aainstitutetext: Laboratoire de Physique des Rayonnements et de leurs
Intéractions avec la Matière
Département de Physique, Faculté des Sciences de la Matière
Université de Batna-1, Batna 05000, Algeriabbinstitutetext: Département de
Physique, Faculté des Sciences de la Matière
Université de Batna-1, Batna 05000, Algeria

# Geodesic equation in non-commutative gauge theory of gravity

Abdellah Touati,111Corresponding author. b Slimane Zaim<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, we construct a non-commutative (NC) gauge theory of gravity for
any metric with spherical symmetries, where we use a non-diagonal tetrad
field. The deformed gauge potentials (tetrad fields) and the components of
deformed metric are computed to the second order in the NC parameter
$\Theta^{\mu\nu}$, as the application to the Schwarzschild black hole we show
that the NC geometry removes the singularity at the origin of the black hole,
and increase the event horizon. The non-commutativity correction to the
effective potential of the Schwarzschild metric is also computed and we show
how this geometry affects the stability condition which it found the NC
parameter plays the same role as the mass that can be used to explain the dark
matter and we show that the NC Schwarzschild space-time has new stable
circular orbits appear near the event horizon that is not allowed by
Schwarzschild space-time. The geodesic equations in the NC space and the
corrections to the periastron advance in terms of $\Theta$ are obtained. We
have also specified the problem of Mercury’s perihelion and used the
experimental data to estimate the NC parameter $\Theta$, then we show that
$\Theta$ of the order $10^{-25}s.kg^{-1}$ gives observable corrections to the
movement at a large scale. We show that the NC propriety of the spacetime
appears at the High Energy.

###### Keywords:

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

##  1 Introduction

In general relativity, the study of the geodesic motion of test particles in
curved spacetime is the first probe towards an understanding of the physics
and geometry of gravitational objects. So that she was able to solve the
problem of mercury perihelion and is considered one of the triumphs of the
general relativity theory. Also, the classification of stable and unstable
orbits is done by drawing the effective potentials and checking their behavior
at different points. We find many references that studied in detail the
geodesy movement of black holes 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 .. However, our interest is in
studying the geodesic motion in non-commutative space-time, by imposing
further commutation relations between position coordinates themselves. This
non-commutativity leads to the modification of Heisenberg uncertainty
relations in such a way that prevents one from measuring positions to better
accuracy than the Planck length. Noncommutativity is mainly motivated by
string theory, being a limit in the presence of a background field hawking1 ;
hawking2 ; hod ; vaz ; harms ; gour ; zaim1 ; mebarki1 . This idea results in
the concept of quantum gravity since quantifying space-time leads 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 non-commutative space-time the coordinate
and momentum 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 which determines the
fundamental cell discretization of space-time much in the same way as the
Planck constant $\hbar$ discretizes the phase space and $x^{\mu}$ are the
coordinate operator on a non-commutative space-time is defined by the
following transformations:

$\hat{x}^{\mu}=x^{\mu}-\Theta^{\mu\nu}p_{\nu}$ (2)

In the non-commutative 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
modifications introduced by a non-commutativity on the geodesic structure in
black hole nozari1 ; nozari2 ; kuniyal ; mirza ; rome1 ; iftikhar ; ulh1 ;
larranaga ; nozari ; nicolini1 ; bhar ; rahaman . Here one uses a gauge
gravity theory in non-commutative space-time with star products and Seiberg-
Witten maps seiberg1 . The non-commutative gauge gravity is a theory of
general relativity in curved space-time with preservation of non-commutative
space-time and based partly on implementing symmetries on flat non-commutative
space-time. In gauge gravity theory the action transforms under ordinary
Lorentz transformation of the ordinary fields, since these ordinary
transformations, via the Seiberg-Witten map, induce the non-commutative
canonical transformations of non-commutative fields under which the non-
commutative action is invariant chai1 ; haranas1 ; haranas2 . Our work will be
in this context, with the aim of writing the geodesic equation that arises
from the metric tensor which has been corrected using star product between
tetrads fields and Seiberg-Witten maps. In addition, we hope to obtain
corrections in terms of the non-commutative for each of the effective
potential and the deviation angle per revolution and we also discussed the
issue of the stability of circular orbits in a non-commutative Schwarzschild
geometry. The article is organized as follows. In Sect. 2 we present the non-
commutative corrections to the metric field using star product between tetrads
fields and Seiberg-Witten maps. In Sect. 3 we calculate the non-commutative
geodesic equation and we also obtained non-commutative effective potentials up
to the second-order from the non-commutativity parameter and the condition for
the stability of the circular orbitals of the particles in non-commutative
Schwarzschild space-time has also been determined. We then calculated the non-
commutative adjustment to the value of the perihelion rotation and gave an
estimate for the non-commutative parameter. In the last section, we present
our concluding remarks.

