#  Can different black holes cast the same shadow?

Haroldo C. D. Lima Junior<EMAIL_ADDRESS>Faculdade de Física,
Universidade Federal do Pará, 66075-110, Belém, PA, Brazil  Luís C. B.
Crispino<EMAIL_ADDRESS>Faculdade de Física, Universidade Federal do Pará,
66075-110, Belém, PA, Brazil  Pedro V. P. Cunha<EMAIL_ADDRESS>Departamento de
Matemática da Universidade de Aveiro and Centre for Research and Development
in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro,
Portugal  Carlos A. R. Herdeiro<EMAIL_ADDRESS>Departamento de Matemática da
Universidade de Aveiro and Centre for Research and Development in Mathematics
and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal

(February 2021)

###### Abstract

We consider the following question: may two different black holes (BHs) cast
exactly the same shadow? In spherical symmetry, we show the necessary and
sufficient condition for a static BH to be shadow-degenerate with
Schwarzschild is that the dominant photonsphere of both has the same impact
parameter, when corrected for the (potentially) different redshift of
comparable observers in the different spacetimes. Such shadow-degenerate
geometries are classified into two classes. The first shadow-equivalent class
contains metrics whose constant (areal) radius hypersurfaces are isometric to
those of the Schwarzschild geometry, which is illustrated by the Simpson and
Visser (SV) metric. The second shadow-degenerate class contains spacetimes
with different redshift profiles and an explicit family of metrics within this
class is presented. In the stationary, axi-symmetric case, we determine a
sufficient condition for the metric to be shadow degenerate with Kerr for far-
away observers. Again we provide two classes of examples. The first class
contains metrics whose constant (Boyer-Lindquist-like) radius hypersurfaces
are isometric to those of the Kerr geometry, which is illustrated by a
rotating generalization of the SV metric, obtained by a modified Newman-Janis
algorithm. The second class of examples pertains BHs that fail to have the
standard north-south $\mathbb{Z}_{2}$ symmetry, but nonetheless remain shadow
degenerate with Kerr. The latter provides a sharp illustration that the shadow
is not a probe of the horizon geometry. These examples illustrate that
nonisometric BH spacetimes can cast the same shadow, albeit the lensing is
generically different.

##  I Introduction

Strong gravity research is undergoing a golden epoch. After one century of
theoretical investigations, the last five years have started to deliver long
waited data on the strong field regime of astrophysical black hole (BH)
candidates. The first detection of gravitational waves from a BH binary merger
[1] and the first image of an astrophysical BH resolving horizon scale
structure [2, 3, 4] have opened a new era in testing the true nature of
astrophysical BHs and the hypothesis that these are well described by the Kerr
metric [5].

In both these types of observations a critical question to correctly interpret
the data is the issue of degeneracy. How degenerate are these observables for
different models? This question, moreover, is two-fold. There is the practical
issue of degeneracy, due to the observational error bars. Different models may
predict different gravitational waves or BH images which, however, are
indistinguishable within current data accuracy. But there is also the
theoretical issue of degeneracy. Can different models predict the same
phenomenology for some (but not all) observables? For the case of
gravitational waves this is reminiscent of an old question in spectral
analysis: can one hear the shape of a drum [6]? (See also [7]). Or, in our
context, can two different BHs be isospectral? For the case of BH imaging,
this is the question if two different BHs capture light in the same way,
producing precisely the same silhouette. In other words, can two different BH
geometries have precisely the same shadow [8]? The purpose of this paper is to
investigate the latter question.

The BH shadow is not an imprint of the BH’s event horizon [9]. Rather, it is
determined by a set of bound null orbits, exterior to the horizon, which, in
the nonspherical cases include not only light rings (LRs) but also non-planar
orbits; in Ref. [10] these were dubbed fundamental photon orbits. In the Kerr
case these orbits are known as spherical orbits [11], since they span
different latitudes at constant radius, in Boyer-Lindquist coordinates [12].

It is conceivable that different spacetime geometries can have, in an
appropriate sense, equivalent fundamental photon orbits without being
isometric to one another. In fact, one could imagine extreme situations in
which one of the spacetimes is not even a BH.111A rough, but not precise,
imitation of the BH shadow by a dynamically robust BH mimicker was recently
discussed in [13]. It is known that equilibrium BHs, in general, must have LRs
[14], and, consequently, also non-planar fundamental photon orbits [15]. But
the converse is not true: spacetimes with LRs need not to have a horizon -
see, $e.g.$, Ref. [16].

In this paper we will show such shadow-degenerate geometries indeed exist and
consider explicit examples. For spherical, static, geometries, a general
criterion can be established for shadow-degenerate geometries. The latter are
then classified into two distinct equivalence classes. Curiously, an
interesting illustration of the simplest class is provided by the ad hoc
geometry introduced by Simpson and Visser (SV) [17], describing a family of
spacetimes that include the Schwarzschild BH, regular BHs and wormholes. In
the stationary axisymmetric case we consider the question of shadow degeneracy
with the Kerr family within the class of metrics that admit separability of
the Hamilton-Jacobi (HJ) equation. We provide two qualitatively distinct
classes of examples. The first one is obtained by using a modified Newman-
Janis algorithm [18] proposed in [19]; we construct a rotating version of the
SV spacetime, which, as shown here, turns out to belong to a shadow-degenerate
class including the Kerr spacetime. A second class of examples discusses BHs
with the unusual feature that they do not possess the usual north-south
$\mathbb{Z}_{2}$ symmetry present in, say, the Kerr family, but nonetheless
can have the same shadow as the Kerr spacetime. This provides a nice
illustration of the observation in [9] that the shadows are not a probe of the
event horizon geometry.

This paper is organized as follows. In Sec. II we discuss a general criterion
for shadow-degeneracy in spherical symmetry and classify the geometries
wherein it holds into two equivalence classes. Section III discusses
illustrative examples of shadow-degenerate metrics for both classes. For class
I, the example is the SV spacetime. We also consider the lensing in shadow-
degenerate spacetimes, which is generically non-degenerate. Section IV
discusses shadow degeneracy for stationary geometries admitting separability
of the HJ equation. Section V discusses illustrative examples of shadow-
degenerate metrics with Kerr, in particular constructing a rotating
generalization of the SV spacetime and a BH spacetime without $\mathbb{Z}_{2}$
symmetry, and analysing their lensing/shadows. We present some final remarks
in Sec. VI.

##  II Shadow-degeneracy in spherical symmetry

Let us consider a spherically symmetric, asymptotically flat, static BH
spacetime.222The class of metrics (1) may describe non-BH spacetimes as well.
Its line element can be written in the generic form:

$ds^{2}=-V(R)A(R)dt^{2}+\frac{dR^{2}}{B(R)V(R)}+R^{2}d\Omega_{2}\ .$ (1)

Here, $V(R)\equiv 1-2m/R$ is the standard Schwarzschild function, where $m$ is
a constant that fixes the BH horizon at $R=2m$. The constant $m$ needs not to
coincide with the ADM mass, denoted $M$. The two arbitrary radial functions
$A(R),B(R)$ are positive outside the horizon, at least $C^{1}$ and tend
asymptotically to unity:

$\lim_{R\rightarrow\infty}A(R),B(R)=1\ .$ (2)

In the line element (1),
$d\Omega_{2}\equiv\left[d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right]$ is
the metric on the unit round 2-sphere.

Due to the spherical symmetry of the metric (1), we can restrict the motion to
the equatorial plane $\theta=\pi/2$. Null geodesics with energy $E$ and
angular momentum $j$ have an impact parameter

$\lambda\equiv\frac{j}{E}\ ,$ (3)

and spherical symmetry allows us to restrict to $\lambda\geqslant 0$.333The
case $\lambda=0$ has no radial turning points.

