# General Approach on Shadow Radius and Photon Spheres in Asymptotically Flat
Spacetimes and the Impact of Mass-Dependent Variations

Vitalii Vertogradov<EMAIL_ADDRESS>Physics department, Herzen state
Pedagogical University of Russia, 48 Moika Emb., Saint Petersburg 191186,
Russia  SPB branch of SAO RAS, 65 Pulkovskoe Rd, Saint Petersburg 196140,
Russia  Ali Övgün<EMAIL_ADDRESS>Physics Department, Eastern
Mediterranean University, Famagusta, 99628 North Cyprus, via Mersin 10,
Turkiye

###### Abstract

Recent observations of black hole shadows have revolutionized our ability to
probe gravity in extreme environments. This manuscript presents a novel
analytic method to calculate, in leading-order terms, the key parameters of
photon sphere and shadow radius. This method offers advantages for complex
metrics where traditional approaches are cumbersome. We further explore the
impact of black hole mass on the photon sphere radius, providing insights into
black hole interactions with their surroundings. Our findings contribute
significantly to black hole physics and gravity under extreme conditions. By
leveraging future advancements in observations, such as the next-generation
Event Horizon Telescope (ngEHT), this work paves the way for even more precise
tests of gravity near black holes.

Black hole; Photon sphere; Mass-dependent; Shadow.

###### pacs:

95.30.Sf, 04.70.-s, 97.60.Lf, 04.50.Kd

##  I INTRODUCTION

The groundbreaking observations of black hole shadows have ushered in a new
era for probing gravity in extreme environments. In the 1970s, physicist James
Bardeen introduced the idea of a black hole shadow, which has since become a
major focus in astrophysics research Bardeen (1968). This shadow forms because
light bends around the black hole, creating a dark area against the background
of bright surrounding matter Luminet (1979); Falcke  _et al._ (2000). By
studying this shadow, scientists can learn more about black holes, including
their mass, spin, and the structure of spacetime nearby. Recent advances in
observation, like very-long-baseline interferometry (VLBI) and the Event
Horizon Telescope (EHT), have enabled direct imaging of the shadow of the
supermassive black holes at the center of the M87 and and Sgr A* galaxies
Akiyama  _et al._ (2019, 2022). This has opened up new opportunities to test
theories about gravity and the nature of black holes Bronzwaer and Falcke
(2021); Vagnozzi  _et al._ (2023); Vertogradov and Övgün (2024); Perlick  _et
al._ (2015); Virbhadra (2024); Adler and Virbhadra (2022); Virbhadra (2022);
Virbhadra and Keeton (2008); Virbhadra and Ellis (2000); Virbhadra  _et al._
(1998); Yang  _et al._ (2024); Zakharov (2022, 2023); Alloqulov  _et al._
(2024); Abdujabbarov  _et al._ (2016); Atamurotov  _et al._ (2013);
Abdujabbarov  _et al._ (2015); Abdikamalov  _et al._ (2019); Abdujabbarov  _et
al._ (2015); Vagnozzi  _et al._ (2020); Claudel  _et al._ (2001); Hod (2020,
2013a, 2013b, 2023a, 2023b); Khoo and Ong (2016); Decanini  _et al._ (2010);
Pantig  _et al._ (2022); Shoom (2017); Konoplya and Zhidenko (2021); Kuang and
Övgün (2022); Bronnikov  _et al._ (2021); Paithankar and Kolekar (2023);
Konoplya  _et al._ (2016); Johannsen and Psaltis (2011); Cardoso  _et al._
(2014); Konoplya and Zhidenko (2019); Qiao (2022); Kokkotas  _et al._ (2017);
Hennigar  _et al._ (2018); Konoplya  _et al._ (2020); Aratore  _et al._
(2024); Tsupko (2022); Cederbaum and Galloway (2016); Johannsen (2013); Teo
(2003); Okyay and Övgün (2022); Lu and Lyu (2020); Younsi  _et al._ (2016);
Tsupko and Bisnovatyi-Kogan (2020); Tsukamoto  _et al._ (2014); Ghosh  _et
al._ (2023); Övgün  _et al._ (2024); Adair  _et al._ (2020); Tsukamoto (2018);
Gómez and Valageas (2024); Shaikh  _et al._ (2019); Lambiase  _et al._ (2023);
Karshiboev  _et al._ (2024); Rayimbaev  _et al._ (2023); Mustafa  _et al._
(2022); Pantig  _et al._ (2023); Pantig and Övgün (2023); Kumaran and Övgün
(2023); Çimdiker  _et al._ (2023); Puliçe  _et al._ (2023); Övgün  _et al._
(2018); Heydarzade and Vertogradov (2023); Allahyari  _et al._ (2020);
Pedrotti and Vagnozzi (2024).