##  2 Non-commutative gauge gravity for spherical symmetric metric

Using the tetrad and spin connection formalism in the gauge theory of gravity
is unavoiable because of the requirement to describe the spinor fields in this
theory. We denote the tetrad fields by $e^{a}_{\mu},a=0,1,2,3$ and the spin
connection $\omega^{ab}_{\mu}(x)=-\omega^{ba}_{\mu}(x)$
[ab]=[01],[02],[03],[12],[13],[23]. Then the Ricci scalar is given by:

$R=e_{a}^{\mu}e_{b}^{\nu}R^{ab}_{\mu\nu}$ (4)

Where $e_{a}^{\mu}$ denotes the inverse of $e^{a}_{\mu}$ which satisfying the
usual properties:

$e^{a}_{\mu}e_{b}^{\mu}=\delta^{a}_{b},\quad
e^{a}_{\mu}e_{a}^{\nu}=\delta^{\nu}_{\mu}$ (5)

And $R^{ab}_{\mu\nu}$ denote the curvature tensor:

$R^{ab}_{\mu\nu}=\partial_{\mu}\omega^{ab}_{\nu}-\partial_{\nu}\omega^{ab}_{\mu}+\left(\omega^{ac}_{\mu}\omega^{db}_{\nu}-\omega^{ac}_{\nu}\omega^{db}_{\mu}\right)\eta_{cd}$
(6)

So then the action of the pur gravity in the gauge theory, are define read:

$S_{g}=\frac{1}{16\pi G}\int d^{4}xeR=\frac{1}{16\pi G}\int
d^{4}xee_{a}^{\mu}e_{b}^{\nu}R^{ab}_{\mu\nu}$ (7)

where $e=det(e^{a}_{\mu})$. Using the variational principle $\delta S=0$ for
the action in (7) respect to $e^{a}_{\mu}$, we can get the field equation for
the gravitational potentials $e^{a}_{\mu}$ in the vacuum:

$R^{a}_{\mu}=0$ (8)

where $R^{a}_{\mu}=R^{ab}_{\mu\nu}e^{\nu}_{b}$ the Ricci tensor.

We consider the general solution of the equation of the gravitational field
(8) in the case of static and spherical symmetry, the following metric:

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})$ (9)

where the $A(r)$ and $B(r)$ are functions related only to the radius $r$. The
tetrad formulation of General Relativity allows to write the tetrad components
with this relation:

$g_{\mu\nu}=e^{a}_{\mu}e_{a\nu}$ (10)

With the choose a particular form of non diagonal tetrad fields satisfies the
relation (10) as follows:

$e^{a}_{\mu}=\left[\begin{array}[]{cccc}A(r)&0&0&0\\\ 0&B(r)sin\theta
cos\phi&rcos\theta cos\phi&-rsin\theta sin\phi\\\ 0&B(r)sin\theta
sin\phi&rcos\theta sin\phi&rsin\theta cos\phi\\\
0&B(r)cos\theta&-rsin\theta&0\end{array}\right]$ (11)

We note that, this particular form of tetrad field can be used for a
stationary observer at spatial infinity ulh1 .

The non-zero component of the spin connection for this tetrad field:

$\displaystyle\omega^{01}_{\mu}$
$\displaystyle=\left(\frac{A^{\prime}(r)}{B(r)}sin\theta
cos\phi,0,0,0\right),\omega^{02}_{\mu}=\left(\frac{A^{\prime}(r)}{B(r)}sin\theta
sin\phi,0,0,0\right),$ (12) $\displaystyle\omega^{03}_{\mu}$
$\displaystyle=\left(\frac{A^{\prime}(r)}{B(r)}cos\theta,0,0,0\right),\omega^{12}_{\mu}=\left(0,0,0,[1-\frac{1}{B(r)}]sin^{2}\theta\right),$
(13) $\displaystyle\omega^{13}_{\mu}$
$\displaystyle=\left(0,0,-[1-\frac{1}{B(r)}]cos\phi,[1-\frac{1}{B(r)}]sin\theta
cos\theta sin\phi\right),$ (14) $\displaystyle\omega^{23}_{\mu}$
$\displaystyle=\left(0,0,-[1-\frac{1}{B(r)}]sin\phi,-[1-\frac{1}{B(r)}]sin\theta
cos\theta cos\phi\right),$ (15)

Using the relations (6), the spin connection, and the tetrads fields, we find
the nonzero component of the curvature tensor $R^{ab}_{\mu\nu}$, which we need
to use further with the spin connection and tetrads fields, in the derivation
of the expressions of the deformed tetrads fields:

