# Timelike Geodesics in Naked Singularity and Black Hole Spacetimes II

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

We derive here the orbit equations of particles in naked singularity
spacetimes, namely the Bertrand (BST) and Janis-Newman-Winicour (JNW)
geometries, and for the Schwarzschild black hole. We plot the orbit equations
and find the Perihelion precession of the orbits of particles in the BST and
JNW spacetimes and compare these with the Schwarzschild black hole spacetime.
We find and discuss different distinguishing properties in the effective
potentials and orbits of particle in BST, JNW and Schwarzschild spacetimes,
and the particle trajectories are shown for the matching of BST with an
external Schwarzschild spacetime. We show that the nature of perihelion
precession of orbits in BST and Schwarzschild spacetimes are similar, while in
the JNW case the nature of perihelion precession of orbits is opposite to that
of the Schwarzschild and BST spacetimes. Other interesting and important
features of these orbits are pointed out.

$\bm{keywords}$ : Naked singularity, Black hole, Perihelion precession,
Galaxy, Universe.

##  I Introduction

In General theory of Relativity (GR), many predictions have been tested by
observational evidence, such as the precession of perihelion of orbits of
mercury precession , the bending of light near the sun dyson , gravitational
red shift rebka , and more recently the discovery of gravitational waves ligo
. In such a context, the gravitational collapse of a massive star is the most
interesting and fascinating phenomenon in the universe. What is the final fate
of the massive star? Dynamical evolution of massive stars within the framework
of Einstein theory of gravity predicts that a spacetime singularity must arise
as a collapse end state, and ultra-strong gravity regions form Joshi:2004sf ;
OppenheimerSnyder39 ; Datt ; Joshi:2011zm ; JMN . Such very strong gravity
regions also exist at the center of a galaxy where so much matter is compacted
in a very small region.

As for predicting the final fate of massive stars in Einstein gravity, in
1939, Oppenheimer and Snyder, and Dutt, gave for the first time a dynamic
collapse model (OSD) OppenheimerSnyder39 ,Datt , where the star finally
terminates into a black hole as the end state of gravitational collapse. For
simplicity they took non-realistic assumptions such as the density
distribution being entirely homogeneous, the gas pressure being totally
neglected within the star, and such others. In this case, the event horizon
and apparent horizon formed before the formation of central strong singularity
which is hidden within the black hole. However, if we consider a more
realistic inhomogeneous collapse, with density higher at center and decreasing
slowly as we move away, the central strong singularity forms before the
formation of event horizon and the apparent horizon, so the singularity is at
least locally visible, also called a naked singularity Joshi:1993zg ;
Singh:1994tb ; Mena:1999bz . More generally, many recent works have shown that
small inhomogeneity or pressures inside the collapsing matter cloud allow for
strong curvature central singularity in Einstein gravity, which is locally or
globally visible.

The black hole as well as naked singularities, if they occur in nature, would
be physically and causally very different entities, and may have quite
different and distinct astrophysical signatures. To explore such possibilities
for some distinguishable observational signatures, we consider here two
specific naked singularity spacetimes as test cases, namely the Bertrand and
JNW geometries. In Dey:2013yga ,Dey+15  astrophysical importance of Bertrand
spacetimes was discussed. The gravitational lensing and shadow due to JNW
naked singularity is discussed in Gyulchev:2019tvk , which closely resembles
with the shadow of a Schwarzschild black hole. Gravitational lensing and
accretion disk properties for these objects are investigated in Joshi:2013dva
, Kovacs:2010xm . Also, recently the black hole shadow was discovered M87 . It
was shown that similar shadow can be created by the JMN naked singularity for
some cases Shaikh:2018lcc . In shaikh1 ; shaikh2 ; shaikh3 ; sunny ,
gravitational lensing and shadow due to the ultra compact objects, wormholes
and superspinars, is investigated. These are all interesting and useful
theoretical predictions in the context of the recent observation of shadow of
the M87 galactic center Akiyama:2019fyp .

It follows that the theoretical predictions of possible observational
signatures of black hole and naked singularity spacetimes, and their
differences and similarities are worth exploring. What we need is to
understand carefully these compact region geometries and their properties. In
such a perspective, understanding the perihelion precession of a particle in
such spacetimes could be a challenging and useful issue in GR, which is
usually treated in terms of the timelike geodesics that the particles move
along in a given spacetime.

As we know, Einstein investigated the timelike geodesics of a particle in
Schwarzschild spacetime schprecession , and he found the well-known perihelion
precession formula,

$\Delta\phi=\frac{6\pi GM}{c^{2}~{}a(1-e^{2})}\,\,,$

where $G$ is the gravitational constant, $M$ is the total mass of the central
object, $a$ is a semi-major axis and $e$ is the eccentricity of orbit. To
calculate the perihelion precession, approximation method for weak
gravitational field was suggested by Kerner et.al.kerner . Higher order
geodesic deviations and orbital precession in Kerr-Newman spacetime was
investigated in heydari . Charged particles moving along circular orbits
around the central body were examined to distinguish the black hole and naked
singularity remo1 . Also, circular stable and unstable orbits around
configurations describing either black holes or naked singularities were
considered in remo2 . In a recent work we also presented a study involving
timelike geodesics in so called JMN naked singularity spacetimes, comparing it
with black hole case Parth .

From an observational perspective, it may be worth noting that recently the
UCLA galactic center group demonstrated that short-period stars (e.g. S0-102
and S0-2) orbiting around the super-massive black hole in our galactic center
can be successfully used to probe the gravitation theory in a strong field
regime UCLA . Over past 17 years, the W.M. Keck observatory was used to image
the galactic center at the highest angular resolution possible today. They
have detected S0-102, a star orbiting our galaxy’s super-massive black hole,
with a period of just 11.5 shortest  years. Also SINFONI gave the galactic
center data, and updates on monitoring stellar orbits in the galactic center
of our milky way are given UCLA ; shortest ; center1 ; Eisenhauer:2005cv .

This observational information can help us reveal the nature of the central
object SGR-A* of our Milky way galaxy, which is considered to be a super
massive black hole with mass of about $10^{6}M_{\odot}$. From such a context,
it is important to do a comparative study of the nature of the timelike
geodesics and perihelion precession in different singular spacetimes. Since
the BST and JNW spacetimes are static, non-vacuum solutions of Einstein
equations, timelike orbits in these spacetimes should be distinguishable from
the timelike orbits in the vacuum black hole spacetimes, namely the
Schwarzschild geometry, and this difference is reported in the present work.
So, we discuss here the timelike geodesics in the BST and JNW spacetimes, both
of which have a central naked singularity.

The plan of the paper is as follows. In section (II), we discuss the timelike
geodesics in BST and Schwarzschild spacetimes in detail. We match the interior
BST spacetime with exterior Schwarzschild spacetime on a hypersurface
$\Sigma$, at a boundary $r=R_{b}$. We then compare the perihelion precession
of a particle in these two spacetimes. In section (III), we discuss the
timelike geodesics in JNW spacetime and compare the same with Schwarzschild
spacetime. As we know, the JNW spacetime is asymptotically flat, so one need
not match JNW with Schwarzschild spacetime. JNW spacetime has a scalar field
effect, and we discuss the how this scalar field changes the nature of orbits.
In section (IV), we derive an approximation solution of orbit equations of
BST, JNW, and Schwarzschild spacetimes and discuss the different
distinguishable properties of particles orbits in those spacetimes. In section
(V), we discuss the results obtained here. Finally, in section (VI), we
discuss conclusion and possible future work. Throughout the paper we take
$G=c=1$.

##  II Timelike geodesics in Schwarzschild and BST spacetimes

General expression of metric for a spherically symmetric, static spacetime can
be written as,

$ds^{2}=-g_{tt}(r)dt^{2}+g_{rr}(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\,\,,$ (1)

where $g_{tt}(r)$, $g_{rr}(r)$ are the metric components of the spherical,
static spacetime. The spherically symmetric, static, vacuum solution of the
Einstein equations can be uniquely represented by a Schwarzschild spacetime
Hartle . One can write down the Schwarzschild spacetime in the following form,

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

where $M_{0},R_{b}$ are two constant parameters of this spacetime, and
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the line element on
sphere.

