# Photon sphere and shadow of a time-dependent black hole
described by a Vaidya metric

Jay Solanki<EMAIL_ADDRESS>Sardar Vallabhbhai National Institute of
Technology,
Surat - 395007, Gujarat, India  Volker Perlick<EMAIL_ADDRESS>ZARM, University of Bremen, 28359 Bremen, Germany

###### Abstract

In this paper we derive exact analytical formulas for the evolution of the
photon sphere and for the angular radius of the shadow in a special Vaidya
space-time. The Vaidya metric describes a spherically symmetric object that
gains or loses mass, depending on a mass function $m(v)$ that can be freely
chosen. Here we consider the case that $m(v)$ is a linearly increasing or
decreasing function. The first case can serve as a simple model for an
accreting black hole, the second case for a (Hawking) radiating black hole.
With a linear mass function the Vaidya metric admits a conformal Killing
vector field which, together with the spherical symmetry, gives us enough
constants of motion for analytically calculating the light-like geodesics.
Both in the accreting and in the radiating case, we first calculate the light-
like geodesics, the photon sphere, the angular radius of the shadow, and the
red-shift of light in coordinates in which the metric is manifestly
conformally static, then we analyze the photon sphere and the shadow in the
original Eddington-Finkelstein-like Vaidya coordinates.

black hole, shadow, Vaidya metric, photon sphere

##  I Introduction

As light cannot escape from black holes, the only possible way of observing
them is to study their influence on light or matter in their neighborhood.
When a light ray passes near a black hole, it can be deflected so strongly
that it travels in a circular orbit. In the case of a spherically symmetric
black hole these circular light rays fill a sphere around the black hole which
is known as a _photon sphere_. An observable feature, intimately related to
the existence of photon spheres, is the so-called _shadow_ of a black hole. A
major breakthrough in view of observing black holes was obtained when the
Event Horizon Telescope Collaboration published an image of a black hole in
2019 article1 ; article2 ; article3 ; article4 ; article5 ; article6 . It
shows a black disk at the center which is interpreted as the shadow. The
shadow is of great interest because its special features can be used for
distinguishing different types of black holes from each other and black holes
from other compact objects.

On the theoretical side, what we now call the shadow was first calculated for
a Schwarzschild black hole by Synge 10.1093/mnras/131.3.463  and independently
by Zeldovich and Novikov ZeldovichNovikov1966 . In a Schwarzschild space-time
with mass parameter $m$ there is a horizon at $r=2m$ and a photon sphere at
$r=3m$. An observer will see the shadow if there are light sources everywhere
around the black hole but not between the observer and the black hole. Then
all past-oriented light rays that start at the observer position can be
divided into two classes: Light rays of the first class are deflected by the
black hole and meet on their way to infinity one of the light sources, so we
associate brightness with their initial directions. Light rays of the second
class go to the horizon without meeting one of the light sources, so we
associate darkness with them. The borderline case consists of light rays that
asymptotically spiral towards a circular light ray in the photon sphere. So
the observer will see the shadow in the sky as a black circular disk whose
angular radius corresponds to the angle between light rays that spiral towards
the photon sphere and the radial direction. Neither Synge nor Zeldovich and
Novikov used the word “shadow” which became common only later. Synge called
the complement of the shadow the _escape cone of light_ ; this term is still
sometimes used today.

Based on the work by Synge or Zeldovich and Novikov, it is easy to generalize
the calculation of the shadow to an arbitrary spherically symmetric and static
metric. This was first worked out by Pande and Durgapal PandeDurgapal1986 .
The possibility of giving an analytical formula for the angular radius of the
shadow relies on the fact that in a spherically symmetric and static space-
time the equation for light-like geodesics is completely integrable, i.e.,
that there are enough constants of motion to reduce the geodesic equation to
first-order form. All one has to do is to calculate the constants of motion
for the circular light rays; as any light ray that asymptotically spirals
against such a circular light ray must have the same constants of motion, this
allows to analytically determine the boundary of the shadow in the observer’s
sky. Here it is important that a light ray can spiral towards a circular
light-like geodesic only if the latter is unstable with respect to radial
perturbations. Therefore, the determination of unstable photon spheres is of
crucial relevance for determining the shadow. For general properties of photon
spheres in spherically symmetric and static space-times we refer to Claudel et
al. ClaudelEtAl2001 .

The situation is more complicated in space-times with less symmetries. In the
case of a rotating object the space-time is only axially symmetric and
stationary, so the equation for light-like geodesics is not in general
completely integrable because there is one constant of motion less than
required. However, in some axially symmetric and static space-times there is
another constant of motion, known as a _Carter constant_ , which is not
related to a Killing vector field of the space-time metric. When a Carter
constant exists, the shadow can be analytically calculated. For a rotating
object the shadow is flattened on one side due to the dragging of light-like
geodesics by the rotating source. The most important example of a space-time
that admits a Carter constant is the Kerr metric. For a stationary observer at
infinity, the shape of the shadow of a Kerr black hole was first calculated by
Bardeen Bardeen1973TimelikeAN . There are a few other axially symmetric and
stationary space-times for which the shadow can be analytically calculated
because they admit a Carter constant. For reviews on black hole shadows, in
particular in axially symmetric and stationary space-times, we refer to Cunha
and Herdeiro CunhaHerdeiro2018  and to Perlick and Tsupko
perlick2021calculating ; related ray-tracing methods are detailed e.g. in a
book by Zink Zink2008 .

In this paper we want to consider a different generalization of spherically
symmetric and static space-times: We want to keep spherical symmetry but drop
the assumption of time-independence. More specifically, we want to calculate
the shadow of an accreting or radiating black hole. As spherical symmetry
alone does not give us enough constants of motion for complete integrability,
we will consider a special case where an additional symmetry exists. The
space-time around a spherically symmetric body that gains or loses mass can be
described by the Vaidya metric Vaidya1951 . This is a solution to Einstein’s
field equation with a null dust as the source. This null dust is either
radially ingoing, causing an increase of mass, or radially outgoing, causing a
decrease of mass. The dependence of the mass on time is coded in a mass
function $m(v)$ that can be freely chosen. In this paper we will restrict
ourselves to the case that $m(v)$ is a _linearly_ increasing or decreasing
function. The first case gives us an idealized model for an accreting black
hole, the second case for a (Hawking) radiating black hole. If $m(v)$ is
linear, the space-time admits a conformal Killing vector field (cf. Nielsen
Nielsen2014 ) which, together with the spherical symmetry, gives us enough
constants of motion for complete integrability of the equation for _light-
like_ geodesics. This allows us to analytically calculate the (time-dependent)
area of the photon sphere and the angular radius of the shadow for an
accreting and for a radiating black hole.

For related material we refer to Example 10 in Claudel et al. ClaudelEtAl2001
where the photon sphere, but not the shadow, in a piecewise defined Vaidya
metric is considered. The photon sphere and the shadow in time-dependent
space-times was also discussed in a recent paper by Mishra, Chakraborty and
Sarkar 2019 . Their study includes the Vaidya metric as a special example.
However, they did not consider the particular Vaidya metrics for which the
problem can be solved analytically. To the best of our knowledge, there are
only two previous works where an exact analytical formula for the angular
radius of the shadow was derived in a time-dependent situation: Schneider and
Perlick 2018  calculated the shadow of a collapsing dark star, and Perlick,
Tsupko and Bisnovatyi-Kogan PhysRevD.97.104062  calculated the shadow in an
expanding universe where the expansion is driven by a cosmological constant
alone. There are, however, several papers where the visual appearance of a
collapsing star was calculated without giving an analytical formula of the
shadow: Based on the pioneering work by Ames and Thorne AmesThorne1968
several authors calculated the redshift of the surface of a collapsing star.
More recent papers by Kong et al. KongMalafarinaBambi2014 ;
KongMalafarinaBambi2015  and by Ortiz et al. OrtizSarbachZannias2015a ;
OrtizSarbachZannias2015b  investigated the frequency shift of light passing
through a collapsing transparent star, thereby contrasting the collapse to a
black hole with the collapse to a naked singularity. Moreover, we also want to
direct the reader’s attention to a very recent paper by Koga et al.
KogaEtAl2022  where the shadow of an accreting black hole is calculated
numerically.

We organize the paper as follows. In section II, we investigate the evolution
of the photon sphere and the shadow of a Vaidya black hole with linearly
increasing mass function. In subsection II.1, it is our goal to determine the
light-like geodesics in this space-time where, because of the spherical
symmetry, we may restrict ourselves to the equatorial plane. To that end we
perform a specific coordinate transformation that makes the space-time
manifestly conformally static and allows us to analytically calculate the
light-like geodesics. Based on these results, we determine in subsection II.2
the location of the photon sphere in the conformally static coordinates. In
subsection II.3, we find an analytical formula for the angular radius of the
shadow as seen by a conformally static observer, i.e., by an observer whose
worldline is an integral curve of the conformal Killing vector field. We
calculate the red-shift under which one conformally static observer sees
another one in subsection II.4. Finally, in the last subsection II.5, we
analyze the results in the original coordinates. In particular, we calculate
the angular radius of the shadow as seen by an observer adapted to the
original coordinates; with respect to the conformally static observers, such
an observer is moving towards the black hole. – In section III we proceed
along the same lines for the black hole with linearly decreasing mass
function. Finally, in the discussion section IV, we summarize the results
obtained in this paper.

##  II Photon sphere and shadow of a black hole with increasing mass

The Vaidya metric Vaidya1951  describes spherically symmetric (i.e., non-
rotating) stars or black holes, emitting or absorbing a null dust. Thus, their
mass will be accordingly decreasing or increasing, in contrast to the
Schwarzschild space-time where the mass is a constant. As a consequence, the
Vaidya metric describes a dynamical space-time instead of a static space-time
as the Schwarzschild metric does. In the real world, astronomical bodies gain
mass when they absorb radiation and they lose mass when they emit radiation
which means that the space-time around them is time-dependent. The Vaidya
metric provides a comparatively simple but interesting setting for studying
the properties of such dynamical space-times, in particular the existence of a
photon sphere and the formation of a shadow.

