# Shadows of a generic class of spherically symmetric, static spacetimes

Md. Golam Mafuz<EMAIL_ADDRESS>Department of Physics, Jadavpur
University, Kolkata, 700032, India  Rishank Diwan<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Kharagpur, 721 302,
India  Laboratory for Space Research, The University of Hong Kong, Hong Kong
Soumya Jana<EMAIL_ADDRESS>(Corresponding author) Department of
Physics, Sitananda College, Nandigram, 721631, India  Department of Physics,
Indian Institute of Technology Kharagpur, 721 302, India  Sayan Kar
<EMAIL_ADDRESS>(Corresponding author) Department of Physics, Indian
Institute of Technology Kharagpur, 721 302, India

###### Abstract

We explore the characteristics of shadows for a general class of spherically
symmetric, static spacetimes, which may arise in general relativity or in
modified theories of gravity. The chosen line element involves a sum (with
constant but different coefficients) of integer powers of $\frac{1}{\text{r}}$
in $\text{g}_{\text{tt}}$ and $\text{g}_{\text{rr}}$, in the Schwarzschild
gauge. We begin our discussion by motivating the line element through a study
of the energy conditions (null and weak) and the extent to which they are
satisfied/violated for diverse choices of the parameters appearing in the
metric functions. Subsequently, we construct the circular shadows and analyse
the dependence of the shadow radius on the metric parameters. We find that
with specific choices of the metric parameters (within the ranges allowed by
the energy conditions) one can, in principle, obtain values that conform with
recent observations on shadows, as available in the literature. We also
mention where such metrics may arise (i.e., in which theory of gravity and the
physical scenario therein), thereby proposing that the observed shadows may be
representative signatures of different theoretical contexts.

##  I Introduction

The recent observations of shadows of M87* Collaboration  _et al._ (2019a);
Akiyama  _et al._ (2019a, b, c); Collaboration  _et al._ (2019b) and SgrA*
Collaboration (2022a, b, c, d, e, f) by the EHT collaboration have opened up
the new possibility of probing a highly dynamical region of strong gravity in
the vicinity of a supermassive black hole/ compact object . A black hole’s
shadow Falcke  _et al._ (2000) (also known as a silhouette Broderick and Loeb
(2009); Dokuchaev and Nazarova (2020)) is the apparent (i.e., gravitationally
lensed) image of the photon sphere (a region of spacetime where photons move
along unstable orbits), appearing as a two-dimensional dark zone in the
observer’s sky Cunha and Herdeiro (2018); Perlick and Tsupko (2022); Dokuchaev
and Nazarova (2020). Strong gravitational lensing of light rays near the
photon sphere is important for distinguishing between the spacetime structures
of black holes and other compact objects Virbhadra and Ellis (2000, 2002);
Bozza (2002); Tsukamoto (2017, 2023); Shaikh  _et al._ (2019). Synge’s (1966)
seminal study defined the angular radius of the photon capture areas bordering
the Schwarzschild black hole Synge (1966). Later, Bardeen explored the Kerr
black hole’s shadow and observed that the spin would deform the shape of the
shadow Bardeen (1973). Luminet (1979) Luminet (1979) established through
simulations that the photon capture radius of a black hole surrounded by a
geometrically thin, optically thick accretion disc would appear to a distant
observer as a thin emission ring inside a lensed image. Since then, much
effort has gone into expanding our knowledge of how to quantify black hole
shadows in known spacetimes employing general relativity (GR) Young (1976);
Perlick (2004); Abdujabbarov  _et al._ (2013); Takahashi (2004); Hioki and
Maeda (2009); Shipley and Dolan (2016); Yumoto  _et al._ (2012); Atamurotov
_et al._ (2015). Furthermore, the notion of shadows has also been analysed for
black holes in extended gravity theories Vagnozzi  _et al._ (2023); Amarilla
_et al._ (2010); Amarilla and Eiroa (2013); Atamurotov  _et al._ (2013);
Vagnozzi and Visinelli (2019); Cunha  _et al._ (2017); Olmo  _et al._ (2023),
for other compact objects Rosa and Rubiera-Garcia (2022); Rosa  _et al._
(2022, 2023), for black hole surrounded by dark matter Konoplya (2019), etc.
Observational results on black hole shadows are being used to validate
theoretical/computational findings on the shadow radius, shadow profile as
well as astrophysical processes in the strong-field regime (see Johannsen and
Psaltis (2010); Psaltis  _et al._ (2015); Broderick  _et al._ (2014); Mizuno
_et al._ (2018)). For reviews one may look at Zakharov (2014) and Perlick and
Tsupko (2022). As mentioned before, one may constrain the parameters in the
black hole solutions in different alternative theories of gravity using
concrete facts and numbers emerging from shadow images Falcke and Markoff
(2013); Eiroa and Sendra (2014). Apart from the Schwarzschild solution, the
Reissner-Nordstrom (RN) Reissner (1916); Nordström (1918) metric is the next
most basic example within the class of metrics to be discussed in this
article. Articles in Alexeyev  _et al._ (2019); Alexeyev and Prokopov (2020)
investigated the effect of including a quadratic in the $\frac{1}{\text{r}}$
term as found in the standard RN solution. Moreover, in many alternative
theories of gravity, there exist static, spherically symmetric solutions
involving a sum (with constant but distinct coefficients) of integer powers of
$\frac{1}{\text{r}}$ in $\text{g}_{\text{tt}}$ and $\text{g}_{\text{rr}}$, in
the Schwarzschild gauge. Some examples appear in Horndeski (1974); Babichev
_et al._ (2017) (see also Kobayashi (2019) for consequences of extensions of
this theory) Myung and Zou (2019); Mannheim (2012), Kalb and Ramond (1974);
Banerjee  _et al._ (2020); Kar  _et al._ (2003), Kiselev (2003)(see Visser
(2020) for a recent revisit on this metric), and Randall and Sundrum (1999);
Dadhich  _et al._ (2000). The above-mentioned examples of such line elements
motivates us to investigate, in a broad sense, a general static spherically
symmetric metric in the Schwarzschild gauge, involving a sum (with constant
but different coefficients) of integer powers of
$\frac{1}{\text{r}^{\text{n}}}$ upto $\text{n}=4$, in
$\text{g}_{\text{tt}}=-{\text{g}_{\text{rr}}^{-1}}$. We address (i) the
violation/satisfaction of the energy conditions for the ‘matter’ (assuming
$GR-$like equations) required to support these spacetimes and (ii) the shadow
profiles for the above static, spherically symmetric geometries. In both (i)
and (ii) our main goal is to find how the choice of the range of values of the
metric parameters influence the consequences. We also use current
observational/imaging results to set bounds on the parameters, which, in turn,
can constrain the theory (when other than GR). In the end, we provide a couple
of specific results for metric functions which may contain a sum of $N$ terms
(arbitrary but finite $N$). While discussing the energy conditions, we have
chosen to use the Weak and Null Energy Conditions, abbreviated as WEC and NEC.
It is assumed at the outset that our generic line elements are ‘solutions’ in
General Relativity or any modified theory in which the Einstein or Einstein-
like equations
$\text{G}_{\alpha\beta}=\kappa\text{T}_{\alpha\beta}^{\text{eff}}$ hold. As a
result, the ‘required matter’ that can be found from $\text{G}_{\alpha\beta}$
must necessarily be examined for the satisfaction/violation of the energy
condition inequalities. In a way, we are actually looking at the ‘convergence
conditions’ (timelike/null). What we broadly call ‘matter’ in our analysis,
may include ‘other’ quantities eg. an ambient scalar (like in Brans-Dicke
theory).

Our article is divided into six sections. Section II introduces the metrics
and comments on the parameters that appear in them. Section III is devoted to
the energy conditions (both Weak and Null) for the metrics under
consideration. Section IV delves into the nature of black hole shadows.
Following this, in Section V, we investigate the nature of values for the
metric parameters, in the context of current shadow observations. In section
VI we discuss some results on a more general metric (arbitrary but finite
$n$). Finally, in Section VII, we summarise our findings and discuss potential
consequences. Throughout the paper we assume $\text{G}=\text{c}=1$ and the
metric signature convention $(-,+,+,+,)$.

##  II The class of static, spherically symmetric spacetimes

As mentioned before, we will be concerned with a generic line element of the
following form:

$\displaystyle\text{ds}^{2}=-\left[1+\frac{\mathcal{P}}{\text{r}}+\frac{\mathcal{Q}}{\text{r}^{2}}+\frac{\mathcal{S}}{\text{r}^{3}}+\frac{\mathcal{T}}{\text{r}^{4}}\right]\text{dt}^{2}+\left[1+\frac{\mathcal{P}}{\text{r}}+\frac{\mathcal{Q}}{\text{r}^{2}}+\frac{\mathcal{S}}{\text{r}^{3}}+\frac{\mathcal{T}}{\text{r}^{4}}\right]^{-1}\text{d}\text{r}^{2}+\text{r}^{2}\,\text{d}{\Omega_{2}}^{2}$
(1)

where ${\cal P,Q,S,T}$ are constants. We could have kept powers upto say
$\frac{1}{\text{r}^{\text{N}}}$, for a general N. However, it is difficult to
say anything concrete for arbitrary N – hence our restriction to $\text{N}=4$.
Numerous examples of metrics of the above form appear in GR or modified
theories of gravity. We mention some of these below in order to motivate our
approach and calculations. We will return to some comments on general N in the
penultimate section of the article.

