial_{r}(fF)\propto\frac{1}{2}\mu-(x-\frac{1}{2}a)^{2}-\frac{1}{2}\hat{\nu}y^{2}$,
where $Q^{2}=q^{2}\Gamma_{0}^{2}D\cal R$,
$\mu=\frac{1}{2}(fD^{\prime}r^{2}+a^{2})=r^{3}-3r^{2}+2r+\frac{1}{2}a^{2}$ and
$\hat{\nu}=\frac{1}{2}(fQ^{2})^{\prime}=q^{2}\Gamma_{0}^{2}(r^{3}-2r^{2}+a^{2}-fDr^{3}/r_{+}^{2})\equiv
q^{2}\Gamma_{0}^{2}\nu$. By combining equations $F(r_{0},r_{0},b)=0$
(definition of $B_{\pm}$) and $\partial_{r}F(r_{0},r_{0},b)=0$, we obtain

$\Phi_{\pm}\equiv\tfrac{1}{2}\mu_{0}-(X_{\pm}-\tfrac{1}{2}a)^{2}-\tfrac{1}{2}\nu_{0}(1-X_{\pm}^{2}/D_{0})=0,$
(D.2)

where $X_{\pm}=x_{0}(B_{\pm})=f_{0}B_{\pm}+2au_{0}$. The desired radii are
found by solving equations $\Phi_{\pm}=0$ for $r_{0}$; for the parameters used
in Fig. 4 the radii are $R_{0+}=1.965$ and $R_{0-}=3.593$.

Points $P_{\pm}$ are not just common points of lines ${\cal B_{\pm}}$ and
$B_{\pm}$; they are points of the tangency of those lines. This is clear from
the very fact that there is just one common point for each pair of lines, but
it can be seen also from the behavior of functions ${\cal B}_{\pm}(r_{0})$ and
$B_{\pm}(r_{0})$ close to $R_{0\pm}$. Consider functions with index $+$. Since
${\cal B}_{+}(r_{0})$ is the minimum of function $b_{+}(r_{0},r)$, there must
hold $B_{+}>{\cal B}_{+}$ in the neighborhood of $R_{0+}$; however, this can
be satisfied for $r_{0}>R_{0+}$ only if $dB_{+}/dr_{0}\geq d{\cal
B_{+}}/dr_{0}$ at $R_{0+}$, and for $r_{0}<R_{0+}$ only if $dB_{+}/dr_{0}\leq
d{\cal B}_{+}/dr_{0}$ at $R_{0+}$. Thus, the derivatives of $B_{+}$ and ${\cal
B}_{+}$ coincide at $R_{0+}$.

Equation $\Phi_{\pm}=0$ has, in addition to solutions $r_{0}=R_{0\pm}$
depending on parameters $a$, $r_{\mbox{\scriptsize c}}$ and $q$, also the
solution $r_{0}=2$ valid for all values of parameters. Indeed, for $r_{0}=2$
we have $\mu_{0}=\frac{1}{2}a^{2}$, $X_{\pm}=a$ and $D_{0}=a^{2}$, so that
$\Phi_{\pm}=\frac{1}{4}a^{2}-(a-\frac{1}{2}a)^{2}=0$. However, this does not
mean that the radii $R_{\pm}$ merge with $r_{0}$ twice, once at the static
limit, provided it falls into the interval $(\rho_{0A},\rho_{0B})$, and once
outside it. The reason is that the coefficient of proportionality in the
expression for $\partial_{r}F$ contains factor $f^{-2}$, so that the equation
to be solved is actually $f_{0}^{-2}\Phi_{\pm}=0$; and since for $r_{0}\sim 2$
it holds $f_{0}=\epsilon_{0}/2$ and $\Phi_{\pm}=O(\epsilon_{0}^{2})$, where
$\epsilon_{0}=r_{0}-2$, the equation is in the generic case not solved by
$r_{0}=2$. If we introduce constants $k_{\pm}=(p\mp 1)/(p\pm 1)$, where
$p=1/\sqrt{1-q^{2}}$, and denote $l=2/r_{+}(2)=2/(4+r_{\mbox{\scriptsize
c}}^{2})^{1/2}$, we can write the coefficients in
$\Phi_{\pm}=K_{\pm}\epsilon_{0}^{2}$ as

