# A STUDY OF BLACK HOLES AND BEYOND: SHADOWS AND RELATIVISTIC ORBITS

This thesis is dedicated to my beloved Alpa Talreja and Mihir Ramanuj

Certificate

This is to certify that the thesis entitled “A Study of Black Holes and
Beyond: Shadows and Relativistic Orbits” submitted by Parthraj G. Bambhaniya
(18DRNST003) to the Charotar University of Science and Technology for the
award of the Degree of Doctor of Philosophy is a bonafide record of research
work carried out by him under my/our supervision. The contents of this thesis,
in full or in parts, have not been submitted to any other Institute or
University for award of any degree, diploma or titles.

Research Supervisor
Signature with Date:
Name : Prof. Pankaj S. Joshi
Designation  : Distinguished Professor, Founding Director of ICSC.
Organization  : International Center for Space and Cosmology (ICSC),
Ahmedabad University, Ahmedabad.

Dean
Signature with Date:
Name : Dr. Gayatri Dave
Organization : Faculty of Applied Sciences,
Charotar University of Science and Technology, Changa.

Declaration

I hereby declare that the thesis “A Study of Black Holes and Beyond: Shadows
and Relativistic Orbits” submitted by me to Charotar University of Science and
Technology for the degree of Doctor of Philosophy is the record of work
carried out by me during the period from December 2018 to September 2023 under
the supervision of Prof. Pankaj S. Joshi. I also declare that this work has
not formed the basis for the award of any degree, diploma or titles by any
other Institution or University.

Signature :
Date : 20th September 2023
Name of the Research Scholar : Parthraj G. Bambhaniya
ID Number : 18DRNST003

Acknowledgement

The journey of my doctoral study would not have been possible without some
incredible people who have encouraged, supported, and mentored me throughout.
The foremost person to be grateful to is my supervisor, Prof. Pankaj S. Joshi
for providing me with such a nice opportunity to work under his guidance. His
immense knowledge and profuse experience has encouraged me throughout my
academic research and daily life. His generosity and flexibility gave me the
freedom to work on a variety of exciting projects. Additionally, I’m very much
grateful for his invaluable advice, continuous support, and patience during my
Ph.D. study. I am highly thankful for the lab facility, and academic and
financial support received from the Charotar University of Science and
Technology under the Charusat Ph.D. student Research Fellowship (CPSF) scheme.

I would like to give very special thanks to my ideals: Mihir Ramanuj, Alpa
Talreja, Khushboo Dhameja and Tushar Kapdi who planted the seed of curiosity
about physics in me. If they had not been there, I would have never taken a
turn toward Astrophysics. I would also like to express my gratitude to Dr.
Dipanjan Dey for his valuable guidance to keep me interested and for
encouragement at various stages of my training period. I would like to thank
all the members and colleagues of the International Center for Cosmology
(ICC), Dr. Rucha Desai, Dr. Kanwarpreet Kaur, Dr. Tapobroto Bhanja, Ashok,
Vishva, Shailee, Jay, Divyesh Vithani for healthy discussions and support. I
would also like to thank Prof. A. Hasmani, Prof. C. K. Sumesh, Prof. R. M.
Patel, Prof. R. Rangarajan, and Prof. R. V. Upadhyay who have been evaluating
my research constantly.

More than four years is a very long duration away from home and can make any
person out of one’s mind. But friends are an amazing gift to us that can
transform and chanelise any bad energy into the best energy. I would like to
give my sincere thanks to my little world, Tushar Kapdi, Anand, Divyesh
Solanki, Siddharth Madan, Saurabh, Aadarsh, Anurag, Vishva, Kauntey, Kshitij,
Ankit who “were always there for discussions on anything that I was unsure of”
and “who have offered invaluable advice that will benefit me throughout my
life. It is their kind care and support that has made my study and life on the
Charusat campus a wonderful time. I also convey my special thanks to Dixitbhai
and Bhavnaben for their constant help and good food. A very special thanks to
my badminton team, Dr. Sunny, Dr. Mrugesh, Samruddhi, Stuti, Deep, Dweep,
Ravi, Simmy, and Richi for making my research stress free. I would also like
to thank visitors of ICC and well-wishers, Meet, Viraj, Suresh, Rittick,
Prashant, Rajibul, Saikat, Karim, Koushiki, Cherian, Divya, Ameya, Nupur,
Abhijeet, Chinmay, Vijay, Hari, Ashwathi, Sagar Baraiya, Nazir, Sagar Jethva,
Hima, Bittu, Bina, Debayan, Shaileshbhai, Sonal Desai, Vittali, Dr. Krunal,
Dr. Palash, Dr. Gayatri, Dr. Kinnari, Dr. Anjana, Dr. Manan, Dr. Shweta,
Mukesh Yadav, Yatin Talati, Rakesh Dave, Sid Ramanuj, Mahipal, Yavkrit,
Aditya, Chirag, Dhruv, Milan, and other MSc students for always being so kind
and supportive.

Science is never the result of an individual endeavor. My all publications are
the result of collaborations with amazing people, whom I have been extremely
fortunate to be able to work along with.

Finally, I would like to express my gratitude to my parents, my little
brother, my lovely sisters, Yogesh, Jagdish, Darshita and my family members.
Without their tremendous understanding and encouragement over the past few
years, it would be impossible for me to complete my study. At last, I would
like to thank all my academic teachers: Prof. S. P. Bhatnagar, Prof. N. K.
Bhatt, Prof. G. M. Sutariya, Prof. Dipika Pandya, Prof. Umesh Dodiya, Prof.
Maniyar, Prof. Bharat, Prof. Shilpa, Minal ma’am, Dr. Kanti, Krunal sir,
Vishal sir, Dr. Virbhadrasinh, Dharmendra sir and others for believing in me
and constant encouragement towards my goal.

Parthraj G. Bambhaniya

Abstract

Key-Words: Black holes, Naked singularities, Milky Way galaxy, Sagittarius A*,
Accretion disk, Shadows of compact objects, Orbital precession.

This doctoral thesis is organized into seven chapters. The first chapter
introduces readers to the formation of black holes and naked singularities as
an end state of continuous gravitational collapse. The physical and
geometrical properties of the Schwarzschild black hole, Joshi-Malafarina-
Narayan-1 (JMN-1) naked singularity, and Janis-Newman-Winicour (JNW) naked
singularity are summarised. The motivation and objectives to be derived are
based on the literature reviews. The second chapter deals with the shadows of
the mentioned compact objects. The equations of motion are explicitly
calculated for general spherically symmetric and static spacetimes using the
ray-tracing formalism and the null geodesics. The third chapter focuses on the
construction of rotating naked singularity using the Newman-Janis-Algorithm
(NJA). The NJA is used without complexification method and obtain rotating JNW
naked singularity spacetime. The general formalism of the shadow shape is
derived for rotating spacetime and obtain the shadow shapes for the rotating
JNW, Kerr and deformed Kerr spacetimes. In the fourth chapter, the precession
of timelike-bound orbits is investigated in the Schwarzschild, JMN-1, and JNW
spacetimes. The fully relativistic orbit equations are derived for the
provided models. The approximate solutions of the orbit equations are used to
characterize the nature of orbital precession. The next chapter is on the
precession of timelike bound orbits in the rotating Kerr and JNW spacetimes.
The sixth chapter deals with the relativistic orbits of S-stars and discusses
the orbital parameters of the real and apparent orbits. Astrometric data of
the S2 star has been adopted from the available literature and use numerical
techniques to study the relativistic orbits of the S2 star in the presence of
a scalar field. The final chapter aims at summarising the results followed by
some futuristic scopes that probe the nature of Sagittarius A* (Sgr A*) with a
possible black hole mimicker.

###### Contents

  1. 1 Introduction
    1. 1 Black hole
    2. 2 Schwarzschild black hole
    3. 3 Naked singularity
    4. 4 JMN-1 naked singularity
    5. 5 JNW naked singularity
    6. 6 Shadows of compact objects
    7. 7 Precession of timelike bound orbits
  2. 2 Shadows cast by non-rotating black hole and naked singularities
    1. 8 Null geodesics in spherically symmetric and static spacetime
    2. 9 Observable intensity
    3. 10 Radiative Modelling
    4. 11 Ray-Tracing Formalism
      1. 11.1 Geodesic equations of motion
      2. 11.2 Initial Conditions
    5. 12 EHT VLBI Observations
      1. 12.1 Qualitative analysis
      2. 12.2 Quantitative Analysis
    6. 13 Discussions and Conclusions
  3. 3 Shadows cast by rotating black holes and naked singularities
    1. 14 Construction of the rotating JNW spacetime
      1. 14.1 JNW metric and the Newman-Janis Algorithm
      2. 14.2 Newman-Janis Algorithm without Complexification
      3. 14.3 Rotating JNW metric
    2. 15 The Kerr spacetime
    3. 16 General formalism of the shadow shapes for rotating spacetimes
      1. 16.1 Shadows in the rotating JNW spacetime
      2. 16.2 Shadows in the deformed Kerr spacetime
    4. 17 Discussions and conclusions
  4. 4 Precession of timelike bound orbits in static spacetimes
    1. 18 Timelike geodesics in spherically symmetric and static spacetime
    2. 19 Fully relativistic orbit equations
      1. 19.1 Schwarzschild orbit equation
      2. 19.2 JMN-1 orbit equation
      3. 19.3 JNW orbit equation
    3. 20 Periastron precession of the timelike bound orbits
    4. 21 Approximate solutions of the orbit equations
      1. 21.1 Approximate solution of the Schwarzschild orbit equation
      2. 21.2 Approximate solution of the JMN-1 orbit equation
      3. 21.3 Approximate solution of the JNW orbit equation
    5. 22 Discussions and conclusions
  5. 5 Precession of timelike bound orbits in rotating spacetimes
    1. 23 Timelike geodesics in Kerr spacetime
    2. 24 An approximate solution of orbit equation
    3. 25 Timelike geodesics in rotating JNW spacetime
    4. 26 Discussion and Conclusion
  6. 6 Relativistic orbits of S-stars
    1. 27 S-stars near Milky Way galactic center
    2. 28 Orbital parameters of the real and apparent orbits
    3. 29 Numerical techniques
    4. 30 Discussion and Conclusion
  7. 7 Concluding remarks and future scopes
    1. 31 Concluding remarks from shadow properties
    2. 32 Concluding remarks from relativistic orbits
    3. 33 Future scopes

###### List of Figures

  1. 1 Shadow image of M87*
  2. 2 Shadow image of Sgr A*
  3. 3 S-stars near Milky Way galactic center
  4. 4 Model images for Sgr A*, parameters details are given in the Table (4).
  5. 5 Baseline coverage of Sgr A⋆ for the array configuration of EHT mentioned in Table 2.
  6. 6 Reconstructed images of Sgr A* for models described in section-I with different array configurations of EHT (top to bottom: EHT-I, EHT-II, EHT-III).
  7. 7 Figure shows the shape of the shadow cast by the rotating JNW naked singularity for various values of scalar field charge $q$ and spin parameter $a$. The values of $M=1$ and $\theta=\pi/2$ are considered for the shape of shadows in celestial coordinates. The details of this figure are discussed in the text.
  8. 8 Event horizon position for $\epsilon=0$ (black), $\epsilon=1$ (Red), $\epsilon=-1$ (Blue) with different spin and 3D view of directional naked singularity for $\epsilon=1$, $a=0.8$.
  9. 9 The shaded region in the figure shows the existence of a naked singularity in deformed Kerr spacetime.
  10. 10 Figure shows the shape of the shadow cast by the deformed Kerr spacetime for various values of deformation parameter $\epsilon$ and spin parameter $a$. The details of this figure are discussed in the text.
  11. 11 Figure shows how the deformation of shadow shape ($D$) changes with the value of spin parameter $a$, where the red, blue and brown lines correspond to the deformed Kerr spacetime having $\epsilon=-1,1$ and $18$ respectively.
  12. 12 Positive orbital precession in Schwarzschild spacetime
  13. 13 Negative and Positive orbital precession in the JMN-1 spacetime
  14. 14 Positive and negative orbital precession in the JNW spacetime
  15. 15 Regions plots for $m>1$ and $m<1$ in JNW spacetime.
  16. 16  Orbits of a test particle in the Kerr spacetime for $L=12$.
  17. 17 Orbits of a test particle in the Kerr spacetime for $L=6$.
  18. 18 The solution points for the timelike bound orbits in $m$ and $a$ coordinates.
  19. 19 The region plots of $f>0$ and $V_{min}\leq E<0$ for M=1 and $L=6$ (left) and $L=12$ (right).
  20. 20 The timelike bound orbits in the rotating JNW naked singularity spacetime.
  21. 21 Real orbit projection into the sky plane Becerra-Vergara:2020xoj .
  22. 22 Left: The best fitting orbit of the JNW model (black) along with the observed astrometric position (Blue) of the S2 star from 1995 to 2018. The red dot shows the position of the Sgr A* at (0,0). Right: R.A. (right ascension $\alpha$ on top) and Dec. (declination $\delta$ on bottom) offset of S2 star with the orbital period.
  23. 23 1-$\sigma$ and 2-$\sigma$ best-fit regions and the posterior distributions of the parameters for the JNW metric and Sgr A* derived using MCMC. The lowest $\chi^{2}$ value obtained is 4.71.

###### List of Tables

  1. 1 Model details
  2. 2 Locations of Existing/Future Sites in the Event Horizon Telescope Array.
  3. 3 Normalized Cross-Correlation coefficient for the comparison between the reconstructed of various models with the Schwarzschild BH.
  4. 4 Estimated best-fit values of the parameters for the JNW metric.
  5. 5 Details of the Gaussian priors of different parameters.

## CHAPTER 1 Introduction

Ever since the dawn of human civilization, the quest to determine the origin
of the universe, has been the most curious one for individuals. With the
invention of scientific logic to approach the physical realm of the universe,
humanity has moved strides into the unknown like no other time in history. To
be more philosophical, in the words of Carl Sagan, "we are a way for the
universe to know itself".

From the current understanding, it is believed that people have descended from
stardust, the remains of a star that has run out of its nuclear fuel. This
stardust is orbiting a second-generation star, i.e. the sun. There are
billions of stars in the Milky Way galaxy. Like any other living entity, these
stars have a life cycle with a beginning and an end. These stars arose from
the clouds of dust and gas known as molecular clouds, which collapsed under
their own gravity to form protostars. Researchers have classified stars into
three categories based on their initial mass: small stars, average stars, and
massive stars. All of these stars undergo the process of nuclear fusion in
which they use their nuclear fuel to survive during their life cycle. At
first, the fusion of hydrogen atoms takes place, resulting in the formation of
helium atoms and production of an enormous amount of energy in the form of an
exothermic process that balances the star’s gravitational pull. When these
stars exhaust hydrogen atoms in their core, the fusion of heavier elements
takes place, such as carbon, oxygen, neon, silicon, and iron. These stars are
also known as the main sequence stars depending on the stages of their nuclear
fusion. The fusion process in these main sequence stars cannot fuse atoms
heavier than iron, as the pressure from the thermal energy produced in this
process can no longer defy its gravitational pull, and the star starts
collapsing under its own gravity (Joshi:2011rlc, ). The small and average
stars behave stable under their own quantum pressures, i.e. electron and
neutron degeneracy pressures to form white dwarfs and neutron stars,
respectively.

On the other hand, more massive stars cannot balance their gravitational pull
by any known quantum pressures. As Einstein’s general theory of relativity
predicts, the continuous gravitational collapse of a massive star leads to a
spacetime singularity where the matter density and gravitational attraction
become infinite penrose . At this point, all physical laws break down. This
ultra-dense compact region can be hypothesized by describing various
astrophysical compact objects, e.g., the black hole, naked singularity, boson
star, and other types of objects Oppenheimer:1939ue , psj1 , psj2 , jnw , bst
, Visinelli:2021uve , Schunck:2003kk . These compact objects have fascinating
physical and geometrical properties, and it would be observationally
significant to distinguish them.

### 1 Black hole

The term "Black hole" is used for a compact astrophysical object from which
nothing can escape to a faraway observer. Even light cannot get away from its
boundary which is an event horizon, due to the high gravitational field
attraction. Therefore, one cannot see that dark region within event horizon
where things can fall, but nothing will come out. The distinct signature of a
black hole is an "Event Horizon" that indicates presence of trapped surfaces.
Within the general theory of relativity, the spacetime singularity can form at
the center of such a black hole which is the actual boundary of the spacetime
manifold penrose . The spacetime singularity is the caustic point where all
the geodesics are incomplete. All the physical quantities, such as matter
densities, pressures, and spacetime curvatures, take arbitrarily large and
diverging values near the singularity. Generally, when an event horizon covers
the singularity, it is known as a black hole.

In the general theory of relativity, there are mainly two well-known non-
rotating black holes, namely (i) the Schwarzschild black hole and (ii) the
Reissner-Nordstrom black hole. The Schwarzschild black hole is a vacuum
solution of the Einstein equations, which has no charge and no rotation.
Whereas the Reissner-Nordstrom black hole is a static solution of the
Einstein-Maxwell field equations with electric charge but no rotation. The
rotating generalizations of these static black holes are given as the Kerr
(with zero charge) and Kerr-Newman (with non-zero charge) black holes,
respectively Hartle:2021pel . In this thesis, the focus is on the
Schwarzschild and Kerr black holes for the comparative studies with the naked
singularities.