Our first example is the ‘4-component’ generalized Kiselev spacetime Kiselev
(2003) given by the following line element

$\displaystyle\text{ds}^{2}=-\,\Bigg{[}1-\frac{\sum_{\text{i}=1}^{4}\,\text{K}_{\text{i}}\,\text{r}^{-3\,\text{w}_{\text{i}}}}{\text{r}}\Bigg{]}\,\text{dt}^{2}+\Bigg{[}1-\frac{\sum_{\text{i}=1}^{4}\,\text{K}_{\text{i}}\,\text{r}^{-3\,\text{w}_{\text{i}}}}{\text{r}}\Bigg{]}^{-1}\,\text{d}\text{r}^{2}+\text{r}^{2}\,\text{d}{\Omega_{2}}^{2}$
(2)

where
$\text{d}{\Omega_{2}}^{2}=\text{d}\theta^{2}+\text{sin}^{2}\,\theta\,\text{d}\phi^{2}$
is the line element corresponding to a $2$-sphere. In the above metric,
$\text{K}_{\text{i}}$’s are constants containing the parameters appearing in
the metrics considered here. We absorb any Schwarzschild mass term present
into one of the $\text{K}_{\text{i}}$ by setting the corresponding
$\text{w}_{\text{i}}$ to zero. Thus, essentially, following Jacobson (2007);
Visser (2020), we define a position-dependent mass function m(r) of the form

$\displaystyle
2\,\text{m}(\text{r})=\sum_{\text{i}=1}^{4}\,\text{K}_{\text{i}}\,\text{r}^{-3\,\text{w}_{\text{i}}}$
(3)

Comparing the two line elements in equation (1) and (2), we have
$\text{w}_{\text{i}}$’s as follows:
$\text{w}_{1}=0,\text{w}_{2}=\frac{1}{3},\text{w}_{3}=\frac{2}{3},\text{w}_{4}=1$,
and $\text{K}_{\text{i}}$’s as:
$\mathcal{P}=-\text{K}_{1},\mathcal{Q}=-\text{K}_{2},\mathcal{S}=-\text{K}_{3},\mathcal{T}=-\text{K}_{4}$.
Originally, the Kiselev spacetime was obtained as a solution in GR for
quintessence matter surrounding a black hole. However, the Kiselev-type
solutions also exist in other contexts such as GR coupled with nonlinear
electrodynamics Dariescu  _et al._ (2022) and black holes surrounded by fluids
in $f(R,T)$ gravity Santos  _et al._ (2023).

The second example stems from early work in braneworld gravity. An interesting
braneworld solution is the tidal Reissner-Nordström (tidal-RN) braneworld
black hole solution Dadhich  _et al._ (2000); Whisker (2008)

$\displaystyle\text{ds}^{2}=-\left[1-\frac{2\text{M}}{\text{r}}\,+\frac{\text{Q}}{\text{r}^{2}}\right]\,\text{dt}^{2}+\left[1-\frac{2\text{M}}{\text{r}}\,+\frac{\text{Q}}{\text{r}^{2}}\right]^{-1}\,\text{dr}^{2}+\text{r}^{2}\,\text{d}{\Omega_{2}}^{2}$
(4)

The above solution has the form of the Reissner-Nordström (RN) solution of GR
with no electric field present on the brane. Q is a tidal charge parameter
arising from the Weyl tensor of the bulk. Unlike in the RN case, Q can be both
positive and negative Dadhich  _et al._ (2000).

The static black hole solutions of the field equations for the quartic
Horndeski square root Lagrangian Babichev  _et al._ (2017) constitute our
third example. The line element is given as:

$\displaystyle\text{ds}^{2}=-\left[1-\frac{\mu}{\text{r}}-\frac{\beta^{2}}{2\zeta\eta\text{r}^{2}}\right]\,\text{dt}^{2}+\left[1-\frac{\mu}{\text{r}}-\frac{\beta^{2}}{2\zeta\eta\text{r}^{2}}\right]^{-1}\,\text{dr}^{2}+\text{r}^{2}\,\text{d}{\Omega_{2}}^{2}$
(5)

where $\mu$ is a free integration constant, $\eta$ and $\beta$ are
dimensionless parameters, and $\zeta=\frac{\text{M}_{\textbf{Pl}}}{16\pi}$.
The above solution describes a black hole with mass $\frac{\mu}{2}$.

Our fourth example is that of a magnetically charged black hole solution
obtained in Extended-Scalar-Tensor-Gauss-Bonnet gravity coupled with nonlinear
electrodynamics Cañate and Bergliaffa (2020). The black hole spacetime is
given by

$\text{ds}^{2}=-\left[1-\frac{2M}{r}-\frac{q^{3}}{r^{3}}\right]\text{dt}^{2}+\left[1-\frac{2M}{r}-\frac{q^{3}}{r^{3}}\right]^{-1}\text{dr}^{2}+r^{2}\text{d}\Omega^{2}_{2},$
(6)

where $M$ is the mass of the black hole and $q$ is the scalar charge and
related to the magnetic charge of the black hole $Q_{m}=\sqrt{2}q$. The
spacetime is completely different from the Riessner-Nordström- type black
holes but falls under the general class that we consider here. The presence of
the $\frac{1}{r^{3}}$ term in the metric functions is a point to note here.

Finally, consider the metric for the black hole surrounded by dust in Rastall
theory of gravity given by Heydarzade and Darabi (2017),

$\displaystyle\text{ds}^{2}=-\left[1-\frac{2\text{M}}{\text{r}}\,+\frac{\text{Q}}{\text{r}^{2}}+\frac{\text{N}_{\text{d}}}{\text{r}^{\frac{1-6\kappa\lambda}{1-3\kappa\lambda}}}\right]\,\text{dt}^{2}+\left[1-\frac{2\text{M}}{\text{r}}\,+\frac{\text{Q}}{\text{r}^{2}}+\frac{\text{N}_{\text{d}}}{\text{r}^{\frac{1-6\kappa\lambda}{1-3\kappa\lambda}}}\right]^{-1}\,\text{dr}^{2}+\text{r}^{2}\,\text{d}{\Omega_{2}}^{2}$
(7)

One can realize that in GR, i.e. in the limit of $\lambda\to 0$ and
$\kappa=8\pi\text{G}$, the black hole in the dust background appears as a
charged black hole with an effective mass
$\text{M}^{\textbf{eff}}=\text{M}-\text{N}_{\text{d}}/2$. Thus, we see that
for $\kappa\lambda\neq 0$, the geometric parameters $\kappa$ and $\lambda$ of
the Rastall theory can play an important role leading to distinct solutions
relative to GR. One can realize that for $\kappa\lambda\neq 0$, the Rastall
correction term never behaves as the mass or charge terms, and introduces a
new character to the black hole.

We have provided some examples of scenarios in GR or modified gravity where a
metric of the form similar to equation (1) arises. There are many more which
one can find in the literature. Each of these line elements can be viewed as
‘solutions’ of equations like
$\text{G}_{\alpha\beta}=\kappa\,\text{T}^{\textbf{eff}}_{\alpha\beta}$, where
the R.H.S. is an effective matter stress energy which contains usual matter
plus an ‘effective’ stress energy whose origins could be geometric or anything
else. We now move on to discuss the energy conditions of this
$\text{T}^{\textbf{eff}}_{\alpha\beta}$ which can be written down from the
Einstein tensor for the general line element in equation (1).

##  III Energy Conditions

The Weak Energy Condition (WEC) Wald (1984); Misner  _et al._ (2017) states
that the matter energy density measured by any observer in a spacetime is
never negative and we may represent it in the following form

$\displaystyle\text{T}_{\alpha\beta}\,\text{v}^{\alpha}\,\text{v}^{\beta}\geq
0$ (8)

where $\text{v}^{\alpha},\text{v}^{\beta}$ are any future-directed timelike
vectors. For a diagonal stress-energy tensor $\text{T}_{\alpha\beta}$, the WEC
in a static observer’s frame implies

$\displaystyle\rho\geq 0\quad\mbox{and}\quad\rho+\tau_{\text{i}}\geq
0,\quad\text{i}=1,2,3$ (9)

The Null Energy Condition (NEC) refers to the following inequality

$\displaystyle\text{T}_{\alpha\beta}\,\text{k}^{\alpha}\,\text{k}^{\beta}\geq
0$ (10)

where $\text{k}^{\alpha},\text{k}^{\beta}$ are arbitrary, future-directed null
vector. The NEC for a diagonal stress-energy tensor in a static observer’s
frame corresponds to

$\quad\rho+\tau_{\text{i}}\geq 0,\quad~{}\text{i}=1,2,3$ (11)