In Eddington-Finkelstein-like coordinates, the Vaidya metric is given by the
equation

$ds^{2}=-\left(1-\frac{2m(v)}{r}\right)dv^{2}\pm
2dvdr+r^{2}(d\theta^{2}+\mathrm{sin}^{2}\theta\,d\phi^{2})\,.$ (1)

For constant $m(v)$ this reduces to the Schwarzschild metric. For non-constant
$m(v)$ the metric describes the gravitational field around a central object
with increasing or decreasing mass, where the mass depends on the coordinate
$v$. Note that the hypersurfaces $v=\mathrm{const.}$ are spanned by the light-
like vector field $\partial_{r}$ and the space-like vector fields
$\partial_{\theta}$ and $\partial_{\phi}$, so these hypersurfaces are light-
like. If the metric (1) is inserted on the left-hand side into Einstein’s
field equation, the right-hand side gives us an energy-momentum tensor of the
form

$T^{\rho\nu}=\pm\dfrac{8\pi G}{c^{4}}\,m^{\prime}(v)\,K^{\rho}K^{\nu}\,,\qquad
K^{\rho}\partial_{\rho}=\mp\,\partial_{r}\,.$ (2)

This is the energy-momentum tensor of a null dust. In (1) and (2) we have to
choose always the upper sign or always the lower sign. Note that then the
vector field $K^{\rho}\partial_{\rho}$ is always future-oriented, assuming
that the time-orientation has been chosen such that the coordinate $v$ is
increasing towards the future. If we choose the upper sign, the coordinates
are referred to as “ingoing Eddington-Finkelstein-like coordinates”. Then the
null dust is moving in the direction of decreasing $r$ and the energy-density
is positive if $m^{\prime}(v)>0$ which means that the central object is
accreting by absorbing infalling matter of positive energy density. This is
the case we will consider in this section. More specifically, we will consider
the special case that $m(v)$ is a _linearly_ increasing function, $m(v)=\mu v$
with a positive constant $\mu$. Then the metric reads

$ds^{2}=-\left(1-\frac{2\mu
v}{r}\right)dv^{2}+2dvdr+r^{2}(d\theta^{2}+\mathrm{sin}^{2}\theta\,d\phi^{2})$
(3)

and, by Einstein’s field equation,

$T^{\rho\nu}=\dfrac{8\pi G}{c^{4}}\,\mu\,K^{\rho}K^{\nu}\,,\qquad
K^{\rho}\partial_{\rho}=-\,\partial_{r}\,.$ (4)

We will see that for $0<16\mu<1$ the space-time features two horizons; it can
then be interpreted as a simple and idealized but useful model for an
accreting _black hole_. The coordinates $\theta$ and $\phi$ have their usual
range on the sphere, whereas $-\infty<v<\infty$ and $0<r<\infty$. On this
domain the metric (3) is obviously regular. As the Kretschmann scalar equals
$48\mu^{2}v^{2}/r^{6}$, there is a curvature singularity at $r=0$, i.e., it is
impossible to extend the metric beyond this range. On the domain $-\infty<v<0$
the mass is negative which should be considered as unphysical. Therefore, we
restrict our consideration to the domain $0<v<\infty$ and we assume that to
the past of this domain the space-time is given by some other metric. The
latter has not to be specified because it is irrelevant for the construction
of the shadow, as we will see below. One possibility is to assume that
$m(v)=0$ (Minkowski space-time) for $v<0$. Such models, where a singularity
forms at $v=0$, have been considered first by Kuroda Kuroda1984b  and by
Papapetrou Papapetrou1986  and are sometimes called _Vaidya-Kuroda-Papapetrou_
models. For a more detailed discussion of Vaidya metrics and the nature of
their horizons and singularities we refer the reader to Section 9.5 of the
book by Griffiths and Podolský griffiths_podolsky_2009 .

The dimensionless number $\mu$ is a measure of the accretion rate of the black
hole. An observer on a $v$-line assigns the mass $m(v_{0})$ to the black hole
when crossing the hypersurface $v=v_{0}$. Here and in the following, by an
“observer on a $v$-line” we mean an observer whose worldline has constant $r$,
$\theta$ and $\phi$ coordinates. From the metric we read that such worldlines
are indeed time-like if $r>2\mu v$ and that proper time $\tau$ is related to
the coordinate $v$ by $c^{2}d\tau^{2}=(1-2\mu v/r)dv^{2}$ where $c$ is the
vacuum speed of light. This demonstrates that $v/c$ can be identified with
proper time if the observer is far away from the black hole. As in SI units
the mass of the black hole is $M=c^{2}\mu v/G$, the accretion rate, as seen by
such a distant observer, is

$\dfrac{dM}{d\tau}\approx c\dfrac{dM}{dv}=c^{3}\mu/G\,.$ (5)

Here $G$ is Newton’s gravitational constant. For a very rough order-of-
magnitude estimate of the highest value of $\mu$ that can be expected in an
astrophysical situation, we may consider a supermassive black hole of
$10^{10}M_{\odot}$ and an accretion rate that exceeds the Eddington accretion
rate by two orders of magnitude; this is not totally unrealistic, see e.g. Pu
et al. SEAMBH2015 . This would give an accretion rate of

$\dfrac{dM}{d\tau}\approx 10^{3}M_{\odot}/\mathrm{yr}\,,\quad\mu\approx
10^{-8}\,.$ (6)

###  II.1 Light-like geodesics in the equatorial plane

To understand the behavior of light traveling in the metric (3), we formulate
the equations of motion for light-like geodesics in this metric. The metric
(3) is obviously spherically symmetric, but this symmetry alone does not
provide us with sufficiently many constants of motion for reducing the
geodesic equation to first order. Apparently there is no additional symmetry.
Actually, the general Vaidya metric (1) does not have any additional symmetry.
The more special metric (3), however, admits an additional conformal Killing
vector field, as was already discussed by Nielsen Nielsen2014 , which allows
us to analytically determine the _light-like_ geodesics. We mention that, more
precisely, a Vaidya metric (1) admits an additional conformal Killing vector
field _only_ if $m(v)$ is a linear function. This follows from the work of
Ojako et al. OjakoEtAl2020  who determined the existence of a conformal
Killing vector field in a class of space-times that includes the traditional
Vaidya metrics. As we are free to add a constant to the coordinate $v$, (3) is
indeed the most general form of a Vaidya metric with increasing mass function
where an additional conformal Killing vector field exists. To make the
existence of this conformal Killing vector field for the metric (3) explicit,
we perform the coordinate transformation

$v=r_{0}e^{cT/r_{0}}\quad\mathrm{and}\quad r=Re^{cT/r_{0}}$ (7)

where $r_{0}$ is a positive constant with the dimension of a length. In the
new coordinates the metric (3) takes the following form:

$ds^{2}=e^{2cT/r_{0}}\left\\{-\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)c^{2}dT^{2}+2cdTdR+R^{2}(d\theta^{2}+\mathrm{sin}^{2}\theta
d\phi^{2})\right\\}\,.$ (8)

We read that in this representation the metric is regular on the domain
$-\infty<T<\infty$ and $0<R<\infty$. By (7), this corresponds to the domain
$0<v<\infty$ and $0<r<\infty$ in the original coordinates. This is precisely
the domain to which we want to restrict our consideration.

From (8) it is obvious that $\partial/\partial T$ is a conformal Killing
vector field. This vector field is time-like if

$1-\frac{2\mu r_{0}}{R}-\frac{2R}{r_{0}}>0,$ (9)

i.e., on the domain

$R_{-}<R<R_{+}$ (10)

where

$R_{\pm}=\frac{r_{0}}{4}(1\pm\sqrt{1-16\mu})$ (11)

are the positions of two horizons. Obviously these are conformal Killing
horizons. In the following we assume that $16\mu<1$ to make sure that we have
a region where $\partial/\partial T$ is time-like. The inner horizon at
$R_{-}$ is then to be interpreted as a black hole horizon.

We will now determine the light-like geodesics in the metric (8). Later we
will then transform back the results to the original coordinates
$(v,r,\theta,\phi)$. The light-like geodesics in the equatorial plane obey the
following equation of motion derived from equation (8)

$0=-\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)c^{2}\dot{T}^{2}+2c\dot{T}\dot{R}+R^{2}\dot{\phi}^{2}\,.$
(12)

The crucial observation is that $\partial/\partial T$ is a conformal Killing
vector field. This gives us along every light-like geodesic the following
constant of motion $E$:

$-\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)e^{2cT/r_{0}}c\dot{T}+e^{2cT/r_{0}}\dot{R}=E\,.$
(13)

Another constant of motion $L$ derives from the fact that
$\partial/\partial\phi$ is a Killing vector field:

$e^{2cT/r_{0}}R^{2}\dot{\phi}=L\,.$ (14)

From (12) we read that _radial_ light rays ($\dot{\phi}=0$) satisfy

$c\,dT=0\,,\quad c\,dT=\frac{2dR}{1-\frac{2\mu r_{0}}{R}-\frac{2R}{r_{0}}}$
(15)

where the first equation is for ingoing and the second for outgoing radial
light rays. Integration of these equations results in

$c\,T=\mathrm{const.}\,,\quad
c\,T=\frac{r_{0}}{R_{+}-R_{-}}\left\\{R_{-}\mathrm{log}\left(\frac{R}{R_{-}}-1\right)-R_{+}\mathrm{log}\left(1-\frac{R}{R_{+}}\right)\right\\}+\mathrm{const.}$
(16)

These radial light rays are plotted in Fig. 1. The little arrows indicate the
future direction.

Figure 1: Radial light rays

For non-radial light rays we divide equations (12) and (13) by
$\dot{\phi}^{2}$ and $\dot{\phi}$, respectively, and use equation (14) to find

$-\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)c^{2}\left(\frac{dT}{d\phi}\right)^{2}+2c\frac{dT}{d\phi}\frac{dR}{d\phi}+R^{2}=0\,,$
(17) $-\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)c\frac{dT}{d\phi}+\frac{dR}{d\phi}=\frac{ER^{2}}{L}\,.$
(18)

By solving equations (17) and (18) for $\frac{dR}{d\phi}$ and
$\frac{dT}{d\phi}$, we get the following equations describing light-like
geodesics in the equatorial plane for the metric (8):

$\frac{dR}{d\phi}=\pm\sqrt{\frac{E^{2}R^{4}}{L^{2}}-R^{2}+2\mu
r_{0}R+\frac{2R^{3}}{r_{0}}}\,,$ (19)
$c\frac{dT}{d\phi}=\frac{-\frac{ER^{2}}{L}\pm\sqrt{\frac{E^{2}R^{4}}{L^{2}}-R^{2}+2\mu
r_{0}R+\frac{2R^{3}}{r_{0}}}}{1-\frac{2\mu r_{0}}{R}-\frac{2R}{r_{0}}}\,.$
(20)

Eq. (19) determines $R$ as a function of $\phi$. Thereupon, (20) can be used
to determine $T$ as a function of $\phi$.