Following Ref. [16], we consider the Hamiltonian
$\mathscr{H}=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}$, to study the null geodesic
flow associated to the line element (1). One introduces a potential term
$\mathcal{V}(R)\leqslant 0$ such that

$2\mathscr{H}=\mathcal{V}+g_{RR}\dot{R}^{2}=0\ .$ (4)

The dot denotes differentiation regarding an affine parameter. Clearly, a
radial turning point $(\dot{R}=0)$ is only possible when $\mathcal{V}=0$ for
null geodesics ($\mathscr{H}=0$). One can further factorize the potential
$\mathcal{V}$ as

$\mathcal{V}=j^{2}g^{tt}\left(\frac{1}{\lambda}-\mathcal{H}\right)\left(\frac{1}{\lambda}+\mathcal{H}\right)\leqslant
0\,,$ (5)

which introduces the effective potential $\mathcal{H}(R)$:

$\mathcal{H}(R)=\frac{\sqrt{A(R)\,V(R)}}{R}\geqslant 0\,.$ (6)

In spherical symmetry, the BH shadow is determined by LRs. In the effective
potential (6) description, a LR corresponds to a critical point of
$\mathcal{H}(R)$, and its impact parameter is the inverse of $\mathcal{H}(R)$
at the LR [16]:

$\mathcal{H}^{\prime}(R_{\rm LR})=0\ ,\qquad\lambda_{\rm
LR}=\frac{1}{\mathcal{H}(R_{\rm LR})}\ ,$ (7)

where prime denotes radial derivative.

The BH shadow is an observer dependent concept. Let us therefore discuss the
observation setup. We consider an astrophysically relevant scenario. First,
the BH is directly along the observer’s line of sight. Second, the observer is
localized at the same areal radius, $R_{\rm obs}$, in the Schwarzschild and
non-Schwarzschild spacetimes,444This is a geometrically significant
coordinate, and thus can be compared in different spacetimes. both having the
same ADM mass $M$. Finally, the observer is sufficiently far away so that no
LRs exist for $R>R_{\textrm{obs}}$ in both spacetimes, but $R_{\textrm{obs}}$
needs not to be at infinity.555 If a LR exists for $R>R_{\textrm{obs}}$ a more
complete analysis can be done, possibly featuring a BH shadow component in the
opposite direction to that of the BH, from the viewpoint of the observer.
However, this will not be discussed here in order to focus on a more
astrophysical setup. The connection between the impact parameter $\lambda$ of
a generic light ray and the observation angle $\beta$ with respect to the
observer-BH line of sight is [20]:

$\lambda=\frac{\sin\beta}{\mathcal{H}\big{(}R_{\textrm{obs}}\big{)}}\,.$ (8)

The degeneracy condition, that is, for the shadow edge to be the same, when
seen by an observer at the same areal radius $R_{\rm obs}$ in comparable
spacetimes ($i.e.$, with the same ADM mass, $M$), is that the observation
angle $\beta$ coincides in both cases, for both the metric (1) and the
Schwarzschild spacetime. This implies that the impact parameter of the shadow
edge in the generic spacetime must satisfy:

$\lambda_{\rm LR}=\frac{\sqrt{27}M}{\sqrt{A_{\textrm{obs}}}}\ \ ,$ (9)

where we have used that, for Schwarzschild, the LR impact parameter is
$\lambda^{\rm Schw}_{{}_{\textrm{\scriptsize LR}}}=\sqrt{27}M$ and
$A_{\textrm{obs}}\equiv A(R_{\rm obs})$. Hence, only for the cases wherein
$A=1$ does shadow degeneracy amounts to having the same impact parameter in
the Schwarzschild and in the non-Schwarzschild spacetimes, for an observer
which is not at spatial infinity. In general, the different gravitational
redshift at the "same" radial position in the two spacetimes must be accounted
for, leading to (9).

In the next two subsections we will distinguish two different classes of
shadow degenerate spacetimes (class I and II), using the results just
established. It is important to remark that the fact that two non-isometric
spacetimes are shadow-degenerate does not imply that the gravitational lensing
is also degenerate. In fact, it is generically not.

This will be illustrated in Sec. III with some concrete examples for the two
classes of shadow-degenerate spacetimes. An interesting example of shadow, but
not lensing, degeneracy can be found in [21], albeit in a different context.

###  II.1 Class I of shadow-degenerate spacetimes

Specializing (7) for (1) yields

$\lambda_{\rm LR}=\frac{R_{\rm LR}}{\sqrt{A(R_{\rm LR})V(R_{\rm LR})}}\ .$
(10)

and

$R_{\rm LR}=3m+\frac{R_{\rm LR}^{4}}{2\lambda_{\rm
LR}^{2}}\frac{A^{\prime}(R_{\rm LR})}{A^{2}(R_{\rm LR})}\ ,$ (11)

where $A^{\prime}\equiv dA/dR$. The notorious feature is that regardless of
$B(R)$, if $A(R)=1$, then it holds for the background (1) that $m=M$ and

$R_{\rm LR}=3M\ ,\qquad\lambda_{{}_{\textrm{\scriptsize LR}}}=\sqrt{27}M\ ,$
(12)

which are precisely the Schwarzschild results. The condition $A=1$ is thus
sufficient to have the same shadow as Schwarzschild, since (9) is obeyed.
Spacetimes (1) with $A=1$ define the equivalence Class I of shadow degenerate
spacetimes.

If the spacetime (1) with $A(R)=1$ but $B(R)$ non trivial describes a BH, it
will have precisely the same shadow as the Schwarzschild spacetime. Such
family of spacetimes is not fully isometric to Schwarzschild. But its constant
(areal) radius hypersurfaces are isometric to those of Schwarzschild and thus
have overlapping $R=$constant geodesics, which explains the result. This
possibility will be illustrated in Sec. III.1.

These observations allow us to anticipate some shadow-degenerate geometries
also for stationary, axially symmetric spacetimes. If the fundamental photon
orbits are “spherical”, not varying in some appropriate radial coordinate, as
for Kerr in Boyer-Lindquist coordinates, any geometry with isometric constant
radial hypersurfaces will, as in the static case, possess the same fundamental
photon orbits, and will be shadow-degenerate with Kerr. We shall confirm this
expectation in an illustrative example in Sec. V.

###  II.2 Class II of shadow-degenerate spacetimes

If $A(R)\neq 1$ the solution(s) of (7) will not coincide, in general, with
(12). In particular, there can be multiple critical points of
$\mathcal{H}(R)$, $i.e.$ multiple LRs around the BH. This raises the question:
if multiple LRs exist, which will be the one to determine the BH shadow edge?

We define the dominant LR as the one that determines the BH shadow edge.666See
[22, 23] for a related discussion. To determine the dominant LR first observe
that:

  1. (i)

Since $g_{tt}<0$ outside the horizon, the condition
$1/\lambda\geqslant\mathcal{H}(R)$ must be satisfied along the geodesic
motion, see Eq. (5). In particular, at LRs, the smaller the impact parameter
$\lambda$ is, the larger the potential barrier $\mathcal{H}(R_{\rm LR})$ is.

  2. (ii)

The radial motion can only be inverted ($i.e.$ have a turning point) when
$\mathcal{V}=0\iff 1/\lambda=\mathcal{H}$.

  3. (iii)

The function $\mathcal{H}(R)$ vanishes at the horizon,
$\left.\mathcal{H}\right|_{R=2m}=0$.

The BH’s shadow edge is determined by critical light rays at the threshold
between absorption and scattering by the BH, when starting from the observer’s
position $R_{\textrm{obs}}$. Considering points 1-3 above, these critical null
geodesics occur at a local maximum of $\mathcal{H}$, $i.e.$, a LR. The
infalling threshold is provided by the LR that possesses the largest value of
$1/\lambda$, since this corresponds to the largest potential barrier in terms
of $\mathcal{H}(R)$. Any other photonspheres that might exist with a smaller
value of $1/\lambda$ do not provide the critical condition for the shadow
edge, although they might play a role in the gravitational lensing. Hence, the
dominant photonsphere has the smallest value of the impact parameter
$\lambda$, and shadow degeneracy with Schwarzschild is established by
constraining the smallest LR impact parameter $\lambda$, via Eq. (9).

Combining Eq. (9) and $\lambda\,\mathcal{H}(R)\leqslant 1$, yields the
necessary and sufficient conditions on $A(R)$ in order to have shadow-
degeneracy with Schwarzschild. Explicitly, these are

  1. (i)

$A(R)\leqslant\frac{R^{3}}{\,\left(R-2m\right)}\,\left(\frac{A_{\textrm{obs}}}{27M^{2}}\right)\,,$
(13)

  2. (ii)

the previous inequality must saturate at least once outside the horizon for
some $R_{\rm LR}^{\rm dom}<R_{\textrm{obs}}$. At such dominant LR, located at
$R_{\rm LR}^{\rm dom}$, (9) is guaranteed to hold.

Observe that $3M<R_{\textrm{obs}}$ so that the observer is outside the LR in
the Schwarzschild spacetime. Spacetimes (1) with $A\neq 1$ obeying the two
conditions above define the equivalence Class II of shadow-degenerate
spacetimes. One example will be given in Sec. III.2. We remark that class I of
shadow-degenerate spacetimes is a particular case of class II.

