# Lense-Thirring effect and precession of timelike geodesics in slowly
rotating black hole and naked singularity spacetimes

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

###### Abstract

In this paper, we investigate the properties of the Lense-Thirring precession
frequency and periastron precession of the relativistic bound orbits in slowly
rotating naked singularity spacetimes. We show that the precession frequencies
of a stationary test gyroscope are quite unique in slowly rotating naked
singularity spacetimes, and therefore, one can easily distinguish them from
the slowly rotating Kerr black hole spacetime. On the other hand, we also show
that in slowly rotating Joshi-Malafarina-Narayan (JMN1) naked singularity
spacetime, negative precession of the timelike bound orbits is allowed which
is not possible in Kerr and slowly rotating null naked singularity spacetimes.

Keywords: Blackhole, naked singularity, Lense-Thirring effect, Periastron
precession.

##  I Introduction

It is well known that the end state of the continual gravitational collapse of
matter cloud is the spacetime singularity hawking ; penrose . However, there
is no such theorem on the causal nature of spacetime singularity formed during
the gravitational collapse. In 1969, Penrose gave the cosmic censorship
conjecture (CCC) which does not allow a spacetime singularity to be causally
connected with other spacetime points penrose2 . There are two versions of the
cosmic censorship conjecture: weak cosmic censorship conjecture and strong
cosmic censorship conjecture. The weak cosmic censorship conjecture does not
allow any globally visible singularity tiplerclarkeellis , whereas its strong
counter-part states that a spacetime singularity should not be locally visible
as well geroch ; hawking2 ; penrose3 . One can show that a globally hyperbolic
spacetime is necessary for the validity of strong cosmic censorship conjecture
psjoshi2 ; joshinarlikar ; bruhat ; klainerman ; ringstrom ; sbierski . It can
be shown that the singularity formed during the homogeneous dust (i.e.
pressure is zero) collapse is always locally hidden. However, if one
introduces some non-zero pressure or inhomogeneity in density, then with some
suitable initial data, it can be shown that the singularity can be locally as
well as globally visible. In many literature, the formation of the visible or
naked singularity during the gravitational collapse of matter cloud is shown
eardley ; christodoulou ; joshi ; goswami ; joshi2 ; vaz ; jhingan0 ;
deshingkar ; mena ; magli1 ; magli2 ; giambo1 ; harada1 ; harada2 ; joshi7 ;
mosani1 ; mosani2 ; mosani3 ; mosani4 ; mosani5 . In Joshi:2011zm , authors
show that a spacetime with a central naked singularity can be formed as an
equilibrium end state of the gravitational collapse of general matter cloud.
Therefore, in the regime of classical general relativity, there are many
theoretical scenarios where a visible singularity exists. Now, the important
question is whether they really exist in nature and if they really exist then
what would be their physical signature.

In several papers, the possible physical signatures of the ultra-high density
compact objects are discussed in detail Shaikh:2019hbm ; Gralla:2019xty ;
Abdikamalov:2019ztb ; Yan:2019etp ; Vagnozzi:2019apd ; Gyulchev:2019tvk ;
Shaikh:2019fpu ; Dey:2013yga ; Dey+15 ; Shaikh:2018lcc ; Joshi2020 ; Paul2020
; Dey:2020haf ; Dey:2020bgo ; abdujabbarov_2015a ; atamurotov_2015 ;
abdujabbarov_2017 ; abdujabbarov_2015b ; younsi_2016 ; papnoi_2014 ;
bambi_2013a ; ohgami_2015 ; stuchlik_2018 ; stuchlik_2019 ; Martinez:2019nor ;
Eva ; Eva1 ; Eva2 ; tsirulev ; Joshi:2019rdo ; Bhattacharya:2017chr ;
Bambhaniya:2019pbr ; Dey:2019fpv ; Bam2020 ; Bambhaniya:2021ybs ; Lin:2021noq
; Deng:2020yfm ; Deng:2020hxw ; Gao:2020wjz ; aa4 . In Shaikh:2019hbm ;
Gralla:2019xty ; Abdikamalov:2019ztb ; Yan:2019etp ; Vagnozzi:2019apd ;
Gyulchev:2019tvk ; Shaikh:2019fpu ; Dey:2013yga ; Dey+15 ; Shaikh:2018lcc ;
Joshi2020 ; Paul2020 ; Dey:2020haf ; Dey:2020bgo ; abdujabbarov_2015a ;
atamurotov_2015 ; abdujabbarov_2017 ; abdujabbarov_2015b ; younsi_2016 ;
papnoi_2014 ; bambi_2013a ; ohgami_2015 ; stuchlik_2018 ; stuchlik_2019 ,
authors elaborately discuss the shadow properties of different types of
compact objects. With respect to an asymptotic observer, the image of a
compact object may consist of a central dark region surrounded by a high
intensity or bright region, where the intensity maximizes at the boundary of
the dark region and smoothly decreases as radial distance increases. In the
asymptotic observer’s sky, the central dark region is the shadow of the
compact object. In several recent papers Shaikh:2018lcc ; Joshi2020 ; Paul2020
; Dey:2020haf ; Dey:2020bgo , it is shown that the shadow is not only the
property of the black hole but it can also be cast by other horizon less
compact objects. In Shaikh:2018lcc , authors show that a naked singularity
spacetime can cast shadow of similar size to what an equally massive
Schwarzschild black hole can cast. In Joshi2020 ; Dey:2020bgo , the shadow
properties of the nulllike and timelike naked singularities are elaborately
discussed. In Dey:2020haf , a spacetime configuration having a central naked
singularity is considered and it is shown that a thin shell of matter can cast
shadow in the absence of photon sphere as well as event horizon. All these
theoretical studies on the shadow of ultra-compact objects are very much
important in the context of recent observation of the shadow of M87 glactic
center and the upcoming image of Milky-way galactic center Sgr-A*.