### 2 Schwarzschild black hole

In 1916, Karl Schwarzschild found the first exact vacuum solution of
Einstein’s field equations Schwarzschild:1916uq . This solution gives the
spacetime metric outside a spherically symmetric and static massive body that
does not contain an electric charge and angular momentum. The Schwarzschild
black hole spacetime in spherical coordinates reads,

$ds^{2}_{SCH}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}\,\,,$
(1)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ represents the metric
of the two spheres, and $M$ is the total mass of the black hole. The signature
convention $(-,+,+,+)$ is used for spacetime metric throughout the thesis. In
the above Schwarzschild spacetime, the singularity occurs at $r=0$, which is
covered by a null surface, an event horizon which occurs at $r=2M$, also known
as the Schwarzschild radius. As it is a vacuum solution of Einstein field
equations, the energy density and pressures of this spacetime are,

$\rho=0,\;\;\;p_{r}=p_{\theta}=p_{\phi}=0.$ (2)

where $\rho$ is the energy density of the matter, $p_{r}$ is the radial
pressure while $p_{\theta}$ and $p_{\phi}$ are tangential pressures. The
following conditions can easily verify the weak, strong and null energy
conditions of this spacetime, respectively as,

$\rho\geq 0,\;\;\;\rho+p_{r}\geq 0,\;\;\;\rho+p_{\theta}\geq
0,\;\;\;\rho+p_{\phi}\geq 0,\;\;\;\rho+p_{r}+2p_{\theta}\geq 0.\;\;\;$ (3)

In 1939, the first dynamical collapse solution of a spherically symmetric and
homogeneous dust cloud was derived by Oppenheimer, Snyder, and Datt
Oppenheimer:1939ue , datt , which was identified as OSD collapse later. It has
been shown that the final state of a spherically symmetric and homogeneous
dust collapse will always be a Schwarzschild black hole. Moreover, Birkhoff’s
theorem suggests that the Schwarzschild spacetime is the most general vacuum
solution of Einstein’s equations Hartle:2021pel . However, the OSD collapse
model depends on the ideal assumption, as the density should be homogeneous
within the massive collapsing star. They also neglected gas pressure and took
it to be zero to avoid dealing with the relatively complex Einstein equations
Joshi:2011rlc . In this work they have shown that an event horizon develops as
collapse progresses in a way in which no material, particle, or light rays,
can escape to a faraway observer. So, one must think beyond this idealistic
scenario and consider more physical situations for the collapsing body.

### 3 Naked singularity

In general, an astrophysical star is not supposed to have a homogeneous
density profile within the star’s surface. It can have high density at the
core area, which could decrease as a function of radius as one go from the
core area to the star’s surface. Under a relatively more physically realistic
situation, if inhomogeneity is introduced into the matter distribution profile
and non-zero pressures are considered, then the end state of the dynamical
collapse could result in a strong curvature singularity within the general
theory of relativity Joshi:2011rlc , psj1 , psj2 , mosani1 , mosani2 , mosani3
, Joshi:2013dva . However, the end state of the dynamical gravitational
collapse depends on the classes of initial conditions of the collapsing
matter. In principle, theory of general relativity predicts that the spacetime
singularity forms necessarily when large enough masses collapse under their
own gravity. Nevertheless, it does not necessarily simultaneously enforce the
formation (or otherwise) of an event horizon. If a singularity is formed
without an event horizon, it is known as a naked or visible singularity
Joshi:2008zz , Joshi:2018exh .

Similar to a black hole, this is a region of an ultra-high-density where all
the physical quantities will diverge arbitrarily. Naked singularity is an
intriguing horizonless compact object. Although the cosmic censorship
conjecture (CCC) prohibits the formation of strong horizonless singularities
penrose , several research studies reveal that the naked singularities can
develop during the continuous gravitational collapse of an inhomogeneous
matter cloud. In psj2 , Joshi, Malafarina and Narayan (JMN) have shown that a
non-zero tangential pressure can prevent the formation of trapped surfaces
around the core of high-density region of a collapsing matter cloud, resulting
in a central naked singularity in large co-moving time. In this thesis, the
focus is on the first type of Joshi-Malafarina-Narayan (JMN-1) and Janis-
Newman-Winicour (JNW) naked singularity spacetimes psj2 , jnw . The details of
these spacetimes are given as follows.

### 4 JMN-1 naked singularity

The equilibrium configurations of the collapsing fluid within the general
theory of relativity are extensively studied in psj2 . The JMN-1 naked
singularity can be formed as an end state of gravitational collapse with zero
radial and non-zero tangential pressures and the corresponding spacetime is
given as,

$ds^{2}_{JMN-1}=-(1-M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(1-M_{0})}}dt^{2}+\frac{dr^{2}}{(1-M_{0})}+r^{2}d\Omega^{2}\,\,,$
(4)

where, $M_{0}$ and $R_{b}$ are positive constants. Here, $R_{b}$ represents
the boundary radius of the distributed matter around the central singularity,
and $M_{0}$ should be within the range $0<M_{0}<4/5$. Note that the upper
limit of $M_{0}$ is defined by the fact that the sound speed cannot exceed
unity. The stress-energy tensor of the JMN-1 spacetime gives the energy
density $\rho$ and pressures $p$ as,

$\rho=\frac{M_{0}}{r^{2}},\;\;\;p_{r}=0,\;\;\;p_{\theta}=\frac{M_{0}}{4(1-M_{0})}\rho.$
(5)

This collapsing fluid is supported only by tangential pressure, and it can be
verified that all energy conditions are satisfied by JMN-1 spacetime.

The spacetime metric is modelled by considering a high-density compact region
in a vacuum, which means that the spacetime configuration should be
asymptotically flat. However, if it is not asymptotically flat, the interior
spacetime must be matched to asymptotically flat exterior spacetime with a
particular radius. On the other hand, asymptotically flat spacetimes need not
be matched with any external spacetime. Two spacetimes can be matched at a
particular space-like or timelike hypersurface. For that, one need to follow
two junction conditions parth1 :

  1. 1.

The induced metrics of internal and external spacetimes should be identical on
the matching hypersurface.

  2. 2.

The extrinsic curvatures ($K_{ab}$) of internal and external spacetimes should
be matched at the hypersurface, where the extrinsic curvatures are expressed
in terms of the covariant derivative of normal vectors on the hypersurface:

$K_{ab}=e^{\alpha}_{a}e^{\beta}_{b}\nabla_{\alpha}\eta_{\beta},$ (6)

where $e^{\alpha}_{a}$ and $e^{\beta}_{b}$ are the tangent vectors on the
hypersurface and $\eta^{\beta}$ is the normal to that hypersurface.

The JMN-1 naked singularity spacetime is not asymptotically flat. The interior
JMN-1 spacetime to exterior Schwarzschild spacetime can be smoothly matched at
$r=R_{b}$ as,

$ds^{2}_{SCH}=-\left(1-\frac{M_{0}R_{b}}{r}\right)dt^{2}+\left(1-\frac{M_{0}R_{b}}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(7)

where $M=\frac{1}{2}M_{0}R_{b}$ is the Schwarzschild mass of the compact
object. The extrinsic curvatures of JMN-1 and Schwarzschild spacetimes are
automatically smoothly matched at $r=R_{b}$ parth1 . Since the JMN-1 spacetime
has zero radial pressure.

### 5 JNW naked singularity

Janis, Newman, and Winicour obtained a minimally coupled mass-less scalar
field solution of the Einstein field equations jnw , and independently by
Wyman found identical solution Wyman . The Lagrangian density of the minimally
coupled scalar field is given as,

$\mathcal{L}=\sqrt{-g}\left(\frac{1}{2}\partial^{\mu}\Phi\partial_{\mu}\Phi-V(\Phi)\right)\,\,,$
(8)

where $\Phi$ is the scalar field and $V(\Phi)$ is the corresponding scalar
field potential. The minimal coupling conditions are defined as,

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa T_{\mu\nu},$ (9)
$\Box\Phi(r)=V^{\prime}(\Phi(r)),$ (10)

where $R$ is the Ricci scalar, $R_{\mu\nu}$ is the Ricci tensor, $T_{\mu\nu}$
is the energy-momentum tensor, and $g_{\mu\nu}$ is the metric tensor. $\kappa$
is a constant parameter, defined by $8\pi G/c^{4}$, where $G=c=1$. The Energy-
momentum tensor ($T_{\mu\nu}$) for the minimally coupled scalar field can be
written as,

$T_{\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi-g_{\mu\nu}\mathcal{L}.$ (11)

and the line element for the JNW spacetime is,

$ds^{2}_{JNW}=-\left(1-\frac{b}{r}\right)^{n}dt^{2}+\left(1-\frac{b}{r}\right)^{-n}dr^{2}+r^{2}\left(1-\frac{b}{r}\right)^{1-n}d\Omega^{2}\,\,,$
(12)

where, $b=2\sqrt{M^{2}+q^{2}}$ and $n=\frac{2M}{b}$. Here, $M$ and $q$
represent ADM mass and scalar field charge, respectively. As $b$ is positive
and greater than $2M$, one can write $0<n<1$. The massless scalar field can be
written as,

$\Phi=\frac{q}{b\sqrt{4\pi}}ln\left(1-\frac{b}{r}\right)\,\,.$ (13)

Further as, the JNW spacetime is asymptotically flat, it is not required to
match with any other external spacetime. However, for $n=1$ or $q=0$, there
are no effects of the scalar field ($\Phi$) on the spacetime, and JNW
spacetime becomes Schwarzschild spacetime. Therefore, the JNW spacetime
considers an extension of the Schwarzschild spacetime when the minimally
coupled massless scalar field is included.

Now, the question arises what could be the end state of the massive collapsing
star? This question remains an open challenge. According to the Cosmic
Censorship Conjecture (CCC), the end state of the massive collapsing body must
be a black hole penrose . However, as divulged before, several studies have
shown that naked singularities can form due to the continual gravitational
collapse of an inhomogeneous matter cloud. As pointed above, naked
singularities form under physically realistic astrophysical situations and
scenarios, their observational consequences become significant in the
understanding of the universe such as the shadows cast by these objects, the
strong gravitational lensing around them, the orbital precession of the stars
in their vicinity and many others. In this thesis, the emphasis is mainly on
the formation of a shadow by compact objects and the periastron precession of
the star’s orbits. From these studies, it can be predicted or probed that the
possible observational signatures which would reveal the nature of the causal
structure of any compact object.

### 6 Shadows of compact objects

A supermassive black hole is believed to exist at the center of almost every
galaxy Kormendy:2013dxa . However, there is no conclusive evidence at this
time about the nature of the compact object. In light of this, in April of
2019, the Event Horizon Telescope (EHT) acquired the first shadow image of a
supermassive black hole at the heart of the M87 galaxy (see fig. 1) M87 .
Recently, the EHT collaboration has announced a major breakthrough in the
imaging of an ultra-compact object at the centre of the Milky way galaxy (see
fig. 2) EventHorizonTelescope:2022xnr , EventHorizonTelescope:2022vjs ,
EventHorizonTelescope:2022wok , EventHorizonTelescope:2022exc ,
EventHorizonTelescope:2022urf , EventHorizonTelescope:2022xqj . A bright
emission ring around a core brightness depression in VLBI horizon-scale images
of Sgr A*, with the latter linked to the shadow of black hole. The shadow
boundary of the Sgr A* marks the visual image of the photon region and
differentiates capture orbits from scattering orbits on the plane of a distant
observer. The radius of the bright ring can be used as an approximation for
the black hole shadow radius under some specific conditions and after proper
calibration, with little reliance on the details of the surrounding accretion
flux. While there is strong evidence that there is a massive concentration of
mass in the center of the Milky Way galaxy, the question of whether or not it
is a black hole is still open. The EHT shadow images of the M87 and Sgr A*
galactic centers do not prove the notion of an event horizon and hence a
supermassive black hole. Other compact objects can cast similar shadows
Vagnozzi:2022moj . The shadows created by compact objects, including black
holes, naked singularities, gravastars, and wormholes, have received
considerable attention shaikh1 , gyulchev , null , Abdikamalov:2019ztb ,
Dey:2020haf , ohgami_2015 , stuchlik_2019 , Kaur , Sakai , Shaikh:2021cvl ,
Bambhaniya:2021ugr . The Event Horizon Telescope (EHT) is now being upgraded
to the next-generation Event Horizon Telescope (ngEHT) in order to monitor the
galactic center of the Milky Way (Sgr A*) and other prospective radio sources.

Fig. 1: Shadow image of M87* Fig. 2: Shadow image of Sgr A*

As discussed earlier, the gravitational collapse with zero radial and non-zero
tangential pressures can form the JMN-1 naked singularity. It could be
achieved as a quasi-equilibrium configuration of the collapsing fluid
employing the general theory of relativity. The shadows of JMN-1 naked
singularity spacetimes are explored and compared to a Schwarzschild black hole
shadow shaikh1 . It is generally speculated that the existence of a photon
sphere causes shadows. Given that there is no photon sphere and hence no
shadow in JMN-1 naked singularity for the range of characteristic parameter
$0<M_{0}<2/3$, an exciting full moon image is produced. JMN-1 naked
singularity, on the other hand, can create a similar shadow for $2/3<M_{0}<1$,
just as the Schwarzschild black hole produces. Furthermore, another naked
singularity model introduced in 1968 by Janis-Newman-Winicour (JNW) can create
a substantially similar shadow for the characteristic parameter value
$0.5<n<1$ as the Schwarzschild black hole does gyulchev , Saurabh:2022jjv .

In the recent work, it is illustrated that a new spherically symmetric naked
singularity solution of the Einstein field equation, despite the absence of a
photon sphere, can cast a shadow null . The general prerequisites for a shadow
to arise without a photon sphere are then defined for null and timelike naked
singularities, where each satisfies all of the energy conditions Dey:2020bgo .
Thus, when two distinct sorts of equally massive compact objects create
shadows of the same size, it becomes challenging to identify them. However,
because their spacetime geometries are distinct, the nature of the null and
timelike geodesics around the two compact objects would be different. The
existence of an upper bound of the effective potential of null geodesics
induces the formation of a shadow, according to these studies null ,
Dey:2020bgo , implying that if a spacetimes’ effective potential of null
geodesics has an upper bound, that spacetime can cast a shadow, such that the
existence of a shadow does not require the presence of an event horizon and a
photon sphere around a singularity. Consequently, these models can provide
viable alternatives for testing or investigating the nature of the galactic
center’s supermassive compact object (e.g. Milky Way).

### 7 Precession of timelike bound orbits

Half of the Nobel prize in physics 2020 was awarded to Prof. Roger Penrose for
discovering that black hole formation is a robust prediction of the general
theory of relativity. At the same time, the half prize was bestowed to Prof.
Reinhard Genzel and Prof. Andrea Ghez for discovering a supermassive compact
object (Sgr A*) at the center of the Milky Way galaxy Do , GRAVITY:2018ofz ,
Hees:2017aal , GRAVITY:2020gka . The Sgr A* is assumed to be a supermassive
black hole (SMBH) that exists at the core of the Milky Way galaxy. There is,
however, no concrete evidence for the same, i.e. the Sgr A* is indeed a SMBH.
The nature of Sgr A* is still unknown. It is expected that the mass of Sgr A*
is about four million times the mass of the sun, which is located at a
distance of 8.2 kpc (1 kpc = $3*10^{16}km$) from the Earth. In recent days,
the analysis of the nature of Sgr A* has become a subject of great interest.
To envisage the nature of the Sgr A*, it is vital to investigate the different
possible observational features, such as the shadow properties, the stellar
orbits of the S-stars around the Sgr A*, accretion matter properties, etc.
These observational features would be helpful in verifying the nature of the
Sgr A*, whether it is a supermassive black hole or a naked singularity.

In the second chapter, the study of observational appearance at $230$ GHz is
covered with probing the nature of Sagittarius-A* (Sgr A*) as a naked
singularity Saurabh:2022jjv . The JMN-1 and JNW naked singularity spacetimes
which are anisotropic fluid solutions of the Einstein field equations are
considered. Motivated by radiatively inefficient accretion flows (RIAF)
Pu:2018ute , Pandya:2016qfh , An analytical model for emission and absorption
coefficients is used to solve the general relativistic radiative transfer
equation. The resulting emission is then utilized to generate images to
predict the nature of the Sgr A* with synthetic Very Long Baseline
Interferometry (VLBI) images from current and future Event Horizon Telescope
(EHT) arrays. Three different EHT array configurations are being used to
simulate the models of naked singularities and a black hole. This may have
little effect on the baseline, but it would increase the u-v plane gridding,
making it feasible to capture a better-resolved image. Therefore, it is pretty
exciting and valuable for probing the shadow image of the Sgr A* to predict
whether it is a supermassive black hole or a naked singularity.

In the third chapter, the rotating Janis-Newman-Winicour (JNW) naked
singularity spacetime is constructed using Newman-Janis Algorithm (NJA) as the
rotating spacetimes are more physically realistic for investigating
observational properties solanki . There is an analysis of NJA with and
without complexification methods and found that the energy conditions that
were satisfied when the complexification step was skipped. The shadows cast by
rotating JNW were studied deeply along with deformed Kerr spacetimes then
compared them with those cast by the Kerr black hole deformed .