###  II.2 The photon sphere

We will now show with the help of (19) that the metric (8) with $\mu<1/16$
admits exactly one photon sphere, and that it is located between the two
horizons. We first evaluate the condition $\frac{dR}{d\phi}=0$ which will give
us the extremum points of light paths, $R=R_{m}$. In a second step we will
then determine the radius coordinate $R_{p}$ of the photon sphere by requiring
in addition that $\frac{d^{2}R}{d\phi^{2}}=0$ at $R_{p}$. By (19) the
condition $\frac{dR}{d\phi}=0$ holds at $R=R_{m}$ if and only if

$\frac{E^{2}R_{m}^{3}}{L^{2}}-R_{m}+2\mu r_{0}+\frac{2R_{m}^{2}}{r_{0}}=0\,.$
(21)

From equation (21), we can write $\frac{E^{2}}{L^{2}}$ in terms of $R_{m}$ as
follows,

$\frac{E^{2}}{L^{2}}=\frac{R_{m}-2\mu
r_{0}-\frac{2R_{m}^{2}}{r_{0}}}{R_{m}^{3}}\,.$ (22)

Now we compute $\frac{d^{2}R}{d\phi^{2}}$ at $R=R_{m}$ from (19):

$\frac{d^{2}R}{d\phi^{2}}\bigg{|}_{R=R_{m}}=\frac{4E^{2}R_{m}^{3}}{L^{2}}-2R_{m}+2\mu
r_{0}+\frac{6R_{m}^{2}}{r_{0}}\,.$ (23)

By (22), eq. (23) results in

$\frac{d^{2}R}{d\phi^{2}}\bigg{|}_{R=R_{m}}=-\frac{2}{r_{0}}(R_{m}^{2}-r_{0}R_{m}+3\mu
r_{0}^{2})$ (24)

which can be rewritten as

$\frac{d^{2}R}{d\phi^{2}}\bigg{|}_{R=R_{m}}=-\frac{2}{r_{0}}\left\\{R_{m}-\frac{r_{0}}{2}(1-\sqrt{1-12\mu})\right\\}\left\\{R_{m}-\frac{r_{0}}{2}(1+\sqrt{1-12\mu})\right\\}\,.$
(25)

On the other hand, we can rewrite eq. (21) in the following form:

$\frac{E^{2}}{L^{2}}=\frac{2\left(R_{m}-\frac{r_{0}}{4}(1-\sqrt{1-16\mu}\right)\left(\frac{r_{0}}{4}(1+\sqrt{1-16\mu})-R_{m}\right)}{R_{m}^{3}}\,.$
(26)

Comparison of this equation with (11) shows that $\frac{E^{2}}{L^{2}}>0$ only
for $R_{-}<R_{m}<R_{+}$, i.e., turning points occur only in the region between
the two horizons where the conformal Killing vector field $\partial/\partial
T$ is time-like. Thus, light travelling inside of both the horizons will move
towards the central singularity without any extremum point and light
travelling outside of both the horizons will move towards infinity without
having any extremum point.

This observation implies that a photon sphere can exist only in the region
between the two horizons. To find the exact location of a photon sphere we
need to find a value $R_{m}$ with $R_{-}<R_{m}<R_{+}$ such that the right-hand
side of (25) vanishes. The only possible solution of this problem is the
following value of $R_{m}=R_{p}$:

$R_{p}=\frac{r_{0}}{2}(1-\sqrt{1-12\mu})\,.$ (27)

Figure 2: Light rays spiraling towards the photon sphere ($\mu=1/32$)

Thus, there is one photon sphere, located at the radius $R_{p}$ given by (27),
for the metric (8) with $0<16\mu<1$ as shown in Fig. 2. In this figure, the
two horizons are shown as black circles and lengths are given in units of
$r_{0}$.

For $R_{m}=R_{p}$ we find from (22) that

$\dfrac{E}{L}=\dfrac{\pm\sqrt{1-\dfrac{2\mu
r_{0}}{R_{p}}-2\dfrac{R_{p}}{r_{0}}}}{R_{p}}\,,$ (28)

so (20) gives us the following relation between $T$ and $\phi$ for an
equatorial light-like geodesic in the photon sphere :

$cT=\frac{\pm\,R_{p}\,\phi}{\sqrt{1-\frac{2\mu
r_{0}}{R_{p}}-\frac{2R_{p}}{r_{0}}}}\,.$ (29)

As $T$ runs over all of $\mathbb{R}$, so does $\phi$. By (14), this implies
that the light-like geodesics in the photon sphere are non-extendable: They
start at the singularity and the affine parameter then runs up to $\infty$.
The latter remark is important, in particular in view of the fact that the
coordinates $(T,R,\phi,\theta)$ cover only half of the space-time.

Light-like geodesics which have a ratio of constants $\frac{E}{L}$ given by
equation (28) are either completely contained in the photon sphere or they
spiral asymptotically towards the photon sphere as shown in Fig. 2. Here it is
important to realize that, although the metric (8) depends on $T$, the photon
sphere and the two horizons exist at _fixed_ $R$-coordinates for all $T$. Also
note that the area of the photon sphere is _not_ equal to $A=4\pi R_{p}^{2}$
with $R_{p}$ from (27); from (8) we read that this area rather depends on $T$
and is equal to

$A=4\pi r_{p}^{2}=4\pi e^{2cT/r_{0}}R_{p}^{2}=\pi
e^{2cT/r_{0}}r_{0}^{2}\big{(}1-\sqrt{1-12\mu}\,\big{)}^{2}\,,$ (30)

so it goes to 0 for $T\to-\infty$ and to $\infty$ for $T\to\infty$. Again
according to (8), along a $T$-line at the photon sphere proper time $\tau$ is
related to $T$ by

$d\tau=e^{cT/r_{0}}\sqrt{1-\dfrac{2\mu
r_{0}}{R_{p}}-\dfrac{2R_{p}}{r_{0}}}\,dT\,.$ (31)

This implies that the area radius $r_{p}$ of the photon sphere increases with
proper time at a constant rate,

$\dfrac{dr_{p}}{d\tau}=\dfrac{R_{p}c}{r_{0}\sqrt{1-\dfrac{2\mu
r_{0}}{R_{p}}-\dfrac{2R_{p}}{r_{0}}}}=\sqrt{27}\,\mu\,c\Big{(}1+O\big{(}\mu\big{)}\Big{)}\,.$
(32)

For the accretion rate according to (6) this gives a velocity of approximately
$10^{-7}c$ for the expansion of the photon sphere.

###  II.3 The angular radius of the shadow

Here we find the angular radius of the black hole shadow seen by an observer
on a $T$-line in the region between the two horizons. We will see that this
angular radius is time-independent, although the area of the photon sphere
increases with $T$. To find the angular radius of the shadow, we use the
following tetrad field for $R_{-}<R<R_{+}$.

$e_{0}=\frac{e^{-cT/r_{0}}}{\left(\sqrt{1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}}\right)c}\frac{\partial}{\partial T}\,,$ (33)
$e_{1}=\frac{e^{-cT/r_{0}}}{\left(\sqrt{1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}}\right)c}\frac{\partial}{\partial
T}+\left(\sqrt{1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}}\right)e^{-cT/r_{0}}\frac{\partial}{\partial R}\,,$
(34)
$e_{2}=\frac{e^{-cT/r_{0}}}{R}\frac{\partial}{\partial\theta}\quad\mathrm{and}\quad
e_{3}=\frac{e^{-cT/r_{0}}}{R\,\mathrm{sin}\,\theta}\frac{\partial}{\partial\phi}\,.$
(35)

In the equatorial plane we consider a light-like geodesic
$(T(\lambda),R(\lambda),\phi(\lambda))$, where $\lambda$ is an affine
parameter. We then expand the light-like geodesic’s tangent vector with
respect to the tetrad (33) to (35). Since the tangent vector is light-like,
the expansion can be written in terms of an angle $\alpha$ as follows:

$\dot{T}\frac{\partial}{\partial T}+\dot{R}\frac{\partial}{\partial
R}+\dot{\phi}\frac{\partial}{\partial\phi}=\xi\left(e_{0}+e_{1}\mathrm{cos}\,\alpha-
e_{3}\mathrm{sin}\,\alpha\right)$ (36)

where $\xi$ is a scalar factor and $\alpha$ is the angle between the light-
like geodesic and the radial direction in the rest system of the observers
governed by the tetrad (33) to (35). Comparing the coefficients of
$\frac{\partial}{\partial R}$ and $\frac{\partial}{\partial\phi}$ in (36)
yields

$\dot{R}=\mathrm{cos}\,\alpha\left(\sqrt{1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}}\right)e^{-cT/r_{0}}\,,$ (37)
$\dot{\phi}=-\frac{(\mathrm{sin}\,\alpha)e^{-cT/r_{0}}}{R}\,.$ (38)

From equations (19), (37) and (38), we have

$\frac{1}{\frac{E^{2}R^{4}}{L^{2}}-R^{2}+2\mu
r_{0}R+\frac{2R^{3}}{r_{0}}}=\frac{\mathrm{sin}^{2}\alpha}{\mathrm{cos}^{2}\alpha\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)R^{2}}\,.$ (39)

From equations (22) and (39), we can write $\mathrm{sin}\,\alpha$ in terms of
$R_{m}$ as follows:

$\mathrm{sin}\,\alpha=\sqrt{\frac{R_{m}^{3}\left(R-2\mu
r_{0}-\frac{2R^{2}}{r_{0}}\right)}{R^{3}\left(R_{m}-2\mu
r_{0}-\frac{2R_{m}^{2}}{r_{0}}\right)}}\,.$ (40)

Now the photon sphere is at the radius
$R_{p}=\frac{r_{0}}{2}(1-\sqrt{1-12\mu})$. Thus, the angular radius of the
shadow of the photon sphere for an observer at $R=R_{0}$ becomes

$\mathrm{sin}\,\alpha_{sh}=\sqrt{\frac{R_{p}^{3}\left(R_{0}-2\mu
r_{0}-\frac{2R_{0}^{2}}{r_{0}}\right)}{R_{0}^{3}\left(R_{p}-2\mu
r_{0}-\frac{2R_{p}^{2}}{r_{0}}\right)}}=\sqrt{\frac{R_{p}^{3}(R_{0}-R_{-})(R_{+}-R_{0})}{R_{0}^{3}(R_{p}-R_{-})(R_{+}-R_{p})}}\,.$
(41)

Of course, the expression under the square-root must be between 0 and 1 which
is indeed the case for $R_{-}<R_{0}<R_{+}$, i.e., in the region where the
conformally static observers exist. Eq. (41) must be supplemented with the
information that $\alpha_{\mathrm{sh}}$ is in the interval between 0 and
$\pi/2$ for $R_{p}<R_{0}<R_{+}$ and in the interval between $\pi/2$ and $\pi$
for $R_{-}<R_{0}<R_{p}$. We have $\alpha_{\mathrm{sh}}\to 0$ (bright sky) for
$R_{0}\to R_{+}$ and $\alpha_{\mathrm{sh}}\to\pi$ (dark sky) for $R_{0}\to
R_{-}$. At $R_{0}=R_{p}$ the shadow covers half of the sky which is obvious
without any calculation. In Fig. 3, we have plotted the angular radius of the
black hole shadow $\alpha_{sh}$ as a function of $R_{0}$. The solid black
lines labeled $R_{-}$ and $R_{+}$ mark the inner and outer horizon,
respectively, and the dashed black line labeled $R_{p}$ represents the photon
sphere. $R_{0}$ is given in units of $r_{0}$.

As an alternative, the shadow formula (41) can be rewritten as

$\mathrm{cos}\,\alpha_{sh}=\dfrac{(R_{0}-R_{p})}{R_{0}}\sqrt{1+\frac{2\mu
r_{0}R_{p}}{R_{0}(4\mu r_{0}-R_{p})}}\,.$ (42)

This version gives a unique value for $\alpha_{\mathrm{sh}}$ in the interval
between 0 and $\pi$, for all $R_{-}<R_{0}<R_{+}$.

Figure 3: Angular radius of the shadow ($\mu=1/24$)

With the shadow known for a conformally static observer, the shadow can be
calculated for an observer in arbitrary motion with the help of the aberration
formula. We will exemplify this in Section II.5 below. As the aberration
formula maps circles in the sky onto circles in the sky, _any_ observer in the
region between the two horizons sees a circular shadow.

