# Lyapunov exponents and geodesic stability of Schwarzschild black hole in the
non-commutative gauge theory of gravity

Abdellah Touati<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Faculty of Sciences of Matter, University of Batna-1,
Batna 05000, Algeria  Zaim Slimane<EMAIL_ADDRESS>Department of Physics,
Faculty of Sciences of Matter, University of Batna-1, Batna 05000, Algeria

###### Abstract

In this paper, we study the stability of geodesic motion for both massive and
massless particles using Lyapunov exponents in the non-commutative (NC)
Schwarzschild black hole (SBH) via the gauge theory of gravity. As a first
step, we investigate the both time-like and null radial motion of particles,
the mean result in NC geometry shows that the particles take infinity proper
time to reach the NC singularity (infinite time affine parameter framework for
photons). The proper/coordinate time of Lyapunov exponents and their ratio of
time-like geodesic for the circular motion of this black hole shows a new
behavior, which describes a new range of stable circular orbits between
unstable ones. Then we analyze the circular motion of photons, where the
result shows a new photon sphere near the event horizon which is not allowed
in the commutative case, and the Lyapunov exponent is expressed in this
geometry, where this confirms the instability of the outer photon sphere and
the stability of the inner one. Finally, we check the effect of the non-
commutativity on the shadow radius of the SBH, where we show the similarity
between the non-commutativity and the black hole mass.

Non-commutative geometry, Gauge gravity, Schwarzschild space-time, Geodesic
equation, Black hole shadow.

##  I Introduction

Studying the geodesic motion of a test body around a compact object has great
importance in astrophysics to understand the physics of compact objects, such
as neutron stars, black holes,..etc. The motion of particles around the black
hole reveals a lot to understand their nature and geometrical propriety, such
as the motion of stars around the galactic center for example the
S-Sagittarius A* in the Milky Way galaxy Eckart ; gillessen1 ; Ri ; peissker .
There are two interesting kinds of geodesic motion in the study of black
holes, which are radial and circular geodesic. There are many references that
studied in detail the geodesic movement around black holes, such as Chandr1 ;
gibbons1 ; islam1 ; jaklit ; stuchli ; kran1 ; kran2 ; cruz ; hackmann1 ;
kagraman ; hackmann2 ; hackmann3 ; levin ; grunau ; belbrun ; barack ;
pugliese1 ; pugliese2 ; virb1 ; virb2 ; pradhan ; olivares ; villanueva ;
pugl1 ; pugl2 . Also, the classification of stable and unstable orbits is
interesting to describe the geometrical proprieties of spacetime around
compact objects. Among the various methods for analyzing the stability of
geodesics, the Lyapunov exponent Lyapunov1  has a great application in this
context, in which this exponent is considered as a bridge between the non-
linear general relativity and the non-linear dynamics. The Lyapunov exponent
measures the average rate of separation between two nearby geodesics in phase
space, in which the unstable geodesic can be expressed by the real Lyapunov
exponent and the stable and marginal stable one are described by the imaginary
and zero Lyapunov exponent respectively geostab12 ; geostab11 ; geostab3 ;
geostab8 ; geostab9 ; geostab4 ; geostab10 ; geostab6 ; geostab1 ; geostab2 ;
geostab7 .

However, our interest is to investigate the geodesic stability via the
Lyapunov exponent in NC spacetime. The NC theory is mainly motivated by string
theory, being a limit in the presence of a background field hawking1 ;
hawking2 ; hod ; vaz ; harms ; gour ; zaim1 ; mebarki1 . The main idea results
in the concept of quantifying spacetime leading to quantifying gravity, these
quantum gravity effects can be neglected in low energy limits, but in the
strong gravitational field of a black hole, one has to consider these effects.
In the NC spacetime, the coordinate operators satisfy the following
commutation relations:

$[x^{\mu},x^{\nu}]=i\Theta^{\mu\nu}$ (1)

where $\Theta^{\mu\nu}$ is an anti-symmetric real matrix that determines the
fundamental cell discretization of space-time. This commutation relation
between coordinates themselves leads to the modification of Heisenberg
uncertainty relations in such a way that prevents one from measuring positions
with better accuracy than the Planck length.

$\Delta\hat{x}^{\mu}\Delta\hat{x}^{\nu}\geq\frac{1}{2}|\Theta^{\mu\nu}|$ (2)

In the NC theory, the ordinary product has changed to the star product (Moyal
product) ”$*$” between two arbitrary functions $f(x)$ and $g(x)$ defined over
this space-time:

$(f*g)(x)=f(x)e^{\frac{i}{2}\Theta^{\mu\nu}\overleftarrow{\partial_{\mu}}\overrightarrow{\partial_{\nu}}}g(x)$
(3)

Recently there has been interest in studies that investigated the geodesic
motion affected by the non-commutativity nozari1 ; nozari2 ; kuniyal ; mirza ;
rome1 ; iftikhar ; ulh1 ; larranaga ; nozari ; nicolini1 ; bhar ; rahaman ;
abdellah1 , and its stability using Lyapunov exponent geostab14 ; geostab13 ;
geostab5 . Here one uses a gauge gravity theory in NC spacetime with star
products and Seiberg-Witten (SW) map seiberg1 . The NC gauge gravity is
considered a theory beyond general relativity, which is an extended theory to
the NC spacetime. In the NC gauge theory of gravity, the transformation of the
action under the NC canonical transformations of NC fields via the SW map is
invariant action1 ; action2 ; action3 . Our work will be in this context, to
write the effective potential and Lyapunov exponent that arises from the
deformed metric tensor which has been obtained using the star product between
tetrads fields and the SW map. In addition, we hope to obtain corrections in
terms of the NC parameter $\Theta$ for each of the effective potentials and
the Lyapunov exponent and we also discussed the issue of the stability of
circular orbits in the NC Schwarzschild geometry. The paper is organized as
follows. In Sect. 2 we present the NC corrections to the metric field using
star product and SW maps. In Sect. 3 we calculate the NC geodesic equation and
we also obtain NC effective potentials up to the second-order in $\Theta$. In
Sect. 4 we investigate the radial and circular motion of massive and massless
particles in this geometry, then we investigate the stability of orbits using
the Lyapunov exponent. We then present the NC correction to the shadow radius
of SBH. In the last section, we present our concluding remarks.

##  II Non-commutative corrections for Schwarzschild black hole

In our previous works abdellah1 ; abdellah2 , we used the tetrad formalism and
both the star $\ast-$product and the SW map to construct a non-commutative
gauge theory for a static metric with a spherical symmetry. One can use a
perturbation form for the SW map to describe the deformed tetrad fields
$\hat{e}^{a}_{\mu}(x,\Theta)$ as a development in the power of $\Theta$ up to
the second-order, which can be obtained by following the same approach in Ref.
cham1 :

$\displaystyle\hat{e}_{\mu}^{a}=\,$ $\displaystyle
e_{\mu}^{a}-\frac{i}{4}\Theta^{\nu\rho}[\omega^{ac}_{\nu}\partial_{\rho}e^{d}_{\mu}+(\partial_{\rho}\omega^{ac}_{\mu}+R^{ac}_{\rho\mu})e^{d}_{\nu}]\eta_{cd}$
$\displaystyle+\frac{1}{32}\Theta^{\nu\rho}\Theta^{\lambda\tau}\left[2\\{R_{\tau\nu},R_{\mu\rho}\\}^{ab}e^{c}_{\lambda}-\omega^{ab}_{\lambda}(D_{\rho}R_{\tau\nu}^{cd}+\partial_{\rho}R_{\tau\nu}^{cd})e^{m}_{\nu}\eta_{dm}\right.$
$\displaystyle\left.-\\{\omega_{\nu},(D_{\rho}R_{\tau\nu}+\partial_{\rho}R_{\tau\nu})\\}^{ab}e^{c}_{\lambda}-\partial_{\tau}\\{\omega_{\nu},(\partial_{\rho}\omega_{\mu}+R_{\rho\mu})\\}^{ab}e^{c}_{\lambda}\right.$
$\displaystyle\left.-\omega^{ab}_{\lambda}\left(\omega^{cd}_{\nu}\partial_{\rho}e^{m}_{\mu}+\left(\partial_{\rho}\omega_{\mu}^{cd}+R_{\rho\mu}^{cd}\right)e^{m}_{\nu}\right)\eta_{dm}+2\partial_{\nu}\omega_{\lambda}^{ab}\partial_{\rho}\partial_{\tau}e^{c}_{\lambda}\right.$
$\displaystyle\left.-2\partial_{\rho}\left(\partial_{\tau}\omega_{\mu}^{ab}+R_{\tau\mu}^{ab}\right)\partial_{\nu}e^{c}_{\lambda}-\\{\omega_{\nu},(\partial_{\rho}\omega_{\lambda}+R_{\rho\lambda})\\}^{ab}\partial_{\tau}e^{c}_{\mu}\right.$
$\displaystyle\left.-\left(\partial_{\tau}\omega_{\mu}+R_{\tau\mu}\right)\left(\omega^{cd}_{\nu}\partial_{\rho}e^{m}_{\lambda}+\left((\partial_{\rho}\omega_{\lambda}+R_{\rho\lambda})\right)e^{m}_{\nu}\right)\eta_{dm}\right]\eta_{cb}+\mathcal{O}\left(\Theta^{3}\right),$
(4)

where $\hat{e}_{a}^{\mu}$ and $\omega^{ab}_{\mu}$ are the tetrad field and the
spin connection (gauge field), and:

