# Tidal force effects and periodic orbits in null naked singularity spacetime

Siddharth Madan<EMAIL_ADDRESS>International Center for
Cosmology, Charusat University, Anand, GUJ 388421, India  Parth Bambhaniya
<EMAIL_ADDRESS>International Center for Cosmology, Charusat University,
Anand, GUJ 388421, India

###### Abstract

Naked singularities form during the gravitational collapse of inhomogeneous
matter clouds. The final nature of the singularity depends on the initial
conditions of the matter properties and types of matter profiles. These naked
singularities can also be divided into two types: null-like and timelike
singularities. The spacelike singularity of the Schwarzschild black hole can
be distinguished from the null and timelike naked singularity spacetimes. In
light of this, we investigate the precession of timelike bound orbits in the
null naked singularity spacetime, as well as tidal force effects and geodesic
deviation features. As a result, we find that the orbital precession of the
timelike bound orbits in null naked singularity spacetime could be
distinguished from the Schwarzschild precession case. The radial component of
the tidal force has an intriguing profile, whereas the angular component has a
profile which is comparable to that of a Schwarzschild black hole scenario.
The geodesic deviation equation is then solved numerically, yielding results
that resemble a Schwarzschild black hole. These characteristic features can
then be used to discern amongst these singularities.

Key words: Black hole, Naked singularity, Tidal force, Periodic orbits.

##  I Introduction

The general relativistic effects of Einstein’s gravity have been observed by
various observations such as the great discoveries of the gravitational waves
LIGOScientific:2016aoc , the first-ever black hole shadow image of the M87
galactic center EventHorizonTelescope:2019dse , stellar motions of the S-stars
around the Milky-way galactic center GRAVITY:2020gka , etc. These discoveries
throughout have sparked a lot of interest in looking into the causal structure
and the dynamics of spacetime around the galactic center.

The supermassive black hole and hence a spacetime singularity are thought to
exist in the core regions of most galaxies, created by the catastrophic
continuous gravitational collapse of primordial matter clouds. However, there
is a fundamental mystery surrounding the singularity’s causal structure. It is
a well-established fact that there are three kinds of strong spacetime
singularities in general relativity: spacelike, timelike, and null-like
Dey:2020bgo ; Joshi:2020tlq . The spacelike singularity is causally detached
from the other points of the spacetime manifold, and they may be hidden within
horizons, whereas the null-like and timelike singularities are causally
linked.

According to a significant amount of research Dey:2020bgo ; Joshi:2020tlq ;
Joshi:2011zm ; Joshi:2013dva , null-like and timelike singularities can be
formed during the continual gravitational collapse of physically plausible
matter clouds, and they contradict Roger Penrose’s Cosmic Censorship
Conjecture (CCC) Penrose:1969pc . In Joshi:2020tlq , authors propose a new
spherically symmetric and static null naked singularity spacetime which is a
solution of the Einstein field equations. It lacks a photon sphere, and they
demonstrate that the singularity casts a shadow in the absence of a photon
sphere. It has been determined that the event horizon and photon sphere are
not essential for the formation of a shadow. From these studies, the existence
of the upper bound of the effective potential of null-geodesics causes the
formation of a shadow. Therefore, a spacetime can cast a shadow if its
effective potential of null-geodesics has an upper bound. It has been shown
that this null naked singularity spacetime satisfies strong, weak, and null
energy conditions and verified that the Kretschmann scalar and Ricci scalar
blow up at the center $r=0$. Moreover, they point out that if any null
geodesic emanates from the past null infinity, the photon would be infinitely
red-shifted with respect to the asymptotic observer.

