We analyze the generalized spin precession of a test gyroscope
around a stationary spacetime i.e. for Kerr-MOG black hole (BH) in scalar-tensor-vector
gravity or modified gravity (MOG). A detailed study of generalized spin frequency has been
done for non extremal Kerr-MOG BH, extremal Kerr-MOG BH and naked singularity (NS)
in comparison to non-extremal BH, extremal BH and NS  of Kerr spacetime. The generalized spin frequency
that we have computed could be expressed in terms of the BH mass parameter, the angular momentum
parameter and the MOG parameter. Moreover,  we differentiate the non extremal BH,
extremal BH and NS via computation of the said precession frequency. The Lense-Thirring (LT) frequency
can obtain from generalized spin frequency by taking the limit as $\Omega=0$ i. e. when the angular
frequency is set to zero limit. Furthermore, we compute the LT frequency for various values of
angular coordinates i.e.  starting from polar to the equatorial plane. We show that the LT frequency
diverges at the horizon for extremal BH. Finally, we study the accretion disk physics by computing
three epicyclic frequencies namely the Keplerian frequency, the radial epicyclic frequency and
the vertical epicyclic frequency. We also compute the periastron frequency and nodal frequency.
With the aid of these frequency profiles, one can distinguish three compact objects i. e.
non-extremal BH, extremal BH and NS.

§ INTRODUCTION:

The de-Sitter precession [1] and the Lense-Thirring (LT) precession [2] are two
extraordinary effects  predicted by Einstein's general theory of
relativity [The possible new experimental test of
Einstein's general theory of relativity was first proposed by Schiff in 1960 [3].].
They are directly measured by Gravity Probe-B experiment:  a space experiment to test the
general theory of relativity (GTR) [5]. It was initiated and launched by NASA in 2004.
The satellite consists of four gyroscopes and a telescope orbiting 642 km or around 400 mile
above the Earth [4]. Another important satellite i. e. Rossi X-ray
Timing Explorer (RXTE) [6] was launched by NASA in 1995 which help
us to confirm the existence of the frame-dragging effect which was
predicted by Einstein's gravity. It was also detected the X-rays from different
compact objects like BHs, neutron stars and X-ray pulsars etc. The effects of
frame dragging effect on the galactic-center stars and kinematic properties
was investigated in [8]. The measurement of the gyroscopic precession
and its implications was discussed in [9].

There are several important tests [10, 11, 12] of GTR
which was carried out by different space mission experiment.
For example, (a) the weak equivalence principle: the basis of geometrical theories of
gravity  which was tested by Lunar laser ranging of accuracy in the order of $10^{-13}$.
(b) The strong equivalence principle: the cornerstone of the GTR which was  tested
by the lunar laser ranging of accuracy  $<10^{3}$. The gravitational time dilation or
gravitational red-shift which was experimentally verified by Gravity Probe-A.
The Shapiro time-delay which was also experimentally tested by very long baseline
interferometry (VLBI) [10].

Moreover, the perihelion advance of mercury which was  tested by the mercury
radar ranging method and the accuracy of the order of $10^{-3}$. The periastron advance,
time dilation, time delay and the rate of change of orbital period which was verified
by binary pulsar PSR 1913+16. The geodetic precession or de-Sitter effect which was
tested by the lunar laser ranging of accuracy in the order of $6\times 10^{-3}$ and
using Gravity Probe-B. Finally, the LT effect or the frame-dragging effect was
experimentally confirmed by using laser geodynamics satellite (LAGEOS) [13]
[It was launched by NASA in 1976.]  and
LAGEOS2 [This satellite was launched in 1992 by Italian space agency and NASA.]
of accuracy in the order of $10^{-1}$.

The de-Sitter precession or geodetic effect is the dragging of a gyroscope due to its motion in a static
gravitational field. Whereas the LT precession or frame-dragging effect of a gyroscope is due
to the rotation of a massive body. In the weak field limit, such LT precession frequency is defined
as [14]

\begin{eqnarray}
\vec {\Omega}_{LT} &=& \frac{G}{c^2r^3} \left[3 \left(\vec{J}.\hat{r}\right) \hat{r}-\vec{J}\right]
\end{eqnarray}

where, $\hat{r}$ is the unit vector along  agular momentum $\vec{J}$ direction,
$G$ is Newton's gravitational constant and $c$ is the speed of light.  Earlier derivations
of the LT frequency [14] in the literature was considered on the weak field approximation
by assuming $r>M$ ($M$ is Arnowitt-Deser-Misner (ADM) mass of any compact object) i. e.
large distances for the test gyroscope. Now the question should be arise naturally what would
be the LT precession frequency in the strong gravity regime? This is  one of the prime
motivation behind this work.

In the strong gravity limit, the LT frequency for Kerr BH and Kerr-Taub-NUT (Newman-Unti-Tamburino)
BH was explicitly derived in  [17]. In this work, the authors showed the role of NUT parameter in
inertial frame dragging effect. It was shown there that  the LT frequency does not vanish for Taub-NUT
spacetime when the value of angular momentum vanishes. Similarly for axisymmetric NUT spacetime, the
result becomes more prominent when the value of ADM mass parameter does vanish. The LT frequency
for more generalized class of spacetimes like Plebański and Demiański in the strong field
limit was explicitly discussed in [18].  The authors in [19] studied
the inertial frame dragging effect of a rotating traversable wormhole.  In his work, he
also described the behavior of a test gyroscope when it moves towards a spinning traversable wormhole.
Moreover, they  derived the LT frequency for this wormhole and showed that the LT frequency diverges
on the ergosphere. Along the pole, the LT frequency is inversely proportional to the spin parameter
of the wormhole.

In [20], the authors derived the LT frequency  inside a rotating neutron star. Where the authors
showed that the LT frequency rate  along the pole decreases from the center to the surface of the
neutron star. Along the equatorial plane the LT frequency rate decreases initially away from the center
and approaches a small value in the surface.  Morsink and Stella [22] first showed that the precession
frequency of a rotating neutron star could be expressed in terms of orbital frequency which was
observed at infinity. They also computed the precession frequencies of circular orbits around
rapidly spinning neutron stars for a variety of masses and the equation of state.
One should consult a good review on inertial frame-dragging effect which would
be found in Ref. [12]. Also one could found the history of the
Lense-Thirring effect in Ref. [15]. Gyroscope precession along
equatorial plane of stable orbits for Kerr BH could be found in [16].

However, the LT precession frequency of various axisymmetric BHs including the analogue
spacetime and neutron star [17, 18, 19, 20, 24] are computed so far
but till date the LT frequency and the generalized spin frequency are not
considered for Kerr-MOG(KMOG) BH. In the present manuscript, we wish to derive the
genralized spin frequency of KMOG BH. It is a kind of BH solution in the
scalar-tensor-vector gravity (STVG) or simply it is known as MOG.
This theory was first proposed by Moffat [25, 27, 28, 29, 30, 31].
The fundamental features of this theory as follows. It correctly explains
various kind of astronomical observations: dynamics of galaxies and cluster
of galaxies, bullet clusters, galaxy rotation curves, the amount of luminous
matter, the exotic dark matter and the acceleration of the universe etc.

The MOG gravity is a type  of  alternative theory of gravity. It contains scalar
field and massive vector field. The action in STVG theory consists of scalar action
and  vector action hence it modifies the Einstein-Hilbert action. The MOG theory is
formulated via MOND phenomenology in the weak field approximation. Most importantly,
this theory correctly explains the observations of the solar system [26].
It could also be used to describe the growth of the structure of the universe,
the power spectrum of the matter and the acoustical power spectrum of the cosmic
microwave background (CMB) data. It should be noted that various features of MOG
like superradiance, quasinormal modes, thermodynamics, geodesics properties, orbital
and vertical epicyclic frequencies, Penrose process, gravitational bending of light,
BH shadow  and BH merger estimates etc. are studied
in [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] respectively.

The fundamental postulate in MOG theory is that the BH charge
parameter is proportional to the Komar mass i. e.
${Q}=\sqrt{\alpha G_{N}}M$ [30]. Where  $\alpha=\frac{G-G_{N}}{G_{N}}$
should be measured deviation of MOG from GR. Using this criterion
the MOG action is given by

\begin{eqnarray}
{\cal I}_{MOG} &=& \frac{1}{16\pi G}\int \sqrt{-g}\, R\, d^4x-
\frac{1}{16\pi}\int \sqrt{-g}\, B_{ab}B^{ab}\, d^4x+{\cal I}_{M}
\end{eqnarray}

where $R$ is the Ricci scalar, $B_{ab}$ is the generalized field strength tensor of the vector
field $\phi^{a}$, $B_{ab}=\partial_{a}\phi_{b}-\partial_{b}\phi_{a}$ and ${\cal I}_{M}$ is the
matter action.

The MOG field equations for vanishing matter electro-magnetic tensor ${\cal T}_{Mab}=0$ are

\begin{eqnarray}
R_{ab} &=& -8\pi G\, {\cal T}_{\phi ab}  ~. \label{mgl}
\end{eqnarray}

\begin{eqnarray}
{\cal T}_{\phi ab} &=& -\frac{1}{4\pi} \left(B_{a}^{c} B_{bc}-
\frac{1}{4} g_{ab} B^{ef}B_{ef} \right)  ~. \label{mgl1}
\end{eqnarray}

Thus vacuum field equations are

\begin{eqnarray}
\frac{1}{\sqrt{-g}} \partial_{a}\left(\sqrt{-g}B^{ab}\right) &=& 0~, \label{mgl2}
\end{eqnarray}

\begin{eqnarray}
\nabla_{a}B_{bc}+\nabla_{b}B_{ca}+\nabla_{c}B_{ab} &=& 0~. \label{mgl3}
\end{eqnarray}

where $\nabla_{a}$ denotes covariant derivative with respect to the metric tensor $g_{ab}$. Using these
criterion one obtains static spherical symmetric metric in MOG

\begin{eqnarray}
\frac{dr^2}{\left[1-\frac{2G_{N}(1+\alpha)M}{r}+\frac{G_{N}^2M^2\alpha(1+\alpha)}{r^2}\right]}
- r^{2}d\Omega^2 ~. \label{mgl4}
\end{eqnarray}

where $G_{N}$ is modified Newtonian constant which is related to the Newton's constan by
$G=G_{N} (1+\alpha)$ and modified charge parameter is ${\cal Q}=\sqrt{\alpha G_{N}}M$,
where $\alpha$ is a MOG parameter.  The above metric can be obtained by substituting
these values in usual Reissner-Nordström BH solution.  The metric of
the rotating BH solution in MOG depends on the spin parameter  $a=J/M$, where $J$ is the Komar angular
momentum of the asymptotically flat, axisymmetric, stationary spacetime. By applying Boyer-Lindquist
coordinates one obtains the metric form which is written in Eq. (<ref>) and it is quite similar
structure of Kerr-Newman BH.

One of the main goals of this work is to explore  the difference between
BH and NS via computation of generalized spin frequency in MOG.
For NS, the motivation comes from the works
[23, 36, 37, 38, 39, 40, 41, 42]. How to differentiate
a BH from a NS, this is a  prime aim of the present work. What is a NS? A NS
is a type of gravitational singularity  without an event horizon. Whereas a BH is a type of
gravitational compact object having an event horizon. Although the thermodynamic properties of
BH is an established subject but the thermodynamics of NS is completely unknown [40].
Recently, the image of a BH has been observed in EHT telescope [7] while for NS
there is no evidence could be seen till date.

However in the present work,  we  would like to provide a detailed analysis of inertial
frame dragging effect for non-extremal situation, extremal situation and NS in
KMOG BH. Earlier mentioned that the KMOG BH is a new class of spinning BH
proposed by Moffat [25] and it is constructed by the ADM mass parameter (${\cal M}$),
spin parameter ($a$) and a deformation parameter or MOG parameter ($\alpha$) in comparison
to Kerr BH which is defined only by the ADM mass parameter and the spin parameter.

First, we derive the generalized spin precession of a test gyroscope around the KMOG
spacetime. Using this frequency, we differentiate the behavior of three compact objects:
non-extremal BH, extremal BH amd NS. We point out a clear distinction between these three
compact objects graphically.  Moreover, this result is compared with Kerr BH. Using
generalized spin frequency versus radial diagram, one can distinguish three compact
objects for various spin limits. Also, we investigate the generalized spin frequency
for various angular coordinte values i. e. $\theta=0$, $\theta=\frac{\pi}{6}$,
$\theta=\frac{\pi}{4}$, $\theta=\frac{\pi}{3}$ and  $\theta=\frac{\pi}{2}$.
Furthermore, we examine  the generalized spin frequency for ring singularity.
Lastly, we compare these results with the result of vanishing angular angular
velocity $\Omega=0$. This is exactly the LT frequency when the other frequencies are excluded.
From the LT frequency vs. radial diagram, we show  that  the LT frequency is
influenced by the MOG parameter . When  $\Omega=0$, we also analyze the LT
frequency for angular values i.e.  $\theta=0$, $\theta=\frac{\pi}{6}$, $\theta=\frac{\pi}{4}$,
$\theta=\frac{\pi}{3}$ and $\theta=\frac{\pi}{2}$. Each diagram clearly exhibits
the key difference between three compact objects.

Moreover, we compute three fundamental frequencies i.e.  the Keplerian frequency ($\Omega_{\phi}$),
the radial epicyclic frequency ($\Omega_{r}$) and the vertical epicyclic frequency ($\Omega_{\theta}$).
Using these frequencies we can find the difference between three compact objects. In  [49],
it was mentioned that the epicyclic frequencies are the main  ingredients for the geodesic models of
quasi-periodic-oscillations (QPO). Again these QPOs help us to testify the strong gravity in a novel
way. The geodesic models were described by relativistic precession model (RPM) [50] and
epicyclic resonance model (ERM) [51]. These models indicates that there exist
low frequency (LF) QPO and twin high frequency (HF) QPO. From RPM, we can find that the
upper and lower HF QPOs meets with the azimuthal frequency, $\Omega_{per}=\Omega_{\phi}-\Omega_{r}$.
While the LF QPOs are computed by the nodal precession  frequency, $\Omega_{nod}=\Omega_{\phi}-\Omega_{\theta}$.
These three QPO frequencies $(\Omega_{\phi}, \Omega_{per}, \Omega_{nod})$ could generate
at the same orbital radius.  Moreover these frequencies could serve as a tool in our investigation
to study the crucial differences between three compact objects.

