# Conic Section on the Sky: Shadows of Linearly Superrotated Black Holes

Feng-Li Lin<EMAIL_ADDRESS>Department of Physics, National Taiwan
Normal University, Taipei, Taiwan 11676  Avani Patel<EMAIL_ADDRESS>Department of Physics, National Central University, Taoyuan, Taiwan 320317
Institute of Physics, Academia Sinica, Taipei, Taiwan 11529  Department of
Physics, National Taiwan Normal University, Taipei, Taiwan 11676  Jason Payne
<EMAIL_ADDRESS>Department of Physics, National Taiwan Normal
University, Taipei, Taiwan 11676

###### Abstract

Soft hairs are intrinsically infrared features of a black hole, which may also
affect near-horizon physics. In this work, we study one of the primary
observables through which we can study the effects of a particular type of
soft hair in Einstein’s gravity: the shadow of a non-rotating black hole
implanted with linear regular superrotational hair. We show that such soft
hairs deform the shadow from the characteristic circular shape of bald
Schwarzschild black holes into ellipses. Such a unique effect can be observed
in future black hole imaging projects. This starkly contrasts with the
supertranslated counterpart, for which the only changes are the static shift
of the circular shadow’s position and thus cannot be detectable. Our results
demonstrate the richness of observable effects due to the infrared structures
of Einstein’s gravity.

Introduction. The discovery of Hawking radiation Hawking (1975, 1974)
supported Bekenstein’s proposal Bekenstein (1972, 1973) that black holes
behave as thermal systems and, at the same time, asked for the microscopic
origin of the area-law entropy of a generic black hole. Amongst many
possibilities, a promising one is to attribute them to some degrees of freedom
living in the near-horizon region that does not violate the no-hair theorem,
i.e., do not affect asymptotic quantities such as mass, charge, and angular
momentum. Within the context of Einstein’s gravity, such a possibility can be
realized by soft hairs Hawking  _et al._ (2017), which can be thought of as
the conserved charges associated with asymptotic Bondi-Metzner-Sachs (BMS)
symmetries Bondi  _et al._ (1962); Sachs (1962a, b). Indeed, this has been
recently proposed by Hawking, Perry, and Strominger as a way of resolving the
information paradox Hawking  _et al._ (2016). As there is no need to introduce
ingredients beyond Einstein’s gravity, this resolution is the most minimal and
thus reasonable one, according to Occam’s razor. If so, it raises the question
of detecting these soft hairs.

Although the soft hairs do not change a black hole’s mass, charge, and angular
momentum, they will change its horizon structure. Therefore, some observables
relevant to the horizon structure may be tools for detecting soft hairs. One
such observable is the black hole’s shadow formed by the marginally escaping
light rays reaching a celestial observer. Indeed, the shadows of supermassive
black holes such as the one located at the center of galaxy M87 have been
observed by Event Horizon Telescope (EHT) et al. (2019a, b, c, d, e, f). The
shadow is closely related to the so-called light rings Synge (1966); Bardeen
(1973) or fundamental photon orbits Cunha  _et al._ (2017); Cunha and Herdeiro
(2020), which are unstable critical closed photon orbits around the black hole
and serve as the dividing boundary between the escaping and in-falling light
rays. The shadows of black holes with supertranslation hairs Hawking  _et al._
(2017); Compère and Long (2016) have been studied by both analytical Lin  _et
al._ (2022); Zhu  _et al._ (2023) and numerical ray-tracing methods Lin  _et
al._ (2022), and the results show that the effect of supertranslation hairs is
just a shift in the center of the shadows of Schwarzschild black holes, but it
does not change their shapes. This implies no observable imprint of
supertranslation hairs on the black hole’s shadow.

Besides supertranslation hairs, a black hole can also be dressed by
superrotation hairs Hawking  _et al._ (2017, 2016), which are associated with
the recently discovered superrotation symmetries arising as an extension of
BMS symmetries to conformal symmetries of the celestial sphere Barnich and
Troessaert (2010a, b, 2011); Cachazo and Strominger (2014); Kapec  _et al._
(2014); Campiglia and Laddha (2015). In this work, we will study the effect of
superrotation hairs on the black hole shadow and show that they can be
observable. There are some subtleties, however, to this problem. First, no
exact solution exists for superrotated black holes, unlike the supertranslated
one found in Compère and Long (2016). Thus, we can only study the black holes
with linear superrotation hairs implemented by performing linear BMS
transformations on a bald Schwarzschild black hole background Strominger
(2017). Second, the resultant metrics from the above procedure are generally
time-dependent and possess seemingly pathological horizon structures. This
will yield difficulties when constructing the deformed light rings and, thus,
the black hole shadows. However, there is a time-independent and pathology-
free subset, which we denote as the linear regular superrotation hairs. In
this paper, we will study the shadow of black holes dressed by them. Our
results demonstrate that the linear regular superrotation hairs deform the
circular shadow of a bald Schwarzschild black hole into elliptic shapes with
their detailed characteristics determined by the superrotation hairs. We argue
that this kind of elliptical shadow is a unique feature of the superrotation
hairs and can be detected in future EHT-like observations to confirm the
infrared structure of gravity.

Black holes with linear (regular) superrotation hairs. To consider the effect
of implanting soft hair on a black hole’s shadow, we shall first construct the
metric of the corresponding black hole. Since soft hair is associated with the
asymptotic isometries of asymptotically flat spacetimes, extending these
symmetries into the interior, or even when the spacetime possesses a black
hole, is not straightforward. As noted in Hawking  _et al._ (2016, 2017), this
extension generally depends on a gauge choice. It is, however, useful for
obtaining an intuitive picture to considering the soft hairs as a leading
order perturbation around a Schwarzschild black hole. This can be done by
implementing the linear soft hairs via large gauge (BMS) transformations on a
bald Schwarzschild metric in the Bondi frame. For the supertranslated case,
the resultant metric agrees with the expansion of the full supertranslated
black hole metric Compère and Long (2016); Lin and Takeuchi (2020); Lin  _et
al._ (2022); Zhu  _et al._ (2023) up to the first order. In contrast, there is
no construction for the full metric of a superrotated black hole. In this
work, we will consider only a linearly superrotated black hole.