However, if such type of null naked singularity exists in reality, then they
must have distinct physical signatures. Many research studies have been
published in which this topic has been investigated Joshi:2020tlq ;
Joshi:2011zm . One might come across abundant research literature concerning
the aim to distinguish a naked singularity spacetime from any other spacetime
which is engulfed by a null hypersurface or an event horizon. To distinguish
these naked singularities, significant work has been done in the recent
literature Bambhaniya:2019pbr ; Joshi:2019rdo ; Dey:2019fpv ;
Bambhaniya:2020zno ; Bambhaniya:2021ybs ; Bambhaniya:2021jum ; Solanki:2021mkt
; Dey:2020haf ; Bambhaniya:2021ugr , where authors investigate the shadows and
orbital precession properties. Similarly, in this paper, we obtain an orbit
equation of a test particle for the null-like naked singularity spacetime and
we show the periastron precession of the timelike bound orbits for the same.

Now, it is a well-known fact that a body in free-fall toward the center of
another body is known to be stretched in the radial direction while being
compressed in the angular direction. The stretching and compression are caused
by gravity’s tidal effect, which is caused by a variation in gravity’s
strength between two adjacent points MTW:1973 ; DInverno:1992 ; Carroll:2004 ;
Hobson:2006 . The presence of tidal force phenomena is extremely common in our
universe and has been a topic of popular scientific exploration for a major
part of the twentieth century. To substantiate it further, John Wheeler
proposed the possible disintegration of a star in the ergosphere of the Kerr
black hole due to tidal interaction which results in the emission of jets
formed from the debris of that star Goswami:2019fyk ; Wheeler:1971 . Since
then, the tidal disruption of stars in the presence of a massively gravitating
object has been a topic for extensive research in the astrophysical research
community Hills:1975 ; Carter:1982 ; Rees:1988 ; komossa:2015 ; Auchettl:2017
; Rossi:2020rvv ; Crispino:2016pnv ; Shahzad:2017vwi ; Hong:2020bdb .

In Crispino:2016pnv , authors studied the effects of tidal force on a freely
falling particle in Reissner-Nordström (RN) black hole and concluded that the
radial and angular components of the tidal force change their signs at a
boundary called the null hypersurface or the event horizon, which is not the
case in the Schwarzschild black hole. In Gad2010 , the authors compared the
tidal force effects in a stringy charged black hole with the Schwarzschild and
the RN black holes. Shahzad and Jawad Shahzad:2017vwi  studied both the radial
and angular components of the tidal force in the Kiselev black hole. In that
paper, they considered the pervasive vicinity of radiation and dust fluids,
also pointed out the change in sign between the event and the Cauchy horizons
just as in the case of RN black holes. In Goel:2015zva , authors examined the
properties of tidal force in naked singularity spacetimes. The effects of
tidal force have been analyzed for myriad other black holes Mahajan:1981 ;
AbdelMegied:2004ni ; Chan:1995fc ; Nandi:2000gt ; Cardoso:2012zn ;
Harko:2012ve ; Uniyal:2014oaa ; Sharif:2018a ; Sharif:2018gzj ; Junior:2020yxg
; Junior:2020par . Motivated by this, in this paper, we study the tidal force
effects, geodesic deviation, and periodic orbits in the null naked singularity
spacetime. We emphasize our results and compare them with the Schwarzschild
black hole case.

The overview of this paper is given as: In the Section (II), we have discussed
the intricacies of this spacetime i.e. a brief on the preceding work on this
metric and its characteristics. In Section (III), we check the periodic orbits
for a particle by varying a couple of parameters. In the section (IV), we
present the calculations for tidal force effects in this spacetime. We have
been able to obtain some riveting results with the aid of tetrad formalism for
the radial component of tidal force in this metric. In section (V), we have
compared the numerical solution of the geodesic deviation equation for the
null naked singularity metric with the Schwarzschild black hole metric.
Conclusions are drawn in Section (VI). Throughout this paper, we consider
gravitational constant (G) and speed of light (c) as unity.