Furthermore, our technique suggests that a comparative study of stationary, axisymmetric
KMOG spacetime for various values of spin parameter. The main findings of the present work
is to highlight crucial differences between non-extremal KMOG BH, extremal KMOG BH
and NS by using generalized spin precession frquency ($\Omega_{p}$) and LT precession
frequency ($\Omega_{LT}$) of a test gyro. Also other frequencies like  radial epicyclic
frequency ($\Omega_{r}$), vertical epicyclic frequency ($\Omega_{\theta}$),
Keplerian frequency ($\Omega_{\phi}$), periastron precession frequency ($\Omega_{per}$),
nodal precession frequency ($\Omega_{nod}$), the ratio  $\frac{\Omega_{r}}{\Omega_{\phi}}$,
the ratio $\frac{\Omega_{r}}{\Omega_{\theta}}$ and the ratio $\frac{\Omega_{\theta}}{\Omega_{\phi}}$
are also help us to support this result in strong gravity regime.

The paper is organized as follows: in  sec. 2,
we derive the generalized spin frequency in the background of KMOG spacetime for non-extremal,
extremal and NS cases. A detailed study has been done and visualize the results for all cases.
Using the spin frequency expression, we differentiate the non extremal BH,
extremal BH and NS. Moreover in Sec. 3,  we compute the LT frequency  by using the result of
generalized spin frequency where we have  taken the value of angular frequency $\Omega=0$. Also
in the subsequent sub Sec., we have specialized
the result for extremal spacetime. Furthermore, we study the accretion
disk physics in Sec. 4,  by deriving the epicyclic frequencies.
We also derive the periastron frequency and nodal
frequency. By using the frequency profile, we can distinguish three spacetimes namely the non-extremal
spacetime, extremal spacetime and NS. In Sec. 5, we have discussed the result.

§ GENERALIZED SPIN PRECESSION OF A TEST GYROSCOPE AROUND A STATIONARY SPACETIME

In this section we shall provide the basic formalism (following the Ref. [43, 18]) of spin
precession of a test gyroscope which is attached to a stationary observer and moves along a Killing
path in a stationary spacetime with a timelike Killing vector field $K$. Then the spin of such a
test gyroscope undergoes a Fermi-Walker transport along

\begin{equation}
\end{equation}

where $K$ is the timelike Killing vector field. For a stationary observers, the four-velocity
can be defined as $u^{\alpha}=(u^{t}, 0, 0, \Omega\,u^{t})$. Where $t$ is time coordinate and
$\Omega$ is the angular velocity of the observer.
It is well-known that in the special case the frequency of the gyroscope may coincide with the
vorticity field associated with the Killing congruence. This indicates that the said gyroscope is
rotating with respect to a corotating frame along with an angular velocity. This effect is said to
be a gravitomagnetic precession in gravitational physics because the vorticity vector behaves as the
magnetic field in the $3+1$ dimension spacetime [46]. Since our moto is to derive the spin
precession frequency of a test gyroscope in a strong gravity regime hence the general spin precession
frequency of a test gyro $\Omega_{s}$ which is actually the rescaled vorticity field of the stationary
observer can be derived as

\begin{eqnarray}
\tilde \Omega_{s} &=& \frac{1}{2K^2}*(\tilde K \wedge d\tilde K)\label{wf1}
\\
\mbox{or} &&  \nonumber
\\
\left(\Omega_{s}\right)_{a} &=&\frac{1}{2K^2} \eta_{a}^{~bcd} K_{b}\partial_{c} K_{d}~, ~\label{wf2}
\end{eqnarray}

where $\eta^{abcd}$ denotes the component of the volume-form in the spacetime,
$*$ denotes the Hodge dual,
and $\tilde K$ & $\tilde \Omega_{s}$ represent the one-form of $K$ & $\Omega_{s}$, respectively. The
parameter $\tilde \Omega_{s}$ will be vanish if and only if $(\tilde K \wedge d\tilde K)$ does vanish.
This will be happen only in case of a static spacetime.

It has already been derived in [20, 21] the LT  precession frequency of a test
gyro due to the rotation of any stationary and axisymmetric spacetime

\begin{equation}
\vec{\Omega}_p|_{\Omega=0}=\frac{1}{2\sqrt {-g}} \times
\left[-\sqrt{g_{rr}}\left(g_{0\phi,\theta}
-\frac{g_{0\phi}}{g_{00}} g_{00,\theta}\right)\hat{r}
\end{equation}

This result is valid for outside the ergoregion and $\Omega=0$. For arbitrary value
of $\Omega$ the formalism is derived in Ref. [23].
The timelike Killing vector field in generalized spacetime can be written as

\begin{eqnarray}
K &=& \partial_{0}+\Omega\, \partial_{s}~\label{K1}
\end{eqnarray}

where $\partial_{s}$ is spacelike Killing vector of the said stationary spacetime
and $\Omega$ is  angular velocity of an observer moving along integral curves of $K$.
The metric of above stationary  spacetime is independent of $x^{0}$ and $x^{s}$ coordinates.
Thus the corresponding co-vector of $K$ is

\begin{eqnarray}
\tilde{K} &=& g_{0\alpha}~dx^{\alpha}+\Omega~g_{\beta s}\,dx^{\beta},
\end{eqnarray}

where $\beta, \alpha=0,s,2,3$ in four dimension spacetime. Now one can
write $\tilde{K}$ by separating into space and time components as

\begin{eqnarray}
\tilde{K}=(g_{00}\,dx^0+g_{0s}\,dx^s+g_{0i}\,dx^i)+\Omega\,\left(g_{0s}\,dx^0+g_{ss}\,dx^s+g_{is}\,dx^i\right)~\label{kt}
\end{eqnarray}

where $i=2, 3$. Since we are interested in ergoregion of the said stationary, axisymmetric spacetime
thus we are neglecting the terms $g_{0i}$ and $g_{is}$ then one obtains

\begin{eqnarray}
\tilde{K}=(g_{00}\,dx^0+g_{0s}\,dx^s)+\Omega\,\left(g_{0s}\,dx^0+g_{ss}\,dx^s\right)~\label{kt1}
\end{eqnarray}

\begin{eqnarray}
d\tilde{K}=\left(g_{00,k}\,dx^k \wedge dx^{0}+g_{0s,k}\,dx^k \wedge dx^s \right)+
\Omega (g_{0s,k}\,dx^k \wedge dx^0+g_{ss,k}\,dx^k \wedge dx^s).~\label{kto}
\end{eqnarray}

Now, Eq. (<ref>) can be re-writeen as

\begin{eqnarray}
\tilde{\Omega}_{p}=\frac{1}{2K^2}*(\tilde{K} \wedge d\tilde{K})~\label{be}
\end{eqnarray}

Substituting the values of $\tilde{K}$ and $d\tilde{K}$ in Eq. (<ref>),
one finds the one-form of the precession frequency

\tilde{\Omega}_{p} = \frac{\varepsilon_{skl}\,g_{l\mu}\,dx^{\mu}}{2\sqrt {-g}\left(1+2\Omega\,\frac{g_{0s}}{g_{00}}
+\Omega^2\,\frac{g_{ss}}{g_{00}}\right)} \times

\begin{eqnarray}
\left[\left(g_{0s,k}-\frac{g_{0s}}{g_{00}}\, g_{00,k}\right)+\Omega\,\left(g_{ss,k}-\frac{g_{ss}}{g_{00}}\,g_{00,k}\right)
+\Omega^2\,\left(\frac{g_{0s}}{g_{00}}\,g_{ss,k}-\frac{g_{ss}}{g_{00}}\, g_{0s,k}\right)\right] \nonumber
\end{eqnarray}

where we have used $*\left(dx^0 \wedge dx^k \wedge dx^s \right)=\eta^{0ksl}\,g_{l\mu}\,dx^{\mu}
=-\frac{1}{\sqrt{-g}}\varepsilon_{ksl}\,g_{l\mu}dx^{\mu}$ and $K^2=g_{00}+2\Omega\, g_{0s}+\Omega^2\, g_{ss}$.
The corresponding vector ($\Omega_p$) of the co-vector $\tilde{\Omega}_p$ is

\Omega_{p} = \frac{\varepsilon_{skl}}{2\sqrt {-g}\left(1+2\Omega\,\frac{g_{0s}}{g_{00}}
\times

\begin{eqnarray}
\left[\left(g_{0s,k}
-\frac{g_{0s}}{g_{00}} g_{00,k}\right)+\Omega\left(g_{ss,k}
-\frac{g_{ss}}{g_{00}} g_{00,k}\right)+ \Omega^2 \left(\frac{g_{0s}}{g_{00}}g_{ss,k}
-\frac{g_{ss}}{g_{00}} g_{0s,k}\right) \right] \nonumber\\
\end{eqnarray}

For a stationary and axisymmetric spacetime with coordinates $t,r,\theta,\phi$, the above
equation becomes

\begin{eqnarray}
\vec{\Omega}_{p} &=&  \frac{-\sqrt{g_{rr}}~{\cal X}(r) \hat{r}+\sqrt{g_{\theta\theta}}~{\cal Y}(r) \hat{\theta}}
{2 \sqrt{-g}~{\cal Z}(r)}~\label{n1}
\end{eqnarray}

\begin{eqnarray}
{\cal X}(r)  &=& {\cal A} +{\cal B} \Omega+{\cal C} \Omega^2\\
{\cal Y}(r)  &=& {\cal F}+{\cal G} \Omega+{\cal H} \Omega^2 \\
{\cal Z}(r)  &=& g_{tt}+2 g_{t\phi} \Omega+g_{\phi\phi}\Omega^2
\end{eqnarray}

\begin{eqnarray}
{\cal A} &=& g_{tt}~g_{t\phi,\theta}-g_{t\phi}~g_{tt,\theta}\\
{\cal B}  &=& g_{tt}~g_{\phi\phi,\theta}-g_{\phi\phi}~ g_{tt,\theta}\\
{\cal C} &=& g_{t\phi}~ g_{\phi\phi,\theta}-g_{\phi\phi}~g_{t\phi,\theta}\\
{\cal F} &=& g_{tt}~g_{t\phi,r}-g_{t\phi}~g_{tt,r}\\
{\cal G}   &=& g_{tt}~g_{\phi\phi,r}-g_{\phi\phi}~g_{tt,r}\\
{\cal H} &=& g_{t\phi}~g_{\phi\phi,r}-g_{\phi\phi}~g_{t\phi,r}
\end{eqnarray}

The most striking feature of  Eq. (<ref>) is that it could be applicable to derive the generalized
spin precession for any stationary axisymmetric BH spacetime which is valid for both outside and inside the
ergosphere. In the limit $\Omega=0$, one obtains

\begin{eqnarray}
\vec{\Omega}_{LT} &=&  \frac{-\sqrt{g_{rr}}~{\cal A} \hat{r}+\sqrt{g_{\theta\theta}}~{\cal F} \hat{\theta}}
{2 \sqrt{-g}~g_{tt}}~~\label{n2}
\end{eqnarray}

This formula is applicable only outside the ergoregion. This is the exact LT  precession frequency of a test
gyro due to rotation of any stationary and axisymmetric spacetime as mentioned in Eq. (<ref>). It must be
noted that $\vec{\Omega_{p}}$ is not the $\vec{\Omega}_{LT}$ of test gyro. Actually $\vec{\Omega_{p}}$
describes the overall frequency of test gyro since gyro has non-zero angular velocity while
$\vec{\Omega}_{LT}$ describes LT frequency of test gyro when the test gyro has zero angular velocity i.e.

§.§ Application to KMOG spacetime

In this subsection we will present a detailed analysis of the frame-dragging effect for
KMOG spacetime according to the above formalism. To do this we have to write the  metric
explicitly for KMOG BH as described in Ref. [25]

\begin{eqnarray}
ds^2 = -\frac{\Delta}{\rho^2} \, \left[dt-a\sin^2\theta d\phi \right]^2+\frac{\sin^2\theta}{\rho^2} \,
\left[(r^2+a^2) \,d\phi-a dt\right]^2
+\rho^2 \, \left[\frac{dr^2}{\Delta}+d\theta^2\right] \nonumber\\
\end{eqnarray}

\begin{eqnarray}
\rho^2 & \equiv & r^2+a^2\cos^2~\theta \nonumber\\
\Delta & \equiv & r^2-2G_{N}(1+\alpha)Mr+a^2 + G_{N}^2 \alpha(1+\alpha) M^2 ~.\label{m2.1}
\end{eqnarray}

where $G_{N}$ is Newton's gravitational constant and $M$ is  Komar mass.
We should mentioned that in the metric $c=1$. The above metric is an
axially-symmetric and stationary spacetime.  The ADM mass and angular momentum are computed
in [34]  as ${\cal M}=(1+\alpha)M$ and $J=a{\cal M}$  [One could determine
the relation between the Komar mass and ADM mass as $M=\frac{\cal M}{1+\alpha}$. If one could
consider either the Komar mass or the ADM mass in the LT frequency computation
then the physics will not be change. We consider here the ADM mass througout
the paper for our convenience.].
Substituting these values in Eq. (<ref>) then $\Delta$ becomes

\begin{eqnarray}
\Delta & = & r^2-2G_{N}{\cal M}r+a^2 + \frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2
\end{eqnarray}

The above metric describes a BH with horizon radii

\begin{eqnarray}
r_{\pm} &=& G_{N}{\cal M} \pm \sqrt{\frac{G_{N}^2{\cal M}^2}{1+\alpha}-a^2} ~. \label{mg2.2}
\end{eqnarray}

where $r_{+}$ is called as event horizon and $r_{-}$ is called as Cauchy horizon. Note that $r_{+}>r_{-}$.
The horizon structure could be seen visually in  Fig. <ref>.

The figure implies the variation  of $r_{\pm}$  with $a$ and $\alpha$ for Kerr BH
and Kerr-MOG BH. Left figure for Kerr BH. Right figure for Kerr-MOG BH.
The presence of the MOG parameter is deformed the shape of the horizon radii.

