# Timelike geodesics in Naked Singularity and Black Hole Spacetimes

Parth Bambhaniya<EMAIL_ADDRESS>International Center for Cosmology,
Charusat University, Anand 388421, Gujarat, India  Ashok B. Joshi
<EMAIL_ADDRESS>International Center for Cosmology, Charusat
University, Anand 388421, Gujarat, India  Dipanjan Dey
<EMAIL_ADDRESS>International Center for Cosmology, Charusat
University, Anand 388421, Gujarat, India  Pankaj S. Joshi
<EMAIL_ADDRESS>International Center for Cosmology, Charusat
University, Anand 388421, Gujarat, India

###### Abstract

In this paper, we derive the solutions of orbit equations for a class of naked
singularity spacetimes, and compare these with timelike orbits, that is,
particle trajectories in the Schwarzschild black hole spacetime. The
Schwarzschild and naked singularity spacetimes considered here can be formed
as end state of a spherically symmetric gravitational collapse of a matter
cloud. We find and compare the perihelion precession of the particle orbits in
the naked singularity spacetime with that of the Schwarzschild black hole. We
then discuss different distinguishable physical properties of timelike orbits
in the black hole and naked singularity spacetimes and implications are
discussed. Several interesting differences follow from our results, including
the conclusion that in naked singularity spacetimes, particle bound orbits can
precess in the opposite direction of particle motion, which is not possible in
Schwarzschild spacetime.

Key words: Black hole, naked singularity, Perihelion precession, Timelike
orbits.

##  I Introduction

The Schwarzschild spacetime is unique, static, spherically symmetric vacuum
solution of Einstein equations. It represents a non-rotating, uncharged black
hole. It is generally believed that when a star with mass greater than a
critical mass (greater than about $4M_{\odot}$) exhausts all its nuclear fuel,
it then undergoes a catastrophic, continual gravitational collapse, finally
terminating into a black hole. Using Einstein equations, one can show that a
spherically symmetric, homogeneous, dust collapse always terminates into a
Schwarzschild black hole. Oppenheimer, Snyder, and also Datt, first showed the
dynamical collapse solution (OSD collapse) of a spherically symmetric,
homogeneous, dust cloud OppenheimerSnyder39 ,Datt .

Further to this, more physically realistic gravitational collapse models, such
as those with inhomogeneous matter distribution, or those with non-zero
pressures and other scenarios have been studied in detail. It is seen, for
example, that an inhomogeneous collapse, e.g. in a physically more realistic
case with matter density higher at center, results in a strong curvature
singularity which can be locally or globally visible. A visible or naked
singularity has rather different causality structure as compared to the black
hole case. For the homogeneous dust collapse case, the trapped surfaces and
apparent horizon close to the center are formed before the formation of the
central strong singularity, and therefore the singularity is always locally
and globally covered. On the other hand, for inhomogeneous collapse, there are
classes of initial conditions for which the central strong singularity forms
before the formation of the trapped surfaces and apparent horizon. In such a
case, the singularity is always at least locally naked Joshi:1993zg ;
Mena:1999bz ; Jhingan:1996jb ; Singh:1994tb . With larger values of pressure,
one can show that in asymptotic time, without forming the trapped surfaces,
the collapsing matter cloud can equilibrate itself into a static, singular
spacetime, which include the Joshi-Malafarina-Narayan (JMN) spacetimes,
Bertrand space-times etc, which are the examples of such static, singular
spacetimes formed as an end state or equilibrium state of gravitational
collapse in asymptotic time Joshi:2011zm ; Joshi:2013dva ; Banik:2016qvf .

There are many different catastrophic collapses which are happening in our
universe at different scales. In stellar scale only baryonic matter collapse
is important. On the other hand, in galactic halo scale or in galaxy cluster
scale, along with baryonic matter collapse one has to consider dark matter
collapse as wellwhite ; GunnGott72 ; Hogan:1985bc . In these different
collapsing scenarios, for various initial conditions, different types of final
static spacetimes can be formed. Depending upon initial conditions, the final
spacetime can be a black hole (rotating or non-rotating, charged or
uncharged), JMN spacetimes, Bertrand spacetimes or other possible static
spacetimes.

Based on recent research works such as above, the current emerging scenario
is, a massive star undergoing catastrophic gravitational collapse towards the
end of its life cycle may terminate into a black hole or a naked singularity,
depending on the initial conditions from which the collapse starts. Both the
black holes and naked singularity models would have very different physical
properties and there must be some distinguishable observational signatures of
those different spacetimes. Various researchers have investigated this
possibility and the observational signatures of different naked singularity
spacetimes have been examined in some detail. In Joshi:2013dva , Kovacs:2010xm
the authors investigated the accretion disk properties and gravitational
lensing properties of naked singularity spacetimes. In Shaikh:2018lcc , the
shadows created by a black hole and a JMN naked singularity are compared and
it is shown that for some cases, the naked singularity spacetime casts a
similar shadow compared to what a Schwarzschild black hole does. This is an
important theoretical prediction in the context of recent observation of the
shadow of M87 galactic center Akiyama:2019fyp . In Dey:2013yga ,Dey+15
astrophysical importance of Bertrand spacetimes is discussed. The
gravitational lensing due to the Janis-Newman-Winicour (JNW) naked singularity
is discussed in Gyulchev:2019tvk , where authors have reported that JNW
spacetime can cast a shadow which closely resembles to the shadow of a
Schwarzschild black hole. Therefore, the theoretical predictions of possible
observational signatures of naked singularity spacetimes would be worth
exploring.

From such a perspective, another possibility that we explore in the present
work is, comparing the particle trajectories in certain black hole and naked
singularity spacetimes. We know that freely falling massive particles follow
timelike geodesics. The curvature and causal structure of the spacetime
determines the nature of these timelike geodesics or nature of the particle
trajectories. Therefore, observing and studying timelike geodesics in a given
geometry gives information about the causal structure of that spacetime. There
are many literature where the timelike geodesics in different spacetimes are
investigated Zhou ; Levin:2009sk ; levin1 ; levin ; Glampedakis:2002ya ;
Chu:2017gvt ; Dokuchaev:2015zxe ; Borka:2012tj ; Martinez:2019nor ;
Fujita:2009bp ; Wang:2019rvq ; Suzuki:1997by ; Zhang:2018eau ;
Pugliese:2013zma ; Farina:1993xw ; Dasgupta:2012zf ; Shoom .

It may be useful to note here that recently, GRAVITY and SINFONI, which are
near infrared integral field spectrometers, reported their updated data on the
stellar motions near the center of our milky way M87 ; Eisenhauer:2005cv ;
center1 . These observations can give the information about the nature of the
central object (SGR-A*) in our Milkyway galaxy, which is considered to be a
super massive black hole with the mass of about $10^{6}M_{\odot}$. In such a
context, it would be important to do a comparative study of the nature of the
timelike geodesics in different black hole and singularity geometries, and how
they possibly differ from each other.

In this paper, we discuss the timelike geodesic in the 1st and 2nd kind of JMN
(JMN-1 and JMN-2) spacetimes. JMN-1 spacetime is an anisotropic fluid solution
of Einstein equations with zero radial pressure. On the other hand, JMN-2
spacetime is an isotropic fluid solution of Einstein equations. Both of these
spacetimes have a central naked singularity. We then compare these
trajectories with those for a Schwarzschild black hole.