##  II Null Naked Singularity Spacetime

Figure 1: Penrose diagram of null naked singularity

The line element representation for any geometry can be written as,

$d{s}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu},$ (1)

this can be elaborated further by delineating all the summed over components
as,

$d{s}^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (2)

and the metric for null naked singularity case can be given by Joshi:2020tlq ,

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

where, $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ and $M$ is the
Arnowitt-Deser-Misner (ADM) mass of the above spacetime. Now from the above
two equations (2) and (3) we can write,

$f(r)=\left(1+\frac{M}{r}\right)^{-2},$ (4)

now one can give the binomial expansion for $f(r)$ as,

$f(r)=\left[1-\frac{2M}{r}+3\left(\frac{M}{r}\right)^{2}-...\right],$ (5)

from this binomial expansion, it is evident that in the large r limit, this
metric mimics the Schwarzschild metric and in asymptotically infinite limit
the metric behaves akin to the Minkowski spacetime. Even though the metric
resembles the Schwarzschild metric at a large distance, near the singularity,
the causal structure of this spacetime becomes different from the causal
structure of Schwarzschild spacetime. It is shown in Joshi:2020tlq  that from
the expressions of Kretschmann scalar and Ricci scalar at the center $(r=0)$
there exists a strong curvature singularity and no null surfaces such as the
event horizon engulfs the singularity, hence the name naked singularity. Fig.
1 shows the Penrose diagram of the null naked singularity spacetime. The most
intriguing information to look for in this diagram is that any null geodesic
emanating from the past null infinity would allow the photon to be infinitely
red-shifted in regards to the asymptotic observer Bambhaniya:2021jum . As a
result, at r = 0, it is a nulllike naked singularity. There it is also shown
that this metric satisfies all energy conditions. Using Einstein field
equations, the expressions for the energy density and pressures of this naked
singularity spacetime are obtained. Eventually, it is deduced that this
spacetime is seeded with an anisotropic fluid for which the equation of state
parameter tends to -1/3 when $r\rightarrow{0}$ and it approaches 1/3 as
$r\rightarrow{\infty}$.

(a) Timelike orbits in null-like naked singularity spacetime for $h=4$, where
$r_{min}=8.17$ and $E=-0.008$ (b) Timelike orbits in null-like naked
singularity spacetime for $h=5$, where $r_{min}=13.74$ and $E=-0.008$ (c)
Timelike orbits in null-like naked singularity spacetime for $h=6$, where
$r_{min}=21.58$ and $E=-0.008$ (d) Timelike orbits in null-like naked
singularity spacetime for $h=7$, where $r_{min}=33.55$ and $E=-0.008$

Figure 2: In this figure, relativistic orbits (The red lines) of a test
particle in the null-like naked singularity spacetime are shown. The blue
circle represents the minimum approach of a test particle (or periastron
points) in a naked singularity case. While dotted black orbits indicate the
orbital precession in the Schwarzschild black hole spacetime. The dark black
region at the center defines the Schwarzschild radius. where the total mass
$M=1$.

##  III Periodic orbits

No material particle or light may escape to distant observers if the
singularity is surrounded by a null hypersurface (event horizon or trapped
surface). This singularity is called a space-like singularity, since it is not
causally connected to any other spacetime points. Other sorts of singularities
can also develop as a result of the gravitational collapse of an inhomogeneous
matter cloud joshi ; mosani3 ; mosani4 ; Joshi:2011zm ; JNW . These
gravitationally strong singularities are linked to other spacetime points,
which are referred to as null and time-like singularities Dey:2020bgo ;
Joshi:2020tlq . Now, the conserved energy ($\gamma$) and angular momentum
($h$) per unit rest mass confirm the axial and temporal symmetries in the null
singularity spacetime given in equation (3). Therefore, we can write the
expressions of these conserved quantities as,

$\gamma=-g_{tt}(r)u^{t}=\frac{1}{\left(1+\frac{M}{r}\right)^{2}}\left(\frac{dt}{d\tau}\right)\,\,,\,\,\,$
(6) $h=g_{\phi\phi}(r)u^{\phi}=r^{2}\left(\frac{d\phi}{d\tau}\right)\,\,,$ (7)

where, $\tau$ is the proper time of the particle. Using these symmetries
property and normalisation of the four velocity conditions for the timelike
geodesics ($u^{\alpha}u_{\alpha}=-1$), we can determine the total relativistic
energy for $\theta=\pi/2$ as,

