# The Shadow of Supertranslated Schwarzschild Black Hole

Qing-Hua Zhu<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China  School of Physical Sciences, University of Chinese Academy of Sciences,
No. 19A Yuquan Road, Beijing 100049, China  Yu-Xuan Han<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road,
Beijing 100049, China  Qing-Guo Huang  Corresponding author<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road,
Beijing 100049, China  School of Fundamental Physics and Mathematical Sciences
Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China

###### Abstract

The supertranslated black hole proposed by Hawking, Perry, and Strominger
might provide a resolution to the information paradox, which is usually
defined by a complicated space-time metric with even less space-time
symmetries compared to Kerr black hole. In this paper, we figure out the
shadow for the supertranslated Schwarzschild black hole by making use of
supertranslated 4-velocities and the trajectories of the light rays. Based on
this approach, we find that the photon sphere gets distorted due to the
supertranslation hairs and the position of the shadow on the projection plane
is shifted by the supertranslation vector, but the size and shape of the
shadow remain the same as those of Schwarzschild black hole.

##  I introduction

In 1975, Hawking pointed out that black holes will end theirs lives with
evaporation Hawking (1975), which significantly supported the picture of the
Bekenstein entropy, that a black hole can be reckon as a thermodynamic system
Bekenstein (1972). However, a question soon arose, due to its breaking the law
of information conservation in conflict with the quantum theory, which is the
so-called black hole information paradox. About a half century passed, in
2016, Hawking, Perry, and Strominger (HPS) revisited this problem, and
proposed soft hairy black hole that might provide a resolution to the
information paradox Hawking  _et al._ (2016, 2017).

The studies on the soft hairy black hole would involve the global spacetime
structure from the event horizon to the asymptotically flat zone. Therefore,
it is interesting to examine whether the influence from the soft hair can be
observed. Recently, it becomes more promising, because Event Horizon Telescope
(EHT) presented the images of the black holes in M87 Akiyama  _et al._ (2019a,
b, c, d, e, f), and just recently in our galaxy Akiyama  _et al._ (2022), and
opened a new way to investigate the physics in strong field regime of gravity
for the black holes.

One can interpret the images of the black holes in this way. A black hole can
not only bend lights but also swallow them. As a result, a shaded zone exists
in the field of vision of an observer, and is thus called the black hole
shadow. Theoretically, it was developed based on the works of Synge Synge
(1966) and Bardeen Bardeen (1973). By extracting information of a black hole
from its shadow, one can tell the difference of the spacetime geometry for
different black holes, and provide a way to explore properties of the black
hole, such as reflection coefficients on the horizon Rummel and Burgess
(2020), quantum structure Giddings and Psaltis (2018); Giddings (2019) and
naked singularity Bambhaniya  _et al._ (2021), or test general relativity
Berti  _et al._ (2015); Psaltis (2019); Glampedakis and Pappas (2021) and the
No-hair theorem Broderick  _et al._ (2014); Psaltis  _et al._ (2016); Lin  _et
al._ (2022).

Due to the interests of information paradox, the shadow of the soft hairy
black hole has been explored in Lin  _et al._ (2022); Sarkar  _et al._ (2022).
Since the soft hairy black hole has less space-time symmetries compared to the
parameterized Kerr black hole Johannsen (2013); Johannsen  _et al._ (2016),
the authors simulated the light rays for calculating the shadow Lin  _et al._
(2022) in which the parameterized metric is limited to some specific types of
soft hairy black holes. The supertranslation hair, as a type of soft hair, is
of great interesting, because the supertranslated black hole carries
conservation charges at classical level Hawking  _et al._ (2017), and thus
might have observable effect in classical physics. Ref. Sarkar  _et al._
(2022) investigated the photon sphere of the supertranslated Schwarzschild
black hole in the equatorial plane, which suggested that the changes of the
photon sphere might lead to a distinguishable shadow.

In this paper, we will investigate the shadow of supertranslated Schwarzschild
black hole that is proposed by Hawking, Perry and Strominger Hawking  _et al._
(2016). Different from pioneers’ works in Lin  _et al._ (2022); Sarkar  _et
al._ (2022), we do not limit to the axisymmetric cases. We calculate the
shadow of the supertranslated Schwarzschild black hole by making use of
supertranslated 4-velocities and the trajectories of the light rays, and find
that the photon sphere get distorted due to the supertranslation hairs. The
position of the shadow on the projection plane is shifted by the
supertranslation vectors, while the size and shape of the shadow remain the
same as those of Schwarzschild black hole.