Another important physical signature of a horizon-less ultra-compact object is
the orbital dynamics of stars around it which can distinguish it from the
black holes. As we know, the nature of timelike geodesics in a spacetime
depends upon the geometry of that spacetime. Therefore, the trajectory of
stars around a compact object can reveal the causal structure of that compact
object. Hence, the stellar dynamics close to our Milky-way galactic center can
reveal important properties of Sgr-A*. For this purpose, GRAVITY, SINFONI,
UCLA galactic group are continuously tracking the dynamics of ‘S’ stars which
are orbiting around Sgr-A* with very small (w.r.t galactic scale) pericenter
distances M87 ; Eisenhauer:2005cv ; center1 . In this context, there are many
literature where the timelike geodesics around different types of compact
objects are elaborately discussed Martinez:2019nor ; Eva ; Eva1 ; Eva2 ;
tsirulev ; Joshi:2019rdo ; Bhattacharya:2017chr ; Bambhaniya:2019pbr ;
Dey:2019fpv ; Bam2020 ; Bambhaniya:2021ybs ; Lin:2021noq ; Deng:2020yfm ;
Deng:2020hxw ; Gao:2020wjz ; aa4 . In some of the recent papers, we show that
the timelike bound orbits of a massive particle in the presence of a central
naked singularity, can precess in the opposite direction of the particle
motion and we denote that type of precession as negative precession. It is
shown that in Schwarzschild and Kerr black hole spacetime, the negative
precession of the timelike bound orbits is not allowed. However, there are
some naked singularity spacetimes for which both the positive and negative
precession of bound orbits are possible Bambhaniya:2019pbr ; Dey:2019fpv ;
Bam2020 ; Bambhaniya:2021ybs . Therefore, the nature of the precession of the
bound timelike orbits can give us the information about the causal structure
of the central compact object. Accretion disk around a compact object and its
thermal property is also an important physical phenomenon that can be used for
distinguishing black holes from other compact objects Guo:2020tgv .

In this paper, we discuss the nature of Lense-Thirring precession of a test
gyro and the orbital precession of a test particle in two rotating naked
singularity spacetimes, where we consider slow rotation approximation. We use
the Newman-Janis algorithm Janis Newman ; Drake:1998gf to construct the
rotating counterparts of the static 1st type of Joshi-Malafarina-Narayan
(JMN1) spacetime Joshi:2011zm  and a null naked singularity spacetime
Joshi2020 . We investigate the frame-dragging effect of these two rotating
naked singularity spacetimes on the spin of a test gyro (i.e. the Lense-
Thirring precession ). The Lense-Thirring precession is one of the important
prediction of Einstein’s general relativity and in the past, a huge amount of
efforts were given to verify the prediction lt ; schiff1 . As it is known, on
4th May 2011, NASA and the Standford-based analysis group confirmed that the
Gravity Probe B satellite measured an equal amount of precession that can be
predicted considering the general relativistic frame dragging effect of Earth
ev . In Gravity Probe B, four very sensitive gyroscopes are installed and
initially, they were pointed towards a fixed star HR8703 which is also known
as IM Pegasi. However, due to the earth’s rotation, the frame dragging effect
changes the direction of the gyroscopes. Due to the weak gravitational field
of the earth (compare to ultra-high compact object), the Gravity Probe B
recorded a precession rate $37.2\pm 7.2$ milliarcsecond/ year, whereas general
relativistic prediction is $39.2\pm 0.19$ milliarcsecond/ year. After the
confirmation of Gravity Probe B, the precession of a test gyroscope near the
rotating ultra-high-density compact objects has become an important field of
research, where a pulsar near a compact object plays the role of the gyroscope
Chakraborty:2016ipk ; Chakraborty:2016mhx ; Kocherlakota:2017hkn . Due to the
Lense-Thirring effect of a rotating massive body, the pulse rate of a pulsar
is expected to be varied and therefore, the observational results on the pulse
rate of a pulsar near an ultra-compact object can reveal the causal structure
of that compact object Kocherlakota:2017hkn . Along with the Lense-Thirring
precession, in this paper, we also discuss the orbital precession of a test
particle around the rotating JMN1 and null naked singularity spacetime and we
show that the rotating counterpart of JMN1 spacetime also allows negative
precession of bound orbits which is not possible in Kerr spacetime.

This paper is organised as follows. In Sec. (II), we discuss the Newman-Janis
algorithm and show how one can derive the rotating version of a spherically
symmetric and static spacetime using that algorithm. In that section, we also
discuss why we need to consider the slow rotation approximation to get the
rotating form of a general static, spherically symmetric spacetime using
Newman-Janis algorithm. At last, in that section, we construct the slowly
rotating forms of JMN1 and null naked singularity spacetimes. In Sec. (III),
we investigate the Lense-Thirring effects on a test gyroscope in those two
rotating naked singularity spacetimes and compare the results with that for
Kerr spacetime. In Sec. (IV), we discuss the nature of the precession of the
timelike bound orbits in slowly rotating JMN1 and the null naked singularity
spacetimes. In Sec. (V), we conclude by discussing the important results of
this paper. Throughout this paper, we consider Newton’s gravitational constant
$G_{N}$ and light velocity $c$ as unity.

##  II The general Newman-Janis procedure

In Janis Newman , Janis and Newman develop a procedure by which the Kerr
metric can be obtained from the Schwarzschild metric. In this section, at
first, we briefly discuss the general Newman-Janis procedure (NJP) for the
readers’ convenience and then we construct the rotating versions of JMN1 and
null naked singularity spacetime using that algorithm. The NJP consists of a
four-step method which help us to construct stationary, axially symmetric, and
rotating spacetime from a seed metric. In this paper, we consider the seed
metric as static and spherically symmetric. The general spherically symmetric
and static metric in Boyer-Lindquist coordinates (BLC) can be written as,

$\displaystyle\text{d}s^{2}=-f(r)~{}\text{d}t^{2}+\frac{~{}\text{d}r^{2}}{g(r)}+h(r)\text{d}\Omega^{2},$
(1)

where $\text{d}\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. Now, the
first step is to write the seed metric in outgoing or incoming Eddington-
Finkelstein coordinates ($u,r,\theta,\phi$). Following is the transformation
from BLC to outgoing Eddington-Finkelstein coordinate system,

$du=dt-\frac{dr}{\sqrt{f(r)g(r)}}.$ (2)

Therefore, in these advanced Eddington-Finkelstein coordinates, the metric (1)
is expressed as,