$E=\frac{1}{2}\left[\left(\frac{dr}{d\tau}\right)^{2}+V_{eff}(r)\right],$ (8)

where, $V_{eff}$ is the effective potential of the null naked singularity
spacetime, which is derived as,

$V_{eff}(r)=\frac{\left(\frac{h^{2}}{r^{2}}+1\right)}{\left(1+\frac{M}{r}\right)^{2}}-1.$
(9)

Note that, we have considered the timelike bound orbits only in the equatorial
plane $(\theta=\pi/2)$, since a spacetime metric is spherically symmetric and
static. Moreover, it is a widely acknowledged fact that the inner most stable
circular orbit (ISCO) could be obtain by solving $\frac{dV_{eff}}{dr}=0$ for
the extremum value of $r$. Therefore, one can find ISCO for Schwarzschild
black hole case at $r_{isco}=6M$. While, in the null naked singularity case,
we obtain expression of ISCO to be at $r={h^{2}}/{M}$. This expression
explicitly demonstrates that ISCO for the case of null naked singularity
spacetime extends up to the singularity at $r=0$ as $h\rightarrow{0}$. This
means that for any particle with some angular momentum, the radius of its
inner most stable circular orbit would depend directly on the square of its
own angular momentum and inversely proportional to the mass of that
singularity. It is also evident that for a particle with zero angular momentum
there would be no ISCO and it would eventually plunge into the singularity.
Moving on, the timelike bound orbits can be accomplished by satisfying the
conditions,

$E-V_{eff}(r)>0;\,\,\ V^{\prime}_{eff}(r)=0;\,\,\
V^{\prime\prime}_{eff}(r)>0,$ (10)

where, prime denote for the derivative with respect to ‘r’. Using these
timelike bound orbit conditions, we can derived an expression for the shape of
an orbit as,

$\left(\frac{dr}{d\phi}\right)^{2}=\frac{r^{4}}{h^{2}}\left[\gamma^{2}-\frac{\left(\frac{h^{2}}{r^{2}}+1\right)}{\left(1+\frac{M}{r}\right)^{2}}\right],$
(11)

this above expression implies how the coordinate ‘r’ change as the coordinate
‘$\phi$’ change. Now, one can derive the orbit equation for null naked
singularity metric by differentiating the above Eq. (11) with respect to
$\phi$ and we get,

$\frac{d^{2}u}{d\phi^{2}}-\frac{M(1+h^{2}u^{2})}{h^{2}(1+Mu)^{3}}+\frac{u}{(1+Mu)^{2}}=0,$
(12)

the equation above is known as the orbit equation of a test particle for null
naked singularity spacetime. It is considerably difficult to solve this
equation analytically since it is second order non-linear differential
equation. Therefore, we solve it numerically and shown in the Fig. 2.

In Fig. 2, we show that the periastron precession of the timelike bound orbits
for different angular momentum values (h = 4, 5, 6, 7) in the null naked
singularity spacetime (The red orbits). The blue circle represents the minimum
approach of a test particle (or periastron points) in a naked singularity
case. While dotted black orbits indicate the orbital precession in the
Schwarzschild black hole spacetime. The dark black region at the center
defines the Schwarzschild radius. One can see from Fig. 2, that the precession
angle would decrease as the angular momentum of a test particle increases, and
hence the distance of the minimum approach will increase. In Schwarzschild
spacetime, the precession angle of the orbit is larger than the precession
angle of the orbit in naked singularity spacetime. However, at large distances
from the singularity, the precession angle of the orbit would be nearly
similar as we increase the angular momentum of the particle ‘h’.