##  III Illustrations of shadow-degeneracy (static)

###  III.1 Class I example: the SV spacetime

The SV spacetime is a static, spherically symmetric geometry that generalizes
the Schwarzschild metric with one additional parameter $b$, besides the ADM
mass $M$, proposed in [17]. It is an ad hoc geometry. Its associated energy-
momentum tensor, via the Einstein equations, violates the null energy
condition, and thus all classical energy conditions. Nonetheless, it is a
simple and illustrative family of spacetimes that includes qualitatively
different geometries. It is given by the following line element:

$\displaystyle ds^{2}=$
$\displaystyle-\left(1-\frac{2M}{\sqrt{r^{2}+b^{2}}}\right)dt^{2}+\left(1-\frac{2M}{\sqrt{r^{2}+b^{2}}}\right)^{-1}dr^{2}$
(14) $\displaystyle+\left(r^{2}+b^{2}\right)d\Omega_{2}\ ,$

where the coordinates have the following domains: $-\infty\leqslant
r\leqslant\infty$, $-\infty\leqslant t\leqslant\infty$, $0<\theta<\pi$ and
$-\pi\leqslant\varphi<\pi$. Depending on the value of the additional parameter
$b$, which without loss of generality we assume to be non-negative, the
spacetime geometry describes: (i) the Schwarzschild geometry ($b=0$); (ii) a
regular BH ($0<b<2M$) with event horizon located at

$r_{h}=\pm\sqrt{(2M)^{2}-b^{2}}\ ;$ (15)

(iii) a one-way traversable wormhole geometry with a null throat at $r_{t}=0$
($b=2M$); or (iv) a two-way traversable wormhole geometry with a timelike
throat at $r_{t}=0$, belonging to the Morris-Thorne class [24] ($b>2M$).

The coordinate system in (14) is relevant to observe that $r=0$ is not a
singularity. Thus the geometry can be extended to negative $r$. However, it
hides some other features of the geometry, since the radial coordinate in (14)
is not the areal radius for $b\neq 0$. Introduce the areal radius $R$ as

$R^{2}\equiv r^{2}+b^{2}\ .$ (16)

The SV spacetime reads, in ($t,R,\theta,\varphi$) coordinates,

$\displaystyle ds^{2}=-V(M)\,dt^{2}+\frac{dR^{2}}{V(M)\,B_{\rm
SV}(R)}+R^{2}\,d\Omega_{2}\ ,\ \ \ \ \ \ \ $ (17)

where

$\displaystyle V(M)=1-\frac{2M}{R},$ (18) $\displaystyle B_{\rm
SV}(R)\equiv\left(1-\frac{b^{2}}{R^{2}}\right)\ .$ (19)

The geometry is now singular at

$R=2M\equiv R_{h}\ ,\ \ \ {\rm and}\ \ \ R=b\equiv R_{t}\ .$ (20)

For $0<b<2M$, $R_{h}>R_{t}$, the null hypersurface $R=R_{h}$ is a Killing
horizon and the event horizon of the spacetime. It can be covered by another
coordinate patch and then another coordinate singularity is found at
$R=R_{t}$. This is again a coordinate singularity, as explicitly shown in the
coordinate system (14). It describes a throat or bounce. A free falling
observer bounces back to a growing $R$, through a white hole horizon into
another asymptotically flat region — see Fig. 4 in [17]. Thus, in the
coordinate system ($t,R,\theta,\varphi$), $b\leqslant R<\infty$. The other
coordinate ranges are the same as before.

As it is clear from Eq. (20), the areal radius of the event horizon (when it
exists) is $b$-independent. Moreover, since the SV spacetime is precisely of
the type (1) with $A(R)=1$ and $B(R)=B_{\rm SV}(R)$, it follows from the
discussion of the preceding section that Eq. (12) holds for the SV spacetime.
Thus, whenever the SV spacetime describes a BH ($0<b<2\,M$) it is class I
shadow-degenerate with a Schwarzschild BH, for an equivalent observer. This
result can be also applied to the wormhole geometry if the LR is located
outside the throat, i.e. the LR must be covered by the R coordinate range
($b\leq 3M$). For $b>3M$, the LR is located at the throat and Eq. (12) does
not hold [25].

The LR in this spacetime has the same areal radius as in Schwarzschild,
$R_{\rm LR}=3M$. However, the proper distance between the horizon and the LR,
along a $t,\theta,\varphi=$constant curve is $b$-dependent:

$\Delta R=\int_{R_{h}}^{R_{\rm
LR}}\sqrt{g_{RR}}dR=\left\\{\begin{array}[]{l}\simeq 3.049M\ ,{\rm for}\ b=0\
,\\\ \rightarrow\infty\ \ \ \ \ \ \ ,{\rm for}\ b=2M\ .\end{array}\right.$
(21)

It is a curious feature of the SV spacetime that the spatial sections of the
horizon and the photonsphere have $b$-independent proper areas, respectively
$4\pi(2M)^{2}$ and $4\pi(3M)^{2}$. But the proper distance between these
surfaces is $b$-dependent and diverges as $b\rightarrow 2M$.

Let us now consider the gravitational lensing in the SV spacetime. We set the
observer’s radial coordinate equal to $R_{\rm obs}=15\,M$. In the top panel of
Fig. 1 we plot the scattered angle $\Delta\varphi$ on the equatorial plane, in
units of $2\pi$, as a function of the observation angle $\beta$. We choose
three distinct values of $b$, including the Schwarzschild case ($b=0$). For
large observation angles, the scattered angle is essentially the same for
different values of $b$. A slight difference arises near the unstable LR,
characterized by the divergent peak, as can be seen in the inset of Fig. 1
(top panel). In this region, for a constant $\beta$, the scattered angle
increases as we increase $b$. We note that the LR, and hence the shadow
region, is independent of $b$, as expected. In the bottom panel of Fig. 1 we
show the trajectories of light rays for the observation angle $\beta=0.33$ and
different values of $b$. The event horizon, for the BH cases, is represented
by the dashed circle. We notice that higher values of $b$ lead to larger
scattering angles.

Figure 1: Top panel: scattering angle for null geodesics as a function of the
observation angle, for the Schwarzschild BH ($b=0$), SV BH ($b=1.99M$) and SV
wormhole ($b=2.5M$) cases. The shaded area corresponds to the shadow region in
the observer’s local sky. Bottom panel: the trajectories described by null
geodesics in the Cartesian plane for the same choices of $b$ and with
observation angle $\beta=0.33$. The observer $O$ is located at $(x=15M,y=0)$,
as shown in the bottom panel. The dashed circle represents the event horizon
of the BH cases.

We show in Fig. 2 the shadow and gravitational lensing of the Schwarzschild
BH, SV BH and SV wormhole spacetimes, obtained using backwards ray-tracing.

\put(-260.0,100.0){\footnotesize}

(a) Schwarzschild ($b=0$)

(b) SV BH ($b=1.99M$)

(c) SV wormhole ($b=2.5M$)

Figure 2: The shadow and gravitational lensing of the: Schwarzschild BH (panel
a), SV BH spacetime with $b=1.99M$ (panel b), and SV wormhole spacetime with
$b=2.5M$ (panel c). The observer is located on the equatorial plane and at the
radial coordinate $R_{\rm obs}=15M$. The angle of view is equal to
$45\degree$. We present, in the top right corner of each panel, a zoom of the
region next to the shadow edge.

In the backward ray-tracing procedure, we numerically integrate the light rays
from the observer position, backwards in time, until the light rays are
captured by the event horizon or scattered to infinity. The results were
obtained with two different codes: a C++ code developed by the authors, and
the PYHOLE code [26] which was used as a cross-check. In this work we only
show the ray-tracing results obtained with the C++ code, since they are
essentially the same as the ones obtained with the PYHOLE code. The light rays
captured by the BH are assigned a black color in the observer’s local sky. For
the scattered light rays, we adopt a celestial sphere with four different
colored quadrants (red, green, blue and yellow). A grid with constant latitude
and longitude lines, and a bright spot behind the BH/wormhole are also present
in the celestial sphere. This setup is similar to the ones considered in Refs.
[27, 28, 26]. On the other hand, the light rays captured by the wormhole
throat are assigned with a different color pattern (purple, cyan, dark green
and dark yellow) on the celestial sphere on the other side of the throat, but
without the grid lines and the bright spot.