It should be noted that when $\alpha=0$, one gets the horizon radii of Kerr BH. The BH solution exists
when $\frac{G_{N}^2{\cal M}^2}{1+\alpha} > a^2$. When $\frac{G_{N}^2{\cal M}^2}{1+\alpha}=a^2$, one obtains
extremal BH. When $\frac{G_{N}^2{\cal M}^2}{1+\alpha} < a^2$, one gets the naked singularity case.

For the sake of convenience and to show the variation with deformation parameter
we can define spin parameter for MOG BH as

\begin{eqnarray}
a_{\alpha}=\frac{a}{G_{N}{\cal M}}=\frac{1}{\sqrt{1+\alpha}}
\end{eqnarray}

The new spin parameter $a_{\alpha}$ varies with MOG parameter that could be seen from
Fig. (<ref>). From the plot we can infer that there is a restriction on spin
parameter in MOG as

\begin{eqnarray}
|a_{\alpha}| &<& \frac{1}{\sqrt{1+\alpha}}~~~~~~{\mbox{Non-extremal BH}}\\
|a_{\alpha}| &=& \frac{1}{\sqrt{1+\alpha}}~~~~~~{\mbox{Extremal BH}}\\
|a_{\alpha}| &>& \frac{1}{\sqrt{1+\alpha}}~~~~~~{\mbox{Naked Singularity}}
\end{eqnarray}

Note that the MOG parameter or deformation parameter ($\alpha$) is always positive definite. If we
invert the above inequality then one gets the restriction on $\alpha$.

For a Kerr BH, when $a_{\alpha}=\frac{a}{G_{N}{\cal M}}<1$, there exists a BH solution. While
$a_{\alpha}=\frac{a}{G_{N}{\cal M}}>1$, it is said to be a NS
and $a_{\alpha}=\frac{a}{G_{N}{\cal M}}=1$ then it is said to be an extremal BH solution.
While for a KMOG BH, these limits can be reduced in the following way. For example, if we
take $\alpha=1$ then these limits are defined as $a_{\alpha}=\frac{1}{\sqrt{2}}=0.7$ which
is the extremal limit, $a_{\alpha}>\frac{1}{\sqrt{2}}$ is the
NS situation and $a_{\alpha}<\frac{1}{\sqrt{2}}$ is the BH solution. Similarly, if we take the MOG
parameter $\alpha=2$, then these limits are: $a_{\alpha}=\frac{1}{\sqrt{3}}$ which  is the extremal limit,
$a_{\alpha}>\frac{1}{\sqrt{3}}$ is the NS situation and $a_{\alpha}<\frac{1}{\sqrt{3}}$ is the BH

The figure indicates  the variation  of spin parameter $a_{\alpha}$ with $\alpha$. Where
$a_{\alpha}=\frac{a}{G_{N}{\cal M}}=\frac{1}{\sqrt{1+\alpha}}$. This means that the value of spin
parameter decreases when MOG parameter increases. Maximum value of MOG parameter is unity. Its value
gradually decreases when $\alpha$ increases.

The outer and inner ergosphere are occur at

\begin{eqnarray}
r &=& r_{e}^{\pm}(\theta) = G_{N}{\cal M} \pm \sqrt{\frac{G_{N}^2{\cal M}^2}{1+\alpha}-a^2 \cos^2~\theta}
~. \label{mg2.3}
\end{eqnarray}

and they satisfied the following inequality $r_{e}^{-}(\theta)\leq r_{-}\leq r_{+}\leq r_{e}^{+}(\theta)$.
The structure of the outer and inner ergosphere could be seen visually  from Fig. <ref>.

The figure indicates the variation  of $r_{e}^{\pm}(\theta)$ with $a$ and $\alpha$
for Kerr BH and Kerr-MOG BH. In each plot upper half
corresponds to $r_{e}^{+}(\theta)$ and lower half corresponds to $r_{e}^{-}(\theta)$.

In the extremal limit, the outer horizon and inner horizon are coincident at $r_{+}=r_{-}= G_{N}{\cal M}$.
The  outer and inner ergosphere radius reduces to

\begin{eqnarray}
r_{e}^{\pm}(\theta) &=& G_{N}{\cal M} \left(1 \pm \frac{\sin\theta}{\sqrt{1+\alpha}}\right)
~. \label{mg2.4}
\end{eqnarray}

\begin{eqnarray}
r_{e}^{\pm}(\theta)|_{\theta=0} &=& G_{N}{\cal M}=r_{\pm}, \,\,\, \mbox{(on axis)}\\
r_{e}^{\pm}(\theta)|_{\theta=\frac{\pi}{2}} &=& G_{N}{\cal M} \left(1 \pm \frac{1}{\sqrt{1+\alpha}}\right)
=r_{\pm}|_{a=0}\,\,\,\mbox{(equatorial plane)}~. \label{mg2.5}
\end{eqnarray}

The structure of the equatorial ergosphere could be seen visually from Fig. <ref>.

In the limit $\alpha=0$,

\begin{eqnarray}
r_{e}^{+}(\theta)|_{\theta=\frac{\pi}{2}} &=& 2G_{N}{\cal M}, \,\,\,\, r_{e}^{-}(\theta)|_{\theta=\frac{\pi}{2}}=0
~. \label{mg2.6}
\end{eqnarray}

The figure implies the variation  of $r_{e}^{\pm}(\theta)$ with  $\alpha$
for extremal Kerr-MOG BH in equatorial plane.

This surface is outer to the event horizon or outer horizon and it coincides with the outer horizon at
the poles $\theta=0$ and $\theta=\pi$. The metric components of above BH  in Boyer-Lindquist
coordinate are

\begin{eqnarray}
g_{tt} &=& - \left(\frac{r^2-\Pi_{\alpha}+a^2 \cos^2\theta}{\rho^2}\right)\\
g_{t\phi} &=& -\left(\frac{a \, \sin^2\theta\,\Pi_{\alpha}}{\rho^2}\right)\\
g_{rr} &=& \left(\frac{\rho^2}{\Delta}\right)\\
g_{\theta\theta} &=& \rho^2 \\
g_{\phi\phi} &=& \left(r^2+a^2+\frac{a \Pi_{\alpha} \sin^2\theta}{\rho^2}\right) \sin^2\theta
\end{eqnarray}

\begin{equation}
\sqrt{-g}=\rho^2 \sin\theta.
\end{equation}

\begin{eqnarray}
\Pi_{\alpha} &=& \left(2G_{N}{\cal M}r-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2\right)
\end{eqnarray}

Putting these metric components in  Eq. (<ref>), one obtains the generalized spin precession frequency
of a test gyro for KMOG BH

\begin{eqnarray}
\vec{\Omega}_{p} &=& \frac{\xi(r)~\sqrt{\Delta}\cos\theta~ \hat{r}+\eta(r)~\sin\theta~\hat{\theta}}{\zeta(r)},~\label{gekm}
\end{eqnarray}

\begin{equation}
\begin{split}
\xi (r) & = a \Pi_{\alpha}-\frac{\Omega}{8} \left[8r^4+8a^2r^2+8 a^2\Pi_{\alpha}+3a^4+
4a^2\left(2\Delta-a^2\right)\cos2\theta+a^4\cos4\theta \right]\\
& +\Omega^2a^3\Pi_{\alpha}\sin^4\theta
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\eta(r) &= aG_{N}{\cal M}\left(r^2-a^2 \cos^2\theta\right)-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 ar+\\
& \Omega\,\left(r^5-3G_{N}{\cal M}r^4+2a^2r^3\cos^2\theta-2G_{N}{\cal M}a^2r^2+a^4 \cos^4\theta r\right)\\
& +\Omega\,\left[G_{N}{\cal M}a^4\cos^2\theta\left(1+\sin^2\theta\right)+2\frac{\alpha}{1+\alpha}G_{N}^2 {\cal M}^2
r \left(r^2+a^2\right) \right]\\
& + \Omega^2 G_{N}{\cal M} a\sin^2\theta \left[3r^4+a^2r^2+a^2\cos^2\theta\left(r^2-a^2\right)\right]\\
& - \frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 a\sin^2\theta \Omega^2 r \left[2r^2+a^2 \left(1+\cos^2\theta\right)\right]
\end{split}
\end{equation}

\begin{eqnarray}
\zeta(r) &=& \rho^3
\left[\left(\rho^2-\Pi_{\alpha}\right)
+2a\Omega \Pi_{\alpha} \sin^2\theta-\Omega^2\sin^2\theta
\left\{\rho^2\left(r^2+a^2\right)+a^2 \Pi_{\alpha} \sin^2\theta\right\}\right] \nonumber\\
\end{eqnarray}

This is the expression of generalized spin frequency  which is valid both for outside and inside the
ergosphere. Now we could determine the range of the angular velocity and therefore  the four velocity
must be time-like i.e. $K^2 = g_{\phi\phi}\Omega^2+2g_{t\phi}\Omega+g_{tt}<0$. The allowed
values of $\Omega$ at any fixed $(r,\theta)$ are
$\Omega_-(r,\theta) < \Omega(r,\theta) < \Omega_+(r,\theta)$ where

\begin{eqnarray}
\Omega_{\pm} =\frac{-g_{t\phi}\pm \sqrt{g_{t\phi}^2-g_{\phi\phi}g_{tt}}}{g_{\phi\phi}}. \label{okn}
\end{eqnarray}

For KMOG BH, it should be

\begin{eqnarray}
\Omega_{\pm}&=& \frac{a \Pi_{\alpha}\sin\theta \pm \rho^2\sqrt{\Delta}}
\end{eqnarray}

Now we shall calculate the values of $\Omega$ when an observer closes to the horizon. At the event horizon the
angular velocity becomes $\Omega_{+} =\frac{a}{2G_{N}{\cal M}r_{+}-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2}$
and at the Cauchy horizon the angular velocity becomes
$\Omega_{-} =\frac{a}{2G_{N}{\cal M}r_{-}-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 }$. In the equatorial plane

\begin{eqnarray}
\Omega_{\pm}|_{\theta=\pi/2}=\frac{a\Pi_{\alpha}\pm
r^2\sqrt{\Delta}}{r^2(r^2+a^2)+a^2 \Pi_{\alpha}}. \label{oken}
\end{eqnarray}

Finally at the ring singularity $r=0$ and $\theta=\frac{\pi}{2}$

\begin{eqnarray}
\Omega_{\pm}|_{r=0, \theta=\pi/2} =1.~\label{omkn1}
\end{eqnarray}

This means that two angular frequencies coincide at the ring singularity. Variation of angular frequency ($\Omega$)
of a stationary  gyroscope could be observed from the whale diagram[Figure (<ref>)]. The significance of
whale diagram is that the whale or a fish can move along the $\Omega_{+}$ and $\Omega_{-}$ and it is prominent
when the MOG parameter vanishes i.e. $\alpha=0$.  For details of whale diagram and escape cones one can see an
interesting work by Tanatorov and Zaslavskii [53].
We set the value of
${\cal M}=G_{N}=1$ during plot which is valid througout the work. In Fig. <ref>(a), Fig. <ref>(b) and
Fig. <ref>(c), we  also set the MOG parameter ($\alpha=0$) and will vary the spin parameter value to differentiate
three types of compact objects  namely non-extremal Kerr, extremal Kerr and NS. Three pictures are qualitative different
to differentiate between non-extremal BH, extremal BH  and NS. In Fig. <ref>(d), Fig. <ref>(e) and
Fig. <ref>(f), we set the value of MOG parameter is unity. In this case, one can see the variation of angular
velocity with radial coordinate to differentiate three compact objects. Both $\Omega_{+}$ and $\Omega_{-}$  are
clearly distinct in each case. The distinction is more pronounced in Fig. <ref>(g), Fig. <ref>(h) and
Fig. <ref>(i) and so on for greater value of MOG  parameter.

The figure describes the variation  of $\Omega$  with $r$ for
different values of spin parameter, with MOG parameter and without MOG parameter. The gyroscope has
angular frequency $\Omega$ varies with in the range $(\Omega_{+}, \Omega_{-})$. Each set of row
depicts the variation of $\Omega$ vs. $r$ for non-extremal BH, extremal BH and NS.
There is a qualitative difference between these plots i. e.  when we have taken into
account the MOG parameter and without MOG parameter.

The angular frequency has two values i. e. in the outer horizon it is $\Omega_{+}$ and
in the inner horizon it is $\Omega_{-}$. Now we analyze the behavior of the gyro inside
the ergosphere of a BH and to do this job we should first compute the magnitude of the
precession frequency for various values of $\theta$.
For this purpose, we introduce the parameter $\delta$ to scan the entire range of $\Omega$ as

\begin{eqnarray}
\Omega &=& \delta~\Omega_{+} + (1-\delta)~\Omega_{-} = \omega -(1-2\delta)\sqrt{\omega^2-\frac{g_{tt}}{g_{\phi\phi}}}\\
&=& \frac{a\Pi_{\alpha}\sin\theta - (1-2\delta)~\rho^2\sqrt{\Delta}}{\sin\theta[(r^2+a^2)\rho^2+a^2\sin^2\theta \Pi_{\alpha}]}
\label{q}
\end{eqnarray}

where $0<\delta<1$ and $\omega=-g_{t\phi}/g_{\phi\phi}$. It is evident that the limiting value of
$\delta$ gives the range of $\Omega$ from $\Omega_{+}$ to $\Omega_{-}$. Using  Eq. (<ref>), one
obtains the generalized spin precession frequency in a compact form

\begin{eqnarray}
\vec{\Omega}_{p} &=& \Upsilon(r)
\left[\xi(r)~\sqrt{\Delta}\cos\theta~ \hat{r}+\eta(r)~\sin\theta~\hat{\theta}\right].~\label{gekn1}
\end{eqnarray}

\begin{eqnarray}
\Upsilon(r) &=& \frac{(r^2+a^2)^2-a^2\Delta \sin^2\theta}{4\delta(1-\delta)\Delta~\rho^7}
\end{eqnarray}

The implication of this equation is that one could study the behaviour of generalized spin
frequency $\vec{\Omega}_{p}$ with spin parameter $a$, $\theta$, $\delta$, $r$ and MOG parameter.
It should be noted that the frequency vector diverges at $\delta=0, 1$, $\Delta=0$ and $\rho=0$.
This means that the generalized spin precession frequency becomes arbitrarily large at these
values. Using this frequency one should differentiate between the BH and NS
in a strong gravitational field.