The rest of the paper is organized as follows. In Section II, we list results
of supertranslated Schwarzschild black hole proposed by HPS. In Section III,
we calculate the 4-velocities and trajectories of the bending light rays
around the supertranslated Schwarzschild black hole. In Section IV, we show
the shadow of the supertranslated Schwarzschild black hole. In Section V, the
conclusions and discussions are summarized.

##  II Schwarzschild black hole with supertranslation hair

As proposed by HPS Hawking  _et al._ (2017), the supertranslation hair on
Schwarzschild black hole is given by introducing the supertranslations,

$g_{\mu\nu}^{{(\rm hairy)}}=g_{\mu\nu}^{{(\rm
bald)}}+\mathcal{L}_{\zeta}g_{\mu\nu}^{{(\rm bald)}}~{},$ (1)

which is also the procedure of implanting supertranslation hair on
Schwarzschild black hole. The quantities in Eq. (1) will be illustrated in the
following part of this paper. The metric of bald Schwarzschild black hole
$g_{\mu\nu}^{(\rm bald)}$ in advanced Bondi coordinate
($v$,$r$,$\theta$,$\phi$) is given by

${\rm d}s^{2}=-\left(1-\frac{2M}{r}\right){\rm d}v^{2}+2{\rm d}v{\rm
d}r+r^{2}\gamma_{AB}{\rm d}\Theta^{A}{\rm d}\Theta^{B},$ (2)

where $M$ is the mass of the black hole, $\Theta^{A}$ denotes the angular
coordinates $\theta$ or $\phi$. The $\mathcal{L}_{\zeta}$ is the Lie
derivative along the supertranslation vector $\zeta$ that is formulated as
Sachs (1962); Barnich and Troessaert (2010); Hawking  _et al._ (2017),

$\zeta=f(\theta,\phi)\partial_{v}-\frac{1}{2}D^{2}f(\theta,\phi)\partial_{r}+\frac{1}{r}D^{A}f(\theta,\phi)\partial_{A}~{}.$
(3)

where $D$ is the covariant derivative with respect to the metric on the unit
two sphere, and the $f(\theta,\phi)$ is an arbitrary function that is
proportional to the first order weak-field expansion. Here, the $\zeta$ is
also the asymptotical Killing vector in Schwarzschild space-time. Expanding
the $f(\theta,\phi)$ into spherical harmonics,

$f(\theta,\phi)=\sum_{lm}a_{lm}Y_{lm}(\theta,\phi)~{},$ (4)

one can find infinite number of choices of $l$ and $m$ that can give different
asymptotical Killing vectors. In particular, in the case of $l=0$, $\zeta$ is
generator of the temporal translation. And in the case of $l=1$ and
$m=-1,0,1$, $\zeta$ are generators of the three spatial translations,
respectively. Finally, the supertranslated Schwarzschild metric
$g_{\mu\nu}^{(\rm hairy)}$ on the left hand side of Eq. (1) in advanced Bondi
coordinate takes the form of

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}-\frac{M}{r^{2}}D^{2}f\right){\rm
d}v^{2}+2{\rm d}v{\rm d}r$ $\displaystyle-
D_{A}\left(2f-\frac{4M}{r}f+D^{2}f\right){\rm d}v{\rm d}\Theta^{A}$
$\displaystyle+(r^{2}\gamma_{AB}+2rD_{A}D_{B}f-r\gamma_{AB}D^{2}f){\rm
d}\Theta^{A}{\rm d}\Theta^{B}~{},$

in which the metric depends on the angular coordinates $\theta$ and $\phi$ due
to the function $f(\theta,\phi)$. It seems to bring complexity in the
calculation of the propagation of light.

##  III Propagation of Light and photon sphere

The calculation about the shadow of a black hole based on the solutions of
null geodesic equations. From the metric shown in Eq. (II), it seems to be a
tough task to deal with the geodesic equations. In this section, we will show
that the 4-velocities and the trajectories of light rays can be obtained by
the procedure of implanting the supertranslation hair. Directly solving the
geodesic equations for the supertranslated Schwarzschild black hole is not
necessary.