In black hole physics, while we study different black hole geometries, it is
important to study the underlying physical processes that would give rise to
such black holes. In astrophysical settings, it is the gravitational collapse
of a massive star that would possibly create a black hole. For example, a
spherical homogeneous collapse would create a Schwarzschild black hole, which
has an event horizon and a singularity at the center. A physically more
realistic class of collapse models, with non-zero pressures and
inhomogeneities in the matter distribution included were studied in joshi:1993
; Satin:2014ima ; Joshi:2012mk . The final outcome of the collapse
asymptotically then was obtained as an equilibrium configuration, with a
singularity at the center, but with no event horizon, as trapped surfaces fail
to form as the collapse proceeds. Such a family of naked singularity static
spacetimes was in fact alluded to by Tolman many years ago tolman . The
accretion disk properties as well as the shadow properties for these models
was studied in detail in Joshi:2013dva ,Shaikh:2018lcc , and these were
compared to the black hole spacetimes in order to explore the key possible
theoretical differences, and possible observational implications were explored
and examined. In the next section, we outline the key details of these models,
whereas for a further discussion, we refer to the above references.

The paper is organized as follows. The next section (II) discusses the
relevant properties of JMN spacetimes needed for our purpose here, of studying
the timelike trajectories. In section (III), we discuss the effective
potentials for timelike geodesics in JMN spacetimes and Schwarzschild
spacetime, and also we compare the orbit equations for massive particles in
these two geometries. In section (IV), we start from an approximate solution
of these orbit equations to explain various characteristics of particles
orbits in the Schwarzschild and JMN spacetimes. This comparison of orbits
brings out different distinguishable properties of particle orbits in the
Schwarzschild black hole spacetime and JMN naked singularity spacetimes. We
then conclude with a discussion of the final results and the possible future
pointers.

##  II JMN Spacetimes

The JMN-1 and JMN-2 spacetimes have the following line elements,

$\displaystyle ds^{2}_{JMN-1}$ $\displaystyle=$
$\displaystyle-(1-M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(1-M_{0})}}dt^{2}+\frac{dr^{2}}{(1-M_{0})}+r^{2}d\Omega^{2}\,\,,$
(1) $\displaystyle ds^{2}_{JMN-2}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\lambda^{2}(2-\lambda^{2})}\left[(1+\lambda)^{2}\left(\frac{r}{R_{b}}\right)^{1-\lambda}-(1-\lambda)^{2}\left(\frac{r}{R_{b}}\right)^{1+\lambda}\right]^{2}dt^{2}+(2-\lambda^{2})dr^{2}+r^{2}d\Omega^{2}\,\,,$
(2)

where both the spacetimes match with the Schwarzschild geometry at the
boundary $r=R_{b}$. In JMN-1, the positive constant $M_{0}$ should be always
less than 1, and in JMN-2, $\lambda$ is a positive constant and its value is
less than unity. Here, $d\Omega^{2}$ is $d\theta^{2}+\sin^{2}\theta d\phi^{2}$
due to the spherical symmetry of JMN spacetimes. In this paper, we use the
units where Newton’s gravitational constant ($G_{N}$) and light velocity ($c$)
is equal to one.

Generally, modelling of a compact object is performed by considering a high
density compact region in vacuum. Therefore, the exterior spacetime of a
compact object is considered as the Schwarzschild spacetime. JMN spacetimes
also can be glued with Schwarzschild spacetime at a boundary $r=R_{b}$.
Therefore, we have a spacetime structure which is internally JMN-1 or JMN-2
and externally matched to a Schwarzschild spacetime, where the Schwarzschild
geometry can be written as,

$ds^{2}=-\left(1-\frac{M_{0}R_{b}}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{M_{0}R_{b}}{r}\right)}+r^{2}d\Omega^{2}\,\,,$
(3)

where the Schwarzschild radius ($R_{s}$) is $R_{s}=M_{0}R_{b}$, and as
$0<M_{0}<1$, so we have $R_{b}>R_{s}$. The total Schwarzschild mass
($M_{TOT}$) is $M_{TOT}=\frac{M_{0}R_{b}}{2}$. In general relativity, for the
matching of two spacetimes at a particular spacelike or timelike hypersurface,
we need to satisfy two junction conditions:

  1. 1.

The induced metrics of internal and external spacetimes on the matching
hypersurface should be identical with each other. One can see that this is
satisfied for the above two metrices (1), (2) at the timelike hypersurface
$r=R_{b}$. From the induced metric matching we get,
$M_{0}=\frac{1-\lambda^{2}}{2-\lambda^{2}}$, for JMN-2.

  2. 2.

Another condition is to match the extrinsic curvatures ($K_{ab}$) at the
hypersurface. Extrinsic curvature can be expressed in terms of the covariant
derivative of normal vectors on the hypersurface:
$K_{ab}=e^{\alpha}_{a}e^{\beta}_{b}\nabla_{\alpha}\eta_{\beta}$, where
$e^{\alpha}_{a}$ is the tangent vectors on the hypersurface and $\eta^{\beta}$
is the normal to that hypersurface.

One can show that due to the zero radial pressure of JMN-1 spacetime, the
extrinsic curvatures of JMN-1 and Schwarzschild spacetimes at $r=R_{b}$ are
automatically matched with each other Dey:20192 . On the other hand, JMN-2
spacetime is seeded by an isotropic fluid, therefore, it can be shown that in
order to match extrinsic curvature, the pressure in JMN-2 should vanish at
$r=R_{b}$. Pressure in JMN-2 can be written as,

$P_{JMN-2}=\frac{1}{(2-\lambda^{2})}\frac{1}{r^{2}}\left[\frac{(1-\lambda)^{2}A-(1+\lambda)^{2}Br^{2\lambda}}{A-Br^{2\lambda}}\right]\,\,,$
(4)

where
$A=\frac{(1+\lambda)^{2}R_{b}^{\lambda-1}}{4\lambda\sqrt{2-\lambda^{2}}}$ and
$B=\frac{(1-\lambda)^{2}R_{b}^{-\lambda-1}}{4\lambda\sqrt{2-\lambda^{2}}}$. It
can be checked that $P_{JMN-2}$ becomes zero at $r=R_{b}$.

As we mentioned, JMN spacetimes can be formed as an end state of gravitational
collapse. It can be shown how a general collapsing metric with non-zero
pressure can reach to an equilibrium state in asymptotic time without forming
an apparent horizon. It may be worth noting here that, using Newtonian
mechanics, one can show how a self-gravitating fluid with $N$ number of
particles reaches to a virialized state in asymptotic time. Such a system of
gravitationally interacting particles is considered to be virialized when,

$\langle T\rangle=-\frac{1}{2}\langle V_{T}\rangle,$ (5)

where $\langle T\rangle$ and $\langle V_{T}\rangle$ are the average kinetic
energy and average total potential energy (over a large system time period)
respectively. Analogous to Newtonian virialization process, the equilibrium
process which is mentioned in Joshi:2011zm , also takes an asymptotic time to
equilibrate the collapsing matter cloud. This type of quasi-static collapse
would be of relevance in the context of dark matter halo formation and galaxy
formation as well Dey:20191 ,Dey:20192 . Therefore, the final asymptotic
spacetimes (e.g. JMN-1, JMN-2 or others), which can be formed as an end state
of the pressure supported quasi-static collapse may give some distinguishable
astrophysical signatures between the black hole and naked singularity collapse
final states. This is in the sense that since these spacetimes are static,
non-vacuum solutions of Einstein equations, their timelike orbits should be
distinguishable from the timelike orbits in the vacuum black hole spacetime,
and it is these differences that are studied and reported in this paper.

