Shadows of Kerr-Vaidya-like black holes

Hai Siong Tan

Email<EMAIL_ADDRESS>
University of Pennsylvania, Perelman School of Medicine, Department of
Radiation Oncology, Philadelphia, USA

Abstract

In this work, we study the shadow boundary curves of rotating time-dependent
black hole solutions which have well-defined Kerr and Vaidya limits. These
solutions are constructed by applying the Newman-Janis algorithm to a
spherically symmetric seed metric conformal to the Vaidya solution with a mass
function that is linear in Eddington-Finkelstein coordinates. Equipped with a
conformal Killing vector field, this class of solution exhibits separability
of null geodesics, thus allowing one to develop an analytic formula for the
boundary curve of its shadow. We find a simple power law describing the
dependence of the mean radius and asymmetry factor of the shadow on the
accretion rate. Applicability of our model to recent Event Horizon Telescope
observations of M87∗ and Sgr A∗ is also discussed.

###### Contents

  1. 1 Introduction
  2. 2 A family of rotating Vaidya-like black hole solutions
    1. 2.1 Conformal factors, coordinate charts and the Newman-Janis algorithm
    2. 2.2 Horizons in the solution parameter space
  3. 3 Null geodesics, photon spheres and shadow formulas
    1. 3.1 On null geodesics
    2. 3.2 Shadow formulas from photon region
  4. 4 Portraits of the shadow
    1. 4.1 Scaling laws for variation of $\overline{R}$ and $\mathcal{A}$ with $\mu$
    2. 4.2 On the shadows of M87∗ and Sagittarius A∗ as observed by EHT
  5. 5 Discussion
  6. A Comments on horizons and causal structure
  7. B On the energy momentum tensor
  8. C On the reference frame of the shadow observer
    1. C.1 In the limit of $a=0$
    2. C.2 Observers in the $\\{v,w,\theta,\phi\\}$ chart and aberration formulas

##  1 Introduction

Recent Event Horizon Telescope (EHT) observations of horizon-scale shadow
images of M87∗ [1] and Sgr A∗ [2] have furnished not only a direct visual
evidence of black holes, but have also led to many new constraints on various
potential deviations from General Relativity. The boundary curve of the black
hole shadow emerges from light rays that spiral asymptotically from the photon
region demarcating the boderline between light rays that will eventually be
captured by the black hole and those that escape to infinity [3].iiiThis curve
is termed as the ‘critical curve’ by Gralla et al. in [4] and ‘apparent
boundary’ by Bardeen in [5]. See for example the review of [6] for an
extensive discussion of basic ideas and history.  The geometry of this
boundary curve depends on the background metric which could thus be probed by
EHT observations [7, 8, 9, 10, 11, 12] . For example, the shadow geometry of
Sgr∗ has been used to exclude the central object being a Reissner-Nordstrom-
type naked singularity or a traversable Misner-Thorne wormhole [8].

Surrounding the black hole shadow is an emission ring of which structure is
sensitive to a rich set of astrophysical phenomena, such as radiative
transfer, that characterize the matter-energy accretion process. Typically,
general relativistic magnetohydrodynamic (GRMHD) simulations are used to model
the accretion flow processes [7, 8, 13, 14]. For the EHT experiments, they
have revealed the emission ring properties to be consistent with a number of
accretion models built upon the background of a Kerr black hole [7, 8]. The
spacetime metric in these simulations is assumed to be purely Kerr spacetime
throughout, with the energy-momentum tensor capturing the magnetic field and
average plasma properties [13, 15]. In [16, 17, 18], it was noted that the
shadow size and shape is hardly influenced by the accretion details, and thus
serves as a pristine signature of spacetime geometry. An implicit assumption
is that the backreaction of the GRMHD energy-momentum tensor on the metric has
a negligible influence on the shadow and could thus be ignored in deriving its
geometry.

In this paper, we study the shadow boundary curves of a class of rotating
Kerr-Vaidya-like black hole solutions. Their metrics can be described as
deformations of the Kerr solution parametrized by a small dimensionless
parameter $\mu$. This parameter can be interpreted as the mass accretion rate
constant in natural units. In the limit of vanishing spin, our spacetime
reduces to a well-known model of spherically accreting black hole - the Vaidya
spacetime with a mass function $\mu v$, with $v$ being an ingoing Eddington-
Finkelstein coordinate. The latter solution was studied most recently in [19]
where the authors derived and examined its shadow characteristics
analytically. Most crucially, an analytic treatment of the shadow was possible
by virtue of the existence of a Carter constant leading to separability of its
null geodesic equations. This is related to a conformal Killing symmetry
associated with the linear mass function, and hence its choice, for it enables
the authors of [19] to derive explicit formulas for the radius of the photon
sphere and the shadow angular diameter. One main motivation of our work here
is to seek a rotating generalization of the analytic treatment in [19]. This
would serve as a simple model of a backreacted Kerr-like geometry that is
accreting mass, and for which an analytic derivation of its shadow geometry is
possible. For readers familiar with exact solutions in GR, a natural candidate
would be the Kerr-Vaidya solution [20] which can be obtained by replacing the
constant Kerr mass with a variable mass function in the original Kerr line
element expressed in Eddington-Finkelstein coordinates. Unfortunately, as
we’ll elaborate later, this solution does not offer any additional Carter
constant that could lead to its null geodesic equations being separable.

We construct our solutions by applying the Newman-Janis algorithm [21] to a
spherically symmetric seed metric conformal to the Vaidya solution with the
mass function that is linear in Eddington-Finkelstein coordinates.
Fortunately, this solution-generating technique turns out to preserve the
conformal Killing vector field in the original Vaidya metric, leading to
separability of null geodesics, and ultimately allows us to develop an
analytic formula for the boundary curve of its shadow. The solution space is
parametrized by $\\{a,\mu,M_{s}\\}$ where $\\{a,M_{s}\\}$ are the spin and
mass parameters of Kerr spacetime in the vanishing $\mu$ limit. Like the Kerr
solution, there are regions in the moduli space which do not pertain to black
holes. Motivated by phenomenological interests, we focus on the regime of
parameters where our solution has event horizons like those of Kerr, with the
conformal Killing horizon at a large distance away from the shadow observer
and the outer horizon. Thus, our solution serves as a simple model of an
accreting Kerr-like geometry not globally but for a finite spatial domain
defined by the interior of the conformal Killing horizon. We work in a chart
which reduces to the Kerr spacetime in Boyer-Lindquist coordinates in the
limit $\mu=0$, and the Vaidya spacetime in Eddington-Finkelstein-like
coordinates in the limit $a=0$. Correspondingly, we verified that our shadow
formulas reduce consistently to those of Kerr [22, 23] and Vaidya [19] under
these limits.

As our main goal here is to seek a metric that is Kerr-Vaidya-like in nature
and which also enables us to have an analytic control over its shadow
geometry, our starting point and metric construction are agnostic to the form
of the energy-momentum tensor. In ordinary Einstein gravity, up to second
order in $\mu$, our Kerr-Vaidya-like solution turns out to be sourced by a
Type III energy-momentum tensor in Hawking-Ellis classification [24], some
other examples of which were recently noted in [25, 26] to admit the
interpretation of ‘spinning null fluids’. Another limitation of our work which
deserves further study is that while we could easily identify a null
hypersurface $\mathcal{S}_{e}$ in our Kerr-Vaidya-like solution to be a
putative candidate for the event horizon, we were unable to mathematically
prove through rigidity theorems that $\mathcal{S}_{e}$ is formally the
boundary of the past of future null infinity unlike the case of the ordinary
Kerr solution. However, there are seminal results in the past literature
concerning conformal Killing horizons [27, 28] which allow us to at least
determine $\mathcal{S}_{e}$ to satisfy local conditions of a putative event
horizon. In separate limits of vanishing $a$ or $\mu$, $\mathcal{S}_{e}$
reduces to the event horizons of Kerr and Vaidya solutions respectively.
Collectively, these limitations imply that our Kerr-Vaidya-like solution
should be regarded as a phenomenological toy model of a rotating black
hole(-like) spacetime that is accreting mass. Yet most crucially, it allows us
to derive explicit formulas relating between accretion rate and aspects of the
shadow geometry.

As reviewed in for example [6], analytic derivations in cases that allow them
complement numerical studies of shadow geometry in general. For example, for
Schwarzschild spacetime, the angular diameter of its shadow is $\sim
3\sqrt{3}M_{s}/R_{o}$ for a distant observer located at the radial coordinate
$R_{o}$ [29]. This numerical value has turned out to be very useful as a guide
in the analysis of shadow size and shape in EHT’s recent observations [1, 2].
For our shadow analysis here, we find a simple power law describing the
dependence of the mean radius and asymmetry factor of the shadow on the
accretion rate. The latter describes the departure of the shadow from
circularity and has been constrained in M87∗ studies by EHT team [1]. When
applied to the parameters of M87∗ and Sgr A∗, our analysis of shadow geometry
appears to indicate that the effect of $\mu$ is very small, and thus provides
support for the assumption of using the pure Kerr metric throughout in GRMHD
simulations. Our results, in addition, yield an empirical formula that
parametrizes the variation of mean radius and asymmetry factor with accretion
rate explicitly, and can thus be used to anticipate when backreaction of
accretion on the metric may be significant.

Our paper is organized as follows. In Section 2, we present the construction
of a class of Kerr-Vaidya-like solutions and elaborate on some basic aspects
of its geometry and moduli space, followed by a derivation of some analytical
formulas for shadow geometry in Section 3. In Section 4, we present several
visual plots of the shadow and examine how the mean radius and asymmetry
factor of the shadows vary with various parameters. We also include a brief
discussion on recent EHT observations of M87∗ and Sgr A∗ in relation to our
model geometry. Finally, we end with some concluding remarks in Section 5.
Appendix A discusses some aspects of the causal structure of our solution,
including more detailed discussion on conformal Killing horizons and some
global aspects of the spacetime geometry. Appendix B presents some expressions
for components of the energy-momentum tensor. In Appendix C, for completeness,
we develop an aberration formula for observers in another reference frame
which, in the zero spin limit, reduces to another class of observers discussed
previously in [19] for Vaidya spacetime.

##  2 A family of rotating Vaidya-like black hole solutions

We begin with the Vaidya metric in the coordinatesiiiiiiThe unusual choice of
the symbol $w$ to denote the areal radius for this line element is solely due
to shortage of conventions for the many different radial coordinates that
we’ll use throughout this paper.