The white ring surrounding the shadow region in Fig. 2 corresponds to the
lensing of the bright spot behind the BH, known as Einstein ring [29]. It is
manifest in the Schwarzschild and SV BHs, as well as in the SV wormhole case.
The Einstein ring has a slightly different angular size in the three
configurations shown in Fig. 2. This can be confirmed in Fig. 1, since on the
equatorial plane, the Einstein ring is formed by the light rays scattered with
angle $\Delta\varphi=\pi$, which corresponds to close values of $\beta$ for
the three cases. Due to the spherical symmetry, a similar analysis holds for
light rays outside the equatorial plane, which explains the formation of the
Einstein ring with similar angular size in Fig. 2.

Inside the Einstein ring, the whole celestial sphere appears inverted. Next to
the shadow edge, there is a second Einstein ring (corresponding to a circular
grid ring best seen in the inset of the figures between the yellow and green
patches), in this case corresponding to the point behind the observer,
$\Delta\varphi=2\pi$ in Fig. 1. A slight difference between the three
configurations is also observed in this region, as can be seen in the inset of
Fig. 2. In between the second Einstein ring and the shadow edge there is an
infinite number of copies of the celestial sphere.

###  III.2 Class II example: A spacetime with $A\neq 1$

As a concrete example of the class II of shadow-degenerate BH spacetimes with
respect to the Schwarzschild BH, as seen by an observer at radial coordinate
$R_{\rm obs}$, we consider (using $m=M=1$ units) the following $A(R)$
function:

$A=1+\left(\frac{R_{\rm
LR}-3}{b_{1}+b_{2}}\right)\\!\left(a_{0}\left[\frac{R_{\rm
LR}}{R}\right]^{2}+a_{1}\left[\frac{R_{\rm LR}}{R}\right]^{4}\right),$ (22)
$\displaystyle a_{0}=-2a_{1}-R_{\rm LR}^{3}\left(1-2u^{2}+u^{4}\right),$
$\displaystyle a_{1}=-36+24R_{\rm LR}-R_{\rm LR}^{2}+R_{\rm
LR}^{3}\left(u^{2}-2\right),$ $\displaystyle b_{1}=27(R_{\rm LR}-2)^{2},$
$\displaystyle b_{2}=R_{\rm LR}^{3}u^{2}\left[(7-3R_{\rm LR})+(2R_{\rm
LR}-5)u^{2}\right].$

We have here introduced a free parameter $R_{\rm LR}$ which sets the radial
position of the dominant photonsphere (after suitably restricting the range of
$R_{\rm LR}$). This spacetime is also modified by the quantity $u\equiv R_{\rm
LR}/R_{\rm obs}$, which depends on the observer location. In particular, the
choice $u=0$ corresponds to setting the observer at spatial infinity. This
spacetime reduces to the Schwarzschild case for $R_{\rm LR}=3$ (provided than
$B=1$), since it implies $A(R)=1$. However, for $R_{\rm LR}\neq 3$ the
spacetime is not Schwarzschild.

Not every parameter combination $\\{R_{\rm LR},R_{\rm obs}\\}$ yields an
acceptable spacetime for our analysis. Considering the discussion in Sec.
II.2, the spacetime outside the horizon must satisfy both $A>0$ and Eq. (13),
together with $2<R_{\rm LR}<R_{\rm obs}$ and $3<R_{\rm obs}$. These conditions
fix the allowed range of parameters.

For concreteness, we can set $R_{\rm obs}=15$, which leads to the following
allowed range for the parameter $R_{\rm LR}$:

$R_{\rm LR}\in[2.22\,,\,3.26]\,.$ (23)

In contrast, the function $B(R)$ can be left fairly unconstrained. Curiously,
for some values of $R_{\rm LR}$ in the range (23) there can be three
photonspheres (two unstable, and one stable) outside the horizon. However, the
LR at $R=R_{\rm LR}$ is always the dominant one by construction, as can be
seen in Fig. 3, where the horizontal dashed lines correspond to
$1/\lambda_{\rm LR}$ \- see Eq. (9). Each line intersects the maximum point of
the associated potential $\mathcal{H}$, that determines the dominant LR
location.

Figure 3: The effective potential $\mathcal{H}$(R) for the class II of shadow-
degenerate BHs, with $A(R)$ given by Eq. (22), for different dominant LR
radius $R_{\rm LR}$. The horizontal lines correspond to the critical impact
parameters for each $R_{\rm LR}$, see Eq. (9).

Let us now consider the gravitational lensing in this spacetime, assuming for
simplicity $B(R)=1$. In the top panel of Fig. 4 we plot the scattered angle,
in units of $2\pi$, as a function of the observation angle $\beta$, for the
two parameter values $R_{\rm LR}=\\{2.227\,,\,2.37\\}$. The Schwarzschild case
$R_{\rm LR}=3$ is also included in Fig. 4, for reference. First taking $R_{\rm
LR}=2.227$, there are two diverging peaks on the scattering angle, related to
the existence of two unstable LRs (there exists also a stable LR which leaves
no clear signature in the scattering plot). In contrast to the first case,
$R_{\rm LR}=2.37$ contains only a single diverging peak. However, it presents
a local maximum of the scattering angle next to $\beta=0.4$, as can be seen in
the inset of Fig. 4 (top panel). This local maximum leads to a duplication of
lensed images. Importantly, the shadow region, corresponding to the shaded
area in Fig. 4, is the same for the different values of $R_{\rm LR}$, as
expected.

Figure 4: Top: the scattered angle of null geodesics, as a function of the
observation angle $\beta$, for $R_{\rm LR}=\\{2.227,2.37,3\\}$. The shaded
area corresponds to the shadow region in the observer’s local sky. The
horizontal dotted line represents the light rays scattered at
$\Delta\varphi=2\pi$ (located behind the observer). Since there is a black dot
in the celestial sphere right behind the observer, the gravitational lensing
displays black Einstein rings (see Fig. 5). Bottom: the corresponding
trajectories described by null geodesics in the equatorial plane in Cartesian-
like coordinates with an observation angle $\beta=0.33$. The observer $O$ is
located at $(x=15,y=0)$. The dashed circle represents the event horizon of
these class II shadow degenerate examples.

On the bottom panel of Fig. 4 we show the trajectories described by null
geodesics, for different values of the parameter $R_{\rm LR}$, given the same
observation angle $\beta=0.33$. Curiously, when $R_{\rm LR}=2.227$, the
trajectory followed by the light ray presents two inflection points when
represented in the $(x=R\cos\varphi,y=R\sin\varphi)$ plane, cf. Fig. 4 (bottom
panel). Such inflection points are determined by $d^{2}y/dx^{2}=0$, which
yields

$R^{2}-R\,\frac{d^{2}R}{d\varphi^{2}}+2\,\left(\frac{dR}{d\varphi}\right)^{2}=0\
\ \ \Leftrightarrow\ \ \ \ddot{R}=\frac{j^{2}}{R^{3}}\ ,$ (24)

where the last equivalence holds for $j\neq 0$. This is also the condition for
the curve $R=R(\varphi)$ to have vanishing curvature. On the other hand, the
equations of motion (4) and $\dot{\varphi}=j/R^{2}$, give, for $B(R)=1$,

$\ddot{R}=\frac{j^{2}}{R^{3}}+\frac{E^{2}}{2}\frac{d}{dR}\left[\frac{1}{A(R)}+\frac{2\lambda^{2}}{R^{3}}\right]\
.$ (25)

Equating the last two results yields

$3+\frac{R^{4}}{2\lambda^{2}}\frac{A^{\prime}(R)}{A(R)^{2}}=0\ ,$ (26)

as the condition at an inflection point. Observe the similarity with (11)
(recall $m=1$ here); the latter has $\ddot{R}=0$, unlike (24). For the
Schwarzschild case, $A(R)=1$, and there are no inflection points. But for
$A(R)$ given by (22) and $R_{\rm LR}=2.227$ it can be checked that the
function in square brackets in (25) has a local maximum, which explains the
existence of inflection points.

To further illustrate the effect of different choices of the parameter $R_{\rm
LR}$, we display the shadows and gravitational lensing in Fig. 5, obtained
numerically via backwards ray-tracing. Despite identical shadow sizes, the
gravitational lensing can be quite different for each value of the parameter
$R_{\rm LR}$. For instance, although Einstein rings are present in all cases
depicted, they have different angular diameters. This is best illustrated by
looking at the white circular rings, which are mapping the point in the
colored sphere directly behind the BH.