Now consider an observer moving with a four velocity $u$ in a stationary and axisymmetric
spacetime. If the angular momentum is zero for a particular situation which is defined by

\begin{eqnarray}
p_{\phi} & \equiv & \frac{\partial {\cal L}}{\partial \dot{\phi}}
=g_{t\phi} \dot{t}+g_{\phi\phi} \dot{\phi}=\ell =0
\end{eqnarray}

(where ${\cal L}$ is the Lagrangian and $\ell$ is angular momentum)
such an observer is called zero angular momentum observer (ZAMO) which was first
observed by Bardeen [47, 54]. Bardeen et al. [48] proved that the
ZAMO frame is a powerful tool to analyze the physical processes near astrophysical
object. What happens in case of Newtonian gravity? The angular momentum $\ell$ and
angular velocity $\Omega$ are satisfied by the relation $\ell =r^2\Omega$. It
implies that there is no problem when we taking into consider the non-rotating
frame i.e. $\ell=\Omega=0$. The problem should arise when we consider Einstein's
gravity: in this case the angular momentum is satisfied the following relation
$\ell \propto (\Omega-\omega)$, where $\omega=-\frac{g_{t\phi}}{g_{\phi\phi}}$.
Here we would not obtain $\ell=0$ when $\Omega=0$. It indicates that there must
exist two different observers i.e.  one should be zero angular momentum observer (ZAMO)
and the other one should be zero angular velocity observer (ZAVO).
In ZAMO frame, the value of angular momentum is $\ell =0$ while in ZAVO frame, the
value of $\Omega=0$. The frame-dragging angular velocity of these two frames is
$\omega=-\frac{g_{t\phi}}{g_{\phi\phi}}$. One should define the gravitational
potential in the ZAMO frame as $\Phi=-\frac{1}{2} \ln|g^{tt}|$. Where $g^{tt}$ is
contravariant component of stationary axisymmetric spacetime metric. This potential
is again related to the  gravitational acceleration (accelaration due to gravity)
as felt by an observer in space $g_{\mu}=(a_{\mu})_{ZAMO}=\nabla \Phi$. The
gravitational accelaration $a_{\mu}$ is a kinematic invariant quantity. Using
Eq. (<ref>), one could easily see that for a particular value of
$\delta=\frac{1}{2}$ the value of angular velocity for KMOG BH in
ZAMO frame as

\begin{eqnarray}
\Omega &=&  \omega  =
\frac{a~\Pi_{\alpha}\sin\theta}{\sin\theta[(r^2+a^2)\rho^2+a^2\sin^2\theta~\Pi_{\alpha}]}
\label{q1}
\end{eqnarray}

It implies that the ZAMOs angular velocity is a function of spin parameter and
charge parameter. In  $\theta=\frac{\pi}{2}$ plane,  the ZAMOs angular
velocity should be

\begin{eqnarray}
\Omega|_{\theta=\frac{\pi}{2}} &=&  \omega|_{\theta=\frac{\pi}{2}}  =
\frac{a~\Pi_{\alpha}}{(r^2+a^2)r^2+a^2~\Pi_{\alpha}}~\label{q2}
\end{eqnarray}

The variation of $\Omega$ and $\omega$ in the equatorial plane
could be seen from Fig. (<ref>) and Fig. (<ref>) respectively.
It must be noted that both these parameters are depend on MOG parameter. From
Fig. (<ref>) and Fig. (<ref>), it
is observed that the three cases namely non-extremal, extremal and NS both in the presence of MOG
parameter and in the absence of MOG parameter are qualitatively different. This means that three geometries
are quite distinguished.

The figure depicts the variation  of $\omega$  with $r$
for different values of spin parameter in  $\theta=\frac{\pi}{2}$ plane. This
frequency is a special class of frequency called ZAMO frequency when $\delta=\frac{1}{2}$.
The first figure is considered without MOG parameter ($\alpha=0$) and  for various values of spin
parameter to differentiate between non-extremal BH, extremal BH and NS. The second and third figure
is considered with MOG parameter $\alpha=1$ and $\alpha=2$. In first figure, the ZAMO frequency
decreases for each spin values $a<1$, $a=1$ and $a>1$ for increasing the value of $r$. While in the
second figure, the ZAMO frequency first increases then at certain point it diverges then it further
increases to a certain peak then decreases (for each spin values).

§.§ Behaviour of $\vec{\Omega}_{p}$ at $r=0$

In this section we will discuss various cases of LT precession frequency. Also we will see the
LT precession frequency structure for various useful limits. First we have taken the limit
$r\rightarrow 0$ (i. e. at the ring singularity $r=0$, $\theta=\frac{\pi}{2}$). It should
be noted that $\vec{\Omega}_p$ is not valid at the ring singularity ($r=0, \theta=\pi/2$).
Yet, we should investigate its behavior in its vicinity i.e. in the region
$r=0$, $0 \leq \theta < 90^{\circ}$. It must be noted that this region is completely outside the
ergoregion since the ergosurface meets the ring singularity. At the $r=0$, the LT precession
frequency becomes

\begin{eqnarray}
\vec{\Omega}_p|_{r= 0}= \frac{\xi(\theta)~\hat{r}+\eta(\theta)~\hat{\theta}}{\zeta(\theta)},
\label{gekn2}
\end{eqnarray}

and the magnitude of this vector is thus

\begin{eqnarray}
{\Omega}_p|_{r= 0}= \frac{\sqrt{\xi^2(\theta)+\eta^2(\theta)}}{\zeta(\theta)}
\end{eqnarray}

\xi(\theta)=\sqrt{a^2+\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2}\times

\begin{eqnarray}
\left[-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2
-\frac{\Omega}{8}\left\{\left(3+4\cos2\theta+\cos4\theta\right)a^3-8\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2
~a\left(1-\cos2\theta \right)\right\}
-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2a^2\Omega^2\sin^4\theta\right]\nonumber\\
\end{eqnarray}

\begin{eqnarray}
\eta(\theta) &=& G_{N} {\cal M}~a^2\cos\theta \sin\theta \left[-1+a\Omega(1+\sin^2\theta)-a^2\Omega^2\sin^2\theta\right]
\end{eqnarray}

\zeta(\theta) = a^2\cos^2\theta \times

\begin{eqnarray}
\left[\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2+a^2\cos^2\theta
-2\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2~a\Omega\sin^2\theta-a^2\Omega^2\sin^2\theta
\left(a^2\cos^2\theta-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2~\sin^2\theta\right)\right]\nonumber\\
\end{eqnarray}

The valid regime of $\Omega$ is

\begin{eqnarray}
\Omega_{-}(\theta)< \Omega <\Omega_{+}(\theta)
\end{eqnarray}

\begin{eqnarray}
\Omega_{-}(\theta) &=&-\frac{\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2~\sin\theta+\sqrt{D}}
{a\sin\theta\left[a^2-\left(a^2+\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2\right)\sin^2\theta\right]} \nonumber \\
\Omega_{+}(\theta) &=& \frac{-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2~\sin\theta
+\sqrt{D}}{a\sin\theta\left[a^2-\left(a^2+\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2\right)\sin^2\theta\right]}\nonumber \\
\end{eqnarray}

D=\left(\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2\right)^2\sin^2\theta
+\left\{a^2-\left(a^2+\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2\right)\sin^2\theta\right\}
\left(\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2+a^2\cos^2\theta\right)

For a static observer outside the ergosphere we have to put $\Omega=0$ therefore
we get from Eq. (<ref>)

\begin{eqnarray}
|\vec{\Omega}_{p}|=\frac{\sqrt{\left(\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2\right)^2
\left(a^2+\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2\right)+{\cal M}^2a^4\sin^2\theta\cos^2\theta}}
{a^2\cos^2\theta\left(\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2+a^2\cos^2\theta\right)}
\label{gekn3}
\end{eqnarray}

It must be noted that $\Omega_{p}$ varies from $0 \leq \Omega_{p} < \infty$
for $0 \leq \theta < 90^{\circ}$ at $r=0$. This indicates that it diverges only
at the ring singularity and in cartesian Kerr-Schild coordinates it is described
by $x^2+y^2=a^2,~z=0$ while it is finite inside the ring singularity i. e.

§.§ Behaviour of $\vec{\Omega}_{p}$ at $\theta=0$

In the limit $\theta=0$, the LT frequency is given by

\begin{eqnarray}
\vec{\Omega}_p|_{\theta= 0} &=&
\frac{\xi(r)|_{\theta= 0}}{\zeta(r)|_{\theta= 0}}~\sqrt{\Delta}~\hat{r},~\label{t0}
\end{eqnarray}

\begin{eqnarray}
\xi(r)|_{\theta= 0} &=&   a \Pi_{\alpha}-\frac{\Omega}{8} \left[8r^4+8a^2r^2+8 a^2\Pi_{\alpha}+3a^4+
4a^2(2\Delta-a^2)+a^4\right] \\
\zeta(r)|_{\theta= 0} &=& \rho^3 \left(\rho^2-\Pi_{\alpha}\right)
\end{eqnarray}

Using the identity $r^2+a^2-\Pi_{\alpha}=\Delta$, the above equations reduced to

\begin{eqnarray}
\xi(r)|_{\theta= 0} &=& a \Pi_{\alpha}-\Omega \left(r^2+a^2\right)^2\\
\zeta(r)|_{\theta= 0} &=& \left(r^2+a^2 \right)^{3\over 2} \Delta
\end{eqnarray}

The magnitude of this vector is calculated to be

\begin{eqnarray}
\Omega_p|_{\theta= 0} &=& \frac{a\Pi_{\alpha}-\Omega \left(r^2+a^2\right)^2}
{ \left(r^2+a^2\right)^\frac{3}{2}\sqrt{\Delta}}~\label{t1}
\end{eqnarray}

From the above equation one could see that the precession frequency is arbitrarily large when $\Delta=0$. The
precession frequency is positive when $a\Pi_{\alpha}>\Omega(r^2+a^2)^2$ and negative when
$a\Pi_{\alpha}<\Omega(r^2+a^2)^2$ and becomes zero when $a\Pi_{\alpha}=\Omega(r^2+a^2)^2$.
From Fig. (<ref>),  one could see that the variation of precession frequency with
radial coordinate without MOG parameter and with MOG parameter.
From this figure, one can distinguish three geometrical structure of the spacetime via
precession frequency.

The figure describes the variation  of precession frequency $\Omega_{p}$
with $r$ for $\theta=0$. In this plot we have chosen the value of $\Omega=0.1 $. We
have varied the spin parameter values for three cases namely non-extremal BH, extremal BH and NS.
Each plot shows the difference between precession frequency $\Omega_{p}$ having MOG parameter
and without MOG parameter.

§.§ Behaviour of $\vec{\Omega}_{p}$ at $\theta=\frac{\pi}{6}$

In the limit $\theta=\frac{\pi}{6}$, the LT frequency is computed as

\begin{eqnarray}
\vec{\Omega}_{p}|_{\theta= \frac{\pi}{6}} &=&
\frac{\sqrt{3 \Delta}\xi(r)|_{\theta=\frac{\pi}{6}}\hat{r}+\eta(r)|_{\theta=\frac{\pi}{6}} \hat{\theta}}
{2\zeta(r)|_{\theta= \frac{\pi}{6}}}~,~\label{t2}
\end{eqnarray}

The magnitude of this vector is turned out to be

\begin{eqnarray}
\Omega_p|_{\theta=\frac{\pi}{6}} &=& \frac{\sqrt{3\Delta \xi^2 (r)|_{\theta=\frac{\pi}{6}}+\eta^2(r)|_{\theta=\frac{\pi}{6}}}}
{2\zeta(r)|_{\theta= \frac{\pi}{6}}} ~\label{t3}
\end{eqnarray}

\begin{eqnarray}
\xi~(r)|_{\theta= \frac{\pi}{6}} &=& a \Pi_{\alpha}-\frac{\Omega}{16}\left(16r^4+24a^2r^2+8a^2\Pi_{\alpha}+9a^4 \right)+
\frac{\Omega^2}{16}a^3\Pi_{\alpha}
\end{eqnarray}

\eta(r)|_{\theta= \frac{\pi}{6}} = aG_{N}{\cal M}\left(r^2-\frac{3}{4}a^2\right)
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 ar+\Omega\times

\left[r^5-3G_{N}{\cal M}r^4+\frac{3}{2} a^2r^3-2G_{N}{\cal M}a^2r^2+\frac{9}{16}a^4 r
+\frac{15}{16} G_{N}{\cal M}a^4+\frac{2\alpha}{1+\alpha}G_{N}^2{\cal M}^2r \left(r^2+a^2\right)\right]

\begin{eqnarray}
+\frac{a\Omega^2}{4}\left[G_{N}{\cal M}\left\{3r^4+a^2r^2+\frac{3}{4}a^2\left(r^2-a^2\right)\right\}
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 r\left(2r^2+\frac{7}{4}a^2\right)\right]
\end{eqnarray}

\zeta(r)|_{\theta= \frac{\pi}{6}} = \left(r^2+\frac{3}{4}a^2\right)^\frac{3}{2}\times

\begin{eqnarray}
\left[\left(r^2+\frac{3}{4}a^2\right)-\Pi_{\alpha} +\frac{a\Omega \Pi_{\alpha}}{2}-\frac{\Omega^2}{4}
\left\{(r^2+\frac{3}{4}a^2)(r^2+a^2)+\frac{a^2\Pi_{\alpha}}{4}\right\}\right]
\end{eqnarray}

Variation of spin precession frequency with radial coordinate with MOG parameter and without MOG parameter
could be seen from  Fig. (<ref>). In Fig. (<ref>-a), Fig. (<ref>-b) and Fig. (<ref>-c),
we have plotted the precession frequency with radial variable for non-extremal BH, extremal BH and NS of Kerr
spacetime. While in Fig. (<ref>-d), Fig. (<ref>-e) and Fig. (<ref>-f), we have plotted the
precession frequency for  non-extremal BH, extremal BH and NS of KMOG spacetime.
In upper panel [(<ref>-a), Fig. (<ref>-b), Fig. (<ref>-c)], we can see that for BH spacetime
the precession frequency has one minimum value while for NS it has two minima. In lower panel,
[(<ref>-d), Fig. (<ref>-e), Fig. (<ref>-f)], the generalized  spin frequency has one minima for
BH spacetime and one peak for NS.