$\displaystyle\\{\alpha,\beta\\}^{ab}=\left(\alpha^{ac}\beta^{db}+\beta^{ac}\alpha^{db}\right)\eta_{cd},\quad$
$\displaystyle[\alpha,\beta]^{ab}=\left(\alpha^{ac}\beta^{db}-\beta^{ac}\alpha^{db}\right)\eta_{cd}$
(5) $\displaystyle
D_{\mu}R_{\rho\sigma}^{ab}=\partial_{\mu}R^{ab}_{\rho\sigma}+$
$\displaystyle\left(\omega_{\mu}^{ac}R^{db}_{\rho\sigma}+\omega_{\mu}^{bc}R^{da}_{\rho\sigma}\right)$
(6)

where the $\hat{e}_{a}^{\mu}$ is the of the vierbein $\hat{e}_{\mu}^{a}$
defined as:

$\hat{e}_{\mu}^{b}\hat{e}_{a}^{\mu}=\delta_{a}^{b},\quad\hat{e}_{\mu}^{a}\hat{e}_{a}^{\nu}=\delta_{\mu}^{\nu}\,.$
(7)

In the following, we consider a symmetric metric $\hat{g}_{\mu\nu}$, so that:

$\hat{g}_{\mu\nu}=\frac{1}{2}(\hat{e}_{\mu}^{b}\ast\hat{e}_{\nu
b}+\hat{e}_{\nu}^{b}\ast\hat{e}_{\mu b})\,.$ (8)

To compute the deformed metric $\hat{g}_{\mu\nu}$, we choose the following NC
anti-symmetric matrix $\Theta^{\mu\nu}$:

$\Theta^{\mu\nu}=\left(\begin{matrix}0&0&0&0\\\ 0&0&0&\Theta\\\ 0&0&0&0\\\
0&-\Theta&0&0\end{matrix}\right),\qquad\mu,\nu=0,1,2,3$ (9)

We follow the same steps outlined in Ref. abdellah1 , we choose the following
general tetrads field:

$\displaystyle\underline{e}_{\mu}^{0}$
$\displaystyle=\left(\begin{array}[]{cccc}\left(1-\frac{2m}{r}\right)^{\frac{1}{2}},&0,&0,&0\end{array}\right),$
(11) $\displaystyle\underline{e}_{\mu}^{1}$
$\displaystyle=\left(\begin{array}[]{cccc}0&\left(1-\frac{2m}{r}\right)^{-\frac{1}{2}}sin\theta
cos\phi&rcos\theta cos\phi&-rsin\theta sin\phi\end{array}\right),$ (13)
$\displaystyle\underline{e}_{\mu}^{2}$
$\displaystyle=\left(\begin{array}[]{cccc}0&\left(1-\frac{2m}{r}\right)^{-\frac{1}{2}}sin\theta
sin\phi&rcos\theta sin\phi&rsin\theta cos\phi\end{array}\right),$ (15)
$\displaystyle\underline{e}_{\mu}^{3}$
$\displaystyle=\left(\begin{array}[]{cccc}0&\left(1-\frac{2m}{r}\right)^{-\frac{1}{2}}cos\theta&-rsin\theta&0\end{array}\right).$
(17)

The non-zero components of the NC tetrad fields $\hat{e}^{a}_{\mu}$ are
calculated in Ref. abdellah1 . Then, using the definition (8), we obtain the
following non-zero components of the non-commutative metric $\hat{g}_{\mu\nu}$
up to second order of $\Theta$, we take the case $\theta=\pi/2$:

$\displaystyle-\hat{g}_{tt}=$
$\displaystyle\left(1-\frac{2m}{r}\right)+\left\\{\frac{m\left(88m^{2}+mr\left(-77+15\sqrt{1-\frac{2m}{r}}\right)-8r^{2}\left(-2+\sqrt{1-\frac{2m}{r}}\right)\right)}{16r^{4}(-2m+r)}\right\\}\Theta^{2}+\mathcal{O}(\Theta^{4})$
(18) $\displaystyle\hat{g}_{rr}=$
$\displaystyle\left(1-\frac{2m}{r}\right)^{-1}+\left\\{\frac{m\left(12m^{2}+mr\left(-14+\sqrt{1-\frac{2m}{r}}\right)-r^{2}\left(5+\sqrt{1-\frac{2m}{r}}\right)\right)}{8r^{2}(2m-r)^{3}}\right\\}\Theta^{2}+\mathcal{O}(\Theta^{4})$
(19) $\displaystyle\hat{g}_{\theta\theta}=$ $\displaystyle
r^{2}+\left\\{\frac{m\left(m\left(10-6\sqrt{1-\frac{2m}{r}}\right)-\frac{8m^{2}}{r}+r\left(-3+5\sqrt{1-\frac{2m}{r}}\right)\right)}{16(-2m+r)^{2}}\right\\}\Theta^{2}+\mathcal{O}(\Theta^{4})$
(20) $\displaystyle\hat{g}_{\phi\phi}=$ $\displaystyle
r^{2}+\left\\{\frac{5}{8}-\frac{3}{8}\sqrt{1-\frac{2m}{r}}+\frac{m\left(-17+\frac{5}{\sqrt{1-\frac{2m}{r}}}\right)}{16r}+\frac{m^{2}\sqrt{1-\frac{2m}{r}}}{(-2m+r)^{2}}\right\\}\Theta^{2}+\mathcal{O}(\Theta^{4})$
(21)

It is clear that, for $\Theta\rightarrow 0$, we obtain the commutative
Schwarzschild solution. For such a black hole, we find that the static limit
surface and the event horizon in the non-commutative spacetime is where the NC
metric (18)-(19) satisfies the following conditions:

$\hat{g}_{00}=0\,,\quad 1/\hat{g}_{rr}=0.$ (22)

The solution to this equations gives us respectively the NC static limit
surface and the event horizon of the SBH abdellah1 ; abdellah2 :

$r_{sls}^{\mathrm{NC}}=r_{h}\left[1+\left(\frac{\Theta}{r_{h}}\right)\left(\frac{4\sqrt{5}+1}{32\sqrt{5}}\right)+\left(\frac{\Theta}{r_{h}}\right)^{2}\left(\frac{10+\sqrt{5}}{128}\right)\right],$
(23)

and

$r_{h}^{NC}=r_{h}\left[1+\frac{3}{8}\left(\frac{\Theta}{r_{h}}\right)^{2}\right].$
(24)

where $r_{h}=2m$ is the event horizon in commutative case when $\Theta=0$. The
effect of non-commutativity is small, which is reasonable to expect since at
large distances it can be neglected.

##  III Geodetic motion and its stability around a non-reciprocal
Schwarzschild black hole

In this section we investigate the stability of circular orbit in the
equatorial plane (i.e. $\theta=\frac{\pi}{2}$) in the deformed Schwarzschild
background, which is given by the following line element

$\displaystyle d\hat{s}^{2}=$
$\displaystyle\hat{g}_{tt}(r,\Theta)c^{2}dt^{2}+\hat{g}_{rr}(r,\Theta)dr^{2}+\hat{g}_{\phi\phi}(r,\Theta)d\phi^{2}$
(25)

According to the above deformed line element, the Lagrangian for test
particles motion in NC spacetime is given by

$\displaystyle
2\mathcal{L}=\hat{g}_{tt}(r,\Theta)c^{2}\dot{t}^{2}+\hat{g}_{rr}(r,\Theta)\dot{r}^{2}+\hat{g}_{\phi\phi}(r,\Theta)\dot{\phi}^{2}$
(26)

where the dot refers to the differentiation with respect to the affine
parameter $\tau$.

Using the Euler-Lagrange equation to compute the generalized momenta:

$\displaystyle p_{t}=c^{2}\hat{g}_{tt}(r,\Theta)\dot{t}=E_{0}=const,$ (27)
$\displaystyle p_{\phi}=\hat{g}_{\phi\phi}(r,\Theta)\dot{\phi}=l=const,$ (28)
$\displaystyle p_{r}=\hat{g}_{rr}(r,\Theta)\dot{r}$ (29)

The Lagrangian $\mathcal{L}$ is independent of $t$ and $\phi$ so we have two
conserved quantities $p_{t}$ (energy $E_{0}$) and $p_{\phi}$ (angular momentum
$l$), and that gives us:

$\displaystyle\dot{t}=\frac{E_{0}}{c^{2}\hat{g}_{tt}(r,\Theta)},\quad\dot{\phi}=\frac{l}{\hat{g}_{\phi\phi}(r,\Theta)}.$
(30)

Substitute these two quantities into the invariant111where
$U^{\mu}=c^{-1}\frac{dx^{\mu}}{d\tau}$ denotes the 4-velocity. of
$\hat{g}_{\mu\nu}U^{\mu}U^{\nu}\equiv-h$, after some algebra we get the
explicit equation for $\dot{r}^{2}$:

$\dot{r}^{2}=-\frac{E_{0}^{2}}{c^{2}\hat{g}_{tt}(r,\Theta)\hat{g}_{rr}(r,\Theta)}-\frac{1}{\hat{g}_{rr}(r,\Theta)}\left(\frac{l^{2}}{\hat{g}_{\phi\phi}(r,\Theta)}+hc^{2}\right)$
(31)

where we shall consider $h=1$ for massive particles and $h=0$ for massless
ones.