##  III The Particle orbit equations in Schwarzschild, JMN-1 and JMN-2
spacetimes

###  III.1 Timelike Geodesics in General Spherically Symmetric, Static
Spacetime

We can write a spherically symmetric, static spacetime as,

$ds^{2}=-g_{tt}(r)dt^{2}+g_{rr}(r)dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})\,\,,$ (6)

where $g_{tt}$,$g_{rr}$ are the functions of $r$ only, and the azimuthal part
of the spacetime shows the spherical symmetry. For a particle which is freely
falling in this type of spacetime, the angular momentum ($h$) and the energy
($\gamma$) per unit of particle’s rest mass are always conserved, where the
angular momentum and energy conservation are the direct consequence of
spherical and temporal symmetry of the above spacetime (6). The conserved $h$
and $\gamma$ of a freely falling particle in the static, spherically symmetric
spacetime can be written as,

$\displaystyle
h=r^{2}\frac{d\phi}{d\tau}\,\,,\,\,\,\gamma=g_{tt}(r)\frac{dt}{d\tau}\,\,,$
(7)

where $\tau$ is the proper time of the particle and we consider
$\theta=\frac{\pi}{2}$. We know that freely falling particles always follow
timelike geodesics, for which $v_{\mu}v^{\mu}=-1$, where $v^{\mu}$ is the
particle’s four-velocity. From this normalization of four-velocity we can
write an effective potential which plays a crucial role on particles’
trajectories in a spacetime. For the static and spherically symmetric
spacetime which is mentioned in eq. (6), the effective potential can be
written as,

$V_{eff}=\frac{1}{2}\left[g_{tt}(r)\left(1+\frac{h^{2}}{r^{2}}\right)-1\right]\,\,,$
(8)

where the total energy ($E$) can be written as,

$E=\frac{g_{rr}(r)g_{tt}(r)}{2}\left(\frac{dr}{d\tau}\right)^{2}+V_{eff}(r)\,\,,$
(9)

where $E=\frac{\gamma^{2}-1}{2}$. To find the stable circular trajectories of
particles we need $V_{eff}(r_{c})=E$, $V_{eff}^{\prime}(r_{c})=0$ and
$V_{eff}^{\prime\prime}(r_{c})>0$, where $r_{c}$ is the radius of the circular
orbit. If we change $h$ or $\gamma$ keeping all the other parameter of the
metric constant, then the radius of the stable circular orbit also changes.
Using $V_{eff}(r)=E$, $V_{eff}^{\prime}(r)=0$, one can write down the required
$h$ and $\gamma$ for a circular orbit at a given radius,

$h^{2}=\frac{r^{2}}{\left(\frac{2g_{tt}(r)}{rg_{tt}^{\prime}(r)}-1\right)}\,,\,\,\,\gamma^{2}=\frac{2g_{tt}^{2}(r)}{rg_{tt}^{\prime}(r)}\frac{1}{\left(\frac{2g_{tt}(r)}{rg_{tt}^{\prime}(r)}-1\right)}\,\,,$
(10)

Figure 1: Here the region where $(V^{\prime\prime}_{eff})_{JMN-2}>0$ is shown,
where we have taken $R_{b}=1000$. We can see that
$(V^{\prime\prime}_{eff})_{JMN-2}>0$ is always valid inside the region
$0\leq\lambda<1$ and $0\leq r\leq R_{b}$.

where the above conditions are true for both stable and unstable circular
orbits. For stable circular orbits we have to consider another constrain
$V_{eff}^{\prime\prime}(r)>0$. Bound non-circular orbits can be describe by
solving $V_{eff}(r)=E$ which gives the radius of minimum approach ($r_{min}$)
and maximum approach ($r_{max}$) of the particle towards the center.
Therefore, one can define bound orbits of the freely falling particles in the
following way,

$\displaystyle V_{eff}(r_{min})=V_{eff}(r_{max})=E\,,\,\,$ $\displaystyle
E-V_{eff}(r)>0\,,\,\,\,\forall r\in(r_{min},r_{max}).$ (11)

Now if we want to describe the shape of an orbit of a particle with some
conserved value of $h$ and $E$, we then need to describe how the radial
coordinate $r$ changes with azimuthal coordinate $\phi$. Using eq. (9) we get,

$\frac{d\phi}{dr}=\frac{h}{r^{2}}\frac{\sqrt{g_{rr}(r)g_{tt}(r)}}{\sqrt{2(E-V_{eff})}}$
(12)

For some simplification, the above equation can be written in terms of
$u=\frac{1}{r}$, and then one can write a second order differential equation
by differentiating the above equation with respect to $\phi$,

$\frac{d^{2}u}{d\phi^{2}}+\frac{u}{g_{rr}(u)}-\left[\frac{u^{2}}{2g_{rr}^{2}(u)}+\frac{1}{2g_{rr}^{2}(u)h^{2}}-\frac{\gamma^{2}}{2g_{tt}(u)g_{rr}^{2}(u)h^{2}}\right]\frac{dg_{rr}(u)}{du}+\frac{\gamma^{2}}{2g_{tt}^{2}(u)g_{rr}(u)h^{2}}\frac{dg_{tt}(u)}{du}=0\,\,.$
(13)

From the above orbit equation, it can be seen that the shape of the orbit
depends totally on the spacetime where the test particle is freely falling.

###  III.2 Orbit equations for Schwarzschild, JMN-1 and JMN-2 Spacetime

While the Schwarzschild spacetime is a static, spherically symmetric, vacuum
solution of Einstein equations, it has necessarily a strong singularity at the
center which is covered by a null surface, the event horizon. As we stated in
previous subsection, due to the temporal and spherical symmetry, Schwarzschild
spacetime also has two conserved quantities: the energy ($\gamma_{SCH}$) and
angular momentum ($h_{SCH}$) per unit rest mass of the freely falling particle
in this spacetime. Using eq. (7), the conserved quantities $\gamma_{SCH}$ and
$h_{SCH}$ for Schwarzschild spacetime can be written as,

$\displaystyle\gamma_{SCH}$ $\displaystyle=$
$\displaystyle\left(1-\frac{M_{0}R_{b}}{r}\right)\left(\frac{dt}{d\tau}\right)\,\,,$
(14) $\displaystyle h_{SCH}$ $\displaystyle=$ $\displaystyle
r^{2}\left(\frac{d\phi}{d\tau}\right)\,\,,$ (15)

Using the general expression of the effective potential in the eq. (8), the
effective potential for Schwarzschild spacetime can be written as,

$(V_{eff})_{SCH}=\frac{1}{2}\left[\left(1-\frac{M_{0}R_{b}}{r}\right)\left(1+\frac{h_{SCH}^{2}}{r^{2}}\right)-1\right]\,\,.$
(16)