While General Relativity (GR) has successfully described many astronomical
observations with high precision, there are compelling reasons to consider
extending this framework. The theory has faced challenges in predicting
phenomena such as the accelerating expansion of the Universe, the existence of
dark matter, and its compatibility with quantum field theory. On the other
hand, the direct observation of supermassive black holes offers a unique
opportunity to test and constrain modifications to GR in their vicinity.
Future projects like the Next Generation Event Horizon Telescope (ngEHT),
which plans to incorporate satellites to increase baseline and image
resolution, will provide even more precise measurements of supermassive black
holes properties Tiede  _et al._ (2022); Ayzenberg  _et al._ (2023), such as
spin. These observations will be crucial for testing alternative theories of
gravity and improving our understanding of the fundamental forces governing
the universe.

The photon sphere, also referred to as the circular photon orbit or light
ring, and the shadow radius play crucial roles in the field of black hole
physics Perlick and Tsupko (2022). These quantities are central to the
examination of particle trajectories, gravitational lensing effects Ishihara
_et al._ (2016); Kudo and Asada (2022), and optical phenomena related to black
holes. Following the EHT Collaboration’s landmark observations of highly
detailed black hole images, interest in studying photon spheres and black hole
shadows has surged within the physics and astronomy communities.
Traditionally, these values are determined through calculations involving the
effective potential governing the motion of test particles within a black
hole’s spacetime. However, in certain scenarios, deriving analytic results can
be exceedingly challenging, necessitating the use of numerical calculations.
In this paper, we present a novel approach for determining the photon sphere
and shadow radius in leading order terms.

On the other hand, comprehending the impact of mass on black hole
characteristics, especially the radius of the photon sphere, holds paramount
importance for advancing our understanding of black hole physics in realistic
scenarios, particularly in the context of accretion processes around black
holes where accretion refers to the process by which matter, such as gas,
dust, or other material, is drawn in and accumulates around a massive object
due to the object’s gravitational pull Tan (2023); Koga  _et al._ (2022);
Solanki and Perlick (2022); Mishra  _et al._ (2019); Rodríguez  _et al._
(2024); Bambi (2017); Shcherbakov  _et al._ (2012); Psaltis (2019); Cunha  _et
al._ (2018); Ayzenberg and Yunes (2018); Wang  _et al._ (2018); Cunha and
Herdeiro (2018); Cunha  _et al._ (2015, 2017a, 2017b); Lima  _et al._ (2021);
Cunha  _et al._ (2017c). When it comes to black holes, accretion is a crucial
phenomenon that occurs when matter from a surrounding disk or cloud falls
toward the black hole. As this matter spirals inward, it forms an accretion
disk—a structure of rapidly orbiting material that emits energy, often in the
form of X-rays, as it is heated by friction and other processes. The mass of a
black hole determines its gravitational pull, dictating where light is trapped
in spacetime. Studying how the photon sphere radius changes with mass can
reveal valuable insights into how black holes interact with their
surroundings.

This paper has two primary aims. The first is to introduce a novel analytic
method for calculating the photon sphere and shadow radius in leading-order
terms. This method is particularly valuable for understanding the properties
of metrics where standard methods are not applicable. The second aim is to
investigate how the mass of a black hole influences the radius of its photon
sphere. By studying the variation of the photon sphere radius with mass, we
seek to comprehend how a black hole’s gravitational field is impacted by its
mass-dependent characteristics. This work has the potential to enhance our
understanding of black hole physics and to contribute significantly to our
knowledge of gravity in extreme environments.

This work is structured as follows: In Section II, we introduce a novel
general approach for calculating the photon sphere and shadow radius. We then
provide several illustrative examples. Section III outlines a method for
estimating the change in the radius of a photon sphere with mass and applies
this method to a selection of examples. Finally, we conclude with a summary in
Section IV.

##  II GENERAL APPROACH FOR ASYMPTOTICALLY FLAT METRIC

We consider the asymptotically flat spherically-symmetric spacetime in the
form

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ (1)

where $f(r)$ is a lapse function and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta
d\varphi^{2}$ is the metric on the unit two-sphere. We can extend the concept
of expanding the photon sphere, where we express it as a Schwarzschild
solution with small corrections, to the metric function. By considering a
metric that approaches flat spacetime at infinity, we can expand it using
negative powers of a small parameter

$f(r)=1-\frac{2M}{r}+\sum_{i=2}^{n}\frac{\alpha_{i}}{r^{i}}+\mathcal{O}\left(\frac{1}{r^{n+1}}\right)$
(2)

The parameters $\alpha_{i}$ represent the differences between a given metric
and the Schwarzschild metric. The spacetime described by Equation (1) is
static and spherically symmetric. Along null geodesics, which describe the
paths of light rays, there are two constants of motion: the energy per unit
mass $E$ and the angular momentum per unit mass $L$. In the equatorial plane,
where $\theta=\frac{\pi}{2}$, these constants of motion can be expressed as
follows:

$\displaystyle E$ $\displaystyle=$ $\displaystyle f(r)\frac{dt}{d\lambda},$
$\displaystyle L$ $\displaystyle=$ $\displaystyle
r^{2}\frac{d\varphi}{d\lambda},$ (3)

where $\lambda$ is an affine parameter. By using condition for null geodesics
$g_{ik}\frac{dx^{i}}{d\lambda}\frac{dx^{k}}{d\lambda}=0$, one obtains the
radial component $\frac{dr}{d\lambda}$ in the form