$K_{\pm}=-(1+\tfrac{1}{2}a^{2})\big{[}2(1+\tfrac{1}{2}a^{2})a^{-2}q^{-2}-l^{2}-\tfrac{1}{2}\big{]}k_{\pm}+\tfrac{3}{2}+\tfrac{1}{8}a^{2}.$

The coefficients are both $\propto-a^{-2}$ for $a\sim 0$, and as $a$ rises to
1, $K_{-}$ stays negative, while $K_{+}$ crosses 0 at some
$a_{\mbox{\scriptsize crit}}<1$, provided $q$ is not too close to 1. If
parameters $r_{\mbox{\scriptsize c}}$ and $q$ are such as in Fig. 4,
$r_{\mbox{\scriptsize c}}=1$ and $q=2/3$, the critical value of $a$ is
$a_{\mbox{\scriptsize crit}}=0.784$. From the behavior of functions
$\Phi_{\pm}$ at $r_{0}=r_{\mbox{\scriptsize h}}$ and $r_{0}\gg 1$ it follows
that they cross 0 at $r_{0}>2$ if they touch the $r_{0}$ axis from below at
$r_{0}=2$, and at $r_{0}<2$ if they touch the $r_{0}$ axis from above. Thus,
$R_{0-}>2$ for all $a$’s and $R_{0+}>2$ for $a<a_{\mbox{\scriptsize crit}}$,
but $R_{0+}<2$ for $a>a_{\mbox{\scriptsize crit}}$. The value of $a$ used in
Fig. 4 is a bit greater than $a_{\mbox{\scriptsize crit}}$, that is why point
$P_{+}$ is shifted a bit to the left of $r_{0}=2$.

The derivatives of $B_{\pm}$ and ${\cal B}_{\pm}$ at $r_{0}=R_{0\pm}$ are
proportional to the function

$(\partial_{r_{0}}F)_{0}=-2Q_{0}^{2}Yy,\quad
Y=[(r_{0}-a^{2}u_{0}^{2})r_{0}/A_{0}-(r_{0}-1)/D_{0}+r_{0}/r_{0+}^{2}]y+2a(3r_{0}^{2}+a^{2})b/A_{0}^{2},$

evaluated at $b=B_{\pm}$. The derivatives seem to be zero in Fig. 4, but in
fact they are only almost zero. To see this, compare the radii $R_{0\pm}$ to
the radii $\hat{R}_{0\pm}$ at which $Y=0$: for the parameters used in Fig. 4
it holds $\hat{R}_{0+}=1.944$ and $\hat{R}_{0-}=3.580$; hence, $\hat{R}_{0+}$
is close to $R_{0+}$ and $\hat{R}_{0-}$ is close to $R_{0-}$. By combining the
expression for $(\partial_{r_{0}}F)_{0}$ given above with the formula
$\partial_{b}F_{0}=-2(x_{0}-Q_{0}^{2}\Omega_{0}y)$, we find that $d{\cal
B}/dr_{0}=-0.0794$ at $r_{0}=R_{0+}$ and $d{\cal B}/dr_{0}=0.0074$ at
$r_{0}=R_{0-}$.