In Eqs. (9) and (11), $\rho$ is the density of mass-energy, $\tau_{1}$ is
radial pressure, $\tau_{2}$ and $\tau_{3}$ are tangential pressures. Since the
Einstein field equations demand that the stress-energy tensor be proportional
to Einstein tensor, in the static observer’s frame the stress-energy tensor
$\text{T}_{\hat{\alpha}\hat{\beta}}$ should have the same algebraic structure
as the Einstein tensor $\text{G}_{\hat{\alpha}\hat{\beta}}$. Thus,
$\text{T}_{\hat{\text{t}}\hat{\text{t}}}=\rho$,
$\text{T}_{\hat{\text{r}}\hat{\text{r}}}=\tau_{1}$ and
$\text{T}_{\hat{\theta}\hat{\theta}}=\text{T}_{\hat{\phi}\hat{\phi}}=\tau_{2}=\tau_{3}$,
are the only non zero components of the stress-energy tensor Wald (1984);
Misner  _et al._ (2017); Morris and Thorne (1988) into a perfect fluid
Boonserm  _et al._ (2016). The energy conditions (assuming
$\text{T}_{\alpha\beta}=\frac{1}{\kappa}\text{G}_{\alpha\beta}$) in the proper
reference frame for this metric are the following in-equalities:

$\displaystyle\rho=\frac{\text{r}^{2}\,\mathcal{Q}+2\,\text{r}\,\mathcal{S}+3\,\mathcal{T}}{\text{r}^{6}}\geq
0$ (12)
$\displaystyle\rho+\tau_{2}=\frac{2\,\text{r}^{2}\,\mathcal{Q}+5\,\text{r}\,\mathcal{S}+9\,\mathcal{T}}{\text{r}^{6}}\geq
0$ (13)

A crucial point to note here is that $\rho+\tau_{1}=0$ for all $r$ and all
values of the parameters. We also infer the following points from the
inequalities:
(a) The inequalities are independent of the parameter $\cal P$.
(b) For ${\cal Q}>0$, ${\cal S}>0$, ${\cal T}>0$ both inequalities hold for
all r. Thus WEC and NEC hold for all r.
(c) For ${\cal Q}>0$, ${\cal S}<0$, ${\cal T}>0$ both inequalities hold, for
all r, if ${\cal{S}}^{2}<\frac{72}{25}\cal{Q}\cal{T}$. This requirement (i.e.
${\cal{S}}^{2}<\frac{72}{25}\cal{Q}\cal{T}$) follows from the condition for
which the roots of the equalities are complex conjugate pairs. One can check
that the minimum value of the numerator factor in the R. H. S. of both the
inequalities (subject to appropriate choices of ${\cal Q}$, $\cal R$ and $\cal
T$) is positive definite.
(d) Other combinations of ${\cal Q}$, ${\cal S}$ and ${\cal T}$ may lead to
the WEC, NEC being satisfied over restricted ranges. For example one can write
both the numerators as products of the form
$(\text{r}-\text{r}_{+})(\text{r}-\text{r}_{-})$. Among the four values of the
roots, if one chooses the largest, then if r is greater than this largest
root, both WEC, NEC will hold in that domain of r.

(e) For ${\cal Q}<0$ the energy conditions are violated for large $r$
irrespective of any values of ${\cal S}$, ${\cal T}$. For ${\cal Q}<0$, ${\cal
S}>0$, ${\cal T}<0$ violation of the energy conditions occur for all $r$ when
${\cal{S}}^{2}<\frac{72}{25}\cal{Q}\cal{T}$, whereas if
${\cal{S}}^{2}>\frac{72}{25}\cal{Q}\cal{T}$ there exists only the intermediate
region of space $r_{-}<r<r_{+}$ where the energy conditions are satisfied.
Also, for ${\cal Q}<0$, ${\cal S}<0$, ${\cal T}<0$ the energy conditions are
violated at any $r$.

In Table 1, we summarise the status of NEC and WEC for all possible positive
or negative combinations of ${\cal Q}$, ${\cal S}$, ${\cal T}$. Fig. 1
demonstrates the status of the energy conditions with reference to the
available parameter space. This status is shown via coloured regions in the
plots for $s$ vs. $t$ (where $s={\cal S}/M^{3}$ and $t={\cal T}/M^{4}$) for
${\cal Q}>0$ and ${\cal Q}<0$. In the green region, NEC and WEC are satisfied
for all $r$. In the yellow region the energy conditions are partially
satisfied for restricted $r$, whereas the grey region indicates the violation
of the energy conditions at all $r$. Thus, the figure provides an overall
summary of the details shown in Table 1.

${\cal Q}$ |  ${\cal S}$ |  ${\cal T}$ | NEC and WEC status
---|---|---|---
$+$ |  $+$ |  $+$ | satisfied for all $r$
$+$ |  $+$ |  $-$ | satisfied for $r>r_{+,max}$
$+$ |  $-$ |  $+$ | satisfied for all $r$ when ${\cal S}^{2}\leq\frac{72}{25}{{\cal Q}{\cal T}}$; violation occur at $r_{-,min}<r<r_{+,max}$ when ${\cal S}^{2}>\frac{72}{25}{{\cal Q}{\cal T}}$
$+$ |  $-$ |  $-$ | violation in between $r_{-,min}<r<r_{+,max}$, satisfied elsewhere
$-$ |  $+$ |  $+$ | satisfied only for $r_{-,2nd}<r<r_{+,2nd}$
$-$ |  $+$ |  $-$ | violation for all $r$ when ${\cal S}^{2}\leq\frac{72}{25}{{\cal Q}{\cal T}}$; satisfied only at $r_{-,2nd}<r<r_{+,2nd}$ when ${\cal S}^{2}>\frac{72}{25}{{\cal Q}{\cal T}}$
$-$ |  $-$ |  $+$ | violation occur at $r\geq r_{+,2nd}$
$-$ |  $-$ |  $-$ | violation for all $r$
Table 1:  NEC and WEC validity/violation are tabulated for different
signatures of ${\cal Q}$, ${\cal S}$, ${\cal T}$. Here $r_{+,max}$ and
$r_{-,min}$ denote the largest and the smallest among all roots of the
numerators of the L.H.S. of Eqs. (12) and (13) and $r_{+,2nd}$ and $r_{-,2nd}$
are the second largest and second smallest among all the roots.

(a) $Q>0$, $q=0.2$

(b) $Q<0$, $q=-0.2$

Figure 1: $t-s$ region plot (for a fixed $q$) showing the status of NEC and
WEC. In the left panel $(a)$ $q={\cal Q}/M^{2}>0$. We choose a particular
value $q=0.2$ to specify the boundary $s^{2}=\frac{72}{25}qt$. However, the
nature of the plot does not change for any other positive value of $q$. In the
right panel $(b)$ $q={\cal Q}/M^{2}<0$. We choose a particular value $q=-0.2$

Let us now move on to illustrate the L. H. S. of the energy condition
inequalities for different sets of values for the parameters
{$\mathcal{P},\mathcal{Q},\mathcal{S},\mathcal{T}$}. We need not fix ${\cal
P}$ since it does not appear in the inequalities. We will rename ${\cal
P}=-2\text{M}$. The remaining ${\cal Q}$, ${\cal S}$, ${\cal T}$ are chosen to
satisfy the constraints mentioned above. In order to deal with dimensionless
quantities we write

${\bar{r}}=\frac{r}{M}\quad~{};\quad~{}{\mathcal{Q}}=qM^{2}\quad~{};\quad~{}{\mathcal{S}}=sM^{3}\quad~{};\quad~{}{\mathcal{T}}=tM^{4}$
(14)

where $q,s,t$ are numbers. The two inequalities rewritten in terms of the new
variables become

$\rho=\frac{q{\bar{r}}^{2}+2s{\bar{r}}+3t}{M^{2}{\bar{r}}^{6}}\geq
0\quad~{};\quad~{}\rho+\tau_{2}=\frac{2q{\bar{r}}^{2}+5s{\bar{r}}+9t}{M^{2}{\bar{r}}^{6}}\geq
0$ (15)

We plot (Figure 2) only the numerators of the above inequalities in order to
verify their positivity for all r. The triplet $(q,s,t)$ are chosen as
$(2,-1,3)$ (energy condition satisfying) and $(1,-2,1)$ (energy condition
violating) – these values are considered with the sole aim of illustration. In
general, as mentioned earlier, if required conditions are met, one is sure of
satisfying the energy conditions. Later we will try to correlate the shadow
radius with scenarios of metrics for which the required matter satisfies WEC,
NEC.

(a) $q=2$, $s=-1$, $t=3$

(b) $q=1$, $s=-2$, $t=1$

Figure 2: Numerators of energy condition inequalities: $\rho\geq 0$ inequality
(blue), $\rho+\tau_{2}\geq 0$ inequality (yellow) are plotted as the function
of $\bar{r}$. Note the violation for values not satisfying
$s^{2}<\frac{72}{25}qt$ (figure on the right).

##  IV Analysis of critical null geodesics and the radius of the shadow

When light from a distant astrophysical object or the accretion disk
surrounding the black hole arrives in the vicinity of the event horizon, a
part of it gets trapped inside the horizon while another part escapes to
infinity Synge (1966); Chandrasekhar (1983); Young (1976). This results in the
black hole shadow Falcke  _et al._ (2000) – the two dimensional closed curve
in the observer’s sky separating capture and scattering orbits Bardeen (1973);
Luminet (1979); Gralla  _et al._ (2019). However, the shadow may not be a
direct consequence of the existence of an event horizon Cunha  _et al._
(2018); Abdikamalov  _et al._ (2019). Rather, it is determined by a set of
bound null orbits, exterior to the horizon, dubbed as the photon sphere in
Perlick and Tsupko (2022) (light rings in Cunha and Herdeiro (2018), photon
ring in Johannsen and Psaltis (2010); Johnson  _et al._ (2020) and critical
curve in Gralla  _et al._ (2019)).