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2m(v)}{w}\right)dv^{2}+2dvdw+w^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (1)

with the domains
$v\in(0,\infty),\,w\in(0,\infty),\theta\in(0,\pi),\phi\in(0,2\pi)$. We note
that $m(v)$ is a mass function that can be used to model a time-dependent
black hole of which exterior is described by (1). The solution (1) solves the
field equations in ordinary GR with the energy momentum tensor
$T^{\mu\nu}=m(v)K^{\mu}K^{\nu},K^{\nu}\partial_{\nu}=\partial_{w}$ which is
typically interpreted as that of a null dust moving in the direction of
decreasing $w$, with the black hole accreting (radiating) mass if
$m^{\prime}(v)$ is positive (negative).

###  2.1 Conformal factors, coordinate charts and the Newman-Janis algorithm

In this work, we restrict ourselves to the case where $m(v)=\mu v$, where
$\mu$ is a positive constant. In this case, the geometry admits a conformal
Killing vector field. To make this manifest, we make a coordinate
transformation as follows.

$v=r_{0}e^{T/r_{0}},\qquad w=re^{T/r_{0}},$ (2)

where $r_{0}$ is a positive constant with dimension of length. This brings (1)
to

$ds^{2}=e^{2T/r_{0}}\left(-\left(1-\frac{2\mu
r_{0}}{r}-\frac{2r}{r_{0}}\right)dT^{2}+2dTdr+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right),$ (3)

with $T\in(-\infty,\infty),r\in(0,\infty)$. In this chart, we can see that a
translation in $T$ generates a Weyl rescaling of the metric with
$\partial/\partial_{T}$ defined as the conformal Killing vector. In this form,
the metric is conformal to a manifestly static spacetime which can be taken to
generate a rotating solution via Newman-Janis algorithm. But first, we seek a
temporal coordinate such that constant time slices are 3-dimensional spatial
manifolds. Defining

$T=t+\Upsilon(r),\qquad\Upsilon(r)=\int^{r}d\tilde{R}\,\,\left(1-\frac{2\mu
r_{o}}{\tilde{R}}-\frac{2\tilde{R}}{r_{0}}\right)^{-1},$ (4)

the line element then reads

$ds^{2}=e^{\frac{2(t+\Upsilon(r))}{r_{0}}}\left(-\left(1-\frac{2\mathcal{M}(r)}{r}\right)dt^{2}+\left(1-\frac{2\mathcal{M}(r)}{r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right)\equiv\Omega^{2}(t,r)ds^{2}_{static},$ (5)

where $t\in(-\infty,\infty)$ and

$\mathcal{M}(r)\equiv\mu r_{0}+\frac{r^{2}}{r_{0}}.$

If we restrict ourselves to the spacetime patch where the conformal Killing
vector field $\frac{\partial}{\partial t}$ is timelike, then letting
$1-2\mathcal{M}/r>0$ leads to the domain

$r\in(R_{h},R_{c}),\qquad
R_{h,c}=\frac{r_{0}}{4}\left(1\pm\sqrt{1-16\mu}\right),\qquad\mu<1/16,$

where $R_{h}$ is the black hole horizon and $R_{c}$ denotes the conformal
Killing horizoniiiiiiiiiA more formal definition can be found in [30]. – the
null hypersurface at which the norm-squared of the conformal Killing vector
field vanishes. We note that $r=R_{h}$ is a conformal Killing horizon too, and
its interpretation as an event horizon was recently discussed in [30] based on
an earlier series of results in [27, 28, 31]. In [19], their interpretations
were also briefly discussed in the authors’ derivation of the shadow formula
for Vaidya spacetime.

Now, the Schwarzschild limit can be obtained as a double scaling limit as
follows.iviviv The limits that we consider in this paper are smooth limits of
metric functions defined specifically in some coordinate systems. See for
example [32] for a limiting procedure that covariantizes the process by
lifting the spacetime to a higher-dimensional one with the metric parameter as
an auxiliary dimension, and then taking its boundary.

$\mu\rightarrow 0,\,\,\,r_{0}\rightarrow\infty,\,\,\,\mu r_{o}=M_{s},$ (6)

where $M_{s}$ is a finite mass parameter equivalent to the ADM mass of the
limiting Schwarzschild black hole. We now apply the Newman-Janis
algorithmvvvThis algorithm is typically applied with the assumption of
asymptotic flatness in the generated metric which doesn’t hold for our
solution though.  to the metric $ds^{2}_{static}$ which leads to a metric
endowed with angular momentum

$\displaystyle ds^{2}_{static}\rightarrow ds^{2}_{rotating}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2\mathcal{M}(r)r}{\Sigma}\right)dt^{2}-\frac{4\mathcal{M}(r)ar\sin^{2}\theta}{\Sigma}d\phi
dt$ (8)
$\displaystyle\qquad+\left(r^{2}+a^{2}+\frac{2\mathcal{M}(r)a^{2}r\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta
d\phi^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma d\theta^{2},$

where $a$ is the spin parameter and

$\Sigma=r^{2}+a^{2}\cos^{2}\theta,\qquad\Delta=r^{2}-2\mathcal{M}(r)r+a^{2}.$

We will also modify the exponential argument of the conformal factor
$\Omega(t,r)$ in (5) as follows

$t+\Upsilon(r)\rightarrow
t+\Upsilon_{a}(r),\,\,\,\,\Upsilon_{a}(r)\equiv\int^{r}dr\,\,\,\frac{r^{2}+a^{2}}{r^{2}-2\mathcal{M}(r)r+a^{2}}.$
(9)

The full metric then reads

$ds^{2}=e^{\frac{2(t+\Upsilon_{a}(r))}{r_{0}}}ds^{2}_{rotating},$ (10)

with $ds^{2}_{rotating},\Upsilon_{a}(r)$ being defined in (8) and (9)
respectively. In the scaling limit of (6), the line element (10) reduces to
Kerr spacetime in Boyer-Lindquist coordinates.

Now, like the Kerr solution where $\mathcal{M}$ is instead just a constant,
the metric (10) has singularities at the roots of $\Delta=0$. To extend the
spacetime beyond these singularities, we can perform a coordinate
transformation

$\tilde{T}=t+\Upsilon_{a}(r),\qquad\tilde{\phi}=-\phi-a\int^{r}dr\,\,\frac{1}{r^{2}-2\mathcal{M}(r)r+a^{2}}$
(11)

which leads to

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
e^{\frac{2\mu}{M_{s}}\tilde{T}}\Bigg{[}-\left(1-\frac{2\mathcal{M}(r)r}{r^{2}+a^{2}\cos^{2}\theta}\right)(d\tilde{T}+a\sin^{2}\theta
d\tilde{\phi})^{2}+2(d\tilde{T}+a\sin^{2}\theta
d\tilde{\phi})(dr+a\sin^{2}\theta d\tilde{\phi})$ (13)
$\displaystyle\qquad\qquad+(r^{2}+a^{2}\cos^{2}\theta)d\Omega^{2}\Bigg{]},$

In the $a=0$ limit, we recover Vaidya spacetime in the conformally static
coordinates of (3), whereas the $\mu=0$ limit (as in (6)) takes the metric to
that of Kerr in ingoing Eddington-Finkelstein coordinates. We note that in
(13), replacing $\mathcal{M}(r)\rightarrow\mu\tilde{T}$ and removing the
conformal factor $e^{\frac{2\mu}{M_{s}}\tilde{T}}$ yields the Kerr-Vaidya
solution [20] which evidently isn’t equipped with the conformal Kiling
symmetry. This leads to non-separability of null geodesics which would not
allow us to solve for the shadow boundary curve analytically.

Our main interest in this class of time-dependent solutions lies in its
property of being locally deformable to the Kerr geometry in Boyer-Lindquist
coordinates in the $\mu\rightarrow 0,\mu r_{0}\rightarrow M_{s}$ limit, and,
in the limit of $a=0$, to the Vaidya solution in a chart where it’s
conformally static. _This gives us a model of local geometry that approximates
both spacetimes in a coordinate system suitable for deriving the analytical
form of the black hole shadow._ For this specific purpose, we work in the
$\\{t,r,\theta,\phi\\}$ chart (line element in (10)), where the spacetime is
conformal to a Kerr-like solution in Boyer-Lindquist coordinates. In Section
2.2, we explore the parameter space $\\{a,\mu,M_{s}\\}$ in greater detail.

###  2.2 Horizons in the solution parameter space

In the ordinary Kerr solution expressed in Boyer-Lindquist coordinates, one
can attempt to determine the event horizon as the hypersurface $r=R_{e}$,
where $R_{e}$ is the (larger) root of $g^{rr}=0$, since this equation is
equivalent to setting the Minkowski norm of the normal vector to zero thereby
establishing it as a null hypersurface – a necessary condition. Further, the
Killing vector field

$\xi_{K}=\frac{\partial}{\partial
t}+\Omega_{K}\frac{\partial}{\partial\phi},\qquad\Omega_{K}=\frac{a^{2}}{a^{2}+R^{2}_{e}},$
(14)

has vanishing norm on $r=R_{e}$ (with $\Omega_{K}$ being the angular velocity
of the black hole), and thus $r=R_{e}$ is a Killing horizon. These properties
satisfy the local conditions for $r=R_{e}$ to be the event horizon of Kerr,
while we can further use rigidity theorems to show that $r=R_{e}$ satisfies
the global notion of the event horizon being the boundary of the causal past
of future null infinity. In our case, since we are dealing with dynamical
black holes, the event horizon cannot be the Killing horizon defined as above.
However, in [27, 28, 31], for certain classes of dynamical black holes which
admit conformal Killing vector fields, the authors showed that one can
similarly locate event horizons through conformal Killing horizons. These were
used most recently in [30] to elucidate black hole thermodynamics in the
Vaidya spacetime which is the $a=0$ limit of our Kerr-Vaidya-like solution. In
Appendix A, we discuss more in detail these notions connecting between
conformal Killing horizons and event horizons.

In a similar vein to finding null hypersurfaces in the Kerr solution of the
form $r=$ constant, in (10), setting $g^{rr}=0$ yields the following cubic
equation in $r$.

$-\frac{2\mu}{M_{s}}r^{3}+r^{2}-2M_{s}r+a^{2}=0.$ (15)

In certain regimes of the parameter space of $\\{\mu,a\\}$, one could find
putative event horizons that necessarily have to satisfy (15) as null
hypersurfaces of the form $r=$ constant. Consider the root space of the cubic
equation (15) of which discriminant reads (henceforth, we define $a\rightarrow
a/M_{s}$ to be a dimensionless parameter)

$\mathfrak{D}=4\left(1-a^{2}-16\mu+18a^{2}\mu-27a^{4}\mu^{2}\right).$

The sign of $\mathfrak{D}$ determines the number of real roots to $g^{rr}=0$.
As a quadratic equation in $\mu$, we can derive the curves along which
$\mathfrak{D}=0$, which read

$\mu_{\pm}=\frac{-8+9a^{2}\pm\left(4-3a^{2}\right)^{3/2}}{27a^{4}}.$ (16)

They enclose the region which pertains to three roots $R_{i}\leq R_{e}\leq
R_{a}$ which are smoothly connected (in the limits of vanishing $a,\mu$) to
the inner Cauchy horizon of Kerr spacetime($R_{i}$), the event horizon of Kerr
spacetime ($R_{e}$) and the conformal Killing horizon of Vaidya spacetime
($R_{a}$) respectively.