$\displaystyle\left(\frac{dr}{d\lambda}\right)^{2}+V_{eff}(r)=0,$
$\displaystyle V_{eff}=f(r)\frac{L^{2}}{r^{2}}-E^{2},$ (4)

where $V_{eff}(r)$ is an effective potential. The radius of a photon sphere
$r_{ph}$ should satisfy the following two conditions

$V_{eff}(r_{ph})=0,~{}~{}V^{\prime}_{eff}(r_{ph})=0.$ (5)

The second condition (5) give for the lapse function (2)

$2r^{n}-6Mr^{n-1}+\sum_{i=2}^{n}(i+2)\alpha_{i}r^{n-i}=0.$ (6)

This algebraic equation has $n$ roots. However, we are interested only in
maximal real root of the equation (6) which has small diviation from
$r_{1}=3M$ and we will consider only this case111We assume, that
$\alpha_{i}\ll 1$..

Theorem (Photon Sphere): In order to estimate the influence $\alpha_{i}$ on
the radius of a photon sphere, we assume the radius of a photon sphere
$r_{ph}$ to be in the form
$r_{ph}=r_{1}+\sum_{i=2}^{n}\alpha_{i}r_{i}+\mathcal{O}\left(\alpha_{n+1}\right)$
(7) substituting it into the defining equation (6) and expanding it up to
order $\mathcal{O}(\alpha_{i}^{2})$, we obtain for $r_{i}$ the relation:
$r_{i}=-\frac{i+2}{6(3M)^{i-2}}$ (8) Therefore, whether the radius increases
or decreases depends on the sign of $\alpha_{i}$: a positive sign leads to a
decrease (as in the Reissner-Nordstrom case), while a negative sign
corresponds to an increase.

Now, we estimate how the radius of a shadow changes. We can write

$b_{cr}^{2}=\frac{r_{ph}^{2}}{f\left(r_{ph}\right)}.$ (9)

By using the fact that at $r=r_{ph}$ the condition $f^{\prime}r-2f=0$ is held,
we can rewrite (9) as

$b_{cr}^{2}=\frac{2r_{ph}}{f^{\prime}\left(r_{ph}\right)}.$ (10)

Assuming lapse function $f(r)$ in the form (2), using the radius $r_{ph}$ in
the form (7), neglecting terms of order
$\mathcal{(}O)\left(\alpha_{I}^{2}\right)$, we arrive, after some algebraic
calculations at

Theorem (Shadow Radius):
$b_{cr}^{2}=27M^{2}+27M\sum_{i=2}^{n}\alpha_{i}r_{i}-\frac{\left(27M^{2}+27M\sum_{i=2}^{n}\alpha_{i}r_{i}\right)\xi}{2Mr_{ph}^{n-1}-\xi},$
(11) where $\xi=\sum_{i=2}^{n}i\alpha_{i}r_{ph}^{n-i}.$ (12)

The equation (11) defines the radius of a black hole shadow for an arbitrary
asymptotically flat spacetime. The visible angular size of a shadow which can
be seen by an observer far from black hole, i.e. $r_{o}\gg M$ ($r_{o}$ is a
distance between an observer and black hole.) is given by (for asymptotically
flat spacetimes $f(r_{o})\to 1$ Vagnozzi  _et al._ (2023)

$\sin^{2}\omega_{sh}=b_{cr}^{2}\frac{f(r_{o})}{r_{o}}$ (13)

$R_{sh}\approx b_{cr}.$ (14)

B employing the first-order WKB formula and expanding the values for $\omega$
as a series of $\alpha_{i}$, it is also possible to determine the quasinormal
modes in the eikonal regime Churilova (2019):

$\begin{array}[]{c}\omega=\frac{\left(\ell+\frac{1}{2}\right)}{3\sqrt{3}M}\left(1+\frac{\alpha_{2}}{6M^{2}}+\frac{\alpha_{3}}{18M^{3}}+\frac{\alpha_{4}}{54M^{4}}+\frac{\alpha_{5}}{162M^{5}}\right)\\\
-i\frac{\left(n+\frac{1}{2}\right)}{3\sqrt{3}M}\left(1+\frac{\alpha_{2}}{18M^{2}}-\frac{\alpha_{3}}{27M^{3}}-\frac{\alpha_{4}}{27M^{4}}-\frac{11\alpha_{5}}{486M^{5}}\right)\\\
\quad+\mathcal{O}\left(\frac{1}{\ell+\frac{1}{2}}\right).\end{array}$ (15)

###  II.1 Example: Reissner-Nordstrom Black Hole

We first give the example of Reissner-Nordstrom black hole. The metric
function for Reissner-Nordstrom black hole solutions is:

$\displaystyle f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}.$ (16)

where $Q$ is the black hole’s charge. Using the Eq.2, we can obtain the
$\alpha$ coefficient for for magnetically charged black hole as

$\displaystyle\alpha_{2}=Q^{2}.$ (17)

In order to estimate the influence $\alpha_{i}$ on the radius of a photon
sphere, we assume the radius of a photon sphere $r_{ph}$ using the general
approach:

$r_{ph}=3M-\frac{2Q^{2}}{3}.$ (18)

Note that analytic result of photon sphere of Reissner-Nordstrom black hole is

$r_{ph}=\frac{1}{2}\left(\sqrt{9M^{2}-8Q^{2}}+3M\right).$ (19)

In the Fig. 1, it is seen how change of $Q$ affect the photon sphere. It is
evident that for small values of $Q$, the two approaches align closely.