$\displaystyle R^{01}_{01}$
$\displaystyle=-\left[\frac{A^{\prime\prime}(r)}{B(r)}-\frac{A^{\prime}(r)B^{\prime}(r)}{B^{2}(r)}\right]sin\theta
cos\phi,\quad R^{01}_{02}=-\frac{A^{\prime}(r)}{B^{2}(r)}cos\theta cos\phi,$
(16) $\displaystyle R^{01}_{03}$
$\displaystyle=\frac{A^{\prime}(r)}{B^{2}(r)}sin\theta sin\phi,\quad
R^{02}_{01}=-\left[\frac{A^{\prime\prime}(r)}{B(r)}-\frac{A^{\prime}(r)B^{\prime}(r)}{B^{2}(r)}\right]sin\theta
sin\phi,$ (17) $\displaystyle R^{02}_{02}$
$\displaystyle=-\frac{A^{\prime}(r)}{B^{2}(r)}cos\theta sin\phi,\quad
R^{02}_{03}=-\frac{A^{\prime}(r)}{B^{2}(r)}sin\theta cos\phi,\quad
R^{03}_{02}=\frac{A^{\prime}(r)}{B^{2}(r)}sin\theta,$ (18) $\displaystyle
R^{03}_{01}$
$\displaystyle=-\left[\frac{A^{\prime\prime}(r)}{B(r)}-\frac{A^{\prime}(r)B^{\prime}(r)}{B^{2}(r)}\right]cos\theta,\quad
R^{12}_{23}=\left[1-\frac{1}{B^{2}(r)}\right]sin\theta cos\theta$ (19)
$\displaystyle R^{12}_{13}$
$\displaystyle=\frac{B^{\prime}(r)}{B^{2}(r)}sin^{2}\theta,\quad
R^{13}_{12}=-\frac{B^{\prime}(r)}{B^{2}(r)}cos\phi,\quad
R^{13}_{13}=\frac{B^{\prime}(r)}{B^{2}(r)}sin\theta cos\theta sin\phi,$ (20)
$\displaystyle R^{13}_{23}$
$\displaystyle=-\left[1-\frac{1}{B^{2}(r)}\right]sin^{2}\theta sin\phi,\quad
R^{23}_{13}=-\frac{B^{\prime}(r)}{B^{2}(r)}sin\theta cos\theta cos\phi,$ (21)
$\displaystyle R^{23}_{12}$
$\displaystyle=-\frac{B^{\prime}(r)}{B^{2}(r)}sin\phi,\quad
R^{23}_{23}=\left[1-\frac{1}{B^{2}(r)}\right]sin^{2}\theta cos\phi,$ (22)

where $A^{\prime}(r)$, $B^{\prime}(r)$ and $A^{\prime\prime}(r)$ denote the
derivatives of first and second-order with respect to the r-coordinate.

In the non-commutative space-time, for finding the deformed tetrad fields
$\hat{e}^{a}_{\mu}(x,\Theta)$ we use the Seiberg-Witten map, which describes
the tetrad fields as a development in the power of $\Theta$ up to the second-
order cham1 :

$\hat{e}^{a}_{\mu}(x,\Theta)=e^{a}_{\mu}(x)-i\Theta^{\nu\rho}e^{a}_{\mu\nu\rho}(x)+\Theta^{\nu\rho}\Theta^{\lambda\tau}e^{a}_{\mu\nu\rho\lambda\tau}(x)+\mathcal{O}(\Theta^{3})$
(23)

where:

$\displaystyle e^{a}_{\mu\nu\rho}$
$\displaystyle=\frac{1}{4}[\omega^{ac}_{\nu}\partial_{\rho}e^{d}_{\mu}+(\partial_{\rho}\omega^{ac}_{\mu}+R^{ac}_{\rho\mu})e^{d}_{\nu}]\eta_{cd}$
(24) $\displaystyle e^{a}_{\mu\nu\rho\lambda\tau}$
$\displaystyle=\frac{1}{32}\left[2\\{R_{\tau\nu},R_{\mu\rho}\\}^{ab}e^{c}_{\lambda}-\omega^{ab}_{\lambda}(D_{\rho}R_{\tau\nu}^{cd}+\partial_{\rho}R_{\tau\nu}^{cd})e^{m}_{\nu}\eta_{dm}\right.$
$\displaystyle\left.-\\{\omega_{\nu},(D_{\rho}R_{\tau\nu}+\partial_{\rho}R_{\tau\nu})\\}^{ab}e^{c}_{\lambda}-\partial_{\tau}\\{\omega_{\nu},(\partial_{\rho}\omega_{\mu}+R_{\rho\mu})\\}^{ab}e^{c}_{\lambda}\right.$
$\displaystyle\left.-\omega^{ab}_{\lambda}\left(\omega^{cd}_{\nu}\partial_{\rho}e^{m}_{\mu}+\left(\partial_{\rho}\omega_{\mu}^{cd}+R_{\rho\mu}^{cd}\right)e^{m}_{\nu}\right)\eta_{dm}+2\partial_{\nu}\omega_{\lambda}^{ab}\partial_{\rho}\partial_{\tau}e^{c}_{\lambda}\right.$
$\displaystyle\left.-2\partial_{\rho}\left(\partial_{\tau}\omega_{\mu}^{ab}+R_{\tau\mu}^{ab}\right)\partial_{\nu}e^{c}_{\lambda}-\\{\omega_{\nu},(\partial_{\rho}\omega_{\lambda}+R_{\rho\lambda})\\}^{ab}\partial_{\tau}e^{c}_{\mu}\right.$
$\displaystyle\left.-\left(\partial_{\tau}\omega_{\mu}+R_{\tau\mu}\right)\left(\omega^{cd}_{\nu}\partial_{\rho}e^{m}_{\lambda}+\left((\partial_{\rho}\omega_{\lambda}+R_{\rho\lambda})\right)e^{m}_{\nu}\right)\eta_{dm}\right]\eta_{cb}$
(25)

and

$\displaystyle\\{\alpha,\beta\\}^{ab}=\left(\alpha^{ac}\beta^{db}+\beta^{ac}\alpha^{db}\right)\eta_{cd},\quad$
$\displaystyle[\alpha,\beta]^{ab}=\left(\alpha^{ac}\beta^{db}-\beta^{ac}\alpha^{db}\right)\eta_{cd}$
(26) $\displaystyle
D_{\mu}R_{\rho\sigma}^{ab}=\partial_{\mu}R^{ab}_{\rho\sigma}+$
$\displaystyle\left(\omega_{\mu}^{ac}R^{db}_{\rho\sigma}+\omega_{\mu}^{bc}R^{da}_{\rho\sigma}\right)$
(27)

The complex conjugate $\hat{e}^{a\dagger}_{\mu}(x,\Theta)$ of the deformed
tetrad fields is obtien from the hermitian conjugate of the relation (23):

$\hat{e}^{a\dagger}_{\mu}(x,\Theta)=e^{a}_{\mu}(x)+i\Theta^{\nu\rho}e^{a}_{\mu\nu\rho}(x)+\Theta^{\nu\rho}\Theta^{\lambda\tau}e^{a}_{\mu\nu\rho\lambda\tau}(x)+\mathcal{O}(\Theta^{3})$
(28)

and the real deformed metric is given by the formula chai1 :

$\tilde{g}_{\mu\nu}(x,\Theta)=\frac{1}{2}\left[\hat{e}^{a}_{\mu}*\hat{e}^{b\dagger}_{\nu}+\hat{e}^{a}_{\nu}*\hat{e}^{b\dagger}_{\mu}\right]\eta_{ab}$
(29)

Using the Seiberg-Witten map (25), we can easily get the deformed tetrad
fields $\hat{e}^{a}_{\mu}(x,\Theta)$ and he’s hermitian conjugate
$\hat{e}^{a\dagger}_{\mu}(x,\Theta)$ given by the relations (23) and (28). To
simplify the calculations, we took only space-space non-commutativity,
$\Theta_{0i}=0$ (due to the known problem with unitary), so we choose the
following the metric for the non-commutativity parameter $\Theta^{\mu\nu}$:

$\Theta^{\mu\nu}=\left(\begin{matrix}0&0&0&0\\\ 0&0&0&\Theta\\\ 0&0&0&0\\\
0&-\Theta&0&0\end{matrix}\right),\qquad\mu,\nu=0,1,2,3$ (30)

where $\Theta$ is a real positive constant.

The non-zero components of the non-commutative tetrad fields
$\hat{e}^{a}_{\mu}$ are:

$\displaystyle\hat{e}^{0}_{0}=$ $\displaystyle
A(r)+\frac{\Theta^{2}}{32B^{4}(r)}\left\\{-4B(r)(4rB^{\prime}(r)A^{\prime\prime}(r)+A^{\prime}(r)(2B^{\prime}(r)+rB^{\prime\prime}(r)))\right.$
$\displaystyle\left.+16rA^{\prime}(r)B^{\prime
2}(r)+B^{3}(r)(A^{\prime}(r)B^{\prime}(r)+4A^{\prime\prime}(r))+B^{2}(r)(-3A^{\prime}(r)B^{\prime}(r)\right.$
$\displaystyle\left.+4(A^{\prime\prime}(r)+rA^{\prime\prime\prime}(r)))\right\\}sin^{2}\theta+\mathcal{O}(\Theta^{3})$
(31) $\displaystyle\hat{e}^{1}_{1}=$ $\displaystyle B(r)sin\theta
cos\phi+\frac{i\Theta}{4}B^{\prime}(r)sin\theta
sin\phi+\frac{\Theta^{2}}{64B^{3}(r)}\left\\{8(2B^{\prime}(r)-B(r)B^{\prime\prime}(r))sin^{2}\theta\right.$
$\displaystyle\left.+B^{3}(r)B^{\prime\prime}(r)(3+cos2\theta)+B(r)(B^{\prime
2}(r)-B(r)B^{\prime\prime}(r))(1+3cos2\theta)\right\\}sin\theta cos\phi$
$\displaystyle+\mathcal{O}(\Theta^{3})$ (32) $\displaystyle\hat{e}^{2}_{1}=$
$\displaystyle B(r)sin\theta sin\phi-\frac{i\Theta}{4}B^{\prime}(r)sin\theta
cos\phi+\frac{\Theta^{2}}{64B^{3}(r)}\left\\{8(2B^{\prime}(r)-B(r)B^{\prime\prime}(r))sin^{2}\theta\right.$
$\displaystyle\left.+B^{3}(r)B^{\prime\prime}(r)(3+cos2\theta)+B(r)(B^{\prime
2}(r)-B(r)B^{\prime\prime}(r))(1+3cos2\theta)\right\\}sin\theta sin\phi$
$\displaystyle+\mathcal{O}(\Theta^{3})$ (33) $\displaystyle\hat{e}^{3}_{1}=$
$\displaystyle\frac{\Theta^{2}sin^{2}\theta}{32B^{3}(r)}\left\\{(8-3B(r))B^{\prime
2}(r)-B(r)B^{\prime\prime}(r)(4+(-3+B(r))B(r))\right\\}cos\theta$
$\displaystyle+B(r)cos\theta+\mathcal{O}(\Theta^{3})$ (34)
$\displaystyle\hat{e}^{1}_{2}=$ $\displaystyle rcos\theta
cos\phi-\frac{i\Theta}{4}\left[B(r)-1\right]cos\theta
sin\phi+\frac{\Theta^{2}}{32B^{4}(r)}\left\\{B^{4}(r)B^{\prime}(r)(-3+cos2\theta)\right.$
$\displaystyle\left.+sin^{2}\theta\left[16rB^{\prime
2}(r)-B^{2}(r)(B^{\prime}(r)-4rB^{\prime\prime}(r))-4B(r)(2B^{\prime}(r)+2rB^{\prime
2}(r)+rB^{\prime\prime}(r))\right]\right.$
$\displaystyle\left.-\frac{1}{2}B^{3}(r)B^{\prime}(r)(-9+5cos2\theta)\right\\}cos\theta
cos\phi+\mathcal{O}(\Theta^{3})$ (35) $\displaystyle\hat{e}^{2}_{2}=$
$\displaystyle rcos\theta
sin\phi+\frac{i\Theta}{4}\left[B(r)-1\right]cos\theta
cos\phi+\frac{\Theta^{2}}{32B^{4}(r)}\left\\{B^{4}(r)B^{\prime}(r)(-3+cos(2\theta))\right.$
$\displaystyle\left.+sin^{2}\theta\left[16rB^{\prime
2}(r)-B^{2}(r)(B^{\prime}(r)-4rB^{\prime\prime}(r))-4B(r)(2B^{\prime}(r)+2rB^{\prime
2}(r)+rB^{\prime\prime}(r))\right]\right.$
$\displaystyle\left.-\frac{1}{2}B^{3}(r)B^{\prime}(r)(-9+5cos(2\theta))\right\\}cos\theta
sin\phi+\mathcal{O}(\Theta^{3})$ (36) $\displaystyle\hat{e}^{3}_{2}=$
$\displaystyle-
rsin\theta+\frac{\Theta^{2}sin\theta}{64B^{4}(r)}\left\\{sin^{2}\theta\left[4B(r)B^{\prime}(r)(4+B^{3}(r))-32rB^{\prime
2}(r)+8rB(r)B^{\prime\prime}(r)\right]\right.$
$\displaystyle\left.B^{2}(r)B^{\prime}(r)(5-B(r)+(-1+5B(r))cos(2\theta))+8rB(r)(B(r)B^{\prime\prime}(r)-2B^{\prime
2}(r))cos^{2}\theta\right\\}$ $\displaystyle+\mathcal{O}(\Theta^{3})$ (37)
$\displaystyle\hat{e}^{1}_{3}=$ $\displaystyle-rsin\theta
sin\phi-\frac{i\Theta}{4}\left[\left(B(r)-1\right)cos^{2}\theta-((1-\frac{1}{B(r)})+2\frac{B^{\prime}(r)}{B^{2}(r)}r)sin^{2}\theta\right]sin\theta
cos\phi$
$\displaystyle+\frac{\Theta^{2}}{32B^{4}(r)}\left\\{[+3B^{2}(r)B^{\prime}(r)+36rB^{\prime
2}(r)+8rB^{2}(r)B^{\prime\prime}(r)-B(r)(7B^{\prime}(r)+16rB^{\prime
2}(r)\right.$
$\displaystyle\left.+12rB^{\prime\prime}(r))]sin^{2}\theta+2B^{3}(r)B^{\prime}(r)-2B^{4}(r)B^{\prime}(r)cos^{2}\theta\right\\}(-sin\theta
sin\phi)+\mathcal{O}(\Theta^{3})$ (38) $\displaystyle\hat{e}^{2}_{3}=$
$\displaystyle rsin\theta
cos\phi+\frac{i\Theta}{4}\left[\left(B(r)-1\right)cos^{2}\theta-((1-\frac{1}{B(r)})+2\frac{B^{\prime}(r)}{B^{2}(r)}r)sin^{2}\theta\right](-sin\theta
sin\phi)$
$\displaystyle+\frac{\Theta^{2}}{32B^{4}(r)}\left\\{[+3B^{2}(r)B^{\prime}(r)+36rB^{\prime
2}(r)+8rB^{2}(r)B^{\prime\prime}(r)-B(r)(7B^{\prime}(r)+16rB^{\prime
2}(r)\right.$
$\displaystyle\left.+12rB^{\prime\prime}(r))]sin^{2}\theta+2B^{3}(r)B^{\prime}(r)-2B^{4}(r)B^{\prime}(r)cos^{2}\theta\right\\}(sin\theta
cos\phi)+\mathcal{O}(\Theta^{3})$ (39) $\displaystyle\hat{e}^{3}_{3}=$
$\displaystyle\frac{i\Theta}{4B(r)^{2}}\left[(-B(r)+B(r)^{3}+2rB^{\prime}(r))\right]sin^{2}\theta
cos\theta$ (40)