From the above expression of effective potential, we can derive the expression
of $h_{SCH}$ and $\gamma_{SCH}$ for circular geodesics,

$\displaystyle\gamma_{SCH}^{2}=\frac{2\left(r-M_{0}R_{b}\right)^{2}}{r\left(2r-3M_{0}R_{b}\right)}\,,\,h_{SCH}^{2}=\frac{M_{0}R_{b}r^{2}}{\left(2r-3M_{0}R_{b}\right)}\,\,,$
(17)

where we use the general expressions of $\gamma$ and $h$ in eq. (10). From the
above expression of conserved quantities for circular geodesics, it can be
seen that no circular orbit is possible in the range: $0\leq
r\leq\frac{3M_{0}R_{b}}{2}$. In terms of the total mass ($M_{TOT}$) of the
black hole, this range can be written as $0\leq r\leq 3M_{TOT}$, where
$M_{TOT}=\frac{M_{0}R_{b}}{2}$. However, for stable circular orbits, along
with above two conditions in eq. (17), we need
$(V_{eff}^{\prime\prime})_{SCH}>0$. One can write the expression of
$(V_{eff}^{\prime\prime})_{SCH}$ as,

$\displaystyle(V_{eff}^{\prime\prime})_{SCH}$ $\displaystyle=$
$\displaystyle\frac{2M_{TOT}}{r}\left(\frac{6M_{TOT}-r}{3M_{TOT}-r}\right)\,\,.$
(18)

Therefore, for stable circular orbits we need $\gamma$ and $h$ as stated in
eq. (17) and $r\geq 6M_{TOT}$. There cannot be any stable circular orbits
below $r=6M_{TOT}$ in Schwarzschild spacetime, therefore, $r=6M_{TOT}$ is
known as the Innermost Stable Circular Orbit (ISCO). For bound orbits in this
spacetime, the conditions which are stated in eq. (11), should be fulfilled.
Using the eq. (13), we can write the shape of particle’s orbits in
Schwarzschild spacetime,

$\frac{d^{2}u}{d\phi^{2}}+u=\frac{3M_{0}R_{b}}{2}u^{2}+\frac{M_{0}R_{b}}{2h^{2}}\,\,.$
(19)

(a) particle orbits in Schwarzschild where $M_{0}=0.09$,$h=200$,$E=-0.0268$
(b) Particle orbit in jmn-1 where $M_{0}=0.09$,$h=200$, $E=-0.0268$ (c)
Particle orbit in jmn-1 where $M_{0}=0.55$,$h=1100$, $E=-0.004$ (d) Particle
orbit in JMN-2 where $\lambda=0.9$, $h=290$, $E=-0.0445$ (e) Particle orbit in
JMN-2 where $\lambda=0.1$, $h=892$, $E=-0.04885$

Figure 2: In this figure, particle orbits in Schwarzschild, JMN-1 and JMN-2
are shown. It can be seen that for $M_{0}=0.09$ (JMN-1) and $\lambda=0.9$
(JMN-2), the angular distance travelled by the particle to reach one
perihelion point to another perihelion point is less than $2\pi$, whereas for
$M_{0}=0.55$ (JMN-1) and $\lambda=0.1$ (JMN-2) it reaches after $2\pi$
rotation. The black dark region in the diagram of orbit in Schwarzschild
spacetime shows the position of the black hole and the brown circle shows the
minimum approach of the particle towards the center. The black circle outside
the orbit in JMN-2, shows the position of matching radius $R_{b}=1000$.

As we mentioned before, JMN-1 and JMN-2 spacetimes are spherically symmetric,
static, non-vacuum solutions of Einstein equations. These spacetimes are
seeded by anisotropic fluid (with zero radial pressure) and isotropic fluid
respectively. Both these spacetimes can be formed as an asymptotic end state
of a quasi-static gravitational collapse process Joshi:2011zm ,Joshi:2013dva .
Both these spacetimes have strong curvature singularities at the center. There
are no event horizons in these spacetimes, therefore, the quantum gravity
effects around the singularities at the center can be visible by asymptotic
observer, at least in principle. Using previously discussed technique, we can
derive now the effective potential for JMN-1 and JMN-2 spacetimes,

$\displaystyle(V_{eff})_{JMN-1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(1-M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(1-M_{0})}}\left(1+\frac{h_{JMN-1}^{2}}{r^{2}}\right)-1\right]\,\,,$
(20) $\displaystyle(V_{eff})_{JMN-2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{16\lambda^{2}(2-\lambda^{2})}\left[(1+\lambda)^{2}\left(\frac{r}{R_{b}}\right)^{1-\lambda}-(1-\lambda)^{2}\left(\frac{r}{R_{b}}\right)^{1+\lambda}\right]^{2}\left(1+\frac{h_{JMN-2}^{2}}{r^{2}}\right)-1\right]\,\,,$
(21)

where $h_{JMN-1}$ and $h_{JMN-2}$ are the conserved angular momentum per unit
rest mass of the test particles in JMN-1 and JMN-2 spacetimes respectively.
For JMN-1 spacetime, the conditions for stable circular orbits are,

$\displaystyle\gamma^{2}_{JMN-1}$ $\displaystyle=$
$\displaystyle\frac{2(1-M_{0})^{2}\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{1-M_{0}}}}{\left(2-3M_{0}\right)}\,,\,$
(22) $\displaystyle h^{2}_{JMN-1}$ $\displaystyle=$
$\displaystyle\frac{r^{2}M_{0}}{2-3M_{0}}\,\,,$ (23)
$\displaystyle(V_{eff}^{\prime\prime})_{JMN-1}$ $\displaystyle=$
$\displaystyle\frac{M_{0}}{R_{b}^{2}}\left(\frac{r}{R_{b}}\right)^{\frac{3M_{0}-2}{1-M_{0}}}>0\,\,.$
(24)

From the above three equations, it can be understood that for
$M_{0}<\frac{2}{3}$, JMN-1 spacetime has stable circular orbits of any radius.
Therefore, unlike Schwarzschild spacetime, in JMN-1 spacetime, for
$M_{0}<\frac{2}{3}$, there is no such region where unstable circular orbits
exist. On the other hand, for $M_{0}>\frac{2}{3}$ there is no solution for
stable circular orbits at any value of $r$, as $h_{JMN-1}$ and
$\gamma_{JMN-1}$ become imaginary. As we previously mentioned, the $\lambda$
parameter in JMN-2 space-time should be inside the range $0\leq\lambda<1$ and
$M_{0}=\frac{1-\lambda^{2}}{2-\lambda^{2}}$. Now, if we consider the matching
radius $R_{b}=nM_{TOT}$ where $n$ is a positive number, we can write,
$R_{b}=n\frac{M_{0}R_{b}}{2}=\frac{nR_{b}(1-\lambda^{2})}{2(2-\lambda^{2})}$,
which implies,

$\lambda=\sqrt{\frac{n-4}{n-2}}\,\,.$ (25)