Figure 1: The Figure shows the comparison between general approach and
analytic method for the photon sphere of Reissner-Nordstrom black hole.

Now, we estimate how the radius of a shadow changes. The shadow radius for the
Reissner-Nordstrom black hole is calculated using the general approach as:

$R_{sh}=-\frac{2Q^{2}\left(27M^{2}-18MQ^{2}\right)}{2M\left(3M-\frac{2Q^{2}}{3}\right)-2Q^{2}}+27M^{2}-18MQ^{2}.$
(20)

Note that analytic result of shadow radius for the Reissner-Nordstrom black
hole is

$R_{sh}=\frac{1}{2}\sqrt{-\frac{\left(\Lambda\right)^{4}}{4M\left(\Lambda\right)+4Q^{2}-6\left(\Lambda\right)}}.$
(21)

where $\Lambda=\sqrt{9-8Q^{2}}+3.$

In the Fig. 2, it is seen how change of $Q$ affect the shadow radius of
Reissner-Nordstrom black hole using two different method. It is observed that
for small values of $Q$, the two approaches are in agreement.

Figure 2: The Figure shows the comparison between general approach and
analytic method for the shadow radius of Reissner-Nordstrom black hole.

###  II.2 Example: Bardeen Black Hole

Bardeen introduced an exact black hole solution in 1968 Bardeen (1968);
Vagnozzi  _et al._ (2023) that does not exhibit singularities at its origin,
meaning that the curvature scalar does not tend to infinity. The line element
describing the Bardeen regular black hole can be expressed as follows:

$\displaystyle f(r)=1-\frac{2Mr^{2}}{\left(e^{2}+r^{2}\right)^{3/2}},$ (22)

Here, $M$ represents the mass, and $e$ denotes the monopole charge of a self-
gravitating magnetic field sourced by non-linear electrodynamics.

The function $f(r)$ can be approximated asymptotically as

$f(r)\approx
1-\frac{2M}{r}+\frac{3e^{2}M}{r^{3}}-\frac{15e^{4}M}{4r^{5}}+O\left(e^{5}\right)$
(23)

Using the Eq.2, we can obtain the $\alpha$ coefficient for Bardeen black hole
as

$\displaystyle\alpha_{3}=3e^{2}M,\alpha_{5}=-\frac{15e^{4}M}{4},$ (24)

In order to estimate the influence $\alpha_{i}$ on the radius of a photon
sphere, we assume the radius of a photon sphere $r_{ph}$

$r_{ph}=3M+\frac{35e^{4}}{216M^{2}}-\frac{5e^{2}}{6}+O\left(e^{5}\right).$
(25)

In the Fig. it is seen how change of $e$ affect the photon sphere.

Now, we estimate how the radius of a shadow changes. The shadow radius for the
Bardeen black hole is calculated using the general approach as:

$\displaystyle
R_{sh}=3\sqrt{3}M-\frac{\sqrt{3}e^{2}(5M+3)}{4M}+O\left(e^{5}\right).$ (26)

The shadow radius of the Bardeen black hole decreases as the monopole charge
$e$ increases.

###  II.3 Example: Magnetically charged Einstein-Euler-Heisenberg Black Hole

Another example is the magnetically charged black hole solutions derived from
the Einstein-Euler-Heisenberg nonlinear electrodynamics, which represents a
low-energy limit of Born-Infeld electrodynamics. The metric function for
magnetically charged black hole solutions within this theory is given by
Allahyari  _et al._ (2020):

$\displaystyle
f(r)=1-\frac{2M}{r}+\frac{q_{m}^{2}}{r^{2}}-\frac{2\mu}{5}\frac{q_{m}^{4}}{r^{6}}\hskip
1.69998pt,$ (27)

where $q_{m}$ signifies the black hole’s magnetic charge, a defining
characteristic, while $\mu$ represents the coupling parameter in the Einstein-
Euler-Heisenberg nonlinear electrodynamics (NLED), as outlined in Equation
(27). Notably, for $\mu\neq 0$, magnetic charges exceeding $q_{m}>1$ are
allowable.