###  II.4 The red-shift of light

In this section we compute the red-shift in the metric (8). We assume two
observers, labeled $A$ and $B$ respectively, who are both moving on $T$-lines.
This is of course only possible in the region between the two horizons. The
red-shift $z=\frac{\nu_{A}}{\nu_{B}}-1$ is determined by the frequency-ratio

$\frac{\nu_{A}}{\nu_{B}}=\sqrt{\frac{g_{00}(T_{B},R_{B})}{g_{00}(T_{A},R_{A})}}$
(43)

where $\nu_{A}$ and $\nu_{B}$ are the frequency of the light at
$(T_{A},R_{A},\theta_{A},\phi_{A})$ and $(T_{B},R_{B},\theta_{B},\phi_{B})$,
respectively. From (8), equation (43) becomes

$\frac{\nu_{A}}{\nu_{B}}=\frac{\sqrt{1-\frac{2\mu
r_{0}}{R_{B}}-\frac{2R_{B}}{r_{0}}}}{\sqrt{1-\frac{2\mu
r_{0}}{R_{A}}-\frac{2R_{A}}{r_{0}}}}\,e^{c(T_{B}-T_{A})/r_{0}}\,.$ (44)

As $\partial/\partial T$ is a conformal Killing vector field, $T_{B}-T_{A}$
has the same value for all light rays from $A$ to $B$, i.e., the red-shift is
time-independent. Note that this result comes from the fact that the conformal
factor in front of the metric (8) is an exponential function of $T$. If this
factor would depend on $T$ in any other form, $T_{B}$ and $T_{A}$ would enter
not only in the form of their difference and the red-shift between two
observers on $T$-lines would depend on $T$.

We now specify to the case that the considered light rays travel in the radial
direction which requires $(\theta_{A},\phi_{A})=(\theta_{B},\phi_{B})$. Then
we find from (16) that (44) becomes

$\frac{\nu_{A}}{\nu_{B}}=\frac{\sqrt{1-\frac{2\mu
r_{0}}{R_{B}}-\frac{2R_{B}}{r_{0}}}}{\sqrt{1-\frac{2\mu
r_{0}}{R_{A}}-\frac{2R_{A}}{r_{0}}}}$ (45)

for ingoing and

$\frac{\nu_{A}}{\nu_{B}}=\frac{\sqrt{1-\frac{2\mu
r_{0}}{R_{B}}-\frac{2R_{B}}{r_{0}}}}{\sqrt{1-\frac{2\mu
r_{0}}{R_{A}}-\frac{2R_{A}}{r_{0}}}}\left\\{\left(\frac{R_{B}-R_{-}}{R_{A}-R_{-}}\right)^{\frac{R_{-}}{R_{+}-R_{-}}}\left(\frac{R_{+}-R_{A}}{R_{+}-R_{B}}\right)^{\frac{R_{+}}{R_{+}-R_{-}}}\right\\}$
(46)

for outgoing light rays.

###  II.5 Analysis in the original coordinates

Now we transform back the obtained results to the original coordinate system
$(v,r,\theta,\phi)$. This allows us to analyze the behavior of the photon
sphere as seen by observers on $v$-lines (lines of constant $r,\theta,\phi$).

From (3) we read that the $v$-lines are time-like on the domain where

$2\mu v<r\,,\quad 2\mu r_{0}<R\,.$ (47)

This domain includes the region between the horizons. The $v$-lines are
plotted as dashed curves in Fig. 5. The domain where the $v$-lines fail to be
time-like (i.e., where (47) is violated) is shown in dark gray. The other
features of Fig. 5 will be explained below. We see that the observers on
$v$-lines come from infinity, first cross the outer horizon at $R_{+}$, then
the photon sphere at $R_{p}$ and then the inner horizon at $R_{-}$.

For analyzing the situation in the coordinates $(v,r,\theta,\phi)$ it is
important to realize that the hypersurfaces $v=\mathrm{const.}$ are light-like
everywhere and that $r$ is an _area coordinate_ , i.e., that the sphere
$(v,r)=\mathrm{const.}$ has area $4\pi r^{2}$, as can be read from (3).

We have found above that in the $(T,R,\theta,\phi)$ coordinates light paths in
the photon sphere are at $R=R_{p}=\frac{r_{0}}{2}(1-\sqrt{1-12\mu})$ and that
in the equatorial plane their $T$ and $\phi$ coordinates are related by (29).
Transforming these two equations back to the original coordinates
$(v,r,\theta,\phi)$ with the help of (7) results in

$r=\frac{R_{p}v}{r_{0}}\quad\mathrm{and}\quad\phi=\pm\left(\frac{r_{0}}{R_{p}}\sqrt{1-\frac{2\mu
r_{0}}{R_{p}}-\frac{2R_{p}}{r_{0}}}\right)\mathrm{log}\left(\frac{v}{r_{0}}\right)\,.$
(48)

These equations give us the path of the geodesic parametrized by $v$. We see
that in these coordinates the light-like geodesics in the photon sphere spiral
outwards, see Fig. 4, i.e., that the photon sphere expands. In this figure
lengths are given in units of $r_{0}$. The radius coordinate $r$ is linearly
increasing with $v$. The angle $\phi$ is a logarithmic function of $v$, so the
angular speed $\frac{d\phi}{dv}$ is inversely proportional to $v$.

Figure 4: Light path in the photon sphere in $(v,r,\theta,\phi)$ coordinates
($\mu=1/24$)

Whereas $v$ is appropriate for parametrizing individual light rays, we have to
keep in mind that the hypersurfaces $v=\mathrm{const.}$ are light-like, i.e.,
that they cannot be interpreted as equal-time slices. For describing the
expansion of the entire photon sphere, as observed by observers on $v$-lines,
we have to introduce an appropriate time function. We define such a function
$t$ in the $(T,R,\theta,\phi)$ coordinates by the equation

$c\,dt=c\,dT-\dfrac{dR}{1-\dfrac{2\mu r_{0}}{R}-\dfrac{R}{r_{0}}}\,.$ (49)

With the help of the relation

$\dfrac{\partial}{\partial
v}=e^{-cT/r_{0}}\dfrac{1}{c}\Big{(}\dfrac{\partial}{\partial
T}-\dfrac{cR}{r_{0}}\dfrac{\partial}{\partial R}\Big{)}$ (50)

which follows from (7), it is easy to verify that the hypersurfaces
$t=\mathrm{const.}$ are orthogonal to the $v$-lines, i.e., that events in such
a hypersurface happen simultaneously for the observers on $v$-lines.
Integration of (49) results in

$c\,t=c\,T-r_{0}\big{(}F(R)-F(R_{p})\big{)}$ (51)

with

$F(R)=\dfrac{1}{2}\,\mathrm{log}\dfrac{\Big{(}\sqrt{1-8\mu}+\dfrac{2R}{r_{0}}-1\Big{)}^{(1-8\mu)^{-1/2}-1}}{\Big{(}\sqrt{1-8\mu}-\dfrac{2R}{r_{0}}+1\Big{)}^{(1-8\mu)^{-1/2}+1}}\,.$
(52)

In (51) we have chosen the integration constant such that $t$ coincides with
$T$ on the photon sphere.

The hypersurfaces $t=\mathrm{const.}$ are shown as solid curves in Fig. 5. The
time function $t$ runs from $-\infty$ to $+\infty$ on the domain
$\frac{r_{0}}{2}\big{(}1-\sqrt{1-8\mu}\big{)}<R<\frac{r_{0}}{2}\big{(}1+\sqrt{1-8\mu}\big{)}$.
Outside of this domain there is a region where $v$ is time-like but $t$ is
running backwards; this region is shown in light gray in Fig. 5. Also note
that $t$ does _not_ give proper time along the $v$-lines: The vector field
$\partial/\partial v$ is synchronizable but not proper-time synchronizable.

Figure 5: $v$-lines (dashed) and hypersurfaces $t=\mathrm{const.}$ (solid)

It is now easy to calculate the expansion of the photon sphere as observed by
the observers on $v$-lines. If we restrict (51) to the hypersurface $R=R_{p}$
we find

$c\,t=c\,T=r_{0}\,\mathrm{log}\dfrac{r}{R_{p}}$ (53)

where the second equality follows from (8). Solving for $r$ results in

$r=R_{p}\,e^{ct/r_{0}}$ (54)

which demonstrates that the observers on $v$-lines see the photon sphere
exponentially expanding with time $t$. As $r$ is an area coordinate, this
result is in agreement with (30).

Similarly, we can calculate the expansion of any other sphere
$R=\mathrm{const.}$ in the domain (47). We can do this in particular for the
horizons at $R_{\pm}$ which results in

$r=R_{\pm}\,e^{F(R_{\pm})-F(R_{p})}\,e^{ct/r_{0}}\,.$ (55)

So the horizons expand exponentially, as any other sphere $R=\mathrm{const.}$
does. We have plotted the expansion of the photon sphere and of the horizons
in Fig. 6, where the inner dark region is bounded by the inner horizon, the
inner (orange) circle represents the photon sphere and the outer (black)
circle represents the outer horizon. Lengths are given in units of $r_{0}$ and
times are given in units of $r_{0}/c$. In our theoretical model the initial
radius of the photon sphere is zero at $t=-\infty$ and its radius becomes
infinite at $t=\infty$.

Figure 6: Evolution of photon sphere as seen by observer on a $v$-line
($\mu=1/24$)

We will now calculate the angular radius of the shadow as seen by an observer
on a $v$-line. We call this angular radius $\tilde{\alpha}{}_{\mathrm{sh}}$,
and we relate it to the angular radius of the shadow as seen by a conformally
static observer, $\alpha_{\mathrm{sh}}$, by the special-relativistic
aberration formula

$\mathrm{tan}^{2}\Big{(}\frac{\tilde{\alpha}{}_{\mathrm{sh}}}{2}\Big{)}=\dfrac{(c-V)}{(c+V)}\,\mathrm{tan}^{2}\Big{(}\frac{\alpha_{\mathrm{sh}}}{2}\Big{)}=\dfrac{(c-V)}{(c+V)}\,\dfrac{\big{(}1-\mathrm{cos}\,\alpha_{\mathrm{sh}}\big{)}^{2}}{\mathrm{sin}^{2}\alpha_{\mathrm{sh}}}\,.$
(56)

Here $\alpha_{\mathrm{sh}}$ is given by (41) or (42), and $V$ is the momentary
3-velocity of the observer on a $v$-line with respect to the observer on a
$T$-line. The latter has 4-velocity

$U=\dfrac{e^{-cT/r_{0}}}{\sqrt{1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}}}\,\frac{\partial}{\partial T}\,,\quad
g(U,U)=-c^{2}$ (57)

whereas, by (50), the former has 4-velocity

$\tilde{U}=\dfrac{e^{-cT/r_{0}}}{\sqrt{1-\frac{2\mu
r_{0}}{R}}}\Big{(}\frac{\partial}{\partial
T}-\frac{cR}{r_{0}}\,\frac{\partial}{\partial R}\Big{)}\,,\quad
g\big{(}\tilde{U},\tilde{U}\big{)}=-c^{2}\,.$ (58)