Substituting Eqs. (18), (19) and (21) into (31), we find the :

$\dot{r}^{2}+V_{eff}(r,\Theta)=0$ (32)

where $V_{eff}(r,\Theta)$ represents the deformed effective potential, and it
is developed up to the second order in $\Theta$, which reads:

$\displaystyle V_{eff}(r,\Theta)$
$\displaystyle=\left(1-\frac{2m}{r}\right)\left(\frac{l^{2}}{r^{2}}+hc^{2}\right)-E^{2}+\Theta^{2}\left\\{-\frac{l^{2}}{r^{4}}\left(1-\frac{2m}{r}\right)\left(\frac{5}{8}-\frac{3}{8}\sqrt{1-\frac{2m}{r}}+\frac{m\left(-17+\frac{5}{\sqrt{1-\frac{2m}{r}}}\right)}{16r}\right.\right.$
$\displaystyle\left.\left.+\frac{m^{2}\sqrt{1-\frac{2m}{r}}}{(-2m+r)^{2}}\right)+E^{2}\left(\frac{m(64m^{2}+m(-49+13\sqrt{1-\frac{2m}{r}})r+2(13-3\sqrt{1-\frac{2m}{r}})r^{2})}{16r^{5}(1-\frac{2m}{r})^{2}}\right)\right.$
$\displaystyle\left.+\left(\frac{l^{2}}{r^{2}}+hc^{2}\right)\left(\frac{m(12m^{2}+m(-14+\sqrt{1-\frac{2m}{r}})r-(5+\sqrt{1-\frac{2m}{r}}))r^{2}}{8r^{5}(1-\frac{2m}{r})}\right)\right\\}+\mathcal{O}(\Theta^{4})$
(33)

It is clear, for $\Theta\rightarrow 0$, we obtain the commutative effective
potential for the Schwarzschild metric:

$V_{eff}(r,\Theta=0)=\left(1-\frac{2m}{r}\right)\left(\frac{l^{2}}{r^{2}}+hc^{2}\right)-E^{2}.$
(34)

###  III.1 Radial motion of massive particles

To investigate the radial motion of massive particle in the NC SBH spacetime
we must use the above effective potential (33) with $l=0$ and $h=1,\quad c=1$,
which give us the effective potential for a radial motion of massive particle
in NC spacetime

$\displaystyle\left(\frac{dr}{d\tau}\right)^{2}$
$\displaystyle=E^{2}-\left(1-\frac{2m}{r}\right)-\Theta^{2}\left\\{E^{2}\left(\frac{m(64m^{2}+m(-49+13\sqrt{1-\frac{2m}{r}})r+2(13-3\sqrt{1-\frac{2m}{r}})r^{2})}{16r^{5}(1-\frac{2m}{r})^{2}}\right)\right.$
$\displaystyle\left.+\left(\frac{m(12m^{2}+m(-14+\sqrt{1-\frac{2m}{r}})r-(5+\sqrt{1-\frac{2m}{r}}))r^{2}}{8r^{5}(1-\frac{2m}{r})}\right)\right\\}+\mathcal{O}(\Theta^{4})$
(35)

It is clear that the above equation (35) reduces the commutative radial
equation of massive particle in SBH when $\Theta=0$. Let’s consider now a free
fall of massive particle in the NC SBH, which is initially at rest i.e.,
$\dot{r}=0$ and located at $r=r_{0}$ when $\tau=0$. The proper time in NC
spacetime can be defined using the above equation (35) with $E=1$

$\displaystyle\hat{\tau}$
$\displaystyle=-\int_{r_{0}}^{r}\left(\frac{2m}{r^{\prime}}-\Theta^{2}\left\\{\left(\frac{m(64m^{2}+m(-49+13\sqrt{1-\frac{2m}{r^{\prime}}})r^{\prime}+2(13-3\sqrt{1-\frac{2m}{r^{\prime}}})r^{\prime
2})}{16r^{\prime 5}(1-\frac{2m}{r^{\prime}})^{2}}\right)\right.\right.$
$\displaystyle\left.\left.+\left(\frac{m(12m^{2}+m(-14+\sqrt{1-\frac{2m}{r^{\prime}}})r^{\prime}-(5+\sqrt{1-\frac{2m}{r^{\prime}}}))r^{\prime
2}}{8r^{\prime
5}(1-\frac{2m}{r^{\prime}})}\right)\right\\}\right)^{-1/2}dr^{\prime}\,,$ (36)

the proper time at the leading order of $\Theta$, is given by :

$\displaystyle\hat{\tau}$
$\displaystyle=\frac{2}{3}\left(\sqrt{\frac{r_{0}^{3}}{2m}}-\sqrt{\frac{r^{3}}{2m}}\right)+\Theta^{2}\left\\{\sqrt{\frac{r_{0}}{2m}}\left(\frac{32m^{2}+4m(-52+\sqrt{1-\frac{2m}{r_{0}}})r_{0}+131r_{0}^{2}}{128(2m-r_{0})r_{0}^{2}}\right)\right.$
$\displaystyle\left.-\sqrt{\frac{r}{2m}}\left(\frac{32m^{2}+4m(-52+\sqrt{1-\frac{2m}{r}})r+131r^{2}}{128(2m-r)r^{2}}\right)+\frac{1}{8m}\left(ArcSin\left(\sqrt{\frac{2m}{r_{0}}}\right)-ArcSin\left(\sqrt{\frac{2m}{r}}\right)\right)\right.$
$\displaystyle\left.+\frac{67}{256m}\left(ArcTanh\left(\sqrt{\frac{r_{0}}{2m}}\right)-ArcTanh\left(\sqrt{\frac{r}{2m}}\right)\right)\right\\}.$
(37)

the commutative expression of proper time can be obtained by setting
$\Theta=0$.

Figure 1: The behavior of the proper time of free fall for massive particle in
the NC SBH as a function of $r$, with $m=1$ and the initial position
$r_{0}=8$.

Fig. 1 shows the behavior of proper time $\hat{\tau}$ for a free fall of a
massive particle into NC SBH. As we observe, the effect of the non-
commutativity on the proper time is located near the event horizon, where its
effect is to increase the proper time of a massive particle near the event
horizon with the increase of $\Theta$, and in this geometry, the massive
particles take an infinite time to reach the NC singularity (see Ref.
abdellah1 ) contrary to the commutative behavior Chandr1 , and this effect
disappears when we move away from the event horizon.

The coordinate time for the free fall in the NC SBH can be obtained using the
relation (30) with (35) and setting $E=1$, we get:

$\displaystyle\hat{t}$
$\displaystyle=-\int_{r_{0}}^{r}\left(\hat{g}_{tt}\right)^{-1}\left(\frac{2m}{r^{\prime}}-\Theta^{2}\left\\{\left(\frac{m(64m^{2}+m(-49+13\sqrt{1-\frac{2m}{r^{\prime}}})r^{\prime}+2(13-3\sqrt{1-\frac{2m}{r^{\prime}}})r^{\prime
2})}{16r^{\prime 5}(1-\frac{2m}{r^{\prime}})^{2}}\right)\right.\right.$
$\displaystyle\left.\left.+\left(\frac{m(12m^{2}+m(-14+\sqrt{1-\frac{2m}{r^{\prime}}})r^{\prime}-(5+\sqrt{1-\frac{2m}{r^{\prime}}}))r^{\prime
2}}{8r^{\prime
5}(1-\frac{2m}{r^{\prime}})}\right)\right\\}\right)^{-1/2}dr^{\prime}\,,$ (38)

the above integral can be evaluated at the leading order in $\Theta$,

$\displaystyle\hat{t}$
$\displaystyle=\frac{2}{3}\left(\sqrt{\frac{r_{0}^{3}}{2m}}-\sqrt{\frac{r^{3}}{2m}}\right)+4m\left(\sqrt{\frac{r_{0}}{2m}}-\sqrt{\frac{r}{2m}}\right)-4m\left(ArcTanh\left(\sqrt{\frac{r_{0}}{2m}}\right)-ArcTanh\left(\sqrt{\frac{r}{2m}}\right)\right)$
$\displaystyle+\Theta^{2}\left\\{\sqrt{\frac{r_{0}}{2m}}\left(-\frac{25}{16r_{0}}-\frac{37m}{128(r_{0}-2m)^{2}}+\frac{15\sqrt{1-\frac{2m}{r_{0}}}}{32(r_{0}-2m)^{2}}-\frac{r_{0}\sqrt{1-\frac{2m}{r_{0}}}}{4(r_{0}-2m)^{2}}-\frac{61}{512(r_{0}-2m)}\right)\right.$
$\displaystyle\left.-\sqrt{\frac{r}{2m}}\left(-\frac{25}{16r}-\frac{37m}{128(r-2m)^{2}}+\frac{15\sqrt{1-\frac{2m}{r}}}{32(r-2m)^{2}}-\frac{r\sqrt{1-\frac{2m}{r}}}{4(r-2m)^{2}}-\frac{61}{512(r-2m)}\right)\right.$
$\displaystyle\left.+\frac{1}{4m}\left(ArcSin\left(\sqrt{\frac{2m}{r_{0}}}\right)-ArcSin\left(\sqrt{\frac{2m}{r}}\right)\right)+\frac{605}{1024m}\left(ArcTanh\left(\sqrt{\frac{r_{0}}{2m}}\right)-ArcTanh\left(\sqrt{\frac{r}{2m}}\right)\right)\right\\}.$
(39)

the commutative expression of time coordinate can be obtained by setting
$\Theta=0$.