E. Allowed regions for freely falling observers

We are interested in allowed regions in the $(r_{0},b)$ plane for rays seen by
freely falling observers. Function ${\cal F}=Fu^{2}$ determining these regions
has again, as in the problem with LNRF observers analyzed in Appendix D, the
form ${\cal F}=\mathring{\cal F}-{\cal D}\zeta^{2}$; however, $\zeta$ is now
given by equation (11). Let us write the additional term appearing in $\cal F$
due to the presence of plasma as ${\cal
D}\zeta^{2}=\gamma(\kappa_{0}-\xi_{0}\eta_{0})^{2}$, where
$\kappa_{0}=1-\Omega_{0}b$ and $\gamma=q^{2}\hat{\Gamma}_{0}^{2}{\cal D}{\cal
R}$. (If we wrote function $\cal F$ for a LNRF observer as quadratic
polynomial in $b$, there would appear, in the coefficients instead of function
$Q^{2}$, function ${\cal Q}^{2}=Q^{2}u^{2}=q^{2}\Gamma_{0}^{2}{\cal D}{\cal
R}$. Since $\gamma$ is obtained from this function by replacing
$\Gamma_{0}\to\hat{\Gamma}_{0}$, we can write it as $\gamma={\hat{\cal
Q}}^{2}$.) To make formulas describing radiation seen by the two classes of
observers more similar to each other, define an “effective LNRF parameter”
$q_{\mbox{\scriptsize NR}}=q\hat{\Gamma}_{0}/\Gamma_{0}=q\Gamma_{0}$; then,
$\gamma=q_{\mbox{\scriptsize NR}}^{2}\Gamma_{0}^{2}{\cal DR}$ and
$\gamma_{0}=q_{\mbox{\scriptsize NR}}^{2}{\cal A}_{0}$.

A new feature of the theory in the case with freely falling observers is the
identity ${\cal F}_{0}=\eta_{0}^{2}$. The difference between the two sides of
the equation can be written as the polynomial
$K_{0}+2L_{0}\eta_{0}+M_{0}\eta_{0}^{2}$, where

$K_{0}=-\mathring{\cal F}_{0}+\gamma_{0}\kappa_{0}^{2},\quad
L_{0}=-\gamma_{0}\xi_{0}\kappa_{0},\quad M_{0}=1+\gamma_{0}\xi_{0}^{2}.$ (E.1)

The polynomial has two roots
$\eta_{0}^{(\pm)}=(-L_{0}\pm\sqrt{\delta_{0}})/M_{0}$, where
$\delta_{0}=L_{0}^{2}-K_{0}M_{0}$. Note that the coefficient in front of
$\eta_{0}^{2}$ can be written as $M_{0}=1+q_{\mbox{\scriptsize
NR}}^{2}(1-\Gamma_{0}^{-2})$.

By plugging expressions for $K_{0}$, $L_{0}$, $M_{0}$ into the definition of
$\delta_{0}$ we find $\delta_{0}=M_{0}\mathring{\cal
F}_{0}-\gamma_{0}\kappa_{0}^{2}$. If we define a “modified effective LNRF
parameter” $\tilde{q}_{\mbox{\scriptsize NR}}=M_{0}^{-1/2}q_{\mbox{\scriptsize
NR}}$ and introduce functions $\tilde{\gamma}=\tilde{q}_{\mbox{\scriptsize
NR}}^{2}\Gamma_{0}^{2}{\cal DR}$ and $\tilde{\cal F}_{\mbox{\scriptsize
NR}}=\mathring{\cal F}-\tilde{\gamma}\kappa_{0}^{2}$, we can write
$\delta_{0}\propto\tilde{\cal F}_{{\mbox{\scriptsize NR}},0}$; thus, rays can
reach the observer at $r=r_{0}$ with the desired frequency only if
$\tilde{\cal F}_{{\mbox{\scriptsize NR}},0}\geq 0$. From the analysis of
allowed regions in the $(r_{0},b)$-plane for LNRF observers (see Appendix D)
we know that parameter $\tilde{q}_{\mbox{\scriptsize NR}}$ must be $<1$ in
order that there exists a non-empty class of rays that arrive at the observer
with the desired frequency, and the rays must have $b$ from the interval(s)
delimited by functions
$\tilde{B}_{\pm}=B_{\pm}(q\to\tilde{q}_{\mbox{\scriptsize NR}})$ (see equation
(D.1)), in order that they fall into that class. Which rays are allowed
depends on whether $r_{0}$ is greater or less than radius
$\tilde{r}_{0{\mbox{\scriptsize a}}}$ at which function $\tilde{B}_{-}$ has an
asymptote: for $r_{0}>\tilde{r}_{0{\mbox{\scriptsize a}}}$ the rays must have
$\tilde{B}_{-}<b<\tilde{B}_{+}$, while for
$r_{0}<\tilde{r}_{0{\mbox{\scriptsize a}}}$ they need to have either
$b<\tilde{B}_{+}$ or $b>\tilde{B}_{-}$. Thus, similarly to the problem with
LNRF observers, there are two distinct allowed regions in the $(r_{0},b)$
plane: region $\tilde{\cal O}_{\mbox{\scriptsize I}}$ extended over all
$r_{0}$’s, with $\tilde{B}_{-}<b<\tilde{B}_{+}$ for
$r_{0}>\tilde{r}_{0{\mbox{\scriptsize a}}}$ and $b<\tilde{B}_{+}$ for
$r_{0}<\tilde{r}_{0{\mbox{\scriptsize a}}}$; and region $\tilde{\cal
O}_{\mbox{\scriptsize II}}$ extended over the interval
$r_{0}<\tilde{r}_{0{\mbox{\scriptsize a}}}$ only, with $b>\tilde{B}_{-}$.