The above form of Schwarzschild spacetime can be used also when there exist a
spacetime structure which has internal non-vacuum spacetime and external
static vacuum, which is Schwarzschild spacetime. Using the above form of
Schwarzschild spacetime, the Schwarzschild radius ($R_{s}$) can be written as
$R_{s}=M_{0}R_{b}$, therefore, total Schwarzschild mass
$M_{TOT}=\frac{M_{0}R_{b}}{2}$. The radial distance $R_{b}$ is the matching
radius, where a static, spherically symmetric, non-vacuum spacetime can be
glued with an external Schwarzschild geometry. For this type of spacetime
structure, we need $R_{b}>R_{s}$, and therefore, $0<M_{0}<1$. When a
collapsing matter cloud virializes before the formation of a black hole, then
the above mentioned spacetime structure can be formed. There are many
literature where the final state of collapsing matter cloud is discussed
elaborately Jhingan:1996jb ; Joshi:2015mwa ; Satin:2014ima ; Malafarina:2010xs
; Joshi:2013dva ; Bhattacharya:2017chr ; Dey:2019fja ; Dey:2019wwh . In
Banik:2016qvf , it is shown that Bertrand spacetimes (BST) can be formed as an
equilibrium state of gravitational collapse. The line element of BST can be
written as,

$\displaystyle ds^{2}_{BST}$ $\displaystyle=$
$\displaystyle-\left(\frac{2\beta^{2}}{1+\frac{Rb}{r}}\right)dt^{2}+\frac{dr^{2}}{\beta^{2}}+r^{2}d\Omega^{2}\,\,,$
(3)

where, $\beta$ and $R_{b}$ are the constant parameters of this spacetime.
Perlick (perlick, ) first discovered this spacetime which admits closed,
stable, circular orbits passing through each point of the spacetime. This
spacetime has a central strong curvature singularity which is not covered by
an event horizon.

From the above metric (eq. (3)), it can be seen that the BST is not
asymptotically flat. Therefore, to describe an internal spacetime of a compact
object by BST spacetime, we need to match this spacetime with an external
Schwarzschild spacetime at a timelike hypersurface. In general relativity,
maintaining two junction conditions Eric , one can smoothly glue two
spacetimes at a spacelike or timelike hypersurface. The first condition is
that the induced metric from the two sides (internal and external) should be
identical on a matching hypersurface, whereas the second condition states that
the extrinsic curvature from the two sides should be identical on that
matching hypersurface. Extrinsic curvature of a hypersurface, which is
embedded in higher dimensional manifold, can be expressed in terms of the
covariant derivative of normal vectors on that hypersurface:
$K_{ab}=e^{\alpha}_{a}e^{\beta}_{b}\nabla_{\alpha}\eta_{\beta}$, where
$e^{\alpha}_{a}$ is the tangent vector on the hypersurface and $\eta^{\beta}$
is the normal to that hypersurface. Now, if we want to glue BST with
Schwarzschild spacetime (eq. (3),(2)) smoothly at the timelike hypersurface,
$\Psi=r-R_{b}=0$, we need to satisfy the following conditions Dey:2013yga ,

$\beta^{2}=1-M_{0}\,\,,\,\,M_{0}=\frac{1}{3}\,\,,$ (4)

where the first condition comes from induced metric matching and the second
condition comes from the extrinsic curvature matching. One can verify that
with the above junction conditions, the radial pressure of BST becomes zero at
the matching hypersurface $\Psi$.

###  II.1 Analytic solution of orbit equation in Schwarzschild and BST
spacetimes

The angular part of the line elements, in eq. (3),(2), shows spherical
symmetry of the Schwarzschild and BST spacetimes. Therefore, the angular
momentum of a freely falling particle is conserved. The conservation of energy
of the particle implies temporal symmetries in static BST and Schwarzschild
spacetimes. The conserved angular momentum ($h$) and energy ($\gamma$) of a
particle per unit mass is given by,

$\displaystyle
h_{SCH}=r^{2}\frac{d\phi}{d\tau}\,\,,\,\,\,\gamma_{SCH}=\left(1-\frac{M_{0}R_{b}}{r}\right)\frac{dt}{d\tau}\,\,,$
(5) $\displaystyle
h_{BST}=r^{2}\frac{d\phi}{d\tau}\,\,,\,\,\,\gamma_{BST}=\left(\frac{2\beta^{2}}{1+\frac{Rb}{r}}\right)\frac{dt}{d\tau}\,\,,$
(6)

where $h_{SCH}$, $\gamma_{SCH}$ and $h_{BST}$, $\gamma_{BST}$ are the
conserved angular momentum and energy per unit rest mass of a freely falling
particle in Schwarzschild and BST spacetimes respectively. Here, $\tau$ is the
proper time of the particle.

We know that the normalization of four-velocity for timelike geodesics is,
$v_{\mu}v^{\mu}=-1$. From the normalization of four-velocity of a freely
falling massive particle, we can write the following effective potentials for
Schwarzschild and BST spacetimes,

$(V_{eff})_{SCH}=\frac{1}{2}\left[\left(1-\frac{M_{0}R_{b}}{r}\right)\left(1+\frac{h_{SCH}^{2}}{r^{2}}\right)-1\right]\,\,$
(7)
$(V_{eff})_{BST}=\frac{1}{2}\left[\left(\frac{2\beta^{2}}{1+\frac{R_{b}}{r}}\right)\left(1+\frac{h_{BST}^{2}}{r^{2}}\right)-1\right]\,\,,$
(8)

where we consider $\theta=\pi/2$ to confine the orbits of a particle on a
plane. The effective potential plays a crucial role on the shape and dynamics
of the trajectories of particles. The total energy ($E$) of the freely falling
massive particle 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 $E=\frac{\gamma^{2}-1}{2}$. For stable circular trajectories of massive
particles, one needs, $V_{eff}(r_{c})=E$, $V_{eff}^{\prime}(r_{c})=0$ and
$V_{eff}^{\prime\prime}(r_{c})>0$, where $r_{c}$ is the radius of the stable
circular orbit. Using $V_{eff}(r)=E$, $V_{eff}^{\prime}(r)=0$, one can write
down the expressions of $h$ and $\gamma$ for a circular timelike geodesic at a
given radius. With $V_{eff}(r)=E$, $V_{eff}^{\prime}(r)=0$, a massive particle
can have both stable and unstable circular orbits. For stable circular orbits
we need another constraint, namely, $V_{eff}^{\prime\prime}(r)>0$. Using
$V_{eff}(r)=E$, we can get perihelion ($r_{min}$: the radius of minimum
approach), and aphelion ($r_{max}$: maximum approach), which are points on the
bound non-circular orbits of a particle. Therefore, we can define bound orbits
of the freely falling massive particles in the following mathematical way,

$\displaystyle V_{eff}(r_{min})=V_{eff}(r_{max})=E\,,\,\,$ $\displaystyle
E-V_{eff}(r)>0\,,\,\,\,\forall r\in(r_{min},r_{max}).$ (10)

From the expression of effective potential in (7), we can derive the
expression of $h_{SCH}$ and $\gamma_{SCH}$ for circular timelike geodesics,

$\displaystyle\gamma_{SCH}^{2}=\frac{2\left(r-M_{0}R_{b}\right)^{2}}{r\left(2r-3M_{0}R_{b}\right)}\,,\,h_{SCH}^{2}=\frac{M_{0}R_{b}r^{2}}{\left(2r-3M_{0}R_{b}\right)}\,\,.$
(11)