Figure 2: The behavior of the time coordinate for a free fall of a massive
particle in the NC SBH as a function of $r$, with $m=1$ and the initial
position $r_{0}=8$.

In Fig. 2 we show the behavior of the coordinate time $\hat{t}$ for a free
fall of a massive particle in the NC SBH. As we see, the effect of the non-
commutativity on the time coordinate is located near the NC event horizon,
which is increasing the time coordinate of a massive particle near the NC
event horizon compared to the commutative one, and that is a similar behavior
to the commutative one Chandr1 .

Figure 3: The behavior of the proper/coordinate time of free fall for massive
particle in the NC SBH as function of $r$, with $m=1$ and the initial position
$r_{0}=8$.

Fig. 3 shows the variation of the proper time and coordinate time of massive
particle falling toward the NC SBH. The massive particles located at $r_{0}$
falling toward the NC SBH take an infinite time with both its own proper time
and in the coordinate time to reach the NC singularity and the NC event
horizon respectively, which is contrary to the commutative case for the proper
time Chandr1 . In this case, the non-commutativity prevents the massive test
particles to reach the NC singularity at a finite time.

###  III.2 Circular motion of massive particles and Lyapunov exponents

The effective potential for a massive test particle is given by (33) with
$h=1,c=1$, which is analyzed in detail in Ref. abdellah1 .

Figure 4: The behavior of the energy $E^{2}_{c}$ (left panel) and angular
momentum $L_{c}^{2}$ (right panel) of a circular orbit for a massive test
particle in NC Schwarzschild spacetime.

The influence of the NC parameters $\Theta$ on the effective potential for a
massive particle is shown in Fig. 4. As we see now a new minimum to the
effective potential appears near the event horizon (whatever the values of
angular momentum $l\geq 0$), where all the extremum of this potential are
located outside the event horizon, which allows us to interpret this new
minimum as a new stable circular orbit abdellah1 , which gives us multiple
stable circular orbits in this geometry. It is worth noting that, the new
minimum of the effective potential shifted away from the event horizon as
$\Theta$ increases. As we can see, the NC effects become insignificant as we
move further from the event horizon ($r\gg r^{NC}_{h}$).

Our aim now is to investigate the geodesic stability of the circular orbits
for a massive particles. For this kind of motion, the effective potential must
satisfy two conditions, which reads:

$V_{eff}(r,\Theta)=0,\frac{dV_{eff}}{dr}=0.$ (40)

using this two conditions, we can compute the energy and the angular momentum
of massive particle for circular orbits:

$\displaystyle E^{2}_{c}$
$\displaystyle\simeq\frac{(-2m+r_{c})^{2}}{r_{c}(r_{c}-3m)}-\left(\frac{B(r_{c})+F(r_{c})\sqrt{1-\frac{2m}{r_{c}}}}{32(r_{c}^{4}(-3m+r_{c})^{2}\sqrt{1-\frac{2m}{r_{c}}})}\right)\Theta^{2}+\mathcal{O}(\Theta^{4}),$
(41) $\displaystyle L^{2}_{c}$
$\displaystyle\simeq\frac{mr_{c}^{2}}{r_{c}-3m}-\left(\frac{S(r_{c})+Q(r_{c})\sqrt{1-\frac{2m}{r_{c}}}}{32(r_{c}(-3m+r_{c})^{2}(r_{c}-2m)\sqrt{1-\frac{2m}{r_{c}}})}\right)\Theta^{2}+\mathcal{O}(\Theta^{4}).$
(42)

where:

$\displaystyle B(r_{c})$
$\displaystyle=-120m^{4}+162m^{3}r_{c}-71m^{2}r_{c}^{2}+4mr_{c}^{3},\quad
F(r_{c})=204m^{4}-174m^{3}r_{c}+41m^{2}r_{c}^{2}-4mr_{c}^{3},$ (43)
$\displaystyle S(r_{c})$
$\displaystyle=-48m^{4}+126m^{3}r_{c}-55m^{2}r^{2},\quad
Q(r_{c})=-120m^{4}+174m^{3}r_{c}-75m^{2}r_{c}^{2}+8mr_{c}^{3}.$ (44)

in the limit of $\Theta\rightarrow 0$ we recover the commutative energy and
angular momentum of the commutative SBH.

Figure 5: The behavior of the effective potential for massive particle with
different values of NC parameters $\Theta$ and fixed: $E=0.998$, $m=1$,
$l=4.2$.

Fig. 5 shows the behaviors of the angular momentum $L^{2}_{c}$ and energy
$E^{2}_{c}$ of circular orbit for massive particles around NC SBH as a
function of $r_{c}$. As we see the NC effect on the energy condition of
circular orbit produces a new behavior above which the result becomes
unphysical ($E^{2}_{c}<0$) (see the left panel) and it’s turned to the
positive values ($E^{2}_{c}>0$) near the event horizon, which is explained by
the new ISCO at this region in the NC geometry. The NC effect on the angular
momentum is shown in the right panel, and a new behavior is observed, as the
observation in the energy, the angular momentum also has positive values near
the event horizon which corresponds to the new stable circular orbit abdellah1
, Where in the commutative case $\Theta=0$, the new behavior of energy and
angular momentum disappears.

####  III.2.1 Time period and orbital velocity

The orbital velocity for a massive particle in NC Schwarzschild spacetime is
given by geostab3 ; geostab5

$\hat{\Omega}_{c}=\frac{\dot{\phi}}{\dot{t}},$ (45)

using the Eq. (30) together with (41) and (42), the orbital velocity at the
second order in the NC parameter $\Theta$ is

$\hat{\Omega}_{c}=\sqrt{\frac{m}{r_{c}^{3}}+\left(\frac{X(r_{c})+W(r_{c})\sqrt{1-\frac{2m}{r_{c}}}}{32r^{6}_{c}(r_{c}-2m)^{3}}\right)\Theta^{2}},$
(46)

where

$\displaystyle X(r_{c})$
$\displaystyle=-1272m^{5}+2304m^{4}r_{c}-1544m^{3}r^{2}_{c}+451m^{2}r^{3}_{c}-48mr^{4}_{c},$
(47) $\displaystyle W(r_{c})$
$\displaystyle=-192m^{4}r+298m^{3}r^{2}_{c}-137m^{2}r^{3}_{c}+24mr^{4}_{c}.$
(48)

when $\Theta=0$, we get the commutative case geostab3 .

The orbital time scale (time period) of the coordinate time in NC spacetime is
given by geostab3

$\hat{T}_{\hat{\Omega}}=\frac{2\pi}{\hat{\Omega}_{c}},$ (49)

Figure 6: The behavior of the NC orbital velocity (left panel) and NC orbital
time scale (right panel) for a massive test particle.

In Fig. 6, we show the behavior of the orbital velocity (left panel) and
orbital time scale of coordinate time (right panel) in the NC spacetime as a
function of $r_{c}$. As we see the effect of the non-commutativity shows a new
behavior in both orbital velocity (left panel) and orbital time scale (right
panel), where the new behavior appears only outside the event horizon
$r_{h}^{NC}$, where the non-commutativity increases the orbital velocity of a
massive particle near the event horizon and at large orbit is negligible. The
orbital time scale in the presence of the non-commutativity is smaller
compared to the commutative case, and as we see the orbital time scale is zero
at the event horizon, where the non-commutativity decreases the orbital time
scale near the event horizon $r_{h}^{NC}$ and at the large orbits $r_{c}>>3$
this effect vanish.