They intersect at the point (see Fig. 1 )

$(a_{e},\mu_{e})=\left(\frac{2}{\sqrt{3}},\frac{1}{12}\right),$ (17)

which represents a generalized extremal limit. At any constant
$\mu\in(0,\frac{1}{12})$, the upper bound on $a$ is given by taking
$\mu=\mu_{-}(a)$ along which the inner Cauchy and event horizons coincide.
Along $\mu=\mu_{+}(a)$, the conformal Killing horizon coincides with the event
horizon.

Figure 1: Graph depicting the parameter space $(\mu,a)$ of our family of
solutions. The curve $\mu_{+}$ begins at $(0,\frac{1}{16})$ and contains
solutions where the outer event horizon and conformal Killing horizon are
degenerate. The curve $\mu_{-}$ contains all the extremal solutions with
degenerate outer and inner horizons and with a finite conformal Killing
horizon radius. Both curves merge at the extremal point
$(\frac{2}{\sqrt{3}},\frac{1}{12})$ at which there is only one horizon at
$r=2M_{s}$. Our family of solutions can be seen as parametric deformations of
the Vaidya solution (vertical axis) and the Kerr solution (horizontal axis).

The small $\mu\ll 1$ region of this enclosed segment in Fig. 1 is of closer
phenomenological interest to us. Let us consider a generic point in this
region. Ordering the roots of (15) as $R_{i}<R_{e}<R_{a}$, we find that up to
first few orders in $\mu,a$ :

$\displaystyle R_{e}$ $\displaystyle=$ $\displaystyle
M_{s}\left(\left(1+\sqrt{1-a^{2}}\right)+\frac{4+4\sqrt{1-a^{2}}-5a^{2}-3a^{2}\sqrt{1-a^{2}}+a^{4}}{2(1-a^{2})}u+\mathcal{O}(\mu^{2})\right)$
(18) $\displaystyle=$ $\displaystyle
M_{s}\left[(2+8\mu)-(2\mu+1/2)a+\ldots\right]$ (19) $\displaystyle R_{i}$
$\displaystyle=$ $\displaystyle
M_{s}\left(\left(1-\sqrt{1-a^{2}}\right)+\frac{4-4\sqrt{1-a^{2}}-5a^{2}+3a^{2}\sqrt{1-a^{2}}+a^{4}}{2(1-a^{2})}\mu+\mathcal{O}(\mu^{2})\right)$
(20) $\displaystyle=$ $\displaystyle
M_{s}\left(\frac{a}{2}+\frac{a^{2}}{8}-\frac{\mu
a^{3}}{16}+\frac{a^{3}}{16}+\ldots\right)$ (21) $\displaystyle R_{a}$
$\displaystyle=$ $\displaystyle\frac{M_{s}}{2\mu}-2M_{s}-8\mu M_{s}+2a\mu
M_{s}+\mathcal{O}(\mu^{2})$ (22)

In the limit of vanishing $\mu$, the radii $R_{e},R_{i}$ are smoothly
connected to their corresponding expressions in the ordinary Kerr solution
while $R_{a}\rightarrow\infty$.

$\displaystyle\lim_{\mu\rightarrow 0}R_{i}$ $\displaystyle=$ $\displaystyle
M_{s}\left(1-\sqrt{1-a^{2}}\right)=R^{(\text{Kerr})}_{\text{Cauchy horizon}},$
(23) $\displaystyle\lim_{\mu\rightarrow 0}R_{e}$ $\displaystyle=$
$\displaystyle
M_{s}\left(1+\sqrt{1-a^{2}}\right)=R^{(\text{Kerr})}_{\text{event horizon}},$
(24) $\displaystyle\lim_{\mu\rightarrow 0}R_{a}$ $\displaystyle\rightarrow$
$\displaystyle\infty.$ (25)

In the $a=0$ limit, $R_{i}$ goes to zero, $R_{e}$ reduces to the event horizon
of the Vaidya solution, whereas $R_{a}$ reduces to the radius of the conformal
Killing horizon associated with the conformal Killing vector $\partial_{t}$ of
the Vaidya solution

$\displaystyle\lim_{a\rightarrow 0}R_{i}$ $\displaystyle=$ $\displaystyle 0,$
(26) $\displaystyle\lim_{a\rightarrow 0}R_{e}$ $\displaystyle=$
$\displaystyle\frac{M_{s}}{4\mu}\left(1-\sqrt{1-16\mu}\right)=R^{(\text{Vaidya})}_{\text{event
horizon}},$ (27) $\displaystyle\lim_{a\rightarrow 0}R_{a}$ $\displaystyle=$
$\displaystyle\frac{M_{s}}{4\mu}\left(1+\sqrt{1-16\mu}\right)=R^{(\text{Vaidya})}_{\text{conformal
Killing horizon}}.$ (28)

In our work, we perform our shadow calculations in the Boyer-Lindquist-like
chart of (10) within the domain $r\in[R_{e},R_{a})$. As shown in Appendix A,
similar to the case for the ordinary Kerr solution, the conformal Killing
vector field

$\xi=\frac{\partial}{\partial
t}+\Omega\frac{\partial}{\partial\phi},\qquad\Omega=\frac{a^{2}}{a^{2}+R^{2}_{e}},$
(29)

has vanishing norm on the null hypersurface $r=R_{e}$. This is identical in
form to Killing vector (14) of Kerr solution which has vanishing norm on
Kerr’s event horizon. As elaborated in Appendix A, we can use the results of
[27, 28, 31] to interpret $r=R_{e}$ as the conformal Killing horizon
associated with $\xi$ in (29). As explained in [27, 28, 31] , such a null
hypersurface satisfies the local conditions of being the putative event
horizon of our solutions. The caveat is that we still lack some rigidity
theorems to rigorously prove that it is an event horizon in the global sense.
For a generic point within our domain of interest (region enclosed by curves
and axes in Fig. 1 ), our shadow observer is a timelike observer located
within $R_{e}<r<R_{a}$, i.e. away from the putative event horizon or conformal
Killing horizon. Since we have been identifying physical interpretations of
our Kerr-Vaidya-like solution through limits of metric functions in the chart
(10), we would also like our observer to lie in a domain free of any
coordinate singularities of (10), in particular, imposing $g_{tt}<0$ for all
polar angles $\theta$ lead to

$R_{e}<R_{h}<R_{o}<R_{c}<R_{a},\qquad
R_{h,c}=\frac{M_{s}}{4\mu}\left(1\mp\sqrt{1-16\mu}\right),$ (30)

where $R_{h,c}$ are the event horizon and conformal Killing horizon of the
Vaidya solution (see [19] for a recent discussion of their physical
interpretations).

##  3 Null geodesics, photon spheres and shadow formulas

###  3.1 On null geodesics

Metrics of the form $ds^{2}_{rotating}$ in (10) admit null geodesics which are
separable. This condition was shown for the Kerr solution in for example [6,
22], and for other well-motivated forms of mass function $\mathcal{M}(r)$ in
[33]. Such a property is preserved within its conformal class, and in
particular by our line element $ds^{2}$ in (10). To appreciate this, we first
recall that for a pair of metrics which are conformally related, say
$g_{\mu\nu}=\Omega^{2}(x)\tilde{g}_{\mu\nu},$ their geodesic equations are
related by

$\frac{d^{2}x^{\alpha}}{d\eta^{2}}+\Gamma^{\alpha}_{\beta\nu}\frac{dx^{\beta}}{d\eta}\frac{dx^{\nu}}{d\eta}=0=\frac{d^{2}x^{\alpha}}{d\eta^{2}}+\tilde{\Gamma}^{\alpha}_{\beta\nu}\frac{dx^{\beta}}{d\eta}\frac{dx^{\nu}}{d\eta}-2\Omega^{-1}\frac{d\Omega}{d\eta}\frac{dx^{\alpha}}{d\eta},$
(31)

where $\tilde{\Gamma}^{\alpha}_{\beta\nu}$ are the Christoffel symbols
associated with the metric $\tilde{g}$. The last term of (31) can be rewritten
as

$\frac{d^{2}x^{\alpha}}{d\lambda^{2}}+\tilde{\Gamma}^{\alpha}_{\beta\nu}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=0,\qquad\frac{d\lambda}{d\eta}=\Omega^{2}(x^{\alpha}(\eta)).$
(32)

Thus, a suitable redefinition of the affine parameter leads to identical forms
of the geodesic equations for the pair of conformally related metrics. In
particular, we expect separability of null geodesic equations just like the
Kerr solution or more generally the Kerr-like solutions studied in [33].