Then, using the definition (29), to obtain the non-zero components of the non-
commutative metric $\tilde{g}_{\mu\nu}$ up to the second-order of $\Theta$,
and we intend to analyze a geodesic movement over a plane
$\theta=\frac{\pi}{2}$, so the new metric will assume a simpler diagonal form:

$\displaystyle\tilde{g}_{00}=$
$\displaystyle-A^{2}(r)-\frac{A(r)\Theta^{2}}{16B^{4}(r)}\left\\{-4B(r)(4rB^{\prime}(r)A^{\prime\prime}(r)+A^{\prime}(r)(2B^{\prime}(r)+rB^{\prime\prime}(r)))\right.$
$\displaystyle\left.+16rA^{\prime}(r)B^{\prime
2}(r)+B^{3}(r)(A^{\prime}(r)B^{\prime}(r)+4A^{\prime\prime}(r))+B^{2}(r)(-3A^{\prime}(r)B^{\prime}(r)\right.$
$\displaystyle\left.+4(A^{\prime\prime}(r)+rA^{\prime\prime\prime}(r)))\right\\}+\mathcal{O}(\Theta^{3})$
(41) $\displaystyle\tilde{g}_{11}=$ $\displaystyle
B^{2}(r)+\frac{\Theta^{2}}{16B^{2}(r)}\left\\{B^{\prime
2}(r)\left(8+B(r)(-1+9B(r))\right)+B^{\prime\prime}(r)B(r)(-4+B(r)\right.$
$\displaystyle\left.+9B^{2}(r))\right\\}+\mathcal{O}(\Theta^{3})$ (42)
$\displaystyle\tilde{g}_{22}=$ $\displaystyle
r^{2}+\frac{\Theta^{2}}{16B^{4}(r)}\left\\{B^{\prime}(r)\left(-B(r)(1+B(r))+(8+B(r)(-5+2B(r)+16rB^{\prime}(r)))\right)\right.$
$\displaystyle\left.-4rB(r)B^{\prime}(r)\right\\}r+\mathcal{O}(\Theta^{3})$
(43) $\displaystyle\tilde{g}_{33}=$ $\displaystyle
r^{2}+\frac{\Theta^{2}}{16B^{4}(r)}\left\\{9B^{4}(r)-2B^{3}(r)(3-rB^{\prime}(r))+40r^{2}B^{\prime
2}(r)-rB(r)(B^{\prime}(r)(11+32rB^{\prime}(r))\right.$
$\displaystyle\left.+12rB^{\prime\prime}(r))+B^{2}(r)(1+27rB^{\prime}(r)+16r^{2}B^{\prime\prime}(r))\right\\}+\mathcal{O}(\Theta^{3})$
(44)

We can clearly see that, if $\Theta\rightarrow 0$, we obtain the commutative
metric (9).