Using the expressions of conserved quantities for circular geodesics, one can
show that no circular orbit is possible in the range: $0\leq
r\leq\frac{3M_{0}R_{b}}{2}$. One can write this range in terms of the total
mass ($M_{TOT}$) as, $0\leq r\leq 3M_{TOT}$, where
$M_{TOT}=\frac{M_{0}R_{b}}{2}$. However, for stable circular orbit, we need to
satisfy $(V_{eff}^{\prime\prime})_{SCH}>0$ along with above two conditions in
eq. (11). In terms of the total mass ($M_{TOT}$), the expression of
$(V_{eff}^{\prime\prime})_{SCH}$ can be written as,

$\displaystyle(V_{eff}^{\prime\prime})_{SCH}$ $\displaystyle=$
$\displaystyle\frac{2M_{TOT}}{r}\left(\frac{6M_{TOT}-r}{3M_{TOT}-r}\right)\,\,.$
(12)

Therefore, a massive particle can have stable circular orbit for $r\geq
6M_{TOT}$ when the two conditions in eq. (11)are satisfied. No stable circular
orbit is possible below $r=6M_{TOT},$ and this minimum radius for stable
circular orbit is known as Innermost Stable Circular Orbit (ISCO). Similarly,
from the effective potential of BST spacetime (8), we can write down the
following conditions for stable, circular geodesics,

$\displaystyle\gamma^{2}_{BST}$ $\displaystyle=$
$\displaystyle\frac{4\beta^{2}r}{2r+R_{b}}\,,\,$ (13) $\displaystyle
h^{2}_{BST}$ $\displaystyle=$ $\displaystyle\frac{R_{b}r^{2}}{2r+R_{b}}\,\,,$
(14) $\displaystyle(V_{eff}^{\prime\prime})_{BST}$ $\displaystyle=$
$\displaystyle\frac{2\beta^{2}R_{b}}{2r^{3}+3r^{2}R_{b}+rR_{b}^{2}}>0\,\,.$
(15)

From the above three equations, it can be understood that for $M_{0}<1$,
unlike Schwarzschild spacetime, BST spacetime has stable circular orbits of
any radius. For bound orbits in this spacetime, the conditions which are
stated in eq. (10), should be fulfilled. Now, with some given conserved value
of $h$ and $E$, one can describe the shape of an orbit of a massive particle,
by showing how radial coordinate $r$ changes with azimuthal coordinate $\phi$.
Using eq. (9) we get,

$\frac{d\phi}{dr}=\frac{h}{r^{2}}\frac{\sqrt{g_{rr}(r)g_{tt}(r)}}{\sqrt{2(E-V_{eff}(r))}}\,\,.$
(16)

Now, using the above equation one can define the following second order
differential equations for Schwarzschild and BST spacetime respectively,

$\frac{d^{2}u}{d\phi^{2}}+u-\frac{3M_{0}R_{b}}{2}u^{2}-\frac{M_{0}R_{b}}{2h^{2}}=0\,\,.$
(17)
$\frac{d^{2}u}{d\phi^{2}}+\beta^{2}u-\frac{\gamma^{2}R_{b}}{4h^{2}}=0\,\,.\\\
$ (18)

where $u=\frac{1}{r}$. From the above orbit equations, one can get the
information about the shape of orbits in Schwarzschild and BST spacetimes and
we can compare them.

(a) $M_{0}=0.05$,$h=0.1$, $E=-0.03$

(b) orbit in Schwarzschild spacetime

(c) orbit in BST spacetime

(d) $M_{0}=0.1$,$h=0.3$, $E=-0.01$

(e) orbit in Schwarzschild spacetime

(f) orbit in BST spacetime

Figure 1: In these figures, effective potential and particle orbits in
Schwarzschild and BST spacetimes are shown. Here, we use a red line for
Schwarzschild, while blue for BST. From the (1(b),1(c),1(e),1(f)), we can see
that to reach one perihelion point to the other perihelion point, the angular
distance travelled by a particle is greater than $2\pi$. The black dark spots
in the figure (1(b),1(e)), show the black hole regions. The brown circles show
the minimum approach of the particle near the center.

(a) $M_{0}=0.1$, $h=0.2$, $E=-0.022$, $R_{b}=1$

(b) orbit in matching of BST with Schwarzschild spacetimes

(c) $M_{0}=0.333$, $h=0.4$, $E=-0.044$, $R_{b}=1$

(d) orbit in matching of BST with Schwarzschild spacetimes

Figure 2: In this figure, we show the nature of the orbit of a particle (fig.
(2(b),2(d))) and corresponding effective potential (fig. (2(a),2(c))) when it
crosses the matching radius $R_{b}$. In the fig. (2(b),2(d)) the dotted blue
line shows what would be the particle trajectory if there is no Schwarzschild
spacetime outside and dotted red line shows what would be the particle orbit
when there is no BST spacetime inside. On the other hand, the solid blue line
shows particle’s actual path in BST spacetime and the solid red line shows
particle’s orbit in Schwarzschild spacetime. The solid red and blue part of
effective potential shows the contribution of exterior Schwarzschild and
interior BST spacetimes respectively.

##  III The Orbit equation in Janis-Newman-Winicour (JNW) spacetime

The JNW spacetime is static, spherically symmetric and massless scalar field
solution of Einstein equations JNW . The Lagrangian density of minimally
coupled scalar field can be written as,

$\mathcal{L}=\sqrt{-g}\left(\frac{1}{2}\partial^{\mu}\Phi\partial_{\mu}\Phi-V(\Phi)\right)\,\,,$
(19)

with the minimal conditions: $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa
T_{\mu\nu}$, $\Box\Phi(r)=V^{\prime}(\Phi(r))$, where the $\Phi$ is the scalar
field, $R,R_{\mu\nu},T_{\mu\nu}$ are the Ricci scalar, Ricci tensor, and
energy-momentum tensor respectively, and $V(\Phi)$ is the scalar field
potential and $\kappa$ is a constant whose value can be considered as 1 for
$G=c=1$ unit system. The Energy-momentum tensor ($T_{\mu\nu}$) can be written
as, $T_{\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi-g_{\mu\nu}\mathcal{L}$.
JNW spacetime is an example of this minimally coupled scalar field which has
zero mass. This spacetime is asymptotically flat, therefore, we need not match
theJNW spacetime with Schwarzschild spacetime. The line element can be written
as,

(a) $M=0.025$,$q=0.06$,$h=0.1$,$E=-0.02$

(b) Particle orbit in Schwarzschild spacetime

(c) Particle orbit in JNW spacetime

(d) $M=0.025$,$q=0.5$,$h=0.1$, $E=-0.02$

(e) Particle orbit in Schwarzschild spacetime

(f) Particle orbit in JNW spacetime

Figure 3: In this figure, we show the effective potentials (fig. (3(a),3(d)))
and the orbits of particles in Schwarzschild and JNW spacetimes (fig.
(3(c),3(f), 3(b),3(e))) for $M=0.025,q=0.1$ and $q=0.5$. In this diagram, the
blue line and red line correspond to JNW and Schwarzschild spacetimes
respectively and the dotted horizontal black line is indicating the total
energy of the freely falling particle. The brown circle shows the perihelion
positions of a particle near the center and dark black region at the center in
diagrams (fig. (3(b),3(e))), shows the positions of the black holes. Here we
consider $R_{b}=1$.