We begin with the Schwarzschild metric in the Bondi frame for a bald black
hole without soft hair,

$\bar{g}_{\rho\sigma}dx^{\rho}dx^{\sigma}=-\left(1-\frac{2M}{r}\right)dv^{2}+2dvdr+r^{2}\gamma_{AB}d\Theta^{A}d\Theta^{B}\,,$
(1)

where $\Theta^{A}=(\theta,\phi)$ and $\gamma_{AB}$ is the metric on the unit
2-sphere. Below, all barred quantities are associated with this bald metric
$\bar{g}_{\rho\sigma}$. Linear superrotation hairs can then be implemented by
acting on (1) with the Lie derivative

$\bar{g}_{\rho\sigma}\rightarrow\bar{g}_{\rho\sigma}+\epsilon\mathcal{L}_{\zeta_{Y}}\bar{g}_{\rho\sigma}$
(2)

along the following superrotation vector Strominger (2017); Barnich and
Troessaert (2010a, 2011),

$\zeta_{Y}=\frac{D\cdot
Y}{2}\big{[}(v-r)\partial_{r}+v\partial_{v}\big{]}+\big{[}Y^{A}+\frac{v}{2r}D^{A}D\cdot
Y\big{]}\partial_{A}$ (3)

where $Y^{A}=Y^{A}(\theta,\phi)$ are the functions characterizing the
superrotation hairs, and $D\cdot Y=D_{A}Y^{A}$ where the covariant derivative
$D_{A}$ is defined w.r.t. the unit-sphere metric $\gamma_{AB}$. In addition, a
small parameter $\epsilon$ is introduced in (2) to indicate the size of the
soft hair. Conventionally we will normalize $Y^{A}$ with unity maximal
amplitude. The resultant metric (2) contains terms subleading in the large $r$
expansion used in deriving the BMS symmetries, which however, can be gauged
away Bondi  _et al._ (1962); Sachs (1962a, b). Keeping only the leading terms,
we arrive at the metric of a non-rotating black hole dressed by linear
superrotation hair as follows Strominger (2017):

$\displaystyle g_{\rho\sigma}dx^{\rho}dx^{\sigma}$ $\displaystyle=$
$\displaystyle\left[-\left(1-\frac{2M}{r}\right)+\frac{3M}{r}\epsilon D\cdot
Y\right]dv^{2}$ (4)
$\displaystyle+2dvdr+r^{2}\left(\gamma_{AB}+\epsilon\kappa_{AB}\right)d\Theta^{A}d\Theta^{B}$

where

$\kappa_{AB}=-\left(1-\frac{v}{r}\right)D\cdot
Y\gamma_{AB}+2D_{(A|}\left[Y^{C}+\frac{v}{2r}\left(D^{C}D\cdot
Y\right)\right]\gamma_{C|B)}\;.$ (5)

We see that this metric, in general, depends on the null-time $v$ through
$\kappa_{AB}$.

We now examine the horizon structures of (4) and any possible associated
pathologies. First, the Killing horizon of (4), which is defined by the zero
norm of the time-like Killing vector
$U^{\rho}=\delta^{\rho}_{v}+\epsilon\mathcal{L}_{\zeta_{Y}}\delta^{\rho}_{v}$,
is still at $r=2M$ up to ${\cal O}(\epsilon)$. Next, the event horizon
$r=r_{\rm H}$ defined by $g_{vv}=0$ is $r_{\rm H}=2M+3M\epsilon D\cdot Y$.
Hence, weak cosmic censorship (WCC) requires

$3\epsilon\;|D\cdot Y|\leq 2\;.$ (6)

This is easily satisfied for small $\epsilon$ and trivially for $D\cdot Y=0$.
Besides these, there is an additional coordinate singularity, called
superhorizon $r=r_{\rm SH}$ at which ${\rm det}[g_{\rho\sigma}]=0$, with

$r_{\rm SH}=-\epsilon v\Big{[}\partial_{\theta}D^{\theta}+\cot\theta
D^{\theta}+\partial_{\phi}D^{\phi}+2\Big{]}\big{(}D\cdot Y\big{)}\;.$ (7)

Note $r_{\rm SH}$ grows linearly with $v$ and can yield pathology at large
$|v|$ or at $\theta=0$ or $\pi$ if $D\cdot Y\neq 0$.

The aforementioned WCC and superhorizon pathologies can be lifted
simultaneously if we impose the following constraint

$D\cdot
Y:=\partial_{\theta}Y^{\theta}+\cot\theta\;Y^{\theta}+\partial_{\phi}Y^{\phi}=0\;.$
(8)

This restricts to a subset of the superrotation hairs. Given an arbitrary
function $Y^{\theta}(\theta,\phi)$, we can determine $Y^{\phi}=h(\theta)-\int
d\phi\big{[}\partial_{\theta}Y^{\theta}+\cot\theta\;Y^{\theta}\big{]}$, with
$h(\theta)$ an arbitrary function of $\theta$. We can see that the condition
(8) still retains quite a rich structure of superrotation hairs but avoids the
aforementioned pathology. More importantly, the resultant metric is now
$v$-independent with a deformed $2$-sphere, i.e.,

$g_{\rho\sigma}dx^{\rho}dx^{\sigma}=\bar{g}_{\rho\sigma}dx^{\rho}dx^{\sigma}+2\epsilon
r^{2}D_{(A|}Y^{C}\gamma_{C|B)}d\Theta^{A}d\Theta^{B}\;.$ (9)

We will see that the features deduced from (8) are quite relevant to obtaining
the deformed light rings for use in constructing the shadow. Therefore, in
this paper, we will mainly consider this subset of superrotation hairs, which
we particularly call linear regular superrotation hairs 111Note that this
metric for a linearly superrotating black hole with $D\cdot Y=0$ is different
from the one of a slowly rotating black hole, which takes the following form:
$ds^{2}_{\rm
slowrot}=\bar{g}_{\rho\sigma}dx^{\rho}dx^{\sigma}-2ar^{2}\sin^{2}\theta
dvd\phi\;.$ (10) The superrotation hair term in (9) mainly deforms the
2-sphere. On the other hand, the last term in the above slowly-rotating black
hole metric introduces nonzero $g_{v\phi}$.  to distinguish them from the
pathological ones with $D\cdot Y\neq 0$. However, we will maintain the $D\cdot
Y$ dependence either implicitly to avoid complicated expressions or explicitly
to discuss some physical issues.

Null geodesics, light rings, and fundamental photon orbits. The shadow of a
black hole arises from the critical circular photon orbits, below which the
light will fall into the black hole and not be observed. For Schwarzschild
black holes, these critical photon orbits are planar and called light rings
(LRs) Bardeen (1973). For more generic black holes such as Kerr black holes or
(4), they can be non-planar and non-circular, and are called fundamental
photon orbits (FPOs) Cunha  _et al._ (2017); Cunha and Herdeiro (2020). Given
the metric (4), one can directly adopt ray-tracing numerical codes to
determine the photon orbits and consequently the shadow. However, as the
metrics (4) or (9) are small perturbations away from the bald metric
$\bar{g}_{\mu\nu}$, we may determine the FPOs and the shadows perturbatively.
This will help to yield an analytical expression for the shapes of shadows.