By applying, at each moment, the special-relativistic formula

$g\big{(}U,\tilde{U}\big{)}=\dfrac{-c^{2}}{\sqrt{1-\Big{(}\frac{V}{c}\Big{)}^{2}}}$
(59)

we find that the relative 3-velocity $V$ of the two observers is given by

$\dfrac{V}{c}=\dfrac{R_{0}^{2}}{R_{0}^{2}+2(R_{+}-R_{0})(R_{0}-R_{-})}$ (60)

at $R=R_{0}$. Plugging (60) and (41) into the aberration formula (56) results
in

$\mathrm{tan}^{2}\Big{(}\frac{\tilde{\alpha}{}_{\mathrm{sh}}}{2}\Big{)}=\dfrac{R_{0}^{3}(4\mu
r_{0}-R_{p})}{R_{p}^{3}(R_{0}-2\mu
r_{0})}\,\big{(}1-\mathrm{cos}\,\alpha_{\mathrm{sh}}\big{)}^{2}\,.$ (61)

This formula is valid for $R_{-}<R_{0}<R_{+}$., i.e., on the domain where both
the $T$-lines and the $v$-lines are time-like. Note that on this domain
$R_{0}>2\mu r_{0}$ and that, because of our assumption $0<\mu<1/16$, also
$4\mu r_{o}>R_{p}$. We read from (61) that for $R_{0}\to R_{+}$, where
$\alpha_{\mathrm{sh}}\to 0$, also $\tilde{\alpha}{}_{\mathrm{sh}}\to 0$. For
$R_{0}\to R_{-}$, where $\alpha_{\mathrm{sh}}\to\pi$, however,
$\tilde{\alpha}{}_{\mathrm{sh}}$ approaches a finite value. By inserting (42)
we see that (61) can be analytically extended to all values $2\mu
r_{0}<R_{0}<R_{+}$, i.e., to the entire domain where the $v$-lines are time-
like,

$\mathrm{tan}^{2}\Big{(}\frac{\tilde{\alpha}{}_{\mathrm{sh}}}{2}\Big{)}=\dfrac{R_{0}^{3}(4\mu
r_{0}-R_{p})}{R_{p}^{3}(R_{0}-2\mu
r_{0})}\,\Bigg{(}1-\dfrac{(R_{0}-R_{p})}{R_{0}}\sqrt{1+\dfrac{2\mu
r_{0}R_{p}}{R_{0}\big{(}4\mu r_{0}-R_{p}\big{)}}}\;\Bigg{)}^{2}\,.$ (62)

In Fig. 7 $\tilde{\alpha}{}_{\mathrm{sh}}$ is plotted against $R_{0}$, with
the latter given in units of $r_{0}$. So an observer on a $v$-line sees the
shadow come into existence when crossing the outer horizon. The shadow
gradually grows while the observer moves inwards. It covers less than half of
the sky at $R_{0}=R_{p}$, which reflects the fact that aberration has a
demagnifying effect in the forward direction. Nothing peculiar happens when
the observer crosses the inner horizon. The shadow covers the entire sky in
the limit $R\to 2\mu r_{0}$, i.e., when the observer reaches the speed of
light.

Figure 7: Angular radius of shadow as seen by observer on $v$-line
($\mu=1/24$)

##  III Photon sphere and shadow of a black hole with decreasing mass

We will now consider the case that the mass function $m(v)$ is decreasing
where we will again assume that $m(v)$ is a linear function, $m(v)=-av$ with a
positive constant $a$. A Vaidya metric with a decreasing mass function may be
viewed as a simplified model for a black hole that evaporates by emitting
Hawking radiation, as was suggested already in the 1980s, see Hiscock
Hiscock1981a ; Hiscock1981b , Kuroda Kuroda1984a  and Beciu Beciu1984 . There
are two different ways in which an evaporating black hole can be modeled by a
Vaidya metric. One possibility is to use (1) and (2) with the lower sign,
i.e., to use outgoing Eddington-Finkelstein-like coordinates. Then the central
object would lose mass by emitting a null dust of positive energy density. For
a linearly decreasing mass function the space-time region covered by the
$(v,r,\theta,\phi)$ coordinates would be just the time-reversed version of the
space-time region considered in the previous section. That is to say, all
results would literally carry over if we just replace $v$ by $-v$ and $T$ by
$-T$ everywhere. In particular, we would again have two horizons in the space-
time region covered by our coordinates, with the inner one now being a _white-
hole_ horizon.

Here we will consider another possibility. We will use (1) and (2) again with
the upper sign. Doing so with a _decreasing_ function $m(v)$ means that the
null dust is again ingoing but with a negative energy density. We will view
such an ingoing null dust with negative energy density as a model for Hawking
radiation. Of course, this model is (over-)simplified, in particular because
Hawking radiation consists of a superposition of ingoing negative energy
density and outgoing positive energy density. Unfortunately, such a
superposition cannot be modeled by a single Vaidya metric. (The simplest model
that describes ingoing negative energy density and outgoing positive energy
density requires patching together two Vaidya metrics, see Hiscock
Hiscock1981b .) In any case, we believe that studying the case of a black hole
that loses mass by absorbing negative energy density is of some conceptual
interest. We will see that this situation is considerably different from the
above-mentioned case which is just the time-reversed version of the accreting
black hole treated in the previous section. In particular, we will see that
now we have only one horizon instead of two.

According to this plan, we consider the metric

$ds^{2}=-\left(1+\frac{2av}{r}\right)dv^{2}+2\,dv\,dr+r^{2}(d\theta^{2}+\mathrm{sin}^{2}\theta\,d\phi^{2})$
(63)

where the corresponding energy-momentum tensor (2) is

$T^{\mu\nu}=-\dfrac{8\pi G}{c^{4}}\,a\,K^{\mu}K^{\nu}\,,\qquad
K^{\mu}\partial_{\mu}=-\partial_{r}\,.$ (64)

The metric is regular for $-\infty<v<\infty$ and $0<r<\infty$, whereas
$\theta$ and $\phi$ have their usual range on the sphere as before. However,
as the mass is negative for $0<v<\infty$, we consider the metric only on the
domain $-\infty<v<0$. That is to say, the central object starts with an
infinite mass (which is admittedly an idealization) and then radiates until
the mass has gone down to zero. From this moment on we think of the space-time
as being of some other form which we need not specify because it is irrelevant
for the shadow.

We have already emphasized that, in our view, the calculations in this Section
are of some conceptual interest. In view of astrophysical observations, the
effect is so small that there is no chance to observe it with stellar or
supermassive black holes: It is known that for a black hole of one Solar mass
Hawking radiation produces a mass loss of
$dM/d\tau\approx-10^{-67}M_{\odot}/\mathrm{yr}$. As the mass loss scales with
$M^{-2}$, it is even 20 orders of magnitude smaller for a supermassive black
hole of $10^{10}$ Solar masses.

###  III.1 Light-like geodesics in the equatorial plane

As in the case of increasing mass, the metric admits a conformal Killing
vector field, i.e., we can make a cordinate transformation $(v,r,\theta,\phi)$
$\rightarrow$ $(T,R,\theta,\phi)$ such that the metric becomes manifestly
conformally static. In the case at hand, this transformation reads

$v=-r_{0}\,e^{-cT/r_{0}}\quad\mathrm{and}\quad r=R\,e^{-cT/r_{0}}$ (65)

where $r_{0}$ is a constant and $c$ is the speed of light. In the new
coordinates the metric (63) takes the following form:

$ds^{2}=e^{-2cT/r_{0}}\left\\{-\left(1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}\right)c^{2}dT^{2}+2\,c\,dT\,dR+R^{2}(d\theta^{2}+{\mathrm{sin}}^{2}\theta\,d\phi^{2})\right\\}\,.$
(66)

The metric (66) is regular on the domain $-\infty<T<\infty$ and $0<R<\infty$.
From equation (65) it can be seen that on this domain the original coordinates
have the range of $-\infty<v<0$ and $0<r<\infty$. This is precisely the domain
to which we restrict our consideration.

The conformal Killing vector field $\partial/\partial T$ is time-like if

$1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}>0$ (67)

i.e., on the domain

$R>R_{h}$ (68)

where

$R_{h}=\frac{r_{0}}{4}(-1+\sqrt{1+16a})$ (69)

is the position of the horizon. Again, it is obvious that this is a conformal
Killing horizon. Thus, in this case there is only one horizon, for any $a>0$,
instead of two as in the previous case.

The light-like geodesics in the equatorial plane obey the following condition
derived from (66):

$0=-\left(1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}\right)c^{2}\dot{T}^{2}+2c\dot{T}\dot{R}+R^{2}\dot{\phi}^{2}\,.$
(70)

The conformal Killing vector field $\partial/\partial T$ and the Killing
vector field $\partial/\partial\phi$ give us the following two constants of
motion $E$ and $L$:

$-\left(1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}\right)e^{-2cT/r_{0}}c\dot{T}+e^{-2cT/r_{0}}\dot{R}=E\,,$
(71) $L=e^{-2cT/r_{0}}R^{2}\dot{\phi}\,.$ (72)

From (70) we read that _radial_ light rays ($\dot{\phi}=0$) satisfy

$c\,dT=0\,,\quad c\,dT=\frac{2dR}{1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}}$ (73)

where the first equation is for ingoing and the second for outgoing radial
light rays. Integration of these equations results in

$c\,T=\mathrm{const.}\,,\quad
c\,T=\frac{r_{0}}{k+R_{h}}\left\\{k\,\mathrm{log}\left(\frac{R}{k}+1\right)+R_{h}\mathrm{log}\left(\frac{R}{R_{h}}-1\right)\right\\}+\mathrm{const.}$
(74)

with $R_{h}$ from (69) and

$k=\frac{r_{0}}{4}\left(1+\sqrt{1+16a}\right)\,.$ (75)

The ingoing and outgoing radial light-like geodesics are plotted in Fig. 8.
Again, the little arrows indicate the future direction.

Figure 8: Radial light rays

For non-radial light rays we divide equations (70) and (71) by
$\dot{\phi}^{2}$ and $\dot{\phi}$, respectively, and use equation (72) to find

$-\left(1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}\right)c^{2}\left(\frac{dT}{d\phi}\right)^{2}+2c\frac{dT}{d\phi}\frac{dR}{d\phi}+R^{2}=0\,,$
(76)

$-\left(1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}\right)c\frac{dT}{d\phi}+\frac{dR}{d\phi}=\frac{ER^{2}}{L}\,.$
(77)

By solving equations (76) and (77) for $\frac{dT}{d\phi}$ and
$\frac{dR}{d\phi}$, we obtain the following equations describing light-like
geodesics in the equatorial plane for the metric (66):

$\frac{dR}{d\phi}=\pm\sqrt{\frac{E^{2}R^{4}}{L^{2}}-R^{2}+2ar_{0}R-\frac{2R^{3}}{r_{0}}}\,,$
(78)

$c\frac{dT}{d\phi}=\frac{-\frac{ER^{2}}{L}\pm\sqrt{\frac{E^{2}R^{4}}{L^{2}}-R^{2}+2ar_{0}R-\frac{2R^{3}}{r_{0}}}}{1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}}\,.$
(79)

Eq. (78) determines $R$ as a function of $\phi$. Thereupon, (79) can be used
to determine $T$ as a function of $\phi$.