$\displaystyle ds^{2}_{JNW}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{b}{r}\right)^{n}dt^{2}+\left(1-\frac{b}{r}\right)^{-n}dr^{2}+r^{2}\left(1-\frac{b}{r}\right)^{1-n}d\Omega^{2}\,\,,$
(20)

(a) $M=0.25$,$q=3$,$h=3$, $E=-0.002$

(b) Particle orbit in Schwarzschild spacetime

(c) Particle orbit in JNW spacetime

(d) $M=0.25$,$q=10$,$h=3$, $E=-0.002$

(e) Particle orbit in Schwarzschild spacetime

(f) Particle orbit in JNW spacetime

Figure 4:  Here, the effective potential (fig. (4(a),4(d))) and particle
orbits in Schwarzschild and JNW spacetimes (fig. (4(b),4(e),4(c),4(f))) are
shown for $M=0.25,q=3$ and $q=10$. It can be seen that for scalar field charge
$q=3$, perihelion precession direction in JNW spacetime is Schwarzschild like,
while, for scalar field charge $q=10$, perihelion precession direction in JNW
spacetime is opposite to the direction of precession in the Schwarzschild
spacetime. Here, again we consider $R_{b}=1$.

where, $b=2\sqrt{M^{2}+q^{2}}$ and $n=\frac{2M}{b}$. $M$ and $q$ are the two
constant parameters of JNW spacetime, which represent total mass and scalar
field charge respectively. From the expressions of $b$ and $n$, we can write
$0<n<1$. The massless scalar can be written as,

$\Phi=\frac{q}{b\sqrt{4\pi}}ln\left(1-\frac{b}{r}\right)\,\,.$ (21)

If we consider $n=1$ or $q=0$, there will not be any scalar field ($\Phi$)
effects and JNW spacetime becomes Schwarzschild spacetime. As we know JNW
spacetime is static and spherically symmetric, so that conserved angular
momentum ($h_{JNW}$) and energy ($\gamma_{JNW}$) per unit rest mass of the
freely falling particle in JNW spacetime are given by,

$\displaystyle
h_{JNW}=r^{2}\left(1-\frac{b}{r}\right)^{1-n}\frac{d\phi}{d\tau}\,,\,\,$
$\displaystyle\gamma_{JNW}=\left(1-\frac{b}{r}\right)^{n}\frac{dt}{d\tau}\,\,,$
(22)

where $\tau$ is the proper time of the particle and we consider
$\theta=\frac{\pi}{2}$. From normalization of four velocity of a freely
falling massive particle, we can write the following effective potential for
JNW spacetime,

$\displaystyle(V_{eff})_{JNW}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(1-\frac{b}{r}\right)^{n}\left(1+\frac{h_{JNW}^{2}}{r^{2}}\left(1-\frac{b}{r}\right)^{n-1}\right)-1\right]\,\,.$
(23)

Using the condition for stable circular orbits, we can write down the
expressions of $h$ and $\gamma$ for JNW spacetime as,

$\displaystyle\gamma_{JNW}^{2}$ $\displaystyle=$
$\displaystyle\left(1-\frac{b}{r}\right)^{n}\left[\frac{2r-b(n+1)}{2r-b(2n+1)}\right]\,\,,$
(24) $\displaystyle h_{JNW}^{2}$ $\displaystyle=$ $\displaystyle
r^{2}\left[\frac{bn\left(1-\frac{b}{r}\right)^{1-n}}{2r-b(2n+1)}\right]\,\,.$
(25)

For stability of circular orbits, as we know, we need
$(V^{\prime\prime}_{eff})>0$. However, for JNW spacetime,
$(V^{\prime\prime}_{eff})_{JNW}$ is not always positive Gyulchev:2019tvk
,Zhou:2014jja . For $n<0.447$, at any point in JNW spacetime, we have
$(V^{\prime\prime}_{eff})_{JNW}>0$ and, therefore a freely falling particle
can have stable, circular orbit at any point in JNW spacetime. However, for
$0.447<n<0.5$, there exists one certain radial interval ($r_{-},r_{+}$) inside
which no stable circular orbits are possible, where $r_{-}$ and $r_{+}$ can be
written as,

$\displaystyle r_{-}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(b~{}(2+6n)-4.472~{}b\sqrt{n^{2}-0.2}\right)\,\,,$
(26) $\displaystyle r_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(b~{}(2+6n)+4.472~{}b\sqrt{n^{2}-0.2}\right)\,\,.$
(27)

In $r_{-}<r<r_{+}$, we always have $(V^{\prime\prime}_{eff})_{JNW}<0$ and,
therefore, stable circular orbits are not possible in that interval of radial
distance. However, outside this radial interval stable circular orbits are
possible, as there we have $(V^{\prime\prime}_{eff})_{JNW}>0$. Now, for
$0.5<n<1$, there exists the inner most circular orbit at $r_{ISCO}$, below
which no stable circular orbits are possible. Therefore, unlike BST spacetime,
JNW spacetime can have ISCO for $0.5<n<1$. Using general method of finding
orbit equation Parth , one can write the orbit equation of a particle freely
falling in JNW spacetime as,

$\displaystyle\frac{d^{2}u}{d\phi^{2}}+u+\frac{b\gamma^{2}}{2h^{2}}(2-2n)(1-bu)^{1-2n}-\frac{b(2-n)}{2h^{2}}(1-bu)^{1-n}-\frac{3bu^{2}}{2}$
$\displaystyle=$ $\displaystyle 0\,\,$ (28)

(a) $h=3$,$E=-0.002,R_{b}=1$

(b) $h=0.1$,$E=-0.02,R_{b}=1$

Figure 5: Here, we show the regions of $m>1$ and $m<1$ in the parameter space
of the scalar charge $q$ and ADM mass $M$ of the JNW spacetime.

##  IV Approximate solution of orbit equations in Schwarzschild, BST and JNW
spacetimes

The approximate solution of the orbit equations can be obtained by considering
the small values of eccentricity ($e$). In Parth ; Struck:2005hi ;
Struck:2005oi , this method is extensively discussed. With this approximation,
one can get important information about the nature and shape of bound orbits
which are difficult to get from orbit equations.

We can write the approximate solution for the orbit eq. (17) of Schwarzschild
spacetime,

$\tilde{u}=\frac{1}{p}\left[1+e\cos(m\phi)+O(e^{2})\right]\,\,,$ (29)

where $m$ and $p$ are positive real numbers and $\tilde{u}=R_{b}u$. As we
know, the solution of Newtonian orbit equation can be written as
$\tilde{u}=\frac{1}{p}\left[1+e\cos(\phi)\right]$, therefore, with the above
approximation we can get modified expression of $p$ and $m$. In Newtonian
mechanics, $p=\frac{2h^{2}}{M_{0}R_{b}^{2}}$ and $m=1$. Therefore, perihelion
precession cannot be possible in Newtonian mechanics. Using the approximate
solution of orbit eq. (29), we can get the following expression of $p$ and $m$
for Schwarzschild spacetime,

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

where, $0<m<1$ and $m>1$ implies that, starting from a perihelion point, a
particle reaches another perihelion point after travelling an angular distance
of greater or less than $2\pi$ respectively. For Schwarzschild case, $m$ is
always less than one Parth  and therefore, in Schwarzschild spacetime, a
particle has to travel an extra angular distance ($\delta\phi_{prec}$:
precession angle) in between two successive perihelion points. With the weak
field approximation Parth , in Schwarzschild case, the precession angle
$\delta\phi_{prec}$ can be written as,

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

As we previously mentioned, in this paper we compare the bound orbits in JNW
and Schwarzschild spacetimes, and the bound orbits in a spacetime structure
where it is internally BST and externally Schwarzschild spacetime. We
basically show how a freely falling particle, with a particular angular
momentum and total energy, moves in Schwarzschild spacetime, JNW spacetime and
in the BST spacetime structure. In Fig. (1), it is shown that the minimum
value of effective potentials in Schwarzschild and BST spacetimes are inside
the matching radius $R_{b}=1$. It is possible to have particles’ whole
trajectory inside the matching radius $R_{b}$ when the following inequality
holds,

$\frac{R_{b}}{h}>3/2\,\,.$ (32)

One can obtain the above inequality by considering the minimum value of
effective potential of BST 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 BST spacetime. From the above condition, it can be
understood that $\frac{R_{b}}{h}$ can have arbitrary large values which gives
bound trajectories of particles inside the BST spacetime. However, to compare
with bound orbits in Schwarzschild spacetime, we need angular momentum per
unit rest mass of the particle $h>\sqrt{3}M_{0}R_{b}$, which is explained in
(Parth, ).