There are also some curious features of the lensing that can be anticipated
from the scattering angle plot in Fig. 4 (top panel). For example, for a
parameter $R_{\rm LR}=2.227$ there are multiple copies of the celestial sphere
very close to the shadow edge that are not easily identifiable in Fig. 5(a).
This is due to light rays scattered with angles greater than $\pi$ having an
observation angle $\beta$ very close to the shadow edge. The diverging peak in
the scattering angle also has a clean signature in the image, in the form of a
very sharp colored ring which is just a little smaller in diameter than the
white circle. Additionally, taking the parameter $R_{\rm LR}=2.37$, we can
further expand our previous remark on the effect of the local maximum of the
scattering angle, which introduces an image duplication of a portion of the
colored sphere directly behind the observer. This feature is best seen in Fig.
5(b) as an additional colored ring structure that does not exist in Fig. 5(c).

(a) Class II example ($R_{\rm LR}=2.227$)

(b) Class II example ($R_{\rm LR}=2.37$)

(c) Schwarzschild ($R_{\rm LR}=3$)

Figure 5: The shadow and gravitational lensing for the class II example of
shadow degenerate BHs, with $A(r)$ given in Eq. (22), for $R_{\rm LR}=2.227$
(panel a), $R_{\rm LR}=2.37$ (panel b), and $R_{\rm LR}=3$ (panel c). The
observer position and angle of view are the same as in Fig. 2.

##  IV Shadow-degeneracy in stationary BHs

Let us now investigate possible rotating BHs with Kerr degenerate shadows. We
assume that the spacetime is stationary, axi-symmetric and asymptotically
flat. In contrast to the spherically symmetric case, the general motion of
null geodesics in rotating BH spacetimes is not constrained to planes with
constant $\theta$. This introduces an additional complication in the analysis
of the geodesic motion, and in many cases it is not possible to compute the
shadows analytically. For Kerr spacetime, there is an additional symmetry that
allows the separability of the HJ equation, thus the shadow problem can be
solved analytically. This symmetry is encoded in the so-called Carter
constant, that arises due to a non-trivial Killing tensor present in the Kerr
metric [30]. Here, we shall investigate shadow degeneracy specializing for
rotating BH spacetimes that admit separability of the null geodesic
equation.777Although this strategy implies a loss of generality it seems
unlikely (albeit a proof is needed) that a spacetime without separability can
yield precisely the same shadow as the Kerr one, which is determined by
separable fundamental photon orbits. A relevant previous analysis in the
literature regarding shadow degeneracy for general observers can be found in
[31], albeit restricted to the Kerr-Newman-Taub-NUT BH family of solutions.

###  IV.1 Spacetimes admitting HJ separability

Following the strategy just discussed, we need the general form of the line
element for a rotating BH spacetime that admits separability of the HJ
equation. It is known that rotating spacetimes obtained through a Newman-Janis
and modified Newman-Janis algorithms allow separability of the HJ equation
[19, 32, 33]. Since it is not guaranteed, however, that every spacetime
allowing separability of the HJ equation can be obtained by such algorithms,
we pursue a different strategy. In Ref. [34], Benenti and Francaviglia found
the form of the metric tensor for a spacetime admitting separability of null
geodesics in n-dimensions (see also [35]). This result is based on the
following general assumptions:

  1. (i)

There exist (locally) $z$ independent commuting Killing vectors
$\textbf{X}_{\alpha}$ .

  2. (ii)

There exist (locally) $n-z$ independent commuting Killing tensors
$\textbf{K}_{a}$, satisfying

$[\textbf{K}_{a},\textbf{X}_{\alpha}]=0.$ (27)

  3. (iii)

The Killing tensors $\textbf{K}_{a}$ have in common $n-z$ commuting
eigenvectors $\textbf{X}_{a}$ such that

$g(\textbf{X}_{a},\textbf{X}_{\alpha})=0,\qquad[\textbf{X}_{a},\textbf{X}_{\alpha}]=0.$
(28)

We are interested in four-dimensional spacetimes ($n=4$) admitting two Killing
vectors ($z=2$), associated to the axi-symmetry and stationarity. Hence the
metric tensor that admits separability of the HJ equation is given by [34]:

$\displaystyle\partial^{2}_{s}=$ $\displaystyle
g^{ab}\partial_{a}\partial_{b}=\frac{1}{\tilde{A}_{1}(R)+\tilde{B}_{1}(\theta)}\left[\left(\tilde{A}_{2}(R)+\tilde{B}_{2}(\theta)\right)\partial_{t}^{2}\right.$
$\displaystyle+2\left(\tilde{A}_{3}(R)+\tilde{B}_{3}(\theta)\right)\partial_{t}\partial_{\varphi}+\tilde{A}_{4}(R)\partial_{R}^{2}+\tilde{B}_{4}(\theta)\partial_{\theta}^{2}$
$\displaystyle\left.+\left(\tilde{A}_{5}(R)+\tilde{B}_{5}(\theta)\right)\partial_{\varphi}^{2}\right].$
(29)

For our purpose, it is convenient to rewrite the functions $\tilde{A}_{i}(R)$
and $\tilde{B}_{i}(\theta)$ as

$\displaystyle\tilde{A}_{1}=R^{2}A_{1},\quad\tilde{B}_{1}=a^{2}\cos^{2}\theta
B_{1},\quad\tilde{A}_{2}=-\frac{(R^{2}+a^{2})^{2}}{\Delta}{A_{2}},$
$\displaystyle\tilde{B}_{2}=a^{2}\sin^{2}\theta{B_{2}},\quad\tilde{A}_{3}=-\frac{2amR}{\Delta}{A_{3}},\quad\tilde{B}_{3}={B_{3}}-1,$
$\displaystyle\tilde{A}_{4}=\Delta{A_{4}},\quad\tilde{B}_{4}={B_{4}},\quad\tilde{A}_{5}=-\frac{a^{2}}{\Delta}{A_{5}},\quad\tilde{B}_{5}=\frac{1}{\sin^{2}\theta}{B_{5}},$

since we can recover the Kerr spacetime by simply taking ${A}_{i}=1$ and
${B}_{i}=1$. The function $\Delta$ is given by

$\Delta=R^{2}-2mR+a^{2},$ (30)

where $m$ and $a$ are constants that fix the BH event horizon at

$R_{h}=m+\sqrt{m^{2}-a^{2}}.$ (31)

Similarly to the spherically symmetric case, $m$ and $a$ need not to coincide
with the ADM mass and total angular momentum per unit mass, respectively. The
metric tensor in terms of ${A}_{i}$ and ${B}_{i}$ assumes the following form:

$\displaystyle\partial^{2}_{s}=\frac{1}{\widetilde{\Sigma}}\left[-\left(\frac{(R^{2}+a^{2})^{2}}{\Delta}A_{2}-a^{2}\sin^{2}\theta
B_{2}\right)\partial_{t}^{2}\right.$
$\displaystyle-2\left(\frac{2amR}{\Delta}A_{3}+1-B_{3}\right)\partial_{t}\partial_{\varphi}+\Delta
A_{4}\partial_{R}^{2}+B_{4}\partial_{\theta}^{2}$
$\displaystyle\left.+\left(-\frac{a^{2}}{\Delta}A_{5}+\frac{1}{\sin^{2}\theta}B_{5}\right)\partial_{\varphi}^{2}\vphantom{\frac{(r^{2}+a^{2})^{2}}{\Delta}}\right],$
(32)

In Eq. (32), we have

$\widetilde{\Sigma}=R^{2}A_{1}+a^{2}\cos^{2}\theta B_{1}.$ (33)

We assume that $A_{i}(R)$ are positive outside the event horizon, and at least
$C^{1}$. Nevertheless, the general result found by Benenti and Francaviglia
may also describe non-asymptotically flat spacetimes. Hence, we need to impose
constraints on the 10 functions present in the metric tensor (32), in order to
describe asymptotically flat BHs. For this purpose it is sufficient that they
tend asymptotically to unity, namely:

$\lim_{R\rightarrow\infty}A_{i}(R)=1.$ (34)

The metric far away from the BH is given by [36]:

$d\tilde{s}^{2}=-\left(1-\frac{2M}{R}\right)dt^{2}-\frac{2J\sin^{2}\theta}{R}dtd\varphi+dR^{2}+R^{2}d\Omega_{2},$
(35)

where $J$ denotes the ADM angular momentum. Expanding the metric tensor (32)
for $R\gg m$ and $R\gg a$, and comparing with the BH metric far away from the
BH, we find that