In fourth chapter, the fully relativistic general orbit equation is derived
for any spherically, symmetric and static spacetime. The precession of
timelike bound orbits is studied in the JMN-1 and JNW naked singularity
spacetimes and compared them with the Schwarzschild black hole case parth1 ,
Dey:2019fpv , Joshi:2019rdo . The approximate solutions are find for the given
models to analyze the nature of orbital precession. Several exciting
differences emerge from those results, including the conclusion that particle-
bound orbits in naked singularity spacetimes can precess in the opposite
direction of particle motion, which is impossible in Schwarzschild spacetime.
Such reverse precession and other distinctions may help distinguish a naked
singularity from a black hole observationally.

The fifth chapter focuses on, precession of the timelike bound orbits in the
rotating Kerr, and JNW spacetimes solanki , Bambhaniya:2020zno . Using
approximation, it is shown that the negative precession is not possible in
Kerr spacetime. While rotating, JNW spacetime can admit negative precession
for the particular range of scalar field charge $q$ and spin parameter $a$.
This distinct feature of the timelike bound orbits in rotating JNW spacetime
can be observationally significant to distinguish from the Kerr and other
rotating compact objects.

Several S-stars are orbiting around the Milky Way galactic center (See fig.
3). Since they are very close to the Sgr A*, a general relativistic effect can
be seen in their dynamics. Hence, their dynamics may distinguish a naked
singularity from a black hole. GRAVITY, SINFONI, and UCLA galactic center
groups shared their most recent data on the stellar motions of the S2 star
around the Sgr A*, which is one of the crucial stars in the S-stars family Do
, GRAVITY:2018ofz , Hees:2017aal , GRAVITY:2020gka . The S2 star’s orbit can
give the essential information about the nature of the Sgr A* and its
dynamics. Therefore, one can verify the precession of the S star’s orbit
within the background of a black hole and naked singularity spacetimes.

Fig. 3: S-stars near Milky Way galactic center

Recently, the GRAVITY collaboration published a work on the S2 star’s orbital
precession by considering the Sgr A* as a black hole, where they have found
that the precession angle of the S2 star’s orbit is about 12 arc-minute which
is too small to detect whether the orbital shift is positive or negative
GRAVITY:2020gka . More orbital data points with precise locations are required
to make a concrete conclusion on the nature of Sgr A*. Hence, any evidence of
negative precession of any ‘S’ star can raise a big question on the existence
of a black hole at the Milky Way galaxy center. In this context, the best-
fitting orbit of the S2 star in the static JNW spacetime is obtained and
investigated the scalar field effect on the orbital dynamics of that star in
the sixth chapter Bambhaniya:2022xbz . The best fitting parameters are
estimated for the JNW metric using the Monte-Carlo-Markov-Chain (MCMC)
technique and obtain the lowest Chi-squared value is 4.71. Here, it is
predicted that the nature of the Sgr A* using the available observed
astrometric data of the S2 star. It can be inferred that JNW naked singularity
could be a plausible candidate which might represent the spacetime structure
of Sgr A* which mimics the Schwarzschild black hole spacetime.

## CHAPTER 2 Shadows cast by non-rotating black hole and naked singularities

The study of null geodesics is useful for describing the formation of the
shadows cast by ultra-compact objects. The study is about an observational
appearance of the shadow image of the Sgr A* to predict whether it is a
supermassive black hole or a naked singularity. The shadow properties are
examined in JMN-1 and JNW naked singularity spacetimes. The novel features of
the results are discussed and compared with the non-rotating black hole case,
i.e. Schwarzschild black hole.

### 8 Null geodesics in spherically symmetric and static spacetime

Null geodesics represent the light ray orbits. To describe the world line of
light rays, one has to consider the coordinates $x^{\alpha}$ as a function of
an affine parameter $\lambda$. The general spherical symmetric and static
spacetime can be written as,

$ds^{2}=-g_{tt}(r)dt^{2}+g_{rr}(r)dr^{2}+g_{\theta\theta}(r)d\theta^{2}+g_{\phi\phi}(r)\sin^{2}\theta
d\phi^{2}\,\,,$ (14)

where $g_{tt}$, $g_{rr}$, $g_{\theta\theta}$ and $g_{\phi\phi}$ are functions
of $r$ only. The conserved quantities for spherically symmetric and static
spacetime (14) corresponding to light ray can be defined as,

$\gamma=-\xi\cdot k=g_{tt}(r)\left(\frac{dt}{d\lambda}\right)\,\,,\,\,\,$ (15)
$h=\eta\cdot k=g_{\phi\phi}(r)\left(\frac{d\phi}{d\lambda}\right)\,\,,$ (16)

where $\lambda$ is an affine parameter and $\xi$ and $\eta$ are killing vector
corresponding to temporal and spherical symmetries. Here, $\gamma$ and $h$
represent the conserved energy and angular momentum per photon. The
normalization of null vectors defines as,

$k\cdot
k=g_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}=0,$
(17)

now, using the conserved quantities and normalization condition into general
spherically symmetric and static spacetime (14) for null geodesics, the
following expression can be taken into consideration,

$-\frac{\gamma^{2}}{g_{tt}(r)}+g_{rr}(r)\left(\frac{dr}{d\lambda}\right)^{2}+\frac{h^{2}}{g_{\phi\phi}(r)}=0.$
(18)

For simplicity, the light ray geodesics are considered in an equatorial plane
$\theta=\pi/2$. By multiplying $g_{tt}(r)/h^{2}$ term to the above equation,

$\frac{1}{b^{2}}=\frac{g_{tt}(r)g_{rr}(r)}{h^{2}}\left(\frac{dr}{d\lambda}\right)^{2}+W_{eff}(r),$
(19)

where, $b=h/\gamma$ is the impact parameter and $W_{eff}(r)$ is the effective
potential for photon orbits given as,

$W_{eff}(r)=\frac{g_{tt}(r)}{g_{\phi\phi}(r)}.$ (20)

The effective potential is useful to determine the stability of the light
orbits. Unstable circular null orbits are observed when the following
conditions are satisfied,

$W^{\prime}_{eff}(r_{ph})=0,\;\;\;W^{\prime\prime}_{eff}(r_{ph})<0,$ (21)

the above conditions imply that the effective potential has a maximum value at
$r=r_{ph}$ for unstable circular photon orbits, where $r_{ph}$ is the radius
of the photon sphere. The path of the light ray can change due to the
spacetime curvature. Using the equations (15), (16) and (19), one can derive
an expression to define the shape of the light orbits as,

$\left(\frac{dr}{d\phi}\right)^{2}=\pm\frac{g_{\phi\phi}^{2}(r)}{g_{tt}(r)g_{rr}(r)}\left[\frac{1}{b^{2}}-W_{eff}(r)\right].$
(22)

Here, the $\pm$ signs represent the outgoing and incoming light ray paths,
respectively. An orbit equation of the light ray (photon) can be obtained by
differentiating above eq. (22) with respect to $\phi$,

$\frac{d^{2}u}{d\phi^{2}}+\frac{g_{\phi\phi}^{2}(u)u^{4}}{2g_{tt}^{2}(u)g_{rr}(u)b^{2}}\left(\frac{dg_{tt}(u)}{du}\right)-D(u)\left(\frac{dg_{rr}(u)}{du}\right)+E(u)\left(\frac{dg_{\phi\phi}(u)}{du}\right)-F(u)=0\,\,,$
(23)

where $u=\frac{1}{r}$ and,

$D(u)=\left[\frac{g_{\phi\phi}^{2}(u)u^{4}}{2g_{tt}(u)g_{rr}^{2}(u)b^{2}}-\frac{g_{\phi\phi}(u)u^{4}}{2g_{rr}^{2}(u)}\right],$
(24)
$E(u)=\left[\frac{u^{4}}{2g_{rr}(u)}-\frac{g_{\phi\phi}(u)u^{4}}{g_{tt}(u)g_{rr}(u)b^{2}}\right]\,\,,$
(25)

and

$F(u)=\left[\frac{2g_{\phi\phi}^{2}(u)u^{3}}{g_{tt}(u)g_{rr}(u)b^{2}}-\frac{2g_{\phi\phi}(u)u^{3}}{g_{rr}(u)}\right].$
(26)

The above equation (23) represents the light ray orbit equation in the general
spherically symmetric and static spacetime which can be solved numerically for
a given spacetime.

### 9 Observable intensity

It is widely studied that the shadows of compact objects can be formed due to
a photon sphere. The photon sphere corresponds to the unstable circular light
ray orbits. In addition to the existence of a photon sphere, one more
condition has to be satisfied: an effective potential of the spacetime for the
null geodesic should have an upper bound at $r=r_{ph}$. The behavior of the
light rays near the compact object can be understood by the condition of
impact parameter $b$. From the equation (19), one can write the impact
parameter corresponding to the turning point of the light ray,

$b_{tp}=\frac{r_{tp}}{\sqrt{g_{tt}(r_{tp})}}.$ (27)

where $dr/d\lambda=0$ at the turning point and $r_{tp}$ is the radius of the
turning point from the compact object. Note that, for null geodesics, if only
one extremum value of effective potential exists and that extremum value is
the unstable photon orbit or maximum value of potential, then the minimum
impact parameter at the turning point of null geodesics is,

$b_{tp}=b_{ph},$ (28)

where $b_{ph}$ is the impact parameter corresponding to the photon sphere. The
light rays coming from the background source with the impact parameter
$b<b_{ph}$ could not reach an asymptotic observer, they would be trapped
inside the photon sphere. Hence, the shadow with radius $b_{ph}$ would form in
the observer’s sky. There are numbers of literature where authors show that
the shadow can form without a photon sphere in particular types of geometries
null , Dey:2020haf , Dey:2020bgo . However, if the photon sphere is absent in
a given spacetime and the effective potential diverges near the center, then
the shadow will not occur in that spacetime shaikh1 .

Now, it is crucial to visualize the shadow cast by any compact object in a
physical situation. This can be done by calculating the intensity distribution
of the light rays coming from the accretion matter around the compact object.
The radially freely falling thin accretion matter radiates monochromatic
light, where the emission rate of that light per unit volume describe as,

$j(\nu_{e})\propto\frac{\delta(\nu_{e}-\nu_{*})}{r^{2}}\,\,,$ (29)

where $\nu_{e}$ is the emitted frequency of a photon as measured in the rest
frame of the emitter. Note that, for simplicity, in this section, the case in
which the thin accretion matter is in spherically symmetric form is
considered. The observed intensity of the light coming from the accretion
matter could be obtained in the asymptotic observer’s sky with the coordinates
$X,Y$ as,

$I_{\nu_{o}}(X,Y)=\int_{\gamma\prime}g^{3}j(\nu_{e})dl_{prop}\,\,,$ (30)

where the infinitesimal proper length ($dl_{prop}$) of the emitter in the rest
frame can be defined as,

$dl_{prop}=-k_{\alpha}u^{\alpha}_{e}d\lambda,$ (31)

where $k_{\alpha}$ is the null four velocities of photons emitted by emitter
and $u^{\alpha}_{e}$ is the time-like 4-velocity of emitter. $\lambda$ is the
affine parameter and $g=\nu_{o}/\nu_{e},$ is the gravitational red-shift
factor defined by,

$g=\frac{k_{\alpha}u^{\alpha}_{o}}{k_{\beta}u^{\beta}_{e}}.$ (32)

The subscripts $0$ and $e$ represent the observed and emitted frequencies,
respectively. Note that in the above equation (30), the integration is carried
out along the null trajectory $\gamma\prime$. One can write the four-velocity
components of the radially freely falling accretion matter for general
spherically symmetric and static spacetime as,

$u^{t}_{e}=\frac{1}{g_{tt}(r)},\;\;\;\;\;\;u^{r}_{e}=-\sqrt{\frac{(1-g_{tt}(r))}{g_{tt}(r)g_{rr}(r)}},\;\;\;\;\;\;u^{\theta}_{e}=u^{\phi}_{e}=0,$
(33)

using these four velocity components, one can define the gravitational red-
shift factor as,

$g=\frac{1}{\frac{1}{g_{tt}(r)}-\frac{k_{r}}{k_{t}}\sqrt{\frac{(1-g_{tt}(r))}{g_{tt}(r)g_{rr}(r)}}}\,\,,$
(34)

where

$\frac{k^{r}}{k^{t}}=\sqrt{\frac{g_{tt}(r)}{g_{rr}(r)}\left(1-\frac{g_{tt}(r)b^{2}}{r^{2}}\right)}\,\,.$
(35)

Now, the intensity distribution in the plane of observer’s sky $(X,Y)$ can be
obtained by integrating equation (30) for all observed frequencies as shaikh1
,

$I_{o}(X,Y)\propto-\int_{\gamma\prime}\frac{g^{3}k_{t}dr}{k^{r}r^{2}}\,\,,$
(36)

where $X^{2}+Y^{2}=b^{2}$. The above equation (36) represents the observed
intensity of the light coming from the thin accretion matter around the
compact object. Which can be used to simulate the shadow of a compact object
in the observer’s sky. This equation can also show how the intensity varies
with the impact parameter $b$.

### 10 Radiative Modelling

In this section, an analytic model of emissivity and absorptivity coefficients
are considered for radiative transfer calculations. In what follows, the
prescription for radiative transfer is described Gold:2020iql . The BH mass
(M) for Sgr A* to be $4.3$ x $10^{6}M_{\odot}$ and source distance (D) from
the Earth is $8200pc$ Do  are adopted for radiative transfer calculations. The
covariant form of the general relativistic radiative transfer is expressed as

$\frac{d\mathcal{I}}{d\tau_{\nu}}=-\mathcal{I}+\frac{{\eta}}{\chi},$ (37)

where $\mathcal{I}$ is the Lorentz-invariant intensity and is related to the
specific intensity via $\mathcal{I}=I_{\nu}/\nu^{3}=I_{\nu_{o}}/\nu_{0}^{3}$
where the subscript ‘o’ denotes quantities in the local rest frame.
$\tau_{\nu}$ is defined as the optical depth. $\chi$ and $\eta$ are the
invariant absorptions and emission coefficients at frequency $\nu$. The number
density of the fluid is given by

$\displaystyle
N=n_{0}\exp{\left\\{-0.5\left[\left(\frac{r}{10}\right)^{2}+z^{2}\right]\right\\}},$
(38)

where $z=h\cos\theta$. $n_{0}$ is the reference number density, and $h$ is the
vertical scale height. The Keplerian angular momentum profile
$\Omega=1/r^{3/2}$ is used and the fluid four-velocity is thus given by

$u_{\mu}=\bar{u}(-1,0,0,\Omega),$ (39)

with $\bar{u}=\sqrt{-(g^{tt}+g^{\phi\phi}\Omega^{2}-2g^{t\phi}\Omega)}$ such
that $u_{\mu}u^{\mu}=-1$. The specific emissivity and absorption coefficients
are given by

$\displaystyle j_{\nu}$ $\displaystyle=$
$\displaystyle\mathcal{C}N\left(\frac{v}{v_{obs}}\right)^{-\alpha},$ (40)
$\displaystyle\alpha_{\nu}$ $\displaystyle=$
$\displaystyle\mathcal{A}\mathcal{C}N\left(\frac{v}{v_{obs}}\right)^{-(\alpha+\beta)},$
(41)

where $\mathcal{C}$ and $\mathcal{A}$ are constants controlling the effect of
absorption and emission coefficients. The Lorentz-invariant coefficients are
then

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle\frac{j_{\nu}}{\nu^{2}},$
(42) $\displaystyle\chi$ $\displaystyle=$ $\displaystyle\nu\alpha_{\nu}.$ (43)

Table (1) shows the values of different parameters for the accretion and the
models of naked singularities. Note that JMN-1 spacetime without a photon
sphere is defined as model-I, JMN-1 spacetime with photon sphere is defined as
model II, and JNW spacetime is defined as model III.

Table 1: Model details Parameter | Model-I | Model-II | Model-III | Sch. BH
---|---|---|---|---
$M(M_{\odot})$ |  $4.3$ x $10^{6}$ |  $4.3$ x $10^{6}$ |  $4.3$ x $10^{6}$ |  $4.3$ x $10^{6}$
$M_{0}$ | 0.63 | 0.70 | - | -
$R_{b}$ | 2.857 | 3.175 | - | -
$\gamma$ | - | - | 0.51 | -
$\mathcal{A}$ | 100 | 100 | 100 | 100
$\mathcal{C}n_{0}$ (x$10^{-17}$) | 2.5 | 2.0 | $10^{-3}$ | 2.0
$\alpha$ | -2 | -2 | -2 | -2
$\beta$ | 2.5 | 2.5 | 2.5 | 2.5
$h$ | 10/3 | 10/3 | 10/3 | 10/3

### 11 Ray-Tracing Formalism

In order to calculate the shadow image of a black hole or naked singularity,
one must first solve the geodesic equations in the background spacetime under
consideration. The approach for ray-traced images of the models described in
the previous section is considered. A system of six differential equations
($\dot{t},\dot{r},\dot{\theta},\dot{\phi},\dot{p_{r}},\dot{p_{\theta}})$ is
solved.