##  IV Tidal force effects

Now in this section, we dive into the intricacies of the equations involving
tidal forces in our spacetime i.e. null naked singularity. In order to examine
the equation for the distance between two infinitesimally close and free
falling particles, we take the equation for the spacelike components of
geodesic deviation vector $\eta^{\mu}$ to be:

$\frac{D^{2}\eta^{\mu}}{D\tau^{2}}-R^{\mu}_{\nu\rho\sigma}v^{\nu}v^{\rho}\eta^{\sigma}=0,$
(13)

where $R^{\mu}_{\nu\rho\sigma}$ is the Riemann curvature tensor and $v^{\mu}$
is the unit tangent vector to the geodesic Hobson:2006 . In order to solve for
the Riemann curvature tensor we take the aid of tetrad formalism. Tetrads are
geometric objects that are orthonormal and form a set of local coordinate
bases i.e. a locally defined set of four linearly independent vector fields
called tetrads or vierbien De . At every point on some geodesic, there is a
tetrad frame that forms a local inertial reference frame which is valid in the
region nearby. In this region, the laws of special relativity apply. An
orthonormal basis, unless it is also a coordinate basis, does not have enough
information to provide the line element (or the connection). There can be
infinitely many orthonormal bases at a particular point related to each other
by Lorentz transformations. But to determine those orthonormal bases which
carry the information of the metric, we must find a linear transformation from
the orthonormal basis to a coordinate basis:

$\vec{e}_{\mu}=\hat{e}^{\hat{\mu}}_{{\mu}}\vec{e}_{\hat{\mu}}$ (14)

where, $\vec{e}_{\mu}$ represents the coordinate basis, $\vec{e}_{\hat{\mu}}$
represents the orthonormal basis and $\hat{e}^{\hat{\mu}}_{{\mu}}$ are the
coefficients also known as the tetrad components. These tetrad components are
the coefficients that contain the information of a metric and eventually
enable these orthonormal basis to extract physical, measurable quantities from
geometric, coordinate-free objects in curved manifolds. These tetrads may be
inverted in the obvious ways:

$\vec{e}_{\hat{\mu}}=\hat{e}_{\hat{\mu}}^{{\mu}}\vec{e}_{\mu}$ (15)

where,
$\hat{e}_{\hat{\mu}}^{{\mu}}\hat{e}_{\nu}^{{\hat{\mu}}}=\delta_{\nu}^{\mu}$.
The metric components in the coordinate basis follow from the tetrad
components:

$g_{\mu\nu}=\vec{e}_{\mu}.\vec{e}_{\nu}=\eta_{\hat{\mu}\hat{\nu}}\hat{e}^{\hat{\mu}}_{{\mu}}\hat{e}^{\hat{\nu}}_{{\nu}}$
(16)

The equation above is the key result allowing us to use these orthonormal
bases in curved spacetime. With the aid of this formalism the parallel
transport of tangent vectors along the geodesics can be convened and hence,
this tetrad formalism satisfies geodesic equations in that curved spacetime
Bert:2002 . The tetrad components for freely falling reference frames can be
obtained from Eq. (14) and Eq. (16) to be:

$\hat{e}^{\mu}_{\hat{0}}\equiv\left\\{\frac{E}{f},-\sqrt{E^{2}+f},0,0\right\\},$
(17)
$\hat{e}^{\mu}_{\hat{1}}\equiv\left\\{-\frac{\sqrt{E^{2}+f}}{f},E,0,0\right\\},$
(18) $\hat{e}^{\mu}_{\hat{2}}\equiv\left\\{0,0,\frac{1}{r},0\right\\},$ (19)
$\hat{e}^{\mu}_{\hat{3}}\equiv\left\\{0,0,0,\frac{1}{r\sin{\theta}}\right\\},$
(20)

where, $(x^{0},x^{1},x^{2},x^{3})=(t,r,\theta,\phi)$. These tetrad components
satisfy the following orthonormality condition,

$\hat{e}^{\
\alpha}_{\hat{\mu}}\hat{e}_{\hat{\nu}\alpha}=\eta_{\hat{\mu}\hat{\nu}},$ (21)

where, $\eta_{\hat{\mu}\hat{\nu}}$ is the Minkowski metric DInverno:1992 . We
have that $\hat{e}^{\mu}_{\hat{0}}=v^{\mu}$. The geodesic deviation vector,
also called the separation vector, can be expanded as,

$\eta^{\mu}=\hat{e}_{\hat{\nu}}^{\ \mu}\eta^{\hat{\nu}}.$ (22)