###  III.2 The photon sphere

We will now show with the help of (78) that the metric (66) admits exactly one
photon sphere, located outside of the horizon. As mentioned in the previous
case, the condition $\frac{dR}{d\phi}=0$ gives the extremum points of light
paths, $R=R_{m}$. Then the radius coordinate $R_{p}$ of the photon sphere can
be calculated from the additional condition $\frac{d^{2}R}{d\phi^{2}}=0$. By
(78) the condition $\frac{dR}{d\phi}=0$ holds at $R=R_{m}$ if

$\frac{E^{2}R_{m}^{3}}{L^{2}}-R_{m}+2ar_{0}-\frac{2R_{m}^{2}}{r_{0}}=0\,.$
(80)

This equation allows us to express $\frac{E^{2}}{L^{2}}$ in terms of $R_{m}$,

$\frac{E^{2}}{L^{2}}=\frac{R_{m}-2ar_{0}+\frac{2R_{m}^{2}}{r_{0}}}{R_{m}^{3}}\,.$
(81)

Now we compute $\frac{d^{2}R}{d\phi^{2}}$ at $R=R_{m}$ from (78):

$\frac{d^{2}R}{d\phi^{2}}\bigg{|}_{R=R_{m}}=\frac{4E^{2}R_{m}^{3}}{L^{2}}-2R_{m}+2ar_{0}-\frac{6R_{m}^{2}}{r_{0}}\,.$
(82)

From (81), equation (82) can be written as

$\frac{d^{2}R}{d\phi^{2}}\bigg{|}_{R=R_{m}}=\frac{2}{r_{0}}\left\\{R_{m}-\frac{r_{0}}{2}(-1-\sqrt{1+12a})\right\\}\left\\{R_{m}-\frac{r_{0}}{2}(-1+\sqrt{1+12a})\right\\}\,.$
(83)

To find the location of the photon sphere, we have to equate the right-hand
side of (83) to zero. As $R_{m}>0$, the only solution $R_{m}=R_{p}$ is

$R_{p}=\frac{r_{0}}{2}(-1+\sqrt{1+12a})\,.$ (84)

This is the only possible value for a photon sphere. We still have to check if
the condition $\frac{E^{2}}{L^{2}}>0$ is satisfied at $R_{m}=R_{p}$. By
rewriting equation (81) as

$\frac{E^{2}}{L^{2}}=\frac{2}{r_{0}}\frac{\left(R_{m}-\frac{r_{0}}{4}(-1-\sqrt{1+16a})\right)\left(R_{m}-\frac{r_{0}}{4}(-1+\sqrt{1+16a})\right)}{R_{m}^{3}}\,,$
(85)

we see that this condition requires $R_{m}>\frac{r_{0}}{4}(-1+\sqrt{1+16a})$
which is indeed satisfied by $R_{m}=R_{p}$ . We summarize these findings in
the following way: For any choice of the parameter $a>0$, there is one
horizon, at radius $R_{h}$ given by (69), and one photon sphere, at radius
$R_{p}$ given by (84) which is always between the horizon and infinity, see
Fig. 9.

Figure 9: Light rays spiraling towards the photon sphere ($a=1/12$)

To find the relation between $T$ and $\phi$ for an equatorial light-like
geodesics in the photon sphere we insert $R_{m}=R_{p}$ into (81) which yields

$\frac{E}{L}=\frac{\pm\sqrt{1-\frac{2ar_{0}}{R_{p}}+2\frac{R_{p}}{r_{0}}}}{R_{p}}\,.$
(86)

By (79) gives us the desired relation,

$cT=\frac{\pm\,R_{p}\,\phi}{\sqrt{1-\frac{2ar_{0}}{R_{p}}+\frac{2R_{p}}{r_{0}}}}\,.$
(87)

As $T$ runs over all of $\mathbb{R}$, so does $\phi$. By (72), this implies
that also in this case the light-like geodesics in the photon sphere are non-
extendable: The affine parameter starts at $-\infty$ and then runs up to a
value where the singularity is reached. We repeat that this observation is
non-trivial because the coordinates $(T,R,\phi,\theta)$ cover only a part of
the original space-time (63).

Equation (86) gives us the ratio of constants $\frac{E}{L}$ for equatorial
light-like geodesics that are completely contained in the photon sphere, and
also of such geodesics that spiral asymptotically towards the photon sphere,
see Fig. 9. In this figure the dashed line represents the horizon and lengths
are given in units of $r_{0}$. As in the previous case, the $R$-coordinate of
the photon sphere is fixed (as is the $R$-coordinate of the horizon), but by
(66) the area $A$ of the photon sphere depends on $T$,

$A=4\pi e^{-2cT/r_{0}}R_{p}^{2}=\pi
e^{-2cT/r_{0}}r_{0}^{2}\big{(}-1+\sqrt{1+12a}\,\big{)}^{2}\,,$ (88)

so it goes to $\infty$ for $T\to-\infty$ and to $0$ for $T\to\infty$.

###  III.3 The angular radius of the black-hole shadow

In this Section we calculate the angular radius of the black-hole shadow as it
is seen by an observer on a $T$-line. Of course, the observer must be between
the horizon and infinity. We will see that the angular radius is time-
independent, although the area of the photon sphere decreases with $T$. We use
the following tetrad field on the domain $R>R_{h}.$

$e_{0}=\frac{e^{cT/r_{0}}}{\left(\sqrt{1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}}\right)c}\frac{\partial}{\partial
T}\,,$ (89)

$e_{1}=\frac{e^{cT/r_{0}}}{\left(\sqrt{1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}}\right)c}\frac{\partial}{\partial
T}+\left(\sqrt{1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}}\right)e^{cT/r_{0}}\frac{\partial}{\partial
R}\,,$ (90)

$e_{2}=\frac{e^{cT/r_{0}}}{R}\frac{\partial}{\partial\theta}\quad\mathrm{and}\quad
e_{3}=\frac{e^{cT/r_{0}}}{Rsin\theta}\frac{\partial}{\partial\phi}\,.$ (91)

Expanding the tangent vector of an equatorial light-like geodesic
$(T(\lambda),R(\lambda),\phi(\lambda))$, where $\lambda$ is an affine
parameter, with respect to this tetrad results in

$\dot{T}\frac{\partial}{\partial T}+\dot{R}\frac{\partial}{\partial
R}+\dot{\phi}\frac{\partial}{\partial\phi}=\xi\left(e_{0}+e_{1}\mathrm{cos}\,\alpha-
e_{3}\mathrm{sin}\,\alpha\right)$ (92)

where $\xi$ is a scalar factor and $\alpha$ is the angle between the light-
like geodesic and the radial direction in the rest system of the observers
with four-velocity $e_{0}$. Comparing the coefficients of
$\frac{\partial}{\partial R}$ and $\frac{\partial}{\partial\phi}$ in (92)
yields

$\dot{R}=\mathrm{cos}\,\alpha\left(\sqrt{1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}}\right)e^{cT/r_{0}}\,,$
(93) $\dot{\phi}=-\frac{(\mathrm{sin}\,\alpha)e^{cT/r_{0}}}{R}\,.$ (94)

From equations (78), (93) and (94), we have

$\frac{1}{\frac{E^{2}R^{4}}{L^{2}}-R^{2}+2ar_{0}R-\frac{2R^{3}}{r_{0}}}=\frac{\mathrm{sin}^{2}\alpha}{\mathrm{cos}^{2}\alpha\left(1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}\right)R^{2}}$
(95)

With the help of (81) and (95), we write $\mathrm{sin}\,\alpha$ as follows:

$\mathrm{sin}\,\alpha=\sqrt{\frac{R_{m}^{3}\left(R-2ar_{0}+\frac{2R^{2}}{r_{0}}\right)}{R^{3}\left(R_{m}-2ar_{0}+\frac{2R_{m}^{2}}{r_{0}}\right)}}\,.$
(96)

The angular radius $\alpha_{\mathrm{sh}}$ of the shadow for an observer at
$R=R_{0}$ is found by inserting into this equation the radius $R_{m}=R_{p}$ of
the photon sphere,

$\mathrm{sin}\,\alpha_{\mathrm{sh}}=\sqrt{\frac{R_{p}^{3}\left(R_{0}-2ar_{0}+\frac{2R_{0}^{2}}{r_{0}}\right)}{R_{0}^{3}\left(R_{p}-2ar_{0}+\frac{2R_{p}^{2}}{r_{0}}\right)}}=\sqrt{\frac{R_{p}^{3}(R_{0}+k)(R_{0}-R_{h})}{R_{0}^{3}(R_{p}+k)(R_{p}-R_{h})}}$
(97)

with $R_{h}$ from (69) and $k$ from (75). This equation is valid for all
observer positions $R_{0}>R_{h}$ where the expression under the square-root is
indeed between 0 and 1. Eq. (97) must be supplemented with the information
that $\alpha_{\mathrm{sh}}$ is in the interval between 0 and $\pi/2$ for
$R_{0}>R_{p}$ and in the interval between $\pi/2$ and $\pi$ for
$R_{h}<R_{0}<R_{p}$. We have $\alpha_{\mathrm{sh}}\to 0$ (bright sky) for
$R_{0}\to\infty$ and $\alpha_{\mathrm{sh}}\to\pi$ (dark sky) for $R_{0}\to
R_{h}$. At $R_{0}=R_{p}$ the shadow covers half of the sky which is obvious
without any calculation for this case too. Fig. 10 shows the angular radius of
the black-hole shadow $\alpha_{\mathrm{sh}}$ as a function of $R_{0}$. The
solid black line labeled $R_{h}$ marks the horizon, the dashed black line
labeled $R_{p}$ represents the photon sphere and $R_{0}$ is given in units of
$r_{0}$. For this case the domain of interest is outside of the horizon
because inside the horizon no conformally static observer can exist.

As for the accreting black hole, is it also possible to give a formula for the
cosine, rather than the sine, of $\alpha_{\mathrm{sh}}$,

$\mathrm{cos}\,\alpha_{sh}=\dfrac{(R_{0}-R_{p})}{R_{0}}\sqrt{1+\frac{2ar_{0}R_{p}}{R_{0}(4ar_{0}-R_{p})}}\,.$
(98)

This equation determines $\alpha_{\mathrm{sh}}$ uniquely in the interval
between 0 and $\pi$, for all $R_{h}<R_{0}<\infty$.