We start with a photon orbit in the bald metric, which is expressed by the
Hamilton-Jacobi method in terms of the canonical momentum, denoted by
$\bar{p}^{\mu}$. The latter is determined by $x^{\mu}$ and the constants of
motion, i.e.,

$\bar{p}^{\mu}=\bar{p}^{\mu}(x^{\mu};E,\lambda,\kappa)$ (11)

where $E$, $\lambda$, and $\kappa$ are the constants of motion corresponding
to the energy, reduced azimuthal angular momentum, and the Carter constant,
respectively. Once the superrotation hair turns on, the geodesic equations in
the new Bondi frame defined by (4) can be obtained by an active coordinate
transformation, i.e.,

$p^{\mu}(x^{\mu};E,\lambda,\kappa)=\bar{p}^{\mu}(x^{\mu};E,\lambda,\kappa)+\epsilon{\cal
L}_{\zeta_{Y}}\bar{p}^{\mu}(x^{\mu};E,\lambda,\kappa)\;.$ (12)

Explicitly, they are

$\displaystyle p^{v}$ $\displaystyle=$ $\displaystyle
E\Big{(}\frac{r}{r-2M}\Big{)}\Big{(}1+{\cal R}\Big{)}+\cdots\;,$ (13)
$\displaystyle p^{r}$ $\displaystyle=$ $\displaystyle E{\cal R}+\cdots\;,$
(14) $\displaystyle p^{\theta}$ $\displaystyle=$
$\displaystyle\frac{E}{r^{2}}{\cal K}+\cdots$ (15) $\displaystyle+$
$\displaystyle\epsilon\frac{E}{r^{2}}\Big{[}\csc^{2}\theta\;\big{(}\lambda^{2}\frac{\cot\theta}{{\cal
K}}Y^{\theta}-\lambda\partial_{\phi}Y^{\theta}\big{)}-{\cal
K}\partial_{\theta}Y^{\theta}\Big{]}\;,$ $\displaystyle p^{\phi}$
$\displaystyle=$ $\displaystyle\frac{E\lambda}{r^{2}\sin^{2}\theta}+\cdots$
(16) $\displaystyle+$
$\displaystyle\epsilon\frac{E}{r^{2}}\Big{[}\lambda\csc^{2}\theta\big{(}2\cot\theta\;Y^{\theta}+\lambda\partial_{\phi}Y^{\phi}\big{)}+{\cal
K}\partial_{\theta}Y^{\phi}\Big{]}\;,$

where in each equation the first term corresponds to that of the bald metric,
while $\cdots$ denotes the ${\cal O}(\epsilon)$ terms proportional to $D\cdot
Y$ (whose detailed forms are given in the supplementary material), and

${\cal R}:=\sqrt{\frac{r^{3}-r\kappa+2M\kappa}{r^{3}}},\qquad{\cal
K}:=\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}\;.$ (17)

Note that the hairy metric $g_{\mu\nu}$ possesses no explicit isometry once
the superrotation hairs are turned on. Thus, the above constants of motion are
inherited from the photon orbits in the bald metric. We can then view the
photon orbits given by (13)-(16) as deformations of the bald orbits which are
still labeled by the inherited constants of motion. In contrast, if we take
the passive approach to consider the geodesic equation in the new frame, we
will not have the inherited constants of motion to label the photon orbits,
which we later use to construct the shadows analytically.

We first note that if $D\cdot Y=0$, the $\nu$\- and $r$-components of the
photon orbit given in (13) and (14) are the same as the ones in bald metric.
This implies that the FPO defined by $\dot{r}=\ddot{r}=0$ (or
$p^{r}=\dot{p^{r}}=0$ where the over-dot denotes the derivative with respect
to the worldline time) has the same radius $r=3M$ as the light ring in the
bald metric, with $\kappa=\kappa_{\rm FPO}:=27M^{2}$. However, the FPO has
more complicated dynamics along $\theta$\- and $\phi$-directions, i.e., it no
longer lies in the equatorial plane. It can be seen as the deformed light ring
obtained from the corresponding one in the bald metric and labeled by $E$,
$\lambda$, and $\kappa=\kappa_{\rm FPO}$. The explicit form of the FPO can
also be obtained by shifting the light ring in the bald metric, i.e.,
$\bar{x}^{\mu}_{\rm LR}=(3Et,3M,\frac{\pi}{2},-\frac{E\lambda}{9M^{2}}t)$
parameterized by the worldline time $t$, with the superrotation vector
$-\epsilon\zeta^{\mu}$ of (3). Explicitly,

$\bar{x}^{\mu}_{\rm LR}\rightarrow\bar{x}^{\mu}_{\rm
LR}-\Big{(}\cdots,\cdots,\epsilon Y^{\theta}(\bar{x}^{\mu}_{\rm
LR})+\cdots,\epsilon Y^{\phi}(\bar{x}^{\mu}_{\rm LR})+\cdots\Big{)}\;,$ (18)

where $\cdots$ again represents the ${\cal O}(\epsilon)$ terms proportional to
$D\cdot Y$, which can be ignored for the current discussion. We indeed see
that the FPO has a radius of $3M$ but with nontrivial dynamics along
$\theta$\- and $\phi$-directions through $\phi(t)=-\frac{E\lambda}{9M^{2}}t$
contained in $Y^{A}(\bar{x}^{\mu}_{\rm LR})$ 222Similar correspondence can be
found in the case of linear supertranslation hairs characterized by the
coordinate transformation:  $x^{\mu}\rightarrow
x^{\mu}-\epsilon\xi_{f}\qquad\mbox{with}\qquad\xi_{f}:=f\partial_{v}-{1\over
2}D^{2}f\partial_{r}+{1\over r}D^{A}f\partial_{A}\;,$ (19) for a arbitrary
function $f=f(\theta,\phi)$. If $f=f(\theta)$, then $E$ and $\lambda$ are
still constants of motion for photon orbit in the hairy background metric with
supertranslation hairs, and the radius of the light ring can be obtained by
solving the geodesic equations and null condition
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$. The result is Sarkar  _et al._
(2022) $r_{\rm LR}=3M+\frac{\epsilon}{2}f^{\prime\prime}(\frac{\pi}{2})\;.$
(20) This matches with $3M-\epsilon\xi^{r}_{f}\big{|}_{x=\bar{x}^{\mu}_{\rm
LR}}$ from the linear supertranslation of $\bar{x}^{\mu}_{\rm
LR}=(3Et,3M,\frac{\pi}{2},-\frac{E\lambda}{9M^{2}}t)$ of the bald metric light
ring.. For generic $Y^{A}$ satisfying $D\cdot Y=0$, the light ring wobbles
about the equatorial light ring of the bald metric 333Note that the
integration constant function $h(\theta)$ obtained by solving $D\cdot Y=0$ for
$Y^{\phi}$ will not affect the FPO as it just shifts $\bar{\phi}_{\rm LR}(t)$
by a constant $h(\tfrac{\pi}{2})$., as shown in Fig. 1.

Figure 1: A typical deformed light ring/fundamental photon orbit (blue curve)
for a non-rotating black hole with linear regular soft hair given by
$\epsilon=0.5$, $Y^{\theta}=\sin{\theta\over 2}\cos{\phi\over 2}$ and where
$Y^{\phi}$ has been solved from $D\cdot Y=0$. This deformed light ring wobbles
around the (red) circular light ring of the bald metric, and both are labeled
by the same set of constants of motion, especially $\kappa=\kappa_{\rm
FPO}:=27M^{2}$.

Although we will not consider the pathological cases with $D\cdot Y\neq 0$, we
would like to highlight some difficulties with constructing the corresponding
FPOs. Firstly, FPOs are unstable so they can serve as the boundary separating
the unbound orbits from the orbits ending at the black hole curvature
singularity. This instability of the FPO is usually characterized by a saddle
point of the effective radial potential of the reduced null geodesic
Lagrangian. However, the superrotated metric with $D\cdot Y\neq 0$ lacks
Killing symmetries along $v$\- and $\phi$-directions, so it is impossible to
construct the reduced radial action. This prevents the stability of FPOs from
being checked analytically. Secondly, even ignoring the stability issue and
obtaining the deformed light rings by (18) with the $D\cdot Y$ terms included,
the resultant deformed light rings are, in general, time-dependent open
orbits. This is in accordance with the aforementioned no-go in determining the
FPOs by the reduced radial potential. However, all of the above can be lifted
for $D\cdot Y=0$ cases, as discussed above.