In this section, we study the null geodesics of photons in a general
stationary spherically symmetric and asymptotically flat line element and
first work out the equation of the photon sphere. Subsequently, we move on to
calculate the radius of the black hole shadow (following Bardeen’s formalism)
in terms of celestial coordinates Bardeen (1973); Vazquez and Esteban (2003);
de Vries (2000); Cunha and Herdeiro (2018).

Let us begin with the line element

$\displaystyle\text{ds}^{2}=\text{g}_{\alpha\beta}\,\text{dx}^{\alpha}\,\text{dx}^{\beta}=\,-\mathscr{A}(\text{r})\,\text{dt}^{2}+\,\mathscr{B}(\text{r})\,\text{dr}^{2}+\,\text{r}^{2}\,d\theta^{2}+\,\text{r}^{2}\,\text{sin}^{2}\,\theta\,d\phi^{2}$
(16)

such that,
$\underset{\text{r}\to\infty}{\lim}\,\mathscr{A}(\text{r})=\underset{\text{r}\to\infty}{\lim}\,\mathscr{B}(\text{r})=1$.
The Lagrangian describing the motion of photons in this metric is

$\displaystyle
2\,\mathscr{L}(\text{x}^{\alpha},{\dot{\text{x}}}^{\alpha})=\text{g}_{\alpha\beta}\,{\dot{\text{x}}}^{\alpha}\,{\dot{\text{x}}}^{\beta}$
(17)

where $\dot{\text{x}}^{\alpha}$ is the tangent vector along a curve
$\text{x}^{\alpha}=\text{x}^{\alpha}(\lambda)$, where $\lambda$ is the affine
parameter. The conjugate momentum corresponding to the coordinate
$\text{x}^{\alpha}$ is $\text{p}_{\alpha}$. Since the metric is independent of
t and $\phi$, the corresponding constants of motion are the energy E and the
angular momentum L. By solving these two expressions we can get the geodesic
equations corresponding to t and $\phi$ coordinates. The Hamiltonian for the
photons is

$\displaystyle\mathscr{H}=\frac{1}{2}\,\text{g}^{\alpha\beta}\,\text{p}_{\alpha}\,\text{p}_{\beta}=0$
(18)

The geodesic equations corresponding to r and $\theta$ coordinates can be
obtained from the Hamilton-Jacobi equation

$\displaystyle\mathscr{H}(\text{x}^{\alpha},\text{p}_{\alpha})+\frac{\partial\mathscr{S}}{\partial\lambda}=0$
(19)

The Jacobi action $\mathscr{S}$ is integrated with the help of the two known
constants of motion and is of the following form

$\displaystyle\mathscr{S}=-\,\text{E}\,\text{t}+\,\text{L}\,\phi+\,\bar{\mathscr{S}}(\text{r},\theta)$
(20)

For the given static, spherically symmetric metric,
$\bar{\mathscr{S}}(\text{r},\theta)$ happens to be separable in r and
$\theta$, i.e.,
$\bar{\mathscr{S}}(\text{r},\theta)=\mathscr{S}^{\text{r}}(\text{r})+\mathscr{S}^{\theta}(\theta)$.
The ‘r’ and ‘$\theta$’ components of the momentum are, respectively,
$\text{p}_{\text{r}}$ and $\text{p}_{\theta}$. Inserting $\text{p}_{\text{r}}$
and $\text{p}_{\theta}$ in the Hamilton-Jacobi equation, and separating out
the parts corresponding to $r$ and $\theta$ coordinates, we get

$\left[\frac{1}{\mathscr{B}}\left(\frac{\text{d}\mathscr{S}^{\text{r}}}{\text{d}r}\right)^{2}-\frac{\text{E}^{2}}{\mathscr{A}}\right]r^{2}+\text{L}^{2}=-\left(\frac{\text{d}\mathscr{S}^{\theta}}{\text{d}\theta}\right)^{2}-\text{L}^{2}\cot^{2}\theta=-\text{K},$
(21)

where K is a separation constant, also known as the Carter constant Carter
(1968), representing the third constant of motion. Thus, geodesic equations
corresponding to r and $\theta$ coordinates are

$\displaystyle\mathscr{B}(r)\dot{r}$ $\displaystyle=$
$\displaystyle\text{E}\sqrt{\mathscr{B}\left[\frac{1}{\mathscr{A}}-\frac{\ell^{2}+\chi}{r^{2}}\right]}=E\sqrt{-V(r)},$
(22) $\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\frac{\text{E}}{r^{2}}\sqrt{\Theta(\theta)},$ (23)

where $\text{$\ell$}=\frac{\text{L}}{\text{E}}$,
$\chi=\frac{\text{K}}{\text{E}^{2}}$ and
$\Theta(\theta)=\chi-\text{$\ell$}^{2}\,\text{cot}^{2}\,\theta$, while the
effective potential in which the photons move is given as,

$V(r)=\mathscr{B}\left[\frac{\ell^{2}+\chi}{r^{2}}-\frac{1}{\mathscr{A}}\right].$
(24)

The radii of the photon sphere $\text{r}_{\textbf{ph}}$ correspond to the
highest maximum of the effective potential $\text{V}(\text{r})$ subject to the
following conditions

$\displaystyle\text{V}(\text{r}_{\textbf{ph}})=0,\quad\text{V}^{{}^{\prime}}(\text{r}_{\textbf{ph}})=0,\quad\text{V}^{{}^{\prime\prime}}(\text{r}_{\textbf{ph}})<0$
(25)

From Eq.(25) we get the following conditions

$\displaystyle\chi+\text{$\ell$}^{2}=\frac{\text{r}^{2}_{{\textbf{ph}}}}{\mathscr{A}(\text{r}_{\textbf{ph}})}\quad\text{and}\quad\chi+\text{$\ell$}^{2}=\frac{1}{2}\,\frac{\mathscr{A}^{{}^{\prime}}(\text{r}_{\textbf{ph}})}{\mathscr{A}^{2}(\text{r}_{\textbf{ph}})}\,\text{r}^{3}_{\textbf{ph}}$
(26)

where a prime denotes derivative w.r.t. the radial coordinate r. We determine
the photon sphere radius taking the maximum positive root of the following
photon sphere equation.

$\displaystyle\frac{\mathscr{A}^{{}^{\prime}}(\text{r}_{\textbf{ph}})}{\mathscr{A}(\text{r}_{\textbf{ph}})}\,\text{r}_{\textbf{ph}}=2$
(27)

In order to derive the contour of the black hole shadow in the observer’s sky,
one considers the projection of the photon sphere in the image plane. The
locus of the shadow boundary is expressed in terms of two celestial
coordinates. In terms of the tangent to a photon geodesic at the observer’s
position located at a distance $\text{r}_{0}$ from the center of the black
hole, the celestial coordinates are Bardeen (1973); Vazquez and Esteban
(2003); de Vries (2000); Cunha and Herdeiro (2018)

$\alpha=-\frac{\ell}{\sin\theta_{0}},\quad~{}\beta=\pm\sqrt{\Theta(\theta_{0})},$
(28)

where $\theta_{0}$ is the inclination angle of the observer with respect to
$Z$-axis of the black hole. The contour of the shadow describes a circle
$\alpha^{2}+\beta^{2}=\chi+\ell^{2}$. Therefore, the radius of the shadow is
given by

$\displaystyle\text{r}_{\textbf{sh}}=\Bigg{(}\frac{1}{2}\,\frac{\mathscr{A}^{{}^{\prime}}(\text{r}_{\textbf{ph}})}{\mathscr{A}^{2}(\text{r}_{\textbf{ph}})}\,\text{r}^{3}_{\textbf{ph}}\Bigg{)}^{\frac{1}{2}}=\frac{\text{r}_{\textbf{ph}}}{\sqrt{\mathscr{A}(\text{r}_{\textbf{ph}}})}$
(29)

The equation of photon sphere is

$\displaystyle
2\,\text{r}_{\textbf{ph}}^{4}+3\,\mathcal{P}\,\text{r}_{\textbf{ph}}^{3}+4\,\mathcal{Q}\,\text{r}_{\textbf{ph}}^{2}+5\,\mathcal{S}\,\text{r}_{\textbf{ph}}+6\,\mathcal{T}=0$
(30)

The condition for the existence of a horizon is given from the requirement
$\text{g}_{\text{tt}}=0$ which leads to:

$\displaystyle r_{\textbf{hr}}^{4}+{\cal P}r_{\textbf{hr}}^{3}+{\cal
Q}r_{\textbf{hr}}^{2}+{\cal S}r_{\textbf{hr}}+{\cal T}=0$ (31)

We will now rewrite the photon sphere equation and the horizon condition using
dimensionless quantities mentioned before, i.e.