First of all, we should work out the 4-velocities of light rays for the bald
Schwarzschild black hole in the Bondi coordinate. Based on the Hamilton-Jacobi
method for geodesic equations Carter (1968), one can obtain 4-velocities of
light rays in static coordinates $(t,r,\theta,\phi)$ for the bald
Schwarzschild black hole (see Eqs. (10)–(13) in ref. Chang and Zhu (2020a)).
And then we can transform the 4-velocities in static coordinates into those in
Bondi coordinate by using $v=t+r+2M\log(r/2M-1)$, namely

$\displaystyle p^{v,(\rm
bald)}=E\left(\frac{r}{r-2M}\right)\left(1+\sqrt{\frac{r^{3}+2M\kappa-r\kappa}{r^{3}}}\right)~{},$
(6a) $\displaystyle p^{r,(\rm
bald)}=E\sqrt{\frac{r^{3}+2M\kappa-r\kappa}{r^{3}}}~{},$ (6b) $\displaystyle
p^{\theta,(\rm
bald)}=\frac{E}{r^{2}}\sqrt{\kappa-\frac{\lambda^{2}}{\sin^{2}\theta}}~{},$
(6c) $\displaystyle p^{\phi,{(\rm
bald)}}=\frac{E\lambda}{r^{2}\sin^{2}\theta}~{},$ (6d)

where $E$, $\kappa\equiv K/E^{2}$ and $\lambda\equiv L/E$ are three integral
constants corresponding to the energy, the total angular momentum, and the
angular momentum with respect coordinate $\phi$ for a light ray at infinity.

According to the above arguments, the 4-velocities of light rays around the
supertranslated Schwarzschild black hole can be obtained by using the
supertranslations,

$p^{\mu,(\rm hairy)}=p^{\mu,(\rm bald)}+\mathcal{L}_{\zeta}p^{\mu,(\rm
bald)}~{},$ (7)

where the $\zeta$ is the supertranslation vector given in Eq. (3), and the
components of the 4-velocities $p^{\mu,(\rm hairy)}$ are presented as follows,

$\displaystyle p^{v,(\rm hairy)}$ $\displaystyle=$
$\displaystyle\frac{E}{2r^{3}}\Biggl{(}\frac{1}{\left(r-2M\right)^{2}\sqrt{\frac{2\kappa
M+r^{3}-\kappa r}{r^{3}}}}\Bigg{(}\left(-2\kappa
M^{2}+2Mr^{3}\left(\sqrt{\frac{2\kappa M+r^{3}-\kappa
r}{r^{3}}}+1\right)+3\kappa Mr-\kappa r^{2}\right)~{},$ (8a)
$\displaystyle\left(\partial^{2}_{\theta}f+\cot\theta\partial_{\theta}f+\csc^{2}\theta\partial^{2}_{\phi}f\right)\Bigg{)}-2r\partial_{\theta}f\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}-2\lambda
r\csc^{2}\theta\partial_{\phi}f+\frac{2r^{4}\left(\sqrt{\frac{2\kappa
M+r^{3}-\kappa r}{r^{3}}}+1\right)}{r-2M}\Biggl{)}~{},$ $\displaystyle
p^{r,(\rm hairy)}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}\left(2\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}\left(-\partial^{3}_{\theta}f-\cot\theta\partial^{2}_{\theta}f+\csc^{2}\theta\partial_{\theta}f-\csc^{2}\theta\partial_{\theta}\partial^{2}_{\phi}f+2\cot\theta\csc^{2}\theta\partial^{2}_{\phi}f\right)\right)$
(8b)
$\displaystyle+\frac{2\lambda\csc^{2}\theta}{r^{2}}\left(\partial^{2}_{\theta}\partial_{\phi}f+\cot\theta\partial_{\theta}\partial_{\phi}f+\csc^{2}\theta\partial^{3}_{\phi}f\right)+4\sqrt{\frac{2\kappa
M+r^{3}-\kappa r}{r^{3}}}~{},$ $\displaystyle p^{{\theta},(\rm hairy)}$
$\displaystyle=$
$\displaystyle\frac{E}{r^{3}}\Bigg{(}-r\partial^{2}_{\theta}f\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}+\frac{\lambda^{2}r\cot\theta\csc^{2}\theta\partial_{\theta}f}{\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}}-\lambda
r\csc^{2}\theta\partial_{\theta}\partial_{\phi}f$ (8c)
$\displaystyle+\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}\left(\partial^{2}_{\theta}f+\cot\theta\partial_{\theta}f+\csc^{2}\theta\partial^{2}_{\phi}f\right)+r\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}\Bigg{)}~{},$
$\displaystyle p^{{\phi},(\rm hairy)}$ $\displaystyle=$
$\displaystyle\frac{E\csc^{2}\theta}{r^{3}}\Bigg{(}-r\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}\partial_{\theta}\partial_{\phi}f+2r\cot\theta\sqrt{\kappa-\lambda^{2}\csc^{2}\theta}\partial_{\phi}f$
(8d) $\displaystyle-2\lambda
r\cot\theta\partial_{\theta}f+\lambda\left(1-r\right)\csc^{2}\theta\partial^{2}_{\phi}f+\lambda\partial^{2}_{\theta}f+\lambda\cot\theta\partial_{\theta}f+\lambda
r\Bigg{)}~{}.$