Our solution has a conformal Killing vector field $K\sim\partial_{t}$ which
naturally gives a quantity conserved along its null geodesics. In parallel
with the notion of energy in the static case, we call this conserved quantity
$\tilde{E}=K^{\mu}P_{\mu}$, $P_{\mu}$ being the 4-momentum of a test particle
traveling along the geodesic. Also, the metric components are independent of
$\phi$, and we call $L=p_{\phi}$ the associated conserved quantity. These
symmetries motivate the use of the Hamilton-Jacobi formalism for geodesics
where one first defines an auxiliary action $S(\kappa,x^{\mu})$ obeying

$\frac{\partial
S}{\partial\kappa}+\frac{1}{2}g_{\mu\nu}p^{\mu}p^{\nu}=0,\qquad
p^{\mu}=g^{\mu\nu}p_{\nu}=g^{\mu\nu}\frac{\partial S}{\partial x^{\nu}}.$ (33)

By virtue of the nature of conserved quantities, we adopt the following ansatz
for the action $S$:

$S=-\tilde{E}t+L\phi+S_{r}(r)+S_{\theta}(\theta)+\frac{1}{2}\mu^{2}\kappa,\qquad\mu=-p_{\alpha}p^{\alpha}.$
(34)

By construction, the 4-momenta
$p^{\mu}=\frac{dx^{\mu}}{d\eta}=g^{\mu\nu}\partial_{\nu}S(x)$ satisfies the
geodesic equation, with $\kappa=0$ for null geodesics. We find that the
Hamilton-Jacobi equation (33) leads to

$-\Delta(r)(\partial_{r}S_{r})^{2}+[(r^{2}+a^{2})E-aL]^{2}/\Delta(r)=(\partial_{\theta}S_{\theta})^{2}+(L-aE\sin^{2}\theta)^{2}/\sin^{2}\theta=\mathcal{K},$
(35)

where the constant $\mathcal{K}$ indicates separability. For deriving the
shadow formulas, we need the explicit expressions for the 4-momenta which we
find to simplify as

$\displaystyle\frac{dt}{d\eta}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma(r)}e^{-\frac{2(t+\Upsilon_{a}(r))}{r_{0}}}\left(-a(aE\sin^{2}\theta-L)+\frac{(r^{2}+a^{2})P(r)}{\Delta(r)}\right),$
(36) $\displaystyle\frac{dr}{d\eta}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{\Sigma(r)}e^{-\frac{2(t+\Upsilon_{a}(r))}{r_{0}}}\sqrt{\mathcal{R}(r)},$
(37) $\displaystyle\frac{d\theta}{d\eta}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{\Sigma(r)}e^{-\frac{2(t+\Upsilon_{a}(r))}{r_{0}}}\sqrt{\Xi(\theta)},$
(38) $\displaystyle\frac{d\phi}{d\eta}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma(r)}e^{-\frac{2(t+\Upsilon_{a}(r))}{r_{0}}}\left(-\left(aE-\frac{L}{\sin^{2}\theta}\right)+aP(r)/\Delta(r)\right),$
(39)

where

$P(r)\equiv E(r^{2}+a^{2})-aL,\,\,\mathcal{R}(r)\equiv
P(r)^{2}-\mathcal{K}\Delta(r),\,\,\,\Xi(\theta)\equiv\mathcal{Q}+\cos^{2}\theta\left(a^{2}E^{2}-L^{2}/\sin^{2}\theta\right),$

with $\mathcal{Q}$ being conventionally called the Carter constant defined by

$\mathcal{Q}\equiv\mathcal{K}-(L-aE)^{2}.$ (40)

At this point, we note that apart from the specific form of our mass function
$\mathcal{M}(r)=M_{s}+r^{2}/r_{0}$ implicitly contained in
$\Delta(r)=r^{2}-2\mathcal{M}(r)r+a^{2}$, the 4-momenta expressions in (36) -
(39) are identical to those found in [33] up to the conformal factor
$e^{-\frac{2(t+\Upsilon_{a}(r))}{r_{0}}}$. This is consistent with the general
relations governing conformally related metrics as described in eqns (31) and
(32).

###  3.2 Shadow formulas from photon region

For our analysis of the shadow, we define the constants of motion

$\eta\equiv\frac{\mathcal{Q}}{E^{2}},\qquad\xi\equiv\frac{L}{E}.$ (41)

Any null geodesics with $r=R_{p}$ for some constant $R_{p}$ leads to the
condition

$\mathcal{R}(R_{p})=\left(E(R_{p}^{2}+a^{2})-aL\right)^{2}-\mathcal{K}\Delta(R_{p})=0.$

Further, if the orbit is unstable then setting $d^{2}r/d\eta^{2}=0$ yields

$\mathcal{R}^{\prime}(R_{p})=0.$

After some algebra, we find that these couple of equations lead to the
constants of motion

$\displaystyle a\frac{L}{E}/M^{2}_{s}$ $\displaystyle=$
$\displaystyle\frac{4(1+\mu R^{2}_{p})R^{2}_{p}-(3\mu
R^{2}_{p}+R_{p}+1)(R^{2}_{p}+a^{2})}{-3\mu R^{2}_{p}-1+R_{p}},$ (42)
$\displaystyle\frac{\mathcal{K}}{E^{2}}/M^{2}_{s}$ $\displaystyle=$
$\displaystyle\frac{((R^{2}_{p}+a^{2})-aL/E)^{2}}{R^{2}_{p}-2(1+\mu
R^{2}_{p})R_{p}+a^{2}}.$ (43)

These constants of motion also characterize non-spherical geodesics which
asymptotically approach those confined to spheres defined by
$\mathcal{R}(R_{p})=\mathcal{R}^{\prime}(R_{p})=0$. In eqns. (42) and (43),
and henceforth, for notational simplicity, we define $R_{p},a$ in units of
$M_{s}$. Now consider an observer at the position $(R_{o},\theta_{inc})$ and
described by an orthonormal tetrad as follows.

$\displaystyle e_{0}=e^{-\frac{\mu
T}{M_{s}}}\frac{(r^{2}+a^{2})\partial_{t}+a\partial_{\phi}}{\sqrt{\Sigma\Delta}},\,\,\,e_{1}=e^{-\frac{\mu
T}{M_{s}}}\sqrt{\frac{1}{\Sigma}}\partial_{\theta},$ (44) $\displaystyle
e_{2}=-e^{-\frac{\mu
T}{M_{s}}}\frac{\partial_{\phi}+a\sin^{2}\theta\partial_{t}}{\sqrt{\Sigma}\sin\theta},\,\,e_{3}=-e^{-\frac{\mu
T}{M_{s}}}\sqrt{\frac{\Delta}{\Sigma}}\partial_{r}.$ (45)

As explained in for example [6, 22], this choice leads to $e_{0}$ being the
4-velocity of the shadow observer with $e_{0}\pm e_{3}$ being tangential to
the principal null congruences. (The various expressions in (44) differ from
those in [6, 22] by the conformal factor which we need to take into account
for an orthonormalized set of basis vectors.) The tangent vector of each light
ray reaching the observer reads

$\frac{d}{d\eta}=p^{\mu}\partial_{\mu}=\dot{r}\partial_{r}+\dot{\theta}\partial_{\theta}+\dot{\phi}\partial_{\phi}+\dot{t}\partial_{t}=\alpha(-e_{0}+\sin\theta\cos\phi
e_{1}+\sin\theta\sin\phi e_{2}+\cos\theta e_{3}),$ (46)

where $\alpha=g_{\mu\nu}p^{\mu}e^{\nu}_{0}$. Equating the coefficients after
evaluating both sides of (46) at $(R_{o},\theta_{inc})$ yields

$\sin\Phi=\frac{L_{E}(R_{p})-a\sin^{2}\theta_{inc}}{\sqrt{\mathcal{K}_{E}(R_{p})}\sin\theta_{inc}},\,\,\,\sin\Theta=\frac{\sqrt{\Delta(R_{o}})\mathcal{K}_{E}(R_{p})}{R^{2}_{o}-aL_{E}(R_{p})+a^{2}},\,\,\,L_{E}\equiv\frac{L}{M^{2}_{s}E},\,\,\,\mathcal{K}_{E}\equiv\frac{\mathcal{K}}{M^{2}_{s}E^{2}},$
(47)

where we have switched to a different notation for the celestial coordinates
$\phi\rightarrow\Phi,\theta\rightarrow\Theta$ for clarity having derived their
eventual expressions. We note that $\theta_{inc}$ measures the angle between
the black hole’s spin axis as defined by $\theta=0$(the zero locus of
$g_{\phi\phi}$) and the observer, with $e_{3}$ being parallel to the line of
sight connecting the observer to the origin of the Boyer-Lindquist-like chart
in (10).

The photon region consists of spherical orbits with radii bounded in the
domain

$R_{p}\in(R_{p,min},R_{p,max}),$ (48)

where $R_{p,min},R_{p,max}$ are defined by setting $\sin\Phi=+1,-1$
respectively, i.e.

$\displaystyle aL_{E}(R_{p,min})$ $\displaystyle=$ $\displaystyle
a^{2}\sin^{2}\theta_{inc}+a\sqrt{K_{E}}(R_{p,min})\sin(\theta_{inc}),$ (49)
$\displaystyle aL_{E}(R_{p,max})$ $\displaystyle=$
$\displaystyle-a^{2}\sin^{2}\theta_{inc}-a\sqrt{K_{E}}(R_{p,max})\sin(\theta_{inc}).$
(50)

In the vanishing $a$ limit, the width of the photon region
$(R_{p,min},R_{p,max})$ goes to zero, and collapses to a single value of
$R_{p}$ which is the photon sphere radius of Vaidya spacetime. In this limit,
from (49),

$\lim_{a\rightarrow 0}aL_{E}\rightarrow R^{2}_{p}\frac{3+\mu
R^{2}_{p}-R_{p}}{R_{p}-1-3\mu R^{2}_{p}}=0\Rightarrow
R_{p}=\frac{1}{2\mu}\left(1-\sqrt{1-12\mu}\right),$

where we have restricted the root to be smaller than the conformal Killing
horizon radius. This is indeed the expression obtained in [19]. Further taking
the $\mu=0$ limit yields $R_{p}=3$ which is the radius of Schwarzschild photon
sphere. In the $a=0$ limit, our expression for $\sin\Theta$ in (47) reduces to
eqn. (41) of [19] which is the sine of the angular radius of the Vaidya
solution’s shadow.

##  4 Portraits of the shadow

In this Section, we discuss geometrical properties of the shadow in more
details. We first recall that the coordinate system describing the shadow
observer has a conformal Killing horizon $R_{a}$ obtained as the largest root
of $g^{rr}=0$ in the line element (10) which we reproduce explicitly below for
convenience.