Even though there are difficulties in constructing FPOs for the pathological
$D\cdot Y\neq 0$ cases, the construction of shadows can still proceed by
assuming that the FPOs carry the same inherited constants of motion from the
light ring of the bald metric. Therefore, we will keep the $D\cdot Y$ terms
when constructing the shadow and then set $D\cdot Y$ to zero in the resultant
shadow equation. In this way, we can see how the $D\cdot Y$ terms affect the
characteristics of the shadow under the assumption that the deformed light
rings are indeed FPOs.

Conic shapes of the shadows. We now derive the analytic form of the shadow of
the black holes with linear superrotation hairs. We will only consider shadows
measured by a distant observer for the sake of simplicity. In such cases, the
shadows on the observer’s celestial plane with coordinates $(\alpha,\beta)$
are described by Synge (1966); Bardeen (1973); Cunningham and Bardeen (1972)

$\alpha=-r_{O}\sin{\theta}_{O}\left.\frac{p^{\phi}}{p^{r}}\right|_{x^{\mu}_{O}}\;,\qquad\beta=r_{O}\left.\frac{p^{\theta}}{p^{r}}\right|_{x^{\mu}_{O}}$
(21)

where $x^{\mu}_{O}=(\nu_{O},r_{O},\theta_{O},\phi_{O})$ is the observer’s
coordinate in the Bondi frame, and $p^{\mu}$ is given by (12) with the
inherited constants of motion therein equal to the values of some FPO (or
deformed light ring). The shadow equation (21) is constructed by ray-tracing a
null geodesic from $x^{\mu}_{O}$ to a region near some FPO labeled by the
inherited constants of motion $\lambda_{FPO}$ and $\kappa_{FPO}$. If the ray-
traced null geodesic can be merged with the given FPO, it is also labeled by
the same inherited constants of motion. The orbits position $(\alpha,\beta)$
on the celestial plane then delineates the boundary of the shadow 444Due to
the undetermined FPOs for $D\cdot Y\neq 0$ cases, we may try to obtain the
shadows directly by the numerical ray-tracing methods. However, as our
background metric is only valid up to ${\cal O}(\epsilon)$, it is hard to
control the contributions to the shape of the shadow within this order when
implementing the numerical calculations. In contrast, the shadows constructed
in this paper are analytical up to ${\cal O}(\epsilon)$..

Plugging the expressions for $p^{\mu}$ given by (12) into (21), after some
algebraic manipulation, and keeping only the ${\cal O}(\epsilon)$ terms, the
two shadow equations can be put into the following symbolic forms,
respectively

$\displaystyle r_{O}\alpha$ $\displaystyle=$ $\displaystyle\epsilon
A_{0}+(-1+\epsilon A_{1})\lambda_{\rm FPO}\csc\theta_{O}+\epsilon A_{2}{\cal
K}\;,$ (22) $\displaystyle r_{O}\beta$ $\displaystyle=$ $\displaystyle\epsilon
B_{0}{\lambda^{2}_{\rm FPO}\csc^{2}\theta_{O}\over{\cal
K}}+\epsilon(B_{1}\lambda_{\rm FPO}\csc\theta_{O}+B_{2})$ (23)
$\displaystyle+\;(1+\epsilon B_{3}){\cal K}\;,\qquad$

where $\cal K$ is given by (17) with $\kappa=\kappa_{\rm FPO}$, and the
$A_{i}$’s and $B_{i}$’s are independent of both $\lambda_{\rm FPO}$ and
$\kappa_{\rm FPO}$ and are given by

$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle 0+\cdots\;,$ (24)
$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle-\partial_{\phi_{O}}Y^{\phi}-2\partial_{\theta_{O}}Y^{\theta}+\cdots\;,$
(25) $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\sin\theta_{O}\partial_{\theta_{O}}Y^{\phi}\;,$ (26)
$\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle\cot\theta_{O}Y^{\theta}\;,$ (27) $\displaystyle B_{1}$
$\displaystyle=$
$\displaystyle-\csc\theta_{O}\partial_{\phi_{O}}Y^{\theta}\;,$ (28)
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle 0+\cdots\;,$ (29)
$\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle-\partial_{\theta_{O}}Y^{\theta}+\cdots$ (30)

Here $Y^{A}=Y^{A}(\theta_{O},\phi_{O})$ and $\cdots$ again denote the ${\cal
O}(\epsilon)$ terms proportional to $D\cdot Y$ 555Their explicit forms are
given in the supplementary material.. For $\epsilon=0$, (22) and (23) yield a
shadow of circular shape described by
$(r_{O}\alpha)^{2}+(r_{O}\beta)^{2}=\kappa_{\rm FPO}$. Also, note that the
$D\cdot Y$ terms in $A_{0}$ and $B_{2}$ are $v$-dependent, but that of $A_{1}$
is not.

For the cases of nonzero $\epsilon$ and $Y^{A}$, we derive the form of the
shape of the shadow as follows. First, we solve (22) for $\lambda_{\rm
FPO}\csc\theta_{O}$, resulting in

$\lambda_{\rm
FPO}\csc\theta_{O}=-r_{O}\alpha+\epsilon\;\Big{(}A_{0}-A_{1}r_{O}\alpha+A_{2}\sqrt{\kappa_{\rm
FPO}-(r_{O}\alpha)^{2}}\Big{)}\;,$ (31)

which leads to

${\cal K}=\sqrt{\kappa_{\rm FPO}-(r_{O}\alpha)^{2}}+\epsilon
r_{O}\alpha\Big{(}A_{2}+{A_{0}-A_{1}r_{O}\alpha\over\sqrt{\kappa_{\rm
FPO}-(r_{O}\alpha)^{2}}}\Big{)}\;.$ (32)

From there, plugging (31) and (32) into (23) and solving for $\kappa_{\rm
FPO}$, we obtain the shape equation of the shadow as follows (up to ${\cal
O}(\epsilon))$,

$\displaystyle\Big{(}1-2\epsilon(B_{0}-A_{1})\Big{)}(r_{O}\alpha)^{2}+(1-2\epsilon
B_{3})(r_{O}\beta)^{2}$
$\displaystyle-2\epsilon\Big{(}A_{0}r_{O}\alpha+B_{2}r_{O}\beta+(A_{2}-B_{1})r^{2}_{O}\alpha\beta\Big{)}=\kappa_{\rm
FPO}\;.$ (33)

In the above derivation, we have kept only ${\cal O}(\epsilon)$ terms as we
only consider superrotation hair at the linear order 666It is tempting to
include the $\epsilon$-order terms in order to obtain more complicated and
probably interesting shapes for the shadows, as (22) and (23) seems nonlinear
in $\epsilon$. However, the terms of ${\cal O}(\epsilon^{n>1})$ are incomplete
since we only include the superrotation hairs at ${\cal O}(\epsilon)$ in the
metric (4)..