$\displaystyle{\cal P}=-2\text{M}\hskip 14.45377pt;\hskip 14.45377pt{\cal
Q}=q\text{M}^{2}\hskip 14.45377pt;\hskip 14.45377pt{\cal
S}=s\text{M}^{3}\hskip 14.45377pt;\hskip 14.45377pt{\cal T}=t\text{M}^{4}$
(32)

We further define

$\displaystyle x=\frac{\text{r}_{\textbf{hr}}}{\text{M}}\hskip
7.22743pt;\hskip 7.22743pty=\frac{r_{\textbf{ph}}}{\text{M}}$ (33)

With these redefinition the equations for $x$ and $y$ become:

$\displaystyle x^{4}-2x^{3}+qx^{2}+sx+t=0,$ (34)

and

$\displaystyle 2y^{4}-6y^{3}+4qy^{2}+5sy+6t=0.$ (35)

Both these equations now involve dimensionless quantities only. From
elementary algebra we know that the cubic terms in the quartic equations can
be removed by simple transformations. If we write $x=z_{1}+\frac{1}{2}$ and
$y=z_{2}+\frac{3}{4}$, the equations for $z_{1}$ and $z_{2}$ turn out to be

$\displaystyle
z_{1}^{4}+\left(q-\frac{3}{2}\right)z_{1}^{2}+\left(q+s-1\right)z_{1}+\left(\frac{s}{2}+\frac{q}{4}+t-\frac{3}{16}\right)=0$
(36)

and

$\displaystyle
z_{2}^{4}+\left(2q-\frac{27}{8}\right)z_{2}^{2}+\left(\frac{5s}{2}+3q-\frac{27}{8}\right)z_{2}+\left(\frac{15s}{8}+\frac{9q}{8}+3t-\frac{243}{256}\right)=0$
(37)

We will now explore some special case solutions of these equations and obtain
the expressions for the photon sphere, the horizon and the shadow radius as
functions of one or more of the parameters $q$, $s$ and $t$. In general, one
may write

$\displaystyle r_{hr}=U(q,s,t)M\hskip 14.45377pt;\hskip
14.45377ptr_{ph}=V(q,s,t)M\hskip 14.45377pt;\hskip 14.45377ptr_{sh}=W(q,s,t)M$
(38)

where $U(0,0,0)=2$, $V(0,0,0)=3$ and $W(0,0,0)=3\sqrt{3}$ and $M$ in
dimensionful form is $\frac{GM}{c^{2}}$. The general form of the three
functions $U(q,s,t)$, $V(q,s,t)$ and $W(q,s,t)$ can, in principle, be obtained
by solving for the real roots of the quartic equations quoted above. We
confine ourselves below to a study of special cases, largely dictated by (a)
the requirements of simplicity and (b) obvious mathematical conditions which
lead to simpler and tractable scenarios for which we can find some viable
answers.

Case 1: This is the simplest case where we consider the coefficients of
$z_{1}$ and $z_{2}$ as zero. This results in algebraic equations for $q$ and
$s$ which have solutions

$\displaystyle q=\frac{7}{4}\hskip 14.45377pt;\hskip 14.45377pts=-\frac{3}{4}$
(39)

We have seen that the energy conditions will hold provided ${\cal
S}^{2}<\frac{72}{25}{\cal Q}{\cal T}$ which, with the $q$, $s$, $t$
parametrisation becomes $s^{2}<\frac{72}{25}qt$. With $q=\frac{7}{4}$ and
$s=-\frac{3}{4}$ this becomes $t>\frac{25}{224}$. With this choice the
equations for $z_{1}$, $z_{2}$ are reduced to quadratic equations in
$z_{1}^{2}$ and $z_{2}^{2}$. The real solutions of these equations give

$\displaystyle z_{1}=\sqrt{\frac{\sqrt{9-64t}-1}{8}}\hskip 14.45377pt;\hskip
14.45377ptz_{2}=\sqrt{\frac{2\sqrt{25-192t}-1}{16}}$ (40)

Recalling definitions, we finally obtain

$\displaystyle\text{r}_{\textbf{hr}}=\left(\frac{1}{2}+\sqrt{\frac{\sqrt{9-64t}-1}{8}}\right)\text{M}$
(41)
$\displaystyle\text{r}_{\textbf{ph}}=\left(\frac{3}{4}+\sqrt{\frac{2\sqrt{25-192t}-1}{16}}\right)\text{M}$
(42)

Notice that there is a restriction imposed on $t$ in order to have real values
for $z_{1}$, $z_{2}$ and hence, $\text{r}_{\textbf{hr}}$ and
$\text{r}_{\textbf{ph}}$. One can verify easily that within this domain
$\text{r}_{\textbf{hr}}<\text{r}_{\textbf{ph}}$ (see Figure 3(a)). Note that
for $t<0$ energy conditions are satisfied for $r>r_{+,max}$ as we noted
earlier (see Table 1). One can verify that, for the specific choice of $q$ and
$s$ here, the horizon radius $\text{r}_{\textbf{hr}}$ is always less than
$r_{+,max}$, meaning there is always a violation of energy conditions outside
the horizon up to the limit $r_{+,max}$. Further, there is a range of $t>0$
(i.e. $0.061<t<0.1116$) over which one can still satisfy the energy
conditions, at least outside the horizon of the spacetime. For $t\geq 25/224$
the energy conditions are satisfied for all $r$. However, $t>1/8$ correspond
to the naked singularities.

Finally, we need to write down the shadow radius for any $t$. This is a
complicated expression, so we just plot it as a function of $t$ (see Figure
3(b)). Note that photon sphere and the shadow do not exist for the naked
singularities for $t>99/768$.

(a) $\frac{r_{hr}}{M}$(blue) and $\frac{r_{ph}}{M}$ (yellow) as a function of
$t$ ($x$-axis).

(b) Shadow radius $\frac{r_{sh}}{M}$ as a function of $t$.

Figure 3: Horizon radius, photon sphere radius and shadow radius as a function
of $t$ for $q=7/4$, $s=-3/4$. Energy condition satisfying scenario for
$t>\frac{25}{224}=0.111607$. Note that the shadow radius is less than
$3\sqrt{3}M$, i.e. lesser than Schwarzschild.

The radius for a fixed M is smaller than that for Schwarzschild, as is evident
from the graph. We will need to choose other values of the free parameters in
order to make it equal or larger than the Schwarzschild.

Case 2: Another simplifying assumption could be to assume the constant and the
coefficients of the linear in the $z_{1}$, $z_{2}$ terms in the quartic
equations to be zero. Since we have three parameters $q$, $s$ and $t$ one
needs three equations. We can either assume both these coefficients in the
$z_{1}$ equation to be equal to zero and one of them in the $z_{2}$ equation
to be zero. The opposite can also be considered. There will be four such
choices given as:

(a) $q+s-1=0$, $8s+4q+16t=3$, $20s+24q-27=0$.

Here $q=\frac{7}{4}$, $s=-\frac{3}{4}$, $t=\frac{1}{8}$. The energy condition
requirement $s^{2}<\frac{72}{25}qt$ holds. This corresponds to $t=\frac{1}{8}$
in Case 1 given above. Here $z_{1}=0$, hence
$\text{r}_{\textbf{hr}}=0.5\text{M}$ and $\text{r}_{\textbf{ph}}=\text{M}$.
The shadow radius can be read off from the figure.

(b) $q+s-1=0$, $8s+4q+16t=3$, $288q+480s+768t-243=0$.

This set does not have a solution for $q$,$s$, $t$.

(c) $q+s-1=0$, $20s+24q-27=0$, $288q+480s+768t-243=0$.

Here $q=\frac{7}{4}$, $s=-\frac{3}{4}$, $t=\frac{33}{256}$. The energy
condition requirement holds. The photon sphere here will be at
$\text{r}_{\textbf{ph}}=0.75\text{M}$. There is no horizon as the $z_{1}$
equation has no real root. Thus, we have a naked singularity. The shadow
radius is $\text{r}_{\textbf{sh}}=2.75568M$.

(d) $8s+4q+16t=3$, $20s+24q-27=0$, $288q+480s+768t-243=0$.

Here $q=\frac{51}{32}$, $s=-\frac{9}{16}$, $t=\frac{9}{128}$. Here also the
energy conditions hold good. There are two real roots of the $z_{1}$ quartic
equation, i.e. $z_{1}=0$ and $z_{1}=-0.0104$. Thus this describes a black hole
with double horizon, where $z_{1}=0$ for the event horizon and $z_{1}=-0.0104$
for the cauchy horizon. Thus, the event horizon is at
$\text{r}_{\textbf{hr}}=0.5\text{M}$. The photon sphere equation has the
solutions $z_{2}=0$ and $z_{2}=\pm\frac{\sqrt{3}}{4}$ One can verify that only
$z_{2}=\frac{\sqrt{3}}{4}$ corresponds to the unstable photon orbit outside
the event horizon. Thus the radius of the photon sphere is
$\text{r}_{\textbf{ph}}=\frac{(3+\sqrt{3})M}{4}$ and the shadow radius
$\text{r}_{\textbf{sh}}=3.11386M$.

Case 3 (General): We now go back to the quartic equations for $z_{1}$ and
$z_{2}$, i.e. (36) and (37). Is it possible to arrive at some general
analytical results on the roots? Following early work by Arnon Arnon (1988)
(see also another recent article Prodanov (2021)) we are able to extract the
following information which we state below.