The approximate solutions of the orbit eq. (18), corresponding to the BST
spacetime can also be written for small values of eccentricity $e$. We can
write the eq. (18) in the following form,

$\tilde{u}\frac{d^{2}\tilde{u}}{d\phi^{2}}+(1-M_{0})\tilde{u}^{2}=C_{\delta}\tilde{u}^{2\delta}\,\,,$
(33)

where $\delta=\frac{1}{2}$, $C_{\delta}=\frac{\gamma^{2}R_{b}^{2}}{4h^{2}}$
and $\tilde{u}=uR_{b}$. The solution of eq. (33) can be written as,

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

Using eq. (18) and eq. (34), we can get the following expressions of $p$ and
$m$,

$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{4h^{2}\beta^{2}}{\gamma^{2}R_{b}^{2}}\,\,,$ $\displaystyle
m$ $\displaystyle=$ $\displaystyle\beta\,\,.$ (35)

From the above expression of $m$, it can be seen that for $0<M_{0}<1$, $m$ is
less than one. Therefore, in BST spacetime, a freely falling massive particle
always travels greater than $2\pi$ angular distance in between two successive
perihelion points. Therefore, we get Schwarzschild like precession in BST
spacetime. Similarly, one can write the approximate solution of orbit eq. (28)
corresponding to JNW spacetime as,

$\tilde{u}\frac{d^{2}\tilde{u}}{d\phi^{2}}+Q\tilde{u}=R\tilde{u}^{2}+S\tilde{u}^{3}\,\,,$
(36)

where,

$Q=\left[\frac{b^{2}\gamma^{2}(1-n)}{h^{2}}-\frac{b^{2}(2-n)}{2h^{2}}\right]\,\,,$
$R=\left[\frac{b^{2}\gamma^{2}(1-n)(1-2n)}{h^{2}}-\frac{b^{2}(2-n)(1-n)}{2h^{2}}-1\right]\,\,,$
$S=\left[\frac{3}{2}-\frac{b^{2}(2-n)(1-n)n}{4h^{2}}+\frac{b^{2}\gamma^{2}(1-2n)(1-n)n}{h^{2}}\right]\,\,.$

The solution of eq. (36) can be written as,

$\tilde{u}=\frac{1}{p}\left[1+e\cos(m\phi)+O(e^{2})\right]\,\,.$ (37)

where, p and m are positive number. Using eq. (28) and (36) we can write the
expression of p and m by neglecting higher order terms of $e$,

$\displaystyle p_{\pm}=\frac{R\pm\sqrt{R^{2}+4QS}}{2Q}\,\,,$ (38)
$\displaystyle m=\sqrt{Qp-2R-\frac{3S}{p}}\,\,,$ (39)

Using eq. (39), one can show that in JNW spacetime, we have two types of
precession. A freely falling particle, in this spacetime, can travel greater
or less than $2\pi$ angular distance in between two successive perihelion
points.

##  V Results and Discussion

From the approximate solution of the orbit equation of BST, we get $m=\beta$.
As $\beta<1$ in BST spacetimes, a freely falling massive particle have only
Schwarzschild like perihelion precession. In fig. (1(c), 1(f)), the
Schwarzschild like precession of the particle orbit in BST is shown, where for
fig. (1(c)) we take $M_{0}=0.05,h=0.1,E=-0.03$ and for fig. (1(f)) we take
$M_{0}=0.1,h=0.3,E=-0.01$. In fig. (2(b),2(d)), the shape of particle orbit is
shown, where the particle crosses the matching radius $R_{b}$. It can be seen
that due to the effect of external Schwarzschild spacetime, the shape of the
bound orbit changes. In fig. (2(d)), one can see that in Schwarzschild
spacetime no bound orbit is possible. For
$M_{0}=0.333,h=0.4,E=-0.044,R_{b}=1$, we can see that in Schwarzschild
spacetime only the plunge orbit is possible. However, due to the presence of
internal BST spacetime, a particle can have bound trajectory, although it has
no bound orbits in Schwarzschild spacetime.

From the approximation solution of JNW spacetime, we get the expression of $m$
as written in eq. (39). In fig. (5(a),5(b)), using eq. (39), we show the
region of $m>1$ and $m<1$ for the parameter space of scalar charge $q$ and ADM
mass $M$. In fig. (3,4), we show the effective potentials and the orbits of
particles in Schwarzschild and JNW spacetimes for different values of scalar
field charge ($q$) and ADM mass $M$. In the fig. (3(c),3(f)), we can see that
for $M=0.025$, and $q=0.06$, a freely falling massive particle, in JNW
spacetime, has Schwarzschild like precession. On the other hand, if we
increase the scalar field charge to $q=0.5$, orbit starts precessing in
reverse direction of particle motion.

One can verify that the points $M=0.025,q=0.06$ and $M=0.025,q=0.5$ lie inside
the $m<1$ and $m>1$ regions in fig. (5(b)) respectively. In fig. (4(c),4(f)),
it can be seen that for $M=0.25$, $q=3$ we get Schwarzschild like precession ,
and for $M=0.25$, $q=10$ we get the reverse precession of the orbits of a
particle. In fig. (5(a)), one can verify that the point $M=0.25$, $q=3$ lies
inside $m<1$ region and $M=0.25$, $q=10$ lies inside $m>1$ region. From
$b=2\sqrt{M^{2}+q^{2}}$, we can get the corresponding values of $b$ for fig.
(3(c),3(f),4(c),4(f)) as $b=0.206,1.00125,6.00021,20.0001$ respectively.

We know that at $r=b$, JNW spacetime has a strong curvature naked singularity.
Therefore, one can see that in fig. (3(f)), a freely falling particle in JNW
spacetime can have bound orbits very close to the central naked singularity.
For the above mentioned values of $b$, we always have $n<0.5$. Therefore, for
these cases, stable circular timelike orbits, passing through any point in JNW
spacetime, are possible.

##  VI Conclusion

In the present paper, we investigated the timelike trajectories of particles
in the Schwarzschild, BST and JNW spacetimes, in order to investigate the
causal structure of these spacetimes, and to understand the important
distinguishable properties between them. This was mainly to understand the
presence or otherwise of the event horizon, and possible physical implications
in either case. Following are some of the important differences which can be
identified from the nature of the timelike trajectories in the black hole and
naked singularity spacetimes:

  * •

In the Schwarzschild spacetime, the perihelion precession of the bound
timelike trajectories of a particle is always in the direction of the particle
motion. As we know, for BST, we have $m=\beta$ and $\beta<1$. Therefore, like
the Schwarzschild spacetime, in BST, particle orbits precess in the direction
of particle motion. However, for JNW naked singularity spacetimes, $m$ can be
greater or less than one, which is shown in fig. (5(a),5(b)). Therefore,
unlike Schwarzschild spacetime, in JNW spacetime, the particle bound
trajectories can precess opposite to the direction of particle motion.

  * •

In the Schwarzschild spacetime, there exists an Innermost Stable Circular
orbit (ISCO) at $r=6M_{TOT}$, below which no stable circular orbits are
possible. On the other hand, in JNW spacetime, for $0.5<n<1$, there exists an
ISCO. However, for $0.447<n<0.5$, other than $r_{-}<r<r_{+}$ (eq. (27,26)),
stable circular orbits of any radius are possible. For $0<n<0.447$, there are
no constrains on the radius of the stable circular orbits of particles in JNW
spacetime. We also show that stable circular orbits of any radius are possible
in BST. In Parth , it is shown that the Joshi-Malafarina-Narayan (JMN)
spacetimes which have naked singularity at the center, also have no ISCO.
These differences can cause distinguishable properties of accretion disk,
which could be possibly detectable by our modern detectors available today
Joshi:2013dva ,Shaikh:2018lcc .

  * •

In fig. (3(a),3(d),4(d),4(a)) and in fig. (1(d),1(a)), we show that the
effective potentials of JNW and BST spacetimes become positive infinite at the
center. Therefore, in these two spacetimes, a freely falling massive particle
with non-zero angular momentum cannot reach the center and with a certain
amount of total energy a particle can be scattered by the infinite potential
barrier. However, unlike BST and JNW cases, the effective potential in the
Schwarzschild spacetime becomes negative infinity at the center. Therefore, a
massive particle with non-zero angular momentum and suitable total energy can
plunge into the center.