At the boundaries of regions $\tilde{\cal O}_{\mbox{\scriptsize I}}$ and
$\tilde{\cal O}_{\mbox{\scriptsize II}}$ rays do not have turning points as in
the problem with LNRF observers; instead, they have a nonzero rescaled radial
velocity $\eta_{0}$ there, $\eta_{0}<0$ at $\partial\tilde{\cal
O}_{\mbox{\scriptsize I}}$ and $\eta_{0}>0$ at $\partial\tilde{\cal
O}_{\mbox{\scriptsize II}}$. Inside regions $\tilde{\cal O}_{\mbox{\scriptsize
I}}$ and $\tilde{\cal O}_{\mbox{\scriptsize II}}$ we have two values of
$\eta_{0}$ assigned to each point, one ($\eta_{0}^{(-)}$ in $\tilde{\cal
O}_{\mbox{\scriptsize I}}$ and $\eta_{0}^{(+)}$ in $\tilde{\cal
O}_{\mbox{\scriptsize II}}$) with the same sign as $\eta_{0}$ at the boundary
of the region and another ($\eta_{0}^{(+)}$ in $\tilde{\cal
O}_{\mbox{\scriptsize I}}$ and $\eta_{0}^{(-)}$ in $\tilde{\cal
O}_{\mbox{\scriptsize II}}$) with opposite sign in a smaller region. In the
latter case, the sign of $\eta_{0}$ changes at lines $B_{\pm}$ given by
equation ${\cal F}_{{\mbox{\scriptsize NR}},0}=0$, where ${\cal
F}_{\mbox{\scriptsize NR}}=\mathring{\cal F}-\gamma\kappa_{0}^{2}$.