####  III.2.2 Lyapunov exponents

In order to analyze the stability and instability of circular geodesics of a
massive test particle around the NC Schwarzschild black hole we use the
Lyapunov exponents $\hat{\lambda}$, where that is determined by checking the
nature of $\hat{\lambda}$. The circular orbit is stable when
$\hat{\lambda}_{p}$ (or $\hat{\lambda}_{c}$) is imaginary and is unstable for
$\hat{\lambda}_{p}$ (or $\hat{\lambda}_{c}$) real geostab2 ; geostab10 . The
proper time Lyapunov exponents $\hat{\lambda}_{p}$ and coordinate time
Lyapunov exponents $\hat{\lambda}_{c}$ in NC spacetime are given respectively
by geostab3 ; geostab2 ; geostab10 ; geostab1

$\hat{\lambda}_{p}=\pm\sqrt{\frac{-V_{eff}(r,\Theta)^{\prime\prime}}{2}},\quad\hat{\lambda}_{c}=\pm\sqrt{\frac{-V_{eff}(r,\Theta)^{\prime\prime}}{2\dot{t}^{2}}}.$
(50)

using the effective potential (33) for massive particle together with (30),
(41) and (42) we find

$\displaystyle\hat{\lambda}_{p}$
$\displaystyle=\sqrt{-\frac{m(r_{c}-6m)}{(r_{c}-3m)r_{c}^{3}}+\left(\frac{Z(r_{c})+P(r_{c})\sqrt{1-\frac{2m}{r_{c}}}}{32(r_{c}-2m)^{3}r_{c}^{6}(r_{c}-3m)^{2}}\right)\Theta^{2}}\,,$
(51) $\displaystyle\hat{\lambda}_{c}$
$\displaystyle=\sqrt{-\frac{-m(r_{c}-6m)}{r_{c}^{4}}+\left(\frac{Y(r_{c})+N(r_{c})\sqrt{1-\frac{2m}{r_{c}}}}{32(r_{c}-2m)^{3}r_{c}^{7}}\right)\Theta^{2}}\,.$
(52)

where

$\displaystyle Z(r_{c})$
$\displaystyle=16416m^{7}-38544m^{6}r_{c}+34992m^{5}r^{2}_{c}-16116m^{4}r^{3}_{c}+4076m^{3}r^{4}_{c}-551m^{2}r^{5}_{c}+32mr^{6}_{c},$
(53) $\displaystyle P(r_{c})$
$\displaystyle=3564m^{6}r_{c}-8058m^{5}r_{c}^{2}+7070m^{4}r_{c}^{3}-2910m^{3}r_{c}^{4}+517m^{2}r_{c}^{5}-24mr^{6}_{c},$
(54) $\displaystyle Y(r_{c})$
$\displaystyle=8880m^{6}-16568m^{5}r_{c}+11200m^{4}r^{2}_{c}-3470m^{3}r^{3}_{c}+515m^{2}r^{4}_{c}-32mr^{5}_{c},$
(55) $\displaystyle N(r_{c})$
$\displaystyle=1668m^{5}r_{c}-2870m^{4}r^{2}_{c}+1808m^{3}r^{3}_{c}-473m^{2}r^{4}_{c}+24mr^{5}_{c},$
(56)

the commutative expressions of $\hat{\lambda}_{p}$ and $\hat{\lambda}_{c}$ are
recovered when $\Theta=0$ geostab1 ; geostab10 .

Figure 7: The behavior of the proper time (left panel) and coordinate time
(right panel) Lyapunov exponents for massive particle as function of circular
orbits $r_{c}$.

The proper time and coordinate time of Lyapunov exponents in the NC spacetime
for massive particles is shown in Fig. 7 (left and right panel respectively),
which describes the instability of orbits. As we observe, in the NC spacetime
a new behavior appears for both $\hat{\lambda}_{p}$ (left panel) and
$\hat{\lambda}_{c}$ (right panel). The coordinate time Lyapunov exponents in
the NC space time (right panel) shows the instability of the orbits in the
region between the event horizon $r^{NC}_{h}$ and the NC innermost stable
circular orbit (ISCO) $r_{outer}^{ISCO}$, which means that $\hat{\lambda}_{c}$
is real in the region $r^{NC}_{h}<r_{c}\leq r_{outer}^{ISCO}$, and that means
the orbits in this region are unstable, where the non-commutativity increases
the instability of the orbits near the event horizon and its decreases with
increase in the radius of circular orbit $r_{c}$, until reaching zero at the
outer ISCO $r_{outer}^{ISCO}$. Moreover, the proper time Lyapunov exponents in
the NC spacetime (left panel) show a different behavior to the coordinate time
case for the instability of the circular orbit of a massive particle, where
$\hat{\lambda}_{p}$ is real in two regions $r_{h}^{NC}<r_{c}\leq
r_{inner}^{ISCO}$ and $r_{c}^{uns}<r_{c}\leq r_{outer}^{ISCO}$ which mean that
the orbit of the particle is unstable and is imaginary in the region
$r_{inner}^{ISCO}\leq r_{c}\leq r_{c}^{uns}$ and that means the circular orbit
is stable, this result coincides with our previous work abdellah1 , where the
new circular orbit near the event horizon is stable.

Now we compute ratio proper/coordinate time Lyapunov exponent is given by

$\frac{\hat{\lambda}_{p}}{\hat{\lambda}_{c}}=\sqrt{\frac{r_{c}}{(r_{c}-3m)}+\left(\frac{Y(r_{c})(3m-r_{c})-Z(r_{c})+(N(r_{c})(3m-r_{c})-P(r_{c}))\sqrt{1-\frac{2m}{r_{c}}}}{32m(r_{c}-2m)^{3}(r_{c}-3m)^{2}(r_{c}-6m)r_{c}^{2}}\right)\Theta^{2}},$
(57)

in the case of $\Theta=0$ the commutative case is recovered geostab1 ;
geostab10 .

Figure 8: The behavior of the ration proper time to coordinates time of
Lyapunov exponent for a massive test particle in the NC Schwarzschild
spacetime.

The behavior of the ratio proper time to coordinate time of Lyapunov exponent
as a function of the circular orbits $r_{c}$ in the NC spacetime is shown in
Fig. 8. As we observe, the ratio $\hat{\lambda}_{p}/\hat{\lambda}_{c}$ is real
in two region and it’s separated with an imaginary one, as in the case of the
proper time Lyapunov exponent.

###  III.3 Radial motion of massless particles

In case of massless particle the radial motion in the NC SBH spacetime can be
investigated using the effective potential (33) with $l=0$ and $h=0$, which
gives us the radial motion

$\displaystyle\left(\frac{dr}{d\tau}\right)^{2}$
$\displaystyle=E^{2}-\Theta^{2}\left\\{E^{2}\left(\frac{m(64m^{2}+m(-49+13\sqrt{1-\frac{2m}{r}})r+2(13-3\sqrt{1-\frac{2m}{r}})r^{2})}{16r^{5}(1-\frac{2m}{r})^{2}}\right)\right\\}+\mathcal{O}(\Theta^{4})$
(58)

the commutative expression is recovered when we set $\Theta=0$. Now if we
consider a photon in free fall into the NC SBH emitted from the point
$r=r_{0}$ with $E=1$, the affine parameter $\tau$ and coordinate time
$\hat{t}$ in this case is defined by

$\displaystyle\hat{\tau}$
$\displaystyle=-\int_{r_{0}}^{r}\left(1-\Theta^{2}\left\\{\left(\frac{m(64m^{2}+m(-49+13\sqrt{1-\frac{2m}{r}})r+2(13-3\sqrt{1-\frac{2m}{r}})r^{2})}{16r^{5}(1-\frac{2m}{r})^{2}}\right)\right\\}\right)^{-1/2},$
(59) $\displaystyle\hat{t}$
$\displaystyle=-\int_{r_{0}}^{r}(\hat{g}_{00})^{-1}\left(1-\Theta^{2}\left\\{\left(\frac{m(64m^{2}+m(-49+13\sqrt{1-\frac{2m}{r}})r+2(13-3\sqrt{1-\frac{2m}{r}})r^{2})}{16r^{5}(1-\frac{2m}{r})^{2}}\right)\right\\}\right)^{-1/2}.$
(60)

the integration of the both above equations are evaluated at the leading order
in $\Theta$, which give

$\displaystyle\hat{\tau}$
$\displaystyle=r_{0}-r+\Theta\left\\{25Log\left(\frac{r_{0}(r-2m)}{r(r_{0}-2m)}\right)-\frac{1}{128m}\left(\frac{32m^{2}+15mr_{0}+8r_{0}^{2}}{r_{0}^{2}}-\frac{32m^{2}+15mr+8r^{2}}{r^{2}}\right)\right.$
$\displaystyle\left.+\frac{1}{128m}\left(\left(1-\frac{2m}{r}\right)^{-1/2}-\left(1-\frac{2m}{r_{0}}\right)^{-1/2}-14\left(\left(1-\frac{2m}{r}\right)^{1/2}-\left(1-\frac{2m}{r_{0}}\right)^{1/2}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{13}{3}\left(\left(1-\frac{2m}{r_{0}}\right)^{3/2}-\left(1-\frac{2m}{r}\right)^{3/2}\right)-\frac{35}{2}\left(\frac{r_{0}(r-2m)-r(r_{0}-2m)}{(r_{0}-2m)(r-2m)}\right)\right)\right\\},$
(61) $\displaystyle\hat{t}$
$\displaystyle=r_{0}-r+2mLog\left(\frac{r_{0}-2m}{r-2m}\right)+\Theta^{2}\left\\{-\frac{224m^{2}-201mr_{0}+63r_{0}^{2}}{128r_{0}(r_{0}-2m)^{2}}+\frac{224m^{2}-201mr+63r^{2}}{128r_{0}(r-2m)^{2}}\right.$
$\displaystyle\left.-\frac{(17m^{2}-24mr_{0}+8r_{0}^{2})\sqrt{1-\frac{2m}{r_{0}}}}{m(r_{0}-2m)^{2}}+\frac{(17m^{2}-24mr+8r^{2})\sqrt{1-\frac{2m}{r}}}{m(r-2m)^{2}}-\frac{63}{m}Log\left(\frac{r_{0}(r-2m)}{r(r_{0}-2m)}\right)\right\\}.$
(62)

the commutative affine parameter and coordinate time of massless particle are
obtained when $\Theta=0$ Chandr1 .