Note that for a fixed temporal component $\eta^{\hat{0}}=0$ DInverno:1992 .
Therefore, subsequently one can obtain non-vanishing independent components of
the Riemann tensor in spherically symmetric spacetimes, including the null
naked singularity spacetime as Wald ,

Figure 3: (a) Radial tidal effects,
$T_{R}=\frac{1}{\eta^{\widehat{1}}}\frac{d^{2}\eta^{\widehat{1}}}{d\tau^{2}}$
and (b) Angular tidal effects,
$T_{A}=\frac{1}{\eta^{\widehat{i}}}\frac{d^{2}\eta^{\widehat{i}}}{d\tau^{2}}$
for radially freely falling particle in the null naked singularity and
Schwarzschild black hole spacetimes.  Figure 4: Radial tidal force ratio for
the null naked singularity spacetime $(T_{R})_{NNS}$ over Schwarzschild black
hole singularity spacetime $(T_{R})_{Sch}$.

$R^{1}_{\ 212}=-\frac{rf^{\prime}}{2}\,\,;\ R^{1}_{\
313}=-\frac{rf^{\prime}}{2}\sin^{2}\theta,$ (23) $R^{1}_{\
010}=\frac{ff^{\prime\prime}}{2}\,\,;\ R^{2}_{\
323}=\left(1-f\right)\sin^{2}\theta,$ (24) $R^{2}_{\
020}=\frac{ff^{\prime}}{2r}\,\,;\ R^{3}_{\ 030}=\frac{ff^{\prime}}{2r}.$ (25)

Now using the tetrad formalism, we obtain the Riemann curvature tensor in
terms of tetrad basis to be:

$R^{\hat{\alpha}}_{\hat{\beta}\hat{\gamma}\hat{\delta}}=R^{\mu}_{\nu\rho\sigma}\hat{e}_{\mu}^{\hat{\alpha}}\hat{e}_{\hat{\beta}}^{\nu}\hat{e}_{\hat{\gamma}}^{\rho}\hat{e}^{\sigma}_{\hat{\delta}}.$
(26)

Upon obtaining the expressions from Eq. (26) while considering that the
vectors ${\hat{e}_{\hat{\nu}}}^{\ \mu}$ are parallelly transported along the
geodesic, one can find the following equations for tidal forces in radial
free-fall reference frames Crispino:2016pnv ,

$\ddot{\eta}^{\widehat{1}}=-\frac{f^{\prime\prime}}{2}\eta^{\widehat{1}},$
(27) $\ddot{\eta}^{\widehat{i}}=-\frac{f^{\prime}}{2r}\eta^{\widehat{i}},$
(28)

where $i=2,\,3$. Now, we can plug in the second and the first derivatives of
$f$ with respect to $r$ in Eq. (27) and Eq. (28) respectively to obtain,

$\ddot{\eta}^{\widehat{1}}=-\frac{M(M-2r)}{(M+r)^{4}}\eta^{\widehat{1}},$ (29)
$\ddot{\eta}^{\widehat{i}}=-\frac{M}{(M+r)^{3}}\eta^{\widehat{i}}.$ (30)

The above two equations Eq. (29) and Eq. (30) are the tidal force equations
for a radially free-falling frame in the null naked singularity metric, where
Eq. (29) represents the radial component of the tidal force and Eq. (30)
represents the angular component of the same. Using these two equations, one
can generate two separate plots, one each for radial and angular components of
the tidal force as shown in Fig. 3. The process of compression in the angular
direction and stretching in the radial direction is known as
’spaghettification’.

Figure 5: (a) Radial solutions of the geodesic deviation equation for
$\eta^{\hat{1}}(b)=1$, $\frac{d\eta^{\hat{1}}(b)}{d\tau}=0$ with $M=1$, (b)
Angular solutions of the geodesic deviation equation for
$\eta^{\hat{i}}(b)=1$, $\frac{d\eta^{\hat{i}}(b)}{d\tau}=0$, $M=1$.