As $r_{0}$ decreases from $\infty$ to $r_{\mbox{\scriptsize h}}$, $\Gamma_{0}$
increases from 1 to $\infty$, and as a result, $\tilde{q}_{\mbox{\scriptsize
NR}}$ increases for the given $q<1$ from $q$ to 1 (since
$\tilde{q}_{\mbox{\scriptsize NR}}=q_{\mbox{\scriptsize
NR}}/\big{[}1+q_{\mbox{\scriptsize
NR}}^{2}(1-\Gamma_{0}^{-2})\big{]}{}^{1/2}=q/\big{[}(1-q^{2})\Gamma_{0}^{-2}+q^{2}\big{]}{}^{1/2}$).
If the value of $q$ is fixed, the form of regions $\tilde{\cal
O}_{\mbox{\scriptsize I}}$ and $\tilde{\cal O}_{\mbox{\scriptsize II}}$
changes substantially in the vicinity of the horizon in comparison with what
we have obtained for LNRF observers: rather than merge at
$b=b_{\mbox{\scriptsize h}}$ on the horizon, the regions become separated by a
gap between $b_{{\mbox{\scriptsize h}}\pm}=b_{\mbox{\scriptsize
h}}/\big{[}1\pm(1-q^{2})^{-1/2}qb_{\mbox{\scriptsize h}}\big{]}$. For the
parameters used in Fig. 5, $b_{{\mbox{\scriptsize h}}+}=1.045$ and
$b_{{\mbox{\scriptsize h}}-}=-2.188$. (Thus, the gap lays entirely under the
merging point at $b_{\mbox{\scriptsize h}}=4$.) If we want to preserve the
form of regions $\tilde{\cal O}_{\mbox{\scriptsize I}}$ and $\tilde{\cal
O}_{\mbox{\scriptsize II}}$ as they look in the problem with LNRF observers,
we need to fix $\tilde{q}_{\mbox{\scriptsize NR}}$ rather than $q$; for
$r_{0}$ decreasing from $\infty$ to $r_{\mbox{\scriptsize h}}$ we will then
have $q$ decreasing from $\tilde{q}_{\mbox{\scriptsize NR}}$ to 0 (since
$q=\tilde{q}_{\mbox{\scriptsize NR}}/\big{[}(1-\tilde{q}_{\mbox{\scriptsize
NR}}^{2})\Gamma_{0}^{2}+\tilde{q}_{\mbox{\scriptsize
NR}}^{2}\big{]}{}^{1/2}$). Note that $q_{\mbox{\scriptsize NR}}$ increases
with decreasing $r_{0}$ from $\tilde{q}_{\mbox{\scriptsize NR}}$ to
$\tilde{q}_{\mbox{\scriptsize NR}}/(1-\tilde{q}_{\mbox{\scriptsize
NR}}^{2})^{1/2}$, which is $>1$ if $\tilde{q}_{\mbox{\scriptsize
NR}}>1/\sqrt{2}$. For such $\tilde{q}_{\mbox{\scriptsize NR}}$ the constant
$q_{\mbox{\scriptsize NR}}$ surpasses 1 at some $r_{0,{\mbox{\scriptsize
lim}}}>r_{\mbox{\scriptsize h}}$, which means that functions $B_{\pm}$ are
defined only for $r_{0}\geq r_{0,{\mbox{\scriptsize lim}}}$ – there is no
change of sign of $\eta_{0}$ between $r_{\mbox{\scriptsize h}}$ and
$r_{0,{\mbox{\scriptsize lim}}}$. For the value $\tilde{q}_{\mbox{\scriptsize
NR}}=2/3$ used in Fig. 5, functions $B_{\pm}$ are defined for all values of
$r_{0}$.

We want to determine the lines dividing the $(r_{0},b)$-plane into regions
${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$, ${\cal
O}_{\mbox{\scriptsize h}}^{\prime}$ and ${\cal O}_{\mbox{\scriptsize
i}}^{\prime}$. For LNRF observers, we did it by computing functions ${\cal
B}_{\pm}$, equal for each $r_{0}$ to the value of $b$ at the lowest point of
the upper forbidden corner and at the highest point of the lower forbidden
corner in the $(r,b)$-plane. Calculation of ${\cal B}_{\pm}$ was
straightforward, since equation ${\cal F}(r,r_{0},b)=0$ for the boundaries of
the corners, lines $b_{\pm}(r,r_{0})$, was quadratic, and hence could be
solved analytically. Now we have two classes of rays with different values of
$\eta_{0}$ – class I with $\eta_{0}=\eta_{0}^{(+)}$ and class II with
$\eta_{0}=\eta_{0}^{(-)}$, so we have also two functions $\cal F$,