##  3 Geodesic equation in the non-commutative Schwarzschild spacetime

The structure of spacetime in the non-commutative case it has given by the new
line element:

$\displaystyle ds^{2}$
$\displaystyle=\tilde{g}_{00}(r,\Theta)c^{2}dt^{2}+\tilde{g}_{11}(r,\Theta)dr^{2}+\tilde{g}_{22}(r,\Theta)d\theta^{2}+\tilde{g}_{33}(r,\Theta)d\phi^{2}$
(45)

Now, if we insert the Schwarzschild potential
$A(r)=B^{-1}(r)=\left(1-\frac{2m}{r}\right)^{\frac{1}{2}}$ into (41-42-43-44),
then we obtien the deformed Schwarzschild metric with a correction up to the
second order in $\Theta$:

$\displaystyle-\tilde{g}_{00}=$
$\displaystyle\left(1-\frac{2m}{r}\right)+\left\\{\frac{m\left(88m^{2}+mr\left(-77+15\sqrt{1-\frac{2m}{r}}\right)-8r^{2}\left(-2+\sqrt{1-\frac{2m}{r}}\right)\right)}{16r^{4}(-2m+r)}\right\\}\Theta^{2}+\mathcal{O}(\Theta^{3})$
(46) $\displaystyle\tilde{g}_{11}=$
$\displaystyle\left(1-\frac{2m}{r}\right)^{-1}+\left\\{\frac{m\left(12m^{2}+mr\left(-14+\sqrt{1-\frac{2m}{r}}\right)-r^{2}\left(5+\sqrt{1-\frac{2m}{r}}\right)\right)}{8r^{2}(-2m+r)^{3}}\right\\}\Theta^{2}+\mathcal{O}(\Theta^{3})$
(47) $\displaystyle\tilde{g}_{22}=$ $\displaystyle
r^{2}+\left\\{\frac{m\left(m\left(10-6\sqrt{1-\frac{2m}{r}}\right)-\frac{8m^{2}}{r}+r\left(-3+5\sqrt{1-\frac{2m}{r}}\right)\right)}{16(-2m+r)^{2}}\right\\}\Theta^{2}+\mathcal{O}(\Theta^{3})$
(48) $\displaystyle\tilde{g}_{33}=$ $\displaystyle
r^{2}+\left\\{\frac{5}{8}-\frac{3}{8}\sqrt{1-\frac{2m}{r}}+\frac{m\left(-17+\frac{5}{\sqrt{1-\frac{2m}{r}}}\right)}{16r}+\frac{m^{2}\sqrt{1-\frac{2m}{r}}}{(-2m+r)^{2}}\right\\}\Theta^{2}+\mathcal{O}(\Theta^{3})$
(49)

From these expressions, all the non-zero components of the metric acquire a
singularity in the NC correction term at $r=2m$, also in $\tilde{g}_{00}$
component, that’s what we don’t see in Ref. chai1 , this difference was the
result of using a general form of the tetrad filed.

The corresponding event horizon in the non-commutative Schwarzschild black
hole can be obtain by solving $\tilde{g}_{tt}=0$.

$r_{H}^{NC}=r_{H}^{C}+\frac{1}{32}\sqrt{8-2r_{H}^{C}}\Theta-(\frac{13r_{H}^{C}-48}{256r_{H}^{C}})\Theta^{2}$
(50)

where $r_{H}^{C}=2m$ is the event horizon of the Schwarzschild black hole in
the commutative spacetime when $\Theta=0$.