The above equation shows that the matching radius $R_{b}$ for JMN-2 spacetime
should be $4M_{TOT}\leq R_{b}<\infty$. One can see that when
$n\rightarrow\infty$ or $R_{b}\rightarrow\infty$, the value of
$\lambda\rightarrow 1$. Therefore, $\lambda$ close to unity implies a largely
extended compact object which has JMN-2 geometry inside, and outside the
object, the spacetime is Schwarzschild. On the other hand, in JMN-1 spacetime
there is no minimum limit of $R_{b}$, $R_{b}$ can have any value between zero
to infinity. Using eq. (10), we can write the conditions for circular orbits
in JMN-2 spacetime as,

$\displaystyle\gamma_{jmn-2}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left[\frac{r^{2}\left\\{-(1+\lambda)^{2}+(\lambda-1)^{2}\left(\frac{r}{R_{b}}\right)^{2\lambda}\right\\}^{3}\left(\frac{r}{R_{b}}\right)^{-2\lambda}}{\lambda^{3}(\lambda^{2}-2)\left\\{(1+\lambda)^{2}+(\lambda-1)^{2}\left(\frac{r}{R_{b}}\right)^{2\lambda}{2\lambda}\right\\}R_{b}^{2}}\right]\,\,,$
(26) $\displaystyle h_{jmn-2}^{2}$ $\displaystyle=$
$\displaystyle\left[\frac{r^{2}(1-\lambda)(\lambda+1)\left\\{(1+\lambda)-(1-\lambda)\left(\frac{r}{R_{b}}\right)^{2\lambda}\right\\}}{\lambda\left[(\lambda+1)^{2}+(\lambda-1)^{2}\left(\frac{r}{R_{b}}\right)^{2\lambda}\right]}\right]\,\,.$
(27)

We can see from the above expressions of $\gamma_{JMN-2}$ and $h_{JMN-2}$ that
for $0\leq\lambda<1$ we can get circular orbits. For stability of circular
orbits, we need $(V^{\prime\prime}_{eff})_{JMN-2}>0$. Due to the large
analytical expression of $(V^{\prime\prime}_{eff})_{JMN-2}$, in fig. (1) we
numerically show the region where it is greater than zero. One can see that
$(V^{\prime\prime}_{eff})_{JMN-2}>0$ is possible for the region
$0\leq\lambda<1$ and $0\leq r\leq R_{b}$. Therefore, like JMN-1 spacetime,
JMN-2 spacetime also can have circular orbits very close to the center.
Therefore, in both JMN-1 and JMN-2 space-times, particles can have stable
circular orbit at any value of $r\leq R_{b}$. Using general orbit eq. (13) for
spherically symmetric, static spacetime, one can write the following orbit
equations of a particle freely falling in JMN-1 and JMN-2 spacetime
respectively,

$\displaystyle\frac{d^{2}u}{d\phi^{2}}+(1-M_{o})u-\frac{\gamma^{2}}{2h^{2}}\frac{M_{0}}{(1-M_{0})}\left(\frac{1}{u}\right)\left(\frac{1}{uR_{b}}\right)^{\frac{-M_{0}}{(1-M_{0})}}$
$\displaystyle=$ $\displaystyle 0\,\,,$ (28)
$\displaystyle\frac{d^{2}u}{d\phi^{2}}+\frac{u}{(2-\lambda^{2})}-\frac{16\lambda^{2}\gamma^{2}u^{1+2\lambda}}{h^{2}}\left[\frac{(1+\lambda)^{2}}{(R_{b})^{1-\lambda}}(1-\lambda)u^{2\lambda}-\frac{(1-\lambda)^{2}}{(R_{b})^{1+\lambda}}(1+\lambda)\right]\left[\frac{(1+\lambda)^{2}}{(R_{b})^{1-\lambda}}u^{2\lambda}-\frac{(1-\lambda)^{2}}{(R_{b})^{1+\lambda}}\right]^{-3}$
$\displaystyle=$ $\displaystyle 0$ (29)

(a) $M_{0}=0.05$,$h=105.26$, $E=-0.02$, $R_{b}=1000$

(b) Particle orbit in Schwarzschild spacetime

(c) Particle orbit in JMN-1 spacetime

Figure 3:  Here, particle orbits in Schwarzschild and JMN-1 spacetimes are
shown, where we take the Schwarzschild mass
($M_{TOT}=\frac{M_{0}R_{b}}{2}=25$), same for both spacetimes. In the
effective potential diagram, the blue line and red line correspond to JMN-1
and Schwarzschild spacetime respectively and the dotted horizontal black line
is indicating the total energy of the freely falling particle. The black
circle outside the orbit in JMN-1, shows the position of matching radius
$R_{b}=1000$, which is greater than the Schwarzschild radius
$R_{s}=M_{0}R_{b}=50$, which is shown by dotted black circle.

(a) $M0=0.008,h=20,E=-0.003$

(b) Particle orbit in Schwarzschild spacetime

(c) Particle orbit partly in JMN-1 and partly in Schwarzschild spacetime.

Figure 4:  In this figure, we show the trajectory of a particle when it
crosses the matching radius $R_{b}$. In the third diagram the dotted blue line
shows what would be the particle trajectory if there is no Schwarzschild
spacetime outside. On the other hand, the solid blue line shows particle’s
actual path in JMN-1 spacetime and the solid red line shows particle’s
trajectory in Schwarzschild spacetime.

##  IV Shape of bound orbits in Schwarzschild and JMN spacetimes

The orbit equations which are written in eq. (19), (28) and (29) can give us
the information about the shape of different orbits of freely falling
particles in Schwarzschild, JMN-1 and JMN2 spacetimes respectively. One can
numerically solve those orbit equations, however, if we obtain the
approximation solution of those orbit equations, then that can be used to
understand some important properties of the orbits in a better way. As we
know, the Schwarzschild spacetime is asymptotically flat, therefore, the orbit
eq. (19) should tends to Newtonian orbit equation in weak field limit. One can
see that in asymptotic limit eq. (19) becomes Newtonian orbit equation,

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

where we neglect the $u^{3}$ term as an asymptotic limit. As we know, the
solution of the above equation is
$u=\frac{M_{0}R_{b}}{2h^{2}}\left[1+e\cos(\phi)\right]$, which suggests that,
in asymptotic limit, the orbiting particle will reach its previous position
after one full $2\pi$ rotation. However, the same is not true when the
particle is near the center and we cannot neglect the $u^{3}$ term in the
orbit equation in the corresponding Schwarzschild spacetime.