${\cal F}^{(\pm)}={\cal F}(\eta_{0}=\eta_{0}^{(\pm)})=\mathring{\cal
F}-\tilde{\gamma}\big{(}M_{0}^{-1}N_{0}\kappa_{0}^{2}+\xi_{0}^{2}\mathring{\cal
F}_{0}\mp 2\xi_{0}M_{0}^{-1/2}\tilde{\cal F}_{{\mbox{\scriptsize
NR}},0}^{1/2}\kappa_{0}\big{)},$ (E.2)

where $N_{0}=1-\gamma_{0}\xi_{0}^{2}=1-q_{\mbox{\scriptsize
NR}}^{2}(1-\Gamma_{0}^{-2})$. Values of $b$ satisfying equations ${\cal
F}^{(\pm)}=0$ are now solutions to quartic equation ${\cal F}^{(+)}{\cal
F}^{(-)}=0$. Even though there exists analytical solution in this case, too,
it is not too practical to determine functions ${\cal B}_{\pm}$ from it.
Instead, we can compute ${\cal B}_{\pm}$ for both $\eta_{0}^{(+)}$ and
$\eta_{0}^{(-)}$ by looking for the values of $b$ for which the minimum of
${\cal F}^{(+)}$ and ${\cal F}^{(-)}$ crosses zero (which may or may not
happen for given $r_{0}$).

The lines we are searching for were, in the problem with LNRF observers, just
two parts of the boundary of region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$, separated by points $P_{\pm}$. Now they are
still parts of that boundary, but they are separated by finite segments of
lines $\tilde{B}_{\pm}$ rather than by points lying on them. At the endpoints
of these segments, points $P_{A+}$ and $P_{B+}$ on the line $\tilde{B}_{+}$
and points $P_{A-}$ and $P_{B-}$ on the line $\tilde{B}_{-}$, there are rays
whose value of $b$ is either maximum (in points $P_{A+}$, $P_{B+}$) or minimum
(in points $P_{A-}$, $P_{B-}$) among all rays coming to the observer with the
desired frequency. Rays in points $P_{A\pm}$ arrive from and rays in points
$P_{B\pm}$ head for the peak of the forbidden region in the $(r,b)$-plane.
Thus, radial coordinates of the peaks, radii $R_{A\pm}$ and $R_{B\pm}$, differ
from radial coordinates of the observers, radii $R_{0A\pm}$ and $R_{0B\pm}$:
radii $R_{A\pm}$ are greater than $R_{0A\pm}$ and radii $R_{B\pm}$ are less
than $R_{0B\pm}$. For the parameters used in Fig. 5, we have
$(R_{0A+},R_{A+})=(1.708,1.900)$, $(R_{0B+},R_{B+})=(2.405,1.919)$,
$(R_{0A-},R_{A-})=(2.859,3.644)$ and $(R_{0B-},R_{B-})=(4.548,3.624)$.

Line ${\cal B}_{+}^{(+)}$ lying on the border of the upper part of region
${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}^{(+)}$, adjacent to
the arc $P_{A+}P_{B+}$, line ${\cal B}_{-}^{(+)}$ forming the boundary of the
lower part region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize
i}}}^{(+)}$, adjacent to the arc $P_{A+}P_{B+}$, as well as lines ${\cal
B}_{\pm}^{(-)}$ demarcating region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}^{(-)}$, are all tangent to the boundaries of
region $\tilde{\cal O}_{\mbox{\scriptsize I}}$, lines ${\cal B}_{+}^{(\pm)}$
to $\tilde{B}_{+}$ and lines ${\cal B}_{-}^{(\pm)}$ to $\tilde{B}_{-}$, just
as in the problem with LNRF observers lines ${\cal B}_{\pm}$ were tangent to
lines $B_{\pm}$. To see why, consider lines in the vicinity of point $P_{B+}$.
In the vertical band of the diagram for rays of class I, where the boundary of
region ${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ crosses the
lines $r_{0}=const$ three times, we have, in addition to forbidden corner, a
separate forbidden region bounded from both sides in the form of a slant
cigar. Rays with $b=\tilde{B}_{+}$ on the right of the point $P_{B+}$, at
$r_{0}>R_{0B+}$, bounce back from the lower edge of a forbidden corner after
passing by the observer, while those on the left of the point $P_{B+}$, at
$r_{0}<R_{0B+}$, hit the upper edge of a forbidden cigar. This leads, just as
in the problem with LNRF observers, to inequalities $dB_{+}/dr_{0}\geq d{\cal
B_{+}}^{(-)}/dr_{0}$ for $r_{0}\to R_{0B+}^{+}$ and $dB_{+}/dr_{0}\leq d{\cal
B}_{+}^{(+)}/dr_{0}$ for $r_{0}\to R_{0B+}^{-}$. The difference is that now
for $r_{0}$ passing through $R_{0B+}$ from values $>R_{0B+}$, forbidden corner
first shrinks to a point and then stretches to a forbidden cigar, while
before, as $r_{0}$ passed through $R_{0+}$, forbidden corner just shifted to
the left without changing its form. However, numerical calculation suggests
that the transformation from a corner to a cigar, even though it seems
discontinuous, is smooth enough to preserve the value of $d{\cal B}/dr_{0}$.