$\text{d}s^{2}=-f(r)~{}\text{d}u^{2}-2\sqrt{\frac{f(r)}{g(r)}}~{}\text{d}u~{}\text{d}r+h(r)\text{d}\Omega^{2}.$
(3)

In step two, we introduce the inverse metric $g^{\mu\nu}$ using the null
tetrad $(l^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu})$ as,

$g^{\mu\nu}=-l^{\mu}n^{\nu}-l^{\nu}n^{\mu}+m^{\mu}\bar{m}^{\nu}+m^{\nu}\bar{m}^{\mu},$
(4)

where over-line indicates complex conjugation and these null tetrads can be
written as,

$\displaystyle l^{\mu}=\delta^{\mu}_{r},\hskip 5.69046ptn^{\mu}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{g(r)}{f(r)}}\delta^{\mu}_{u}-\frac{g(r)}{2}\delta^{\mu}_{r},$
$\displaystyle m^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}r}\left(\delta^{\mu}_{\theta}+\frac{i}{\sin\theta}\delta^{\mu}_{\phi}\right),$
$\displaystyle\bar{m}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}r}\left(\delta^{\mu}_{\theta}-\frac{i}{\sin\theta}\delta^{\mu}_{\phi}\right),$
(5)

where they satisfy the following relations,

$l_{\mu}l^{\mu}=n_{\mu}n^{\mu}=m_{\mu}m^{\mu}=l_{\mu}m^{\mu}=n_{\mu}m^{\mu}=0,$
(6) $l_{\mu}n^{\mu}=-m_{\mu}\bar{m}^{\mu}=-1.$ (7)

In order to obtain a new axially symmetric and rotating spacetime, one need to
do complexification of the functions as, $f(r),g(r),h(r)$
$\rightarrow\tilde{f}(r^{\prime}),\tilde{g}(r^{\prime}),\tilde{h}(r^{\prime})$.
Therefore, the third step is the complexification of coordinates in which we
allow the coordinates to take complex values. The radial coordinate r can be
written as a real function of r and its complex conjugate $\bar{r}$. Following
are the examples of that modifications,

$\frac{1}{r}\rightarrow\frac{1}{2}\left(\frac{1}{r^{\prime}}+\frac{1}{\bar{r}^{\prime}}\right),\hskip
8.5359ptr^{2}\rightarrow r^{\prime}\bar{r}^{\prime},$ (8)

where the complex coordinate can be written as,

$\displaystyle
u^{\prime}=u-ia\cos\theta,\,\,r^{\prime}=r+ia\cos\theta,\,\,\theta^{\prime}=\theta,\,\,\phi^{\prime}=\phi\,.$
(9)

The reason to choose this particular complexification (Eq. (8)) is that it is
successful in generating the rotating vacuum solutions. In the Drake:1998gf ,
it is shown that this is the unique choice of complexification to generate
Kerr-Newman solution from Reissner–Nordstrom spacetime using the Newman-Janis
procedure. With the complex coordinate system, the null tetrads can be written
as,

$\displaystyle l^{\prime\mu}$ $\displaystyle=$
$\displaystyle\delta^{\mu}_{r},\hskip
8.5359ptn^{\prime\mu}=\sqrt{\frac{\widetilde{g}(r^{\prime})}{\widetilde{f}(r^{\prime})}}\delta^{\mu}_{u}-\frac{\widetilde{g}(r^{\prime})}{2}\delta^{\mu}_{r},$
$\displaystyle m^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\widetilde{h}(r^{\prime})}}\left(ia\sin\theta(\delta^{\mu}_{u}-\delta^{\mu}_{r})+\delta^{\mu}_{\theta}+\frac{i}{\sin\theta}\delta^{\mu}_{\phi}\right),$
$\displaystyle\bar{m^{\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\widetilde{h}(r^{\prime})}}\left(ia\sin\theta(\delta^{\mu}_{u}-\delta^{\mu}_{r})-\delta^{\mu}_{\theta}+\frac{i}{\sin\theta}\delta^{\mu}_{\phi}\right).$

Now, using the new null tetrads, one can write the metric in the advance
Eddington-Finkelstein or null coordinates as,

$\displaystyle ds^{2}=$ $\displaystyle\
-\widetilde{f}(r)du^{2}-2\sqrt{\frac{\widetilde{f}(r)}{\widetilde{g}(r)}}dudr+2a\sin^{2}\theta\left(\widetilde{f}(r)-\sqrt{\frac{\widetilde{f}(r)}{\widetilde{g}(r)}}\right)dud\phi+2a\sqrt{\frac{\widetilde{f}(r)}{\widetilde{g}(r)}}\sin^{2}\theta
drd\phi$ (11) $\displaystyle\
+\widetilde{h}(r)d\theta^{2}+\sin^{2}\theta\left[\widetilde{h}(r)+a^{2}\sin^{2}\theta\left(2\sqrt{\frac{\widetilde{f}(r)}{\widetilde{g}(r)}}-\widetilde{f}(r)\right)\right]d\phi^{2}.$

Here, one can see, the complexification usually produces a metric with several
non-zero off-diagonal components. These can be removed (except $dtd\phi$) by
transforming it in the Boyer-Lindquist coordinates $(t,r,\theta,\phi)$. So,
the final step of the Newman-Janis algorithm is to write down the metric in
Boyer-Lindquist form using the following transformations,

$du=dt^{\prime}+F(r)dr,\hskip 14.22636ptd\phi=d\phi^{\prime}+G(r)dr.$ (12)

Inserting the above transformations in the metric (11) and setting $g_{tr}$
and $g_{r\phi}$ to zero, we obtain

$F(r)=-\frac{\sqrt{\frac{\widetilde{g}(r,\theta)}{\widetilde{f}(r,\theta)}}\widetilde{h}(r,\theta)+a^{2}\sin^{2}\theta}{\widetilde{g}(r,\theta)\widetilde{h}(r,\theta)+a^{2}\sin^{2}\theta},$
(13)
$G(r)=-\frac{a}{\widetilde{g}(r,\theta)\widetilde{h}(r,\theta)+a^{2}\sin^{2}\theta}.$
(14)

(a) Figure for $M_{0}=0.10907099$ and $R_{b}=18.336$.

(b) Figure for $M_{0}=0.9579581$ and $R_{b}=2.08778$.