The radial component of the tidal force as described in Fig. 3.(a) shows clear
differences between the null naked singularity and the Schwarzschild black
hole that can be summarised as follows. One can see from this figure that the
radially free-falling object will stretch initially as it approaches the
singularity at $r=0$. At a finite distance from the singularity, the signature
of the radial tidal force flips and eventually behaves like a compressing
force. This can also be verified from Eq. (29) where the sign of the radial
tidal force remains positive when $2r>M$ and at $r=M$ the radial tidal force
reaches its maximum value. And beyond this maxima i.e. when $2r<M$ the radial
tidal force component becomes negative and as $r\rightarrow{}0$ the radial
tidal force converges to a finite value which is equivalent to $-1/M^{2}$.
This indicates no more stretching of the radially infalling object beyond this
finite maxima and rather the object gets subjected to finite compression until
the plunge into the singularity. This result is contrary to the Schwarzschild
black hole case where the radially infalling object is subjected to infinite
stretching as it approaches the singularity.

For the angular component of the tidal force as shown in Fig. 3.(b), the rate
of change of angular tidal force for both cases is distinguishable from one
another. Interestingly, in this case, the angular tidal force approaches
$-1/M^{2}$ as $r\rightarrow{}0$ from Eq. (30), suggesting compression in the
angular direction but up to this finite value. While in the case of
Schwarzschild black hole the angular tidal force diverges to negative infinity
as $r\rightarrow{}0$ which means infinite compression in the angular direction
for a radially free-falling object.

In Fig. 4, we compare the radial tidal force components of null naked
singularity and Schwarzschild black hole spacetimes. In this plot, we have
considered the ratio in the cases of different $ADM$ mass values i.e.
$M=(1,5,10,20,30,100)$. It is particularly evident from this plot that the
ratio approaches zero near the singularity, indicating the fact that radial
tidal force in null naked singularity settles to a finite value beyond its
maxima and it diverges in the case of Schwarzschild black hole, resulting the
ratio approaching zero near the singularity. The profile of all the curves in
this figure approaches unity for an asymptotical distance from the singularity
and only near the maximal distance for each particular singularity i.e. at
$M=r$, the ratio begins to rapidly converge to zero.

##  V Geodesic deviation

Substituting the explicit form of $f(r)$ in null naked singularity spacetime
into Eqs. (27) and (28), we obtain the tidal force equations for our spacetime
to be:

$\left(E^{2}-\left(1+\frac{M}{r}\right)^{-2}\right){\eta}^{\widehat{1}}{{}^{\prime\prime}}=\frac{Mr}{(M+r)^{3}}{\eta}^{\widehat{1}}{{}^{\prime}}-\frac{M(M-2r)}{(M+r)^{4}}\eta^{\widehat{1}},$
(31)
$\left(E^{2}-\left(1+\frac{M}{r}\right)^{-2}\right){\eta}^{\widehat{i}}{{}^{\prime\prime}}=\frac{Mr}{(M+r)^{3}}{\eta}^{\widehat{i}}{{}^{\prime}}-\frac{M}{(M+r)^{3}}\eta^{\widehat{i}}.$
(32)

From the above two equations, it can be inferred that the tidal forces in this
spacetime depend on the mass and the radial distance from the singularity. We
note that the expressions of the tidal forces, given by Eqs. (27) and (28),
are identical to the Newtonian tidal forces with the force $-f^{\prime}/2$ in
the radial direction. Using the above equations (31) and (32), we compute
their numerical solutions for $\eta^{\hat{1}}(r)$ and $\eta^{\hat{i}}(r)$ as
depicted in Fig. 5. Both the plots Fig. 5.(a) and 5.(b) represent how the
initial radial and angular separations, respectively, of two nearby geodesics
are changing while falling towards the central singularity in both the cases
of a null naked singularity and Schwarzschild black hole. It is appropriate to
comment that $\eta^{\hat{1}}(b)$ and $\eta^{\hat{i}}(b)$ are initial
separation distances between two nearby geodesics at $r=b$ to the radial and
angular directions, respectively, and in the case of both the plots we have
considered the impact parameter $b=100M$. Additionally,
$\frac{d\eta^{\hat{1}}(b)}{d\tau}$ and $\frac{d\eta^{\hat{i}}(b)}{d\tau}$ are
initial velocities at $r=b$ to the radial and angular directions,
respectively. We have considered the case when $\eta^{\hat{\alpha}}(b)\neq 0$,
$\frac{d\eta^{\hat{\alpha}}(b)}{d\tau}=0$, where $(\alpha=1,~{}2,~{}3)$.