#### 11.1Geodesic equations of motion

For a given metric $g_{\alpha\beta}$, the Lagrangian can be written as

$2\mathcal{L}=g_{\alpha\beta}\,\dot{x}^{\alpha}\dot{x}^{\beta},$ (44)

where an overdot denotes differentiation with respect to the affine parameter,
$\lambda$. From the Lagrangian, the covariant four-momenta of geodesics are:

$\displaystyle
p_{\alpha}=\frac{\partial\mathcal{L}}{\partial\dot{x}^{\alpha}}.$ (45)

Using the conservation of energy and angular momentum, this can be broken down
to write $p_{t}=-E$ and $p_{\phi}=L$ where $E$ is the particle’s total energy
and $L$ is the angular momentum in the direction of $\phi$. The final set of
geodesic equations that will be solved is then given by

$\displaystyle\dot{t}$ $\displaystyle=$ $\displaystyle\frac{E}{g_{tt}}$ (46)
$\displaystyle\dot{r}$ $\displaystyle=$ $\displaystyle\frac{p_{r}}{g_{rr}}$
(47) $\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\frac{p_{\theta}}{g_{\theta\theta}}$ (48)
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{L}{g_{\phi\phi}}$ (50) $\displaystyle\dot{p_{r}}$
$\displaystyle=$
$\displaystyle\frac{1}{2g_{tt}}\left[-{p_{r}}^{2}\left(\frac{g_{tt}}{g_{rr}}\right)^{\prime}-(Q+L^{2})\left(\frac{g_{tt}}{g_{rr}}\right)^{\prime}\right]$
(51) $\displaystyle\dot{p_{\theta}}$ $\displaystyle=$
$\displaystyle\frac{L^{2}}{g_{\theta\theta}}\frac{\cos{\theta}}{{\sin^{3}{\theta}}}$
(52)

where $(^{\prime})$ denotes differentiation with respect to $r$ and $Q$ is
Carter’s constant, which is the third constant of motion.

#### 11.2Initial Conditions

The initial conditions which are required to solve the system of an ordinary
differential equation (ODE) are described in the previous section. In
literature, different authors have assumed different initial conditions based
on the type of work and ODEs which are needed to be solved. The formalism
described in Younsi:2016azx  is used. An observer is at some distance from the
source. The observer is placed far away from the Black hole
($r_{\mathrm{obs}}=10^{3}\,M$), where the spacetime is assumed to be flat
(asymptotic flatness). The observer’s position is specified in Boyer-Lindquist
(oblate spheroidal) coordinates as
$(r_{\mathrm{obs}},\theta_{\mathrm{obs}},\phi_{\mathrm{obs}})$ where
$\theta_{obs}$ and $\phi_{obs}$ are the inclination and azimuthal angles of
the observer. The transformation from Cartesian coordinates to Boyer-Lindquist
coordinates is given by

$\displaystyle\ \\!r^{2}$ $\displaystyle=$
$\displaystyle\sigma+\sqrt{\sigma^{2}+a^{2}Z^{2}}\,,$ (53)
$\displaystyle\cos\theta$ $\displaystyle=$ $\displaystyle Z/r\,,$ (54)
$\displaystyle\tan\phi$ $\displaystyle=$ $\displaystyle Y/X\,,$ (55)

where

$\displaystyle X$ $\displaystyle\equiv$
$\displaystyle\mathcal{D}\cos\phi_{\mathrm{obs}}-x\sin\phi_{\mathrm{obs}}\,,$
(56) $\displaystyle Y$ $\displaystyle\equiv$
$\displaystyle\mathcal{D}\sin\phi_{\mathrm{obs}}+x\cos\phi_{\mathrm{obs}}\,,$
(57) $\displaystyle Z$ $\displaystyle\equiv$ $\displaystyle
r_{\mathrm{obs}}\cos\theta_{\mathrm{obs}}+y\sin\theta_{\mathrm{obs}}\,,$ (58)

and

$\displaystyle\sigma$ $\displaystyle\equiv$
$\displaystyle\left(X^{2}+Y^{2}+Z^{2}-a^{2}\right)/2\,,$ (59)
$\displaystyle\mathcal{D}$ $\displaystyle\equiv$
$\displaystyle\sin\theta_{\mathrm{obs}}\sqrt{r_{\mathrm{obs}}^{2}+a^{2}}-y\cos\theta_{\mathrm{obs}}\,.$
(60)

Finally, to obtain the ray’s velocity components in Boyer- Lindquist
coordinates one has to differentiate Eqs.(53)–(55), yielding:

$\displaystyle-\Sigma\ \dot{x}^{r}$ $\displaystyle=$ $\displaystyle
r\mathcal{R}\sin\theta\sin\theta_{\mathrm{obs}}\cos\Phi+\mathcal{R}^{2}\cos\theta\cos\theta_{\mathrm{obs}}\,,$
(61) $\displaystyle-\Sigma\ \dot{x}^{\theta}$ $\displaystyle=$
$\displaystyle\mathcal{R}\cos\theta\sin\theta_{\mathrm{obs}}\cos\Phi-r\sin\theta\cos\theta_{\mathrm{obs}}\,,$
(62) $\displaystyle\mathcal{R}\ \dot{x}^{\phi}$ $\displaystyle=$
$\displaystyle\sin\theta_{\mathrm{obs}}\sin\Phi\ \\!\mathrm{cosec}\theta\,,$
(63)

where

$\displaystyle\Sigma$ $\displaystyle\equiv$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta\,,$ (64) $\displaystyle\mathcal{R}$
$\displaystyle\equiv$ $\displaystyle\sqrt{r^{2}+a^{2}}\,,$ (65)
$\displaystyle\Phi$ $\displaystyle\equiv$
$\displaystyle\phi-\phi_{\mathrm{obs}}\,.$ (66)

Using the above formalism, the images for various models described in the
previous section and parameter values in Table 1 can be generated. Different
columns in Figure 4 depict these ray-traced models. Contrary to Model II and
Schwarzschild BH, Model-I produces a particular image with multiple Einstein
rings without casting a shadow. The authors previously produced a similar
full-moon type image in shaikh1 . On the other hand, Model III casts a
relatively smaller shadow than Model-II and Schwarzschild BH.

Fig. 4: Model images for Sgr A*, parameters details are given in the Table
(4).

### 12 EHT VLBI Observations

Continuing the analysis of the previous section, the next step of visualizing
these images with VLBI is applied. For this purpose, Three different array
configurations are utilised, including current and future observations. The
label EHT-I, EHT-II and EHT-III for 2017, 2021 and 2025 observations are
given, respectively, as mentioned in Table (2). It should be noted that EHT-
III is one of the possible configurations for future (2025) observations.
Meanwhile, EHT-II completed the observations for 2021 in early April. The
images produced in the preceding section are infinite-resolution in that they
are produced by a pristine source that encounters no interstellar matter on
its way to a perfect detector that can detect every photon that passes through
it. In reality, interstellar scattering will occur, and the baselines will
only cover a portion of the observing region.

Table 2: Locations of Existing/Future Sites in the Event Horizon Telescope
Array.

EHT-I | Latitude | Longitude | SEFD.†
---|---|---|---
| ($\deg$) | ($\deg$) | (Jy)
ALMA | -23.03 | -67.75 | 90
APEX | -23.01 | -67.76 | 3500
JCMT | 19.82 | -155.48 | 6000
LMT | 18.98 | -97.31 | 600
IRAM⋆ | 36.88 | -3.39 | 1400
SMA | 19.82 | -155.48 | 4900
SMT | 32.70 | -109.89 | 5000
SPT | -90.00 | 45.00 | 5000
EHT-II, (Additional) | Latitude | Longitude | SEFD.†
| ($\deg$) | ($\deg$) | (Jy)
GLT | 76.54 | -68.69 | 10000
KP | 31.96 | -111.61 | 10000
NOEMA | 44.63 | 5.91 | 700
EHT-III, (Additional) | Latitude | Longitude | SEFD.†
| ($\deg$) | ($\deg$) | (Jy)
BAJA | 30.87 | -115.46 | 10000
BOL | -16.25 | -68.13 | 10000
CARMA | 37.1 | -118.14 | 10000
DRAK | -29.3 | 29.27 | 10000
GAM | 23.25 | 16.17 | 10000
HAY | 42.43 | -71.49 | 2500
KAUAI | 21.79 | -159.51 | 10000
KEN | -0.15 | 37.31 | 10000
PDB | 44.44 | 5.91 | 1500
PIKES | 38.65 | -105.04 | 10000
VLT | -24.48 | -70.4 | 10000

$\star$ sites used for the observation only in EHT-I, III.
†System Equivalent Flux Density

Fig. 5: Baseline coverage of Sgr A⋆ for the array configuration of EHT
mentioned in Table 2. Fig. 6: Reconstructed images of Sgr A* for models
described in section-I with different array configurations of EHT (top to
bottom: EHT-I, EHT-II, EHT-III).

The synthetic radio images are generated with ehtim Chael:2018oym  using the
array configurations mentioned in Table (2). The following parameters were
used in the simulations: $\Delta\nu=4$ GHz bandwidth, $t=24$ hours,
corresponding to a full day, at a central frequency of $230$ GHz. The total
observation time is one of the most important parameters in the imaging
process, and this unusually long observation time allows to present the ideal-
case scenario. The (u, v) coverage grows larger as the observation time
increases, and reconstruct a better image. The Maximum Entropy Method (MEM) is
used to reconstruct static images from synthetic VLBI data. Following this
procedure, the synthetic observations at the Galactic center are performed.
The visibility amplitudes are calculated by Fourier transforming the images
and sampling them over the projected baselines of different arrays (EHT-I,
EHT-II and EHT-III). The effects of thermal noise and phase errors are
included to mimic realistic observations during the simulations. As it can be
seen in Fig. (5), the (u,v) coverage is subsequently increasing as more
antennas are added to the configuration. Images reconstructed for different
models and arrays are presented in Fig. (6). It should also be noted that the
same imaging script was used for all the reconstructions used here. In the
next sections, a qualitative and quantitative analysis of the results produced
in this section would discuss.

#### 12.1Qualitative analysis

Overall visual inspection of the images clearly shows the difference in the
images of different models with different arrays. Starting from the left-hand
side, as discussed in the previous section Model-I produces an interesting
concentric rings type image. So the corresponding synthetic image for
different arrays does not give a shadow-type structure. Due to the spherically
symmetric nature of the model, the image in the third row for EHT-III shows a
nearly circular structure. Those multiple rings are not visible in the
reconstruction. This aberration is due to the resolution limit, making it very
difficult to resolve them.

Model II casts a shadow and is clearly visible in all the arrays. Due to
additional emissions in the line of sight, the shadow structure appears to be
divided. Although, it should be noted that it is a consequence of the
radiative transfer modelling and raytracing that can be seen such emission
along with the particular inclination. There are differences between Model-I
and Model II in the image structure. Comparing Model-II with Schwarzschild BH
is evident that significant difference does not appear due to the same shadow
size and similar emission regions. Hence, it can be stated that Model II and
Schwarzschild BH mimic each other, and one cannot tell the difference between
them from the VLBI images.

For Model III, the shadow size is smaller than the other models prescribed
here, which can be seen from the ray-traced images. So, in the EHT-I image,
the shadow region is not as noticeable. The same may be said for EHT-II. The
situation improves substantially with EHT-III, and the shadow region becomes
more apparent. Model III appears to be distinct from the Black Hole case. Even
so, the change is not all that substantial. This coincides with the previous
case as well for Model II. On a relative scale, Model III is more distinct
than Model II.

Table 3: Normalized Cross-Correlation coefficient for the comparison between the reconstructed of various models with the Schwarzschild BH. Sch. BH | Model-I | Model-II | Model-III
---|---|---|---
EHT-I | 0.88 | 0.96 | 0.74
EHT-II | 0.90 | 0.98 | 0.73
EHT-III | 0.92 | 0.99 | 0.75

#### 12.2Quantitative Analysis

Image-comparison measures, such as the Normalized Cross-correlation
coefficient (NCC), structural dissimilarity index etc., can be used to make a
more quantitative evaluation of the degree of similarity among the various
images under consideration. For this purpose, NCC has been utilised to compare
the resultant images. If two images are similar to each other, the maximum
value NCC can have 1, and the maximum dissimilarity will give NCC value to be
around 0. This can be incorporated with three different arrays: EHT-I, EHT-II
and EHT-III. NCC is then computed between the corresponding reconstructed
Schwarzschild BH image and the other models, as seen in Figure 6. Lower NCC
values suggest that these models can be differentiated from the Schwarzschild
BH case. Given that, Model-I and Model-III produce distinct features (as
explained in the previous section), it is expected to deduce a similar result
from NCC. Table 3 summaries these results. Model-I and Model III give
relatively lower values than Model II, suggesting that the images are
distinguishable. Model-I having no shadow, and Model-III having a smaller
shadow size as shown in Figure 4, are responsible for various structures in
the reconstructed images resulting in relatively lower values of NCC than
Model-II, which has the same shadow size as Schwarzschild BH.

### 13 Discussions and Conclusions

In this chapter, Sgr A* as the JMN-1 or JNW naked singularity has been
modelled with the possibility of having either a shadow-like structure or a
pristine source (concentric ring-like structure) emitting from all the
regions. The raytracing algorithm is applied to solve the geodesic equations
and then solve the general relativistic radiative transfer equation. An
analytic model for emission and absorption coefficients is utilised for
generating emission maps. Afterwards, the synthetic observations have been
produced for the prescribed models of the source considering three different
array configurations, namely EHT-I, EHT-II and EHT-III and reconstruct the
images.

From the overall analysis, it is seen that the naked singularity models are
distinguishable from a Black hole with VLBI imaging. Both qualitative and
quantitative analyses of the images provide a rough estimate of the
differences between the models. Model-I and Model-II exhibit different
accretion structures and imaging artifacts, which can be seen in the image
reconstructions. Meanwhile, Model II remains a Black hole mimicker and is
almost similar to a Black hole in size and emission region. The NCC quantifies
this inference and has values near $1$ for only Model II. However, for other
models, the values are relatively smaller, differentiating the models
quantitatively. The NCC can be used to explore these potential differences to
some extent.

The synthetic VLBI images for the EHT-I array for Model-II, Model-III and
Schwarzschild BH give shadow-like features. However, on the other hand,
Model-I does not hint at that since it does not theoretically produce a
shadow. As observed further the number of arrays increase in the
configuration, the Fourier domain fills up significantly, generating a clear
image. So, when the models are compared based on increased arrays, they retain
their original state in shadow.

The above analysis concludes that when Sgr A* is observed on these baselines
and an image is reconstructed, if the image showcase a shadow-like structure,
one cannot distinguish between a Black hole and a naked singularity. Moreover,
if Sgr A* does not contain a BH but a naked singularity (Model-I), then one
can expect an image without shadow as seen in Fig (6). No matter how much the
baseline is increased or the number of arrays are added, there will not be any
image having a shadow feature. Indeed, the absence of the shadow in the JMN-1
spacetime introduces essential and fundamental differences in the flow
dynamics and images. It can act as a possible observable to test the presence
of a naked singularity. This can thus allow to probe the nature of Sgr A* with
shadows and VLBI imaging.

Although the analysis presented in this work is based on the images, there are
a few caveats that require specific attention. Firstly, the radiative
modelling used for the accretion region is a simple analytical model. The
number density can be considered analogous to the electron number density in
RIAF modelling. RIAF spatial distributions of the electron temperature $T_{e}$
and thermal electron density $n_{e}$ are usually described by a hybrid
combination Pu:2018ute . Different eDF’s (electron distribution functions) are
also incorporated in the emission and absorption coefficients to model the
emission regions more accurately Pandya:2016qfh . The radiative cooling time-
frame for RIAF is significantly longer than the accretion timescale for a
cold, geometrically thin, Keplerian revolving disk, resulting in a hot,
geometrically thick flow with a sub-Keplerian rotation Narayan:1994is ,
Narayan:1996wu . Introducing different flow dynamics can produce some
interesting and distinguishable features in the resulting images Pu:2016qak
as well. To conclude, these following caveats would be interesting topics to
follow up the current work in the future. Additionally, they would provide a
more robust way to probe the nature of Sgr A* for several observable. Recent
results from the 2017 and 2021 observational campaigns of EHT may provide more
insight into the argument. Concerning the general relativity (and not the
other modified gravity theories), these naked singularity solutions present an
excellent alternative to the conventionally used Black hole. Since, rotation
is pervasive in astrophysical systems, the rotating solution of the compact
body is believed to be the most realistic for studying any physical process
that occurs in the vicinity of a black hole. Therefore, the next chapter,
would cover the shadow properties in the rotating compact objects.