If we consider the case when the eccentricity of a particle’s orbit is small
enough to neglect the second and higher order contributions of eccentricity
($e$), we can write the following approximate solution for the orbit eq. (19)
of Schwarzschild spacetime,

$u=\frac{1}{p}\left[1+e\cos(m\phi)+O(e^{2})\right]\,\,,$ (30)

(a) $M0=0.62,h=1250,E=-0.02$

(b) Particle orbits in Schwarzschild spacetime

(c) Particle orbits in JMN-1 spacetime

Figure 5: In this diagram, we take $M_{0}=0.62$. As $M_{0}>0.6$, the minimum
value of effective potential in Schwarzschild spacetime is always outside the
matching radius $R_{b}$. In the effective potential diagram, the blue line is
for JMN-1 spacetime, whereas the red line corresponds to Schwarzschild
spacetime. The blue line ends at the matching radius $R_{b}=1000$.

where $m$ and $p$ are positive real numbers. As the orbit equation in
Schwarzschild spacetime asymptotically approaches to Newtonian orbit equation,
one can write the above form of solution. Using eq. (30),(19), one can get the
following expression of $p$ and $m$ by neglecting second order and higher
order terms,

$\displaystyle
p=\frac{1+\sqrt{1-\frac{3M_{0}^{2}R_{b}^{2}}{h^{2}}}}{\frac{M_{0}R_{b}^{2}}{h^{2}}}\,\,,$
(31) $\displaystyle m=\sqrt{1-\frac{3M_{0}}{p}}\,\,,$ (32)

where we do not write the another solution of
$p=\frac{1-\sqrt{1-\frac{3M_{0}^{2}R_{b}^{2}}{h^{2}}}}{\frac{M_{0}R_{b}^{2}}{h^{2}}}$,
as $m$ becomes imaginary when $h>\sqrt{3}M_{0}R_{b}$, which is a necessary
condition for real values of $p$. When $h>\sqrt{3}M_{0}R_{b}$, we always have
$p>3M_{0}$ and $0<m<1$, which implies that starting from the closest approach
to the center (perihelion position), after a full $2\pi$ rotation particle
would not reach its previous perihelion position. To reach the perihelion
position again, the particle needs some extra angle of rotation which depends
upon the value of $m$. This precession of the closest approach of the orbit is
known as perihelion precession. In Schwarzschild case, as $0<m<1$, the orbit
precesses in the direction of particle rotation. In weak field limit, we can
take $h\gg\sqrt{3}M_{0}R_{b}$, which implies
$p=\frac{2h^{2}}{M_{0}R_{b}^{2}}$.and $p\gg 3M_{0}$. With this limit, we can
write the approximate expression of $m$ as,
$m=\left(1-\frac{3M_{0}^{2}R_{b}^{2}}{4h^{2}}\right)$. Therefore, in weak
field limit, if a particle starts from perihelion position, after a rotation
of angle $\frac{2\pi}{\left(1-\frac{3M_{0}^{2}R_{b}^{2}}{4h^{2}}\right)}$ it
will again reach the perihelion position. Therefore, the particle has to
travel extra

$\delta\phi_{prec}=\frac{6\pi M_{0}^{2}R_{b}^{2}}{4h^{2}}=\frac{6\pi
M_{TOT}^{2}}{h^{2}}\,\,.$

This $\delta\phi_{prec}$ is the weak field limit of the precession angle of a
particle’s orbit in Schwarzschild spacetime.

Coming now to the JMN case, the approximate solution of the orbit equation
(28) corresponding to the JMN-1 spacetime can also be written considering
small value of eccentricity $e$. If we transform the $u\rightarrow uR_{b}$, we
can write the eq. (28) in the following form,

$\tilde{u}\frac{d^{2}\tilde{u}}{d\phi^{2}}+\frac{1}{1+2\delta}\tilde{u}^{2}=C_{\delta}\tilde{u}^{2\delta}\,\,,$
(33)

(a) Effective potential for $Rb/h=0.1$

(b) Effective potential for $Rb/h=0.4$

(c) Effective potential for $Rb/h=0.9$

Figure 6: This diagram shows how the behaviour of effective potential changes
with $h$, where we take $M_{0}=0.65$ and $R_{b}=1000$.

where $\delta=\frac{M_{0}}{2(1-M_{0})}$,
$C_{\delta}=\frac{\gamma^{2}R_{b}}{h^{2}}\delta$ and $\tilde{u}=uR_{b}$. In
Struck:2005hi ,Struck:2005oi , the author introduces an approximate solution
for this type of differential equations. The solution can be written as,

$\tilde{u}=\frac{1}{p}\left[1+e\cos(m\phi)+O(e^{2})\right]^{\frac{1}{2}+\delta}\,\,.$
(34)

Using eq. (28) and eq. (34) we can get the following expressions of $p$ and
$m$,

$\displaystyle p$ $\displaystyle=$
$\displaystyle\left[C_{\delta}(1+2\delta)\right]^{-\frac{1}{2(1-\delta)}}\,\,,$
$\displaystyle m$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2(1-\delta)}{2\delta+1}}=\sqrt{2-3M_{0}}\,\,.$ (35)

From the above expression of $m$, it can be seen that for
$0<M_{0}<\frac{1}{3}$, $m$ is greater than one, whereas for
$\frac{1}{3}<M_{0}<\frac{2}{3}$, $m$ is less than one. Therefore, in JMN-1
spacetime, starting from the perihelion position, a particle can reach the
same position before (for $0<M_{0}<\frac{1}{3}$ ) and after (for
$\frac{1}{3}<M_{0}<\frac{2}{3}$) one $2\pi$ rotation. Consequently, the nature
of the orbits also changes across $M_{0}=\frac{1}{3}$. For
$M_{0}<\frac{1}{3}$, the orbit precesses in the opposite direction of particle
motion, whereas, for $M_{0}>\frac{1}{3}$ we get Schwarzschild like precession.
In JMN-2, also we can have two type of precession. In fig. (2), we show all
these properties of orbits in JMN-1 and JMN-2 and compare it with the orbits
in Schwarzschild spacetime. In that figure, it can be seen that the particle
starts form its closest approach $r_{c}=558.941$ in Schwarzschild spacetime
and again reaches to that radius after a $2\pi$ rotation. On the other hand,
in that figure, it is shown that in JMN-1 and JMN-2 spacetimes, particle can
reach the perihelion position before and after the $2\pi$ rotation. It follows
that the particle orbits and the precession behaviours are significantly
different in these black hole and naked singularity spacetimes.

Using eq. (35), one can easily show that the orbits in JMN-1 spacetime have no
precession when $M_{0}=\frac{1}{3}$ or $m=1$. When $M_{0}$ is very small
compared to $\frac{2}{3}$, we can write the eq. (33) as,

$\tilde{u}\frac{d^{2}\tilde{u}}{d\phi^{2}}+\tilde{u}^{2}=C_{\delta}\,\,,$ (36)

where we consider the value of $R_{b}$ to be large, so that
$C_{\delta}=\frac{\gamma^{2}R_{b}}{h^{2}}\delta$ cannot be neglected. In
Newtonian mechanics, the above orbit equation appears when one takes a
logarithmic potential. For $0<M_{0}\ll\frac{2}{3}$, $m$ approaches $\sqrt{2}$.
Therefore, in this limit, in JMN-1 spacetime, particle reaches to the
perihelion point after 254.56 degrees of rotation.