We see that the linear superrotation hair deforms the circular shadow of a
Schwarzschild black hole into a conic section described by (33). In general,
this conic curve is an ellipse or hyperbola and (33) can be put into the
canonical form,

${X^{2}\over\kappa_{\rm FPO}a_{+}}+{Y^{2}\over\kappa_{\rm FPO}a_{-}}=1$ (34)

with

$a_{\pm}=1\pm\epsilon\sqrt{(A_{2}-B_{1})^{2}+(A_{1}-B_{0}+B_{3})^{2}}+\cdots$
(35)

and

$(X,Y)^{T}=R(\Psi)\;(r_{O}\alpha-\epsilon A_{0},r_{O}\beta-\epsilon
B_{2})^{T}$ (36)

where $R(\Psi)$ is a $2\times 2$ rotation matrix of the $v_{O}$-independent
angle $\Psi$ satisfying $\tan 2\Psi=2\epsilon(B_{1}-A_{2})$, and $\cdots$
again denoting the ${\cal O}(\epsilon)$ $D\cdot Y$ term. Given the regular
soft hair functions $Y^{A}$ satisfying $D\cdot Y=0$, the shape of the shadow
depends on the observer’s angular position $(\theta_{O},\phi_{O})$ in the
Bondi frame. Note that $A_{1}-B_{0}+B_{3}$ can be reduced to
$-2\partial_{\theta_{O}}Y^{\theta}$, and similarly
$A_{2}-B_{1}=\csc\theta_{O}\big{(}\partial_{\phi_{O}}Y^{\theta}+\sin^{2}\theta_{O}\partial_{\theta_{O}}Y^{\phi}\big{)}$,
by exploiting $D\cdot Y=0$. In Fig. 2, we show some shadows associated with
the deformed light ring of Fig. 1.

Figure 2: Examples of shadows associated with the deformed light ring of Fig.
1, with $r_{O}$ as the length unit and $(\theta_{O},\phi_{O})=({\pi\over
2},0)$ (blue), $(0,\pi)$ (orange), and $({\pi\over 3},{2\pi\over 3})$ (green).
The dashed red circle is the shadow of a bald Schwarzschild black hole.

Some remarks about this conic section shape of the shadow are in order. First,
the discriminant of (33) for a non-degenerate conic turns out to be
$\Delta=1-4\epsilon D\cdot Y$, which can be used to classify the conics into
ellipses ($\Delta>0$), parabolas ($\Delta=0$), and hyperbolas ($\Delta<0$).
For the black holes with linear regular superrotation hairs considered in this
paper, the shadows take the form of ellipses.

Second, even though the metric (4) depends on the null time $v_{O}$, the
shape, size, and orientation of the shadow are, however, time-independent,
i.e., the lengths of major and minor axes, $\sqrt{\kappa_{\rm FPO}|a_{\pm}|}$,
and $\Psi$ are $v_{O}$-independent. Alternatively, the shape of (33) can be
characterized by its $v_{O}$-independent eccentricity $e$,

$e=\sqrt{1-{a_{-}\over
a_{+}}}=\sqrt{2\epsilon}\;\Big{[}(A_{2}-B_{1})^{2}+(A_{1}-B_{0}+B_{3})^{2}\Big{]}^{1/4}+{\cal
O}(\epsilon^{3/2})\;.$ (37)

On the other hand, the center of the conic curve is located at $(\epsilon
A_{0},\epsilon B_{2})$. Recall that $A_{0}=B_{2}=0$ if $D\cdot Y=0$;
otherwise, they linearly grow with $v_{O}$.

In summary, our results demonstrate the richness of observables due to the
infrared structure of Einstein’s gravity, which leaves imprints of the linear
regular superrotation hair on black hole shadows via the deformation of
circular shapes into conics. An EHT-like project can observe these effects.
This contrasts sharply with the shadow of (linearly) supertranslated holes
considered in Lin and Takeuchi (2020); Zhu  _et al._ (2023), which showed that
the supertranslation hairs only cause static shifts of the shadow’s center
without any shape-changing; thus yielding no observable effect on the shadow
777The methods used in Lin and Takeuchi (2020) and Zhu  _et al._ (2023) are
quite different. In Lin and Takeuchi (2020), a full numerical ray-tracing
method is adopted to obtain the shadow for the fully nonlinear supertranslated
black holes. On the other hand, the method adopted in Zhu  _et al._ (2023) is
the same as the one used in this paper for the linearly supertranslated black
holes.. As for the irregular superrotation hairs with $D\cdot Y\neq 0$, we can
not ascertain the existence of FPOs. Assuming their existence, the resultant
shadow’s center position is $v_{O}$-dependent but not its shape. Also, its
shape can be parabolic or hyperbolic if $D\cdot Y\geq\frac{1}{4\epsilon}$,
i.e., $\Delta\leq 0$.

Discussion. Concerning the detection of the superrotation hair through black
hole shadows, we can ask how unique linear superrotation hairs are to produce
conic shadows. There is a possibility for a small deformation of the shape of
the shadow to be caused by modified gravity beyond Einstein’s theory. However,
in such cases the shadow should have a unique pattern fixed by the given
gravity theory and will be devoid of the richness provided by the arbitrary
soft hair functions considered here. On the other hand, our result seems to
suggest that coordinate transformations may generate detectable changes in the
shadow. In the supplementary material, we construct the shadow for metrics
obtained by the most general linear coordinate transformations of the bald
metric and find that the resultant shape of the shadow can go beyond quadratic
curves. However, this is only true if the corresponding FPO exists. Based on
the uniqueness of black holes and the peculiarity of BMS symmetries for the
soft hairs, we believe there are no FPOs for the general coordinate
transformations. Thus, the regular superrotation hairs considered in this
paper are unique in their simplicity, with their reduced radial dynamics for
FPOs and the resultant elliptic shadows, for future detection.

Acknowledgements. We thank Ling-Wei Luo for his earlier participation in this
project. FLL and JP are supported by Taiwan’s National Science and Technology
Council (NSTC) through Grant No. 109-2112-M-003-007-MY3 and
112-2112-M-003-006-MY3. AP is supported by NSTC grant No.
110-2811-M-003-507-MY2.

##  I References

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Supplementary Material for Conic Section on the Sky: Shadows of Linearly
Superrotated Black Holes

###### Contents

  1. I References
  2. .1 Linearly superrotated black hole: metric, horizons, photon orbits, and tentative shadows
  3. .2 Tentative shadows for the hairy metrics from general linear coordinate transformations

In this supplementary material, we first give the details of the geodesic
equations for the photon orbits of black holes with the generic linear
superrotation hairs and then determine the corresponding shadows by assuming
the existence of FPOs. Continuing with the same assumption, we construct the
shadows for the hairy metric by the most general linear coordinate
transformations acting on the bald Schwarzschild metric. The result shows that
the shape of the shadow can go beyond quadratic curves.

However, we shall caution the readers that the shadows discussed here for the
generic hairs other than the azimuthally symmetric supertranslation hairs and
the linear regular superrotation hairs are only tentative, as the
corresponding FPOs remain elusive. This is due to the lack of enough
isometries required to obtain a reduced radial potential for determining the
(in)stability of photon orbits. Most probably, there are no FPOs for such
hairs and thus there are no associated shadows. Otherwise, it would seem that
the uniqueness of black holes is violated heavily by arbitrary linear
coordinate transformations.