## CHAPTER 3 Shadows cast by rotating black holes and naked singularities

The inclusion of rotation in the non-rotating spacetime geometry makes it
physically more realistic spacetime. In 1965, Newman and Janis showed that the
Kerr metric could be obtained from the Schwarzschild metric using a complex
coordinate transformation Newman:1965tw . Using the same method for the static
Reissner-Nordström metric, one can derive the Kerr-Newman metric, which
represents a spacetime geometry of the electrically charged and rotating black
hole Newman:1965my . The application of the NJA for generating interior
solutions which match smoothly to the external Kerr metric is studied in
Drake1997 . In this algorithm, authors have used two different types of
complexification with no apparent reason. There are many ways to complexify
the coordinates. However, in the literature, as mentioned above, the only
reason given for choosing those particular complexifications is that it is
successful in generating the Kerr and Kerr-Newman metric. However, the success
of the NJA is limited to the spacetimes which at least satisfy the reciprocal
condition, $g_{tt}g_{rr}=-1$. Later, it was proved by S.P. Drake, and P.
Szekeres that the only perfect fluid solution generated by the NJA is the Kerr
metric, and the only Petrov typed D solution to the Einstein-Maxwell equation
is the Kerr-Newman metric Drake2000 .

In this chapter, the rotating JNW naked singularity spacetime is constructed
using NJA. The NJA is analyzed with and without complexification methods and
find that the energy conditions do satisfied when the complexification step is
skipped. Then the shadows cast by rotating JNW naked singularity is studied
and compared with those cast by the Kerr black hole. The shadow of the
deformed Kerr metric is also studied and discussed.

### 14 Construction of the rotating JNW spacetime

In this section, the rotating JNW naked singularity spacetime is determined
using the NJA. The final result of the NJA is obtained in the null
coordinates, which must be transformed into the Boyer-Lindquist coordinates
(BLC). The BLCs are more convenient to work with the rotating spacetime
metric. It is analysed that it is not possible to perform the BLC
transformation when considering the JNW spacetime metric (67) as a seed
metric. The complexification step in NJA Mustapha3  is dropped to resolve this
problem.

#### 14.1JNW metric and the Newman-Janis Algorithm

The JNW spacetime metric is given by,

$\displaystyle
ds^{2}=-\left(1-\frac{2M}{r\nu}\right)^{\nu}dt^{2}+\left(1-\frac{2M}{r\nu}\right)^{-\nu}dr^{2}+r^{2}\left(1-\frac{2M}{r\nu}\right)^{1-\nu}d\Omega^{2}$
(67)

where, $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. It is
straightforward to show that the following expression is the final form of
NJA, considering the JNW metric (67) as a seed metric.

$\displaystyle
ds^{2}=-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}du^{2}-2dudr-2a\sin^{2}\theta\left[1-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]dud\Phi$
$\displaystyle+\,\rho^{2}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{1-\nu}d\Omega^{2}+2a\sin^{2}\theta
drd\Phi+a^{2}\sin^{4}\theta\left[2-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]d\Phi^{2}$
(68)

The above form of the spacetime metric seems rather complicated. It can be
made simpler and symmetrical by transforming it into the Boyer-Lindquist
coordinates (BLC). The BLC $(t,r,\theta,\phi)$ represents the rotating black
hole spacetime metrics because all the off-diagonal terms of the metric,
except $dtd\Phi$, vanish; and its axial symmetry becomes apparent. The
transformation into the BLC requires the following:

$\displaystyle du=dt-\xi(r)dr,$ (69) $\displaystyle d\Phi=d\phi-\chi(r)dr.$
(70)

It is noted that the transformation functions, i.e., $\xi(r)$ and $\chi(r)$,
strictly depend on coordinate $r$ only; otherwise eq. (69) and (70) will not
be integrable. To write eq. (68) into the BLC like form, the transformation
functions must take the following form,

$\displaystyle\xi(r,\theta)=\frac{1}{\Delta}\left[\rho^{2}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{1-\nu}+a^{2}\sin^{2}\theta\right],$
(71) $\displaystyle\chi(r)=\frac{a}{\Delta},$ (72)
$\displaystyle\Delta=r^{2}-\frac{2Mr}{\nu}+a^{2}.$ (73)

It can be seen that the transformation function $\xi$ also depends on
coordinate $\theta$, and therefore the eq. (69) is not integrable. In these
coordinates, eq. (68) is written as

$\displaystyle
ds^{2}=-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}dt^{2}+\rho^{2}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{1-\nu}\left(\frac{dr^{2}}{\Delta}+d\Omega^{2}\right)$
$\displaystyle-\,2a\sin^{2}\theta\left[1-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]dtd\phi+a^{2}\sin^{4}\theta\left[2-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]d\phi^{2}.$
(74)

It can be seen that the above metric reduces to the JNW spacetime metric (67)
when $a=0$. However, the transformation into the BLC was improper in obtaining
eq. (74), one can check the energy conditions because the NJA is just an
algorithm that successfully works for Kerr and Kerr-Newman metric. Thus, it
cannot be commented whether the final result eq. (74) is a valid solution of
the Einstein field equation. It can only be answered once it is known whether
or not the energy conditions for the metric (74) are satisfied. The energy
conditions for the above spacetime metric are discussed in Appendix A. It can
be concluded that the rotating JNW spacetime metric (74) was obtained using
NJA with complexification violates the energy conditions. Now, the $\theta$
dependency of the transformation function $\xi$ is dropped by considering a
slow rotation approximation in which the second and higher-order terms of
rotation parameter ($a$) are ignored. Thus, eq. (74) reduces to the following
form,

$\displaystyle ds^{2}=-\left(1-\frac{2M}{\nu
r}\right)^{\nu}dt^{2}+\left(1-\frac{2M}{\nu
r}\right)^{-\nu}dr^{2}-2a\sin^{2}\theta\left[1-\left(1-\frac{2M}{\nu
r}\right)^{\nu}\right]dtd\phi$ $\displaystyle+\,r^{2}\left(1-\frac{2M}{\nu
r}\right)^{1-\nu}d\Omega^{2}$ (75)

And, the transformation functions $\chi$ and $\xi$ become

$\displaystyle\xi(r)=\left(1-\frac{2M}{\nu r}\right)^{-\nu},$ (76)
$\displaystyle\chi(r)=\frac{a}{r^{2}}\left(1-\frac{2M}{\nu r}\right)^{-1}.$
(77)

Eq. (75) is the slow rotation form of the JNW spacetime metric. The rotating
JNW spacetime metric (74) obtained using the NJA with complexification is not
properly transformed into the BLC because of the complexification of the
coordinates. This issue can be solved by skipping the complexification step,
which is discussed in the next section.

#### 14.2Newman-Janis Algorithm without Complexification

Consider a general form of the static and spherically symmetric spacetime
metric as,

$ds^{2}=-G(r)dt^{2}+\frac{dr^{2}}{F(r)}+H(r)d\Omega^{2}.$ (78)

To apply the algorithm, transform it into the null coordinates
$\\{u,r,\theta,\phi\\}$ using the coordinate transformation,

$du=dt-\frac{dr}{\sqrt{F(r)G(r)}}$ (79)

In null coordinates, the spacetime metric (78) takes the form,

$ds^{2}=-G(r)du^{2}-2\sqrt{\frac{G(r)}{F(r)}}dudr+H(r)d\Omega^{2}.$ (80)

The null tetrads of the above spacetime metric are,

$\displaystyle l^{\mu}=\delta^{\mu}_{1}\,,$ (81) $\displaystyle
n^{\mu}=\sqrt{\frac{F(r)}{G(r)}}\delta^{\mu}_{0}-\frac{1}{2}F(r)\delta^{\mu}_{1}\,,$
(82) $\displaystyle
m^{\mu}=\frac{1}{\sqrt{2H(r)}}\left(\delta^{\mu}_{2}+\frac{i}{\sin\theta}\delta^{\mu}_{3}\right).$
(83)

The next step of the NJA is the complexification, which generalize the
function (i.e. $G(r),F(r),H(r)$) as the real function of radial coordinate $r$
and it’s complex conjugate $\bar{r}$. The trick is to skip this step and
proceed directly to the next step, which is the complex coordinate
transformation given as,

$\displaystyle r^{\prime}=r+ia\cos\theta,$ (84) $\displaystyle
u^{\prime}=u-ia\cos\theta.$ (85)

Under the coordinate transformation, by considering that the components of the
metric tensor (78) are transformed as

$\\{G(r),F(r),H(r)\\}\to\\{A(r,\theta,a),B(r,\theta,a),\psi(r,\theta,a)\\},$

where the following conditions are satisfied.

$\displaystyle\lim\limits_{a\to 0}A(r,\theta,a)=G(r)\,,$ (86)
$\displaystyle\lim\limits_{a\to 0}B(r,\theta,a)=F(r)\,,$ (87)
$\displaystyle\lim\limits_{a\to 0}\psi(r,\theta,a)=H(r).$ (88)

Now, perform the complex coordinate transformation (84), (85) on the null
tetrad given in equations (81), (82), (83),

$\displaystyle l^{\mu}=\delta^{\mu}_{1}\,,$ (89) $\displaystyle
n^{\mu}=\sqrt{\frac{B}{A}}\delta^{\mu}_{0}-\frac{1}{2}B\delta^{\mu}_{1}\,,$
(90) $\displaystyle
m^{\mu}=\frac{1}{\sqrt{2\psi}}\left(\delta^{\mu}_{2}+ia\sin\theta(\delta^{\mu}_{0}-\delta^{\mu}_{1})+\frac{i}{\sin\theta}\delta^{\mu}_{3}\right).$
(91)

The fourth null vector $\bar{m}^{\mu}$ is just a complex conjugate of
$m^{\mu}$. These are the components of the null tetrad of the final metric
tensor in null coordinates $(u,r,\theta,\Phi)$.

$\displaystyle
ds^{2}=Adu^{2}+2\sqrt{\frac{A}{B}}dudr+2a\sin^{2}\theta\left(\sqrt{\frac{A}{B}}-A\right)dud\Phi-2a\sin^{2}\theta\sqrt{\frac{A}{B}}drd\Phi$
$\displaystyle-\,\psi
d\theta^{2}-\sin^{2}\theta\left[\psi+a^{2}\sin^{2}\theta\left(2\sqrt{\frac{A}{B}}-A\right)\right]d\Phi^{2}.\,\,\,\,\,\,\,\,\,\,\,$
(92)

The spacetime metric is determined in the null coordinates
$\\{u,r,\theta,\Phi\\}$, which can be transformed into the BLC
$\\{t,r,\theta,\phi\\}$ as,

$\displaystyle du=dt-\lambda(r)dr,$ (93) $\displaystyle
d\Phi=d\phi-\chi(r)dr.$ (94)

The transformation functions $\chi$ and $\lambda$ should strictly depend on
the radial coordinate ($r$) only for eq. (93), (94) to be integrable, which is
not possible in NJA with complexification. In the Boyer-Lindquist form, the
spacetime metric contains only one off-diagonal term, which is $dtd\phi$.
Therefore, using the above coordinate transformation (93) and (94), and
dropping all off-diagonal terms except $dtd\phi$, the following two
expressions are derived as,

$\displaystyle
A\lambda(r)-\sqrt{\frac{A}{B}}+a\sin^{2}\theta\left(\sqrt{\frac{A}{B}}-A\right)\chi(r)=0,$
(95) $\displaystyle
a\left(\sqrt{\frac{A}{B}}-A\right)\lambda(r)+a\sqrt{\frac{A}{B}}+\left[\psi-a^{2}\sin^{2}\theta\left(2\sqrt{\frac{A}{B}}-A\right)\chi(r)\right]=0.$
(96)

Solving these expressions for $\lambda$ and $\chi$,

$\displaystyle\chi(r,\theta)=\frac{a}{\psi B+a^{2}\sin^{2}\theta}\,,$ (97)
$\displaystyle\lambda(r,\theta)=\frac{\psi\sqrt{\frac{B}{A}}+a^{2}\sin^{2}\theta}{\psi
B+a^{2}\sin^{2}\theta}.$ (98)

The BLC transformation functions $\chi$ and $\lambda$ depends on $\theta$, but
this time there are three unknown functions $A(r,\theta,a),B(r,\theta,a)$ and
$\psi(r,\theta,a)$ which play an important role in dropping the $\theta$
dependency of $\chi$ and $\lambda$. The following form of the transformation
functions is chosen because it successfully works in Kerr and Kerr-Newman
case. Therefore above equations reduces into,

$\displaystyle\chi(r)=\frac{a}{FH+a^{2}},$ (99)
$\displaystyle\lambda(r)=\frac{K+a^{2}}{FH+a^{2}},$ (100)

where, $K=H\sqrt{\frac{F}{G}}$. Substitute the above expression of $\chi$ and
$\lambda$ into the equations (95) and (96), and solve them for the unknown
functions $A(r,\theta,a)$ and $B(r,\theta,a)$,

$\displaystyle
A(r,\theta,a)=\frac{(FH+a^{2}\cos^{2}\theta)\psi}{(K+a^{2}\cos^{2}\theta)^{2}},$
(101) $\displaystyle B(r,\theta,a)=\frac{FH+a^{2}\cos^{2}\theta}{\psi}.$ (102)

The function can be verified by $A(r,\theta,a)$ and $B(r,\theta,a)$
respectively, reduce to the function $G(r)$ and $F(r)$ when $a\to 0$ as,

$\displaystyle\lim\limits_{a\to 0}A(r,\theta,a)=G(r),$ (103)
$\displaystyle\lim\limits_{a\to 0}B(r,\theta,a)=F(r).$ (104)

Therefore, under the BLC coordinate transformation, the metric (92) becomes,

$\displaystyle
ds^{2}=-\frac{(FH+a^{2}\cos^{2}\theta)\psi}{(K+a^{2}\cos^{2}\theta)^{2}}dt^{2}+\frac{\psi}{FH+a^{2}}dr^{2}-2a\sin^{2}\theta\left(\frac{K-FH}{(K+a^{2}\cos^{2}\theta)^{2}}\right)\psi
dtd\phi$ $\displaystyle+\,\psi d\theta^{2}+\psi
sin^{2}\theta\left[1+a^{2}\sin^{2}\theta\left(\frac{2K-FH+a^{2}\cos^{2}\theta}{(K+a^{2}\cos^{2}\theta)^{2}}\right)\right]d\phi^{2},$
(105)

where, $\psi(r,\theta,a)$ is still unknown. It must satisfy the following
constraint, which corresponds to the vanishing component of the Einstein
tensor $G_{r\theta}=0$,

$3a^{2}\sin{2\theta}\psi^{2}K_{,r}+(K+a^{2}\cos^{2}\theta)^{2}(3\psi_{,\theta}\psi_{,r}-2\psi\psi_{,r\theta})=0.$
(106)

This non-linear partial differential equation can be solved for $\psi$. One
can verify that a solution of the above partial differential equation is,

$\psi(r,\theta,a)=K(r)+a^{2}\cos^{2}\theta.$ (107)

For the Kerr and Kerr-Newman metric, $K(r)=r^{2}$, and the above expression
reduces to $\psi=r^{2}+a^{2}\cos^{2}\theta$. After substituting equation (107)
into the general form of the rotating spacetime metric (105), if the static
spacetime metric (78) can be obtained in the limit $a\to 0$, the rotating
solution is known as normal fluid. Otherwise, it is known as conformal fluid.
The normal fluid solution requires,

$\lim\limits_{a\to 0}\psi(r,\theta,a)=H(r),$ (108)

which implies $F(r)=G(r)$. The metric (105) can be written in Kerr-like form
as,

$\displaystyle
ds^{2}=-\left(1-\frac{2f}{\rho^{2}}\right)dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{\Sigma\sin^{2}\theta}{\rho^{2}}d\phi^{2}-\frac{4af\sin^{2}\theta}{\rho^{2}}dtd\phi,\,\,\,\,\,\,\,\,\,\,$
(109)

where,

$\displaystyle\rho^{2}=K(r)+a^{2}\cos^{2}\theta\,,$ (110) $\displaystyle
f=\frac{K(r)-F(r)H(r)}{2}\,,$ (111) $\displaystyle\Delta=F(r)H(r)+a^{2}\,,$
(112)
$\displaystyle\Sigma=\big{(}K(r)+a^{2}\big{)}^{2}-a^{2}\Delta\sin^{2}\theta.$
(113)

When $a\to 0$, the rotating solution (271) reduces to the static, and
spherically symmetric spacetime metric if $F(r)=G(r)$.