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
e^{\frac{2(t+\Upsilon_{a}(r))}{r_{0}}}\Bigg{[}-\left(1-\frac{2\mathcal{M}(r)r}{\Sigma}\right)dt^{2}-\frac{4\mathcal{M}(r)ar\sin^{2}\theta}{\Sigma}d\phi
dt$ (52)
$\displaystyle\qquad+\left(r^{2}+a^{2}+\frac{2\mathcal{M}(r)a^{2}r\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta
d\phi^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma d\theta^{2}\Bigg{]},$

where $\mathcal{M}(r)=M_{s}+\frac{r^{2}}{r_{0}}$ and
$\Upsilon_{a}(r)\equiv\int^{r}dr\,\,\,\frac{r^{2}+a^{2}}{r^{2}-2\mathcal{M}(r)r+a^{2}}$.
For our shadow observer located at some radial distance $R_{o}$, we would also
like $g_{tt}<0$ for all values of $\theta$, leading to the condition

$R_{h}<R_{o}<R_{c}<R_{a},\qquad
R_{h,c}=\frac{r_{0}}{4}\left(1\mp\sqrt{1-16\mu}\right),$ (53)

where $R_{h},R_{c}$ are the event and Killing horizons of the Vaidya solution
in the $a=0$ limit. This also implies that for some fixed $R_{o}$, we have an
upper bound on

$\mu<\mu_{b}\equiv M_{s}\frac{R_{o}-2M_{s}}{2R^{2}_{o}},$ (54)

since we wish to have $R_{o}<R_{c}$. For the visual representation of the
shadow, we follow the convention used by Johannsen and Psaltis in [16]. This
is essentially the orthonormal tetrad we used in deriving the shadow formula,
with the $x,y$ coordinates being

$\displaystyle x$ $\displaystyle=$ $\displaystyle
R_{o}\sin\left(\Theta(R_{p},R_{o})\right)\sin\left(\Phi(R_{p},\theta_{inc})\right),$
(55) $\displaystyle y$ $\displaystyle=$ $\displaystyle\pm
R_{o}\sin\left(\Theta(R_{p},R_{o})\right)\cos(\Phi(R_{p},\theta_{inc})).$ (56)

These coordinates parametrize the observer’s plane upon which the shadow is
projected.viviviAnother set of projection coordinates used for plotting the
black hole shadow is the $(\alpha,\beta)$ parameters of Bardeen which would
not be entirely suitable in our case since our solution is not asymptotically
flat. See for example the review of [6] which discusses the relations between
Bardeen’s impact parameters and others such as stereographic coordinates, etc.
From (47), noting that $L_{E}$ and $\mathcal{K}_{E}$ are odd and even in the
spin parameter $a$ respectively, one can straightforwardly identify the
discrete symmetry

$a\rightarrow-a,\,\,\,\theta_{inc}\rightarrow-\theta_{inc},$ (57)

which implies in particular that $a<0$ shadows can be obtained from their
$a>0$ counterparts by a reflection in the $y$-axis (for all our shadow plots,
we take $a>0$).viiviiviiIn [34], continuous symmetries of the shadow modulo
Mobius transformations of the celestial sphere were studied for the Kerr-
Newman family of black holes. For the shadow of our black hole solution, we
are unable to identify other continuous symmetries apart from the discrete
reflection in (57).  In quantifying the shape of the shadow, we follow the
work of Johannsen and Psaltis in [16] who introduced the asymmetry parameter
$\mathcal{A}$ to describe departure from circularity.

$\mathcal{A}=2\sqrt{\frac{\int^{2\pi}_{0}d\alpha\,\,\,(R-\overline{R})^{2}}{\int^{2\pi}_{0}d\alpha}},\,\,\,\tan\alpha=\frac{y}{x},\,\,\,R\equiv\sqrt{(x-D)^{2}+y^{2}},\,\,\,D\equiv\frac{|x_{max}+x_{min}|}{2},$
(58)

and $\overline{R}=\int^{2\pi}d\alpha\,\,R/2\pi$ is the averaged radius
projected upon the observer’s plane. These quantities were mentioned in the
EHT paper [1] for M87∗, and we checked that our shadow geometries for the
background Kerr solution (with $\mu=0$) yield comparable features obtained
previously in the work of Johannsen and Psaltis [16]. In Figure
LABEL:fig:PortraitRow, we picked a few values of $a,\theta_{inc}$ for a black
hole located at $R_{o}/M_{s}\sim 5.4\times 10^{10}$ (estimated for M87∗) to
demonstrate how the shadow curve changes with $\mu$.

###  4.1 Scaling laws for variation of $\overline{R}$ and $\mathcal{A}$ with
$\mu$

The parameter space for the shadow geometry is spanned by
$\\{a,\theta_{inc},R_{o},\mu\\}.$ Computing the shadow’s mean radius and
asymmetry factor for a range of parameters, we find a simple empirical scaling
law that describes the variation of $\overline{R},\mathcal{A}$ with the
accretion rate parameter $\mu$ and other parameters as follows.viiiviiiviiiThe
data-to-model fitting and standard errors of the model parameters was computed
using the ‘NonlinearModelFit’ function of Wolfram Mathematica 12.1.1.0 which
uses the quasi-Newton method.

$\displaystyle\overline{R}\approx\overline{R}_{o}\left(a,\theta_{inc},R_{o}\right)\left(1-\frac{\mu}{\mu_{b}(R_{o})}\right)^{\kappa_{R}},\,\,\,\kappa_{R}\approx
0.51\pm 0.01$ (59)
$\displaystyle\overline{\mathcal{A}}\approx\overline{\mathcal{A}}_{o}\left(a,\theta_{inc},R_{o}\right)\left(1-\frac{\mu}{\mu_{b}(R_{o})}\right)^{\kappa_{A}},\,\,\,\kappa_{R}\approx
0.495\pm 0.005$ (60)

where $\mu_{b}(R_{o})$ is the upper bound (54) on the accretion rate allowed
by our model for some fixed observer distance $R_{o}$, obtained by setting the
conformal Killing horizon to be the observer distance. The dependence on $\mu$
appears as a separate factor independent of the other shadow parameters, with
the functions
$\overline{R}_{o}\left(a,\theta_{inc},R_{o}\right),\overline{\mathcal{A}}_{o}\left(a,\theta_{inc},R_{o}\right)$
describing the radius and asymmetry factor at $\mu=0$. The form of (59) and
(60) implies that for $\mu\ll 1,R_{0}\gg M_{s}$, the fractional decrease in
the mean radius and the asymmetry factor that is induced by a non-zero $\mu$
scales approximately as

$\frac{\delta\overline{R}}{\overline{R}}\approx\frac{\delta\mathcal{A}}{\mathcal{A}}\approx-\mu\frac{R_{o}}{M_{s}}.$
(61)

In Figure LABEL:fig:ZeroQ, we plot these functions for a few values of spin at
a fixed $R_{0}$ and $\theta_{inc}$. These plots are expectedly similar to the
corresponding ones presented in [16] and [17]. At higher spin values, the
asymmetry factor and mean radius exhibit a greater range of values over the
$\theta_{inc}$ domain. At any $\theta_{inc}$, increasing $a$ increases
$\overline{\mathcal{A}}_{o}$ but decreases $\overline{R}_{o}$. The form of
(59), (60) also implies that the asymmetry factor expressed in units of the
mean radius is approximately independent of $\mu$, with

$\frac{\mathcal{A}}{\overline{R}}\approx\frac{\overline{\mathcal{A}}_{o}}{\overline{R}_{o}}\left(a,\theta_{inc},R_{o}\right).$
(62)

In Figures LABEL:fig:Varya, LABEL:fig:VaryT and LABEL:fig:VaryR, we plot the
empirical fitting curves (described by (59) and (60) ) that depict how
$\overline{R},\mathcal{A}$ vary with the accretion parameter $\mu$ and each of
the parameters $\\{a,\theta_{inc},R_{o}\\}$ separately.

###  4.2 On the shadows of M87∗ and Sagittarius A∗ as observed by EHT

The M87∗ black hole was found to be about 16.8 Mpc away, with a mass
$M_{s}\simeq 6.5\times 10^{9}M_{\odot}$[1]. The ensemble of accretion models
used by the EHT team [1, 7] involved mass accretion rates that ranged from
about $2\times 10^{-7}$ to $4\times 10^{-4}$ times the Eddington rate
$\dot{M}_{\text{Edd}}$. In their work, $\dot{M}_{\text{Edd}}\sim
137M_{\odot}/\text{yr}$, and we find that this translates intoixixix We note
that $\mu$ is the dimensionless mass accretion rate in natural units. To
restore its dimension to that of mass/time, rescale it by $\mu\rightarrow
c^{3}\mu/G$, where $G$ is the Newtonian gravitational constant and $c$ the
speed of light.

$\mu_{M87}=\dot{M}\times\frac{GM_{\odot}}{\text{Yr}\times
c^{3}}\sim\dot{M}\times 1.56\times 10^{-13}\in\left(4\times 10^{-18},9\times
10^{-15}\right),$

where $\dot{M}$ is mass accretion rate in units of $M_{\odot}/\text{yr}$. This
is smaller than the upper bound $\mu_{b}\sim 9.2\times 10^{-12}$ for
$R_{o}\sim 16.8$ Mpc. Equivalently, for this range of $\mu$, the conformal
Killing horizon size falls within

$R_{c}/M_{s}\sim\left(5.9\times 10^{13},1.16\times 10^{17}\right),$

which lies beyond $R_{o}/M_{s}\sim 5.4\times 10^{10}$. Thus, the observed
distance to M87∗ and estimates of mass accretion are well within the domains
of validity of our simple model geometry. Now it was estimated in [35] that
the angle of inclination is around $17^{\circ}$. Corresponding to this value,
in Figure 7, we plot the variation of the shape parameters
$\overline{R},\mathcal{A}$ and their ratio with the spin parameter $a$. The
range of values of $\overline{R}$ translates into the shadow angular diameter
$\sim(36.9\mu\text{as},39.6\mu\text{as})$ which is comparable to the measured
emission ring diameter in the EHT experiment [1]. The maximum
$\mathcal{A}/\overline{R}$ ratio is about 0.01 which is within limits of the
upper bound of $10\%$ indicated in [1]. The highest $\mu\sim 8.5\times
10^{-15}$ in the ensemble of models considered in [1] translates only to a
fractional shift of 0.05 $\%$ in $\overline{R},\mathcal{A}$.

Figure 7: Graphs showing how the mean radius $\overline{R}$, asymmetry factor
$\mathcal{A}$ and their ratio vary with $a$, with $R_{o}/M_{s}=5.4\times
10^{10}$ (pertaining to EHT observation of M87∗), $\theta_{inc}=17^{\circ}$.

Sgr A∗ has been observed to located near the dynamical center of our galaxy at
a distance $R_{o}\sim 8$ kpc away, with a dense concentration of mass
$M_{s}\sim 4\times 10^{6}M_{\odot}$. In contrast to M87∗ where its prominent
jet provides robust constraints on source orientation with respect to the line
of sight, fixing it to be $\sim 17^{\circ}$, there is no such constraint
unfortunately on Sgr A∗ [2]. However, GRMHD models appeared to have favored
$\theta_{inc}<50^{\circ}$, with accretion rate of order-of-magnitude
$10^{-9}-10^{-8}M_{\odot}\,\text{yr}^{-1}$. These models were equipped with
spin parameter values of $a=0.5,0.94$ [2]. As mentioned in [2], in the earlier
works of Quataert [36] and Baganoff [37], the captured accretion rate was
estimated to be $10^{-6}-10^{-5}M_{\odot}\,\text{yr}^{-1}$ from Chandra
observations of thermal bremsstrahlung emission at the vicinity of the gas
capture radius. Most recently, in [14], a promising model was identified in
which $\theta_{inc}\leq 30^{\circ}$, and accretion models of $\dot{M}\sim
5.2-9.5\times 10^{-9}M_{\odot}/$yr were examined.