Let us assume the quartic equation in the form

$\displaystyle u^{4}+au^{2}+bu+c=0$ (43)

We further define two quantities

$\displaystyle\delta(a,b,c)=256c^{3}-128a^{2}c^{2}+144ab^{2}c+16a^{4}c-27b^{4}-4a^{3}b^{2},$
(44) $\displaystyle L(a,b,c)=8ac-9b^{2}-2a^{3}.$ (45)

Real roots of the quartic will exist for specific conditions on $\delta$ and
$L$. In fact, the multiplicity of the real root can also be obtained. The list
appears in Arnon (1988) and Prodanov (2021). We mention the conditions of
interest to us:

(a) The condition for no real roots is ($\delta>0$ and ($L\leq 0$ or $a>0$))
OR ($\delta=0$ and $L=0$ and $a>0$).

(b) The condition for two distinct real roots is just $\delta<0$.

Six other conditions are stated in Arnon (1988) and Prodanov (2021).

We will use (a) for $z_{1}$ to find examples of spacetimes without horizons,
i.e. those with a naked singularity. We can use (a) for $z_{2}$ to show there
are no photon spheres and hence no shadow. With (b) we can find horizons and
photon spheres. In particular, if we have one positive and one negative root
we can use the obvious fact that $r>0$ in order to throw away the negative
root. That will leave us, from (b), with one horizon and one photon sphere.

In Fig. 4, we show $s$ vs. $q$ region plots for the chosen values of $t$. Each
panel of the figure shows three shaded regions with uniform grey colour
density but different boundary colours, i.e. black, blue, and red, signifying
three different characteristics as mentioned in the figure. The region with
black boundary satisfies the NEC and WEC, whereas the region outside do not
satisfy the energy conditions. The shaded region under the blue boundary
correspond to the naked singularities and the outside region is for the black
holes. For the shaded region under the red boundary, photon spheres as well as
the shadows do not exist. Note that the red boundary never crosses the blue
boundary, meaning the non-existence of photon spheres only correspond to the
naked singularities. However, there are overlaps of the parts of these three
shaded regions–in the darker patches the overlap is more. There are the
overlapping regions A for the black holes and B for the naked singularities,
where the photon spheres exist and also the energy conditions are satisfied.
In C, where parts of all the three shaded regions overlap, photon spheres and
shadows do not exist although the energy conditions are satisfied. As the $t$
values are increased as shown in the different panels of the figure, the
region A and B become thinner in the direction of $q$ but lengthier along $s$,
i.e. for a given choice of $q$ the allowed range of $s$ increases, but for a
given choice of $s$ the allowed range of $q$ decreases. In Table 2, we
demonstrate these facts with specific choices of $q$, $s$, and $t$. We take
all possible combinations of $q=\\{\frac{1}{2},\frac{4}{3}\\}$,
$s=\\{-\frac{1}{3},-\frac{1}{10}\\}$, and $t=\\{\frac{1}{100},\frac{1}{10}\\}$
and compute the horizon radius, shadow radius, and check the energy conditions
explicitly, thereby verifying their correspondence with the Fig. 4.

Figure 4: $q$ vs. $s$ region plot for fixed $t>0$. Note that we choose the domain $q>0$ and $s<0$. The shaded region with blue boundary correspond to naked singularities (the rest is for the black holes), the shaded region under the black boundary satisfies the energy conditions at all $r$, i.e. $s^{2}<\frac{72}{25}qt$, and for the shaded region with red boundary, photon spheres do not exist. The overlapping regions A and B correspond to allowed black holes and naked singularities respectively for which the energy conditions are satisfied and also the photon spheres exist. In the region C the photon spheres do not exist though energy conditions are satisfied. $(q,s,t)$ | $(\delta_{1},L_{1})$ | $(\delta_{2},L_{2})$ | $(z_{1},z_{2})$ |  $r_{H}$ (in M) |  $r_{ph}$ (in M) |  $r_{sh}$ (in M) | $s^{2}<\frac{72}{25}qt?$
---|---|---|---|---|---|---|---
$(\frac{1}{2},-\frac{1}{10},\frac{1}{100})$ | $<0,\,<0$ | $<0,\,<0$ | $(1.24434,1.90751)$ | $1.74434$ | $2.65751$ | $4.74945$ | $Yes$
$(\frac{1}{2},-\frac{1}{10},\frac{1}{10})$ | $<0,\,<0$ | $<0,\,<0$ | $(1.22413,1.8909)$ | 1.72413 | 2.6409 | 4.73556 | $Yes$
$(\frac{1}{2},-\frac{1}{3},\frac{1}{100})$ | $<0,\,<0$ | $<0,\,<0$ | $(1.32444,1.99484)$ | 1.82444 | 2.74484 | 4.83878 | $No$
$(\frac{1}{2},-\frac{1}{3},\frac{1}{10})$ | $<0,\,<0$ | $<0,\,<0$ | $(1.30854,1.98082)$ | 1.80854 | 2.73082 | 4.82672 | $Yes$
$(\frac{4}{3},-\frac{1}{10},\frac{1}{100})$ | $>0,\,<0$ | $>0,\,<0$ | $(None,None)$ | $None$ | $None$ | $None$ | $Yes$
$(\frac{4}{3},-\frac{1}{10},\frac{1}{10})$ | $>0,\,<0$ | $>0,\,<0$ | $(None,None)$ | $None$ | $None$ | $None$ | $Yes$
$(\frac{4}{3},-\frac{1}{3},\frac{1}{100})$ | $<0,\,>0$ | $<0,\,>0$ | $(0.465475,0.982887)$ | 0.965475 | 1.73289 | 3.63773 | $No$
$(\frac{4}{3},-\frac{1}{3},\frac{1}{10})$ | $>0,\,<0$ | $<0,\,<0$ | $(None,0.775662)$ | $None$ | 1.52566 | 3.53269 | $Yes$
Table 2:  Specific illustrative examples of the general case. Each entry can
be easily correlated with the Fig. 4 and Fig. 4 and the conditions on the Eqs.
(44) and (45); $(\delta_{1},L_{1})$ and $(\delta_{2},L_{2})$ correspond to the
quartic equations of $z_{1}$ and $z_{2}$ respectively.

##  V Link with observational results from M87* and SgrA*

It is known from the EHT observations that the shadow radius for the
supermassive compact object (black hole) in the M87 galaxy Collaboration  _et
al._ (2019a) and Sgr A* (in the center of our galaxy) Collaboration (2022a),
correspond to angular diameters $42\pm 3$ $\mu$as and $48.7\pm 7$ $\mu$as,
respectively. Assuming the Schwarzschild black hole model, we can obtain the
shadow radius using the formula $r_{sh}=\frac{3\sqrt{3}GM}{c^{2}}$ for the
given masses and distances of these objects. The angular diameter value for
the black hole in M87 turns out to be $39.6$ $\mu$as (considering the mass
$M=6.5\times 10^{9}\,M_{\odot}$ and the distance $D=16.8\,\text{Mpc}$ ). For
Sgr A*, the Schwarzschild value is around $51.2$ $\mu$as (considering the mass
$M=4\times 10^{6}\,M_{\odot}$ and the distance $D=8\,\text{Kpc}$). We will now
see if the $r_{sh}$ for the line element proposed here can match the
observations within error bars. In other words, we ask– what values of $q,s,t$
can yield a shadow radius close or equal to the value found in observations.

For M87*, the angular diameter can be close to the observed value provided the
$3\sqrt{3}$ factor is replaced by numerical values ranging from $4.94$ (lower
bound $39$ $\mu$) as to $5.71$ (upper bound $45$ $\mu$as). In the case of
SgrA* the corresponding values are: 4.1 (for $41.7$ $\mu$as) and $5.46$ (for
$55.7$ $\mu$as). We show below that there does exist ($q,s,t)$ values for
which one can safely say that observations match with theory.

We first consider the case with $s$ negative and $q=t=-s$. The energy
conditions obviously hold good without any further restrictions. The roots
$z_{1}$, $z_{2}$ are found as functions of $q$ and thereafter $r_{hr}$,
$r_{ph}$ and $r_{sh}$ are also obtained as functions of $q$. The expressions
are long and complicated and are therefore not quoted here. However, one can
check that $q\leq 1.1169$ for black holes, otherwise they are naked
singularities.

The figures below (Fig. 5 and Fig. 6) show the possibilities and the $q$
values for which a match with M87* and SgrA* observations is feasible. We
note, that the horizon radius is always smaller than the photon sphere radius.
The metrics in general are:

$\displaystyle
g^{rr}=-g_{tt}=\left(1-\frac{2M}{r}+\frac{qM^{2}}{r^{2}}-\frac{qM^{3}}{r^{3}}+\frac{qM^{4}}{r^{4}}\right)$
(46)

One can identify the above line element with a Kiselev black hole with
appropriately chosen $K_{i}$. The range of $q$ in the graphs is chosen to
avoid any jumps in the curves which may occur due to pathologies in the roots
for certain $q$ values. The allowed $q$ values are the ones for which the
shadow radius lies in the observational green band in the figures.

For the black hole M87∗, observational limit is $q\leq 0.35$ (see Fig. 5(a))
obtained from the earlier version of the observational range of the angular
shadow diameter. However, the new improved (PRIMO) version Medeiros  _et al._
(2023) of the range of angular diameter puts a more stringent limit $q\leq
0.0145$.

Figure 5: Shadow radius ($r_{sh}$) in $M$ units vs. $q$ plot (black solid
curve) for the case $q=-s=t$. The green band is the observational range for
the black hole M87∗. In the plot $(a)$, we take earlier version of the angular
shadow diameter range $42\pm 3$ $\mu\text{as}$. In the plot $(b)$, we take the
new updated (PRIMO) version of the range $41.5\pm 0.6$ $\mu\text{as}$. The
dashed lines are the upper and lower limits of the observational range.