Based on the diffeomorphisms, it is not difficult to find that the
4-velocities of the light rays in Eqs. (8) are exactly the solutions of
geodesic equations with respect to the metric in Eq. (II).

From the 4-velocities given in Eqs. (8), the worldlines of the light rays
$\gamma^{\mu}(\lambda)$ in supertranslated Schwarzschild spacetime can be
easily obtained. The geodesic equations for the worldlines of light rays
$\bar{\gamma}^{\mu}$ in bald Schwarzschild space-time are given by

$\displaystyle\frac{\mathrm{d}\bar{\gamma}^{\mu}}{\mathrm{d}\lambda}$
$\displaystyle=$ $\displaystyle p^{\mu,(\rm bald)}~{}.$ (9)

After implanting the supertranslation hair, the geodesic equations take the
form of

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\lambda}(\bar{\gamma}^{\mu}+\delta{\gamma}^{\mu})$
$\displaystyle=$ $\displaystyle p^{\mu,(\rm
bald)}+\mathcal{L}_{\zeta}p^{\mu,(\rm bald)}~{},$ (10)

where $\delta\gamma^{\mu}$ indicates a small change for the $\bar{\gamma}$
ascribed from the supertranslation hair. Using Eqs. (9) and (10), we obtain
the solution $\delta\gamma^{\mu}=\zeta^{\mu}$. Therefore the worldline of a
light ray around the supertranslated Schwarzschild black hole is formulated as

$\displaystyle\gamma^{\mu}=\bar{\gamma}^{\mu}+\zeta^{\mu}~{}.$ (11)

It shows that each event on the null geodesic is shifted by the
supertranslation vector. Since a finite $\zeta^{\mu}$ can not shift the
“endpoint” of a null geodesic from the singularity inside the black hole to
spatial infinity, the fact that the light rays finally escape away from or
fall into the black hole will not be changed after implanting supertranslation
hair.

The photon sphere for Schwarzschild black hole is the assemble of the unstable
circle orbits of the light rays, which determines whether a bending light ray
approaching the black hole can escape to infinity again. In this sense, it
describes critical escape orbits for a light ray. Since implanting
supertranslation hair does not change whether a light ray can escape to
infinity or not, the 4-velocities of the light rays in critical escape orbits
for the supertranslated Schwarzschild black hole can be formulated as

$p^{\mu,(\rm hairy)}_{c}=p^{\mu,(\rm
bald)}|_{\kappa=27M^{2}}+\mathcal{L}_{\zeta}(p^{\mu,(\rm
bald)}|_{\kappa=27M^{2}})~{},$ (12)

where the subscript ‘$c$’ denotes the light rays in the critical escape
orbits.

We will further illustrate Eq. (12) with ray-trace simulations. In Fig. 1, we
show the trajectories of the light rays $\gamma^{\mu}$ in Eq. (11) for the
supertranslated Schwarzschild black hole by numerically solving the Eqs. (6),
and then associating it with Eq. (11). It shows that the photon sphere, which
locates at $r=3M$ (the black circle) for the bald black hole, is distorted
(the red circle) for the supertranslated Schwarzschild black hole. The orange
curves represent the light rays that are almost tangent to the photon sphere.
One can observe that these light rays are determined by integrating constant
$\kappa=27M^{2}$, which are the same as those in bald Schwarzschild space-
time.

Based on Eq. (11), one can also obtain the distorted photon sphere for other
types of supertranslation hairs. In Fig. 2, we plot the distorted photon
spheres for selected $l$ and $m$ in Eq. (4). It shows that the photon sphere
get more distorted for the larger $l$. Because the positive and negative $m$
in Eq. 4 share the same shape of the photon sphere, we only show the cases of
$m\geq 0$ for illustration.