Even for the accretion rate $\dot{M}=10^{-5}M_{\odot}\,\text{yr}^{-1}$ this
translates into merely

$\mu_{sgr}\sim 1.6\times 10^{-18}.$

Like the case of M87∗, this turns out to be smaller than the upper bound
$\mu_{b}\sim 1.2\times 10^{-11}$ for $R_{o}\sim 8$ kpc. Equivalently, taking
this value of $\mu_{sgr}$, the conformal Killing horizon size is

$R_{c}/M_{s}\sim 3.2\times 10^{17},$

which lies beyond $R_{o}/M_{s}\sim 4.2\times 10^{10}$. Thus again, both
observer distance and (estimated) mass accretion rates are well within the
domains of validity of our simple model geometry.

In Figure 8, we plot the variation of the shape parameters
$\overline{R},\mathcal{A}$ and their ratio with the spin parameter $a$, at a
few representative values of $\theta_{inc}$. Over the domain of
$\theta_{inc}\in(10^{\circ},50^{\circ})$, the range of shadow angular
diameters is $\sim(47.6\mu\text{as},51.2\mu\text{as})$ which is comparable to
the shadow diameter estimate $48.7\pm 7.0\mu$as in the EHT experiment [2]. The
asymmetry factor-to-mean radius ratio $\mathcal{A}/\overline{R}$ ratio
increases with $\theta_{inc}$, and can be as high as $\sim 10\%$ for
$\theta_{inc}=50^{\circ}$. It would be interesting to study this geometrical
signature for the EHT’s Sgr A∗ shadow image. In [8], the EHT team mentioned in
passing that the sparse interferometric coverage of 2017 observations led to
significant uncertainties in circularity measurements which were thus not
quantified yet, but future EHT observations with additional telescopes may
place constraints on the circularity. Finally, let us mention that the effect
of $\mu$ on $\overline{R},\mathcal{A}$ is even smaller for Sgr A∗, inducing
only a fractional change of $10^{-6}$ in these quantities.

Figure 8: Graphs showing how the mean radius $\overline{R}$, asymmetry factor
$\mathcal{A}$ and their ratio vary with $a$, with $R_{o}/M_{s}=4.2\times
10^{10}$ (pertaining to EHT observation of Sgr A∗),
$\theta_{inc}=10^{\circ},30^{\circ},50^{\circ}$.

##  5 Discussion

We have presented a study of the shadow geometry for a class of spacetime
metric that is Kerr-Vaidya-like in nature. The family of time-dependent black
hole solutions we constructed in this work has well-defined Kerr and Vaidya
limits. Agnostic to the source and the underlying theory, we had conceived of
its form starting from the Vaidya solution with a mass function that is linear
in Eddington-Finkelstein coordinates since this particular class of solutions
furnishes a model for accretion and is equipped with a conformal Killing
isometry that leads to separability of null geodesics. After expressing it as
being conformal to a solution that is Schwarzschild-like with a radial
coordinate-dependent mass function, the Newman-Janis algorithm was applied to
obtain a Kerr-like solution that reduces to the Kerr solution in the limit of
vanishing accretion parameter $\mu$. In real-life applications, the
dimensionless accretion rate $\mu$ is expected to be very small, for instance,
for the recent M87∗ and Sgr A∗ observations, the highest model estimates of
$\mu$ are of the order $10^{-15},10^{-18}$ respectively. Thus, our model
geometry can be considered as a small $\mu$-deformation of the Kerr solution
that preserves separability of null geodesics, or from a different
perpsective, a rotating generalization of the Vaidya solution. For a finite
spatial domain, it could act as a simple model of a Kerr-like geometry that
takes into account the backreaction of accretion. The existence of a conformal
Killing vector field allows us to solve for the shadow geometry
straightforwardly yet also brings with it the subtlety of a horizon that
should be located beyond the shadow observer. Equivalently, at any fixed
observer distance, there exists an upper bound to $\mu$ (eqn.(54)) for
applicability of our model.

In our study of the variation of the mean radius $\overline{R}$ and asymmetry
factor $\mathcal{A}$ with regards to various parameters —
$\\{\mu,a,\theta_{inc},R_{o}\\}$, we found a simple empirical scaling law of
the form in (59) and (60). In particular, this implies that at small $\mu\ll
1$ and large observer distance $R_{o}\gg M_{s}$, the fractional change induced
by turning on the accretion rate parameter $\mu$ reads simply as

$\frac{\delta\overline{R}}{\overline{R}}\approx\frac{\delta\mathcal{A}}{\mathcal{A}}\approx-\mu\frac{R_{o}}{M_{s}}.$
(63)

To our knowledge, we have not encountered any previous descriptive relations
between shadow geometrical features and accretion rate, black hole and
observer parameters of the form similar to (59) or (63)xxxIn [38], it was
found that increasing the flux of infalling gas into a Schwarzschild black
hole (and hence the accretion rate) by increasing an axion-plasmon coupling
parameter decreases the size of the shadow which qualitatively agrees with the
variation of $\overline{R}$ with $\mu$ of our model. It would be interesting
to study if the results in [38] could be cast in a similar form as (59) or
(63)..GRMHD simulations (e.g. [13, 15, 39]) typically assume the validity of
the background metric being purely Kerr in some suitable coordinate system,
with the complicated astrophysics of accretion contained within the choice of
the energy-momentum tensor. Even for GRMHD simulations involving spacetime
metrics motivated by beyond-GR theories (e.g. [8]), the accretion parameter is
rarely involved in describing the background spacetime. In [16, 17] and most
recently in [18], it was found that the accretion details do not appear to
influence the shadow geometry which is sensitive only to the background
metric. The assumption here is that backreactions of accretion on the metric
are insignificant. Indeed, in applying our model to EHT observations of M87∗
and Sgr A∗, we found that the most generous estimates of $\mu$ in [1] and [2]
yield fractional changes of $\overline{R}$ and $\mathcal{A}$ of the order of
$10^{-4}$ and $10^{-6}$ respectively, consistent with such an assumption. In
addition, our model yields explicit relations in (59) and (60) that describe
how, at least in principle, the shadow geometry changes with accretion rate.
From (63), it may seem that for situations where the (fractional) experimental
uncertainties
$\delta\overline{R}_{e}/\overline{R}_{e},\delta\mathcal{A}_{e}/\mathcal{A}_{e}$
are of the order $\mu R_{o}/M_{s}$, then metric backreactions due to accretion
may be important in analyzing geometrical details of the black hole shadow.

A limitation of our model geometry is that it is not asymptotically flat. As a
model of an effective Kerr-like geometry with accretion backreaction, it is
thus valid only for a finite spatial domain. Imposing a stricter condition for
$g_{tt}<0$ in (52), in our exposition of the black hole shadow analysis, we
have taken the observer location $R_{o}$ to lie within the interior of the
sphere defined by the conformal Killing horizon of the limiting Vaidya
spacetime, i.e. $R<R_{c}$ in the coordinate system of (52). (For M87∗ and Sgr
A∗, $R_{c}/R_{o}\sim 10^{3},10^{7}$ respectively.) It would be interesting to
seek a refinement of our model geometry in one that allows for separability of
null geodesics while being asymptotically flat.

## Acknowledgments

I am grateful to Chong-Sun Chu, Ori Ganor, Petr Horava and Neal Snyderman for
sharing with me their insights on various aspects of gravitational physics
over the years. I dedicate this work in loving memory of my Aunt Mdm Tan Siew
Huan to whom I’ll always be indebted for her love and guidance.

##  Appendix A Comments on horizons and causal structure

Our solution is manifestly a hybrid geometry interpolating between Kerr and
Vaidya metrics which can be recovered in separate, smooth limits. It is
characterized by length parameters which reduce to those in Kerr and Vaidya
that have been associated with horizons. In the main narrative, we have
alluded to these parameters as horizon radii as guided by the associated
quantities in the $\mu=0,a=0$ limits. In the following, we furnish more
elaborations on these geometric notions.

For a dynamical black hole, its event horizon cannot be described via a
Killing horizon, yet as explained in [27, 28, 31] and most recently in [30],
for spacetime geometries which admit a timelike conformal Killing vector field
$\xi$, one could, at least locally, describe a putative event horizon by using
the surface generated by the conformal Killing vector field. In [27, 28], the
conformal stationary limit surface $\mathcal{S}$ is defined as the
hypersurface on which $\xi$ is null. If such a surface is also a null surface,
its generators are null geodesics as explained in the generic context in [27,
28] where it was further shown that since it is a null geodesic hypersurface,
it acts as a one-way membrane with the future null cone lying on one side of
it such that furture directed timelike directions cross the null surface in
the same sense. It was also noted that in spacetimes where $\mathcal{S}$ is
not a null surface, one can still have the same construction provided there
exists a ‘mixed’ conformal Killing field $\tilde{\xi}$ that vanishes on
another hypersurface $\tilde{S}$ which is null and interpretable as a
conformal Killing horizon. In [27, 28], it was explained how this is analogous
to the Kerr solution in which the stationary limit surface (or ergosphere) is
separated from the outer event horizon by an ergoregion. The notion of surface
gravity and related notions of black hole thermodynamics on the conformal
Killing horizon was also developed most recently in [30], where it was shown
how thermodynamical properties of dynamical black holes that admit conformal
Killing horizons can be elucidated. Notably, the Vaidya spacetime with linear
mass accretion which is the $a=0$ limit of our solution was adopted in [30] as
the concrete illustration of these ideas.

Let us now see how these previous results are applicable to our case. Consider
our Kerr-Vaidya-like solution in the chart $\\{t,r,\theta,\phi\\}$ of (10),
where the conformal Killing vector can be expressed as
$\xi=\frac{\partial}{\partial t}$. The conformal stationary limit surface
$\mathcal{S}$ is the surface $g_{tt}=0$

$\mathcal{S}\equiv r^{2}+a^{2}\cos^{2}\theta-2\mathcal{M}(r)r=0$ (64)

One can evaluate the norm-squared of the normal vector
$N_{\alpha}=\partial_{\alpha}\mathcal{S}$ and straightforwardly verify that
this is a timelike surface. In the $\mu=0$ limit, one can identify
$\mathcal{S}$ to be the defining equation for the ergosphere of the ordinary
Kerr black hole. This motivates us to consider the constant $r$ surfaces in
this chart. For $r=R_{0}$ where $R_{0}$ is some constant,
$N_{\mu}=\partial_{\mu}r$, and one can identify the locus of vanishing norm.