Utilizing Equation (2), we can derive the $\alpha$ coefficient for
magnetically charged black holes as:

$\displaystyle\alpha_{2}=q_{m}^{2},$ (28)
$\displaystyle\alpha_{6}=\frac{-2\mu}{5}q_{m}^{4}.$ (29)

In order to estimate the influence $\alpha_{i}$ on the radius of a photon
sphere, we assume the radius of a photon sphere $r_{ph}$

$r_{ph}=3M-\frac{8\mu q_{m}^{4}}{1215M^{4}}-\frac{2q_{m}^{2}}{3}.$ (30)

Now, we estimate how the radius of a shadow changes. The shadow radius for
magnetically charged black hole is calculated using the general approach as:

$\displaystyle R_{sh}=\frac{2\sqrt{\frac{2}{5}}M^{3/2}\sqrt{-\frac{\mu
q_{m}^{4}}{M^{3}}}}{3M^{3/2}}-\frac{81M^{3/2}\left(\sqrt{\frac{5}{2}}q_{m}^{2}\right)}{4\left(M^{3/2}\sqrt{-\frac{\mu
q_{m}^{4}}{M^{3}}}\right)}$ $\displaystyle+O\left(M^{5/2}\right).$ (31)

The photon sphere and shadow radius of the magnetically charged black hole
decreases as the magnetic charge $q_{m}$ increases.

##  III ESTIMATION HOW RADIUS OF A PHOTON SPHERE CHANGES WITH MASS

In this section, we find a method to estimate how radius of a photon sphere
changes with mass. For this purpose we consider general static spherically
symmetric spacetime in Eddington-Finkelstein coordinates

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dv^{2}+2dvdr+r^{2}d\Omega^{2},$ $\displaystyle f(r)$
$\displaystyle=$ $\displaystyle 1-\frac{2M(r)}{r}.$ (32)

This spacetime is supported with anisotropic energy-momentum tensor of the
form

$\displaystyle\rho=-T^{0}_{0}=T^{1}_{1}=\frac{2M^{\prime}}{r^{2}},$
$\displaystyle P=T^{2}_{2}=T^{3}_{3}=-\frac{M^{\prime\prime}}{r}.$ (33)

The radial component of the four-vector tangent to null geodesic is given by

$\displaystyle\left(\frac{dr}{d\lambda}\right)^{2}+V_{eff}=0,$ $\displaystyle
V_{eff}=E^{2}-\left(1-\frac{2M}{r}\right)\frac{L^{2}}{r^{2}}.$ (34)

Where $V_{eff}$ is an effective potential and $E$ and $L$ are energy and
angular momentum respectively. To find unstable photon orbit, one needs to
satisfy two conditions (5). The second condition gives us a equation

$3M-M^{\prime}r-r=0.$ (35)

Now, we will find out how a radius of photon sphere $r_{ph}$ changes if we
consider small change of mass of a black hole. For this purpose, we introduce
small dimensionless parameter $|\alpha|\ll 1$ and the mass of black hole
becomes $M+\alpha M$. Where $\alpha>0$ corresponds to accretion and $\alpha<0$
to radiation processes respectively. We write the effective potential in the
form

$E^{2}-\left(1-\frac{2M}{r}\right)\frac{L^{2}}{r^{2}}\left(1-\frac{2\alpha
M}{r-2M}\right).$ (36)

For simplicity, we introduce two new functions

$\displaystyle F(r)$ $\displaystyle\equiv$ $\displaystyle-\frac{r-2M}{r^{3}},$
$\displaystyle G(r)$ $\displaystyle\equiv$ $\displaystyle\frac{2M}{r-2M}.$
(37)

Then the second condition (5) gives

$F^{\prime}(1-\alpha G)-\alpha G^{\prime}F=0.$ (38)

Now we assume that the radius of a photon sphere is given by

$r_{ph}=r_{0}+\alpha r_{1},$ (39)

where $r_{0}$ is the radius of a photon sphere before accretion or radiation
processes. Substituting (39) into (38) and expanding it with respect to alpha
we find

$r_{1}=\frac{F(r_{0})G^{\prime}(r_{0})}{F^{\prime\prime}(r_{0})}.$ (40)

Substituting (III) into (40) and evaluating it on the photon sphere, we find

$r_{1}=\frac{r_{0}}{1-2M^{\prime}(r_{0})+M^{\prime\prime}(r_{0})r_{0}}.$ (41)

The dominant energy condition demands $\rho\geq|P|$. From (III), we can write
$2M^{\prime}\geq|-M^{\prime\prime}r|$ which gives us the dominator of (41) as
$1-4M^{\prime}$. From (35), we know that $M^{\prime}=\frac{3M}{r_{0}}-1$. Thus
we have in the denominator of (41) $5-\frac{12M(r_{0})}{r_{0}}$.

In the paper Hod (2020), it was proved that the radius of a photon sphere
should be at least $\frac{3}{2}r_{h}$, where $r_{h}$ is the event horizon
location. If we substitute it and use the fact that $\frac{2M}{r_{h}}\leq 1$
then we find that denominator of (41) less than 1. This fact allows us to
state that $r_{1}$ is always positive.

Theorem (Photon Sphere and Mass): This implies a direct relationship between
the mass and the radius of the photon sphere: as the mass increases, the
radius of the photon sphere also increases; conversely, if the mass decreases,
so does the radius of the photon sphere $r_{ph}$.