Examples of the variation  of $\Omega_{p}$ with $r$ for $\theta=\frac{\pi}{6}$ in KMOG with variation
of MOG parameter, spin parameter and $\Omega$. The upper panel describes the variation  of $\Omega_{p}$
with $r$ for non-extremal BH, extremal BH and NS without MOG parameter. The lower panel describes the
variation  of $\Omega_{p}$  with $r$ for non-extremal BH, extremal BH and NS with MOG parameter.

§.§ Behaviour of $\vec{\Omega}_{p}$ at $\theta=\frac{\pi}{4}$

In the limit $\theta=\frac{\pi}{4}$, the LT frequency is derived as

\begin{eqnarray}
\vec{\Omega}_{p}|_{\theta= \frac{\pi}{4}} &=&
\frac{\sqrt{\Delta}\xi(r)|_{\theta=\frac{\pi}{4}}\hat{r}+\eta(r)|_{\theta=\frac{\pi}{4}} \hat{\theta}}
{\sqrt{2} \zeta(r)|_{\theta= \frac{\pi}{4}}}~,~\label{t6}
\end{eqnarray}

The magnitude of this vector is

\begin{eqnarray}
\Omega_p|_{\theta=\frac{\pi}{4}} &=& \frac{\sqrt{\Delta \xi^2 (r)|_{\theta=\frac{\pi}{4}}
+\eta^2(r)|_{\theta=\frac{\pi}{4}}}}{\sqrt{2}\zeta(r)|_{\theta= \frac{\pi}{4}}} ~\label{t7}
\end{eqnarray}

\begin{eqnarray}
\xi~(r)|_{\theta= \frac{\pi}{4}} &=& a \Pi_{\alpha}-\frac{\Omega}{8}\left[8r^4+8a^2r^2+8a^2\Pi_{\alpha}+2a^4 \right]+
\frac{\Omega^2}{4}a^3\Pi_{\alpha}~\label{t8}
\end{eqnarray}

\eta(r)|_{\theta= \frac{\pi}{4}} = aG_{N}{\cal M}\left(r^2-\frac{a^2}{2}\right)
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 ar+\frac{\Omega}{4}\times

\left[4r^5-12G_{N}{\cal M}r^4+4 a^2r^3-8G_{N}{\cal M}a^2r^2+a^4 r
+3 G_{N}{\cal M}a^4+\frac{8\alpha}{1+\alpha}G_{N}^2{\cal M}^2r\left(r^2+a^2\right)\right]

\begin{eqnarray}
+\frac{a\Omega^2}{4}\left[G_{N}{\cal M}\left(6r^4+3a^2r^2-a^4\right)
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 r\left(2r^2+a^2\right)\right]
\end{eqnarray}

\zeta(r)|_{\theta= \frac{\pi}{4}} = \left(r^2+\frac{a^2}{2}\right)^\frac{3}{2}\times

\begin{eqnarray}
\left[\left(r^2+\frac{a^2}{2}\right)-\Pi_{\alpha} +a\Omega \Pi_{\alpha} -\frac{\Omega^2}{2}
\left\{\left(r^2+\frac{a^2}{2}\right) \left(r^2+a^2\right)+\frac{a^2\Pi_{\alpha}}{2}\right\}\right]~.\label{t10}
\end{eqnarray}

Variation of spin precession frequency with radial coordinate with MOG parameter and without MOG parameter
may be seen from  Fig. (<ref>). In Fig. (<ref>-a), Fig. (<ref>-b) and Fig. (<ref>-c),
we have plotted the precession frequency with radial variable for non-extremal BH, extremal BH and NS of Kerr
spacetime. While in Fig. (<ref>-d), Fig. (<ref>-e) and Fig. (<ref>-f), we have plotted the
precession frequency for  non-extremal BH, extremal BH and NS of KMOG spacetime.
In upper panel [(<ref>-a), Fig. (<ref>-b), Fig. (<ref>-c)], one can see that for BH spacetime
the precession frequency has one minimum value while for NS it has one minimum value and one maximum value.
In lower panel,  [(<ref>-d), Fig. (<ref>-e), Fig. (<ref>-f)], the generalized  spin frequency
has one minima for BH spacetime and one peak for NS.

Examples of the variation  of $\Omega_{p}$ versus $r$ for $\theta=\frac{\pi}{4}$ in KMOG with variation
of MOG parameter, spin parameter and $\Omega$. The first row describes the variation  of $\Omega_{p}$
with $r$ for non-extremal BH, extremal BH and NS without MOG parameter. The second row describes the variation
of $\Omega_{p}$  with $r$ for non-extremal BH, extremal BH and NS with MOG parameter.

§.§ Behaviour of $\vec{\Omega}_{p}$ at $\theta=\frac{\pi}{3}$

In this limit $\theta=\frac{\pi}{3}$, the LT frequency is computed to be

\begin{eqnarray}
\vec{\Omega}_{p}|_{\theta= \frac{\pi}{3}} &=&
\frac{\sqrt{\Delta}\xi(r)|_{\theta=\frac{\pi}{3}}\hat{r}+\sqrt{3}\eta(r)|_{\theta=\frac{\pi}{3}} \hat{\theta}}
{2\zeta(r)|_{\theta= \frac{\pi}{3}}}~,~\label{t12}
\end{eqnarray}

The magnitude of this vector is given by

\begin{eqnarray}
\Omega_p|_{\theta=\frac{\pi}{3}} &=& \frac{\sqrt{\Delta \xi^2 (r)|_{\theta=\frac{\pi}{3}}+3\eta^2(r)|_{\theta=\frac{\pi}{3}}}}
{2\zeta(r)|_{\theta= \frac{\pi}{3}}} ~\label{t13}
\end{eqnarray}

\begin{eqnarray}
\xi~(r)|_{\theta= \frac{\pi}{3}} &=& a \Pi_{\alpha}-\frac{\Omega}{8}\left(8r^4+4a^2r^2+12a^2\Pi_{\alpha}+\frac{a^4}{2}\right)
+\frac{9}{16}a^3 \Omega^2 \Pi_{\alpha}
\end{eqnarray}

\eta(r)|_{\theta= \frac{\pi}{3}} = aG_{N}{\cal M}\left(r^2-\frac{a^2}{4}\right)
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 ar+\Omega\times

\left[r^5-3G_{N}{\cal M}r^4+\frac{a^2r^3}{2}-2G_{N}{\cal M}a^2r^2+\frac{a^4}{16} r
+\frac{7}{16} G_{N}{\cal M}a^4+\frac{2\alpha}{1+\alpha}G_{N}^2{\cal M}^2r(r^2+a^2)\right]

\begin{eqnarray}
+\frac{3}{4}a\Omega^2\left[G_{N}{\cal M}\left(3r^4+a^2r^2+\frac{a^2}{4}(r^2-a^2)\right)
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 r\left(2r^2+\frac{5}{4}a^2\right)\right]
\end{eqnarray}

\zeta(r)|_{\theta= \frac{\pi}{3}} = \left(r^2+\frac{a^2}{4}\right)^\frac{3}{2}\times

\begin{eqnarray}
\left[\left(r^2+\frac{a^2}{4}\right)-\Pi_{\alpha} +\frac{3}{2}a\Omega \Pi_{\alpha}-\frac{3}{4}\Omega^2
\left\{\left(r^2+\frac{a^2}{4}\right) \left(r^2+a^2\right)+\frac{3a^2\Pi_{\alpha}}{4}\right\}\right]
\end{eqnarray}

Analogously, the variation of spin precession frequency with radial coordinate with MOG parameter
and without MOG parameter may be seen from  Fig. (<ref>).

Examples of the variation  of $\Omega_{p}$ vs. $r$ for $\theta=\frac{\pi}{3}$ in KMOG
with variation of MOG parameter, spin parameter and $\Omega$. The first row describes the variation  of $\Omega_{p}$
with $r$ for non-extremal BH, extremal BH and NS without MOG parameter. The second row describes the variation
of $\Omega_{p}$  with $r$ for non-extremal BH, extremal BH and NS with MOG parameter.

§.§ Behaviour of $\vec{\Omega}_{p}$ at $\theta=\frac{\pi}{2}$

In the equatorial plane the precession frequency vector is given by

\begin{eqnarray}
\vec{\Omega}_{p}|_{\theta=\frac{\pi}{2}} &=&
\frac{\eta(r)|_{\theta=\frac{\pi}{2}}}{\zeta(r)|_{\theta=\frac{\pi}{2}}}\hat{\theta}
\end{eqnarray}

The magnitude of this vector is thus

\begin{eqnarray}
\Omega_p|_{\theta=\frac{\pi}{2}} &=& \frac{\eta(r)|_{\theta=\frac{\pi}{2}}}{\zeta(r)|_{\theta=\frac{\pi}{2}}}  ~\label{t17t}
\end{eqnarray}

\begin{eqnarray}
\eta(r)_{\theta=\frac{\pi}{2}} &=& ar\left(G_{N}{\cal M} r-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2\right)\nonumber\\
+\Omega~\left[r^5-3G_{N} {\cal M}r^4-2G_{N} {\cal M}a^2r^2
+2\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2r\left(r^2+a^2\right)\right]\nonumber\\
+a\Omega^2\left[G_{N} {\cal M}r^2\left(3r^2+a^2 \right)-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2r \left(2r^2+a^2\right)\right] \\
\zeta(r)_{\theta=\frac{\pi}{2}} &=& r^3\,\left[r^2-\Pi_{\alpha}+2a\Omega \Pi_{\alpha}-\Omega^2\left\{r^2\left(r^2+a^2\right)
\end{eqnarray}

Variation of spin precession frequency with radial coordinate with
MOG parameter and without MOG parameter may be seen from  Fig. (<ref>).

Examples of the variation  of $\Omega_{p}$ versus $r$ for $\theta=\frac{\pi}{2}$ in KMOG with variation
of MOG parameter, spin parameter and $\Omega$. The first row describes the variation  of $\Omega_{p}$
with $r$ for non-extremal BH, extremal BH and NS without MOG parameter. The second row describes the variation
of $\Omega_{p}$  with $r$ for non-extremal BH, extremal BH and NS with MOG parameter.

The range of $\Omega$ could be determined by using Eq. (<ref>). As we said earlier
that the outer horizon and the outer ergosurface are located at
$r_{+} = G_{N}{\cal M} + \sqrt{\frac{G_{N}^2{\cal M}^2}{1+\alpha}-a^2}$ and
$r_{e}^{+} = G_{N}{\cal M} + \sqrt{\frac{G_{N}^2{\cal M}^2}{1+\alpha}-a^2 \cos^2~\theta}$. While
in the equatorial plane, the ergosphere is located at $r_{e}^{+} =m+\epsilon$ where $m=G_{N}{\cal M}$
and $\epsilon=\frac{m}{\sqrt{1+\alpha}}$.
Therefore one could obtain the precession frequency at the equatorial
ergosurface $r=r_{e}^{+}=m+\epsilon$

\begin{eqnarray}
{\Omega}_{p}|_{r=r_{e}^{+}} &=& \frac{\eta(r)|_{r=r_{e}^{+}}}{\zeta(r)|_{r=r_{e}^{+}}}
\end{eqnarray}

\begin{eqnarray}
\eta(r)|_{r=r_{e}^{+}} &=& (m+\epsilon)\left[a m_{\epsilon}+\Omega m_{\epsilon\epsilon}
+a\Omega^2m_{\epsilon\epsilon\epsilon}\right] ~\label{t20}
\end{eqnarray}

m_{\epsilon\epsilon}= (m+\epsilon)^2(\epsilon^2-m\epsilon-2m^2+2\alpha\epsilon^2)-2ma^2(m+\epsilon)+2\alpha\epsilon^2a^2

\begin{eqnarray}
\zeta(r)|_{r=r_{e}^{+}} &=& (m+\epsilon)^3\left[ m_{\chi}+2a\Omega m_{\chi\chi}
-\Omega^2m_{\chi\chi\chi}\right] ~\label{t21}
\end{eqnarray}

In the extremal limit $a=\epsilon$, the precession frequency becomes

\begin{eqnarray}
&=& \frac{\epsilon \left[\epsilon-\Omega (3\epsilon^2+2\epsilon m+m^2)
+\Omega^2(\epsilon^3+m^3+4\epsilon m^2+7m \epsilon^2)\right]}
{(\epsilon+m)^3\left[2\epsilon\Omega-\Omega^2(3\epsilon^2+m^2+2\epsilon m)\right]}
\end{eqnarray}

The value of $\Omega$ lies in the range $0 < \Omega < \frac{2\epsilon}{3\epsilon^2+m^2+2\epsilon m}$.
It indicates that if the mass or angular momentum of the central object increases then the
value of $\Omega$ both at the ergosurface and in the $\theta=\frac{\pi}{2}$ plane becomes
decreases. It may be written as in terms of MOG parameter as

= \frac{1+\alpha}{\left(1+\sqrt{1+\alpha} \right)^3}\times

\begin{eqnarray}
\frac{1- G_{N}{\cal M}\Omega\left(2+\sqrt{1+\alpha}+\frac{3}{\sqrt{1+\alpha}} \right)+ G_{N}^2{\cal M}^2\Omega^2
\left(4+\sqrt{1+\alpha}+\frac{7}{\sqrt{1+\alpha}}+\frac{1}{1+\alpha}\right)}
{ G_{N}^2{\cal M}^2\Omega\left[2- G_{N}{\cal M}\Omega\left(2+\sqrt{1+\alpha}+\frac{3}{\sqrt{1+\alpha}}
\right) \right]}
\end{eqnarray}

when the MOG parameter reduces to zero value one gets the result of extremal Kerr BH as
${\Omega}_{p}|_{\theta=\frac{\pi}{2}}^{r=r_{e}^{+}}=\frac{1-6 G_{N}{\cal M}\Omega+13 G_{N}^2{\cal M}^2\Omega^2}
{16  G_{N}^2{\cal M}^2\Omega(1-3 G_{N}{\cal M}\Omega)}$.
One could observe the variation of precession frequency with MOG parameter in the equatorial ergosphere
from the Fig. (<ref>).