Figure 1: In this figure, we show the Penrose diagrams of JMN1 null and
timelike naked singularities. In Figs. (1(a)) and (1(b)), we show the Penrose
diagrams of JMN1 spacetime with $M_{0}<2/3$ and $M_{0}>2/3$ respectively. It
can be seen that for $M_{0}>2/3$, the central singularity is null like and for
$M_{0}<2/3$, the singularity becomes timelike (i.e. shown by the red dashed
line). In the above figures, the matching radius ( $R_{b}$ ) of a junction
between internal JMN1 and external Schwarzschild spacetime is shown by the
pink dashed line, whereas, the black line represents asymptotic regions in
Schwarzschild spacetime. Therefore, the area in between the pink dashed line
and the black line represents Schwarzschild spacetime, and the area in between
the red dashed line and the pink dashed line represents JMN1 spacetime.

Note that the transformation in (12) is possible only when the left hand sides
of Eqs. (13) and (14) are independent of $\theta$. We find the transformation
in (12) is not possible in general. However, to construct a valid
transformation, one needs to consider the slow rotation limit
$(a^{2}<<r^{2})$, where we ignore the $a^{2}$ and higher order terms of $a$.
Therefore, in the slow rotation limit, the metric functions $f(r),g(r)$ and
$h(r)$ remain independent of $\theta$ after complexification. Additionally, to
make the right hand side of Eqs. (13) and (14) independent of $\theta$, we
have to assume the other conditions for the slow rotation limit as below,

$\displaystyle a^{2}\ll\sqrt{\frac{g(r)}{f(r)}}h(r),\hskip 8.5359pta^{2}\ll
g(r)h(r).$ (15)

In the slow rotation limit, by substituting the transformation (12) in (11),
we can write the general form of the slow rotating spacetime as,

$\displaystyle ds^{2}=-f(r)dt^{2}$ $\displaystyle+$
$\displaystyle\frac{dr^{2}}{g(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$ $\displaystyle-$ $\displaystyle
2a\sin^{2}\theta\left(\sqrt{\frac{f(r)}{g(r)}}-f(r)\right)dtd\phi.$

Figure 2: Penrose diagram of null naked singularity

###  II.1 Joshi-Malafarina-Narayan 1 (JMN1) Naked Singularity Spacetime

As we know, the Oppenheimer-Snyder and Datt (OSD) collapse model describes the
continual gravitational collapse of a spherically symmetric, homogeneous dust
cloud. The end state of the OSD collapse is always a black hole. However, as
it was discussed previously, for the matter cloud having inhomogeneous density
and pressures, the final fate of the gravitational collapse can be either a
naked singularity or a black hole depending upon the initial conditions. Apart
from the above mentioned scenarios, such scenarios can arise where the
presence of pressure in matter cloud can equilibrate the collapsing body into
a static configuration in a large co-moving time. In Joshi:2011zm , it is
shown that a static naked singularity spacetime, namely 1st type of Joshi-
Malafarina-Narayan (JMN1) spacetime, can be formed as an end equilibrium state
of gravitational collapse. JMN1 spacetime is a spherically symmetric static
spacetime which has a central naked singularity. This spacetime is seeded by
anisotropic matter which has zero radial and non-zero tangential pressures.
The line element of this metric can be written as,

$\displaystyle
ds^{2}=-(1-M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(1-M_{0})}}dt^{2}+\frac{dr^{2}}{(1-M_{0})}+r^{2}d\Omega^{2}\,\,,$
(17)

where the positive constant $M_{0}$ should always be less than 1, $R_{b}$ is
the radius of the boundary of the JMN1 spacetime where it is smoothly matched
with the external Schwarzschild spacetime.

Now, one can derive the slowly rotating JMN1 naked singularity spacetime by
using eqs. (II) and (17) as,

$\displaystyle
ds^{2}=-(1-M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(1-M_{0})}}dt^{2}+\frac{dr^{2}}{1-M_{0}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$
$\displaystyle-2a\sin^{2}\theta\left[\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{2(1-M_{0})}}-(1-M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(1-M_{0})}}\right]dtd\phi.$
(18)

The above slowly rotating JMN1 spacetime can be matched with the exterior
slowly rotating Kerr spacetime at $r=R_{b}$,

$\displaystyle ds^{2}=$ $\displaystyle-$
$\displaystyle\left(1-\frac{M_{0}R_{b}}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{M_{0}R_{b}}{r}\right)}$
$\displaystyle+$ $\displaystyle r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)-\frac{2aM_{0}R_{b}}{r}\sin^{2}\theta dt~{}d\phi$

where $M=\frac{M_{0}R_{b}}{2}$ is the Schwarzschild mass of the slowly
rotating compact object.

In order to describe the causal structure of JMN1 naked singularity spacetime,
in Fig. (1), we show the Penrose diagram of the same. In Figs. 1(a) and 1(b),
we show Penrose diagrams of JMN1 interior spacetime for $M_{0}=1/3$,
$M_{0}=3/4$ respectively. The pink dashed line in Figs. 1(a) and 1(b), shows
the matching of internal JMN1 spacetime with external Schwarzschild spacetime
at the matching radius $r=R_{b}$. In those Penrose diagrams, the red dashed
line shows the singularity of spacetime and the black line represents the past
and future null infinities in Schwarzschild spacetime. Therefore, the area in
between the pink dashed line and the black line represents Schwarzschild
spacetime, and the remaining area in between the red dashed line and the pink
dashed line represents JMN1 spacetime.

###  II.2 Null naked singularity spacetime

As we know, if the singularity is covered by trapped surfaces, then no
information from the singularity can reach the faraway observer. Since this
singularity is not causally connected to any other spacetime points, it is
known as space-like singularity. There are other types of singularities that
can be formed as the end state of the gravitational collapse of an
inhomogeneous matter cloud, such gravitationally strong singularities are
connected with other spacetime points and these are known as nulllike and
timelike singularities Joshi:2011zm ; Joshi2020 ; Dey:2020bgo .