#### 14.3Rotating JNW metric

A new rotating spacetime metric is formed that reduces to the static JNW naked
singularity spacetime metric (67) in the limit $a\to 0$. The components of the
JNW spacetime metric is substituted into eq. (271) and the following
expression is obtained,

$\displaystyle
ds^{2}=-\left(1-\frac{2f}{\rho^{2}}\right)dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{\Sigma\sin^{2}\theta}{\rho^{2}}d\phi^{2}-\frac{4af\sin^{2}\theta}{\rho^{2}}dtd\phi,\,\,\,\,\,\,\,\,\,\,\,\,$
(114)

where,

$\displaystyle
2f=r^{2}\left(1-\frac{2M}{r\nu}\right)\left[-1+\left(1-\frac{2M}{r\nu}\right)^{-\nu}\right]\,,$
(115)
$\displaystyle\rho^{2}=r^{2}\left(1-\frac{2M}{r\nu}\right)^{1-\nu}+a^{2}\cos^{2}\theta\,,$
(116) $\displaystyle\Delta=r^{2}-\frac{2Mr}{\nu}+a^{2}\,,$ (117)
$\displaystyle\Sigma=(\rho^{2}+a^{2}\sin^{2}\theta)^{2}-a^{2}\Delta\sin^{2}\theta.$
(118)

One can verify that it reduces to the Kerr metric when $\nu=1$. Also, it
reduces to eq. (75) under the slow rotation approximation. To check whether
the spacetime metric is physically valid, one must proceed to check the energy
conditions. Consider the inverse form of the rotating spacetime metric (114),

$\displaystyle(\partial_{s})^{2}=-\frac{(K+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta}{\rho^{2}\Delta}(\partial_{t})^{2}+\frac{\Delta}{\rho^{2}}(\partial_{r})^{2}+\frac{1}{\rho^{2}}(\partial_{\theta})^{2}+\frac{\Delta-a^{2}\sin^{2}\theta}{\rho^{2}\Delta\sin^{2}\theta}(\partial_{\phi})^{2}$
$\displaystyle-\,\frac{4aMK}{r\rho^{2}\Delta}\partial_{t}\partial_{\phi}.\,\,\,$
(119)

It can also be written as,

$\displaystyle(\partial_{s})^{2}=-\left(\frac{K+a^{2}}{\rho\sqrt{\Delta}}\partial_{t}+\frac{a}{\rho\sqrt{\Delta}}\partial_{\phi}\right)^{2}+\frac{\Delta}{\rho^{2}}(\partial_{r})^{2}+\frac{1}{\rho^{2}}(\partial_{\theta})^{2}+\left(\frac{a\sin\theta}{\rho}\partial_{t}+\frac{1}{\rho\sin\theta}\partial_{\phi}\right)^{2}.\,\,\,$
(120)

Now, choose a set of orthonormal basis
$\\{u^{\mu},e_{r}^{\mu},e_{\theta}^{\mu},e_{\phi}^{\mu}\\}$, where $u^{\mu}$
can be considered as a four velocity of the fluid.

$\displaystyle
u^{\mu}=\left(\frac{K+a^{2}}{\rho\sqrt{\Delta}},0,0,\frac{a}{\rho\sqrt{\Delta}}\right)\,,$
(121) $\displaystyle
e_{r}^{\mu}=\left(0,\frac{\sqrt{\Delta}}{\rho},0,0\right)\,,$ (122)
$\displaystyle e_{\theta}^{\mu}=\left(0,0,\frac{1}{\rho},0\right)\,,$ (123)
$\displaystyle
e_{\phi}^{\mu}=\left(\frac{a\sin\theta}{\rho},0,0,\frac{1}{\rho\sin\theta}\right).$
(124)

One can verify that $u^{\mu}u_{\mu}=-1$, $e_{i}^{\mu}(e_{i})_{\mu}=1$,
$u^{\mu}(e_{i})_{\mu}=0$; where, $(i\to r,\theta,\phi)$. In terms of these
orthonormal bases, eq. (120) is written as,

$(\partial_{s})^{2}=(-u^{\mu}u^{\nu}+e_{r}^{\mu}e_{r}^{\nu}+e_{\theta}^{\mu}e_{\theta}^{\nu}+e_{\phi}^{\mu}e_{\phi}^{\nu})\partial_{\mu}\partial_{\nu},$
(125)

where the metric tensor is,

$g^{\mu\nu}=-u^{\mu}u^{\nu}+e_{r}^{\mu}e_{r}^{\nu}+e_{\theta}^{\mu}e_{\theta}^{\nu}+e_{\phi}^{\mu}e_{\phi}^{\nu}.$
(126)

Similarly, the energy-momentum tensor can be decomposed as,

$T^{\mu\nu}=\rho_{e}u^{\mu}u^{\nu}+P_{r}e_{r}^{\mu}e_{r}^{\nu}+P_{\theta}e_{\theta}^{\mu}e_{\theta}^{\nu}+P_{\phi}e_{\phi}^{\mu}e_{\phi}^{\nu},$
(127)

where, $\rho_{e}$ is the energy density, and $P_{i}$ $(i\to r,\theta,\phi)$
are the principal pressure components. The Einstein field equation in the form
$G_{\mu\nu}=T_{\mu\nu}$ requires,

$\displaystyle\rho_{e}=u^{\mu}u^{\nu}G_{\mu\nu}\,,$ (128) $\displaystyle
P_{r}=e_{r}^{\mu}e_{r}^{\nu}G_{\mu\nu}=g^{rr}G_{rr}\,,$ (129) $\displaystyle
P_{\theta}=e_{\theta}^{\mu}e_{\theta}^{\nu}G_{\mu\nu}=g^{\theta\theta}G_{\theta\theta}\,,$
(130) $\displaystyle P_{\phi}=e_{\phi}^{\mu}e_{\phi}^{\nu}G_{\mu\nu}.$ (131)

The components of the energy-momentum tensor of the rotating JNW spacetime are
as follows,

$\displaystyle\rho_{e}=\frac{M^{2}(1-\nu^{2})(\Delta+a^{2})}{r^{6}\nu^{2}}\left(1-\frac{2M}{r\nu}\right)^{-3+\nu},$
(132) $\displaystyle
P_{r}=-P_{\theta}=\frac{M^{2}(1-\nu^{2})}{r^{4}\nu^{2}}\left(1-\frac{2M}{r\nu}\right)^{-2+\nu},$
(133) $\displaystyle
P_{\phi}=P_{\theta}+\frac{2a^{2}}{r^{4}}\left(1-\frac{2M}{r\nu}\right)^{-3+\nu}\bigg{[}\left(1-\frac{2M}{r\nu}\right)^{1+\nu}-\left(1-\frac{M(1+\nu)}{r\nu}\right)^{2}\bigg{]}.\,$
(134)

It can be verified that the rotating JNW spacetime metric (114) satisfies the
weak energy condition ($\rho_{e}\geq 0$, $\rho_{e}+P_{i}\geq 0$), null energy
condition ($\rho_{e}+P_{i}\geq 0$), strong energy condition ($\rho_{e}+\sum
P_{i}\geq 0$, $\rho_{e}+P_{i}\geq 0$), and the dominant energy condition
($\rho_{e}>|P_{i}|$). It is noted that when $\nu=1$, all the components of
(132), (1), (134) vanish, which is evident as it is already seen that the
rotating JNW metric reduces to the Kerr metric when $\nu=1$. Also, $P_{r}\neq
P_{\theta}\neq P_{\phi}$, therefore, it is an anisotropic fluid solution.

### 15 The Kerr spacetime

Using the above NJA, one can get the stationary, axisymmetric and rotating
Kerr spacetime from the Schwarzschild spacetime in Boyer-Lindquist coordinates
as,

$\displaystyle
ds^{2}=-\left(1-\frac{r_{s}r}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}$ $\displaystyle+$
$\displaystyle\left(r^{2}+a^{2}+\frac{r_{s}ra^{2}\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta
d\phi^{2}$ (135) $\displaystyle-$
$\displaystyle\frac{2r_{s}ra\sin^{2}\theta}{\Sigma}dtd\phi\,\,,$

where $r_{s}=2M$ and

$\Sigma=r^{2}+a^{2}\cos^{2}\theta,$ (136) $\Delta=r^{2}+a^{2}-r_{s}r.$ (137)

The spin parameter $a$ in length dimension is defined corresponding to the
total angular momentum $J$ as $a=J/M$. Although, the dimensionless spin
parameter $\tilde{a}$ can be written as $\tilde{a}=J/M^{2}$, where $G=c=1$.
The horizons are determined by the relation $g_{rr}\rightarrow\infty$, which
implies that the solutions of $\Delta=0$ can give us the positions of the
horizons as,

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}\,\,.$ (138)

where $\pm$ indicates the event horizon and Cauchy horizon, respectively.
These horizons coincide at $r=M$ for $a=M$, which is known as an extreme Kerr
black hole. On the other hand, $a>M$ is considered, the horizons disappear,
and hence the Kerr black hole becomes a timelike Kerr naked singularity.

### 16 General formalism of the shadow shapes for rotating spacetimes

In this section, the general formalism of the null geodesics is reviewed using
the Hamilton-Jacobi separation method. The general expressions for finding the
shape and nature of a shadow is obtained. In general relativity, the Hamilton-
Jacobi equation is given by,

$\frac{\partial
S}{\partial\lambda}+\mathcal{H}=0,\;\mathcal{H}=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu},$
(139)

where $\lambda$ is an affine parameter, S is the Jacobi action, $\mathcal{H}$
is the Hamiltonian, and $p_{\mu}$ is the momentum defined by,

$p_{\mu}=\frac{\partial S}{\partial
x^{\mu}}=g_{\mu\nu}\frac{dx^{\nu}}{d\lambda}.$ (140)

In the above equation (139), the Hamiltonian does not depend explicitly on $t$
and $\phi$ coordinates. Therefore there are two constants of motion, the
conserved energy $E=-p_{t}$ and the conserved angular momentum $L=p_{\phi}$
(about the axis of symmetry). The separable solution of the above differential
equation is given (139), the Jacobi action can be written in terms of already
known constants of the motion as,

$S=\frac{1}{2}\mu^{2}\lambda-Et+L\phi+S_{r}(r)+S_{\theta}(\theta),$ (141)

where $\mu$ is the rest mass of the test particle. Therefore, for a photon or
null geodesic, $\mu=0$ is considered. The metric tensor components of general
rotating spacetime (105) are given as,

$g^{tt}=-\frac{H-a^{2}G\sin^{2}\theta+2a^{2}\sqrt{\frac{G}{F}}\sin^{2}\theta}{P},$
(142) $g^{rr}=\frac{FH+a^{2}\sin^{2}\theta}{H},$ (143)
$g^{\theta\theta}=\frac{1}{H},$ (144)
$g^{\phi\phi}=-\frac{G\csc^{4}\theta}{2a^{2}G^{2}-4a^{2}G\sqrt{\frac{G}{F}}+a^{2}\left(\frac{G}{F}\right)-GH\csc^{2}\theta},$
(145) $g^{t\phi}=g^{\phi t}=\frac{a\left(\sqrt{\frac{G}{F}}-G\right)}{P},$
(146)

where,

$P=-GH+2a^{2}G^{2}\sin^{2}\theta-4a^{2}G\sqrt{\frac{G}{F}}\sin^{2}\theta+a^{2}\left(\frac{G}{F}\right)\sin^{2}\theta.$
(147)

Substituting eq. (141) into eq. (139), furthermore, using the components of
the metric tensor, the following expression can be obtain:

$\displaystyle-\left(FH+a^{2}\sin^{2}\theta\right)\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{\left(\left(\sqrt{\frac{F}{G}}H+a^{2}\sin^{2}\theta\right)E-aL\right)^{2}}{FH+a^{2}\sin^{2}\theta}-\left(L-aE\right)^{2}$
$\displaystyle=\,\left(\frac{dS_{\theta}}{d\theta}\right)^{2}+L^{2}\cot^{2}\theta-a^{2}E^{2}\cos^{2}\theta.$
(148)

Note that the left and right-hand sides of the above equation (148) are only
functions of $r$ and $\theta$, respectively. Therefore one can write each side
equal to a separation constant since the Jacobi action principle is the
separable solution,

$-\left(FH+a^{2}\sin^{2}\theta\right)\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{\left(\left(\sqrt{\frac{F}{G}}H+a^{2}\sin^{2}\theta\right)E-aL\right)^{2}}{FH+a^{2}\sin^{2}\theta}-\left(L-aE\right)^{2}=\mathcal{K},$
(149)

$\left(\frac{dS_{\theta}}{d\theta}\right)^{2}+L^{2}\cot^{2}\theta-a^{2}E^{2}\cos^{2}\theta=\mathcal{K},$
(150)

where the separation constant $\mathcal{K}$ is known as the Carter constant.
Using the above differential equations, one can write down the following
separated null geodesic equations for radial ($r$) and tangential ($\theta$)
parts as,

$H^{2}\left(\frac{dr}{d\lambda}\right)^{2}-R(r)=0\,\,,$ (151)
$H^{2}\left(\frac{d\theta}{d\lambda}\right)^{2}-\Theta(\theta)=0\,\,,$ (152)

where $R(r)$ and $\Theta(\theta)$ can be considered as radial and tangential
parts of effective potentials, respectively, which can be defined as,

$R(r)=\left[X(r)-a\xi\right]^{2}-\Delta(r)\left[\eta+(\xi-a)^{2}\right]\,\,,$
(153)
$\Theta(\theta)=\eta+(\xi-a)^{2}-\left(\frac{\xi}{\sin\theta}-a\sin\theta\right)^{2}\,\,$
(154)

where, $\xi=\frac{L}{E}$ and $\eta=\frac{\mathcal{K}}{E^{2}}$. The functions
$X(r)$ and $\Delta(r)$ can be define as,

$X(r)=\sqrt{\frac{F}{G}}H+a^{2}\sin^{2}\theta,$ (155)
$\Delta(r)=FH+a^{2}\sin^{2}\theta.$ (156)

Note that, $R(r)$ and $\Theta(\theta)$ must be non-negative for photon motion.
In a general rotating spacetime, unstable circular orbits exist when the
following conditions hold,

$R(r_{ph})=0,\;\frac{dR(r_{ph})}{dr}=0,\;\frac{d^{2}R(r_{ph})}{dr^{2}}\leq
0\,\,.$ (157)

where $r=r_{ph}$ is the radius of the unstable photon orbit. Using the above
conditions (157), the critical impact parameters can be obtain corresponding
to the maxima of the $R(r)$ as,

$\xi=\frac{X_{ph}\Delta^{\prime}_{ph}-2\Delta_{ph}X^{\prime}_{ph}}{a\Delta^{\prime}_{ph}},$
(158)
$\eta=\frac{4a^{2}X^{\prime}_{ph}\Delta_{ph}-\left[(X_{ph}-a^{2})\Delta^{\prime}_{ph}-2X^{\prime}_{ph}\Delta_{ph}\right]^{2}}{a^{2}\Delta^{\prime
2}_{ph}},$ (159)

where prime denoted for derivative concerning $r$ and subscript “ph" denoted
for the quantities which are evaluated at $r=r_{ph}$. The above expressions
are the general forms of the critical impact parameters $\xi$ and $\eta$ for
the unstable photon orbits, which would be important to find out the shape of
the shadows. However, one can obtain the apparent shape of the shadow by using
the celestial coordinates $\alpha$ and $\beta$, which lie in the line of sight
of the observer direction. The expressions of the celestial coordinates
$\alpha$ and $\beta$ can be written in the following way,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\lim_{r_{0}\to\infty}\left(-r_{0}^{2}\sin{\theta_{0}}\frac{d\phi}{dr}\right)\,\,,$
(160) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\lim_{r_{0}\to\infty}\left(r_{0}^{2}\frac{d\theta}{dr}\right)\,\,,$
(161)

where $r_{0}$ and $\theta_{0}$ are the coordinates of the asymptotic observer.
Note that, if the general rotating metric to be an asymptotically flat in the
limit of $r\to\infty$, then one can consider $G\to 1$, $F\to 1$, $H\to r^{2}$,
$X\to r^{2}$ and $\Delta\to r^{2}$. Therefore, the above expressions of the
celestial coordinates $\alpha$ and $\beta$ becomes,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\frac{\xi}{\sin{\theta_{0}}}\,\,,$ (162) $\displaystyle\beta$
$\displaystyle=$
$\displaystyle\pm\sqrt{\eta+a^{2}\cos^{2}{\theta_{0}}-\xi^{2}\cot^{2}{\theta_{0}}}.\,\,$
(163)

#### 16.1Shadows in the rotating JNW spacetime

From the above expressions (158) and (159), one can obtain the celestial
coordinates $\alpha$ and $\beta$ for the rotating JNW spacetime by using its
metric tensor components. The JNW metric tensor components are used without
adopting any complexification method since the general rotating metric is
obtained by skipping the complexification method. The metric tensor components
of the JNW spacetime are given as,

$G=F=\left(1-\frac{2M}{r\nu}\right)^{\nu}\,\,;\
H=r^{2}\left(1-\frac{2M}{r\nu}\right)^{1-\nu}$ (164)

using the above equations, one can find that the expressions for $X(r)$ and
$\Delta(r)$,

$X(r)=r^{2}\left(1-\frac{2M}{r\nu}\right)^{1-\nu}+a^{2}\sin^{2}\theta,$ (165)
$\Delta(r)=r^{2}\left(1-\frac{2M}{r\nu}\right)+a^{2}\sin^{2}\theta.$ (166)

Therefore, using the above expressions, one can calculate the critical impact
parameters and celestial coordinates for the rotating JNW spacetime. The shape
of the shadows are constructed using the expressions of celestial coordinates
for rotating JNW spacetime, where the inclination angle of the observer is
$\theta_{0}=\pi/2$.

(a) Shadow for $q=0.4$ and $a=0.3$.

(b) Shadow for $q=0.4$ and $a=0.4$.

(c) Shadow for $q=0.4$ and $a=0.5$.

(d) Shadow for $q=0.6$ and $a=0.3$.

(e) Shadow for $q=0.6$ and $a=0.4$.

(f) Shadow for $q=0.6$ and $a=0.5$.

(g) Shadow for $q=0.8$ and $a=0.3$.

(h) Shadow for $q=0.8$ and $a=0.4$.

(i) Shadow for $q=0.8$ and $a=0.5$.