The figure describes the variation  of $\Omega_{p}$
with MOG parameter ($\alpha$) for various values of $\Omega$.

It should be noted that the precession frequency at the outer
horizon $r=r_{+}=m$ of extremal KMOG BH is computed to be

\begin{eqnarray}
=-\frac{\epsilon }{m^2}~\label{t24}
\end{eqnarray}

In terms of MOG parameter the above expression can be written as

\begin{eqnarray}
{\Omega}_{p}|_{\theta=\frac{\pi}{2}}^{r=r_{+}}=-\frac{1}{\sqrt{1+\alpha} G_{N}{\cal M}} ~\label{t25}
\end{eqnarray}

when $\alpha=0$, the precession frequency reduces to extremal BH,
${\Omega}_{p}|_{\theta=\frac{\pi}{2}}^{r=r_{+}}=-\frac{1}{G_{N}{\cal M}}$.

§.§  $\vec{\Omega}_{p}$ in the Schwarzschild-MOG  spacetime

In the limit $a=0$, the KMOG BH reduces to Schwarzschild-MOG (SMOG) spacetime.
In this case, the LT precession frequency becomes

\begin{eqnarray}
\vec{\Omega}_{p}|_{a=0}=\Omega ~ \frac{-(r^2-\Pi_{\alpha})^{\frac{1}{2}}\cos\theta ~\hat{r}+
(r^2-\Pi_{\alpha}-G_{N} {\cal M}r+\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2)\sin\theta ~\hat{\theta}}
\end{eqnarray}

where $\Omega$ might have any value so that $u$ should be timelike.

It must be noted that SMOG spacetime is a spherically symmetric
solution of the  Einstein equation. It denotes a BH of mass ${\cal M}$.
In the equatorial plane $\Omega_{p}|_{\theta=\frac{\pi}{2}}$

\begin{eqnarray}
\Omega_{p} &=& \Omega ~ \frac{r^2-\Pi_{\alpha}-G_{N} {\cal M}r+
\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2}{r^2-\Pi_{\alpha}-\Omega^2 r^4}.~\label{rn1}
\end{eqnarray}

It implies that when a gyroscope moving in the equatorial plane of the
Reissner-Nordström spacetime (static spacetime) it becomes precess.
Now if the gyro moves along circular orbits then $\Omega_{p}$ becomes
Keplerian frequency i.e.

\begin{eqnarray}
\Omega_{p}=\Omega=\Omega_{K}= \sqrt{\frac{G_{N} {\cal M}r-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2}{r^{4}}}.
\end{eqnarray}

This equation indicates the precession frequency in the Copernican
frame which is derived with respect to the proper time $\tau$. The
proper time $\tau$  is computed in the Copernican frame is related to the
coordinate frame $t$ via the relation
$d\tau=\sqrt{1-\frac{3G_{N} {\cal M}}{r}+2\left(\frac{\alpha}{1+\alpha}\right)\frac{ G_{N}^2{\cal M}^2}{r^2}} dt$. Thus
one obtains the precession frequency in the coordinate basis $\Omega^{'}$ as,

\begin{equation}
\Omega^{'} = \left(\frac{G_{N} {\cal M}r-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2}{r^4}\right)^{\frac{1}{2}}
\sqrt{1-\frac{3G_{N} {\cal M}}{r}+2\left(\frac{\alpha}{1+\alpha}\right)\frac{ G_{N}^2{\cal M}^2}{r^2}}.
\end{equation}

Thus one can compute the geodetic precession frequency which is the difference
between $\Omega^{'}$ and $\Omega$, and one obtains

\begin{eqnarray}
\Omega_{\rm geodetic} &=& \left(\frac{G_{N} {\cal M}r-\frac{\alpha}{1+\alpha} G_{N}^2{\cal M}^2}{r^4}\right)^{\frac{1}{2}}
\left(1- \sqrt{1-\frac{3G_{N} {\cal M}}{r}+2\left(\frac{\alpha}{1+\alpha}\right)\frac{ G_{N}^2{\cal M}^2}{r^2}}\right)
\end{eqnarray}

This is called the geodetic precession of a test gyroscope around a nonrotating
spherical object of mass $G_{N} {\cal M}$. In the limit when $\alpha=0$, one gets the
geodetic precession of Schwarzschild BH [52]. One could see the geodetic precession of a
nonrotating spherical object from the Fig. (<ref>) for different values of MOG parameter.

The figure describes the variation  of geodetic precession  with $r$
for different values of spin parameter in the $\theta=\frac{\pi}{2}$ plane.

§ FRAME-DRAGGING EFFECT OF KMOG SPACETIME WITH $\OMEGA=0$

For strengthening our work, in this section now we will focus on particularly LT effect by
calculating the frequency $\Omega_{LT}$ and considering the angular velocity of test gyro
is equal to zero i.e. $\Omega=0$ (We ommit other precession
e.g. geodetic precession etc.). The theoretical prescription of this formalism for $\Omega=0$ was
first introduced by Chakraborty & Majumdar [17] in the strong gravity regime and for arbitrary value
of $\Omega$. The result of the weak field limit of the LT frequency can be easily obtained from the
general formalism. Therefore, one can derive the LT precession frequency for KMOG BH in MOG by
using Eq. (<ref>).

As we have said earlier when $\Omega=0$ then $\vec{\Omega_{p}}=\vec{\Omega_{LT}}$. Thus one can compute
for the metric (<ref>) the LT frequency vector is

\begin{eqnarray}
\vec{\Omega}_{LT} &=& \frac{\chi(r)~\sqrt{\Delta}\cos\theta~\hat{r}+\mu(r)~\sin\theta~\hat{\theta}}{\sigma(r)}
\end{eqnarray}

\chi (r)= a \Pi_{\alpha}

\mu(r) = aG_{N}{\cal M}(r^2-a^2\cos^2\theta)-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 ar

\sigma(r) = \rho^3(\rho^2-\Pi_{\alpha})

The magnitude of this vector is computed to be

\begin{eqnarray}
\Omega_{LT}(r,\theta) &=& \frac{\sqrt{\Delta~\chi^2(r)~\cos^2\theta+\mu^2(r)~\sin^2\theta}}
{\sigma(r)} ~\label{ttl3}
\end{eqnarray}

It clearly shows that the LT frequency is affected by the MOG parameter $\alpha$. This is very interesting in a sense
that the deformation parameter changes the geometry of the spacetime and it also changes the
LT frequency. This could be seen from the graphical plot. Without  MOG parameter the LT frequency
reduces to Kerr BH. The presence of the
MOG parameter decreases the LT frequency. Now we would derive the LT frequency for various values of $\theta$.
One could see the variation of Lense-Thirring frequency with the radial coordinate from the subsequent
figures. From this diagram one can observed how much amount of LT frequency is changed due to the
deformation parameter.

§.§ Behaviour of $\vec{\Omega}_{LT}$ at $\theta=0$

First we consider the case $\theta=0$.  In this case the LT frequency vector becomes

\begin{eqnarray}
\vec{\Omega}_{LT}|_{\theta= 0} &=&
\frac{\chi(r)|_{\theta= 0}}{\sigma(r)|_{\theta= 0}}~\sqrt{\Delta}~\hat{r},~\label{tl0}
\end{eqnarray}

The magnitude of this vector is calculated to be

\begin{eqnarray}
\Omega_{LT}|_{\theta= 0} &=&  \frac{a\Pi_{\alpha}}{(r^2+a^2)^\frac{3}{2}\sqrt{\Delta}}~\label{tl1}
\end{eqnarray}

It follows that the LT frequency depends on the MOG parameter. Using this equation one can differentiate
between BHs and NS in MOG theory. It could be observed from the  Fig. <ref>. In this diagram,
we have plotted the LT frequency $\Omega_{LT}$ with respect
to the radial coordinate for angular coordinate value $\theta=0$. We have speculated that the LT
frequency is diminished due to the presence of the MOG parameter in compared to Kerr BH.

Examples of the variation  of $\Omega_{LT}$ versus $r$ for $\theta=0$ in KMOG with variation
of MOG parameter and  spin parameter. Here  $\Omega=0$. The first figure describes the variation
of $\Omega_{LT}$  with $r$ for non-extremal BH, extremal BH and NS without MOG parameter.
The rest of the figure describes the variation  of $\Omega_{LT}$  with $r$ for non-extremal BH,
extremal BH and NS with MOG parameter.  Using these plots one can easily distinguish
between three compact objects.

§.§ Behaviour of $\vec{\Omega}_{LT}$ at $\theta=\frac{\pi}{6}$

In this limit $\theta=\frac{\pi}{6}$, the LT frequency is computed as

\begin{eqnarray}
\vec{\Omega}_{LT}|_{\theta= \frac{\pi}{6}} &=&
\frac{\sqrt{3 \Delta}\chi(r)|_{\theta=\frac{\pi}{6}}\hat{r}+\mu(r)|_{\theta=\frac{\pi}{6}} \hat{\theta}}
{2\sigma(r)|_{\theta= \frac{\pi}{6}}}~,~\label{tl2}
\end{eqnarray}

The magnitude of this vector is calculated to be

\begin{eqnarray}
\Omega_{LT}|_{\theta=\frac{\pi}{6}} &=& \frac{\sqrt{3\Delta \chi^2 (r)|_{\theta=\frac{\pi}{6}}
+\mu^2(r)|_{\theta=\frac{\pi}{6}}}}{2\sigma(r)|_{\theta= \frac{\pi}{6}}} ~\label{tl3}
\end{eqnarray}

\begin{eqnarray}
\chi~(r)|_{\theta= \frac{\pi}{6}} &=& a \Pi_{\alpha}
\end{eqnarray}

\mu(r)|_{\theta= \frac{\pi}{6}} = aG_{N}{\cal M}\left(r^2-\frac{3}{4}a^2\right)
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 ar

\sigma(r)|_{\theta= \frac{\pi}{6}} = \left(r^2+\frac{3}{4}a^2\right)^\frac{3}{2}
\left[\left(r^2+\frac{3}{4}a^2\right)-\Pi_{\alpha}\right]

From the above equation one can easily seen that the LT frequency is dependent on the MOG parameter
and spin parameter. It could seen from the diagram (<ref>) by plotting the LT precession frequency
with the radial coordinate.

Examples of the variation  of $\Omega_{LT}$ versus $r$ for $\theta=\frac{\pi}{6}$ in
KMOG with variation of MOG parameter and  spin parameter. Here  $\Omega=0$. The first figure
describes the variation  of $\Omega_{LT}$  with $r$ for non-extremal BH, extremal BH and NS
without MOG parameter.
The rest of the figure describes the variation  of $\Omega_{LT}$  with $r$ for
non-extremal BH, extremal BH and NS with MOG parameter.

§.§ Behaviour of $\vec{\Omega}_{LT}$ at $\theta=\frac{\pi}{4}$

In the limit $\theta=\frac{\pi}{4}$, the LT frequency is computed to be

\begin{eqnarray}
\vec{\Omega}_{LT}|_{\theta= \frac{\pi}{4}} &=&
\frac{\sqrt{\Delta}\chi(r)|_{\theta=\frac{\pi}{4}}\hat{r}+\mu(r)|_{\theta=\frac{\pi}{4}}\hat{\theta}}
{\sqrt{2}\sigma(r)|_{\theta= \frac{\pi}{4}}}~,~\label{ttl6}
\end{eqnarray}

The magnitude of this vector is thus

\begin{eqnarray}
\Omega_{LT}|_{\theta=\frac{\pi}{4}} &=& \frac{\sqrt{\Delta \chi^2 (r)|_{\theta=\frac{\pi}{4}}
+\mu^2(r)|_{\theta=\frac{\pi}{4}}}}{\sqrt{2}\sigma(r)|_{\theta= \frac{\pi}{4}}} ~\label{ttl7}
\end{eqnarray}

\begin{eqnarray}
\chi~(r)|_{\theta= \frac{\pi}{4}} &=& a \Pi_{\alpha}~\label{tl8}
\end{eqnarray}

\mu(r)|_{\theta= \frac{\pi}{4}} = aG_{N}{\cal M}\left(r^2-\frac{a^2}{2}\right)
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 ar

\sigma(r)|_{\theta= \frac{\pi}{4}} = \left(r^2+\frac{a^2}{2}\right)^\frac{3}{2}
\left[\left(r^2+\frac{a^2}{2}\right)-\Pi_{\alpha} \right]

One could differentiate the non-extremal BH, extremal BH and NS from the
diagram (<ref>)

Examples of the variation  of $\Omega_{LT}$ versus $r$ for $\theta=\frac{\pi}{4}$ in
KMOG with variation of MOG parameter and  spin parameter. Here  $\Omega=0$. The first figure
describes the variation  of $\Omega_{LT}$  with $r$ for non-extremal BH, extremal BH and NS
without MOG parameter. The rest of the figure describes the variation  of $\Omega_{LT}$
with $r$ for non-extremal BH, extremal BH and NS with MOG parameter.

§.§ Behaviour of $\vec{\Omega}_{LT}$ at $\theta=\frac{\pi}{3}$

Similarly for $\theta=\frac{\pi}{3}$, the LT frequency is

\begin{eqnarray}
\vec{\Omega}_{LT}|_{\theta= \frac{\pi}{3}} &=&
\frac{\sqrt{\Delta}\chi(r)|_{\theta=\frac{\pi}{3}}\hat{r}+\sqrt{3}\mu(r)|_{\theta=\frac{\pi}{3}} \hat{\theta}}
{2\sigma(r)|_{\theta= \frac{\pi}{3}}}~,~\label{tl5}
\end{eqnarray}

The magnitude of this vector is given by

\begin{eqnarray}
\Omega_{LT}|_{\theta=\frac{\pi}{3}} &=&
\frac{\sqrt{\Delta \chi^2 (r)|_{\theta=\frac{\pi}{3}}+3\mu^2(r)|_{\theta=\frac{\pi}{3}}}}
{2\sigma(r)|_{\theta= \frac{\pi}{3}}} ~\label{vt16}
\end{eqnarray}

\begin{eqnarray}
\chi~(r)|_{\theta= \frac{\pi}{3}} &=& a \Pi_{\alpha}
\end{eqnarray}

\mu(r)|_{\theta= \frac{\pi}{3}} = aG_{N}{\cal M}\left(r^2-\frac{a^2}{4}\right)
-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2 ar

\sigma(r)|_{\theta= \frac{\pi}{3}} = \left(r^2+\frac{a^2}{4}\right)^\frac{3}{2}
\left[\left(r^2+\frac{a^2}{4}\right)-\Pi_{\alpha}\right]

Variation of spin precession frequency with radial coordinates for various values
of spin parameter could be seen from the Fig. (<ref>).