In Joshi2020 , we propose a new spherically symmetric and static nulllike
naked singularity spacetime which asymptotically resembles the Schwarzschild
spacetime. The line element of the nulllike naked singularity spacetime can be
written as,

$ds^{2}=-\frac{dt^{2}}{\left(1+\frac{M}{r}\right)^{2}}+\left(1+\frac{M}{r}\right)^{2}dr^{2}+r^{2}d\Omega^{2}\,\,,$
(20)

where $M$ is the ADM (Arnowitt, Deser & Misner) mass and we show that the
Kretschmann scalar and Ricci scalar blow up at the center $r=0$ Joshi2020 .
One can easily verify that the strong, weak and null energy conditions are
satisfied for the above spacetime. The Penrose diagram of the above nulllike
naked singularity spacetime is shown in (Fig. (2)). The main interesting thing
is that if any null geodesic emanates from the past null infinity, the photon
would be infinitely red-shifted with respect to the asymptotic observer. Now
using the general expression of the slowly rotating metric (Eq. II), we can
write the slowly rotating version of the above spacetime as,

$\displaystyle ds^{2}=-\frac{dt^{2}}{\left(1+\frac{M}{r}\right)^{2}}$
$\displaystyle+$
$\displaystyle\left(1+\frac{M}{r}\right)^{2}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$ (21) $\displaystyle-$
$\displaystyle\frac{2aM\left(M+2r\right)}{\left(M+r\right)^{2}}\sin^{2}\theta
dtd\phi.$

Note that, since it is asymptotically flat spacetime, one need not match the
above spacetime to the slowly rotating exterior Kerr spacetime.

In the next two sections, we investigate the nature of the Lense-Thirring
precession of a test gyroscope and the orbital precession of the timelike
bound orbit in the slowly rotating JMN1 (Eq. (18)) and nulllike nake
singularity (Eq. (21)) spacetimes.

##  III Lense-Thirring effects in slowly rotating JMN1 and null naked
singularity spacetimes

In 1918, Josef Lense and Hans Thirring described spin precession of test
gyroscope in the gravitational field of a spinning spherical body. As we
discussed before, one can predict the causal structures of the spacetime
geometries by investigating the Lense-Thirring precession effect. Therefore,
the Lense-Thirring precession phenomenon is very much important for
distinguishing black holes from naked singularities.

Here, we consider a simple scenario where the gyroscopes are held fixed at
every point of the space and we investigate how the precession frequency of
the spin of test gyroscope changes with the change of spatial points. Since
the gyroscope is at rest in a stationary spacetime, it moves along an integral
curve ($\gamma(t)$) of a time like killing vector field $K$. The 4-velocity
($\bar{u}$) of the gyroscope can be written in terms of $\bar{K}$ as,

$\displaystyle\bar{u}=\frac{1}{\sqrt{-K^{\alpha}K_{\alpha}}}\bar{K}.$ (22)

The four velocity of timelike curve $\gamma(\tau)$ can be written as
$u=\dot{\gamma}$, where $\tau$ is the proper time and $u$ is the tangent to
the $\gamma(\tau)$ with the condition $u^{\mu}u_{\mu}=-1$. The component of
Fermi derivative of any vector in the direction of $u$ can be written as,

$\displaystyle
u^{\alpha}F_{\alpha}X^{\beta}=u^{\alpha}\nabla_{\alpha}X^{\beta}-X^{\alpha}a_{\alpha}u^{\beta}+X^{\alpha}u_{\alpha}a^{\beta}\,\,,$
(23)

where $a^{\alpha}$ is the acceleration which can be written as
$a^{\alpha}=u^{\beta}\nabla_{\beta}u^{\alpha}$. Now, if we consider an
orthonormal frame $\\{e_{i}\\}$ ($i=1,2,3$) perpendicular to $e_{0}=u$, then
along with the condition $e^{\alpha}_{\mu}e_{{\nu}{\alpha}}=\eta_{\mu\nu}$,
where $\eta_{\mu\nu}=diag(-1,1,1,1)$, we can write the components of co-
variant derivative of orthonormal basis in the direction of four velocity $u$
as,

$\displaystyle
u^{\alpha}\nabla_{\alpha}e^{\beta}_{i}=e^{\alpha}_{i}a_{\alpha}u^{\beta}+\omega^{j}_{i}e^{\beta}_{j}\,\,,$
(24)

where $\omega^{j}_{i}=u^{\alpha}\nabla_{\alpha}e^{j}_{\beta}e^{\beta}_{i}$.
Therefore, using the above Eqs. (23) and (24), the component of Fermi
derivative of orthonormal basis in the direction of four velocity $u$ can be
written as,

$\displaystyle
u^{\alpha}F_{\alpha}e_{\beta}=\omega^{\alpha}_{\beta}e_{\alpha}.$ (25)

Let S be the expectation value of the spin operator for a particle or the spin
vector of a rigid rotator like gyroscope, then $u^{\alpha}F_{\alpha}S=0$, and
after expanding this equation, we get the following expression of the change
in spin S with respect to proper time $\tau$,

$\displaystyle\frac{dS^{\alpha}}{d\tau}=\omega^{\alpha}_{\beta}S^{\beta}\,\,,$
(26)

which shows that spin vector S would precess with some frequency. The angular
frequency ($\omega_{ij}$) can be defined as,

$\displaystyle\omega_{ij}=\epsilon_{ijk}\Omega^{k}\,\,,$ (27)

where $\Omega^{k}$ is the Lense-Thirring precession frequency and
$\epsilon_{ijk}$ is the levi-civita tensor. In terms of timelike killing
vector field $\bar{K}$, $\omega_{ij}$ can be written as,

$\displaystyle\omega_{ij}=\frac{1}{\sqrt{-K^{\alpha}K_{\alpha}}}e^{\beta}_{j}K^{\delta}{\nabla}_{\delta}e_{i{\beta}},$
(28)

and therefore, $\bar{\Omega}$ can be expressed as,

$\displaystyle\bar{\Omega}=\frac{1}{2}\frac{1}{\sqrt{-K^{\alpha}K_{\alpha}}}*(\bar{K}\land
d\bar{K})\,\,,$ (29)

where $*$ represents Hodge operator and ${\land}$ represents the exterior
product. The above expression of the Lense-Thirring precession frequency for
the spin vector S can be written as,