In contrast to the case of an accreting black hole, we don’t have an outer
horizon, so we can consider conformally static observers that are arbitrarily
far away from the black hole. For $R_{0}\gg R_{p}$ we may rewrite eq. (97) as

$\mathrm{sin}\,\alpha_{\mathrm{sh}}=\sqrt{\dfrac{2\,R_{p}^{2}}{R_{p}r_{0}-2ar_{0}^{2}+2R_{p}^{2}}}\;\sqrt{\dfrac{R_{p}}{R_{0}}}\,\Bigg{(}1+\dfrac{r_{0}}{4R_{p}}\,\dfrac{R_{p}}{R_{0}}+O\Big{(}\dfrac{R_{p}^{2}}{R_{0}^{2}}\Big{)}\Bigg{)}$
(99)

and neglect the terms of second and higher order. We see that the sine of the
angular radius of the shadow falls off with $R_{0}^{-1/2}$ in leading order.
This is in contrast to the Schwarzschild case where it falls off, for a static
observer at Schwarzschild coordinate $R_{0}$, with $R_{0}^{-1}$, see Synge
10.1093/mnras/131.3.463 .

Figure 10: Angular radius of the shadow ($a=1/24$)

###  III.4 The gravitational red-shift of light

In this section we compute the gravitational red-shift of light in the metric
(66). We assume that two observers, labeled $A$ and $B$ respectively, are
moving on $T$-lines in the domain outside of the horizon. Then

$\frac{\nu_{A}}{\nu_{B}}=\sqrt{\frac{g_{00}(T_{B},R_{B})}{g_{00}(T_{A},R_{A})}}$
(100)

where $\nu_{A}$ and $\nu_{B}$ are the frequency of the light at
$(T_{A},R_{A},\theta_{A},\phi_{A})$ and $(T_{B},R_{B},\theta_{B},\phi_{B})$,
respectively. From (66), equation (100) becomes

$\frac{\nu_{A}}{\nu_{B}}=\frac{\sqrt{1-\frac{2ar_{0}}{R_{B}}+\frac{2R_{B}}{r_{0}}}}{\sqrt{1-\frac{2ar_{0}}{R_{A}}+\frac{2R_{A}}{r_{0}}}}e^{c(T_{A}-T_{B})/r_{0}}\,.$
(101)

As in the previous case, the fact that $\partial/\partial T$ is a conformal
Killing vector field implies that $T_{B}-T_{A}$ has the same value for all
light rays traveling from $A$ to $B$, so the red-shift is again time-
independent.

For light rays traveling in the radial direction, which requires
$(\theta_{A},\phi_{A})=(\theta_{B},\phi_{B})$, we find from (74) that (101)
becomes

$\frac{\nu_{A}}{\nu_{B}}=\frac{\sqrt{1-\frac{2ar_{0}}{R_{B}}+\frac{2R_{B}}{r_{0}}}}{\sqrt{1-\frac{2ar_{0}}{R_{A}}+\frac{2R_{A}}{r_{0}}}}$
(102)

for ingoing and

$\frac{\nu_{A}}{\nu_{B}}=\frac{\sqrt{1-\frac{2ar_{0}}{R_{B}}+\frac{2R_{B}}{r_{0}}}}{\sqrt{1-\frac{2ar_{0}}{R_{A}}+\frac{2R_{A}}{r_{0}}}}\left\\{\left(\frac{k+R_{A}}{k+R_{B}}\right)^{\frac{k}{k+R_{h}}}\left(\frac{R_{A}-R_{h}}{R_{B}-R_{h}}\right)^{\frac{R_{h}}{k+R_{h}}}\right\\}$
(103)

for outgoing light rays, where $k=\frac{r_{0}}{4}\left(1+\sqrt{1+16a}\right)$
and $R_{h}=\frac{r_{0}}{4}\left(-1+\sqrt{1+16a}\right)$.

###  III.5 Analysis in the original coordinates

As in the previous case, we now transform back the results to the original
coordinate system $(v,r,\theta,\phi)$. This allows us to analyze the shadow as
seen by observers on $v$-lines for the metric (63). Recall that we restrict to
the domain where $-\infty<v<0$ (and $0<r<\infty$).

From (63), we read that the $v$-lines are time-like on the domain

$-2av<r\,,\quad 2ar_{0}<R\,.$ (104)

This domain includes the photon sphere if $0<a<1/4$; it always lies outside of
the horizon. As in the previous case we observe that the hypersurfaces
$v=\mathrm{const.}$ are light-like everywhere and that $r$ is an _area
coordinate_ , i.e., that the sphere $(v,r)=\mathrm{const.}$ has area $4\pi
r^{2}$, as can be read from (63).

We have found above that in the $(T,R,\theta,\phi)$ coordinates light paths in
the photon sphere are at $R=R_{p}=\frac{r_{0}}{2}(-1+\sqrt{1+12a})$ and that
in the equatorial plane their $T$ and $\phi$ coordinates are related by (87).
Transforming these two equations back to the original coordinates
$(v,r,\theta,\phi)$ with the help of (65) results in

$r=\frac{-R_{p}\,v}{r_{0}}\quad\mathrm{and}\quad\phi=\pm\left(\frac{r_{0}}{R_{p}}\sqrt{1-\frac{2ar_{0}}{R_{p}}+\frac{2R_{p}}{r_{0}}}\right)\mathrm{log}\left(\frac{-v}{r_{0}}\right)\,.$
(105)

These equations give us the path of the geodesic parametrized by $v$. We see
that in these coordinates the light-like geodesics in the photon sphere spiral
inwards, see Fig. 11, i.e., that the photon sphere shrinks. The radius
coordinate $r$, which is again plotted in units of $r_{0}$, is linearly
decreasing with $v$. The angle $\phi$ is a logarithmic function of $-v$, so
the angular speed $\frac{d\phi}{dv}$ is inversely proportional to $-v$
implying that the angular speed increases with time.

Figure 11: Light path in the photon sphere seen in $(v,r,\theta,\phi)$
coordinates ($a=1/24$)

We repeat that the hypersurfaces $v=\mathrm{const.}$ are light-like, i.e.,
that they cannot be interpreted as equal-time slices. For describing the
shrinking of the entire photon sphere, as observed by observers on $v$-lines,
we have to introduce an appropriate time function for this case also. We
define such a function $t$ in the $(T,R,\theta,\phi)$ coordinates by the
equation

$c\,dt=c\,dT-\dfrac{dR}{1-\dfrac{2ar_{0}}{R}+\dfrac{R}{r_{0}}}\,.$ (106)

With the help of the relation

$\dfrac{\partial}{\partial
v}=e^{cT/r_{0}}\dfrac{1}{c}\Big{(}\dfrac{\partial}{\partial
T}+\dfrac{cR}{r_{0}}\dfrac{\partial}{\partial R}\Big{)}\,,$ (107)

which follows from (65), it is easy to verify that the hypersurfaces
$t=\mathrm{const.}$ are orthogonal to the $v$-lines, i.e., that events in such
a hypersurface happen simultaneously for the observers on $v$-lines. As in the
previous case, the vector field $\partial/\partial v$ is synchronizable on the
domain (104) but not proper-time synchronizable, so $t$ does not give proper
time for the observers on $v$-lines.

Integration of (106) results in

$c\,t=c\,T-r_{0}\big{(}F(R)-F(R_{p})\big{)}$ (108)

with

$F(R)=\dfrac{1}{2}\,\mathrm{log}\dfrac{\Big{(}1+\sqrt{1+8a}+\dfrac{2R}{r_{0}}\Big{)}^{(1+8a)^{-1/2}+1}}{\Big{(}1-\sqrt{1+8a}+\dfrac{2R}{r_{0}}\Big{)}^{(1+8a)^{-1/2}-1}}\,.$
(109)

If we restrict (108) to the hypersurface $R=R_{p}$ we find that

$c\,t=c\,T=-r_{0}\,\mathrm{log}\dfrac{r}{R_{p}}$ (110)

where the second equality follows from (65). Solving for $r$ results in

$r=R_{p}\,e^{-ct/r_{0}}$ (111)

which demonstrates that the observers on $v$-lines see the photon sphere
exponentially shrinking with time $t$.

Any other sphere $R=\mathrm{const.}$ in the domain (104) shrinks in a similar
way. In particular, for the horizon we find that

$r=R_{h}\,e^{F(R_{p})-F(R_{h})}\,e^{-ct/r_{0}}\,.$ (112)

We have plotted the shrinking of the photon sphere and of the horizon in Fig.
12. The dark region is bounded by the horizon and the (orange) circle
represents the photon sphere. Lengths are given in units of $r_{0}$ and times
are given in units of $r_{0}/c$.

Fig. 13 shows the $v$-lines and the hypersurfaces $t=\mathrm{const.}$ for the
case $a=1/18$. The region where the $v$-lines fail to be time-like is shaded.
For the chosen value of $a$ the photon sphere is outside of this region.

Figure 12: Evolution of photon sphere as seen by observer on $v$-line
($a=1/18$)

Figure 13: $v$-lines (dashed) and hypersurfaces $t=\mathrm{const.}$ (solid)

We will now calculate the angular radius $\tilde{\alpha}{}_{\mathrm{sh}}$ of
the shadow as seen by an observer on a $v$-line. It is related to the angular
radius $\alpha_{\mathrm{sh}}$ of the shadow as seen by a conformally static
observer by the special-relativistic aberration formula (56). Here for this
case $\alpha_{\mathrm{sh}}$ is given by (97) or (98), and $V$ is the momentary
3-velocity of the observer on a $v$-line with respect to the observer on a
$T$-line. The latter has 4-velocity

$U=\dfrac{e^{cT/r_{0}}}{\sqrt{1-\frac{2ar_{0}}{R}+\frac{2R}{r_{0}}}}\,\frac{\partial}{\partial
T}\,,\quad g(U,U)=-c^{2}$ (113)

whereas, by (107), the former has 4-velocity

$\tilde{U}=\dfrac{e^{cT/r_{0}}}{\sqrt{1-\frac{2ar_{0}}{R}}}\Big{(}\frac{\partial}{\partial
T}+\frac{cR}{r_{0}}\,\frac{\partial}{\partial R}\Big{)}\,,\quad
g\big{(}\tilde{U},\tilde{U}\big{)}=-c^{2}$ (114)

From (59) we find that

$\dfrac{V}{c}=\dfrac{-R_{0}^{2}}{R_{0}^{2}-2ar_{0}^{2}+r_{0}R_{0}}$ (115)

at $R=R_{0}$. Note that here $V$ is negative because, with respect to $U$, the
four-velocity $\tilde{U}$ is directed away from the center.