We can show in the same manner that even we consider the equation of state
$P=w(r)\rho$ then this condition is held for an arbitrary $w(r)$. The mass
function $M(r)$ depends upon several parameters which describe a black hole.

For example in Reissner-Nordstrom case it depends on both constant mass $m$
and electric charge $Q$. It may happen, that by increases the mass of black
hole and electric charge the radius of photon sphere can decrease. As an
example let’s consider the Reissner-Nordstrom spacetime in which mass function
is given by

$M(r,m,Q)=m-\frac{Q^{2}}{2r}.$ (42)

And we consider small positive change in mass and electrical charge by
$m=m+\alpha m$ and $Q+\beta Q$, where $\alpha\ll 1$ and $\beta\ll 1$ are small
positive constants. By substituting these changes into the mass function (42)
and expanding it with respect to $\alpha$ and $\beta$, we arrive at

$\displaystyle M(r,m+\alpha m,Q+\beta Q)$ $\displaystyle\approx
m+\alpha\frac{\partial M(r,m,Q)}{\partial m}+\beta\frac{\partial
M(r,m,Q)}{\partial Q}.$ (43)

By choosing parameters $\alpha,\beta,m$ and $Q$ we can make two last terms be
negative. It means that we increases mass parameter $m$ but decreases the mass
function $M(r)$ which means that the radius of a photon sphere and shadow
decrease.

Now, we shall prove that it is possible only when the weak energy condition is
violated. For this purpose, we should understand the nature of parameters
$\alpha$ and $\beta$. We consider the accretion or radiating processes. It
means that the spacetime should be dynamical one. We write it in the form

$ds^{2}=-\left(1-\frac{2M(v,r)}{r}\right)dv^{2}+2\varepsilon
dvdr+r^{2}d\Omega^{2}.$ (44)

Where $M(v,r)$ is the mass function of both time $v$ and radial coordinate $r$
and $\varepsilon=\pm 1$ depending on ingoing or outgoing flux respectively.
The mass function change is described by $T^{1}_{0}$ component of energy-
momentum tensor, which reads

$T^{1}_{0}=2\varepsilon\frac{\dot{M}}{r^{2}}.$ (45)

In order to satisfy the weak energy condition this flux must be positive which
means $\dot{M}\geq 0$ for accretion $\varepsilon=+1$ and negative $\dot{M}\leq
0$ for radiation $\varepsilon=-1$. However, the mass function depends on
parameters $m$ and $Q$ which should vary in time, i.e. they should be the
functions of time. We consider the accretion process, i.e. $\varepsilon=+1$.
It means

$\displaystyle\dot{M}=\frac{\partial M(r,M(v),Q(v))}{\partial
m}\frac{dM(v)}{dv}$ $\displaystyle+\frac{\partial M(r,M(v),Q(v))}{\partial
Q}\frac{dQ(v)}{dv}.$ (46)

Comparing this expression with , we find

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\frac{dM(v)}{dv},$
$\displaystyle\beta$ $\displaystyle=$ $\displaystyle\frac{dQ(v)}{dv}.$ (47)

If the mass function $M(r)$ decreases it means that $\dot{M}$ decreases which
violate weak energy condition. We prove it only for two parameters but this
can be extended for an arbitrary amount of parameters. We can state it as a
theorem.

Theorem: If the mass of a black hole increases while the weak energy condition
holds, the size of the black hole’s shadow also increases.

###  III.1 Example: Hairy Schwarzschild Black Hole

Our first example is the hairy Schwarzschild black hole, obtained through
gravitational decoupling as described in the paper by Ovalle et al. Ovalle
_et al._ (2021). It has the lapse function and the mass function has the
following form

$\displaystyle f(r)=1-\frac{2M(r)}{r},$ (48)

and

$\displaystyle M(r)$ $\displaystyle=$ $\displaystyle
m+\frac{\bar{\alpha}l}{2}-\frac{\bar{\alpha}r}{2}e^{-\frac{r}{M}}.$ (49)

Here, $m=const.$ is the mass of a black hole, $\bar{\alpha}>0$ is coupling
constant and $l>0$ is related to a primary hair. We consider changes in mass
and primary hair as

$\displaystyle m\rightarrow m+\varepsilon m,~{}~{}\varepsilon\ll
1,~{}~{}\varepsilon>0,$ $\displaystyle l\rightarrow l+\beta l,~{}~{}\beta\ll
1,~{}~{}\beta>0.$ (50)

We can write the mass function up to order
$\mathcal{O}\left(\varepsilon^{2}\right)$, $M(r)\to\mathcal{M}(r)$ as

$\displaystyle\mathcal{M}(r)=\frac{\bar{\alpha}\beta
l}{2}+\frac{\bar{\alpha}l}{2}-\frac{1}{2}\bar{\alpha}re^{-\frac{r}{m}}+m$
$\displaystyle+\epsilon\left(m-\frac{\bar{\alpha}r^{2}e^{-\frac{r}{m}}}{2m}\right)+O\left(\epsilon^{2}\right),$
(51)

and

$\displaystyle\mathcal{M}(r)<M(r)+\frac{\bar{\alpha}\beta
l}{2}+\epsilon\left(m-\frac{\bar{\alpha}r^{2}e^{-\frac{r}{m}}}{2m}\right)+O\left(\epsilon^{2}\right).$
(52)

As one can see, under some conditions, the new mass function $\mathcal{M}(r)$
might be less than $M(r)$, i.e. the radius of a photon sphere decreases
despite the mass of a black hole grows. However, it might happen only if
$\bar{\alpha}>1$.