Figure 9: The behavior of the affine parameter and the coordinate time along a
radial null geodesic of a photon in the NC SBH as function of $r$.

The variation of the affine parameter $\hat{\tau}$ and the coordinate time
$\hat{t}$ along a radial null geodesic is shown in Fig. 9. For a photon at
$r=r_{0}$ falling toward the NC SBH, it is seen that for the case of the NC
spacetime in the affine parameter framework, the photons take an infinite
affine parameter to reach the NC singularity contrary to the commutative case
Chandr1 ; radial1 , and it’s the same case for the time coordinate which takes
an infinite time, which is the same behavior as in the commutative case for
the time coordinate Chandr1 . It is worth noting that, the non-commutativity
prevents the photons from reaching the NC singularity only after an infinite
affine parameter, which means the photons can not reach the singularity for a
finite time, and that is the same observation as the massive particles.

###  III.4 Circular motion of massless particles and Lyapunov exponents

In which follows we focus on the null geodesic motion of photon around the NC
SBH. The effective potential for the photon in this geometry is given by (33)
with $h=0$.

Figure 10: The behavior of the effective potential for massless particle with
different values of NC parameters $\Theta$ and fixed: $E=0.998$, $m=1$,
$l=4.2$ (left panel), and with different angular momentum $l$ and fixed
$E=0.998$, $m=1$, $\Theta=0.2$ (right panel).

In Fig. 10 we show the influence of the NC parameters $\Theta$ (left panel)
and the angular momentum $l$ (right panel) on the effective potential for a
massless particle (photon). As we observe now, there is a new minimum for the
effective potential that appears near the event horizon in this geometry in a
similar way as for the massive particle case, while for the massless one,
there are only two extrema located outside the event horizon, and that allows
us to interpret this new minimum as a new stable circular orbit, which gives a
stable photon sphere in this geometry. It is worth to note that, the new
minimum of the effective potential shifts away from the event horizon as
$\Theta$ increases, and the NC effects become insignificant as we move further
from the event horizon ($r\gg r^{NC}_{h}$). In the right panel, it is clear
that in the NC spacetime, there is a condition in angular momentum in which
the effective potential has an extremum, contrary to the commutative case
where the effective potential has a maximum, whatever the value of the angular
momentum, in our case of $\Theta=0.2$ this potential has always two extremum
for critical angular momentum $l^{crit}\geq 2.55$, and these values depend on
the NC parameter. As a conclusion, in the NC spacetime, there is always a two-
photon sphere, where the inner one is stable and the outer is unstable.

####  III.4.1 Null circular orbit and photon sphere

The circular orbit of photon subjects to the same condition as the massive
particle (40), using the effective potential (33) with $h=0$, which leads to
the ratio between the energy $E_{c}$ and the angular momentum $L_{c}$ by the
following expression:

$\frac{1}{D_{c}}=\frac{E_{c}}{L_{c}}=\sqrt{\frac{r_{c}-2m}{r_{c}^{3}}+\left(\frac{J(r_{c})+G(r_{c})\sqrt{1-\frac{2m}{r_{c}}}}{16r_{c}^{6}(r_{c}-2m)}\right)\Theta^{2}}.$
(63)

where $D_{c}$ is the impact parameter, and

$J(r_{c})=156m^{3}-185m^{2}r_{c}+73mr_{c}^{2}-10r_{c}^{3},\quad
G(r_{c})=33m^{2}r_{c}-37mr_{c}^{2}+6r_{c}^{3}.$ (64)

Table 1: Some numerical values of unstable $r_{uns}$ and new stable circular orbit $r_{sta}$ of photon sphere in the NC spacetime with different parameter $\Theta$ and fixed $E=0.998,\,l=4.2,\,m=1$. $\Theta$ | 0 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30
---|---|---|---|---|---|---
$r_{sta}(new)$ |  | 2.18400 | 2.24624 | 2.30383 | 2.35865 | 2.41183
$r_{uns}$ | 3. | 2.99782 | 2.99504 | 2.99101 | 2.98559 | 2.97855

This table 1 shows some numerical solution of equation (40), which represents
the variation of stable and unstable circular orbits of photon sphere as a
function of $\Theta$, where these two types of circular orbits get closer to
each other as $\Theta$ increases, for $3.29\leq l\leq 4.48$ (the condition on
$l$ depends on $0\leq\Theta\leq 0.3$), where the stable circular orbit
increasing and the unstable one decreasing, while for $l>4.48$ these circular
orbits increase both of them with increasing $\Theta$.

Figure 11: The behavior of the photon sphere radius in the NC spacetime.
stable and unstable circular orbit as function of $\Theta$ and for fixed
$l=4.2$, $E=1$, $m=1$ (two left panels), and as function of $l$ and for fixed
$\Theta=0.3$, $E=1$, $m=1$ (two right panels).

As we see in Fig. 5 (two left panels) that as the NC parameter $\Theta$
increases, the new stable null circular orbit increases, while the unstable
one decreases. Therefore the unstable circular orbital of the photon has a
smaller radius in NC spaces as the parameter increases for the case $l<4.48$
and is greater than the commutative case for $l\geq 4.48$, where this orbit in
the commutative case is independent of the angular momentum $l$, and that is
due to the modification in the gravitational field by the non-commutativity.
For the two right panels Fig. 5, the new stable circular orbit decreases as
the angular momentum $l$ increases in this geometry, while the unstable one
increases.

NC SBH Singularity NC event horizonStable photon sphereUnstable photon sphere
Figure 12: The schematic picture of the stable/unstable photon sphere around
the NC SBH.

####  III.4.2 Lyapunov exponents

As we know, for the null circular orbit of photons, there is no proper time,
and that leaves us with only one Lyapunov exponent, which is the coordinate
time.

Firstly we compute the angular frequency and the time period in coordinate
time, which is given by (45) and (49) respectively,

$\hat{\Omega}_{c}^{Null}=\frac{1}{D_{c}},\quad\hat{T}_{\hat{\Omega}}^{Null}=2\pi
D_{c}$ (65)

Now, we derive the coordinate time Lyapunov exponent for the photons orbit
around NC SBH.

$\hat{\lambda}_{c}^{Null}=\sqrt{-\frac{3((-4m+r_{c})(-2m+r_{c}))}{r_{c}^{4}}+\left(\frac{O(r_{c})+I(r_{c})\sqrt{1-\frac{2m}{r_{c}}}}{32(r_{c}-2m)^{2}r_{c}^{7}}\right)\Theta^{2}}.$
(66)

where

$\displaystyle O(r_{c})$
$\displaystyle=-19488m^{5}+39000m^{4}r_{c}-29076m^{3}r^{2}_{c}+9958m^{2}r^{3}_{c}-1716mr^{4}_{c}+140r^{5}_{c},$
(67) $\displaystyle I(r_{c})$
$\displaystyle=-3171m^{4}r_{c}+5773m^{3}r^{2}_{c}-3908m^{2}r^{3}_{c}+1080mr^{4}_{c}-84r^{5}_{c}.$
(68)

Figure 13: The behavior of the coordinate time Lyapunov exponent for massless
particle as a function of $r_{c}$ (left panel), and as a function of the NC
parameters $\Theta$ (right panel).

The Fig. 13, shows the behavior of the coordinate time Lyapunov exponent for
photon as a function of $r_{c}$ (left panel), and as a function of the NC
parameters $\Theta$ (right panel). As we see in the left panel, the
instability increases rapidly to the maximum (where this maximum decreases as
$\Theta$ increases.), then decreases as $r_{c}$ increases until reaches zero
for $r_{c}\geq 4$. It is worth noting that, in the NC case the instability of
orbit is small compared to the commutative case for the orbit near the event
horizon, and this observation is reflected when we move away from the event
horizon. In the commutative case, the instability of the orbits begins from
the event horizon, but in the NC spacetime this instability is shifted away
from the event horizon which left some stable orbits near the event horizon
and that is what we found in our previous discussion of the effective
potential for a massless particle which correspond to the stable photon sphere
of NC SBH, and that is the same observation in other theory of non-
commutativity as in Ref. geostab5  where in this theory the orbit is located
inside the event horizon, which can’t be seen as stable photon sphere around
NC SBH. Also, we check the instability of some orbits $r_{c}$ as a function of
the NC parameter $\Theta$ (right panel), it can be observed that the orbit
$r_{c}=3$ is also unstable in the NC spacetime where this geometry increases
the instability of this orbit. For others, a new circular orbit near the event
horizon that emerged in the NC spacetime has less instability and decreases as
$\Theta$ increases before going to zero for specified values of $\Theta$, and
that indicates that some circular orbits near the event horizon in NC space-
time are stable.