$N^{2}=g^{rr}=e^{-\frac{2\mu}{M_{s}}\left(t+\Upsilon_{a}(r)\right)}\left.\frac{r^{2}-2\mathcal{M}(r)r+a^{2}}{r^{2}+a^{2}\cos^{2}\theta}\right|_{r=R_{0}}=0,$
(65)

which leads to the roots of the cubic equation
$r^{2}-2\mathcal{M}(r)r+a^{2}=0$ that we ordered earlier as $R_{i}\leq
R_{e}\leq R_{a}$. The chart is valid for the domain $r\in[R_{e},R_{a})$, since
the conformal factor is singular at $R=R_{a}$ and vanishes at $R=R_{e}$. One
can continue past these singularities by using for example the chart in (13).

Now, the surfaces $R=R_{e},R_{a}$ are null hypersurfaces. Following the
prescription of [27, 28, 31], we seek a conformal Killing vector that is a
linear combination of $\partial_{t}$ and $\partial_{\phi}$. Writing it as

$\tilde{\xi}=\partial_{t}+\Omega\partial_{\phi},$

setting $\tilde{\xi}^{\alpha}\tilde{\xi}_{\alpha}=0$ yields

$\Omega=\frac{a^{2}}{a^{2}+R_{e}^{2}}.$ (66)

This expression is identical in form to the angular velocity of rotation for
the Kerr black hole, with $R_{e}$ being the event horizon radius. xixixi In
the chart of (13), we verified that the same expression for the ‘mixed’
conformal Killing vector field is obtained. In this case, the $R_{e}$
appearing in (66) can be replaced by any root of the cubic equation
$r^{2}-2\mathcal{M}(r)r+a^{2}=0$ which includes $R=R_{i},R=R_{a}$.

Since $\tilde{\xi}^{\alpha}\tilde{\xi}_{\alpha}=0$ on the null surface
$R=R_{e}$, the latter is a geodesic null hypersurfaces as explained in [27,
28]. Locally, it satisfies conditions for the event horizon of a dynamical
black hole spacetime. However, to prove that $R=R_{e}$ is the boundary of the
region from which null curves cannot escape to infinity, one requires rigidity
theorems to connect this global notion to the local notion of a conformal
Killing horizon as in the ordinary Kerr geometry. In [27, 28], generalized
versions of strong and weak rigidity theorems were proposed for conformal
Killing horizons. However, the conditions specified in them are not satisfied
by our model spacetime. Specifically, as we discuss in detail in Appendix B,
the energy-momentum tensor violates the weak energy condition at order up to
$\mathcal{O}(\mu^{2})$, and this invalidates the proof for a generalized
strong rigidity theorem in [31]. Thus, we cannot rely on the results of [27,
31] for a rigorous proof of $R=R_{e}$ being the event horizon of our dynamical
solution. Nonetheless, as emphasized in [27, 28, 31], a conformal Killing
horizon is a natural candidate for the event horizon, being a geodesic null
hypersurface that acts locally as a one-way membrane for null cones.

In our work we restrict the observer location $R_{o}$ in a finite spatial
domain of the Boyer-Lindquist-like chart of (10) : $R_{h}<R_{o}<R_{c}$, where
$R_{h},R_{c}$ are the roots of $r^{2}-2\mathcal{M}(r)r+a^{2}=0$ in the limit
of $a=0$. This ensures that we have $g_{tt}<0$ for all polar angle $\theta$.
For this chart, the radial coordinate $r$ lies in the domain $[R_{e},R_{a})$,
and $R=R_{e}$ is the only conformal Killing horizon in this interval. In the
limit of vanishing $\mu$, it reduces to the outer event horizon of the
ordinary Kerr solution. In the limit of vanishing $a$, it reduces to the event
horizon of the Vaidya spacetime as recently identified in [19, 30]. On the
other hand, $R_{a}\rightarrow\infty$ in the vanishing $\mu$ limit, whereas in
the $a=0$ limit, it was identified as a conformal Killing horizon of the
Vaidya spacetime in [19, 30] although its physical meaning was not explicitly
discussed in these papers.

For the Vaidya spacetime, the conformally static chart in (3) has a coordinate
singularity at the conformal Killing horizon. It can be continued beyond that
via (2). Similarly, from (13), we can perform the coordinate transformation
(2)

$v=r_{0}e^{\tilde{T}/r_{0}},\qquad w=re^{\tilde{T}/r_{0}},$ (67)

after which the line element takes the form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-F\left(\frac{w}{v}\right)\left(dv+\frac{av\sin^{2}\theta}{r_{0}}d\tilde{\phi}\right)^{2}+2\left(dv+\frac{av\sin^{2}\theta}{r_{0}}d\tilde{\phi}\right)\left(dw+\frac{av\sin^{2}\theta}{r_{0}}d\tilde{\phi}\right)$
(69)
$\displaystyle-\frac{2aw\sin^{2}\theta}{r_{0}}dvd\tilde{\phi}+\left(w^{2}+\frac{v^{2}a^{2}\cos^{2}\theta}{r^{2}_{0}}\right)d\Omega^{2},$

where

$F\left(\frac{w}{v}\right)=-1+\frac{2\left(\mu+(\frac{w}{v})^{2}\right)\frac{w}{v}}{\left(\frac{w}{v}\right)^{2}+\frac{a^{2}\cos^{2}\theta}{r^{2}_{0}}}-\frac{2w}{v}.$

The Vaidya metric in the chart (1) is obtained in the $a=0$ limit, whereas the
double scaling limit of (6) brings it to Kerr in Eddington-Finkelstein chart.
The metric in the chart $\\{v,w,\theta,\tilde{\phi}\\}$ is convenient for
examining the asymptotic infinity of the spacetime. Taking the limit of
infinite $w$ yields the following asymptotic form

$\lim_{w\rightarrow\infty}ds^{2}\sim-
dv^{2}+2dvdw+w^{2}d\Omega^{2}+\frac{2a\mu}{M_{s}}\sin^{2}\theta\left(wdvd\tilde{\phi}+vdwd\tilde{\phi}\right).$
(70)

At radial infinity there are still non-zero Ricci and Einstein tensor
components which read
$R_{\theta\theta}=-G_{\theta\theta}=-\frac{a^{2}}{r^{2}_{0}}\sin^{2}\theta,\,\,\,R_{\tilde{\phi}\tilde{\phi}}=3G_{\tilde{\phi}\tilde{\phi}}=-3\frac{a^{2}}{r^{2}_{0}}\sin^{4}\theta,$
with other components being zero. The asymptotic form of the spacetime shows
that at infinity, the spacetime geometry is not foliated by two-spheres. For
such metrics which lack sufficient symmetry to be represented by Penrose
diagrams of the two-dimensional spacetimes, it was proposed in [40] that one
can still piece together the global causal structure through a set of 2D
projections. While we will not further attempt to invoke such a theory of
projection diagrams to analyze our spacetime, in the following, as a simple
probe, we consider the $\theta=\pi/2,\tilde{\phi}=0$ slice of the metric (69)
and sketch its Penrose diagram for this 2D spacetime. From this toy Penrose
diagram, one can see that in this auxiiary spacetime with line element

$ds^{2}_{aux}=\left(-1+\frac{2\mu v}{w}\right)dv^{2}+2dvdw,$ (71)

the null curve $\frac{w}{v}=\frac{R_{h}}{r_{0}}$ represents the boundary of
the region containing the entire past of all null rays that could infinity,
$R_{h}=\frac{r_{0}}{4}\left(1-\sqrt{1-16\mu}\right)$. Another null curve
$\frac{w}{v}=\frac{R_{c}}{r_{0}}$ with
$R_{c}=\frac{r_{0}}{4}\left(1+\sqrt{1-16\mu}\right)$ is the conformal Killing
horizon of the ordinary Vaidya spacetime described in [19, 30]. We checked
that the curvature invariants such as $R_{ab}R^{ab}$ of this auxiliary
spacetime are only singular at the $w=0$ locus. We note that in the vanishing
$a$ limit, $R_{e}\rightarrow R_{h}$, $R_{a}\rightarrow R_{c}$. As mentioned
earlier in our main narrative, in the chart (10), we restrict the observer’s
$r$ coordinate to be in the interval $(R_{h},R_{c})$ which is precisely the
interval in our spacetime solution that leads to $g_{tt}<0$ for all polar
angles $\theta$.

Figure 9: Penrose diagram for the auxiliary 2D spacetime (71) which is the
$\theta=\frac{\pi}{2},\tilde{\phi}=0$ slice of our Kerr-Vaidya-like black hole
solution with metric (69) . Right boundaries of the two diamonds indicate that
this auxiliary 2D spacetime is asymptotically flat but the full 4D spacetime
is not. As evident in (70), the asymptotic form of the spacetime is not
foliated by two-spheres, and the full causal structure can only be clearly
elucidated via different 2D projections, possibly via the techniques
introduced in [40]. Nonetheless, this simple auxiliary diagram makes manifest
the event horizon interpretation of the curve $r=R_{h}$ which is the $a=0$
limit of the putative event horizon $r=R_{e}$. All curvature invariants of
this spacetime admit $w=0$ as the only singular locus. For example,
$R^{ab}R_{ab}\sim\frac{8\mu^{2}v^{2}}{w^{6}}.$ Our shadow observer is
restricted to lie within the central diamond bounded by the horizons.