###  III.2 Example: Kiselev Black Hole

Kiselev black hole Kiselev (2003) is explicit example that the radius of a
photon sphere might decrease while the mass of a black hole grows. The Kiselev
solution has the lapse and mass functions of the form

$\displaystyle f(r)=1-\frac{2M(r)}{r},$ (53)

and

$\displaystyle M(r)$ $\displaystyle=$ $\displaystyle
m+\frac{N}{2r^{3\omega}}.$ (54)

Here $m=const.$ is a black hole mass, $N$ is a cosmological parameter and
$\omega$ is a parameter of equation of state. We consider only positive values
of $\omega$. In this case, the cosmological parameter $N$ should be always
negative. We consider the positive changes of mass and cosmological parameters

$\displaystyle m\rightarrow m+\varepsilon m,~{}~{}\varepsilon\ll
1,~{}~{}\varepsilon>0,$ $\displaystyle N\rightarrow N+\beta N,~{}~{}\beta\ll
1,~{}~{}\beta>0.$ (55)

In this case $M(r)\to\mathcal{M}(r)$ as

$\mathcal{M}(r)=m+m\epsilon+\frac{1}{2}\beta
Nr^{-3\omega}+\frac{1}{2}Nr^{-3\omega}$ (56)

and one can write it

$\mathcal{M}(r)<M(r)+m\epsilon+\frac{1}{2}\beta Nr^{-3\omega}.$ (57)

With negative $N$ one can always find such $\varepsilon$ and $\beta$ that
$\mathcal{M}(r)<M(r)$.

###  III.3 Example: Magnetically charged Einstein-Euler-Heisenberg Black Hole

Another example is the magnetically charged black hole solutions within this
theory, as described in Allahyari  _et al._ (2020)

$\displaystyle
f(r)=1-\frac{2m}{r}+\frac{q_{m}^{2}}{r^{2}}-\frac{2\mu}{5}\frac{q_{m}^{4}}{r^{6}}\hskip
1.69998pt,$ (58)

where $q_{m}$ denotes the magnetic charge of the black hole, representing a
distinct attribute, while $\mu$ signifies the coupling parameter in the
Einstein-Euler-Heisenberg nonlinear electrodynamics (NLED), as detailed in Eq.
(58). It’s noteworthy that for $\mu\neq 0$, magnetic charges exceeding
$q_{m}>1$ are admissible.

The metric function for magnetically charged black hole solutions can also be
expressed in the following form:

$\displaystyle f(r)=1-\frac{2M(r)}{r},$ (59)

where the function $m(r)$, in this case, is given by:

$M(r)=\frac{10mr^{5}+2\mu q_{m}^{4}-5q_{m}^{2}r^{4}}{10r^{5}}.$ (60)

On the other hand we can also estimate how radius of a photon sphere changes
with mass

Here $m=const.$ is a black hole mass. and $q_{m}$ is the magnetic charge which
is always positive. We consider the positive changes of mass and charge
parameters

$\displaystyle m\rightarrow m+\varepsilon m,~{}~{}\varepsilon\ll
1,~{}~{}\varepsilon>0,$ $\displaystyle q_{m}\rightarrow q_{m}+\beta
q_{m},~{}~{}\beta\ll 1,~{}~{}\beta>0.$ (61)

In this case $M(r)\to\mathcal{M}(r)$ as

$\displaystyle\mathcal{M}(r)=\left(m\epsilon+m+\frac{\mu
q_{m}^{4}}{5r^{5}}-\frac{q_{m}^{2}}{2r}\right)$ (62)
$\displaystyle+\beta\left(\frac{4\mu
q_{m}^{4}}{5r^{5}}-\frac{q_{m}^{2}}{r}\right)$

which can be written as

$\displaystyle\mathcal{M}(r)<M(r)+m\epsilon+\beta\left(\frac{4\mu
q_{m}^{4}}{5r^{5}}-\frac{q_{m}^{2}}{r}\right).$ (63)

With negative $\mu$ one can always find such $\varepsilon$ and $\beta$ that
$\mathcal{M}(r)<M(r)$.

###  III.4 Example: Black holes in MOdified Gravity (scalar-tensor-vector
gravity)

Last example is the black holes in Modified Gravity (MOG black hole) proposed
by Moffat in Ref. Moffat (2006), commonly known as scalar-tensor-vector
gravity. In this scenario, the metric function describing the resulting black
hole solution is given by Moffat (2015).