$\displaystyle\Omega_{\mu}=\frac{1}{2}\frac{1}{\sqrt{-K^{\alpha}K_{\alpha}}}\eta^{{\nu}{\rho}{\sigma}}_{\mu}K_{\nu}{\partial}_{\rho}K_{\sigma}\,,$
(30)

where ${\eta}$ is the canonical volume corresponding to the metric
($g_{{\mu}{\nu}}$). Since only for rotating spacetime
$u^{\alpha}F_{\alpha}e_{\beta}\neq 0$, the Lense-Thirring precession of a
gyroscope is possible only in rotating spacetimes. For the stationary test
gyroscope, the timelike killing vector ($\bar{K}$) mentioned above can be
expressed as, $\bar{K}=g_{tt}dt+g_{ti}dx^{i}$. Therefore, Eq. (30) becomes,

$\displaystyle\Omega=\frac{g_{tt}}{2\sqrt{-g}}\epsilon_{ijk}\bigg{(}\frac{g_{ti}}{g_{tt}}\bigg{)}_{,j}\bigg{(}\partial_{k}-\frac{g_{kt}}{g_{tt}}\partial_{t}\bigg{)}\,\,,$
(31)

(a) Lense-Thirring precession in slowly rotating Kerr spacetime at
$\theta=0\degree$.

(b) Lense-Thirring precession in slowly rotating Kerr spacetime at
$\theta=90\degree$.

(c) Lense-Thirring precession in slowly rotating JMN1 spacetime at
$\theta=0\degree$.

(d) Lense-Thirring precession in slowly rotating JMN1 spacetime at
$\theta=90\degree$.

(e) Lense-Thirring precession in slowly rotating JMN1 spacetime at
$\theta=0\degree$.

(f) Lense-Thirring precession in slowly rotating JMN1 spacetime at
$\theta=0\degree$.

(g) Lense-Thirring precession in slowly rotating null naked singularity
spacetime at $\theta=0\degree$.

(h) Lense-Thirring precession in slowly rotating null naked singularity
spacetime at $\theta=90\degree$.

Figure 3: Figure shows Lense-Thirring precession frequency ($\Omega$) in Kerr,
JMN1 and null like naked singularity spacetimes for different values of spin
parameter ($a$).

which also can be reduced to the following form

$\displaystyle\Omega=\frac{g_{tt}}{2\sqrt{-g}}\bigg{(}\frac{g_{t{\phi}}}{g_{tt}}\bigg{)}_{,r}\partial_{\theta}-\frac{g_{tt}}{2\sqrt{-g}}\bigg{(}\frac{g_{t{\phi}}}{g_{tt}}\bigg{)}_{,{\theta}}\partial_{r}\,\,,$
(32)

where we consider the general line element of a slowly rotating spacetime as

$\displaystyle ds^{2}=-g_{tt}(r)dt^{2}$ $\displaystyle+$ $\displaystyle
g_{rr}(r)dr^{2}+g_{\theta\theta}(r)d\theta^{2}$ (33) $\displaystyle+$
$\displaystyle g_{\phi\phi}d\phi^{2}-2g_{t\phi}(r)dtd\phi\,\,.$

The magnitude of the spin precession frequency can be written as,

$\displaystyle\Omega=\sqrt{g_{{\theta}{\theta}}(\Omega_{\theta})^{2}+g_{rr}(\Omega_{r})^{2}}\,\,,$
(34)

where
$\Omega_{\theta}=\frac{g_{tt}}{2\sqrt{-g}}\bigg{(}\frac{g_{t{\phi}}}{g_{tt}}\bigg{)}_{,r}$
and
$\Omega_{r}=\frac{g_{tt}}{2\sqrt{-g}}\bigg{(}\frac{g_{t{\phi}}}{g_{tt}}\bigg{)}_{,{\theta}}$.

Now, from the above Eq. (34), we can write the magnitude of the precession
frequency for the slowly rotating Kerr spacetime (Eq. (II.1)) in terms of the
spin parameter $a$, mass $M$, radial distance $r$ and the angle ${\theta}$ as,

$\displaystyle\Omega_{Kerr}=\sqrt{\frac{a^{2}M^{2}((8M-4r)\cos^{2}\theta-r\sin^{2}\theta)}{(2M-r)r^{2}(r^{3}(-2M+r)+4a^{2}M^{2}\sin^{2}\theta)}},$

where the above expression is only valid for the radial distance greater than
the radius of the ergosphere. In Fig. (3), we show Lense-Thirring precession
frequency ($\Omega$) in slowly rotating Kerr, JMN1, and null naked singularity
spacetimes. In Chakraborty:2016ipk , the Lense-Thirring effect in Kerr
spacetime is elaborately discussed. In this paper, we first reproduce the
results of the spin precession of a test gyroscope in Kerr spacetime, and
then, we compare the same with that for JMN1 and nulllike naked singularity
spacetimes. We get novel features of the spin precession frequency of a test
gyroscope in the naked singularity spacetimes and that can be useful to
distinguish them from a black hole. Figs. (5(c),5(d)) show the behaviour of
Lense-Thirring precession in Kerr spacetime for $\theta=0$ and
$\theta=\frac{\pi}{2}$ respectively and we can see that the Lense-Thirring
precession frequency in slowly rotating Kerr spacetime monotonically increases
as radial distance decreases and becomes infinity at $r=2M$, where we consider
$M=1$. On the other hand, the expression of the Lense-Thirring precession
frequency for the Null and JMN1 naked singularities are,

$\displaystyle\Omega_{Null}=\sqrt{{\frac{a^{2}{M}^{2}((2r+{M})^{2}\cos^{2}{\theta}+r^{2}\sin^{2}{\theta})}{r^{2}((r+{M})^{2}r^{4}+a^{2}M^{2}(M+2r)^{2})\sin^{2}\theta)}}}\,\,,$
$\displaystyle\Omega_{JMN-1}=\frac{1}{4}\sqrt{\frac{a^{2}\left(16(-1+M_{0})\left(1-\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{2-2M_{0}}}+M_{0}\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{2-2M_{0}}}\right)^{2}\cos^{2}\theta-
M_{0}^{2}\sin^{2}\theta\right)}{(-1+M_{0})r^{2}\left(-((-1+M_{0})r^{2})+a^{2}\left(1-\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{2-2M_{0}}}+M_{0}\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{2-2M_{0}}}\right)^{2}\sin^{2}\theta\right)}}.$
(37)

(a) Lense-Thirring precession in slowly rotating Kerr spacetime.