In the problem with freely falling observers, regions ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ for rays of classes I and
II look significantly different, each in its own way, from region ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}$ in the problem with LNRF
observers. It is not only that, as we just discussed, they are attached to
forbidden regions at finite segments of their boundaries rather than touching
them at a single point; region ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}$ also shrinks to a point at a pair of points
$Q_{\pm}$ lying on the lines $B_{\pm}$ for rays of class I, and it is attached
to the horizon in a finite interval of values of $b$ for rays of class II.

As seen in Fig. 5, the boundary of ${\cal O}_{{\mbox{\scriptsize
h}}+{\mbox{\scriptsize i}}}^{(-)}$ consists of two parts, arcs $P_{A+}P_{A-}$
and $P_{B+}P_{B-}$ intersecting at points $Q_{\pm}$. The intersection at
$Q_{-}$ has the form of a “straight cross”, the cross at $Q_{+}$ is “inclined”
(the detailed numerics shows that in fact the arc $P_{B+}P_{B-}$ at $Q_{+}$ is
first deflected slightly to the left and only then turns sharply to the
right). In the vicinity of point $Q_{-}$ there are two forbidden regions in
the $(r,b)$-plane, a cigar and an upper corner in a finite height above it,
and as $r_{0}$ passes through the radial coordinate of point $Q_{-}$,
$r_{Q-}=3.494$, the cigar shrinks to a point and then stretches again, leaning
to the other side than before. To the right of point $Q_{+}$, there is a cigar
and a lower corner which approach each other as $r_{0}$ passes through the
radial coordinate of point $Q_{+}$, $r_{Q+}=2.159$, and at $r_{0}<r_{Q+}$ only
the corner remains. Since the left boundary of the central part of region
${\cal O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}^{(+)}$ turns to the
right under the point $Q_{+}$, minimum radii $(r_{0},r)$ in that region are
just $\rho_{0A}^{(+)}=\rho_{A}^{(+)}=r_{Q+}$. For maximum radii numerical
calculation yields $(\rho_{0B}^{(+)},\rho_{B}^{(+)})=(3.966,3.161)$.