Figure 1: Ray-tracing simulations for the light rays (blue and orange curves)
around a supertranslated Schwarzschild black hole with
$f(\theta,\phi)=\frac{1}{5}Y_{20}(\theta,\phi)$ (left panel), and
$f(\theta,\phi)=\frac{1}{5}Y_{30}(\theta,\phi)$ (right panel), respectively.
The black circle represents the photon sphere for the bald black hole, and the
red circle represents the photon sphere for the supertranslated Schwarzschild
black hole. The orange curves represent the light rays that are almost tangent
to the photon sphere, which determine the edge of the shadow on observers’
projection plane.  Figure 2: Distorted photon spheres of the supertranslated
Schwarzschild black holes described by Eq. (3) and (4) with
$a_{lm}=\frac{1}{5}$. For comparison, the photon sphere of the bald
Schwarzschild black hole is plotted in the translucent sphere.

##  IV The shadow of a supertranslated Schwarzschild black hole

For the supertranslated Schwarzschild black hole, it was suggested that the
distorted photon sphere might lead to a distinguishable shadow Sarkar  _et
al._ (2022). In this section, we will calculate the shadow in details. For
distant observers, the shadow described by the coordinates $(\alpha,\beta)$ on
the observers’ projection plane is given by Synge (1966); Bardeen (1973)

$\displaystyle\alpha$ $\displaystyle\equiv$
$\displaystyle-\frac{r\sin\theta\mathrm{d}\phi}{\mathrm{d}r}\bigg{|}_{x^{i}_{\rm
obs}}=-r_{o}\sin\theta_{o}\frac{p^{{\phi},(\rm hairy)}_{c}}{p^{{r},(\rm
hairy)}_{c}}\bigg{|}_{x^{i}_{\rm obs}}$ (13) $\displaystyle=$
$\displaystyle-\frac{\lambda+2\partial_{\phi}f(\theta_{o},\phi_{o})}{r_{o}\sin\theta_{o}}~{},$
$\displaystyle\beta$ $\displaystyle\equiv$
$\displaystyle\frac{r\mathrm{d}\theta}{\mathrm{d}r}\bigg{|}_{x^{i}_{\rm
obs}}=r_{o}\frac{p^{{\theta},(\rm hairy)}_{c}}{p^{{r},(\rm
hairy)}_{c}}\bigg{|}_{x^{i}_{\rm obs}}$ (14) $\displaystyle=$
$\displaystyle\frac{\sqrt{\kappa-\lambda^{2}\csc^{2}\theta_{o}}+2\partial_{\theta}f(\theta_{o},\phi_{o})}{r_{o}}~{},$

where $x^{i}_{\rm obs}=(r_{o},\theta_{o},\phi_{o})$ is the location of an
observer, and we have used Eq. (12) for evaluating the 4-velocities
$p^{\mu,(\rm hairy)}_{c}$. Here, the $\kappa$ and $\lambda$ are determined by
the critical escape orbits, and the values of them are exactly the same as
those for bald Schwarzschild space-time as discussed in Eq. (12).

Based on Eqs. (13) and (14), the shape and size of the shadow remain the same
as those for bald Schwarzschild black hole, but the position of the shadow is
shifted by $\delta\alpha=-2\partial_{\phi}f/(r_{0}\sin\theta_{0})$ and
$\delta\beta=2\partial_{\theta}f/r_{0}$ on the projection plane. It can be
clearly shown by eliminating the $\lambda$ in Eqs. (13) and (14), and then one
can obtain the curve equation in coordinates $(\alpha,\beta)$,

$\displaystyle\left(\alpha+\frac{2\partial_{\phi}f}{r_{o}\sin\theta_{o}}\right)^{2}+\left(\beta-\frac{2\partial_{\theta}f}{r_{o}}\right)^{2}=\frac{\kappa}{r^{2}_{o}}~{}.$
(15)