(b) Lense-Thirring precession in slowly rotating JMN1 spacetime.

(c) Lense-Thirring precession in slowly rotating JMN1 spacetime.

Figure 4: Figure compares Lense-Thirring precession frequency ($\Omega$) at
poles with that at the equatorial plane for Kerr, JMN1 and null like naked
singularity spacetimes with fixed spin parameter $a=0.01$.

Figs. (3(c)) and (3(e)) show the behaviour of the spin precession frequency
($\Omega_{JMN1}$) at the pole (i.e., $\theta=0$) in slowly rotating JMN1
spacetime, where spin parameter $a=0.01$ and $a=0.1$ respectively. On the
other hand, Figs. (3(d)) and (3(f)) depicts the nature of the Lense-Thirring
precession for same values of spin parameters of the slowly rotating JMN1
spacetime at the equitorial plane (i.e., $\theta=\frac{\pi}{2}$). Unlike Kerr
spacetime, in JMN1 spacetime, the precession frequency becomes infinity at the
central singularity. In Figs. (3(g), 3(h)), similar nature of the Lense-
Thirring effect can be seen in null naked singularity spacetime. The spin
precession frequency ($\Omega_{Null}$) in Null naked singularity spacetime
also increases monotonically as radial distance decreases and blows up at the
center. Fig. (4) depicts how
$\Omega_{Kerr}(Fig.~{}(\ref{re66})),\Omega_{JMN1}(Fig.~{}(\ref{re77})),\Omega_{Null}(Fig.~{}(\ref{re88}))$
behave at pole ($\theta=0$) and at equatorial plane ($\theta=\frac{\pi}{2}$),
where we consider $a=0.01$. In Fig. (4(a)), it can be seen that in Kerr
spacetime, near $r=2M$, $\Omega_{Kerr}$ at equatorial plane is greater than
that at pole. However, after a certain radius, we can see an opposite
behaviour of $\Omega_{Kerr}$. On the other hand, in JMN1 spacetime,
$\Omega_{JMN1}$ at pole is always greater than that at equatorial plane ((Fig.
(4(b)))), whereas in null naked singularity spacetime, $\forall
r,~{}\Omega_{Null}|_{\theta=\frac{\pi}{2}}>\Omega_{Null}|_{\theta=0}$ (Fig.
(4(c))).

As we discussed above, in slowly rotating Kerr spacetime, $\Omega_{Kerr}$
blows up near a finite radius $r=2M$. However, $\Omega_{JMN1}$ and
$\Omega_{null}$ blow up only at the central singularity. This distinguishable
feature of the Lense-Thirring precession in the naked singularity spacetimes
can be very useful observationally in order to distinguish a black hole from a
naked singularity.

(a) $M_{0}=0.3$, $R_{b}=6.66$,$E=-0.18$ and $a=0.01$.

(b) $M_{0}=0.4$, $R_{b}=5$,$E=-0.21$ and $a=0.01$.

(c) $M_{0}=0.3$, $R_{b}=6.66$,$E=-0.18$ and $a=0.1$.

(d) $M_{0}=0.4$, $R_{b}=5$,$E=-0.21$ and $a=0.1$.

Figure 5: Relativistic orbits of a test particle in slowly rotating JMN1
spacetime for $L=1$, $M=1$. Where, black circle denotes for the minimum
approach of a particle and the red circle represents the boundary of the JMN1
spacetime $R_{b}$.

(a) $E=-0.013$ and $a=0.01$.

(b) $E=-0.013$ and $a=0.1$.

Figure 6: Relativistic orbits of a test particle in slowly rotating null naked
singularity spacetime for $L=5$, $M=1$. Where, black circle denotes for
minimum approach of a particle.

##  IV Timelike geodesics in slowly rotating spacetimes

In this section, we analyse the timelike geodesics of the bound orbits in the
slowly rotating black hole and naked singularity spacetimes. First, we review
the general procedure for deriving an orbit equation. As it is mentioned in
the previous section, the general form of the stationary, axially symmetric,
and slowly rotating spacetime can be written as,

$\displaystyle ds^{2}=-g_{tt}(r)dt^{2}$ $\displaystyle+$ $\displaystyle
g_{rr}(r)dr^{2}+g_{\theta\theta}(r)d\theta^{2}$ (38) $\displaystyle+$
$\displaystyle g_{\phi\phi}d\phi^{2}-2g_{t\phi}(r)dtd\phi\,\,.$

The above general spacetime is independent of $t$ and $\phi$, therefore, the
energy $e$ and the angular momentum $L$ (about the symmetry axis) per unit
rest mass are conserved. Using these symmetry properties and the four-velocity
normalization condition for the timelike geodesics
($u^{\alpha}u_{\alpha}=-1$), we can derive the general expression of the total
relativistic energy as,

$\displaystyle E=\frac{1}{2}(e^{2}-1)=\frac{1}{2}(u^{r})^{2}+V_{eff}(r)\,\,,$
(39)

where $u^{r}$ is radial component of the four velocity $u^{\mu}$, and the
effective potential can be described as,

$\displaystyle
V_{eff}(r)=\frac{1}{2g_{rr}}+\frac{e^{2}-1}{2}+\frac{L^{2}g_{tt}+2eLg_{t\phi}-e^{2}g_{\phi\phi}}{2g_{rr}(g_{t\phi}^{2}+g_{tt}g_{\phi\phi})}$
(40)

Note that, for simplicity, we have considered the equatorial plane
($\theta=\frac{\pi}{2}$) for the analysis of the timelike bound orbits. Using
the components of the metric tensor $g_{\mu\nu}$ given in Eqs. (18) and (II),
respectively the effective potentials for the JMN1 and null naked singularity
spacetimes can be written by as,