As we previously mentioned, in this paper we pedagogically compare the bound
orbits in Schwarzschild spacetimes and bound orbits in a spacetime structure
where it is internally JMN-1 or JMN-2, externally matched to a Schwarzschild
spacetime. We basically show how a freely falling particle, with a particular
angular momentum and total energy, moves in Schwarzschild spacetime and in a
naked singularity spacetime structure, where the Schwarzschild mass $M_{TOT}$
is same for both Schwarzschild spacetime and the other spacetime structure. In
Fig. (3), it is shown that the minimum value of effective potentials in
Schwarzschild and JMN-1 spacetimes are inside the matching radius
$R_{b}=1000$. It is possible to have particles’ whole trajectory inside the
matching radius $R_{b}$ when the following inequality holds,

$\frac{R_{b}}{h}>\sqrt{\frac{2-3M_{0}}{M_{0}}}\,\,.$ (37)

One can obtain the above relation by considering the minimum value of
effective potential of JMN-1 spacetime inside the matching radius $R_{b}$.
With the above inequality, a particle needs to have certain amount of total
energy to be inside the interior JMN-1 spacetime. Now, maintaining the above
condition, $\frac{R_{b}}{h}$ can have arbitrary large values which gives bound
trajectories of particle inside the JMN-1 spacetime. However, as it was
discussed previously (eq. (31)), in Schwarzschild spacetime there exists a
lower limit of $h$ below which no bound timelike orbits are possible. For
$h>\sqrt{3}M_{0}R_{b}$ or $\frac{R_{b}}{h}<\frac{1}{\sqrt{3}M_{0}}$, particles
in Schwarzschild spacetime can have bound orbits. Therefore, to compare with
bound orbits in Schwarzschild spacetime, we need to satisfy the following
condition,

$\sqrt{\frac{2-3M_{0}}{M_{0}}}<\frac{R_{b}}{h}<\frac{1}{\sqrt{3}M_{0}}\,\,.$
(38)

If the above condition is satisfied, with some suitable total energy, a
massive particle can have entire trajectory inside the interior JMN spacetime.
In fig. (3), we take $M_{0}=0.05$, therefore the range of $\frac{R_{b}}{h}$
is, $6.08<\frac{R_{b}}{h}<11.547$. To maintain this inequality of
$\frac{R_{b}}{h}$, we consider $h=105.26$ and $R_{b}=1000$, where
$\frac{R_{b}}{h}=9.5$ and to get the entire particle trajectory inside the
matching radius $R_{b}$, we take $E=-0.02$. These parameter values which
satisfy the inequality of eq. (38), allow us to compare the bound, timelike
trajectories in a Schwarzschild spacetime with the bound, timelike
trajectories in JMN-1 spacetime. With the above mentioned parameter values, it
can be seen in the fig. (3(b)) and fig. (3(c)) that the orbit precession in
Schwarzschild spacetime is in the direction of particle motion, however, on
the other hand, in JMN-1 spacetime particle’s orbits precesses in the opposite
direction of the particle’s motion.

In fig. (3), it can be seen that a particle with energy $E=-0.015$, cannot
have whole bound orbit inside the matching radius $R_{b}=1000$. This situation
is described in fig. (4), where we take $M0=0.008,h=20,E=-0.003$. With the
total energy $E=-0.003$, one can see in fig. (4(a)) that the particle will
cross the matching radius $R_{b}$. In fig. (4(c)), we show that as particle
crosses the matching radius, the trajectory (shown by solid blue and solid red
line) of the particle changes from what it should be (shown by dotted blue
line) in JMN-1 spacetime.

Keeping the matching radius fixed ar $r=R_{b}$, if we make $h$ greater than
the value $R_{b}\sqrt{\frac{M_{0}}{2-3M_{0}}}$, no bound orbit will be
possible inside the interior JMN-1 spacetime, and only in the external
Schwarzschild spacetime particles can have bound orbits. Therefore, there
exists a maximum limit of angular momentum (per unit rest mass) $h$, below
which particle’s full trajectory can be inside the matching radius $R_{b}$. On
the other hand, if $h<\sqrt{3}M_{0}R_{b}$, there will be no bound orbits in
Schwarzschild spacetime. As
$\sqrt{3}M_{0}R_{b}<R_{b}\sqrt{\frac{M_{0}}{2-3M_{0}}}$, it is possible that
the particle can have full trajectories inside the matching radius. This type
of situation is shown in the fig. (6(c)). Also in JMN-2 spacetime, we have the
following inequality for which it is possible that the particle can have bound
trajectories inside the JMN-2 spacetime and also in Schwarzschild spacetime,

$\sqrt{\frac{1+\lambda^{2}}{1-\lambda^{2}}}<\frac{R_{b}}{h}<\frac{2-\lambda^{2}}{\sqrt{3}\left(1-\lambda^{2}\right)}\,\,.$
(39)

An important point here is, as JMN spacetimes have no event horizon, particles
with some finite angular momentum can go very close to the central strong
singularity. On the other hand, as we know, in Schwarzschild spacetime, a
particle having a bound orbit can only reach up to $r=4M_{TOT}=2M_{0}R_{b}$
and below $r=6M_{TOT}$, stable circular orbit is not possible. In JMN-1
spacetime, particle with $h<M_{0}R_{b}\sqrt{\frac{M_{0}}{2-3M_{0}}}$, can have
bound trajectories inside the Schwarzschild radius. In the fig. (5), with
$M_{0}=0.62,h=1250,E=-0.02$, we show that particle orbits in JMN-1 spacetime
can be inside the Schwarzschild radius $R_{s}$, In fig. (5(a)), we show that
the effective potential has two minimum: one is outside $R_{b}$ and another
one is inside the interior JMN-1 spacetime. One can check that this type of
potential is not possible for JMN-2 spacetime for any value of $\lambda$. In
fig. (6), we show how the nature of effective potential changes when we change
the $h$, where the effective potential corresponds to the spacetime structure
which is internally JMN-1 and externally Schwarzschild spacetime. The fig.
(6(a)) show the situation when
$\sqrt{\frac{2-3M_{0}}{M_{0}}}>\frac{R_{b}}{h}$. In this case, there is no
bound orbit inside the matching radius, however, in the exterior Schwarzschild
spacetime particle’s bound orbits exist. In fig. (6(b)), we show the case
where the inequality in eq. (38) holds. The third case (shown in fig. (6(c)))
is for $\frac{R_{b}}{h}>\frac{1}{\sqrt{3}M_{0}}$. In this case, bound orbit of
the particle can only be possible inside the interior JMN-1 spacetime.

##  V Discussion and Conclusion

In the present investigation, we examined the timelike particle trajectories
in the JMN and Schwarzschild geometry, in order to explore the causal
structure of these spacetimes, and to understand the main characteristic
differences between the two. The study of these orbits bring out several
interesting differences in the causal structure of these black hole and naked
singularity spacetimes, some of these are summarized as below:

  * •

In the Schwarzschild spacetime, the bound trajectories of a massive particle
always precess in the direction of the particle motion. However, in JMN naked
singularity spacetimes, for a range of parameter values, the bound orbits
precess in the opposite direction of particle motion, which is a very novel
feature. In JMN-1 spacetime, for $0<M_{0}<\frac{1}{3}$, the orbits of a
massive particle precess in the opposite direction of particle motion and for
$\frac{1}{3}<M_{0}<\frac{2}{3}$, the orbits precess in the same direction of
particle motion. In fig. (2), we show these two situations and compare it with
the Schwarzschild spacetime. By the eq. (35), we show that there exists a
maximum value of the angle ($\delta\phi_{prec}=254.56$ degree) for the
opposite precession of a massive particle in JMN-1 spacetime.