The instability exponent of null circular orbits in the NC spacetime is
defined by the ratio of Lyapunov exponent to angular frequency in the
coordinate time ($\hat{\lambda}_{c}^{Null}/\hat{\Omega}_{c}^{Null}$), which
give us

$\frac{\hat{\lambda}_{c}^{Null}}{\hat{\Omega}_{c}^{Null}}=\sqrt{\frac{3(4m-r_{c})}{r_{c}}+\left(\frac{O(r_{c})+6J(r_{c})(8m^{2}-6mr_{c}+r^{2}_{c})+(I(r_{c})+6G(r_{c})(8m^{2}-6mr_{c}+r^{2}_{c}))\sqrt{1-\frac{2m}{r_{c}}}}{32(r_{c}-2m)^{3}r_{c}^{4}}\right)\Theta^{2}}\,.$
(69)

Figure 14: The behavior of the ratio of Lyapunov exponent to angular frequency
in the coordinate time as function of $r_{c}$.

The variation of the ratio $\hat{\lambda}_{c}^{Null}/\hat{\Omega}_{c}^{Null}$
as function of $r_{c}$ is depicted in Fig. 14. As we see, this ratio shows the
same behavior as in Fig. 13, except for the commutative case, which means that
the orbits near the event horizon in the NC spacetime are stable contrary to
the results obtained in Ref. geostab5 .

As final analysis of instability we check the critical exponent $\hat{\gamma}$
for the null circular orbit around NC SBH, which is defined as a ratio of
Lyapunov timescale $\hat{T}_{\hat{\lambda}}=\frac{2\pi}{\hat{\lambda}}$ to the
orbital timescale $\hat{T}_{\hat{\Omega}}=\frac{2\pi}{\hat{\Omega}}$ geostab11
; geostab3 ; geostab12 , in this case we get:

$\hat{\gamma}^{Null}=\frac{T_{\hat{\lambda}}^{Null}}{T_{\hat{\Omega}}^{Null}}=\frac{1}{2\pi\,\sqrt{\frac{3(4m-r_{c})}{r_{c}}+\left(\frac{O(r_{c})+6J(r_{c})(8m^{2}-6mr_{c}+r^{2}_{c})+(I(r_{c})+6G(r_{c})(8m^{2}-6mr_{c}+r^{2}_{c}))\sqrt{1-\frac{2m}{r_{c}}}}{32(r_{c}-2m)^{3}r_{c}^{4}}\right)\Theta^{2}}}\,.$
(70)

in limit of $\Theta=0$ we recover the commutative expression.

Table 2: Some numerical values of the critical exponant $\hat{\gamma}^{Null}$ for unstable $r_{uns}$ and new stable circular orbit $r_{sta}$ of photon sphere in the NC spacetime with different parameter $\Theta$ with $m=1$. | $\Theta$ | $r_{c}=3$ | $r_{c}=2.4$ | $r_{c}=2.3$ | $r_{c}=2.2$
---|---|---|---|---|---
| 0.0 | 0.159155 | // | // | //
| 0.1 | 0.158439 | 0.117291 | 0.121293 | 0.251639
$\hat{\gamma}^{Null}$ | 0.2 | 0.156349 | 0.136225 | 0.328429 | 0\. -0.0662942 i
| 0.3 | 0.153042 | 0.210585 | 0\. -0.10626 i | 0\. -0.0397438 i
| 0.4 | 0.148746 | 0.-0.216564 i | 0\. -0.0665877 i | 0\. -0.0288573 i

The Table. 2 we show some numerical values of the critical exponent
$\hat{\gamma}^{Null}$ for the photons, for different range of orbits $r_{c}$
and different NC parameter $\Theta$. As we can see from this table there are
real and imaginary values, where the real positive values of this quantity
$\hat{\lambda}^{Null}$ for the range of $r_{c}$ correspond to unstable
circular orbits geostab11 , and the imaginary ones correspond to the stable
one. It is worth noting that the ordinary orbit of the photon sphere around
SBH $r_{c}=3$ is always unstable, also in the NC spacetime according to the
definition (70) means that the Lyapunov timescale is shorter than the orbital
timescale ($\hat{T}_{\hat{\lambda}}<\hat{T}_{\hat{\Omega}}$) geostab12  and
the same observation for all real values of $\hat{\lambda}^{Null}$ for the
others ranges of $r_{c}$ in this table. Also, we show the effect of the NC
parameter in the values of $\hat{\lambda}^{Null}$, where these values are
decreasing with the increase of $\Theta$ for $r_{c}=3$, and the smallest one
of $\hat{\lambda}^{Null}$ indicates a strong Lyapunov instability geostab3
(see right panel Fig. 13), while the imaginary values which correspond to the
stable circular orbit coincide with the new minimum location of the effective
potential for $l=5.2$ (close values to the values of Table. 1), and that shows
the stability of the new stable photon sphere around NC SBH and prove the
instability of the external photon sphere in both of commutative and non
commutative spacetime.

####  III.4.3 Black hole shadow

In the equatorial plane $\theta=\pi/2$, it is easy to show that, the radius of
the black hole shadow $R_{shadow}$ is related to the impact parameter $D_{c}$
by the relation $R_{shadow}=D_{c}|_{r=r_{ps}}$, where $r^{uns}_{ps}$ is the
radius of the unstable photon sphere, which can be obtained by solving
$\frac{dV_{eff}}{dr}=0$, and the solution to this equation in the leading
order in $\Theta$ and $m$ is given by:

$r^{uns}_{ps}=3m-\left(\frac{-38+540D_{c}^{2}m^{2}}{288\,m}\right)\Theta^{2}\,.$
(71)

It is worth to note that, the new stable photon sphere is only obtained by
numerical solution (see Tab. 2). Using this expression, we can express the
radius of the NC SBH shadow

$R_{shadow}=3\sqrt{3}m\left(1+\frac{(1+2\sqrt{3})}{72\,m^{2}}\Theta^{2}\right)\,.$
(72)

where the commutative expression can be obtained by setting $\Theta=0$.

Figure 15: The SBH shadow in the NC spacetime, for different values of NC
parameter $\Theta$ (left panel) with $m=1$, and for different values of black
hole mass $m$ (right panel) with $\Theta=0.4$.

Fig. 15 shows the NC SBH shadow for different values of $\Theta$ (left panel)
and $m$ (right panel). As we see in the left panel, the shadow radius of NC
SBH increases with the increasing $\Theta$, and it’s the same note in the
right panel for the fixed NC parameter and increasing in the black hole mass
$m$. Moreover, we can see a similar effect of the non-commutativity and the
black hole mass on the shadow radius, where the NC property of spacetime plays
a similar role as a mass, and that’s what was expected in our previous work
abdellah1 , and that means to increase the gravitational field. It is worth to
motioned that, this observation it’s the same one obtained in Hassanabadi2 ,
in which the authors used the NC correction to the black hole mass using gauge
theory of gravity with negligence of the metric correction. However, in our
case, we use the NC Schwarzschild metric to describe the radius of the NC SBH
shadow.

####  III.4.4 Energy emission rate

In order to investigate the unstable photon sphere emission in the NC
spacetime, we compute the energy emission rate energy1 ; energy2 , and it’s
given by:

$\frac{d^{2}\hat{E}}{d\omega
dt}=\frac{2\pi^{2}\hat{\sigma}_{lim}\omega^{3}}{e^{\omega/\hat{T}}-1},$ (73)

where deformed Hawking temperature of this black hole is given by
$\hat{T}=\frac{1}{4\pi r_{h}}\left(1-\frac{\Theta^{2}}{r_{h}^{2}}\right)$ (for
more detail see Ref. abdellah2 ), $\hat{\sigma}$ denoted the frequency of
emitted particles, and $\sigma_{lim}$ is the greybody factor, which is
appreciatively related to the radius of black hole shadow by
$\hat{\sigma}_{lim}=\pi R_{shadow}^{2}$. By substituting this relation into
the above equation, we find

$\frac{d^{2}\hat{E}}{d\omega
dt}=\frac{2\pi^{3}R_{shadow}^{2}\omega^{3}}{e^{\omega/\hat{T}}-1},$ (74)

Figure 16: Behavior of the NC energy emission rate of SBH as a function of
frequency $\omega$.

In Fig. 16, we illustrate the behavior of NC energy emission rate of SBH as a
function of the frequency $\omega$ and for different NC parameter $\Theta$. As
we observe, there is a gaussian distribution peak of energy emission rate also
in the NC spacetime. It is clear that, this geometry decreases the peak and
shifts it toward the low frequency with $\Theta$ increase.

##  IV Conclusions

In this paper, we study the geodesic motion for both massive and massless
particles in NC Schwarzschild spacetime. The background used for our
calculation is the deformed SBH via the SW maps and the star product in the
context of the gauge theory of gravity abdellah1 , where the deformed metric
was obtained in the leading order in $\Theta$, according to the deformed
metric we obtain the correction to the effective potential up to the second
order in $\Theta$.