$\displaystyle B_{3}(\theta)=B_{4}(\theta)=B_{5}(\theta)=1,$ (36)

while $B_{1}(\theta)$ and $B_{2}(\theta)$ are left unconstrained. Hence we
conclude that the metric tensor for a spacetime admitting separability and
asymptotically flatness is given by

$\displaystyle\partial^{2}_{s}=\frac{1}{\widetilde{\Sigma}}\left[-\left(\frac{(R^{2}+a^{2})^{2}}{\Delta}A_{2}-a^{2}\sin^{2}\theta
B_{2}\right)\partial_{t}^{2}\right.$
$\displaystyle-2\frac{2amRA_{3}}{\Delta}\partial_{t}\partial_{\varphi}+\Delta
A_{4}\partial_{R}^{2}+\partial_{\theta}^{2}$
$\displaystyle\left.+\left(-\frac{a^{2}}{\Delta}A_{5}+\frac{1}{\sin^{2}\theta}\right)\partial_{\varphi}^{2}\vphantom{\frac{(r^{2}+a^{2})^{2}}{\Delta}}\right].$
(37)

This line element (37) have been obtained previously in Ref. [37]

###  IV.2 Fundamental photon orbits

We are now able to study the problem of shadow degeneracy in stationary and
axisymmetric BHs, using the concrete metric form (37) for spacetimes with a
separable HJ equation. It is straightforward to compute the following geodesic
equations for the coordinates $\\{R,\theta\\}$:

$\displaystyle\frac{\widetilde{\Sigma}^{2}}{A_{4}}\frac{\dot{R}}{E^{2}}^{2}=\mathcal{R}(R),$
(38)
$\displaystyle\widetilde{\Sigma}^{2}\frac{\dot{\theta}}{E^{2}}^{2}=\Theta(\theta),$
(39)

where

$\displaystyle\mathcal{R}(R)=\left(R^{2}+a^{2}\right)^{2}A_{2}-4\lambda
amRA_{3}+a^{2}\lambda^{2}A_{5}$
$\displaystyle-\Delta\left(\eta+a^{2}+\lambda^{2}\right),$ (40)
$\displaystyle\Theta(\theta)=\eta+a^{2}\left(1-\sin^{2}\theta
B_{2}\right)-\frac{\lambda^{2}}{\tan^{2}\theta}.$ (41)

The constant parameters $\lambda$ and $\eta$ are given by

$\lambda=\frac{j}{E},\qquad\quad\eta=\frac{Q}{E^{2}}.$ (42)

As before, $\\{E,j\\}$ are the energy and angular momentum of the photon
respectively. The quantity $Q$ is a Carter-like constant of motion, introduced
via the separability of the HJ equation.

The analysis of the spherical photon orbits (SPOs) is paramount to determine
the BH’s shadow analytically. These orbits are a generalization of the LR
orbit (photon sphere) that was discussed in the introduction. SPOs have a
constant radial coordinate $R$ and are characterized by the following set of
equations:

$\displaystyle\mathcal{R}(R)=0,$ (43)
$\displaystyle\frac{d\,\mathcal{R}(R)}{dR}=0.$ (44)

In general, the set of SPOs in the spacetime (37) will not coincide with the
Kerr one. However if we set

$\displaystyle A_{2}=A_{3}=A_{5}=B_{2}=1,$ (45)

Eq. (40) is identical to the Kerr case in Boyer-Lindquist coordinates, the
same will hold for the set of equations (43)-(44), regardless of $A_{1}$,
$B_{1}$ and $A_{4}$. One might then have the expectation that (45) is a
sufficient condition to be shadow-degenerate with Kerr. Although this will
turn out to be indeed true, caution is needed in deriving such a conclusion,
since the influence of the observer frame also needs to be taken into account.

The solutions to Eqs. (43)-(44) with (45) are the following two independent
sets:

$\displaystyle\eta^{*}=-\frac{R^{4}}{a^{2}};\ \
\lambda^{*}=\frac{R^{2}+a^{2}}{a},$ (46)

and

$\displaystyle\eta=-\frac{R^{3}\left(R^{3}-6\,M\,R^{2}+9\,M^{2}\,R-4\,a^{2}\,M\right)}{a^{2}\left(R-M\right)^{2}},$
(47)
$\displaystyle\lambda=-\frac{\left(R^{3}-3\,M\,R^{2}+a^{2}\,R+M\,a^{2}\right)}{a\,\left(R-M\right)}.$
(48)

The first set $\\{\lambda^{*},\eta^{*}\\}$ is unphysical because (41) is not
satisfied for real values. In contrast, the second set of Eqs. (47)-(48) is
physically relevant: it defines the constants of motion for the different SPOs
as a function of the parametric radius $R\in[R_{1},R_{2}]$. In the latter,
$R_{1}$ and $R_{2}$ are defined as the roots of Eq. (47), given by

$\frac{R_{k}}{M}=2+2\cos\left\\{\frac{2}{3}\arccos\left[(2k-3)\frac{\mid
a\mid}{M}\right]\right\\},\\\ $ (49)

where $k\in\\{1,2\\}$. Importantly, the physical range $R\in[R_{1},R_{2}]$
follows from the requirement that $\Theta\geqslant 0$ (from Eq. (41)). The set
of equations (47)-(48) have been extensively analyzed in the literature for
the Kerr metric [11, 38]. Remarkably, this set of orbits does not depend on
the functions $A_{1}$, $B_{1}$ and $A_{4}$.

###  IV.3 Shadows

The BH shadow edge in the spacetime (37) is determined by the set of SPOs
discussed above. To analyze the former, it is important to obtain the
components of the light ray’s 4-momentum $p^{\mu}$, as seen by an observer in
a local ZAMO (zero angular momentum observer) frame (see discussion in Ref.
[20]). In the following expressions, all quantities are computed at the
observer’s location:

$\displaystyle
p^{(t)}=\sqrt{\frac{g_{\varphi\varphi}}{g^{2}_{t\varphi}-g_{tt}g_{\varphi\varphi}}}\left(E+\frac{g_{t\varphi}}{g_{\varphi\varphi}}j\right),$
(50) $\displaystyle
p^{(R)}=\frac{p_{R}}{\sqrt{g_{RR}}}=\sqrt{g_{RR}}\,\dot{R},$ (51)
$\displaystyle
p^{(\theta)}=\frac{p_{\theta}}{\sqrt{g_{\theta\theta}}}=\sqrt{g_{\theta\theta}}\,\dot{\theta},$
(52) $\displaystyle p^{(\varphi)}=\frac{j}{\sqrt{g_{\varphi\varphi}}}.$ (53)

One generically requires two observation angles $\\{\alpha,\beta\\}$, measured
in the observer’s frame, to fully specify the detection direction of a light
ray in the local sky. These angles are defined by the ratio of the different
components of the light ray’s four momentum in the observer frame. Following
[20], we can define the angles $\alpha,\beta$ to satisfy:

$\sin\alpha=\frac{p^{(\theta)}}{p^{(t)}}\
,\qquad\tan\beta=\frac{p^{(\varphi)}}{p^{(R)}}\,.$ (54)

In addition, one can expect the angular size of an object to decrease like
$\alpha\sim 1/R_{\rm circ}$ in the limit of far away observers, where the
circumferential radius888The quantity $R_{\rm circ}$ is computed by displacing
the observer to the equatorial plane ($\theta=\pi/2$), while keeping its
coordinate $R$ fixed; $R_{\rm circ}=\sqrt{g_{\varphi\varphi}}$ at that new
location. $R_{\rm circ}$ is a measure of the observer’s distance to the
horizon [20, 39]. Given this asymptotic behavior, it is useful to introduce
the impact parameters:

$X=-R_{\rm circ}\beta,\qquad Y=R_{\rm circ}\alpha\,.$ (55)

By construction, these quantities $\\{X,Y\\}$ are expected to have a well
defined limit when taking $R_{\rm circ}\rightarrow\infty$.
The relation between $\\{X,Y\\}$ and the constants of null geodesic motion
$\\{\lambda,\eta\\}$ can be obtained by considering Eqs. (40)-(41) for
$\dot{R}$ and $\dot{\theta}$, and then combining them with Eqs. (50)-(54).