$\displaystyle f(r)=1-\frac{2M(r)}{r},$ (64)

where the function $M(r)$, in this case, is given by:

$M(r)=-\frac{\alpha^{2}m^{2}}{2r}-\frac{\alpha m^{2}}{2r}+\alpha m+m.$ (65)

Here, $\alpha$ is a parameter governing the strength of the effective
gravitational coupling $G=G_{N}(1+\alpha)$, representing a universal hair.

Here $m=const.$ is a black hole mass. and $\alpha$ is a constant. We consider
the positive changes of mass and charge parameters

$\displaystyle m\rightarrow m+\varepsilon m,~{}~{}\varepsilon\ll
1,~{}~{}\varepsilon>0,$
$\displaystyle\alpha\rightarrow\alpha+\beta\alpha,~{}~{}\beta\ll
1,~{}~{}\beta>0.$ (66)

In this case $M(r)\to\mathcal{M}(r)$ as

$\displaystyle\mathcal{M}(r)=-\frac{\alpha\beta m^{2}}{2r}-\frac{\alpha
m^{2}}{2r}+\alpha\beta m+\alpha m+m$
$\displaystyle+\epsilon\left(-\frac{\alpha\beta m^{2}}{r}-\frac{\alpha
m^{2}}{r}+\alpha\beta m+\alpha m+m\right)$ (67)

then we rewrite it

$\displaystyle\mathcal{M}(r)<M(r)-\frac{\alpha\beta m^{2}}{2r}+\alpha\beta m$
$\displaystyle+\epsilon\left(-\frac{\alpha\beta m^{2}}{r}-\frac{\alpha
m^{2}}{r}+\alpha\beta m+\alpha m+m\right)$ (68)

With negative $\alpha$ one can always find such $\varepsilon$ and $\beta$ that
$\mathcal{M}(r)<M(r)$.

##  IV CONCLUSIONS

A shadow is a critical aspect of a black hole, providing valuable insights
into its structure. Modern imaging techniques have enabled us to observe these
shadows, prompting a deeper investigation into their characteristics,
particularly regarding the influence of various black hole properties like
accretion, radiation, and other parameters. Understanding the shadow is
pivotal for validating or refuting black hole models. Furthermore, shadows are
not exclusive to black holes; other compact objects like wormholes and naked
singularities can also cast shadows, offering a means to probe these enigmatic
entities Heydarzade and Vertogradov (2023); Shaikh  _et al._ (2019). Shadows
also serve as a tool to test the nature of the objects obscured behind them.
However, due to the often complex spacetime structures, analytical
calculations of photon spheres or shadows are frequently unfeasible,
necessitating the use of numerical methods to study their properties.

In this study, we present a novel method for determining the radius of a
photon sphere and shadow in any asymptotically flat spherically symmetric
spacetime. Our method facilitates a straightforward comparison of photon
sphere and shadow radii with those of a Schwarzschild black hole, providing
valuable insights into the gravitational effects of various spacetime
geometries. We validate our approach by applying it to several well-known
examples, demonstrating its efficacy.

Moreover, we address a significant question regarding the shadow of a
dynamical black hole. In scenarios involving accretion and radiation
processes, the mass of the black hole and the surrounding spacetime geometry
evolve dynamically. However, the absence of a timelike Killing vector in
dynamical spacetimes poses challenges in reducing the second-order equations
of motion to first-order ones. Consequently, numerical methods or the
identification of additional symmetries become necessary. In this paper, we
take initial steps towards analytically understanding shadows and photon
spheres in dynamical spacetimes, focusing specifically on the accretion
process.

We show that variations in the mass function of the black hole correlate with
changes in the radius of the photon sphere. However, it is crucial to
distinguish between the mass function and the actual mass of the black hole,
as the former depends on various parameters such as radius and mass. We
illustrate this distinction with an example involving the accretion of charged
particles, where despite an increase in mass, the radius of the photon sphere
decreases due to a higher rate of electrical charge accumulation relative to
mass. We rigorously prove, as a theorem, that such scenarios occur only when
the weak energy condition is violated. This theorem lays the groundwork for
the future development of analytical methods for calculating shadows cast by
dynamical black holes, promising advancements in our understanding of these
complex phenomena.

Looking towards the future, our work lays the groundwork for further
advancements in the field. The next-generation Event Horizon Telescope (ngEHT)
promises even more precise tests of gravity near black holes. By continuously
leveraging advancements in observational technology, we can refine our
understanding of black holes, ultimately leading to a deeper grasp of the
fundamental forces governing the universe.

###### Acknowledgements.

V. Vertogradov thanks the Basis Foundation (grant number 23-1-3-33-1) for the
financial support. A. Ö. would like to acknowledge the contribution of the
COST Action CA21106 - COSMIC WISPers in the Dark Universe: Theory,
astrophysics and experiments (CosmicWISPers) and the COST Action CA22113 -
Fundamental challenges in theoretical physics (THEORY-CHALLENGES). We also
thank TUBITAK and SCOAP3 for their support.

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