###  .1 Linearly superrotated black hole: metric, horizons, photon orbits, and
tentative shadows

We start with the metric of a bald Schwarzschild black hole without soft hair
in the Bondi frame,

$\bar{g}_{\rho\sigma}dx^{\rho}dx^{\sigma}=-\bigg{(}1-\frac{2M}{r}\bigg{)}dv^{2}+2dvdr+r^{2}\gamma_{AB}d\Theta^{A}d\Theta^{B},$
(S1)

where $\Theta^{A}=(\theta,\phi)$ and $\gamma_{AB}$ is the metric of unit
2-sphere. In what follows, all barred quantities are the ones associated with
the bald metric $\bar{g}_{\rho\sigma}$. We can then describe the dynamics of a
light ray (also called a photon) moving through this spacetime via the
corresponding Lagrangian

$L_{\rm bald}:=\frac{1}{2}\bar{g}_{\rho\sigma}\dot{x}^{\rho}\dot{x}^{\sigma},$
(S2)

where the dot indicates a derivative with respect to the worldline time. From
(S2), we can define the 4-momentum of the photon as

$\bar{p}^{\rho}:=\bar{g}^{\rho\sigma}{\delta L_{\rm
bald}\over\delta\dot{x}^{\sigma}}=\dot{x}^{\rho}.$ (S3)

Due to the isometries associated with the Killing vectors $\partial_{v}$ and
$\partial_{\phi}$, there are two corresponding conserved quantities,
$E=-p^{v}$ and $L_{z}=p^{\phi}$ (or $\lambda:=L_{z}/E$). Based on the
Hamilton-Jacobi formalism, from (S2) and by introducing an additional
conserved quantity associated with the spherical symmetry, the Carter constant
$\kappa$, we can obtain the following first-order geodesic equations for the
photon orbit labeled by a given set of $(E,\lambda,\kappa)$,

$\displaystyle\bar{p}^{v}=E\bigg{(}{r\over r-2M}\bigg{)}\Big{(}1+{\cal
R}\Big{)}\;,$ $\displaystyle\qquad\bar{p}^{\phi}={E\lambda\over
r^{2}\sin^{2}\theta}\;,$ (S4a) $\displaystyle\bar{p}^{r}=E{\cal R}\;,$
$\displaystyle\qquad\bar{p}^{\theta}={E\over r^{2}}{\cal K}\;,$ (S4b)

where

${\cal R}:=\sqrt{r^{3}-r\kappa+2M\kappa\over r^{3}},\qquad{\cal
K}:=\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}\;.$ (S5)

The first two equations given by (S4a) are the defining equations for $E$ and
$\lambda$, and the last two equations given in (S4b) describe the reduced
dynamics of photon orbits in the $(r,\theta)$ space.

As is known Strominger (2017), superrotations by themselves are an extension
of Bondi-Metzner-Sachs (BMS) symmetries of an asymptotically flat spacetime
Bondi  _et al._ (1962); Sachs (1962a, b) generated by the superrotation vector
field,

$\zeta_{Y}=\frac{v}{2}D_{A}Y^{A}\;\partial_{v}+\frac{1}{2}(v-r)D_{A}Y^{A}\;\partial_{r}+\left[Y^{A}+\frac{v}{2r}D^{A}D_{B}Y^{B}\right]\;\partial_{A}\;,$
(S6)

where $Y^{A}(\theta,\phi)$, with $A=\theta,\phi$, are the arbitrary functions
of $(\theta,\phi)$ characterizing the superrotations, and $D_{A}$ is the
covariant derivative with respect to $\gamma_{AB}$. Since the bald metric (S1)
is also an asymptotically flat spacetime, we can then implement the
superrotation hair for the bald Schwarzschild black hole by applying the Lie
derivative $\mathcal{L}_{\zeta_{Y}}$ to the bald metric (S1),

$g_{\rho\sigma}=\bar{g}_{\rho\sigma}+\epsilon\mathcal{L}_{\zeta_{Y}}\bar{g}_{\rho\sigma}.$
(S7)

Here, we introduce a small parameter $\epsilon$, i.e., $0<\epsilon\ll 1$, to
characterize the size of the superrotation hair by properly normalizing the
$Y^{A}$’s. Explicitly, we find that

$g_{\rho\sigma}dx^{\rho}dx^{\sigma}=\left[-\left(1-\frac{2M}{r}\right)+\frac{3M}{r}D_{A}Y^{A}\right]\;dv^{2}+2\;dv\;dr+r^{2}\,\left(\gamma_{AB}+\kappa_{AB}\right)d\Theta^{A}d\Theta^{B}\;,$
(S8)

where

$\kappa_{AB}=-\left(1-\frac{v}{r}\right)D\cdot
Y\gamma_{AB}+2D_{(A|}\left[Y^{C}+\frac{v}{2r}\left(D^{C}D\cdot
Y\right)\right]\gamma_{C|B)}\;.$ (S9)

In arriving at (S8), we have dropped the sub-leading terms in (S7) for the BMS
symmetry, which can be gauged away and will not affect any physical
observables. Unlike the bald metric (S1) and the metric for the
supertranslated black holes Compère and Long (2016), this metric depends on
$v$. Moreover, for the readers’ convenience, we also record the following
identities for later usage.

$\displaystyle D_{A}Y^{A}$ $\displaystyle\equiv D\cdot
Y=\partial_{\theta}Y^{\theta}+\partial_{\phi}Y^{\phi}+\cot{\theta}\;Y^{\theta}\;,$
(S10a) $\displaystyle\partial_{\phi}\big{(}D\cdot Y\big{)}$
$\displaystyle=\cot\theta\
\partial_{\phi}Y^{\theta}+\partial_{\phi}^{2}Y^{\phi}+\partial_{\theta}\partial_{\phi}Y^{\theta}\;,$
(S10b) $\displaystyle D^{\theta}\big{(}D\cdot Y\big{)}$
$\displaystyle=\partial_{\theta}^{2}Y^{\theta}+\partial_{\theta}\partial_{\phi}Y^{\phi}+\cot{\theta}\;\partial_{\theta}Y^{\theta}-\csc^{2}{\theta}\;Y^{\theta}\;,$
(S10c) $\displaystyle D^{\phi}\big{(}D\cdot Y\big{)}$
$\displaystyle=\csc^{2}{\theta}\left(\partial_{\phi}\partial_{\theta}Y^{\theta}+\partial_{\phi}^{2}Y^{\phi}+\cot{\theta}\;\partial_{\phi}Y^{\theta}\right)\;,$
(S10d) $\displaystyle\partial_{\phi}D^{\phi}\big{(}D\cdot Y\big{)}$
$\displaystyle=\csc^{2}\theta\big{(}\partial_{\phi}^{2}Y^{\theta}+\partial_{\phi}^{3}Y^{\phi}+\partial_{\theta}\partial_{\phi}^{2}Y^{\theta}\big{)}\;.$
(S10e)

We would now like to consider photon orbits in the superrotated black hole
with the metric (S8). Since (S8) lacks time-translational and spherical
symmetries, and thus the associated conserved quantities, it seems difficult
to label the photon orbits by them. However, (S1) and (S8) are related by a
Lie derivative corresponding to an active coordinate transformation; we can
then transform the photon orbits of (S4a) and (S4b) labeled by a given
$(E,\lambda,\kappa)$ to the photon orbits in (S8) with the same label, i.e.,