The endpoints of the interval of $b$ where region ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}^{(-)}$ is attached to the
horizon coincide with impact parameters $\mathring{b}_{{\mbox{\scriptsize
ph}}\pm}$ of photon orbits in vacuum. To see that, express additional term in
$\cal F$ due to the presence of plasma as $\Delta{\cal F}=-k_{0}^{2}{\cal
D}{\cal R}$, where $k_{0}=q_{\mbox{\scriptsize
NR}}\Gamma_{0}\tilde{\kappa}_{0}$ and
$\tilde{\kappa}_{0}=\kappa_{0}-\xi_{0}\eta_{0}$, write the factor
$\tilde{\kappa}_{0}$ as
$\tilde{\kappa}_{0}=\kappa_{0}-\xi_{0}(-L_{0}-\sqrt{\delta_{0}})/M_{0}=\big{[}\kappa_{0}+\xi_{0}(M_{0}\tilde{\cal
F}_{{\mbox{\scriptsize NR}},0})^{1/2}\big{]}/M_{0}$, insert into the
expression under the square root $\tilde{\cal F}_{{\mbox{\scriptsize
NR}},0}=\mathring{\cal
F}_{0}-\tilde{\gamma}_{0}\kappa_{0}^{2}=(1-\tilde{q}_{\mbox{\scriptsize
NR}}^{2}){\cal A}_{0}-{\cal D}_{0}u_{0}^{2}b^{2}/{\cal A}_{0}$ (this is just
the expression for ${\cal F}_{0}$ obtained at the beginning of Appendix D,
with $q$ replaced by $q_{\mbox{\scriptsize NR}}$) and take the limit $r_{0}\to
r_{\mbox{\scriptsize h}}$. It holds $\xi_{\mbox{\scriptsize h}}=-{\cal
A}_{\mbox{\scriptsize h}}^{-1/2}$, $M_{\mbox{\scriptsize
h}}=1+q_{{\mbox{\scriptsize NR}}}^{2}=1/(1-\tilde{q}_{{\mbox{\scriptsize
NR}},{\mbox{\scriptsize h}}}^{2})^{1/2}$ (since $q_{{\mbox{\scriptsize
NR}},{\mbox{\scriptsize h}}}=(q_{0}\Gamma_{0})(r_{0}\to r_{\mbox{\scriptsize
h}})=\tilde{q}_{\mbox{\scriptsize NR}}/(1-\tilde{q}_{\mbox{\scriptsize
NR}}^{2})^{1/2}$) and $\tilde{\cal F}_{{\mbox{\scriptsize
NR}},{\mbox{\scriptsize h}}}=(1-\tilde{q}_{\mbox{\scriptsize NR}}^{2}){\cal
A}_{\mbox{\scriptsize h}}$, so that $\tilde{\kappa}_{\mbox{\scriptsize
h}}\propto\kappa_{\mbox{\scriptsize h}}-|\kappa_{\mbox{\scriptsize h}}|=0$
(since $\kappa_{\mbox{\scriptsize h}}>0$ for
$b=\mathring{b}_{{\mbox{\scriptsize ph}}\pm}$). Thus, for
$\epsilon_{0}=r_{0}/r_{\mbox{\scriptsize h}}\ll 1$ we have
$\tilde{\kappa}_{0}=O(\epsilon_{0})$, and since
$\Gamma_{0}=O(\epsilon_{0}^{-1/2})$, we have also
$k_{0}=O(\epsilon_{0}^{1/2})$ and $k_{\mbox{\scriptsize h}}=0$. We see that
function ${\cal F}(r,r_{0},b)$ reduces to $\mathring{\cal F}(r,b)$ at
$r_{0}=r_{\mbox{\scriptsize h}}$; therefore, functions ${\cal
B}_{\pm}(r_{0})$, defined by equations ${\cal F}(r,r_{0},b)=\partial_{r}{\cal
F}(r,r_{0},b)=0$, reduce to $\mathring{b}_{{\mbox{\scriptsize ph}}\pm}$ at
$r_{0}=r_{\mbox{\scriptsize h}}$. For Kerr parameter $a=0.8$ used in Fig. 5,
this yields extreme values of $b$ for rays seen by an observer on the horizon
(in the sense of limit) $({\cal B}_{{\mbox{\scriptsize h}}+},{\cal
B}_{{\mbox{\scriptsize h}}-})=(3.237,-6.662)$. Finally, since region ${\cal
O}_{{\mbox{\scriptsize h}}+{\mbox{\scriptsize i}}}^{(-)}$ is adjacent to the
horizon, the radii from which radiation has access to the black hole are
bounded only from above: for the parameters used in Fig. 5, maximum radii
$(r_{0},r)$ are $(\rho_{0B}^{(-)},\rho_{B}^{(-)})=(11.378,3.067)$.

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