For a better understanding of causal structure and the mass and dynamics of
the galactic center, we need to study the timelike and lightlike geodesic
behaviour around the galactic center. As we know, GRAVITY and SINFONI are
continuously eyeing up the Milkyway galactic center to get important
observational data of stellar motion around the center. There are many papers
where the bound timelike orbits of particles around black holes and naked
singularities are investigated levin1 ; 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:2015slu ; Eva ; Eva1 ; Eva2 ; tsirulev
. In such a context, we show in this paper that the timelike geodesics of a
freely falling particle in the JNW and BST naked singularity spacetimes can be
significantly different from the timelike geodesics in Schwarzschild
spacetime.

## References

  * (1) G. M. Clemence Rev. Mod. Phys. 19, 361 – Published 1 October 1947
  * (2) Frank Watson Dyson , Arthur Stanley Eddington and C. Davidson, Royal society, 220 (1920)
  * (3) R. V. Pound and G. A. Rebka, Jr. Phys. Rev. Lett. 4, 337 – Published 1 April 1960
  * (4) B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration) Phys. Rev. Lett. 116, 241103 – Published 15 June 2016
  * (5) P.S.Joshi, 0405132, 61 (2004)
  * (6) J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939).
  * (7) S. Datt, Zs. f. Phys. 108, 314 (1938).
  * (8) P. S. Joshi, D. Malafarina and R. Narayan,  Class. Quant. Grav. 28, 235018 (2011).
  * (9) Rituparno Goswami and Pankaj S. Joshi  Phys. Rev. D 76, 084026 – Published 22 October 2007
  * (10) P. S. Joshi and I. H. Dwivedi, Phys. Rev. D 47, 5357 (1993).
  * (11) T. P. Singh and P. S. Joshi,  Class. Quant. Grav. 13, 559 (1996).
  * (12) F. C. Mena, R. K. Tavakol and P. S. Joshi,  Phys. Rev. D 62, 044001 (2000).
  * (13) D. Dey, K. Bhattacharya and T. Sarkar, Phys. Rev. D 88, 083532 (2013).
  * (14) D. Dey, K. Bhattacharya and T. Sarkar, Phys. Rev. D 87, 10, 103505 (2013).
  * (15) G. Gyulchev, P. Nedkova, T. Vetsov and S. Yazadjiev,  arXiv:1905.05273 [gr-qc].
  * (16) P. S. Joshi, D. Malafarina and R. Narayan,  Class. Quant. Grav. 31, 015002 (2014).
  * (17) Z. Kovacs and T. Harko,  Phys. Rev. D 82, 124047 (2010).
  * (18) Gravity Collaboration and Abuter et al. app. 618 L10
  * (19) R. Shaikh, P. Kocherlakota, R. Narayan and P. S. Joshi,  Mon. Not. Roy. Astron. Soc. 482, no. 1, 52 (2019).
  * (20) Rajibul Shaikh, Pritam Banerjee, Suvankar Paul, and Tapobrata Sarkar Phys. Rev. D 99, 104040 – Published 16 May 2019
  * (21) Rajibul Shaikh Phys. Rev. D 100, 024028 – Published 15 July 2019
  * (22) Rajibul Shaikh, Pritam Banerjee, Suvankar Paul and Tapobrata Sarkar Published 16 July 2019 • © 2019 IOP Publishing Ltd and Sissa Medialab
  * (23) Cosimo Bambi, Katherine Freese, Sunny Vagnozzi, Luca Visinelli https://arxiv.org/abs/1904.12983
  * (24) K. Akiyama et al. [Event Horizon Telescope Collaboration],  Astrophys. J. 875, no. 1, L5 (2019).
  * (25) V Perlick Classical and Quantum Gravity, Volume 9, Number 4
  * (26) Allen I. Janis, Ezra T. Newman, and Jeffrey Winicour Phys. Rev. Lett. 20, 878 – Published 15 April 1968
  * (27) A. Einstein annalen der physik, 354 (1916)
  * (28) R Kerner, J W van Holten and R Colistete Jr, 2001, Classical and Quantum Gravity, 18, 22
  * (29) Heydari-Fard, M., Heydari-Fard, M. and Sepangi, H.R. Gen Relativ Gravit (2019) 51: 77.
  * (30) Daniela Pugliese, Hernando Quevedo, and Remo Ruffini Phys. Rev. D 83, 104052 – Published 27 May 2011
  * (31) Daniela Pugliese, Hernando Quevedo, and Remo Ruffini Phys. Rev. D 83, 024021 – Published 18 January 2011
  * (32) A. Hees, et al, Phys. Rev. Lett. 118, 211101, 2017
  * (33) L. Meyer, A. M. Ghez,*, R. Schödel, S. Yelda, A. Boehle, J. R. Lu, T. Do, M. R. Morris, E. E. Becklin, K. Matthews Vol. 338, Issue 6103, pp. 84-87 DOI: 10.1126/science.1225506
  * (34) S. Gillessen et al., Astrophys. J. 837, 30 (2017).
  * (35) F. Eisenhauer et al.,  Astrophys. J. 628, 246 (2005).
  * (36) Hartle, J. B. Addison-Wesley, (2003).
  * (37) S. Jhingan, P. S.Joshi and T. P. Singh,  Class. Quant. Grav. 13, 3057 (1996).
  * (38) D. Malafarina and P. S. Joshi, Eur. Phys. J. C 75, no. 12, 596 (2015)
  * (39) S. Satin, D. Malafarina and P. S. Joshi,  Int. J. Mod. Phys. D 25, 1650023 (2016)
  * (40) D. Malafarina and P. S. Joshi, Int. J. Mod. Phys. D 20, 463 (2011).
  * (41) K. Bhattacharya, D. Dey, A. Mazumdar and T. Sarkar, arXiv:1709.03798 [gr-qc] (2017).
  * (42) D. Dey, P. Kocherlakota and P. S. Joshi, arXiv:1907.12792 [gr-qc].
  * (43) D. Dey and P. S. Joshi, arXiv:1907.12738 [gr-qc].
  * (44) U. Banik, D. Dey, K. Bhattacharya and T. Sarkar,  Gen. Rel. Grav. 49, no. 9, 116 (2017).
  * (45) P. Bambhaniya, A. B. Joshi, D. Dey and P. S. Joshi, arXiv:1908.07171 [gr-qc].
  * (46) C. Struck,  Astron. J. 131, 1347 (2006)
  * (47) C. Struck,  Monthly Notices of the Royal Astronomical Society, 446, 3139-3149 (2014).
  * (48) E. Poisson, Cambridge University Press. (2004) doi:10.1017/CBO9780511606601
  * (49) S. Zhou, R. Zhang, J. Chen and Y. Wang, Int. J. Theor. Phys. 54, no. 8, 2905 (2015), Int. J. Theor. Phys. 54, no. 8, 2905 (2015).
  * (50) Janna Levin and Gabe Perez-Giz, Phys. Rev. D 77, 103005 (2008).
  * (51) K. Glampedakis and D. Kennefick, Phys. Rev. D 66, 044002 (2002).
  * (52) D. S. Chu et al., Astrophys. J. 863, no. 1, 111 (2018).
  * (53) V. I. Dokuchaev, Y. N. Eroshenko and K. S. Klimkov, Phys. Procedia 74, 292 (2015)  Phys. Procedia 74, 292 (2015).
  * (54) D. Borka, P. Jovanovic, V. B. Jovanovic and A. F. Zakharov,  Phys. Rev. D 85, 124004 (2012).
  * (55) C. Martínez, N. Parra, N. Valdés and J. Zanelli, arXiv:1902.00145 [hep-th],(2019)
  * (56) R. Fujita and W. Hikida,  Class. Quant. Grav. 26, 135002 (2009)
  * (57) Y. Wang, X. Liu, N. Yang and J. Jia,  arXiv:1903.12486v1 [gr-qc]
  * (58) S. Suzuki and K. i. Maeda, Phys. Rev. D 58, 023005 (1998)  Phys. Rev. D 58, 023005 (1998)
  * (59) M. Zhang and W. B. Liu,  Phys. Lett. B 789, 393 (2019)
  * (60) D. Pugliese, H. Quevedo and R. Ruffini,  Phys. Rev. D 88, no. 2, 024042 (2013)
  * (61) C. Farina, J. Gamboa and A. J. Segui-Santonja,  Class. Quant. Grav. 10, L193 (1993)
  * (62) A. Dasgupta, H. Nandan and S. Kar,  Phys. Rev. D 85, 104037 (2012)
  * (63) A. A. Shoom, C. Walsh and I. Booth,  Phys. Rev. D 93, no. 6, 064019 (2016)
  * (64) Eva Hackmann and Claus Lämmerzahl Phys. Rev. Lett. 100, 171101 – Published 2 May 2008
  * (65) E. Hackmann, V. Kagramanova2, J. Kunz2 and C. Lämmerzahl1 Published 13 November 2009 • Europhysics
  * (66) Eva Hackmann, Claus Lämmerzahl, Yuri N. Obukhov, Dirk Puetzfeld, and Isabell Schaffer Phys. Rev. D 90, 064035 – Published 22 September 2014
  * (67) I. M. PotashovJu. V. TchemarinaA. N. Tsirulev The European Physical Journal C, August 2019, 79:709