$p^{\rho}=\bar{p}^{\rho}+\epsilon\mathcal{L}_{\zeta_{Y}}\bar{p}^{\rho}.$ (S11)

Explicitly, the new photon orbits are given by

$\displaystyle p^{v}$
$\displaystyle=\frac{rE(1+\mathcal{R})}{r-2M}+\frac{\epsilon
E}{2r^{3}}\bigg{(}\frac{r^{4}(1+\mathcal{R})D_{A}Y^{A}}{r-2M}+\frac{(v-r)\big{(}2M^{2}\kappa-3M\kappa
r+\kappa
r^{2}-2Mr^{3}(1+\mathcal{R})\big{)}D_{A}Y^{A}}{(r-2M)^{2}}-rv\big{(}\mathcal{K}D^{\theta}D_{A}Y^{A}+\lambda
D^{\phi}D_{A}Y^{A}\big{)}\bigg{)}\;,$ (S12a) $\displaystyle p^{r}$
$\displaystyle=\frac{E\mathcal{K}}{r^{2}}+\frac{\epsilon
E}{2r^{3}}\bigg{(}r^{3}\mathcal{R}D_{A}Y^{A}+\frac{r^{4}(1+\mathcal{R})D_{A}Y^{A}}{r-2M}+\frac{\kappa(v-r)(r-3M)D_{A}Y^{A}}{\mathcal{R}r}-r(v-r)\big{(}\mathcal{K}D^{\theta}D_{A}Y^{A}+\lambda
D^{\phi}D_{A}Y^{A}\big{)}\bigg{)}\;,$ (S12b) $\displaystyle p^{\theta}$
$\displaystyle=E\mathcal{R}+\frac{\epsilon
E}{2r^{3}}\bigg{(}\frac{r^{3}(1+\mathcal{R})D^{\theta}D_{A}Y^{A}}{r-2M}-2(v-r)\mathcal{K}D_{A}Y^{A}+rv\mathcal{R}D^{\theta}D_{A}Y^{A}+\bigg{(}\frac{\lambda^{2}\cot\theta\csc^{2}\theta}{\mathcal{K}}-\lambda\csc^{2}\theta\bigg{)}\big{(}2rY^{\theta}+vD^{\theta}D_{A}Y^{A}\big{)}$
$\displaystyle\hskip
325.215pt+\mathcal{K}\big{(}2r\partial_{\theta}Y^{\theta}+v\partial_{\theta}D^{\theta}D_{A}Y^{A}\big{)}\bigg{)}\;,$
(S12c) $\displaystyle p^{\phi}$
$\displaystyle=\frac{E\lambda\csc^{2}\theta}{r^{2}}+\frac{\epsilon
E}{2r^{3}}\bigg{(}\frac{r^{3}(1+\mathcal{R})D^{\phi}D_{A}Y^{A}}{r-2M}-2\lambda\csc^{2}\theta(v-r)D_{A}Y^{A}+rv\mathcal{R}D^{\phi}D_{A}Y^{A}-\lambda\csc^{2}\theta\big{(}2r\partial_{\phi}Y^{\phi}+v\partial_{\phi}D^{\phi}D_{A}Y^{A}\big{)}$
$\displaystyle\hskip
144.54pt-2\lambda\cot\theta\csc^{2}\theta\big{(}2rY^{\theta}+vD^{\theta}D_{A}Y^{A}\big{)}+\mathcal{K}\big{(}2r\partial_{\theta}Y^{\phi}-v\partial_{\theta}D^{\phi}D_{A}Y^{A}\big{)}\bigg{)}\;,$
(S12d)

where $\cal R$ and $\cal K$ are given by (S5). These photon orbits can only be
labeled by the inherited constants of motion from the ones from the bald
metric as there are no corresponding isometries for the metric (S8). For the
same reason, we cannot obtain the reduced dynamics of the photon orbits, which
is crucial in the conventional method for determining FPOs by extremizing the
effective radial potential. Thus, for generic linearly superrotated black
holes, the FPOs remain elusive. However, we can still adopt the above geodesic
equations to construct the tentative shadows of linearly superrotated black
holes by assuming the existence of corresponding FPOs for the moment.

The basic idea of constructing the shadow is to follow a light ray labeled by
a given set $(E,\lambda,\kappa)$ from the observer’s location to the region
near the black hole. This is the so-called ray-tracing method. In the near
black hole region, we need to find the so-called light rings (LRs) or
fundamental photon orbits (FPOs), which are critical, unstable closed orbits
specified by a particular set of $(E,\lambda,\kappa)$. If the radius of a
closed photon orbit is below that of the FPOs, it will be trapped by the black
hole. Otherwise, it will be stable. In this sense, the LRs/FPOs will define
the boundary of the shadow for a black hole. Therefore, once the values of
$(E,\lambda,\kappa)=(E_{\rm FPO},\lambda_{\rm FPO},\kappa_{\rm FPO})$ are
known, we can construct the boundary of the shadow.

Following the above procedure, we obtain the following two shadow equations,
as given in the main text

$\displaystyle r_{O}\alpha$ $\displaystyle=$ $\displaystyle\epsilon
A_{0}+(-1+\epsilon A_{1})\lambda_{\rm FPO}\csc\theta_{O}+\epsilon A_{2}{\cal
K}\;,$ (S13) $\displaystyle r_{O}\beta$ $\displaystyle=$
$\displaystyle\epsilon B_{0}{\lambda^{2}_{\rm FPO}\csc^{2}\theta_{O}\over{\cal
K}}+\epsilon(B_{1}\lambda_{\rm FPO}\csc\theta_{O}+B_{2})+\;(1+\epsilon
B_{3}){\cal K}\;,\qquad$ (S14)

where $\cal K$ is given by (S5) with $\kappa=\kappa_{\rm FPO}:=27M^{2}$, and
where the $A_{i}$’s and $B_{i}$’s are independent of both $\lambda_{\rm FPO}$
and $\kappa_{\rm FPO}$ and are given by

$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle{1\over
2}(4M+2r_{O}-v_{O})\sin\theta_{O}D^{\phi}D\cdot Y\;,$ (S15) $\displaystyle
A_{1}$ $\displaystyle=$ $\displaystyle{1\over 2}(D\cdot
Y-2\partial_{\phi_{O}}Y^{\phi}-4\partial_{\theta_{O}}Y^{\theta})\;,$ (S16)
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\sin\theta_{O}\partial_{\theta_{O}}Y^{\phi}\;,$ (S17)
$\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle\cot\theta_{O}Y^{\theta}\;,$ (S18) $\displaystyle B_{1}$
$\displaystyle=$
$\displaystyle-\csc\theta_{O}\partial_{\phi_{O}}Y^{\theta}\;,$ (S19)
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle-{1\over
2}(4M+2r_{O}-v_{O})\partial_{\theta_{O}}D\cdot Y\;,$ (S20) $\displaystyle
B_{3}$ $\displaystyle=$ $\displaystyle{1\over 2}(3D\cdot
Y-2\partial_{\theta_{O}}Y^{\theta})$ (S21)

with $Y^{A}=Y^{A}(\theta_{O},\phi_{O})$.