Fig. 7: Figure shows the shape of the shadow cast by the rotating JNW naked
singularity for various values of scalar field charge $q$ and spin parameter
$a$. The values of $M=1$ and $\theta=\pi/2$ are considered for the shape of
shadows in celestial coordinates. The details of this figure are discussed in
the text.

In fig.(10), it is shown that the shape of the shadows for different values of
spin parameter $a$ and scalar field charge $q$, where the blue line indicates
the shadows in rotating JNW spacetime and the red dotted line indicate the
shadows in the Kerr spacetime. It is well known that the shadow shapes in the
Kerr black hole spacetime always be in contour shape, and as the spin
parameter increasing, it will become more and more prolate. One can see in
fig.(10), that the horizontal radius of the red dotted line will decrease from
the left side as the value of a spin parameter increasing. On the other hand,
it is found that the shadows in rotating JNW spacetime change from contour
shape to arc shape as the values of the spin parameter and scalar field charge
or one of them are increasing. In the next section, the shape of the shadow
cast by deformed Kerr spacetime is shown.

#### 16.2Shadows in the deformed Kerr spacetime

The deformed Schwarzschild spacetime is considered a solution in the modified
Quadratic gravity Yunes:2011we . In that paper, the authors show that
parametric deformation to the Schwarzschild metric solves the modified field
equations in the small coupling approximations. The deformed Schwarzschild
spacetime is given as,

$ds^{2}=-\left(1-\frac{2M}{r}\right)\left(1+h(r)\right)dt^{2}+\frac{\left(1+h(r)\right)}{\left(1-\frac{2M}{r}\right)}dr^{2}+r^{2}d\Omega^{2}\,\,,$
(167)

where $M$ is the mass of the deformed Schwarzschild black hole.
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the metric of two
sphere, and the parametric deformation

$\displaystyle h(r)=\frac{\epsilon M^{3}}{r^{3}}.$ (168)

The above deformation represents the deviation from the Schwarzschild
spacetime, where $\epsilon$ is a deformation parameter. When $\epsilon=0$, the
above metric reduces to the original Schwarzschild metric. The specific
deformation is considered here, and the reasons are described as follows: The
general form of the deformation function is given as,

$h(r)=\sum_{k=0}^{\infty}\epsilon_{k}\left(\frac{M}{r}\right)^{k}.$ (169)

In general relativity, it can be shown that the stationary and asymptotically
flat spacetime should be Schwarzschild-like Beig:1980be . Therefore, in the
above equation (169), one need to consider $k\geq 2$. One can show that
asymptotically flat spacetimes with slower fall off ($0\leq k<1$) can generate
gravitational radiation, and hence, the spacetime cannot be stationary
Kennefick:1993zr . From the Lunar Laser Ranging experiment, it is shown that
the value of $\epsilon_{2}$ is $|\epsilon_{2}|\leq 4.6\times 10^{-4}$
Johannsen:2011dh , Williams:2004qba . So that, one can consider
$\epsilon_{2}=0$. In Psaltis:2008bb , it is shown that the value of
$|\epsilon_{3}|$ is unconstrained. Hence, in this work, $\epsilon_{3}\neq 0$
is considered and neglected the higher order contributions (i.e., $k>3$)
Johannsen:2011dh .

(a) Spin $a=0.6$

(b) Spin $a=0.7$

(c) Spin $a=0.8$

(d) Spin $a=0.8$

Fig. 8: Event horizon position for $\epsilon=0$ (black), $\epsilon=1$ (Red),
$\epsilon=-1$ (Blue) with different spin and 3D view of directional naked
singularity for $\epsilon=1$, $a=0.8$.

One can determine the Kerr-like deformed spacetime using the NJA,

$\displaystyle ds^{2}=-\bigg{(}1-\frac{2Mr}{\rho^{2}}\bigg{)}(1+h)dt^{2}$
$\displaystyle+$
$\displaystyle\frac{\rho^{2}(1+h)}{\Delta+a^{2}h\sin^{2}\theta}dr^{2}+\rho^{2}d\theta^{2}-\frac{4Mar}{\rho^{2}}(1+h)\sin^{2}\theta
dtd\phi$ (170) $\displaystyle+$
$\displaystyle\bigg{[}\rho^{2}+a^{2}\sin^{2}\theta(1+h)\bigg{(}1+\frac{2Mr}{\rho^{2}}\bigg{)}\bigg{]}\sin^{2}\theta
d\phi^{2}\,.$

In Johannsen:2011dh , the above metric is defined as deformed Kerr spacetime.
One can see that the above metric reduces to the original Kerr metric when
$\epsilon=0$. Kerr geometry has an oblate shape. However, the deformed Kerr
spacetime is more oblate for $\epsilon<0$ and more prolate for $\epsilon>0$.
One can derive the position of the inner and outer horizon of deformed Kerr
spacetime by solving the following equation,

$g_{t\phi}^{2}+g_{tt}g_{\phi\phi}=0\,\,.$ (171)

The position and shape of an outer event horizon of the deformed Kerr
spacetime is shown for various values of the deformation parameter
($\epsilon$) and spin parameter ($a$). Figs. (8) show the shapes and position
of outer event horizon for spin parameter $a=0.6,0.7,0.8$ respectively. It can
be seen from the above figures that for $\epsilon=1$ and $a=0.8$, the outer
event horizon is not closed, and it has a hole near the equatorial plane of
deformed Kerr spacetime. Therefore, the singularity at the centre becomes
naked for the particular values of $\epsilon$ and $a$.

Fig. 9: The shaded region in the figure shows the existence of a naked
singularity in deformed Kerr spacetime.

In Fig. (9), the parameters’ space in $\epsilon$ and $a$ frame is shown for
which the central singularity of deformed Kerr spacetime becomes naked. The
shaded region in that figure shows the possible values of $\epsilon$ and $a$
for the existence of a naked singularity in the deformed Kerr spacetime. On
the other hand, the unshaded region shows the existence of a deformed Kerr
black hole. From Fig. (9), it can be seen that for $0\leq a\leq 1$, the
central singularity can be naked when the deformation parameter $\epsilon>0$.
However, when $\epsilon<0$ and $0\leq a\leq 1$, the central singularity of the
deformed Kerr spacetime cannot be naked.

(a) Shadow for $\epsilon=-1$ and $a=0.6$.

(b) Shadow for $\epsilon=-1$ and $a=0.7$.

(c) Shadow for $\epsilon=-1$ and $a=0.8$.

(d) Shadow for $\epsilon=1$ and $a=0.6$.

(e) Shadow for $\epsilon=1$ and $a=0.7$.

(f) Shadow for $\epsilon=1$ and $a=0.8$.

(g) Shadow for $\epsilon=18$ and $a=0.6$.

(h) Shadow for $\epsilon=18$ and $a=0.7$.

(i) Shadow for $\epsilon=18$ and $a=0.8$.

Fig. 10: Figure shows the shape of the shadow cast by the deformed Kerr
spacetime for various values of deformation parameter $\epsilon$ and spin
parameter $a$. The details of this figure are discussed in the text.

Now from the definition of the celestial coordinates, one can write down the
following expressions of celestial coordinates for deformed Kerr spacetime,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\frac{-\Delta\xi}{(\Delta+a^{2}h\sin^{2}{\theta_{0}})\sin{\theta_{0}}}\,\,,$
(172) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sqrt{\eta-a^{2}\sin^{2}{\theta_{0}}-\xi^{2}\csc^{2}{\theta_{0}}\,\,.}$
(173)

Using the above expressions of celestial coordinates, it can be verified that
they are not independent of the deformation parameter $\epsilon$. Therefore,
though the behaviour of $R(r)$ is similar for both Kerr and deformed Kerr
spacetime, the shadow shapes of both spacetimes are distinguishable. In Fig.
(10), the shapes of the shadows cast by deformed Kerr spacetime are shown for
various values of $\epsilon$ and $a$, where the shaded region shows the shape
of the shadow cast by deformed Kerr spacetime and the dashed lines show the
shape of the shadow cast by Kerr spacetime. Figs. (10(a)),(10(b)) and (10(c))
show the shadow shape for $a=0.6,0.7$ and $0.8$ respectively, where the
$\epsilon=-1$. On the other hand, in Figs. (10(d)),(10(e)) and (10(f)), the
shadow shape for $a=0.6,0.7$ and $0.8$ are shown respectively, where the
$\epsilon=1$. Those figures show how the shadow shape changes when $\epsilon$
changes its signature. Figs. (10(g)),(10(h)) and (10(i))also show the shadow
shape of deformed Kerr spacetime for the same values of spin parameter, where
$\epsilon=18$ is taken. As discussed above, the central singularity in
deformed Kerr spacetime cannot be naked when $\epsilon<0$ where $0\leq a\leq
1$. Therefore, all the shadow shapes shown in Figs. (10(a)),(10(b)) and
(10(c)) are corresponding to the deformed Kerr black hole. However, though the
Figs. (10(d)),(10(e)) are showing the shape of the shadow cast by deformed
Kerr black hole, Figs. (10(f)),(10(g)),(10(h)) and (10(i)) are showing the
shadow shape of the deformed Kerr naked singularity, which can be verified
from Fig. (9). Therefore, the deformed Kerr naked singularity can cast a
shadow, unlike Kerr naked singularity.

The amount of the deformation ($D$) of the shadow shape from the shadow shape
of the Kerr black hole can be defined as,

$\alpha_{dk}-\alpha_{k}=D\,,$ (174)

where the $\beta=0$ axis to measure the deformation of the shadow shape and
$\alpha_{dk}$ and $\alpha_{k}$ are celestial coordinates for deformed Kerr and
Kerr spacetimes, respectively. One can derive the following equation of $D$
for deformed Kerr spacetime,

$D=\frac{a\epsilon
M^{3}(r^{2}(r-3M)+a^{2}(M+r))}{(M-r)(r^{4}(r-2M)+a^{2}(\epsilon
M^{3}+r^{3}))}\,\,,$ (175)

where

$r=\frac{\left(M+(2a^{2}M-M^{3}+2aM\sqrt{a^{2}-M^{2}})^{1/3}\right)^{2}}{(2a^{2}M-M^{3}+2aM\sqrt{a^{2}-M^{2}})^{1/3}}\,\,.$
(176)

Fig. 11: Figure shows how the deformation of shadow shape ($D$) changes with
the value of spin parameter $a$, where the red, blue and brown lines
correspond to the deformed Kerr spacetime having $\epsilon=-1,1$ and $18$
respectively.

Fig. (11) shows how the deformation of the shadow shape $D$ changes with spin
parameter $a$ for various values of $\epsilon$. The left side figure in Fig.
(11) shows how one can define the deformation in the shadow shape. On the
other hand, in the right side figure in Fig. (11), the red, blue and brown
curves show how the deformation of the shape of the shadow changes with spin
parameter $a$, when the $\epsilon=-1,1$ and $18$ respectively.

### 17 Discussions and conclusions

The conclusions of this chapter are in the following way: the rotating JNW
spacetime is consructed using the NJA without employing the complexification
of coordinates. The rotating JNW spacetime obtained using the NJA, considering
the complexification step, cannot be written into the BLC. If that metric is
written into the BLC-like form by performing an improper coordinate
transformation, it does not satisfy the energy conditions. To resolve this
problem, the complexification step is dropped in the NJA to properly transform
the rotating solution into the BLC. As a result, the rotating JNW spacetime is
obtained into the BLC, which does satisfy the energy conditions.

The rotating Kerr black hole can cast a prolate contour shape shadow (see
dotted red line in fig.(10)). While the spin parameter increase up to $M\geq
a$, it will become more and more prolate contour, where $M$ is the mass of the
black hole. Nevertheless, as it reach the $M<a$ limit, the Kerr black hole
becomes the Kerr naked singularity, which may not cast contour shaped shadow.
To get more insight, in this paper, the shape of the shadows cast by rotating
JNW naked singularity is obtained and compared with the shape of shadows cast
by the Kerr black hole. It is found that the shape of the shadows in rotating
JNW spacetime becomes prolate arc shape from the prolate contour shape as the
scalar field charge $q$ and/or spin parameter $a$ is increases (see blue line
in fig. (10)).

In addition, it is shown that the deformed Kerr naked singularity can cast a
shadow, unlike the Kerr naked singularity. If there exists a finite amount of
deformation of Kerr spacetime, then the effect of the deformation may have
become very significant near the singularity of deformed Kerr spacetime. In
the context of the observations by the EHT/ngEHT group, these studies on
shadows are beneficial. This novel signature of the shadow shapes in rotating
JNW spacetime (prolate arc) and the shadow shape of the deformed Kerr naked
singularity could be observationally significant to differentiate the naked
singularities from the black holes. However, the existence and observation of
a shadows are not sufficient, or may not ensure the existence of an event
horizon necessarily in the galactic center. This necessitates studying
different observables such as accretion spectra, light curves, etc. Since the
main focus is on Sgr A*, the motion of various stars around the central object
are helpful via astrometric observations which can show precession in their
orbits. Therefore, the precession of relativistic orbits in the vicinity of
black hole and its mimickers are discussed.

## CHAPTER 4 Precession of timelike bound orbits in static spacetimes

The term ‘geodesic’ notion represents the extremum distance between two points
on a spherical surface or any other curved surface. The motion of the freely
falling test particle described the shortest possible route. Thus, it always
follows the timelike or null geodesics in a given curvature of spacetime. The
nature of the causal structures and spacetime curvatures can be scrutinized by
analyzing the geodesic motions of the test particles. Timelike geodesics are
essential to describe the orbital motion of a massive test particle.
Therefore, in this chapter, the timelike bound orbits of a test particle is
analysed and discuss the orbital precession (periastron precession) in naked
singularity and black hole spacetimes.

### 18 Timelike geodesics in spherically symmetric and static spacetime

The spherical and temporal symmetry of the general spacetime (14) can be
illustrated by the conservation of angular momentum and energy of a freely
falling particle. Therefore the angular momentum and energy of a particle per
unit rest mass are always conserved. All possible types of orbits are
classified under the most conserved angular momentum and energy of a test
particle. The conserved angular momentum ($h$) per unit rest mass and
conserved energy ($\gamma$) per unit rest mass of a freely falling particle in
the spherically symmetric and static spacetime can be written as,

$h=g_{\phi\phi}(r)\frac{d\phi}{d\tau}\,\,,\,\,\,\gamma=g_{tt}(r)\frac{dt}{d\tau}\,\,,$
(177)

where the proper time of the particle is given by $\tau$. the sign convention
of the metric components $g_{\mu\nu}$ is considered as $(-,+,+,+)$, and
employing this sign convention to the normalization of four velocities for
timelike geodesics, one can have the relation $u^{\mu}u_{\mu}=-1$, where
$u^{\mu}$ is the four velocity of a particle. Now utilizing the conservation
laws given in (177) and normalization of four velocity relation, one can
derive the general expression of the total energy of a freely falling test
particle as,

$E=\frac{g_{rr}(r)g_{tt}(r)}{2}\left(\frac{dr}{d\tau}\right)^{2}+V_{eff}(r)\,\,,$
(178)

where $E$ is given by defining the constant as $E=\frac{\gamma^{2}-1}{2}$ and
radial effective potential can be written as,

$V_{eff}(r)=\frac{1}{2}\left[g_{tt}(r)\left(1+\frac{h^{2}}{g_{\phi\phi}(r)}\right)-1\right]\,\,,$
(179)

note that an equatorial plane ($\theta=\frac{\pi}{2}$) is considered for the
particle’s trajectories since the spacetime configuration is spherically
symmetric. The above equation of effective potential plays an important role
in the motion of the particles in a given general form of spacetime (14). By
defining some conditions in between the total energy and effective potential
as,

$V_{eff}(r)=E,\;\;\;V_{eff}^{\prime}(r)=0,$ (180)

one can get types of bound orbital shapes, for example, circular orbits and
elliptical orbits. For circular orbits, the required expressions for conserved
angular momentum ($h$) and energy ($\gamma$) are calculated by using the above
conditions,

$h^{2}=\frac{g_{\phi\phi}(r)}{\left(\frac{2g_{tt}(r)}{\sqrt{g_{\phi\phi}(r)}g_{tt}^{\prime}(r)}-1\right)}\,,\,\,\,\gamma^{2}=\frac{2g_{tt}^{2}(r)}{\sqrt{g_{\phi\phi}(r)}g_{tt}^{\prime}(r)}\frac{1}{\left(\frac{2g_{tt}(r)}{\sqrt{g_{\phi\phi}(r)}g_{tt}^{\prime}(r)}-1\right)}\,\,,$
(181)

where prime denoted for derivative with respect to $r$ and the above
conditions $V_{eff}(r)=E$, $V_{eff}^{\prime}(r)=0$, are valid for both stable
and unstable circular orbits. The stability of the circular orbits defined by
the second derivatives of effective potential with respect to $r$ as,

$V_{eff}^{\prime\prime}(r)>0\,,\,\,\,V_{eff}^{\prime\prime}(r)<0\,\,,$ (182)

for stable and unstable circular orbits, respectively. One can get bound
elliptical orbits by employing the condition $V_{eff}(r)=E$, in which the
minimum ($r_{min}$) and maximum ($r_{max}$) radial approaches of the freely
falling test particle from the center are established. Therefore, the
necessary conditions to get the stable bound orbits can be define for freely
falling test particles in the following manner,