$\displaystyle
V_{eff}|_{JMN1}=\frac{1}{2}\left[e^{2}-M_{0}+\frac{(-1+M_{0})\left(P(e,a,r,L)\right)\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{-1+M_{0}}}}{-(-1+M_{0})r^{2}+\left(a+a(-1+M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(2-2M_{0})}}\right)^{2}}\right]\,\,,$
(41)

where
$P(e,a,r,L)=e^{2}r^{2}+2aeL\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{(2-2M_{0})}}+L(2ae+L)(-1+M_{0})\left(\frac{r}{R_{b}}\right)^{\frac{M_{0}}{1-M_{0}}}$
and

$\displaystyle
V_{eff}|_{null}=\frac{1}{2}\left[-1+\frac{r^{2}}{(M+r)^{2}}+\frac{(Lr^{2}+2aeM(M+2r))^{2}}{r^{4}(M+r)^{2}+4a^{2}M^{2}(M+2r)^{2}}\right]\,\,.$
(42)

For the timelike bound circular orbits, the following condition is always
satisfied,

$\displaystyle\frac{dV_{eff}}{dr}|_{r_{b}}=0;\frac{d^{2}V_{eff}}{dr^{2}}|_{r_{b}}>0\,\,,$
(43)

where at $r=r_{b}$ the effective potential has a minimum. Therefore, for the
timelike bound orbits, the total energy ($E$) of the particle is greater than
or equal to the minimum of the effective potential $V_{min}\leq E$. By using
the bound timelike orbit conditions, we can determine the shape and nature of
the orbits from the following differential equation of orbit,

$\displaystyle\bigg{(}\frac{dr}{d\phi}\bigg{)}^{2}=-\frac{(g_{t\phi})^{2}+g_{tt}g_{\phi\phi}}{g_{rr}(eg_{t\phi}+Lg_{tt})^{2}}[(g_{t\phi})^{2}-e^{2}g_{\phi\phi}+2eLg_{t\phi}+g_{tt}(L^{2}+g_{\phi\phi})].$
(44)

Now, we can derive the form of an orbit equation by differentiating the above
equation with respect to $\phi$ Bam2020 ; Bambhaniya:2021ybs . One can get the
orbit equations for the slowly rotating JMN1 and null naked singularity
spacetimes given in Eqs. (18) and (II). Since these are non-linear second-
order differential equations, it is very difficult to solve analytically.
Therefore, we solve it numerically and we depict the timelike bound orbits in
Fig. (5) and Fig. (6).

In Fig. (5), we show the relativistic orbits of a test particle in slowly
rotating JMN1 spacetime, where we consider $M_{0}=0.3$ and $M_{0}=0.4$ along
with the different parameters’ values (i.e., $a,E,R_{b}$). For all the cases
the Schwarzschild mass $M=1$. One can see from Figs. (5(a)) and (5(c)), the
nature and shape of the orbits are different compare to the orbits shown in
Figs. (5(b)) and (5(d)). We find the timelike bound orbits of a particle in
slowly rotating JMN1 spacetime can have negative as well as positive
precession. As we discussed previously, for positive precession, orbit
precesses in the direction of the particle motion, whereas for the negative
precession, it precesses in the opposite direction of the particle motion.
Figs. (5(a), 5(c)) and Figs. (5(b), 5(d)) respectively show the negative and
positive precession of bound timelike orbits in slowly rotating JMN1
spacetime. In Bam2020 , it is shown that in Kerr spacetime, the negative
precession of timelike bound orbits is not allowed. Therefore, the nature of
the precession of the timelike bound orbits may be useful for distinguishing a
rotating black hole from a rotating naked singularity. Moreover, we find that
the direction of the orbital shift or precession depends upon the values of
$M_{0}$ and spin parameter $a$. In Bambhaniya:2019pbr , we derive the
analytical expression of the orbital shift by using first-order eccentricity
approximation for the spherically symmetric and static JMN1 spacetime. In that
paper, we show that the negative precession occurs for $M_{0}<1/3$, while for
$M_{0}>1/3$ we get positive precession. Now, if we introduce rotation by
applying the Janis-Newman procedure in the static JMN1 spacetime, the range of
$M_{0}$ for which we get negative or positive precession, would depend upon
the value of the spin parameter $a$. In Fig. (6), we show the relativistic
orbits of a test particle in slowly rotating null naked singularity spacetime
for $a=0.01$ (Fig. (6(a))) and $a=0.1$ (Fig. (6(b))), where we consider $M=1$,
$L=5$, $E=-0.013$. From Figs. (6(a)) and (6(b)), one can see that the positive
precession occurs for the small values of the spin parameter.

##  V Discussion and Conclusion

Followings are the important results what we get in this paper,

  * •

In this paper, we first use the Newman-Janis algorithm to get the rotating
counter-parts of JMN1 and null naked singularity spacetimes. We show that the
algorithm is not suitable to get the rotating version of a general spherically
symmetric spacetime having any arbitrary value of the spin parameter. One has
to use slow rotation approximation to get rid of this problem. Therefore, in
this paper, we derive the metric of slowly rotating JMN1 and null naked
singularity spacetime using Newman- Janis algorithm.

  * •

We investigate the nature of the spin precession of a stationary test
gyroscope in the slowly rotating JMN1 and null naked singularity spacetime. We
show that in both the naked singularity spacetimes, the Lense-Thirring
precession frequency ($\Omega_{JMN1},\Omega_{Null}$) monotonically increases
as the radial distance from the central singularity decreases and it becomes
infinity at the center. However, unlike slowly rotating naked singularity
spacetimes, in Kerr spacetime, the spin precession frequency ($\Omega_{Kerr}$)
blows up near the finite radial distance $r=2M$.

  * •

In the Sec. (IV), we investigate the precession of the time like bound orbits
in slowly rotating JMN1 and null naked singularity spacetimes. The novel
feature of the timelike bound orbits in the rotating JMN1 spacetime is that it
can precesses in the opposite direction of the particle motion (i.e., negative
precession), which is not allowed in Kerr spacetime. On the other hand, in
slowly rotating null naked singularity spacetime, only the positive precession
is allowed.

The Lense-Thirring precession and the orbital precession both are important
physical phenomenon in the strong gravity region. Therefore, this two
phenomena can be used to understand the spacetime geometry near an ultra high-
density compact object. Hence, the distinguishable behaviour of the two types
of precession in naked singularity spacetimes may be useful to differentiate a
black hole from a naked singularity observationally.

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