It is evident from Fig. 5.(a) that the radial component of the geodesic
separation vector in both the cases i.e. for null naked singularity and
Schwarzschild black hole follow a similar trajectory. This implies that the
geodesic separation vector $\eta^{\widehat{1}}$ gets stretched radially and
approaches infinity as the separation vector comes closer to the singularity.
In the case of angular component of the geodesic separation vector
$\eta^{\widehat{i}}$, Fig. 5.(b) illustrates the act of compression as the
separation vector tends to zero when near the singularity and the profile of
the plot is the same in both null naked and Schwarzschild singularities.

##  VI Conclusions

The comparison of null naked singularity spacetime and the Schwarzschild black
hole spacetime was intriguing not only from the very inception of the idea for
the same but instead, it concluded with some very interesting results. The
important remarks and conclusions of this work are as follows:

  * •

The orbit’s precession angle in Schwarzschild spacetime is greater than that
of the naked singularity spacetime orbit for the same parameters value.
However, as the angular momentum of the particle ’h’ increases, the precession
angle of the orbit becomes almost equivalent at larger distances from the
singularity. As we point out before, the null naked singularity asymptotically
resembles the Schwarzschild spacetime, but the nature of the orbital
precession remains unchanged. As a result, the null naked singularity can be
thought of as a Schwarzschild black hole mimicker.

  * •

The most counter-intuitive result that we were able to obtain is the fact that
the radial component of the tidal force in null naked singularity spacetime
also shows a steep rise but, at a later stage when compared to the case of
Schwarzschild black hole as shown in Fig. 3(a). And this steep rise in the
radial tidal force does not culminate into an infinite force when approaching
the singularity like in the case of the Schwarzschild black hole but instead
it flips its signature by turning from a stretching force to a compressing
force at a finite distance from the singularity. The magnitude of which would
depend on the inverse square of the ADM mass $M$ of the spacetime i.e.
$-{1}/{M^{2}}$ when $r\rightarrow{}0$ from Eq. (29). This aspect alone can be
sufficient to distinguish between a null naked singularity spacetime and a
Schwarzschild spacetime.

  * •

To add further we have also compared the results for the angular component of
the tidal force for the two spacetimes and a similar steep rise at a later
stage in the angular tidal force is seen in the case of null naked
singularity, but no change in the signature of the tidal force component. Only
the fact that the magnitude of the angular part of the tidal force in null
naked singularity approaches $-{1}/{M^{2}}$ as $r\rightarrow{}0$ from Eq.
(30), which suggests a finite force of compression which is dependent only on
the inverse square of the ADM mass $M$ of the spacetime. This is another
distinguishing aspect between the two spacetimes, as in the Schwarzschild
spacetime the angular component of the tidal force is solely dependent on the
inverse cube of the radial distance from the singularity Hong:2020bdb .

  * •

We also computed the numerical solution of the geodesic deviation equations
about a radially free-falling geodesic. In particular, we examined the
behavior of the geodesic separation vector for such a geodesic under the
influence of tidal forces created by the null naked singularity spacetime and
compared the results to Schwarzschild black hole spacetime. We conclude that
both spacetimes show similar behavior of the geodesic separation vector.

##  VII Acknowledgments

The authors would like to acknowledge the support and encouragement provided
by all the members and research fellows of ICC (International Centre for
Cosmology). Also would like to express our gratitude towards Pankaj S. Joshi
for all his valuable suggestions and astute insights.

## Data Availability Statement

This manuscript has no associated data or the data will not be deposited.
[Authors’ comment: The results were obtained via analytical and numerical
simulations. So there are no associated data.]

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