Examples of the variation  of $\Omega_{LT}$ versus $r$ for $\theta=\frac{\pi}{3}$ in
KMOG with variation of MOG parameter and  spin parameter. Here  $\Omega=0$. The first figure describes the variation
of $\Omega_{LT}$  with $r$ for non-extremal BH, extremal BH and NS without MOG parameter.
The rest of the figure describes the variation  of $\Omega_{LT}$  with $r$ for non-extremal BH,
extremal BH and NS with MOG parameter.

§.§ Behaviour of $\vec{\Omega}_{LT}$ at $\theta=\frac{\pi}{2}$

On the equatorial plane the precession frequency vector is given by

\begin{eqnarray}
\vec{\Omega}_{LT}|_{\theta=\frac{\pi}{2}} &=&
\frac{\mu(r)|_{\theta=\frac{\pi}{2}}}{\sigma(r)|_{\theta=\frac{\pi}{2}}}\hat{\theta}
\end{eqnarray}

The magnitude of this vector is then

\begin{eqnarray}
\Omega_{LT}|_{\theta=\frac{\pi}{2}} &=& \frac{\mu(r)|_{\theta=\frac{\pi}{2}}}{\sigma(r)|_{\theta=\frac{\pi}{2}}}
\end{eqnarray}

\begin{eqnarray}
\mu(r)|_{\theta=\frac{\pi}{2}} &=& ar\left(G_{N}{\cal M} r-\frac{\alpha}{1+\alpha} G_{N}^2 {\cal M}^2\right)\\
\sigma(r)|_{\theta=\frac{\pi}{2}} &=& r^3\left(r^2-\Pi_{\alpha}\right)
\end{eqnarray}

Variation of spin precession frequency with radial coordinates for various values of spin parameter
could be seen from the Fig. (<ref>).

Examples of the variation  of $\Omega_{LT}$ versus $r$ for $\theta=\frac{\pi}{2}$ in
KMOG with variation of MOG parameter and  spin parameter. Here  $\Omega=0$. The first figure
describes the variation  of $\Omega_{LT}$  with $r$ for non-extremal BH, extremal BH and NS
without MOG parameter. The second and third figure describes the variation  of $\Omega_{LT}$
with $r$ for non-extremal BH, extremal BH and NS with MOG parameter.

Using these plots, one can differentiate between non-extremal BH, extremal BH and NS in KMOG spacetime
and Kerr spacetime.  It could be easily say that the presence of the MOG parameter drastically  changes
the geometry of the BH spacetime. More appropriately, one could say that
the presence of the MOG parameter enoromously changes the shape of the LT frequency diagram in
contrast with the zero MOG parameter  LT frequency diagram. Futhermore, it is observed that in the
NS case the geometric structure drastically different from BH spacetime.

§.§ Lense-Thirring Precession in  Extremal KMOG spacetime

It is very crucial to study the LT precession in case of extremal KMOG BH
in comparison to extremal Kerr BH.  This is because the extremal BH has several important features.
One crucial feature is that it has no Hawking temperature i.e. $T_{H}=0$. It has also no bifurcaton
2-sphere. Moreover, it has no trapped surface. Whereas its near-extremal counterpart possecess all
the said features. Extremal BHs also playing  a major role both in  string theory and quantum gravity.
They have used to count the string states in string theory while in quantum gravity they have used
as a theoretical toy. In supersymmetric theory, extremal BHs satisfied the BPS (Bogomolnyi-Prasad-Sommerfield)
bound and they are invariant under several super charges. They are also stable and do not radiate
Hawking radiation. Their entropy was calculated in string theory [44].
The other definition of extremal BH is that it  is a BH when two horizons are coincident.
The extremal limit of KMOG BH is defined by $a=\frac{G_{N}{\cal M}}{\sqrt{1+\alpha}}$ or
$J=\frac{G_{N}^2{\cal M}^2}{\sqrt{1+\alpha}}$. Thus one gets the extremal horizon is at
$r_{ex}=G_{N}{\cal M}=a\sqrt{1+\alpha}$. So far we have not written the exact expression of
LT precession frequency vector for extremal KMOG BH now we should write this expression as

\vec \Omega_{LT} = \frac{G_{N}{\cal M}}{\sqrt{1+\alpha}} \times

\begin{eqnarray}
\frac{\left[ \Pi_{\alpha}(r-G_{N}{\cal M})\cos\theta~\hat{r}
+G_{N}{\cal M} \left(r^2-\frac{\alpha}{1+\alpha} G_{N}{\cal M} r-\frac{G_{N}^2{\cal M}^2}{1+\alpha}\cos^2\theta \right)
\sin\theta~\hat{\theta}\right]}
{\left(r^2+\frac{G_{N}^2{\cal M}^2}{1+\alpha}\cos^2\theta\right)^\frac{3}{2}
\left(r^2-\Pi_{\alpha}+\frac{G_{N}^2{\cal M}^2}{1+\alpha}\cos^2\theta\right)}
\end{eqnarray}

Taking magnitude of this vector one obtains the LT frequency for extremal KMOG BH as

\Omega_{LT} (r,\theta,\alpha) = \frac{G_{N}{\cal M}}{\sqrt{1+\alpha}}
\times

\begin{eqnarray}
\frac{\left[\Pi_{\alpha}^2(r-G_{N}{\cal M})^2 \cos^2\theta +
G_{N}^2{\cal M}^2\left(r^2-\frac{\alpha}{1+\alpha} G_{N}{\cal M} r-\frac{G_{N}^2{\cal M}^2}{1+\alpha}\cos^2\theta\right)^2
\sin^2\theta\right]^{\frac{1}{2}}}
{\left(r^2+\frac{G_{N}^2{\cal M}^2}{1+\alpha}\cos^2\theta\right)^\frac{3}{2}
\left(r^2-\Pi_{\alpha}+\frac{G_{N}^2{\cal M}^2}{1+\alpha}\cos^2\theta\right) }
\end{eqnarray}

Now we will compute the LT frequency of extremal KMOG BH for various angles starting from polar region to
equatorial plane and the variation of the said frequency could be observed from the plot.

Case I:

First we take the value of $\theta=0$, then one gets the LT frequency as

\begin{eqnarray}
\Omega_{LT}  =  \frac{G_{N} {\cal M}}{\sqrt{1+\alpha}} \frac{\Pi_{\alpha}}
{\left(r^2+\frac{G_{N}^2{\cal M}^2}{1+\alpha}\right)^\frac{3}{2}\left(r-G_{N}{\cal M}\right)}
= f(r, {\cal M}, \alpha)
\end{eqnarray}

It follows that the frequency is a function of  $f(r, {\cal M}, \alpha)$ while for Kerr BH it is a function
of $f(r, {\cal M})$ only. It also should be noted that at the extremal horizon the LT frequency diverges both
for extremal KMOG and extremal Kerr BH. Variation of Lense-Thirring frequency of extremal KMOG BH could be
observed from  Fig. (<ref>) for different spin values.

The first figure
describes the variation  of $\Omega_{LT}$  with $r$ for extremal BH in KMOG
without MOG parameter. The second and third figure describes the variation
of $\Omega_{LT}$  with $r$ for extremal BH with MOG parameter. Here we set $\theta=0$.

Case II:

For $\theta=\frac{\pi}{6}$, the LT frequency is derived to be

\Omega_{LT} (r,\frac{\pi}{6}) = \frac{G_{N} {\cal M}}{2\sqrt{1+\alpha}}  \times

\begin{eqnarray}
\frac{\left[3 \Pi_{\alpha}^2(r-G_{N}{\cal M})^2
+ G_{N}^2{\cal M}^2\left(r^2-\frac{\alpha}{1+\alpha} G_{N}{\cal M} r-\frac{3 G_{N}^2{\cal M}^2}{4(1+\alpha)}\right)^2
\right]^{\frac{1}{2}}}{\left[r^2+\frac{3G_{N}^2{\cal M}^2}{4(1+\alpha)}\right]^\frac{3}{2}
\left[r^2-\Pi_{\alpha}+\frac{3G_{N}^2{\cal M}^2}{4(1+\alpha)}\right]}
\end{eqnarray}

The first figure
describes the variation  of $\Omega_{LT}$  with $r$ for extremal BH in KMOG
without MOG parameter. The second and third figure describes the variation
of $\Omega_{LT}$  with $r$ for extremal BH in KMOG with MOG parameter.
Here we set $\theta=\frac{\pi}{6}$.

Case III:

For $\theta=\frac{\pi}{4}$, the LT frequency is computed to be

\Omega_{LT} (r, \frac{\pi}{4}) = \frac{G_{N} {\cal M}}{\sqrt{2(1+\alpha)}}   \times

\begin{eqnarray}
\frac{\left[ \Pi_{\alpha}^2(r-G_{N}{\cal M})^2
+ G_{N}^2{\cal M}^2\left(r^2-\frac{\alpha}{1+\alpha} G_{N}{\cal M} r-\frac{ G_{N}^2{\cal M}^2}{2(1+\alpha)}\right)^2
\right]^{\frac{1}{2}}}{\left[r^2+\frac{G_{N}^2{\cal M}^2}{2(1+\alpha)}\right]^\frac{3}{2}
\left[r^2-\Pi_{\alpha}+\frac{G_{N}^2{\cal M}^2}{2(1+\alpha)}\right]}
\end{eqnarray}

The first figure describes the variation  of $\Omega_{LT}$  with $r$
for extremal BH in KMOG without MOG parameter. The second and third figure describes the variation
of $\Omega_{LT}$  with $r$ for extremal BH in KMOG with MOG parameter. Here we set $\theta=\frac{\pi}{4}$.

Case IV:

For $\theta=\frac{\pi}{3}$, the LT frequency is derived to be

\Omega_{LT} (r, \frac{\pi}{3}) = \frac{G_{N} {\cal M}}{2\sqrt{1+\alpha}} \times

\begin{eqnarray}
\frac{\left[\Pi_{\alpha}^2 (r-G_{N}{\cal M})^2
+ 3G_{N}^2{\cal M}^2\left(r^2-\frac{\alpha}{1+\alpha} G_{N}{\cal M} r-\frac{ G_{N}^2{\cal M}^2}{4(1+\alpha)}\right)^2
\right]^{\frac{1}{2}}}{\left[r^2+\frac{G_{N}^2{\cal M}^2}{4(1+\alpha)}\right]^\frac{3}{2}
\left[r^2-\Pi_{\alpha}+\frac{G_{N}^2{\cal M}^2}{4(1+\alpha)}\right]}
\end{eqnarray}

The first figure describes the variation  of $\Omega_{LT}$  with $r$
for extremal BH in KMOG without MOG parameter. The second and third figure describes the variation
of $\Omega_{LT}$  with $r$ for extremal BH in KMOG with MOG parameter.
Here we set $\theta=\frac{\pi}{3}$.

Case V:

Finally, for $\theta=\frac{\pi}{2}$, the LT frequency is

\begin{eqnarray}
\Omega_{LT}  =  \frac{G_{N}^2 {\cal M}^2}{\sqrt{1+\alpha}} \frac{\left(r-\frac{\alpha}{1+\alpha} G_{N}{\cal M} \right)}
\end{eqnarray}

It should be noted that in each cases the LT frequency reduces to Kerr BH for $\alpha=0$. The variation
of these frequencies could be observed from following Fig.  (<ref>).

The first figure describes the variation  of $\Omega_{LT}$  with $r$
for extremal BH in KMOG without MOG parameter. The second and third figure describes the variation
of $\Omega_{LT}$  with $r$ for extremal BH in KMOG with MOG parameter. Here we set $\theta=\frac{\pi}{2}$.

§ ACCRETION DISK PROPERTIES IN KMOG SPACETIME

To differentiate non-extremal BH spacetime, extremal BH spacetime and NS, it is essential to study
accretion disk physics  around a stationary, axisymmetric spacetime. To distinguish three geometries of
the compact object, also it is  essential to study three fundamental frequencies of accretion disk. This could
be possible only by studying the geodesic motion of test particle around the compact objects. Again to study
the physics of accretion disk, one should  compute the stable circular orbit of the said compact object. The main
stable orbit is inner-most stable circular orbit (ISCO) or last stable circular orbit (LSCO). This ISCO can
help us to distinguish three geometries. In Kerr geometry, the ISCO radii depends on spin parameter  ($a$) while
in KMOG geometry, it depends on both spin parameter ($a_{\alpha}$) and MOG parameter ($\alpha$). We previously
noticed that in KMOG geometry, the spin parameter decreases as MOG value increases (See Fig. (<ref>)).

Three fundamenta frequencies which are very important
for accretion disk physics of KMOG BH, namely the Keplerian frequency ($\Omega_{\phi}$), the radial
epicyclic frequency ($\Omega_{r}$) and the vertical epicyclic frequency ($\Omega_{\theta}$).  They
can easily be derived using the formulae (<ref>,<ref>,<ref>) given in
Appendix: [See also [35]]
[Note that in Ref. [35] the term $G_{N}$ is missing, here we have included it in
each frequencies formulae]

\begin{eqnarray}
\Omega_{\phi} &=& \pm \frac{\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r}}{r^2\pm a \sqrt{\Pi_{\alpha}-G_{N}{\cal M}r}} \label{ekm}\\
\Omega_{r} &=& \frac{\sqrt{G_{N}{\cal M}r \Delta-4\left(\Pi_{\alpha}-G_{N}{\cal M}r\right)
\left(\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r} \mp a \right)^2}}
{r \left(r^2 \pm a\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r} \right)}~\label{ekm1}\\
\Omega_{\theta} &=& \frac{\sqrt{r^2\left(\Pi_{\alpha}-G_{N}{\cal M}r\right) \mp 2 a \Pi_{\alpha}\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r}
+a^2\left(2\Pi_{\alpha}-G_{N}{\cal M}r \right)}}{r \left(r^2 \pm a\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r} \right)}~\label{ekm2}\\
\end{eqnarray}

It is important to note that the upper sign for corotating (direct) orbit and lower sign for
counter-rotating (retrograde) orbit. When MOG value vanishes, one obtains the epicyclic frequencies of
Kerr BH.