Plugging (115) and (97) into the aberration formula (56) results in

$\mathrm{tan}^{2}\Big{(}\frac{\tilde{\alpha}{}_{\mathrm{sh}}}{2}\Big{)}=\dfrac{R_{0}^{3}(4ar_{0}-R_{p})}{R_{p}^{3}(R_{0}-2ar_{0})}\,\big{(}1-\mathrm{cos}\,\alpha_{\mathrm{sh}}\big{)}^{2}$
(116)

This formula is valid for $R_{0}>2ar_{0}$., i.e., on the domain where both the
$T$-lines and the $v$-lines are time-like. We read from (116) that for
$R_{0}\to\infty$, where $\alpha_{\mathrm{sh}}\to 0$, also
$\tilde{\alpha}{}_{\mathrm{sh}}\to 0$, and that for $R_{0}\to 2ar_{0}$, where
the $v$-lines become light-like, $\tilde{\alpha}_{\mathrm{sh}}\to\pi$. By
inserting (98) into (116) we find that

$\mathrm{tan}^{2}\Big{(}\frac{\tilde{\alpha}{}_{\mathrm{sh}}}{2}\Big{)}=\dfrac{R_{0}^{3}(4ar_{0}-R_{p})}{R_{p}^{3}(R_{0}-2ar_{0})^{2}}\,\Bigg{(}1-\dfrac{(R_{0}-R_{p})}{R_{0}}\sqrt{1+\dfrac{2ar_{0}R_{p}}{R_{0}\big{(}4ar_{0}-R_{p}\big{)}}}\;\Bigg{)}^{2}\,.$
(117)

Figure 14: Angular radius of shadow as seen by observer on $v$-line ($a=1/24$)

The observer on a $v$-line starts at $R_{0}=2ar_{0}$, asymptotically with the
speed of light. The shadow, which in the beginning covers the entire sky,
gradually shrinks while the observer moves outwards. It covers more than half
of the sky at $R_{0}=R_{p}$, which reflects the fact that aberration has a
magnifying effect in the backwards direction. The shadow vanishes for
$R_{0}\to\infty$.

##  IV Conclusions

In this paper we have considered the class of Vaidya metrics which describe
the spherically symmetric and non-stationary space-time around a time-
dependent mass. We have restricted ourselves to the special case that the mass
function increases or decreases in such a way that the space-time admits a
conformal Killing vector field. In this case the equation of light-like
geodesics is completely integrable. In the case of an increasing mass
function, the space-time gives us a model for an accreting black hole, whereas
in the case of a decreasing mass function it can be considered as a (rough)
model for a black hole that loses mass by way of Hawking radiation. In view of
observations, the second case is certainly less relevant than the first
because, for black holes of a Solar mass or more, the mass loss by Hawking
radiation is very, very slow. However, we believe that both cases are of some
interest from a conceptual point of view.

With the help of the conformal Killing vector field, we have explicitly
calculated the light-like geodesics. In the case of a black hole with
increasing mass, there are two horizons with a conformally static region in
between, whereas in the case of a black hole with decreasing mass there is
only one horizon, with the conformally static region between the horizon and
infinity. We have found that in both cases there is a photon sphere in the
conformally static region. In the case of increasing mass, the area of the
horizons and of the photon sphere increases in the course of time, whereas in
the case of decreasing mass the area of the horizon and of the photon sphere
decreases in the course of time. The photon sphere determines the shadow. We
have calculated the angular radius of the shadow for a conformally static
observer (which has to be, of course, in the conformally static region), and
we have found that it is time-independent, in spite of the fact that the area
of the photon sphere is either increasing or decreasing with time. We have
also calculated the red-shift under which one conformally static observer sees
another one and we have found that it is time-independent. This is not in
general true in conformally static metrics, as is exemplified by the well-
known Robertson-Walker metrics, but it is true here because the conformal
factor has a special form. With the angular radius of the shadow known for
conformally static observers, one can easily calculate the angular radius of
the shadow for any other observer with the help of the aberration formula. We
have exemplified this for some ingoing observers in the case of a black hole
with increasing mass and for some outgoing observers in the case of a black
hole with decreasing mass.

What we have presented here is the third example of an exact analytical
calculation of the shadow in a time-dependent situation. The other two
examples are the shadow of an isolated collapsing star surrounded by vacuum
2018  and the shadow of a black hole in an expanding universe modeled by the
Kottler metric PhysRevD.97.104062 .

## ACKNOWLEDGMENTS

VP gratefully acknowledges support from the Deutsche Forschungsgemeinschaft
(DFG) within the Research Training Group 1620 “Models of Gravity”.

## References

  * [1] The Event Horizon Telescope Collaboration.  First M87 Event Horizon Telescope Results. I. The shadow of the supermassive black hole.  Astrophys. J. Lett., 875:L1, 2019.
  * [2] The Event Horizon Telescope Collaboration.  First M87 Event Horizon Telescope Results. II. Array and instrumentation.  Astrophys. J. Lett., 875:L2, 2019.
  * [3] The Event Horizon Telescope Collaboration.  First M87 Event Horizon Telescope Results. III. Data processing and calibration.  Astrophys. J. Lett., 875:L3, 2019.
  * [4] The Event Horizon Telescope Collaboration.  First M87 Event Horizon Telescope Results. IV. Imaging the central supermassive black hole.  Astrophys. J. Lett., 875:L4, 2019.
  * [5] The Event Horizon Telescope Collaboration.  First M87 Event Horizon Telescope Results. V. Physical origin of the asymmetric ring.  Astrophys. J. Lett., 875:L5, 2019.
  * [6] The Event Horizon Telescope Collaboration.  First M87 Event Horizon Telescope Results. VI. The shadow and mass of the central black hole.  Astrophys. J. Lett., 875:L6, 2019.
  * [7] J. L. Synge.  The escape of photons from gravitationally intense stars.  Mon. Not. Roy. Astron. Soc., 131:463, 1966.
  * [8] Yu. B. Zeldovich and I. D. Novikov.  Relativistic astrophysics. II.  Sov. Phys. Usp., 8:522, 1966.
  * [9] A. K. Pande and M. C. Durgapal.  Trapping of photons in spherical static configurations.  Class. Quantum Grav., 3:547, 1986.
  * [10] C.-M. Claudel, K. S. Virbhadra, and G. F. R. Ellis.  The geometry of photon surfaces.  J. Math. Phys., 42:818, 2001.
  * [11] J. M. Bardeen.  Timelike and null geodesics in the Kerr metric.  In C. DeWitt and B. DeWitt, editors, Black Holes, page 215, London, 1973. Gordon and Breach.
  * [12] P. V. P. Cunha and C. A. R. Herdeiro.  Shadows and strong gravitational lensing: a brief review.  Gen. Rel. Grav., 50:42, 2018.
  * [13] V. Perlick and O. Yu. Tsupko.  Calculating black hole shadows: Review of analytical studies.  Phys. Rep., 947:1, 2022.
  * [14] B. Zink.  Ray-tracing Black Holes: Theory and Implementation.  VDM-Verlag Dr. Müller, Saarbrücken, 2008.
  * [15] P. C. Vaidya.  Nonstatic solutions of Einstein’s field equations for spheres of fluids radiating energy.  Phys. Rev., 83:10, 1951.
  * [16] A. B. Nielsen.  Revisiting Vaidya horizons.  Galaxies, 2:62, 2014.
  * [17] A. K. Mishra, S. Chakraborty, and S. Sarkar.  Understanding photon sphere and black hole shadow in dynamically evolving spacetimes.  Phys. Rev. D, 99:104080, 2019.
  * [18] S. Schneider and V. Perlick.  The shadow of a collapsing dark star.  Gen. Rel. Grav., 50:58, 2018.
  * [19] V. Perlick, O. Yu. Tsupko, and G. S. Bisnovatyi-Kogan.  Black hole shadow in an expanding universe with a cosmological constant.  Phys. Rev. D, 97:104062, 2018.
  * [20] W Ames and K. Thorne.  The optical appearance of a star that is collapsing through its gravitational radius.  Astrophys. J., 151:659, 1968.
  * [21] L. Kong, D. Malafarina, and C. Bambi.  Can we observationally test the weak cosmic censorship conjecture?  Eur. Phys. J. C, 74:2983, 2014.
  * [22] L. Kong, D. Malafarina, and C. Bambi.  Gravitational blueshift from a collapsing object.  Phys. Lett. B, 741:82, 2015.
  * [23] N. Ortiz, O. Sarbach, and T. Zannias.  Shadow of a naked singularity.  Phys. Rev. D, 92:044035, 2015.
  * [24] N. Ortiz, O. Sarbach, and T. Zannias.  Observational distinction between black holes and naked singularities: the role of the redshift function.  Class. Quantum Grav., 32:247001, 2015.
  * [25] K. Koga, N. Asaka, M. Kimura, and K. Okabayashi.  Dynamical photon sphere and time evolving shadow around black holes with temporal accretion.  arXiv:2202.00201, 2022.
  * [26] Y. Kuroda.  Naked singularities in the Vaidya spacetime.  Prog. Theor. Phys., 72:63, 1984.
  * [27] A. Papapetrou.  Formation of a singularity and causality.  In N. J. Dadhich, editor, A random walk in relativity and cosmology, page 184, Singapore, 1986. Wiley.
  * [28] J. B. Griffiths and J. Podolský.  Exact Space-Times in Einstein’s General Relativity.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2009.
  * [29] D. Pu et al. (SEAMBH Collaboration).  Supermassive black holes with high accretion rates in Active Galactic Nuclei. IV. H$\beta$ time lags and implications for super-Eddington accretion.  Astrophys. J., 806:22, 2015.
  * [30] S. Ojako, R. Goswami, S. D. Maharaj, and R. Narain.  Conformal symmetries in generalised Vaidya spacetimes.  Class. Quantum Grav., 37, 2020.
  * [31] W. A. Hiscock.  Models of evaporating black holes. I.  Phys. Rev. D, 23:2813, 1981.
  * [32] W. A. Hiscock.  Models of evaporating black holes. II. Effects of the outgoing created radiation.  Phys. Rev. D, 23:2823, 1981.
  * [33] Y. Kuroda.  Vaidya spacetime as an evaporating black hole.  Prog. Theor. Phys., 71:1422, 1984.
  * [34] M. I. Beciu.  Evaporating black hole in Vaidya metric.  Phys. Lett. A, 100:77, 1984.


${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 9 | $24.5\\!\\!\pm\\!\\!6.5$ | $20.6\\!\\!\pm\\!\\!6.8$ | $22\\!\\!\pm\\!\\!50~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | - |  | - | - | 5.4 | CyearlyK31-1R58.dat
| $8.76\\!\\!\times\\!\\!10^{4}$ | $1826.19\\!\\!\pm\\!\\!0.69$ | $1856.3\\!\\!\pm\\!\\!4.3$ | $1875\\!\\!\pm\\!\\!428$ | - |  | - | - | 0.0013 |
$\mathcal{M}$=8 |  $\mathcal{M}$${}_{4,1,-1,\chi^{2}}$ | $7.991\\!\\!\pm\\!\\!0.048$ | $8.819\\!\\!\pm\\!\\!0.070$ | $48.7\\!\\!\pm\\!\\!2.1$ | $2400\\!\\!\pm\\!\\!1700~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ |  | - | - | - |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 12 | $23.0\\!\\!\pm\\!\\!7.0$ | $20.7\\!\\!\pm\\!\\!4.8$ | $21.0\\!\\!\pm\\!\\!4.8$ | $22\\!\\!\pm\\!\\!29{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ |  | - | - | - | CyearlyK41-1R58.dat
| $8.07\\!\\!\times\\!\\!10^{4}$ | $1826.03\\!\\!\pm\\!\\!0.62$ | $1819.3\\!\\!\pm\\!\\!1.1$ | $1824.1\\!\\!\pm\\!\\!5.0$ | $1842\\!\\!\pm\\!\\!433$ |  | - | - | - |