We can compute the shadow edge expression in the limit of far-away observers
($R_{\rm circ}\rightarrow\infty$):

$\displaystyle X=-\frac{\lambda}{\sin\theta_{0}},$ (56) $\displaystyle
Y=\pm\sqrt{\eta+a^{2}\cos^{2}\theta_{0}-\frac{\lambda^{2}}{\tan^{2}\theta_{0}}}\,.$
(57)

The shadow degeneracy occurs when the quantities $\\{X,Y\\}$ coincide in the
non-Kerr and Kerr spacetimes, for an observer located at the same
circumferential radius $R_{\rm circ}$ and polar angle $\theta_{0}$. This will
certainly be the case if the set of SPOs coincides in both Kerr and the non-
Kerr geometry for observers that are very far-away. Thus recalling Eq. (45),
we note that the latter is a sufficient condition for shadow degeneracy at
infinity.

We conclude with the following line element obtained from (37) together with
(45):

$\displaystyle
ds^{2}=-\frac{\left(\Delta-a^{2}\sin^{2}\theta\right)\widetilde{\Sigma}}{\Sigma^{2}}dt^{2}+\frac{\widetilde{\Sigma}}{A_{4}\Delta}dR^{2}+\widetilde{\Sigma}d\theta^{2}$
$\displaystyle-\frac{4amR\sin^{2}\theta\widetilde{\Sigma}}{\Sigma^{2}}dtd\phi$
$\displaystyle+\frac{\left[\left(R^{2}+a^{2}\right)^{2}-a^{2}\Delta\sin^{2}\theta\right]\widetilde{\Sigma}}{\Sigma^{2}}\sin^{2}\theta
d\phi^{2},$ (58)

where

$\displaystyle\Sigma=R^{2}+a^{2}\cos^{2}\theta.$ (59)

This geometry will be shadow degenerate with respect to the Kerr spacetime for
very far-away observers, with very weak constraints on $A_{1},B_{1}$ and
$A_{4}$.

##  V Illustrations of shadow-degeneracy (stationary)

###  V.1 Rotating SV spacetime

An example of a rotating, stationary and axisymmetric BH with a Kerr
degenerate shadow, can be obtained by applying the method proposed in [19] to
the static SV geometry.999As this paper was being completed a rotating version
of the SV spacetime was independently constructed, using the NJA, in [40, 41].
This method consists on a variation of the Newman-Janis algorithm (NJA) [18].
Starting from a static seed metric, we can generate via this modified NJA a
rotating circular spacetime that can be always expressed in Boyer-Lindquist-
like coordinates. This comes in contrast to the standard NJA, for which the
latter is not always possible. However, the method introduces a new unknown
function ($\Psi$ below) that may be fixed by additional (physical) arguments,
for instance, interpreting the component of the stress-energy tensor
$T_{\mu\nu}$ to be those of a fluid [19].

Applying the modified NJA discussed in Ref. [19] to the seed metric (14)
(class I spacetime - SV), we introduce the functions:

$\displaystyle F(r)=\left(1-\frac{2M}{\sqrt{r^{2}+b^{2}}}\right),$ (60)
$\displaystyle K(r)=r^{2}+b^{2}.$ (61)

The latter contains the same functional structure of some of the metric
elements of the seed metric (14). Combining these functions $F,K$ with a
complex change of coordinates (which introduces a parameter $a$), we obtain a
possible rotating version of the SV metric (14), namely:

$\displaystyle
ds^{2}=-\frac{\left(FK+a^{2}\cos^{2}\theta\right)}{\left(K+a^{2}\cos^{2}\theta\right)^{2}}\Psi
dt^{2}+\frac{\Psi}{FK+a^{2}}dr^{2}$
$\displaystyle-2a\sin^{2}\theta\left[\frac{K-FK}{\left(K+a^{2}\cos^{2}\theta\right)^{2}}\right]\Psi
dtd\varphi+\Psi d\theta^{2}$ (62)
$\displaystyle+\Psi\sin^{2}\theta\left[1+a^{2}\sin^{2}\theta\frac{2K-FK+a^{2}\cos^{2}\theta}{\left(K+a^{2}\cos^{2}\theta\right)^{2}}\right]d\varphi^{2},$

where $\Psi$ is an undefined function that can be fixed by an additional
requirement. Assuming a matter-source content of a rotating fluid, we can
check that setting $\Psi=r^{2}+a^{2}\cos^{2}\theta+b^{2}$ leads to an Einstein
tensor that satisfies the suitable fluid equations detailed in Ref. [19]. We
may then rewrite the line element (62) in terms of the radial coordinate
$R=\sqrt{r^{2}+b^{2}}$, which leads to the more compact form:

$\displaystyle
ds^{2}=-\left[1-\frac{2\,M\,R}{\Sigma}\right]\,dt^{2}+\frac{\Sigma}{\Delta}\,\frac{dR^{2}}{B_{\rm
SV}(R)}$
$\displaystyle-\frac{4\,M\,a\,R\,\sin^{2}\theta}{\Sigma}\,dt\,d\varphi+\Sigma\,d\theta^{2}+$
$\displaystyle+\Sigma\left[1+\frac{a^{2}\,\sin^{2}\theta}{\Sigma^{2}}\left(\Sigma+2\,M\,R\right)\right]\,\sin^{2}\theta\,d\varphi^{2},$
(63)

where

$\displaystyle\Sigma=R^{2}+a^{2}\,\cos^{2}\theta,$ (64)
$\displaystyle\Delta=R^{2}-2\,M\,R+a^{2}.$ (65)

For simplicity, we shall designate the geometry (63) as the rotating SV
spacetime. Curiously, the line element (63) is precisely the Kerr one, except
for the extra factor $B_{\rm SV}(R)$ in the $g_{RR}$ component. For $a=0$, the
SV line element (17) is recovered; for $b=0$ we recover the Kerr metric in
Boyer-Lindquist coordinates.

An important remark is in order. Depending on the coordinate choice of the
seed metric (14), the latter might be mapped to a different final geometry.
For instance, had we applied the modified NJA to the seed SV metric in the
coordinates of Eq. (17) rather than those of Eq. (14), then we would have
obtained a rotating spacetime different than that of (63).

Let us now examine some of the properties of the spinning SV geometry (63).

####  V.1.1 Singularities and horizons

The line element (63) presents singularities at:

$\displaystyle R_{\pm}=M\pm\sqrt{M^{2}-a^{2}},$ (66) $\displaystyle R_{t}=b.$
(67)

These are coordinate singularities and the spacetime is regular everywhere. In
particular (66) are Killing horizons that only exist if $R_{\pm}>R_{t}$, or

$\displaystyle M\pm\sqrt{M^{2}-a^{2}}\geqslant b.$ (68)

Adopting the positive sign in Eq. (68), a BH only exists if
$M+\sqrt{M^{2}-a^{2}}>b$ and $a<M$; for $M+\sqrt{M^{2}-a^{2}}=b$ the geometry
describes a wormhole with a throat, which can be nulllike, spacelike or
timelike, depending on the value of $a$ and $b$ [40]. These singularities
$R_{\pm}$ can be removed by writing the line element in Eddington-Finkelstein-
like coordinates. On the other hand, the singularity $R_{t}$ can be removed by
writing the line element in the coordinates given by Eq. (62), in which
$R_{t}$ corresponds to the radial coordinate $r=0$.

In order to inquire if the geometry (63) is regular everywhere, let us
consider some curvature invariants: the Kretschmann scalar
($K=R_{\mu\,\nu\,\beta\,\rho}\,R^{\mu\,\nu\,\beta\,\rho}$), the Ricci scalar
($R^{\mu\,\nu}\,g_{\mu\,\nu}$), and the squared Ricci tensor
($R_{\mu\,\nu}\,R^{\mu\,\nu}$). The full expressions of the curvature
invariants are too large to write down. However, we can write them in a
compact form as:

$\displaystyle K=\frac{\mathcal{P}(R,\theta)}{2R^{6}\Sigma^{6}},$ (69)
$\displaystyle
R^{\mu\nu}g_{\mu\nu}=\frac{b^{2}\mathcal{Q}(R,\theta)}{R^{3}\Sigma^{3}},$ (70)
$\displaystyle
R^{\mu\nu}R_{\mu\nu}=\frac{b^{4}\mathcal{S}(R,\theta)}{2R^{6}\Sigma^{6}},$
(71)

where $\mathcal{P}(R,\theta),\mathcal{Q}(R,\theta)$ and
$\mathcal{S}(R,\theta)$ are polynomials of powers of $R$, $\sin\theta$ and
$\cos\theta$. We observe that the curvature invariants are all finite in the
range of the radial coordinate $R$ ($b<R<\infty$), for $b\neq 0$. The Carter-
Penrose diagram, as well as further properties of this rotating SV geometry
can be found in Ref. [40].