  * •

In Schwarzschild spacetime the perihelion point of a timelike orbit always
lies in $r>2M_{0}R_{b}$. On the other hand, in JMN naked singularity
spacetime, the perihelion point of a timelike orbit lies in $r>0$ region.
Therefore, in JMN spacetimes, a massive particle can go very close to the
central naked singularity. In Schwarzschild spacetime, there exists an inner
most stable circular orbit (ISCO) at $r=3M_{0}R_{b}$, however in JMN
spacetimes, stable circular orbit of a massive particle can exist at any
radius. This difference can create distinguishable accretion disk properties
which can be detectable Joshi:2013dva ,Shaikh:2018lcc .

  * •

In fig. (6), it can be seen that the effective potential of JMN spacetimes
becomes positive infinite at the center. This indicates that in JMN
spacetimes, a massive particle with non-zero angular momentum cannot reach the
center. However, as the effective potential in Schwarzschild spacetime becomes
negative infinity at the center, a massive particle with non-zero angular
momentum and suitable total energy can reach the center.

As we know, GRAVITY and SINFONI are continuously observing the stellar motion
around the Milkyway galaxy center. This observation can give us important
information about the causal structure of galactic center, and on its mass and
dynamics. In this context, we show in this paper that, the timelike geodesics
in JMN naked singularity spacetime can be significantly different from the
timelike geodesics in Schwarzschild spacetime. There are other astrophysical
important naked singularity spacetimes (e.g. the JNW spacetime, Bertrand
spacetime) for which we can do a similar comparison with Schwarzschild
spacetimes, which will be reported separately parth .

## References

  * (1) J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939).
  * (2) S. Datt, Zs. f. Phys. 108, 314 (1938).
  * (3) P. S. Joshi and I. H. Dwivedi,  Phys. Rev. D 47, 5357 (1993).
  * (4) F. C. Mena, R. K. Tavakol and P. S. Joshi,  Phys. Rev. D 62, 044001 (2000).
  * (5) S. Jhingan, P. S.Joshi and T. P. Singh,  Class. Quant. Grav. 13, 3057 (1996).
  * (6) T. P. Singh and P. S. Joshi,  Class. Quant. Grav. 13, 559 (1996).
  * (7) P. S. Joshi, D. Malafarina and R. Narayan,  Class. Quant. Grav. 28, 235018 (2011).
  * (8) P. S. Joshi, D. Malafarina and R. Narayan,  Class. Quant. Grav. 31, 015002 (2014).
  * (9) U. Banik, D. Dey, K. Bhattacharya and T. Sarkar,  Gen. Rel. Grav. 49, no. 9, 116 (2017).
  * (10) S. D. M. White and D. Q. Liu, The Origin And Evolution Of Structures In A Universe Dominated By Cold Dark Matter, Tuxni, Proceedings, Origin, structure and evolution of galaxies, 281 (1987).
  * (11) J. E. Gunn and J. R. Gott III, ApJ 176, 1 (1972).
  * (12) C. J. Hogan, N. Kaiser, M. S. Turner, N. Vittorio and S. D. M. White, FERMILAB-CONF-85-057-A.
  * (13) Z. Kovacs and T. Harko,  Phys. Rev. D 82, 124047 (2010).
  * (14) R. Shaikh, P. Kocherlakota, R. Narayan and P. S. Joshi,  Mon. Not. Roy. Astron. Soc. 482, no. 1, 52 (2019).
  * (15) K. Akiyama et al. [Event Horizon Telescope Collaboration],  Astrophys. J. 875, no. 1, L5 (2019).
  * (16) D. Dey, K. Bhattacharya and T. Sarkar, Phys. Rev. D 88, 083532 (2013).
  * (17) D. Dey, K. Bhattacharya and T. Sarkar, Phys. Rev. D 87, 10, 103505 (2013).
  * (18) G. Gyulchev, P. Nedkova, T. Vetsov and S. Yazadjiev,  arXiv:1905.05273 [gr-qc].
  * (19) S. Zhou, R. Zhang, J. Chen and Y. Wang, Int. J. Theor. Phys. 54, no. 8, 2905 (2015).
  * (20) J. Levin, Class. Quant. Grav. 26, 235010 (2009).
  * (21) Janna Levin and Rebecca Grossman, Phys. Rev. D 79, 043016 (2009).
  * (22) Janna Levin and Gabe Perez-Giz, Phys. Rev. D 77, 103005 (2008).
  * (23) K. Glampedakis and D. Kennefick, Phys. Rev. D 66, 044002 (2002).
  * (24) D. S. Chu et al., Astrophys. J. 863, no. 1, 111 (2018).
  * (25) V. I. Dokuchaev, Y. N. Eroshenko and K. S. Klimkov,  Phys. Procedia 74, 292 (2015).
  * (26) D. Borka, P. Jovanovic, V. B. Jovanovic and A. F. Zakharov,  Phys. Rev. D 85, 124004 (2012).
  * (27) C. Martínez, N. Parra, N. Valdés and J. Zanelli, arXiv:1902.00145 [hep-th],(2019)
  * (28) R. Fujita and W. Hikida,  Class. Quant. Grav. 26, 135002 (2009)
  * (29) Y. Wang, X. Liu, N. Yang and J. Jia,  arXiv:1903.12486v1 [gr-qc]
  * (30) S. Suzuki and K. i. Maeda, Phys. Rev. D 58, 023005 (1998)  Phys. Rev. D 58, 023005 (1998)
  * (31) M. Zhang and W. B. Liu,  Phys. Lett. B 789, 393 (2019)
  * (32) D. Pugliese, H. Quevedo and R. Ruffini,  Phys. Rev. D 88, no. 2, 024042 (2013)
  * (33) C. Farina, J. Gamboa and A. J. Segui-Santonja,  Class. Quant. Grav. 10, L193 (1993)
  * (34) A. Dasgupta, H. Nandan and S. Kar,  Phys. Rev. D 85, 104037 (2012)
  * (35) A. A. Shoom, C. Walsh and I. Booth,  Phys. Rev. D 93, no. 6, 064019 (2016)
  * (36) Gravity Collaboration and Abuter et al. app. 618 L10
  * (37) F. Eisenhauer et al.,  Astrophys. J. 628, 246 (2005).
  * (38) S. Gillessen et al., Astrophys. J. 837, 30 (2017).
  * (39) P. S. Joshi and I. H. Dwivedi, Phys. Rev. D 47, 5357 (1993).
  * (40) S. Satin, D. Malafarina and P. S. Joshi,  Int. J. Mod. Phys. D 25, 1650023 (2016).
  * (41) P. S. Joshi and D. Malafarina, Int. J. Mod. Phys. D 20, 2641 (2011).
  * (42) R. C. Tolman, Proc. Natl. Acad. Sci. USA. , 20(3), 169–176, 1934.
  * (43) D. Dey and P. S. Joshi,  AJM, 2019
  * (44) D. Dey, P. Kocherlakota and P. S. Joshi,  arXiv:1907.12792 [gr-qc].
  * (45) R. Arnowitt, S. Deser, C. W. Misner, Phys. Rev. 116, 1322 (1959).
  * (46) C. Struck,  Astron. J. 131, 1347 (2006)
  * (47) C. Struck,  Monthly Notices of the Royal Astronomical Society, 446, 3139-3149 (2014).
  * (48) A. B. Joshi, P. Bambhaniya, D. Dey, P.S. Joshi, in preparation.