# NFT-K: Non-Fungible Tangent Kernels

###### Abstract

Deep neural networks have become essential for numerous applications due to
their strong empirical performance such as vision, RL, and classification.
Unfortunately, these networks are quite difficult to interpret, and this
limits their applicability in settings where interpretability is important for
safety, such as medical imaging. One type of deep neural network is neural
tangent kernel that is similar to a kernel machine that provides some aspect
of interpretability. To further contribute interpretability with respect to
classification and the layers, we develop a new network as a combination of
multiple neural tangent kernels, one to model each layer of the deep neural
network individually as opposed to past work which attempts to represent the
entire network via a single neural tangent kernel. We demonstrate the
interpretability of this model on two datasets, showing that the multiple
kernels model elucidates the interplay between the layers and predictions.
††This work was supported by NSF grants CCF-1911094, IIS-1838177, and
IIS-1730574; ONR grants N00014-18-12571, N00014-20-1-2787, and
N00014-20-1-2534; AFOSR grant FA9550-18-1-0478; and a Vannevar Bush Faculty
Fellowship, ONR grant N00014-18-1-2047.

Index Terms—  Deep neural network, neural tangent kernel, interpretability,
multi-layer perception

##  1 Introduction

Deep neural networks (DNNs) have become the main parametric models used to
solve machine learning problems as they provide state-of-the-art empirical
performance, e.g., see [brock2018large, karras2020analyzing,
szegedy2016rethinking, simonyan2014very, he2016deep, silver2016mastering].
DNNs can be used for a host of applications, including autonomous driving
[uccar2017object], medical imaging [ronneberger2015u], medical records
[shickel2017deep], and cyber security [dixit2021deep]. These applications all
benefit from DNNs but are also high risk situations.

Despite the extraordinary performance of DNNs, it is still not clear how to
best interpret their predictions and inferences. We direct the reader to this
summary of recent efforts on interpretations [gilpin2018explaining]. Indeed
there are some aspects within DNN interpretability that need more
understanding or are poorly explained by prior work. A core challenge is to
understand which of the layers are responsible for the which aspects of the
decision-making process for each class.

Towards a deeper understanding of the functionality of neural networks, recent
work has demonstrated that some DNNs correspond to kernel machines
[friedman2017elements]. Kernel machines have been studied extensively and can
provide extra interpretability depending on the structure of the kernel.
Similar to their kernel counterparts, DNN architectures typically use a large
number of parameters which can be orders of magnitude more than the training
samples. Overparameterized DNNs generalize unseen data very well despite over-
fitting training data, which is a property of interpolating methods such as
kernel machines [belkin2019reconciling, belkin2018understand]. The exact
relationship between DNNs and kernel methods have been revealed by analysing
the infinite-width DNNs, when the number of units (neurons) in the hidden
layers goes to infinity. It has been shown that infinite-width DNNs trained
with gradient descent converges to a kernel ridge regression predictor with
respect to a kernel known as the neural tangent kernel (NTK)
[jacot2018neural]. In recent years NTKs have been developed for a variety of
architectures [arora2019exact, novak2018bayesian, du2019graph, huang2020deep,
hron2020infinite, alemohammad2020recurrent, alemohammad2020scalable,
yang2020tensor].

Though they live in potentially infinite dimensional spaces, kernel predictors
can be reduced to linear classifiers in the eigenspace of the grammian of the
input data. This is the famed “kernel trick” in machine learning and allows
improved computational efficiency and interpretability of results. NTKs
likewise use a gram matrix, it is easier to interpret the relationship between
the input and the output of the DNN. However, it is not straightforward to use
NTK to interpret which layers the network uses to classify a certain class via
the gram matrix. The relationship between class and layer is not well
understood but can be useful for identifying when misclassifications occur or
if depth should be increased or decreased in a neural network. We begin to
understand the relationship between classes and layers by modelling it using
multiple NTKs, one for each layer.

###  1.1 Contributions

In order to capture the relationship between layers and classes, we combine
NTK with inspiration from another network [lin2017feature]. Our architecture
admits multiple-kernel learning in the infinite-width limit, where each kernel
corresponds to the NTK of each layer. We use the layers of multi-layer
perceptrons (MLPs) and call our model the non-fungible tangent kernel MLP
(NFT-K MLP) for our proposed architecture. This stems from our model learning
to select layers of a DNN which are crucial for classification of certain
classes (Figure 1). As the layers selected by the model are the most
important, they are not fungible with the layers that were not selected, and
this reduces the overall complexity of the model while ensuring strong
empirical performance.

To validate our model, we conduct experiments on the MNIST dataset and the
large movie review dataset to provide interpretable results into how our
network provides insight into which layers are important for classification.
We see that NFT-K MLPs focus on similar layers for classes which visually
appear similar, even without the use of convolutional layers. Moreover, we can
see which layers are the most important for classification; some layers are
not at all important and are dropped from the model. Overall, NFT-K MLPs are
able to model the relationship between classes and layers and provide
insightful information.

##  2 Background

Given an input ${\bm{x}}\in\mathbb{R}^{m}$, an MLP with $L$ hidden layers and
$n$ neurons performs the following recursive relations

$\displaystyle{\bm{g}}^{(1)}({\bm{x}})$
$\displaystyle={\mathbf{W}}^{(1)}{\bm{x}}+{\mathbf{b}}^{\left(1\right)},$ (1)
$\displaystyle{\bm{h}}^{(\ell)}({\bm{x}})$
$\displaystyle=\phi({\bm{g}}^{(\ell)}({\bm{x}})),$ (2)
$\displaystyle{\bm{g}}^{(\ell)}({\bm{x}})$
$\displaystyle={\mathbf{W}}^{(\ell)}{\bm{h}}^{(\ell-1)}({\bm{x}})+{\mathbf{b}}^{\left(\ell\right)},\quad
2\leq\ell\leq L,$ (3)

where ${\mathbf{W}}^{(1)}\in\mathbb{R}^{m\times n}$,
${\mathbf{W}}^{(\ell)}\in\mathbb{R}^{n\times n}$, and
${\mathbf{b}}^{\left(\ell\right)}\in\mathbb{R}^{n}$.
$\phi(\cdot):\mathbb{R}\rightarrow\mathbb{R}$ represents the activation
function that acts coordinate wise on a vector.