The curve equation represents a circle with radius $\sqrt{\kappa}/r_{0}$ and
centre $(-2\partial_{\phi}f/(r_{0}\sin\theta_{0}),2\partial_{\theta}f/r_{0})$
on projection plane. For illustrations, we plot the shadows of the
supertranslated Schwarzschild black hole with
$f(\theta,\phi)=\frac{1}{5}Y_{33}(\theta,\phi)$ and
$f(\theta,\phi)=\frac{1}{5}Y_{31}(\theta,\phi)$ for observers at equatorial
plane $\theta_{0}=\frac{\pi}{2}$ in Figs. 3 and 4, respectively. For observers
located at different $\phi$ ranged from $0$ to $2\pi$, the center positions of
the shadows would move from the red point to the other one shown in the plots.
Comparing the two examples, one might find that the positions of the shadows
with supertranslation hair $f(\theta,\phi)=\frac{1}{5}Y_{33}(\theta,\phi)$ is
shifted by a larger distance than that with
$f(\theta,\phi)=\frac{1}{5}Y_{13}(\theta,\phi)$. In Fig. 5, we plot the
shadows of the supertranslated Schwarzschild black hole with
$f(\theta,\phi)=\frac{1}{5}Y_{32}(\theta,\phi)$ for observers at equatorial
plane. Differed from above two cases, the observer move around the
supertranslated Schwarzschild black hole would view the shadows shifted along
the orbital axis. In addition, the positions of the shadows for
$f(\theta,\phi)=\frac{1}{5}Y_{30}(\theta,\phi)$ are fixed at the origin of the
coordinate $(\alpha,\beta)$ on projection plane. Thus, we did not show the
case here. Generally, for arbitrary types of supertranslation hairs described
in Eqs. (3) and (4), the shadows on projection plane are completed described
by Eq. (15). Here, we only showed the supertranslated Schwarzschild black hole
with $l=3$ for examples.

Figure 3: The shadow of supertranslated Schwarzschild black hole with
$f(\theta,\phi)=\frac{1}{5}Y_{33}(\theta,\phi)$ for distant observers at
equatorial plane $\theta_{0}=\frac{\pi}{2}$, and at different $\phi_{o}$
ranged from $0$ to $2\pi$, respectively. The photon rings on the projection
plane are presented in the black circles, and red point–dashed line–red point
represents the the centre of the shadows.  Figure 4: The shadow of
supertranslated Schwarzschild black hole with
$f(\theta,\phi)=\frac{1}{5}Y_{31}(\theta,\phi)$ for distant observers at
equatorial plane $\theta_{0}=\frac{\pi}{2}$, and at different $\phi_{o}$
ranged from $0$ to $2\pi$, respectively. The photon rings on the projection
plane are presented in the black circles, and red point–dashed line–red point
represents the the centre of the shadows.  Figure 5: The shadow of
supertranslated Schwarzschild black hole with
$f(\theta,\phi)=\frac{1}{5}Y_{32}(\theta,\phi)$ for distant observers at
equatorial plane $\theta_{0}=\frac{\pi}{2}$, and at different $\phi_{o}$
ranged from $0$ to $2\pi$, respectively. The photon rings on the projection
plane are presented in the black circles, and red point–dashed line–red point
represents the the centre of the shadows.

##  V Conclusions and Discussions

We investigate the shadow for supertranslated Schwarzschild black hole
proposed by Hawking, Perry and Strominger Hawking  _et al._ (2016). The
calculation about the shadow of supertranslated Schwarzschild black hole turns
to be much easier, if the supertranslated 4-velocities and the trajectories of
the light rays are adopted. Based on this approach, we showed that the photon
sphere gets distorted due to the supertranslation hairs, and the position of
the shadow on the projection plane is shifted by the supertranslation vector
and depends on the location of the observer, but the size and shape of the
shadow are the same as those of Schwarzschild black hole.

In this paper, we mainly focus on the shadow for distant observers. It might
be non-trivial to consider the observer located at finite distance Grenzebach
_et al._ (2014); Stuchlík  _et al._ (2018); Chang and Zhu (2020b); Tsupko  _et
al._ (2020); Li  _et al._ (2020), in particular for those near the event
horizon Donnay  _et al._ (2018). In this case, the distorted photon sphere
might also have impact on the shape or size of the shadows.

Acknowledgments.  This work is supported by the National Key Research and
Development Program of China Grant No.2020YFC2201502, grants from NSFC (grant
No. 11975019, 11991052, 12047503), Key Research Program of Frontier Sciences,
CAS, Grant NO. ZDBS-LY-7009, CAS Project for Young Scientists in Basic
Research YSBR-006, the Key Research Program of the Chinese Academy of Sciences
(Grant NO. XDPB15).

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