For the black hole SgrA∗, the EHT papers Collaboration (2022a, f) provide two
observational data for the angular diameter. One is mentioned as emission ring
diameter $51.8\pm 2.3$ $\mu\text{as}$ and the other one is mentioned as the
shadow diameter $48.7\pm 7$ $\mu\text{as}$. We consider both as the observed
shadow diameter and obtain the constraints on the $q$ parameter. From Fig.
6.(a) we find that $q\leq 0.472$ whereas $q\leq 1.268$ from Fig. 6.(b). The
latter bound on $q$ implies the possibility of SgrA∗ being a naked singularity
as well. We scrutinize this possibility further using the constraint on
fractional deviation parameter ($\delta$) which signals a deviation from the
Schwarzschild black hole.

Figure 6: Shadow radius ($r_{sh}$) in $M$ units vs. $q$ plot (black solid
curve) for the case $q=-s=t$. The green band is the observational range for
the black hole SgrA∗. In the plot $(a)$, we take the angular (emission) ring
diameter range $51.8\pm 2.3$ $\mu\text{as}$. In the plot $(b)$, we take the
angular shadow diameter range $48.7\pm 7$ $\mu\text{as}$. The dashed lines are
the upper and lower limits of the observational range.

The recent EHT papers on SgrA∗ observations have used the fractional deviation
parameter $\delta$ to constrain the spacetime geometries different from the
Schwarzschild or Kerr black holes. The definition of $\delta$ is as follows,

$\delta=\frac{d_{sh}}{d_{sh,Sch}}-1=\frac{R_{avg}}{3\sqrt{3}M}-1,$ (47)

where the average diameter of the shadow, $d_{sh}=2R_{avg}$. Using the
observations of the shadow of SgrA∗ and two separate sets of prior values of
mass and distance of SgrA∗ from the VLTI and Keck observations, the EHT
collaboration provided the bound on $\delta$ Collaboration (2022a, f) as

$\delta=\begin{cases}-0.08^{+0.09}_{-0.09}&\text{(VLTI)}\\\
-0.04^{+0.09}_{-0.10}&\text{(Keck)}\\\ \end{cases}$ (48)

Therefore, combining both these bounds, we have $-0.14<\delta<0.01$. Using
this constraint we obtain observational limit $q\leq 0.93$ (see Fig. 7).
Considering all the obtained bounds on $q$ together, we get the best possible
one, $q\leq 0.472$, for the black hole SgrA∗, with the present accuracy of
available data. However, it is expected that improved values will appear when
the PRIMO technique is applied to the data for SgrA∗.

Figure 7: $q=-s=t$, $q>0$. $q$ is on the x-axis. The y-axis represents the
fractional deviation parameter $\delta$. The black solid line is the
theoretical. The green band bounded by lower and upper dashed lines is
observational range for the black hole SgrA∗ from VLTI and Keck bounds.

(a) M87∗ (old angular diameter)

(b) M87∗ (new angular diameter)

Figure 8: $q$ vs. $s$ contour plots for the case $t=0$ (for M87*). The area
bounded by the marked contour lines are only allowed in each plot. In the plot
$(a)$, we take earlier version of the angular shadow diameter range $42\pm 3$
$\mu\text{as}$ for the M87*. Contours represent the fixed values of the shadow
radius in the mass unit. The observational limits on the shadow radius are
marked on the specific contours in the plot. In the plot $(b)$, we take the
new updated (PRIMO) version of the range $41.5\pm 0.6$ $\mu\text{as}$ for the
M87*.

(a) SgrA* (ring diameter)

(b) SgrA∗ (fractional deviation parameter)

Figure 9: $q$ vs. $s$ contour plots for the case $t=0$ (for SgrA*). In the
plot $(a)$, the angular (emission) ring diameter range $51.8\pm 2.3$
$\mu\text{as}$ is considered for the SgrA* . In the plot $(b)$, we use the
observational range for the fractional deviation parameter ($\delta$) for the
SgrA*. Here, the contours represent the fixed values of $\delta$ with the
observational ones are marked on the plot.

(a) M87∗ (old angular diameter)

(b) M87∗ (new angular diameter)

Figure 10: $q$ vs. $t$ contour plots for the case $s=0$ (for M87*).

(a) SgrA* (ring diameter)

(b) SgrA∗ (fractional deviation parameter)

Figure 11: $q$ vs. $t$ contour plots for the case $s=0$ (for SgrA*).

In order to facilitate a comparison with the metric that arises in Rastall
gravity we must try and see what happens if either the term in
$\frac{1}{r^{3}}$ or the term in $\frac{1}{r^{4}}$ is nonzero. For the term
$\frac{1}{r^{3}}$ we look at the case $t=0$ but $s\neq 0$, and, for the term
$\frac{1}{r^{4}}$, we look at $s=0$ but $t\neq 0$. We scan the parameter space
$q$ vs $s$ and $q$ vs $t$ respectively for the cases with $\frac{1}{r^{3}}$
and $\frac{1}{r^{4}}$. In Figs. 8, 9, 10, 11, we show the contour plots in the
parameter space. The allowed regions are specified by the contour lines marked
with the observational values. One can see from the metric in Eq. (7) that for
$\kappa\lambda=\frac{2}{3}$ one retains the term in $\frac{1}{r^{3}}$ and for
$\kappa\lambda=\frac{1}{2}$ one retains the term in $\frac{1}{r^{4}}$ . Thus
the shadow corresponding to the metric in Rastall gravity can also be modeled
with a suitable choice of parameters in a special case of our general line
element.

It is also possible to consider more general cases where all of $q$, $s$, and
$t$ take non-zero values. For example, one can choose a specific value of $t$
and then analyze the parameter space $q$ vs. $s$ for the allowed region.

##  VI A more general metric and consequences

Let us now consider a more general metric (arbitrary but finite $N$) for which
the $\text{g}_{\text{tt}}$ and $\text{g}_{\text{rr}}$ are given as:

$\displaystyle\text{g}_{\text{tt}}=-\left(1+\sum_{n=1}^{N}\frac{\text{a}_{\text{n}}}{\text{r}^{\text{n}}}\right)=-\frac{1}{\text{g}_{\text{rr}}}$
(49)

Here $\text{a}_{\text{n}}$ are the generalisations of the ${\cal P},{\cal
Q},{\cal R},{\cal T}$ discussed just above.

We begin with the photon sphere equation in this metric which turns out to be

$\displaystyle
2\,\text{r}_{\textbf{ph}}^{\text{N}}+\sum_{\text{n}=1}^{\text{N}}\text{a}_{\text{n}}(\text{n}+2)\,\text{r}_{\textbf{ph}}^{\text{N-n}}=0$
(50)

On the other hand the horizon equation for the general case is

$\text{r}_{\textbf{hz}}^{\text{N}}+\sum_{n=1}^{N}\text{a}_{\text{n}}\text{r}_{\textbf{hz}}^{\text{N-n}}=0$
(51)

Below we discuss two special solutions of the general case.
$(a)$ First we assume a special choice wherein the general photon sphere
equation (50) reduces to the form

$\displaystyle\left(\text{r}_{\textbf{ph}}-\text{p}\right)^{\text{N}}=\text{r}_{\textbf{ph}}^{\text{N}}-\text{N}\,\text{r}_{\textbf{ph}}^{\text{N}-1}\,\text{p}+.....+(-1)^{\text{N}}\,\text{p}^{\text{N}}=0.$
(52)

This means that the photon sphere is located at
$\text{r}_{\textbf{ph}}=\text{p}$. Of course this will happen only for a
specific set of $\text{a}_{\text{n}}$ in the original metric. Comparing the
two photon sphere equations (i.e. 6.2 and 6.4) one finds that

$\displaystyle\text{a}_{\text{n}}=(-1)^{\text{n}}\,\frac{2\text{N}!}{\text{n}!\,(\text{N-n})!\,(\text{n}+2)}\,\text{p}^{\text{n}}$
(53)

The general metric now takes on a special form with

$\displaystyle\text{g}_{\text{tt}}=-\left(1+\sum_{\text{n}=1}^{\text{N}}\frac{2\,(-1)^{\text{n}}\,\text{N}!}{\text{n}!\,(\text{N-n})!\,(\text{n}+2)}\left(\frac{\text{p}}{\text{r}}\right)^{\text{n}}\right)=-\frac{1}{\text{g}_{\text{rr}}}$
(54)

It is now easy to write down the shadow radius $\text{r}_{\textbf{sh}}$,
which, after some straightforward algebra becomes

$\displaystyle\text{r}_{\textbf{sh}}=\sqrt{\frac{(\text{N}+1)(\text{N}+2)}{2}}\,\text{p}$
(55)

One can check that the two inequalities $\rho\geq 0$ and
$\rho+\tau_{\text{i}}\geq 0$ for satisfying the WEC and NEC, reduce to the
following:

$\displaystyle\sum_{\text{n}=1}^{\text{N}}\frac{2\,(-1)^{\text{n}+1}\,(1-\text{n})\,\text{N}!}{\text{n}!\,(\text{N-n})!\,(\text{n}+2)}\frac{\text{p}^{\text{n}}}{\text{r}^{\text{n}+2}}\geq
0$ (56)
$\displaystyle\sum_{\text{n}=1}^{\text{N}}\frac{(-1)^{\text{n}+2}\,(\text{n}-1)\,\text{N}!}{\text{n}!\,(\text{N-n)}!}\frac{\text{p}^{\text{n}}}{\text{r}^{\text{n}+2}}\geq
0$ (57)

Thus, one will need to check the energy conditions for different N (or, if
possible for a general N directly) and also find out if the spacetime has
horizons and where the horizon is located. Let us quickly check what happens
for $\text{N}=4$. For this case, we find

$\displaystyle\text{g}_{\text{tt}}=-\left(1-\frac{8\text{p}}{3\text{r}}+\frac{3\text{p}^{2}}{\text{r}^{2}}-\frac{8\text{p}^{3}}{5\text{r}^{3}}+\frac{\text{p}^{4}}{3\text{r}^{4}}\right)$
(58)

A straightforward calculations shows that there are no horizons and the
geometry represents a naked singularity. The shadow radius is given as:

$\displaystyle\text{r}_{\textbf{sh}}=\sqrt{15}\,\text{p}$ (59)

The energy condition inequalities lead to the relations

$\displaystyle\frac{15\text{r}^{2}\text{p}^{2}-16\text{p}^{3}\text{r}+5\text{p}^{4}}{\text{r}^{6}}\geq
0$ (60)
$\displaystyle\frac{6\text{p}^{2}\text{r}^{2}-8\text{p}^{3}\text{r}+3\text{p}^{4}}{\text{r}^{6}}\geq
0$ (61)

One may verify (use $\frac{\text{r}}{\text{p}}$ as a variable) that both these
inequalities hold good and the energy conditions are satisfied for all r.

One can define $\text{p}=\nu\frac{\text{GM}}{\text{c}^{2}}$ in order to bring
in the right dimensions and the mass dependence. It is also instructive to
look at what happens for $\text{N}=1$–this is the simple case of the vacuum
Schwarzschild spacetime for which

$\displaystyle\text{g}_{\text{tt}}=-\left(1-\frac{2\text{p}}{3\text{r}}\right)=-\frac{1}{\text{g}_{\text{rr}}}$
(62)

Choosing $\text{p}=\frac{3\text{GM}}{\text{c}^{2}}$ one ends up with the
standard form of the metric and the shadow radius also reduces to the well-
known $3\sqrt{3}\frac{\text{GM}}{\text{c}^{2}}$.

$(b)$ We obtain another special solution by making a choice for the form of
the general horizon equation (51)

$(\text{r}_{\textbf{hz}}-\text{e})^{\text{N}}=0,$ (63)

where the horizon equation has the only root ‘e’ with multiplicity N. Then the
coefficients of the general metric take the form

$\text{a}_{\textbf{n}}=(-1)^{\text{n}}\frac{\text{N!}}{\text{n!}\text{(N-n)}!}\text{e}^{\text{n}},$
(64)

and the resulting metric components are

$\text{g}_{\text{tt}}=-\left(1-\frac{\text{e}}{\text{r}}\right)^{\text{N}}=-\text{g}_{\text{rr}}^{-1}.$
(65)

The spacetime represents a generic version of the extremal Reissner-Nordström
(RN) black holes. For $\text{N}=2$ we recover a RN black hole with mass
$\text{M}=\text{e}$ and the charge $Q=\pm\text{e}$ (note that e must be
positive for a black hole). For $\text{N}=4$, the metric component takes the
form

$\text{g}_{\text{tt}}=-\left(1-\frac{4\text{e}}{\text{r}}+\frac{6\text{e}^{2}}{\text{r}^{2}}-\frac{4\text{e}^{3}}{\text{r}^{3}}+\frac{\text{e}^{4}}{\text{r}^{4}}\right),$
(66)

where the mass of the black hole is then $\text{M}=2\text{e}$ and the
effective charge $Q=\pm\sqrt{6}\text{e}$. The shadow radius for such a class
of extremal black holes is given by

${\text{r}}_{\textbf{sh}}=\text{e}\left(1+\frac{\text{N}}{2}\right)\left(1+\frac{2}{\text{N}}\right)^{\frac{\text{N}}{2}}.$
(67)

The validity of NEC and WEC leads to the inequality

$\left[1+\frac{\text{e}}{\text{r}-\text{e}}\right]^{\text{N}}-\left[1+\frac{\text{e}\text{N}}{\text{r}-\text{e}}\right]\geq
0,$ (68)

which is satisfied for all $\text{r}\geq\text{e}$, i.e. outside the horizon.

##  VII Conclusions

Schwarzschild and Reissner-Nordström spacetimes are the simplest and most
widely used representatives of spherically symmetric, static black holes. In
this paper, we have considered a more general static, spherisymmetric
spacetime in the Schwarzschild gauge, by considering a general series
involving $\frac{1}{r^{k}}$ ($k\geq 0$) terms, with arbitrary constant
coefficients, in the metric functions. Such a metric may be a solution in GR
coupled with matter fields or may be a solution in a modified/ alternative
theory of gravity. It could also represent black holes or other real or
hypothetical objects in astrophysics. We have first analyzed the energy
conditions (NEC and WEC) for the matter required (assuming the Einstein field
equations of GR), if such a general spacetime has to exist. Thereafter, we
restrict the parameter space of our line element by comparing the size of the
circular shadow profile with observational data for M87∗ and SgrA∗.

Our generic metric (1) is parametrized by four (assuming terms upto
$\frac{1}{r^{4}}$) constants ${\cal P},\,\cal{Q},\,\cal{S},\,\cal{T}$, where
the effective mass and charge of the black hole/ naked singularity are related
as ${\cal P}=2\text{M}_{\textbf{eff}}$ and
$\text{Q}_{\textbf{eff}}=\pm\sqrt{{\cal Q}}$. Analysis of the energy
conditions reveal that NEC and WEC are satisfied for all $r$ for any of the
following two conditions only: $(i)$ ${\cal Q},{\cal S},{\cal T}>0$, and
$(ii)$ ${\cal Q}>0$, ${\cal S}<0$, ${\cal T}>0$, and ${\cal
S}^{2}\leq\frac{72}{25}{\cal Q}{\cal T}$. The first condition implies a naked
singularity only and, for the second condition, the spacetime may be a black
hole or a naked singularity. For any other choice of the parameters the energy
conditions are partially or completely violated. The details are provided in
Table 1 and Fig. 1.

Next we study the critical null geodesic (i.e. photon sphere) and the shadow.
We obtain a quartic horizon equation (36) and a quartic photon sphere equation
(37) for the general case. By a root analysis, we scan the parameter space to
identify viable regions for black holes/naked singularities and the existence
of photon spheres, along with the energy conditions (see Fig. 4 and the Table
2 for illustration). We also demonstrate some special solutions of the horizon
and photon sphere equations.

Thereafter, we compare the observed angular diameters of the M87* and SgrA*
with the theoretical prediction for different parameter values satisfying the
energy conditions. In particular, we choose $q=-s=t$ (for which the metric
takes the form (46) ). For this metric, comparing with all observations, we
get a bound on $q\leq 0.0145$ for M87* (with updated PRIMO data) and $q\leq
0.472$ for SgrA*. We also scan $q$ vs. $s$ parameter space ($t=0$ for
$1/r^{3}$ term) in Figs. 8, 9 and $q$ vs. $t$ parameter space ($s=0$ for
$1/r^{4}$ term) in Figs. 10, 11 for observationally allowed regions. Similar
plots may also be obtained for more general cases where $q,\,s,\,t\neq 0$.

Finally, we try to further generalize the metric functions by including higher
order terms $\frac{1}{r^{N}}$ where $N$ is arbitrary but integral, positive,
and finite. However, for arbitrary $N$ and the corresponding constant
coefficients, any known simple solution with viable matter fields is not
really known. We obtain two special solutions for which either the photon
sphere equation or the horizon equation has a single root with multiplicity
$N$. In the first case, for $N=4$, the solution turns out to be naked
singularity whereas in the second case, the solution (for $N=4$) seems to bear
a resemblance (via $M=2\text{e}$) to an extremal Reissner-Nordström black hole
but with higher powers of $\frac{1}{r}$ included.

In the section II, we provided some specific examples which are indeed special
cases of the general metric quoted in Eqn. (1). In particular, we noted how
such special cases of the general metric may represent black holes or naked
singularities surrounded by some fluids, namely the Kiselev type solutions
both in GR and modified gravity or other solutions as known in diverse
contexts. As mentioned before, there do exist many more line elements which
fall within the broad class assumed in (1). Thus, our generic results on
energy conditions and shadows may help in understanding features associated
with specific line elements which have arisen in different theories of
gravity, over the years.

Finally, one possible and natural extension of our work is to include
rotation, i.e. construct a line element with rotation, and proceed in the same
way, as done in this article. In such a scenario, the shadow profiles will be
much more complicated (no longer simple circles) and it is likely that one
will have to use numerical methods to arrive at any meaningful conclusions. We
hope to return to this aspect as well as other, similar questions in our
future endeavours.

## Acknowledgements

Research of SJ is partially supported by the SERB, DST, Govt. of India,
through a TARE fellowship grant no. TAR/2021/000354, hosted by the Department
of Physics, Indian Institute of Technology Kharagpur.

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