The variation of these frequencies for three compact objects namely
non-extremal BH, extremal BH and NS could be seen from the following diagram
[Fig.  (<ref>), Fig.  (<ref>), Fig.  (<ref>)].
From first figure when $\alpha=0$ means that for Kerr BH, it is observed that the  Keplerian frequency
decreases with increasing the radial value starting from maximum value for non-extremal BH ($a=0.5$).
As spin value increases the value of Keplerian frequency decreases. While in second figure where the
MOG parameter is present then the picture is quite different from the former. Here the Keplerian
frequency has gone through a peak for different spin values.

The figure depicts the variation  of $\Omega_{\phi}$  with $r$ for different MOG parameter and
spin parameter. Each figure demonstrates the difference between non-extremal BH, extremal BH and NS. Without MOG
there is no peak while with MOG there exists a peak value in Keplerian frequency.

The figure depicts the variation  of $\Omega_{r}$  with $r$ for different MOG parameter and
spin parameter. Each figure shows the difference between three compact objects namely,  non-extremal BH,
extremal BH and NS.

The figure depicts the variation  of $\Omega_{\theta}$  with $r$ for different MOG parameter and
spin parameter. Each figure characterizes the difference between non-extremal BH, extremal BH and NS.

The other two important frequency namely, the periastron
precession frequency ($\Omega_{per}$) and nodal precession frequency ($\Omega_{nod}$) could
be defined as

\begin{eqnarray}
\Omega_{per} &=& \Omega_{\phi}-\Omega_{r}\\
\Omega_{nod} &=& \Omega_{\phi}-\Omega_{\theta}~\label{ekm3}
\end{eqnarray}

All these frequencies above that we have defined are related to the precession of the orbit and orbital plane.
Periastron frequency occurs due to the precession of the orbit while nodal precession frequency occurs due to
the precession of orbital plane. Sometimes the nodal precession frequency is called as orbital planer precession
frequency or Lense-Thirring precession frequency [2]. The radial variation of these two frequencies could
be observed from Fig.  (<ref>) and Fig.  (<ref>).

The figure depicts the variation  of $\Omega_{per}$  with $r$ for different MOG parameter and
spin parameter. Each figure shows the difference between non-extremal BH, extremal BH and NS.

The figure depicts the variation  of $\Omega_{nod}$  with $r$ for different MOG parameter and
spin parameter. Each figure classifies the difference between non-extremal BH, extremal BH and NS.

The figure depicts the variation  of $\frac{\Omega_{r}}{\Omega_{\theta}}$ and
$\frac{\Omega_{\theta}}{\Omega_{r}}$  with $r$ for different MOG parameter and
spin parameter. Each figure depicts the difference between non-extremal BH,
extremal BH and NS.

The figure depicts the variation  of
$\frac{\Omega_{r}}{\Omega_{\phi}}$ with $r$ for
different MOG parameter and spin parameter.
Each figure depicts the difference between
non-extremal BH, extremal BH and NS.

The figure depicts the variation  of
$\frac{\Omega_{r}}{\Omega_{\phi}}$ with $r$ for
different MOG parameter and spin parameter.
Each figure depicts the difference between
non-extremal BH, extremal BH and NS.

The fact that $\Omega_{r}^2\geq 0$ and $\Omega_{\theta}^2\geq 0$  determined the stability of
circular orbits. From the first condition one can compute radii of ISCO. It is well known that
ISCO of Kerr BH is located at [48]

\begin{eqnarray}
\frac{r_{ISCO}}{\cal M} &=& 3+z_{2}\mp \sqrt{(3-z_{1})(3+z_{1}+2z_{2})}
\end{eqnarray}

\begin{eqnarray}
z_{1} &=& 1+\left(1-\frac{a^2}{{\cal M}^2}\right)^{1/3} \left[(1-\frac{a}{{\cal M}})^{1/3}+(1+\frac{a}{{\cal M}})^{1/3}\right]\\
z_{2} &=& \sqrt{3\frac{a^2}{{\cal M}^2}+z_{1}^2}
\end{eqnarray}

Here, upper sign indicates  direct orbit and lower sign indicates retrograde orbit.
For extremal Kerr BH, the ISCO is situated at $r_{ISCO}={\cal M}$ for direct orbit while
$r_{ISCO}=9{\cal M}$ for retrograde orbit [48]. The non-negativeness of $\Omega_{\theta}$
implies that the geodesic motion is stable under small oscillations in the vertical
direction. While in KMOG BH, the ISCO radii can be calculated via the equation $\Omega_{r}^2=0$:

G_{N}{\cal M}r \Delta-4\left(\Pi_{\alpha}-G_{N}{\cal M}r\right)
\left(\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r} \mp a \right)^2 = 0

In spherically symmetric spacetime where the value of spin parameter $a=0$ means that
$\Omega_{\phi}=\Omega_{\theta}$, which implies that the Lense-Thirring precession is
absent  while in axisymmetric spacetime $\Omega_{\phi} \neq \Omega_{\theta}$. This
will be vanish for a particular value of $r=r_{0}$ i.e.

\Omega_{nod}|_{r=r_{0}} =0

which implies that

4\Pi_{\alpha}^2 \frac{(\Pi_{\alpha}-G_{N}{\cal M}r)}{(2\Pi_{\alpha}-G_{N}{\cal M}r)^2}|_{r=r_{0}}=a^2

when MOG parameter vanishes it reduces to

\frac{16}{9} \frac{r_{0}}{M} =a^2

We can differentiate three compact objects via ratio of two epicyclic frequencies
(plot for direct orbit only) which is defined to be as follows

\frac{\Omega_{r}}{\Omega_{\theta}} =\frac{\sqrt{G_{N}{\cal M}r \Delta-4\left(\Pi_{\alpha}-G_{N}{\cal M}r\right)
\left(\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r} \mp a \right)^2}}
{\sqrt{r^2\left(\Pi_{\alpha}-G_{N}{\cal M}r\right) \mp 2 a \Pi_{\alpha}\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r}
+a^2\left(2\Pi_{\alpha}-G_{N}{\cal M}r \right)}}

\frac{\Omega_{r}}{\Omega_{\phi}} =\pm \frac{\sqrt{G_{N}{\cal M}r \Delta-4\left(\Pi_{\alpha}-G_{N}{\cal M}r\right)
\left(\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r} \mp a \right)^2}}{r\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r}}

\frac{\Omega_{\theta}}{\Omega_{\phi}} =\pm \frac{\sqrt{r^2\left(\Pi_{\alpha}-G_{N}{\cal M}r\right)
\mp 2 a \Pi_{\alpha}\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r}
+a^2\left(2\Pi_{\alpha}-G_{N}{\cal M}r \right)}} {r\sqrt{\Pi_{\alpha}-G_{N}{\cal M}r}}

The variation of these ratio may be seen from the following diagram [(<ref>),
(<ref>), (<ref>)].

§ DISCUSSION

We performed a detailed analysis of the inertial frame-dragging effect of
stationary, axisymmetric KMOG BH.
Specifically, we computed the generalized spin precession of a test gyroscope
due to frame-dragging effect when it is placed in the domain of the BH as well as NS.
By computing
this frequency, we distinguished the behaviour of three astrophysical compact objects
namely non-extremal BH, extremal BH and NS. We showed in principle a clear distinction
between these three compact objects visually. We also compared this result with the Kerr
BH. We studied different features in MOG by using spin precession versus radial profile.
In each spin precession vs. radial diagram, we observed distinguished features of
three compact objects for various spin limits.

Perhaps the most promising result we obtained is that the presence of the MOG parameter deformed
the geometric construction of different essential parameters like
event horizon ($r_{+}$), Cauchy horizon ($r_{-}$), outer ergosphere ($r^{+}_{e}$),
inner ergosphere ($r^{-}_{e}$), generalized spin frequency ($\Omega_{p}$),
LT frequency ($\Omega_{LT}$) etc.  in contrast to zero MOG parameter.

We also investigated the generalized spin precession frequency for various angular
limits: $\theta=0$, $\theta=\frac{\pi}{6}$, $\theta=\frac{\pi}{4}$, $\theta=\frac{\pi}{3}$ and
finally $\theta=\frac{\pi}{2}$. Moreover, we  studied the generalized spin frequency for ring
singularity. Then we studied the frame-dragging effect with vanishing angular velocity.
This means that we derived the LT frequency only by ommiting the other frequencies. We showed
that the LT frequency is affected by the MOG parameter. It is clearly  observed
from LT frequency vs. radial diagram. For vanishing $\Omega=0$, we also studied the LT
frequency for angular values: $\theta=0$, $\theta=\frac{\pi}{6}$, $\theta=\frac{\pi}{4}$,
$\theta=\frac{\pi}{3}$ and $\theta=\frac{\pi}{2}$. Each diagram clearly showed the distinction
between three compact objects namely, non-extremal BH, extremal BH and NS.

Moreover, we studied the LT frequency particularly for extremal KMOG BH in compared with extremal
Kerr BH. In this case we also examined the LT frequency for various angular values. From each diagram
of $\Omega_{LT}$ vs. $r$, we observed a diverging value of LT frequency at $r=G_{N}M$. It has been also
observed that the presence of MOG parameter significantly changes the geometry of the BH spacetime
in contrast to zero MOG parameter.

Finally, we studied the accretion disk properties by computing three fundamental epicyclic frequencies,
namely the Keplerian frequency, radial epicyclic frequency and vertical epicyclic frequency. We also studied
periastron frequency and nodal frequency. Using these properties of epicyclic frequencies, we differentiated
three compact objects. From different frequency diagram, it should be clearly observed
that three geometries are distinct. Furthermore, we calculated the ratio $\frac{\Omega_{r}}{\Omega_{\phi}}$,
$\frac{\Omega_{r}}{\Omega_{\theta}}$ and $\frac{\Omega_{\theta}}{\Omega_{\phi}}$. Using these features, we
differentiated three compact objects.

In summary, to test the strong gravity in MOG and to distinguish three compact objects
namely the non-extremal BH, extremal BH and NS we have computed different essential
parameters i. e. $\Omega_{p}$, $\Omega_{LT}$, $\Omega_{r}$, $\Omega_{\theta}$,
$\Omega_{\phi}$, $\Omega_{per}$, $\Omega_{nod}$, $\frac{\Omega_{r}}{\Omega_{\phi}}$,
$\frac{\Omega_{r}}{\Omega_{\theta}}$ and $\frac{\Omega_{\theta}}{\Omega_{\phi}}$.
By using the features of these parameters, one can distinguish
between BH (non-extremal & extremal) and NS.

§ ACKNOWLEDGEMENTS

I would like to thank Dr. Chandrachur Chakraborty of KIAA, China for helpful suggestions.

§  EPICYCLIC FREQUENCIES IN A GENERAL AXISYMMETRIC AND STATIONARY SPACETIME

We consider a general stationary and axisymmetric spacetime as

\begin{eqnarray}
ds^2=g_{tt}\,dt^2+g_{rr}\,dr^2+g_{\theta\theta}\,d\theta^2+g_{\phi\phi}\,d\phi^2+2g_{t\phi}\,d\phi dt, \label{a1}
\end{eqnarray}

where $g_{\mu\nu}=g_{\mu\nu}(r, \theta)$. For this spacetime the proper angular momentum  ($l$) of
a test particle can be defined as

\begin{eqnarray}
l=-\frac{g_{t\phi}+\Omega_{\phi} g_{\phi\phi}}{g_{tt}+\Omega_{\phi} g_{t\phi}},
\end{eqnarray}

where, $\Omega_{\phi}$ is the orbital frequency of a test particle. Now the $\Omega_{\phi}$
can be defined as

\begin{eqnarray}
\Omega_{\phi} =\frac{\dot{\phi}}{\dot{t}}=\frac{(\frac{d\phi}{d\tau})}{(\frac{dt}{d\tau})}
=\frac{-\partial_{r}g_{t\phi}\pm \sqrt{(\partial_{r}g_{t\phi})^2-(\partial_{r}g_{tt})(\partial_{r}g_{\phi\phi})}}
\label{a2}
\end{eqnarray}

The upper sign is for corotating orbit and the lower sign is for counterrotating orbit.
The general expressions for computing the radial ($\Omega_{r}$) and vertical
($\Omega_{\theta}$) epicyclic frequencies are [45, 55]

\begin{eqnarray}
\Omega_{r}^2 &=& \frac{(g_{tt}+\Omega_{\phi}g_{t\phi})^2}{2~g_{rr}}~\partial_{r}^2~U\nonumber\\
&=& \frac{(g_{tt}+\Omega_{\phi}g_{t\phi})^2}{2~g_{rr}}
\left[\partial_{r}^2\left(\frac{g_{\phi\phi}}{Y}\right) +2l~\partial_{r}^2\left(\frac{g_{t\phi}}{Y}\right)
+l^2~\partial_{r}^2\left(\frac{g_{tt}}{Y}\right)\right]|_{r=const.,~\theta=\frac{\pi}{2}}.~ \label{a3}
\end{eqnarray}

\begin{eqnarray}
\Omega_{\theta}^2 &=& \frac{(g_{tt}+\Omega_{\phi}g_{t\phi})^2}{2~g_{\theta\theta}}~\partial_{\theta}^2~U \nonumber\\
&=& \frac{(g_{tt}+\Omega_{\phi}g_{t\phi})^2}{2~g_{\theta\theta}}\left[\partial_{\theta}^2\left(\frac{g_{\phi\phi}}{Y}\right)
\end{eqnarray}

respectively and $Y$ can be defined as

\begin{eqnarray}
Y=g_{tt}g_{\phi\phi}-g_{t\phi}^2 .
\end{eqnarray}

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\begin{equation}
\begin{split}

\end{split}
\end{equation}