###  .2 Tentative shadows for the hairy metrics from general linear coordinate
transformations

In the main text, we find that the shadows of the linearly superrotated black
holes take the form of conic sections. Among them, those with regular
superrotation hairs have well-defined FPOs which yield definite shadows.
Otherwise, the FPOs remain elusive and the associated shadows can only be
considered tentative. Despite that, we still want to see if this conic shadow
is a unique feature of superrotated black holes. We can answer this question
by extending the FPO assumption to the hairy metrics obtained by acting with
the most general linear coordinate transformations. This is what we will
construct below.

Consider a linear perturbation away from the Schwarzschild case

$\displaystyle g_{\rho\sigma}$
$\displaystyle=\bar{g}_{\rho\sigma}+\epsilon\mathcal{L}_{\zeta}\,\bar{g}_{\rho\sigma}$
(S22) $\displaystyle p^{\mu}$
$\displaystyle=\bar{p}^{\mu}+\epsilon\mathcal{L}_{\zeta}\,\bar{p}^{\mu}$ (S23)

where $\bar{g}_{\rho\sigma}$ and $\bar{p}^{\mu}$ are the bald metric and the
corresponding null geodesic equations relating to canonical momenta as given
in the main text, respectively, and $\zeta$ is an arbitrary vector field

$\zeta^{\rho}=\zeta^{v}(v,r,\theta,\phi)\,\partial_{v}+\zeta^{r}(v,r,\theta,\phi)\,\partial_{r}+\zeta^{\theta}(v,r,\theta,\phi)\,\partial_{\theta}+\zeta^{\phi}(v,r,\theta,\phi)\,\partial_{\phi}\;.$
(S24)

Unlike the supertranslation and superrotation cases which are constrained by
appropriate asymptotic fall-off behaviors on $\zeta^{\mu}$ required to yield
BMS symmetry, here we ignore such constraints as a tentative study of what
possible shadows arise (again, if the corresponding FPOs exist). Thus, the
resultant metric could be pathological or unphysical.

From (S23), we can obtain the null geodesic equations in the deformed metric
(S22), which are labeled by the inherited constants of motion from the bald
metric. Assuming FPOs labeled by these inherited constants of motion $(E_{\rm
FPO},\lambda_{\rm FPO},\kappa_{\rm FPO})$ exist as null geodesics in such a
background metric (S22), we can then follow the same procedure as given in the
main text to construct the shadow equations. The results up to ${\cal
O}(\epsilon)$ are

$\displaystyle r\alpha$ $\displaystyle=$ $\displaystyle\epsilon
A_{0}+(-1+\epsilon A_{1})\lambda_{\rm FPO}\csc\theta+\epsilon A_{2}{\cal
K}+\epsilon A_{3}\kappa_{\rm FPO}\;,$ (S25) $\displaystyle r\beta$
$\displaystyle=$ $\displaystyle\epsilon B_{0}{\lambda^{2}_{\rm
FPO}\csc^{2}\theta\over{\cal K}}+\epsilon(B_{1}\lambda_{\rm
FPO}\csc\theta+B_{2})+(1+\epsilon B_{3}){\cal K}+\epsilon B_{4}\kappa_{\rm
FPO}\;,\qquad$ (S26)

where the inherited $\lambda$ and $\kappa$ inside ${\cal K}$ are also replaced
by $\lambda_{\rm FPO}$ and $\kappa_{\rm FPO}$, respectively, and the
coefficients $A_{i}$ and $B_{i}$ are independent of $\lambda_{\rm FPO}$ and
$\kappa_{\rm FPO}$. The explicit forms of these coefficients are

$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle
4M(r+2M)\sin\theta\left[\partial_{v}\zeta^{\phi}\right]_{u=0}+2(r+2M)\sin\theta\left[\partial_{v}\partial_{u}\zeta^{\phi}\right]_{u=0}-\sin\theta\left[\partial_{u}\zeta^{\phi}\right]_{u=0}+\sin\theta\left[\partial_{v}\partial^{2}_{u}\zeta^{\phi}\right]_{u=0}\;,$
(S27) $\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
2\cot\theta\left[\zeta^{\theta}\right]_{u=0}+\left[\partial_{\phi}\zeta^{\phi}\right]_{u=0}-2\left[\partial_{v}\zeta^{r}\right]_{u=0}\;,$
(S28) $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\sin\theta\left[\partial_{\theta}\zeta^{\phi}\right]_{u=0}\;,$
(S29) $\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sin\theta\left[\partial_{v}\zeta^{\phi}\right]_{u=0}\;,$
(S30) $\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle\cot\theta\left[\zeta^{\theta}\right]_{u=0}\;,$ (S31)
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-\csc\theta\left[\partial_{\phi}\zeta^{\theta}\right]_{u=0}\;,$
(S32) $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-4M(r+2M)\left[\partial_{v}\zeta^{\theta}\right]_{u=0}-2(r+2M)\left[\partial_{v}\partial_{u}\zeta^{\theta}\right]_{u=0}+\left[\partial_{u}\zeta^{\theta}\right]_{u=0}-\left[\partial_{v}\partial^{2}_{u}\zeta^{\theta}\right]_{u=0}\;,$
(S33) $\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle-\left[\partial_{\theta}\zeta^{\theta}\right]_{u=0}+2\left[\partial_{v}\zeta^{r}\right]_{u=0}\;,$
(S34) $\displaystyle B_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\partial_{v}\zeta^{\theta}\right]_{u=0}\;,$
(S35)

where $u=1/r$. For simplicity, in the above, we have omitted the subscript $O$
of $x^{\mu}_{O}=(v_{O},r_{O},\theta_{O},\phi_{O})$. Two additional
coefficients $A_{3}$ and $B_{4}$ arise if $\zeta^{\theta,\phi}$ depend on $v$.
Following the same prescription as in the main text, we can transform the
shadow equations (S25) and (S26) into the following form:

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle(1-2\epsilon(B_{0}-A_{1}))\,(r\alpha)^{2}+(1-2\epsilon
B_{3})\,(r\beta)^{2}-2\epsilon A_{0}\,(r\alpha)-2\epsilon
B_{2}\,(r\beta)-2\epsilon(A_{2}-B_{1})\,(r\alpha)(r\beta)$ (S36)
$\displaystyle+\;2\epsilon\big{(}(r\alpha)^{2}+(r\beta)^{2})\big{)}\big{(}A_{3}(r\alpha)+B_{4}(r\beta)\big{)}\;.$

The first line takes the same form as the quadratic curve that we obtained in
the superrotation hair case. However, the second line yields a higher-order
curve for nonzero $A_{3}$ and $B_{4}$. Thus, we can conclude that the conic
shadow is a unique feature arising from the structure of the superrotation
hairs constrained by BMS symmetry. That said, it is worth highlighting that
the above result is merely tentative, as there is no guarantee for the
existence of the associated FPOs. Judging by the uniqueness of black hole
observables, it is quite probable that there are no FPOs for the above generic
hairs, as there are no appropriate asymptotic BMS symmetries associated with
them. Thus, we shall expect no detectable signature of shadow deviation away
from that of the bald metric.