##  Appendix B On the energy momentum tensor

Although we have constructed our spacetime metric in a way that is agnostic to
the source, it is instructive to examine the form of the associated energy-
momentum tensor in the chart (69). In the coordinate system (69), to first
order in the small dimensionless parameter $\mu$, the only non-vanishing
component of $T_{\mu\nu}$ is

$8\pi\mathcal{G}T_{vv}=\frac{2\mu}{w^{2}}.$ (72)

Unfortunately beyond the first order expansion in $\mu$, evaluating the
Einstein tensor of our metric yields very complicated expressions for the
components of $T_{\mu\nu}$. Up to second-order in $\mu$ , we find the
components to read

$\displaystyle T_{vv}$ $\displaystyle=$
$\displaystyle\frac{2\mu}{w}+\frac{A^{2}\mu^{2}}{w^{4}}\left(v^{2}-2vw-w^{2}+v(v-5w)\cos(2\theta)\right),$
(73) $\displaystyle T_{vw}$ $\displaystyle=$
$\displaystyle\frac{A^{2}\mu^{2}}{2w^{4}}\left((v-w)(2v+w+(2v-w)\cos(2\theta)\right),$
(74) $\displaystyle T_{v\theta}$ $\displaystyle=$
$\displaystyle\frac{A^{2}\mu^{2}}{2w^{2}}\left(-3v-w+(v+w)\cos(2\theta)\right),$
(75) $\displaystyle T_{v\phi}$ $\displaystyle=$
$\displaystyle\frac{6\mu^{2}Av\sin^{2}\theta}{w^{2}},$ (76) $\displaystyle
T_{ww}$ $\displaystyle=$
$\displaystyle\frac{A^{2}\mu^{2}v^{2}\cos^{2}\theta}{w^{4}},$ (77)
$\displaystyle T_{\phi\phi}$ $\displaystyle=$
$\displaystyle-\frac{A^{2}\mu^{2}v(v-2w)\cos^{2}\theta\sin^{2}\theta}{w^{2}},$
(78) $\displaystyle T_{w\phi}$ $\displaystyle\sim$
$\displaystyle\mathcal{O}(\mu^{3}),\,\,\,T_{\theta\theta}\sim\mathcal{O}(\mu^{3}),\,\,\,T_{\theta\phi}\sim\mathcal{O}(\mu^{3}),$
(79)

where $A\equiv a^{2}/M^{2}_{s}$. These components of the energy-momentum
tensor imply that (at least for generic values of $\mu$) the source cannot be
interpreted as a standard null dust of which energy-momentum tensor can
typically be cast in the form

$T^{ab}\sim k^{a}k^{b},\qquad k^{2}=0,\,{T^{a}}_{a}=0$ (80)

From (73) to (79), up to and including $\mathcal{O}(\mu^{2})$, we find the
trace of the energy-momentum tensor to be

${T^{\alpha}}_{\alpha}=-\frac{A^{2}\mu^{2}}{w^{4}}\left(2v^{2}-2vw-w^{2}+(2v^{2}-4vw+w^{2})\cos(2\theta)\right).$
(81)

In Hawking-Ellis classification, the nature of the energy-momentum tensor in
relation to energy conditions can be further elucidated via the eigenvector
equation

${T^{\mu}}_{\nu}n^{\nu}=\lambda n^{\mu}.$ (82)

We find that up to and including $\mathcal{O}(\mu^{2})$, there are three
triply degenerate null eigenvector and one spacelike eigenvector, with the
non-zero eigenvalue being the energy-momentum tensor trace
${T^{\alpha}}_{\alpha}$, i.e.

$\lambda=\\{0,0,0,{T^{\alpha}}_{\alpha}\\}.$ (83)

It can be shown (e.g. [41] ) that a generic Hawking-Ellis type III energy-
momentum tensor violates the energy conditions (NEC,WEC,SEC) and has no known
classical physical origins. More recently in [25, 26], some attempts were made
at furnishing physical interpretations of type III energy-momentum tensor
including examples that the authors argued to behave like ‘spinning null
fluids’.

##  Appendix C On the reference frame of the shadow observer

###  C.1 In the limit of $a=0$

In the limit of $a=0$, the tetrad basis (44) defining the reference frame of
our shadow observer reduces to the following.

$e_{0}=\frac{e^{-\mu\tilde{T}/M_{s}}}{\sqrt{1-\frac{2\mathcal{M}}{r}}}\partial_{t},\,\,\,e_{1}=\frac{e^{-\mu\tilde{T}/M_{s}}}{r}\partial_{\theta},\,\,\,e_{2}=-\frac{e^{-\mu\tilde{T}/M_{s}}}{r\sin\theta}\partial_{\phi},\,\,\,e_{3}=-e^{-\mu\tilde{T}/M_{s}}\sqrt{\left(1-\frac{2\mathcal{M}}{r}\right)}\partial_{r}.$
(84)

In the chart $\\{\tilde{T},r,\theta,\tilde{\phi}\\}$, we replace
$\partial_{t}\rightarrow\partial_{\tilde{T}},\,\,\partial_{r}\rightarrow\frac{\partial\tilde{T}}{\partial
r}\frac{\partial}{\partial\tilde{T}}+\frac{\partial}{\partial r}$ which leads
to

$e_{0}=\frac{e^{-\mu\tilde{T}/M_{s}}}{\sqrt{f}}\partial_{\tilde{T}},\,\,\,e_{1}=\frac{e^{-\mu\tilde{T}/M_{s}}}{r}\partial_{\theta},\,\,\,e_{2}=-\frac{e^{-\mu\tilde{T}/M_{s}}}{r\sin\theta}\partial_{\phi},\,\,\,e_{3}=-\frac{e^{-\mu\tilde{T}/M_{s}}}{\sqrt{f}}\left(\partial_{\tilde{T}}+f\partial_{r}\right),$
(85)

where $f=1-\frac{2\mathcal{M}(r)}{r}$. This is the tetrad basis used in [19]
for Vaidya spacetime’s shadow calculation, relevant for an observer with
4-velocity $e_{0}$, at constant $\theta,\phi,r$.

###  C.2 Observers in the $\\{v,w,\theta,\phi\\}$ chart and aberration
formulas

The coordinate system $\\{v,w,\theta,\phi\\}$ avoids the horizons as
coordinate singularities. In [19], the shadow observed by an observer with
4-velocity $\sim\frac{\partial}{\partial v}$ was derived using an aberration
formula being applied to the shadow angle formula.

In general, for a pair of reference frames $(S,S^{\prime})$, aberration
formulas relating the coordinates of their celestial spheres can be derived
from the expression (46) after we express the 4-velocity $e^{\prime}_{0}$ of
the $S^{\prime}$ reference frame as a linear combination of the original
tetrad basis components, writing

$e^{\prime}_{0}=\frac{e_{0}+V^{k}e_{k}}{\sqrt{1-v^{2}}},$ (86)

where $\vec{V}$ is the relative 3-velocity of $S^{\prime}$ observer. Consider
an observer of which $e^{\prime}_{0}$ is a linear combination of $e_{0}$ and
another basis vector $e_{3}$. Its tetrad basis components read

$e^{\prime}_{0}=\frac{1}{\sqrt{1-V^{2}}}(e_{0}+Ve_{3}),\,\,\,e^{\prime}_{3}=\frac{1}{\sqrt{1-V^{2}}}(e_{3}+Ve_{0}),\,\,\,e^{\prime}_{1,2}=e_{1,2}.$
(87)

Taking the inner product between $e^{\prime}_{0},e^{\prime}_{3}$ and the
tangent vector expression in (46) in both unprimed and primed coordinates, we
obtain the aberration formulas

$\cos\theta^{\prime}=\frac{V+\cos\theta}{1+V\cos\theta},\qquad\phi^{\prime}=\phi,\qquad
V=-\frac{e^{\prime}_{0}\cdot e_{3}}{e^{\prime}_{0}\cdot e_{0}}.$ (88)

In [19], the observer with $e^{\prime}_{0}\sim\frac{\partial}{\partial v}$ was
also considered. After a coordinate transformation from
$\\{\tilde{T},r,\theta,\tilde{\phi}\\}$ used for the tetrad basis in (85), one
can show that

$e^{\prime}_{0}\sim\frac{\partial}{\partial
v}=\frac{1}{\sqrt{1-\frac{2M_{s}}{r}}}e^{-\frac{T}{r_{0}}}\left(\partial_{\tilde{T}}-\frac{r}{r_{0}}\partial_{r}\right)=\frac{1}{\sqrt{1-V^{2}}}(e_{0}+Ve_{3}),\,\,V=\frac{r^{2}}{r^{2}+rr_{0}\left(1-\frac{2\mathcal{M}(r)}{r}\right)}$
(89)

where $e_{0},e_{1},e_{2},e_{3}$ are as defined in (85). In [19], the same
expression for the relative velocity $V$ was obtained with an aberration
relation
$\tan^{2}\frac{\theta^{\prime}}{2}=\frac{1-V}{1+V}\tan^{2}\frac{\theta}{2}$
that we verified to be identical to (88).

Let us now consider an appropriate $S^{\prime}$ observer for our Kerr-Vaidya-
like geometry. We note that for the $S$ observer, its 4-velocity $e_{0}$ is a
linear combination of $\partial_{\tilde{T}}$ and $\partial_{\tilde{\phi}}$, or
in the Boyer-Lindquist-like chart, a linear combination of $\partial_{t}$ and
$\partial_{\phi}$. The angular component is such that $e_{0}\pm e_{3}$ are
tangential to the principal null congruences of the metric. Relating between
$v$ and $t$, we choose the following $S^{\prime}$ observer with

$\displaystyle e^{\prime}_{0}\sim\frac{\partial}{\partial
v}+\frac{r_{0}a}{v(r^{2}+a^{2})}\frac{\partial}{\partial\tilde{\phi}}$
$\displaystyle=$
$\displaystyle\frac{r_{0}}{v}\left(1+\frac{r(r^{2}+a^{2})}{r_{0}(r^{2}-2\mathcal{M}r+a^{2})}\right)\partial_{t}-\frac{r}{v}\partial_{r}-\frac{r_{0}a}{v(r^{2}-2\mathcal{M}r+a^{2})}\partial_{\phi}$
(91)
$\displaystyle+\frac{r_{0}a}{v(r^{2}+a^{2})}\frac{\partial}{\partial\tilde{\phi}}.$

This choice of $e^{\prime}_{0}$ is also uniquely the one that allows us to
write

$e^{\prime}_{0}\sim e_{0}+we_{3},$ (92)

for some relative 3-velocity $w$. Thus the aberration formulas in (88) apply
similarly. Recall that in (44), the relevant basis tetrad components read

$e_{0}=e^{-\mu\tilde{T}/M_{s}}\frac{(r^{2}+a^{2})\partial_{t}+a\partial_{\phi}}{\sqrt{\Sigma\Delta}},\,\,e_{3}=-e^{-\mu\tilde{T}/M_{s}}\sqrt{\frac{\Delta}{\Sigma}}\partial_{r},$
(93)

which allows us to read off the 3-velocity as

$w=\frac{r^{2}+a^{2}}{r^{2}+a^{2}+\frac{r_{0}}{r}(r^{2}-2\mathcal{M}r+a^{2})}.$
(94)

In the limit of vanishing $a$, we recover the 3-velocity $v$ for the observer
in Vaidya spacetime with $e_{0}\sim\frac{\partial}{\partial\tilde{T}}$ as
derived in [19]. In the limit $\mu\rightarrow 0$, up to leading order, we have

$w\approx\mu\left(\frac{r^{2}+a^{2}}{\frac{M_{s}}{r}(r^{2}-2M_{s}r+a^{2})}\right)+\mathcal{O}(\mu^{2}).$
(95)

Thus, this reference frame may be relevant for theoretical situations where
the observer’s velocity is proportional to the strength of the accretion rate,
although arguably not so for realistic EHT observations where the accretion is
hardly expected to backreact on the metric significantly to affect the
3-velocity of the shadow observer in such a manner.

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