As a first step, we investigate the radial motion for both massive particles
and massless ones, where we compute the correction to the affine parameter and
coordinate time in this geometry. Our results show that the massive and
massless particles take an infinite time to reach the NC singularity and NC
event horizon in both of affine parameter framework and coordinate time
respectively, where that’s the effect of the non-commutativity which prevents
particles from reaching the NC singularity at a finite proper time (affine
parameter for photons).

Moreover, we examinate the geodesic stability using the Lyapunov exponent to
establish the instability of circular orbits for both massive and massless
particles. In the case of massive particles, we compute the angular frequency
$\hat{\Omega}_{c}$ and the period time $\hat{T}_{c}$ in NC spacetime, and we
describe both of the proper time Lyapunov exponent $\hat{\lambda}_{p}$ and the
coordinate time one $\hat{\lambda}_{c}$ in this geometry, where the proper
time one shows a new behavior which allows the existence of a stable region
between two unstable ones near the event horizon, and this agrees with our
results in previous work abdellah1 , and the same observation for their ratio.
For the massless particles case, we study the effective potential of this type
of particles, we show a new kind of motion near the event horizon, which is
interpreted as a new stable circular orbit in the NC spacetime, and that is a
new photon sphere near the event horizon that is stable compared to the
commutative one. In this geometry, the NC SBH has two-photon spheres, and the
first one can be seen as the inner photon sphere which is considered stable
while the outer one is unstable. Then the coordinate time Lyapunov exponent
$\hat{\lambda}_{c}^{Null}$ is expressed to describe the instability of the
photon sphere, where our results show that the outer photon sphere is unstable
in both commutative and NC spacetime, while for the inner photon sphere is
stable and only allowed in the NC spacetime. Also the critical exponent
$\hat{\gamma}^{Null}$ is computed to analyze the instability of the null
circular orbit around the NC SBH. It is worth noting that, the all commutative
expression in this paper can be recovered when we put $\Theta=0$. Finally, we
show the effect of the non-commutativity on the shadow of SBH, and show a
similarity between the mass and the non-commutativity based on our results,
which indicates that the NC propriety of spacetime increases the gravitational
field of the SBH. Then show the effect of this geometry on the energy emission
rate of SBH, where the non-commutativity decrease this energy, and that agree
with our remark for the effect on SBH shadow. However, more studies of this
phenomenon may help to understand more about this geometry and its effect on
the black hole theory.

###### Acknowledgements.

This work is supported by project B00L02UN050120230003, Univ. Batna 1,
Algeria.

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traveltime - home to school travel time (numeric: 1 - <15 min., 2 - 15 to 30
min., 3 - 30 min. to 1 hour, or 4 - >1 hour)

  14. 14.

failures - number of past class failures (numeric: n if 1<=n<3, else 4)

  15. 15.

schoolsup - extra educational support (binary: yes or no)

  16. 16.

famsup - family educational support (binary: yes or no)

  17. 17.

activities - extra-curricular activities (binary: yes or no)

  18. 18.

nursery - attended nursery school (binary: yes or no)

  19. 19.

higher - wants to take higher education (binary: yes or no)

  20. 20.

internet - Internet access at home (binary: yes or no)

  21. 21.

romantic - with a romantic relationship (binary: yes or no)

  22. 22.

famrel - quality of family relationships (numeric: from 1 - very bad to 5 -
excellent)

  23. 23.

freetime - free time after school (numeric: from 1 - very low to 5 - very
high)

  24. 24.

goout - going out with friends (numeric: from 1 - very low to 5 - very high)

  25. 25.

Dalc - workday alcohol consumption (numeric: from 1 - very low to 5 - very
high)

  26. 26.

Walc - weekend alcohol consumption (numeric: from 1 - very low to 5 - very
high)

  27. 27.

health - current health status (numeric: from 1 - very bad to 5 - very good)

  28. 28.

absences - number of school absences (numeric: from 0 to 93)

We show the attributes of ID number $1$ in Table 1.

Table 1: Attributes of the ID number $1$ subject. school | sex | age | address | famsize | Pstatus | Medu | Fedu | Mjob | Fjob | reason
---|---|---|---|---|---|---|---|---|---|---
GP | F | 18 | U | GT3 | A | 4 | 4 | at_home | teacher | course
guardian | traveltime | studytime | failures | schoolsup | famsup | paid | activities | nursery | higher | internet
mother | 2 | 2 | 0 | yes | no | no | no | yes | yes | no
romantic | famrel | freetime | goout | Dalc | Walc | health | absences | G1 | G2 | G3
no | 4 | 3 | 4 | 1 | 1 | 3 | 6 | 5 | 6 | 6

###  D.2 Additional Analyses of Application

We give three additional analyses of the four applications in the body of the
paper.

Effect of study time only. First, we evaluate conditional PNS, PN, and PS,
letting ${\boldsymbol{y}}=(6,6,6)$, ${\boldsymbol{x}}_{0}=(2,1)$,
${\boldsymbol{x}}_{1}=(4,1)$, and ${\boldsymbol{c}}_{1}$ in Def. 4.1. The
estimated values of conditional PNS, PN, and PS are

PNS: $\displaystyle 2.491\%$ $\displaystyle(\text{CI}:[0.000\%,7.395\%]),$
(58) PN: $\displaystyle 2.709\%$ $\displaystyle(\text{CI}:[0.000\%,8.060\%]),$
PS: $\displaystyle 25.864\%$ $\displaystyle(\text{CI}:[0.000\%,73.544\%]),$

respectively. Second, we evaluate conditional PNS with evidence
$({\boldsymbol{y}}^{\prime},{\boldsymbol{x}}^{\prime},{\boldsymbol{c}})$,
letting ${\boldsymbol{y}}=(6,6,6)$, ${\boldsymbol{y}}^{\prime}=(6,6,5)$,
${\boldsymbol{x}}_{0}=(2,1)$, ${\boldsymbol{x}}_{1}=(4,1)$,
${\boldsymbol{x}}^{\prime}=(2,1)$, and ${\boldsymbol{c}}_{1}$ in Def. 5.1.
Then, the estimated value of it is

$\text{PNS:}\ \ 0.000\%\ \ \ \ (\text{CI}:[0.000\%,0.000\%]).$ (59)

Third, we evaluate conditional PNS with multi-hypothetical terms, letting
${\boldsymbol{y}}_{1}=(5,5,5)$, ${\boldsymbol{y}}_{2}=(6,6,6)$,
${\boldsymbol{x}}_{0}=(1,1)$, ${\boldsymbol{x}}_{1}=(2,1)$,
${\boldsymbol{x}}_{2}=(4,1)$, and ${\boldsymbol{c}}_{1}$ in Def. 5.2. The
estimated value of it is

$\text{PNS:}\ \ 0.000\%\ \ \ \ (\text{CI}:[0.000\%,0.000\%]).$ (60)

Finally, we evaluate conditional PNS with multi-hypothetical terms and
evidence
$({\boldsymbol{y}}^{\prime},{\boldsymbol{x}}^{\prime},{\boldsymbol{c}})$,
letting ${\boldsymbol{y}}_{1}=(5,5,5)$, ${\boldsymbol{y}}_{2}=(6,6,6)$,
${\boldsymbol{y}}^{\prime}=(6,6,5)$, ${\boldsymbol{x}}_{0}=(1,1)$,
${\boldsymbol{x}}_{1}=(2,1)$, ${\boldsymbol{x}}_{2}=(4,1)$,
${\boldsymbol{x}}^{\prime}=(2,1)$, and ${\boldsymbol{c}}_{1}$ in Def. 5.3 . We
eliminate the results of NA, then the estimated value of it is

$\text{PNS:}\ \ 42.489\%\ \ \ \ (\text{CI}:[0.000\%,100.000\%]).$ (61)

Effect of extra paid classes only. First, we evaluate conditional PNS, PN, and
PS, letting ${\boldsymbol{y}}=(6,6,6)$, ${\boldsymbol{x}}_{0}=(1,1)$,
${\boldsymbol{x}}_{1}=(2,2)$, and ${\boldsymbol{c}}_{1}$ in Def. 4.1. The
estimated values of conditional PNS, PN, and PS are

PNS: $\displaystyle 7.700\%$ $\displaystyle(\text{CI}:[1.072\%,16.614\%]),$
(62) PN: $\displaystyle 8.132\%$
$\displaystyle(\text{CI}:[1.090\%,18.139\%]),$ PS: $\displaystyle 65.398\%$
$\displaystyle(\text{CI}:[37.015\%,89.795\%]),$

respectively. Second, we evaluate conditional PNS with evidence
$({\boldsymbol{y}}^{\prime},{\boldsymbol{x}}^{\prime},{\boldsymbol{c}})$,
letting ${\boldsymbol{y}}=(6,6,6)$, ${\boldsymbol{y}}^{\prime}=(6,6,5)$,
${\boldsymbol{x}}_{0}=(1,1)$, ${\boldsymbol{x}}_{1}=(2,2)$,
${\boldsymbol{x}}^{\prime}=(2,1)$, and ${\boldsymbol{c}}_{1}$ in Def. 5.1.
Then, the estimated value of it is

$\text{PNS:}\ \ 0.009\%\ \ \ \ (\text{CI}:[0.000\%,0.139\%]).$ (63)