$V_{eff}(r_{min})=V_{eff}(r_{max})=E\,,\,\,$ (183)
$E-V_{eff}(r)>0\,,\,\,\,\forall r\in(r_{min},r_{max}).$ (184)

The shape of the timelike bound orbits can describe using eq. (178) with the
value of conserved energy $\gamma$ and angular momentum $h$,

$\frac{dr}{d\phi}=\frac{g_{\phi\phi}(r)}{h}\frac{\sqrt{2(E-V_{eff})}}{\sqrt{g_{rr}(r)g_{tt}(r)}}.$
(185)

The above equation (185) represents how the particle changes its radial
coordinate $r$ with azimuth coordinate $\phi$ in an equatorial plane. One can
derive an orbit equation of a freely falling test particle by differentiating
above eq. (185) with respect to $\phi$,

$\frac{d^{2}u}{d\phi^{2}}+\frac{\gamma^{2}g_{\phi\phi}^{2}(u)u^{4}}{2g_{tt}^{2}(u)g_{rr}(u)h^{2}}\left(\frac{dg_{tt}(u)}{du}\right)-A(u)\left(\frac{dg_{rr}(u)}{du}\right)+B(u)\left(\frac{dg_{\phi\phi}(u)}{du}\right)-C(u)=0\,\,,$
(186)

where $u=\frac{1}{r}$ and,

$A(u)=\left[\frac{\gamma^{2}g_{\phi\phi}^{2}(u)u^{4}}{2g_{tt}(u)g_{rr}^{2}(u)h^{2}}-\frac{g_{\phi\phi}(u)u^{4}}{2g_{rr}^{2}(u)}-\frac{g_{\phi\phi}^{2}(u)u^{4}}{2g_{rr}^{2}(u)h^{2}}\right],$
(187)
$B(u)=\left[\frac{\gamma^{2}g_{\phi\phi}(u)u^{4}}{g_{tt}(u)g_{rr}(u)h^{2}}-\frac{u^{4}}{2g_{rr}(u)}-\frac{g_{\phi\phi}(u)u^{4}}{g_{rr}(u)h^{2}}\right]\,\,,$
(188)

and

$C(u)=\left[\frac{2\gamma^{2}g_{\phi\phi}^{2}(u)u^{3}}{g_{tt}(u)g_{rr}(u)h^{2}}+\frac{2g_{\phi\phi}(u)u^{3}}{g_{rr}(u)}+\frac{2g_{\phi\phi}^{2}(u)u^{3}}{g_{rr}(u)h^{2}}\right].$
(189)

It is very complicated to derive the exact analytical solution of the above
orbit equation (186) since the derived orbit equation is a second-order non-
linear differential equation. However, one can solve it numerically, and one
can get the nature and shape of the timelike bound orbits for the given
conserved energy $\gamma$ and angular momentum $h$. Note that the nature and
shape of the timelike bound orbits depend on the causal structure of spacetime
geometry.

### 19 Fully relativistic orbit equations

The fully general relativistic orbit equation of a freely falling particle can
be derived using the timelike geodesics. In the above section, the general
procedure to derive an orbit equation of a test particle is described in
spherically symmetric and static spacetime. This section analyzes the orbit
equations for the Schwarzschild, JMN-1, and JNW models, which are spherically
symmetric and static spacetimes.

#### 19.1Schwarzschild orbit equation

The Schwarzschild spacetime is defined as given in equation (1); it has a
strong singularity at $r=0$, which is covered by an event horizon. One can see
that the metric tensor components of the Schwarzschild spacetime are
independent of $t$ and $\phi$. Therefore, due to the temporal and spherical
symmetries of the Schwarzschild spacetime, the conserved energy and angular
momentum per unit rest mass can be written of the freely falling test particle
in this spacetime as,

$\gamma_{SCH}=\left(1-\frac{2M}{r}\right)\left(\frac{dt}{d\tau}\right)\,\,,$
(190) $h_{SCH}=r^{2}\left(\frac{d\phi}{d\tau}\right)\,\,,$ (191)

the effective potential for the Schwarzschild spacetime can be derived using
the general expression given in (179),

$(V_{eff})_{SCH}=\frac{1}{2}\left[\left(1-\frac{M_{0}R_{b}}{r}\right)\left(1+\frac{h_{SCH}^{2}}{r^{2}}\right)-1\right].\,\,$
(192)

where $2M=M_{0}R_{b}$. The expressions of the conserved energy and angular
momentum for circular geodesics are defined as from the conditions
$V_{eff}(r)=E$ and $V_{eff}^{\prime}(r)=0$,

$\gamma_{SCH}^{2}=\frac{2\left(r-M_{0}R_{b}\right)^{2}}{r\left(2r-3M_{0}R_{b}\right)}\,,\,,$
(193) $h_{SCH}^{2}=\frac{M_{0}R_{b}r^{2}}{\left(2r-3M_{0}R_{b}\right)}\,\,$
(194)

from the above expressions, one can see that no circular orbit is possible
within the range $0\leq r\leq\frac{3M_{0}R_{b}}{2}$, or in terms of the total
mass of the black hole, one can write $0\leq r\leq 3M$, where
$M=\frac{3M_{0}R_{b}}{2}$. Note that the above range is valid for stable and
unstable circular orbits. Therefore, for stable circular orbits, one has to
use the $(V_{eff}^{\prime\prime})_{SCH}>0$ condition as,

$(V_{eff}^{\prime\prime})_{SCH}=\frac{2M}{r}\left(\frac{6M-r}{3M-r}\right)\,\,.$
(195)

from the above expression, one can see that no stable circular orbits are
possible below $r=6M$ in the Schwarzschild spacetime. Thus, $r=6M$ is the
innermost stable circular orbit (ISCO) in the Schwarzschild spacetime. Note
that for bound orbits, the conditions given in equations (183) and (184)
should be fulfilled. Now, an orbit equation of the test particle can be
written using the general expression of the orbit equation given in (186) for
the Schwarzschild spacetime as,

$\frac{d^{2}u}{d\phi^{2}}+u=\frac{3M_{0}R_{b}}{2}u^{2}+\frac{M_{0}R_{b}}{2h^{2}}\,\,.$
(196)

which is the second-order non-linear differential equation.

#### 19.2JMN-1 orbit equation

The JMN-1 naked singularity is the non-vacuum solution of Einstein equations.
As discussed above, JMN-1 is formed as an end state of quasi-static
gravitational collapse. It has a strong curvature singularity at the center
without an event horizon. As it is a spherically symmetric and static
solution, the conserved energy and angular momentum per unit rest mass is
obtained as,

$\gamma_{JMN-1}=(1-M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{1-M_{0}}}\left(\frac{dt}{d\tau}\right)\,\,,$
(197) $h_{JMN-1}=r^{2}\left(\frac{d\phi}{d\tau}\right)\,\,,$ (198)

now, employing the general formalism of the orbit equation, the effective
potential for the JMN-1 spacetime is derived as,

$(V_{eff})_{JMN-1}=\frac{1}{2}\left[(1-M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(1-M_{0})}}\left(1+\frac{h_{JMN-1}^{2}}{r^{2}}\right)-1\right]\,\,,\\\
$ (199)

where $h_{JMN-1}$ is the conserved angular momentum per unit rest mass of the
test particles in JMN-1. The conserved quantities for circular orbits are
defined as,

$\gamma^{2}_{JMN-1}=\frac{2(1-M_{0})^{2}\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{1-M_{0}}}}{\left(2-3M_{0}\right)}\,,\,\\\
$ (200) $h^{2}_{JMN-1}=\frac{r^{2}M_{0}}{2-3M_{0}}\,\,,\\\ $ (201)

the stable circular orbit can be obtained by using the condition,

$(V_{eff}^{\prime\prime})_{JMN-1}=\frac{M_{0}}{R_{b}^{2}}\left(\frac{r}{R_{b}}\right)^{\frac{3M_{0}-2}{1-M_{0}}}>0\,\,$
(202)

from the above equations, it is seen that for $M_{0}<2/3$, JMN-1 spacetime can
have stable circular orbits of any radius. In JMN-1 spacetime, the unstable
circular orbits do not exist within the region $0<M_{0}<2/3$. Since,
$h_{JMN-1}$ and $\gamma_{JMN-1}$ become imaginary for $M_{0}>2/3$, the stable
circular orbits are not possible at any value of $r$ in JMN-1 spacetime. Now,
an orbit equation of a freely falling test particle in JMN-1 spacetime can
write as,

$\frac{d^{2}u}{d\phi^{2}}+(1-M_{o})u-\frac{\gamma^{2}}{2h^{2}}\frac{M_{0}}{(1-M_{0})}\left(\frac{1}{u}\right)\left(\frac{1}{uR_{b}}\right)^{\frac{-M_{0}}{(1-M_{0})}}=0\,\,,\\\
$ (203)

#### 19.3JNW orbit equation

The JNW spacetime is a minimally coupled mass-less scalar field solution of
the Einstein equations. Unlike JMN-1 naked singularity spacetime, JNW naked
singularity spacetime is asymptotically flat. Therefore, this spacetime need
not match with the external Schwarzschild spacetime. From the equation (12),
the metric tensor components of this spacetime are independent of $t$ and
$\phi$. Thus, one can define the conserved energy and angular momentum of a
test particle per unit rest mass as,

$h_{JNW}=r^{2}\left(1-\frac{b}{r}\right)^{1-n}\frac{d\phi}{d\tau}\,,\,\,$
(204) $\gamma_{JNW}=\left(1-\frac{b}{r}\right)^{n}\frac{dt}{d\tau}\,\,.$ (205)

Similarly, using the general formalism, the effective potential of a freely
falling test particle can be obtain for JNW spacetime as,

$(V_{eff})_{JNW}=\frac{1}{2}\left[\left(1-\frac{b}{r}\right)^{n}\left(1+\frac{h_{JNW}^{2}}{r^{2}}\left(1-\frac{b}{r}\right)^{n-1}\right)-1\right]\,\,,$
(206)

the conserved quantities for circular geodesics can be defined as,

$\gamma_{JNW}^{2}=\left(1-\frac{b}{r}\right)^{n}\left[\frac{2r-b(n+1)}{2r-b(2n+1)}\right]\,\,,\\\
$ (207)
$h_{JNW}^{2}=r^{2}\left[\frac{bn\left(1-\frac{b}{r}\right)^{1-n}}{2r-b(2n+1)}\right]\,\,,$
(208)

where the stable circular orbits would be excess by applying the condition
$(V^{\prime\prime}_{eff})>0$. Although, In JNW spacetime,
$(V^{\prime\prime}_{eff})$ is not always positive parth1 . One can find
$(V^{\prime\prime}_{eff})>0$ at any point in JNW spacetime for $n<0.447$. But,
for $0.447<n<0.5$, one certain radial interval ($r_{-},r_{+}$) is defined, in
which no stable circular orbits are possible, where $r_{-}$ and $r_{+}$ can be
written as,

$r_{-}=\frac{1}{4}\left(b~{}(2+6n)-4.472~{}b\sqrt{n^{2}-0.2}\right)\,\,,\\\ $
(209)
$r_{+}=\frac{1}{4}\left(b~{}(2+6n)+4.472~{}b\sqrt{n^{2}-0.2}\right)\,\,.$
(210)

Note that stable circular orbits are not possible in the certain radial
interval $r_{-}<r<r_{+}$. However, one can have stable circular orbits outside
this radial interval as there is $(V^{\prime\prime}_{eff})_{JNW}>0$. On the
other hand, for $0.5<n<1$, one can get the innermost stable circular orbit
(ISCO) at $r_{(ISCO)}$. Now, employing the general method to derive an orbit
equation for the JNW spacetime, one can get the second-order non-linear
differential equation as,

$\frac{d^{2}u}{d\phi^{2}}+u+\frac{b\gamma^{2}}{2h^{2}}(2-2n)(1-bu)^{1-2n}-\frac{b(2-n)}{2h^{2}}(1-bu)^{1-n}-\frac{3bu^{2}}{2}=0\,\,.$
(211)

### 20 Periastron precession of the timelike bound orbits

Since the fully relativistic orbit equations are second-order non-linear
differential equations, it is solved numerically. The solutions of these
equations are used to get parametric orbital plots. In figure (12), the
orbital precession in the equatorial plane of the Schwarzschild spacetime is
shown for the parameters $M_{0}=0.09,h=200,E=-0.0268$ and $R_{b}=1000$.

Fig. 12: Positive orbital precession in Schwarzschild spacetime

The dark black region indicates the position of the Schwarzschild black hole.
The red line represents the orbit of the freely falling test particle. The
brown circle shows the test particle’s minimum approach (periastron position)
towards the center. Due to the spacetime curvature of the Schwarzschild
geometry, a test particle’s orbit would precess in a particular direction. In
the Schwarzschild black hole case, the orbital precession of a test particle
occurs in the direction of the particle’s angular motion. One can see from the
figure (12) that the arrow indicates the direction of the particle’s angular
movement, which is in the anti-clockwise order. Whenever the particle’s orbit
precesses in the direction of the particle’s orbiting path, the angular
distance travelled by a particle around the center is always found greater
than $2\pi$, known as the positive precession. As seen from the above figure,
the angular distance travelled by a particle between two successive periastron
points remains greater than $2\pi$ for any parametric values. Therefore, the
Schwarzschild black hole spacetime always admits positive precession. the
proof of this statement will discuss in the next section.

Fig. 13: Negative and Positive orbital precession in the JMN-1 spacetime

Figure 13 shows the particle orbits in the JMN-1 spacetime, where the blue
line represents the particle’s orbit in the JMN-1 spacetime. The brown circle
indicates the periastron position. For the comparison, the same parameters is
used in the JMN-1 case as it is used in the Schwarzschild case, where
$M_{0}=0.09,h=200,E=-0.0268$ and $R_{b}=1000$. An interesting result emerged
from the orbital precession of a test particle in the JMN-1 spacetime. In this
spacetime, the angular distance travelled by a test particle in between two
successive periastron points can have less than $2\pi$. In other words, the
precession of the orbits will occur in the opposite direction of the
particle’s angular motion, known as the negative precession. One can see,
Figure 13(a) shows the negative precession of the orbits in the JMN-1
spacetime. On the other hand, JMN-1 spacetime admits the same nature of the
precession as Schwarzschild spacetime does. Figure 13(b), shows the positive
precession of the orbits in JMN-1 spacetime for the parameters
$M_{0}=0.55,h=1100,E=-0.004$ and $R_{b}=1000$. This figure shows that the
angular distance travelled by a particle between two successive periastron
points is greater than $2\pi$.

Fig. 14: Positive and negative orbital precession in the JNW spacetime

Similarly, the orbital precession in the JNW spacetime have been analysed. As
discussed above in the JMN-1 case, JNW spacetime can also allow the negative
precession of the orbits. Figure 14(a) shows the particle orbits in the
Schwarzschild spacetime corresponding to the JNW case. As shown in figure
14(b), the JNW spacetime admits negative precession of the orbits. Figures
14(a) and 14(b) are shows the comparison of the orbital precession in between
the Schwarzschild and JNW spacetime for the same parameters $M=0.25,q=10,h=3$
and $E=-0.002$. The positive precession of the orbits in JNW spacetime is
shown in figure 14(c) for $M=0.25,q=3,h=3$ and $E=-0.002$.

### 21 Approximate solutions of the orbit equations

In the above section (19), The fully relativistic second-order non-linear
differential orbit equations are derived for Schwarzschild, JMN-1, and JNW
spacetimes. The numerical solution of those orbit equations can give the
information about the shape of different orbits of freely falling particles in
the given spacetime. However, the approximate solutions of those orbit
equations are helpful in understanding the nature of orbital precession.

#### 21.1Approximate solution of the Schwarzschild orbit equation

In weak field limits, the fully relativistic Schwarzschild orbit equation
becomes the Newtonian orbit equation since the Schwarzschild metric is
asymptotically flat. The Schwarzschild orbit equation in the asymptotic limit
can be derived by neglecting the $u^{3}$ term as,

$\frac{d^{2}u}{d\phi^{2}}+u=\frac{M_{0}R_{b}}{2h^{2}}\,\,,$ (212)

the well-known solution of the above Newtonian orbit equation is given as,

$u=\frac{M_{0}R_{b}}{2h^{2}}\left[1+e\cos(\phi)\right],$ (213)

This equation implies that the orbiting particle will reach its previous
position after the $2\pi$ rotation. Nevertheless, it is not true for strong
field limits since one cannot neglect $u^{3}$ term from the Schwarzschild
orbit equation. We have considered the case in which a small eccentricity
approximation is valid in both the strong and weak field regimes. An
approximate solution for the orbit equation of Schwarzschild spacetime can be
write as,

$\tilde{u}=\frac{1}{p}\left[1+e\cos(m\phi)+O(e^{2})\right]\,\,,$ (214)

where $\tilde{u}=uR_{b}$, and $m$ and $p$ are positive real numbers. Now, by
substituting the above solution of the orbit equation (214) into the second-
order non-linear differential Schwarzschild orbit equation (196), then one can
get the following expressions of $p$ and $m$ from the zeroth and first order
of eccentricity terms respectively,