The output of an MLP is obtained by a linear transformation of the last layer

$\displaystyle f_{\theta}^{(L)}({\bm{x}})$
$\displaystyle={\mathbf{V}}^{(L)}{\bm{h}}^{(L)}({\bm{x}})\in\mathbb{R}^{d},$
(4)

where $\theta$ is the set of all parameters in the networks. Let $\mathcal{X}$
be the set of training inputs. Let and
$\mathcal{Y}\in\mathbb{R}^{|\mathcal{X}|d}$ be the concatenation of all
labels. Let
$l(\widehat{y},y):\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{+}$
be the MSE loss function, and
$\mathcal{L}(\theta)=\frac{1}{{|\mathcal{X}|}}\sum_{({\bm{x}},{\bm{y}})\in\mathcal{X}\times\mathcal{Y}}l(f_{\theta}({\bm{x}}),{\bm{y}})$
be the the empirical loss. Let $\theta_{0}$ be the networks parameters at
initialization. An MLP trained with infinitesimal step size for Gaussian
distributed $\theta_{0}$ using the following loss

$\displaystyle\min_{\theta}\Big{(}\mathcal{L}(\theta)+\lambda\|\theta-\theta_{0}\|_{2}^{2}\Big{)}$
(5)

is equivalent to a kernel ridge regression predictor with ridge hyperparameter
$\lambda$ and a deterministic kernel known as its neural tangent kernel (NTK).
The NTK is defined as

$\displaystyle\Theta({\bm{x}},{\bm{x}}^{\prime})=\Theta^{(L)}({\bm{x}},{\bm{x}}^{\prime}){\bm{I}}_{d}=\nabla_{\theta}f^{(L)}_{\theta}({\bm{x}})^{\top}\nabla_{\theta}f^{(L)}_{\theta}({\bm{x}}^{\prime})$
(6)

where $\Theta^{(L)}({\bm{x}},{\bm{x}}^{\prime})\in\mathbb{R}$ is the NTK of a
MLP with depth $L$ that depends on the network’s activation function, depth
and initialization hyperparameters [jacot2018neural]. We define
$\Theta^{(L)}({\bm{x}},\mathcal{X})$ as the vector in
$\mathbb{R}^{|\mathcal{X}|}$ whose entries are
$\Theta^{(L)}({\bm{x}},{\bm{x}}^{\prime})$ for
${\bm{x}}^{\prime}\in\mathcal{X}$; similarly,
$\Theta^{(L)}(\mathcal{X},\mathcal{X}^{\prime})\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{X}^{\prime}|}$.

Fig. 1: Diagram of the NFT-K MLP network. Each inner product of the kernel
with the layer is sent as the input to the classifier. The weights of the
classifier can interpret which layers are aiding in the decision-making
process.

##  3 Non-Fungible Tangent Kernels

In this section, we propose our NFT-K architecture. Consider the recursive
relations from Equations (1), (2), and (3) that form the hidden states
${\bm{h}}^{(\ell)}({\bm{x}})$. Instead of obtaining the output by a linear
transformation from the last layer as shown in Equation (4), we obtain the
output $f_{P}({\bm{x}})$:

$\displaystyle f^{(\ell)}({\bm{x}})$
$\displaystyle={\mathbf{V}}^{(\ell)}{\bm{h}}^{(\ell)}({\bm{x}}),$ (7)
$\displaystyle f({\bm{x}})$ $\displaystyle=\sum_{\ell=1}^{L}\omega_{\ell}\odot
f_{P}^{(\ell)}({\bm{x}})\quad\omega_{\ell}\in\mathbb{R}^{d},$ (8)

where $\odot$ is the entry-wise product. Note that ${\mathbf{V}}^{(\ell)}$ for
$\ell=1,\dots,L-1$ are not present in the MLP architecture. Figure 1 displays
a network diagram for visualizing of this architecture.

We are interested in the following optimization problem

$\displaystyle\min_{\Omega}\min_{\theta}\Big{(}\mathcal{L}(\theta,\Omega)+\lambda\|\theta-\theta_{0}\|_{2}^{2}\Big{)}\quad
s.t\quad\Omega\in\mathcal{S}$ (9)

where

$\displaystyle\theta$
$\displaystyle=\mathrm{Vect}[\\{{\mathbf{W}}^{(\ell)},{\mathbf{V}}^{(\ell)},{\mathbf{b}}^{(\ell)}\\}_{\ell=1}^{L}],$
(10) $\displaystyle\Omega$
$\displaystyle=[\omega_{\ell},\dots,\omega_{L}]\in\mathbb{R}^{d\times L},$
(11) $\displaystyle\mathcal{S}$
$\displaystyle=\left\\{\Omega\in\mathbb{R}^{d\times L}|\Omega>0\right\\}.$
(12)

To solve (9), we use alternating minimization by first solving over $\theta$
for a fixed $\Omega$ and then solving over $\Omega$ for a fixed $\theta$. More
precisely, we perform gradient descent on $\theta$ for fixed $\Omega$:

$\displaystyle\theta_{t+1}=\theta_{t}-\eta_{\theta}\nabla_{\theta_{t}}\mathcal{L}(\theta_{t},\Omega)-\eta_{\theta}\lambda(\theta_{t}-\theta_{0})$
(13)

until convergence. In practice, we use stochastic gradient descent with mini-
batches. The following theorem characterizes the output of the network after
each update performed in Equation 13.

###### Theorem 1

After convergence of gradient descent in (13) over $\theta$ for a fixed
$\Omega$ with infinitesimal step size $\eta_{\theta}\rightarrow 0$, the output
of the neural network will be

$\displaystyle
f({\bm{x}})=\Theta_{\Omega}({\bm{x}},\mathcal{X})\left(\Theta_{\Omega}(\mathcal{X},\mathcal{X})+\lambda{\bm{I}}_{d|\mathcal{X}|}\right)\mathcal{Y},$
(14)

where

$\displaystyle\Theta_{\Omega}({\bm{x}},{\bm{x}}^{\prime})=\sum_{\ell=1}^{L}\Theta^{(\ell)}({\bm{x}},{\bm{x}}^{\prime})\mathrm{diag}(\omega_{\ell})$
(15)

and $\Theta^{(\ell)}({\bm{x}},{\bm{x}}^{\prime})$ is the NTK of an MLP with
depth $\ell$ as defined in Equations 1 to 6.

_Proof of Theorem 1._ For a fixed $\Omega$, the output of the network is just
a linear combination of the output at each layers. Since we are using gradient
descent with infinitesimal step size the out put will be kernel ridge
regression with respect to the NTK of network and the NTK can be easily
obtained from the results of [yang2020tensor]. $\square$

In particular this result highlights that in the infinite width limit, solving
over $\theta$ has a closed form solution that reduces to kernel ridge
regression. The corresponding kernel is a linear combination of the NTKs of
each layer of the network, as shown in Equation 15.

In the second step of our alternating minimization procedure, optimizing over
$\Omega\in\mathcal{S}$ is a very well known problem known as multiple kernel
learning [zhang2016fast]. Hence, optimizing the network loss in 9 using
alternating minimization results in multiple-kernel learning in the infinite-
width limit and the correspondingly learned $\omega_{\ell}$ informs us of each
layer’s importance in the given classification task.

To solve over $\Omega$, we use gradient projection:

$\displaystyle\Omega_{t+1}=\mathcal{P}_{\mathcal{S}}\left(\Omega_{t}-\eta_{\Omega}\nabla_{\Omega_{t}}\mathcal{L}(\theta,\Omega)\right).$
(16)

Table 1:  Test accuracy among different datasets showing that MLPs and NFT-K MLP perform similarly. In the MNIST classification task, the 10 layer NFT-K MLP outperforms all other networks. In the large movie review classification problem, MLPs and NFT-K MLP perform within 0.42% of one another. Network | Dataset | Test Acc
---|---|---
10 layer MLP | MNIST | 96.89%
10 layer NFT-K MLP | MNIST |  97.25%
20 layer MLP | MNIST | 97.22%
20 layer NFT-K MLP | MNIST | 95.23%
5 layer MLP | Large Movie Review |  86.73%
5 layer NFT-K MLP | Large Movie Review | 86.4%
10 layer MLP | Large Movie Review | 86.31%
10 layer NFT-K | Large Movie Review | 86.7%