####  V.1.2 Shadow and lensing

The rotating SV geometry is a particular case of Eq. (58), since

$A_{1}=1,\quad B_{1}=1,\quad A_{4}=B_{SV}(R).$ (72)

Hence, this BH geometry is shadow degenerate with Kerr spacetime, regardless
of $b$. A plot of the shadow edge is presented in Fig. 6 (top panel) for the
rotating SV spacetime with different values of $b$. As was previously
mentioned, the shadow does not depend on the parameter $b$. Notwithstanding,
the dependence on $b$ through $B_{\rm SV}(R)$ has a subtle impact on the
gravitational lensing, as can be seen in Fig. 7, where we show the shadow and
gravitational lensing of the rotating SV spacetime, obtained using backwards
ray-tracing.

We remark that applying the modified NJA to the spherically symmetric and
static geometries presented in Sec. II does not generically result in a Kerr
degenerate shadow geometry. In particular, if one applies the modified NJA to
the class II shadow degenerate example (22), the resulting rotating BH
geometry is not shadow degenerate.

(a) Rotating SV (b) Rotating BH without $\mathbb{Z}_{2}$ symmetry

Figure 6: Panel a: shadow of the rotating SV spacetime, for different values
of the parameter $b$ and $a=0.999M$. Panel b: Shadow of a BH without the
$\mathbb{Z}_{2}$ symmetry, for different values of $\alpha_{i}$ and the same
value of the BH spin ($a=0.7M$). The observer is located at spatial infinity.

Figure 7: The shadow and gravitational lensing for the rotating SV spacetime,
for $a=0.999M$ and different values of $b$. In the top panel, we have the Kerr
spacetime ($b=0$). In the middle and bottom panels, we have $b=0.5M$ and
$b=M$, respectively. The observer position and angle of view are the same as
in Figs. 2 and 5. Although the images look the same, there is a subtle
difference in the gravitational lensing observed mainly next to the shadow
edge.

###  V.2 Black holes without $\mathbb{Z}_{2}$ symmetry

It has been pointed out that the BH shadow is not a probe of the event horizon
geometry [9]. As a second example of shadow-degeneracy in rotating, stationary
and axisymmetric BHs, allowing separability of the HJ equation, we shall
provide a sharp example of the previous statement. We shall show that a
rotating, stationary and axisymmetric BH without $\mathbb{Z}_{2}$ symmetry
(i.e. without a north-south hemispheres discrete invariance) can, nonetheless,
be shadow degenerate with the Kerr BH.

Geometries within the family (58) are $\mathbb{Z}_{2}$ symmetric if invariant
under the transformation

$\theta\rightarrow\pi-\theta,$ (73)

which maps the north and south hemispheres into one another. The Kerr [5],
Kerr-Newman [42], Kerr-Sen [43] and the rotating SV spacetimes are examples of
BHs with $\mathbb{Z}_{2}$ symmetry. But BHs without $\mathbb{Z}_{2}$ symmetry
are also known in the literature. One example was constructed in Ref. [44];
this property was explicitly discussed in in Ref. [45], where the
corresponding shadows were also studied.

The general line element displaying shadow degeneracy, under the assumptions
discussed above is given in Eq. (58). It has a dependence on the $\theta$
coordinate through the function $B_{1}$ [see Eq. (33)]. $B_{1}(\theta)$ needs
not be invariant under $\mathbb{Z}_{2}$ reflection:

$B_{1}(\theta)\neq B_{1}(\pi-\theta)\ ;$ (74)

then, the BH geometry is shadow degenerate and displays no $\mathbb{Z}_{2}$
symmetry.

In order to provide a concrete example, $B_{1}(\theta)$ can be chosen to be

$\cos^{2}\theta
B_{1}(\theta)=\alpha_{0}+\alpha_{1}\cos\theta+\alpha_{2}\cos^{2}\theta+\alpha_{3}\cos^{3}\theta,$
(75)

while $A_{1}$ and $A_{4}$ are left unconstrained. In Eq. (75), $\alpha_{0}$,
$\alpha_{1}$ and $\alpha_{2}$ are constant parameters for which
$\widetilde{\Sigma}>0$. If

$\alpha_{0}=\alpha_{1}=\alpha_{3}=0,\quad\text{and}\quad\alpha_{2}=1,$ (76)

we recover the Kerr metric (provided that $A_{1}=A_{4}=1$). In Fig. 8, we show
the Euclidean embedding of the event horizon geometry of this BH spacetime for
$a/m=0.7$.101010For higher values of the spin parameter $a/m$, it may be
impossible to globally embed the spatial sections of the event horizon
geometry in Euclidean 3-space. This is also the case for the Kerr spacetime
[46] — see also [47]. We also show the corresponding result for the Kerr BH
[46] (top panel). In the middle panel, we have chosen the constants
$(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})=(0,\ -2.6,\ 1,\ 0)$, while in
the bottom panel we have chosen
$(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})=(0,\ -4.2,\ 1,\ 2)$. We note
that the $\mathbb{Z}_{2}$ symmetry is clearly violated. Nonetheless, the
shadow is always degenerate with the Kerr BH one, as can be seen in the bottom
panel of Fig. 6.

Figure 8: Euclidean embedding of the (spatial sections of the) event horizon
geometry for the shadow degenerate BHs without $\mathbb{Z}_{2}$ symmetry. In
the top panel, we show the Kerr result, while in the middle and bottom panels
we have chosen $(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})=(0,\ -2.6,\ 1,\
0)$ and $(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})=(0,\ -4.2,\ 1,\ 2)$,
respectively. $a/m=0.7$ for all cases.

##  VI Conclusions

The imaging of the M87 supermassive BH by the Event Horizon Telescope
collaboration [2, 3, 4] has established the effectiveness of very large
baseline interferometry to probe the strong gravity region around
astrophysical BHs. In due time, one may expect more BHs, and with smaller
masses, will be imaged, increasing the statistical significance of the data.
It is then expected these data can be used to test the Kerr hypothesis and
even general relativity, by constraining alternative gravity models. It is
therefore timely to investigate theoretical issues related to the propagation
of light near BHs and, in particular, the properties of BH shadows [8, 39,
48].

In this paper, we investigated the issue of degeneracy: under which
circumstances, two different BH spacetimes cast exactly the same shadow? We
have established generic conditions for equilibrium BHs, both static and
stationary to be shadow degenerate with the Schwarzschild or Kerr geometry,
albeit in the latter case under the restrictive (but plausible) assumption
that precise shadow degeneracy occurs for spacetimes admitting the
separability of the HJ equation and the same SPO structure as Kerr. We have
provided illustrative examples, in both cases, which highlight, among other
properties, that exact shadow degeneracy is not, in general, accompanied by
lensing degeneracy, and that shadow degeneracy can occur even for
qualitatively distinct horizon geometries, for instance, with and without
north-south symmetry.

The examples herein are mostly of academic interest, as a means to understand
light bending near compact objects. These examples could be made more
realistic by considering general relativistic magneto-hydrodynamic simulations
in the BH backgrounds studied here, to gauge the extent to which such precise
shadow degeneracy survives a more realistic astrophysical setup.

Finally, we remark there is also a trivial instance of shadow degeneracy when
the same geometry solves different models. This is not the case discussed
herein.

###### Acknowledgements.

The authors thank Fundação Amazônia de Amparo a Estudos e Pesquisas (FAPESPA),
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) - Finance
Code 001, in Brazil, for partial financial support. This work is supported by
the Center for Research and Development in Mathematics and Applications
(CIDMA) through the Portuguese Foundation for Science and Technology (FCT -
Fundação para a Ciência e a Tecnologia), references No. BIPD/UI97/7484/2020,
No. UIDB/04106/2020 and No. UIDP/04106/2020. We acknowledge support from the
projects No. PTDC/FIS-OUT/28407/2017, No. CERN/FIS-PAR/0027/2019 and No.
PTDC/FIS-AST/3041/2020. This work has further been supported by the European
Union’s Horizon 2020 research and innovation (RISE) programme No. H2020-MSCA-
RISE-2017 Grant No. FunFiCO-777740. The authors would like to acknowledge
networking support by the COST Action No. CA16104.

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