As we see in Fig.1, the event horizon in the NC spacetime is bigger than in
the commutative case $r_{H}^{NC}>r_{H}^{C}$, and new behaviors of
$\tilde{g}_{\mu\nu}$ are showing in this theory, where the singularity of the
Schwarzschild solution at $r=0$ is now shifted by the noncommutativity of
space to the finite radius $r=2m$. Then the interior region ($r<r_{H}$) became
banned from any observer, so the NC structure of the space-time gives a non-
singular black hole. Which is a logical result because we are using a general
form of the tetrad field for a stationary observer at the spatial infinity
(11). This result is not available in the diagonal form of tetrad in the NC
gauge theory as in the literature chai1 ; chai2 ; mukhe1  or in the theory of
non-singularity black hole like the noncommutativity eliminates point-like
gravitational source nozari ; larranaga ; nicolini1 ; bhar ; rahaman , Hayward
black hole hayw ; hayw1 ; hayw2 ; hayw3 . We can see this results in the
theory of quantum-corrected black hole qcbh1 ; qcbh2 ; nozari3 , but just in
the particular case where $a=r_{H}$, and $a$ in this theory represent a
minimal distance and it is expected of the order of the Planck length,
$l_{p}$, when the singularity of the black hole in this theory is shifted to
$r=a=r_{min}\sim l_{p}$, so is not a natural result because we need to fixed
the parameter $a$ for a particular value to observe the same result as in Fig
1. Contrary to our results, which is emerged naturally from the quantum
structure of the space-time itself, when we impose the NC property of the
geometry to the space-time, without the need to impose a particular value to
the NC parameter $\Theta$. Then we conclude that the NC geometry removes the
singularity at the origin of the black hole and increases the radius of the
event horizon.

Figure 1: Behaviors of $\tilde{g}_{tt}$ for a stationary observer at spatial
infinity in the non-commutative spacetime with a given $\Theta$.

The corresponding Lagrangian can be written according to the non-commutative
spacetime structure described by (45), as follows.

$\displaystyle 2L=$
$\displaystyle\tilde{g}_{tt}(r,\Theta)c^{2}\dot{t}^{2}+\tilde{g}_{rr}(r,\Theta)\dot{r}^{2}+\tilde{g}_{\phi\phi}(r,\Theta)\dot{\phi}^{2}$
(51)

where the dots represent the derivative with respect to the affine parameter
$\tau$, along the geodesic.

Use the Euler-Lagrange equation:

$\frac{d}{ds}\left(\frac{\partial
L}{\partial\dot{x}^{\mu}}\right)-\frac{\partial L}{\partial x^{\mu}}=0$ (52)

Since $L$ is independent of $t$ and $\phi$ so we have tow conserved
quantities:

$\displaystyle E_{0}$
$\displaystyle=p_{t}=c^{2}\tilde{g}_{tt}(r,\Theta)\dot{t}\Rightarrow\dot{t}=\frac{E_{0}}{c^{2}\tilde{g}_{tt}(r,\Theta)}$
(53) $\displaystyle l$
$\displaystyle=p_{\phi}=\tilde{g}_{\phi\phi}(r,\Theta)\dot{\phi}\Rightarrow\dot{\phi}=\frac{l}{\tilde{g}_{\phi\phi}(r,\Theta)}$
(54)

Use the invariant222where $U^{\mu}=c^{-1}\frac{dx^{\mu}}{d\tau}$ denote the
4-velocity. of $\tilde{g}_{\mu\nu}U^{\mu}U^{\nu}\equiv-h$, together with the
relations (53) and (54) and with some rearrange we can get the explicit
equation for $\dot{r}^{2}$:

$\dot{r}^{2}=-\frac{E_{0}^{2}}{c^{2}\tilde{g}_{tt}(r,\Theta)\tilde{g}_{rr}(r,\Theta)}-\frac{1}{\tilde{g}_{rr}(r,\Theta)}\left(\frac{l^{2}}{\tilde{g}_{\phi\phi}(r,\Theta)}+hc^{2}\right)$
(55)

where we shall consider $h=m_{0}^{2}$ for massive particles.

Substituting (46), (47) and (49) into (55), and with the developed up to
$\mathcal{O}(\Theta^{3})$ respect to $\Theta$, then the equation (55) can be
written as:

$\dot{r}^{2}+V_{eff}(r,\Theta)=0$ (56)

where:

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

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

$V_{eff}(r,\Theta=0)=\left(1-\frac{2m}{r}\right)\left(\frac{l^{2}}{r^{2}}+hc^{2}\right)-E^{2}$
(58)

(a)(b)(c)(d)

Figure 2: The behaviors of the effective potential for massive particle. (a)
different $\Theta$ and fixed: $E=0.998$, $m=1$, $l=4.2$. (b) different $m$ and
fixed: $E=0.998$, $\Theta=0.2$, $l=4.2m$. (c) different $E$ and fixed: $m=1$,
$\Theta=0.2$, $l=4.2$. (d) different $l$ and fixed: $E=0.998$, $\Theta=0.2$,
$m=1$.

In Figure 2, we show the influence of parameters ($\Theta$, $m$, $E$ and $l$)
on the effective potential for a massive particle. From this figure, we can
see that in the NC space-time all the extremes of the effective potential are
located outside the event horizon whatever the value of the parameter used,
and this deformed geometry adds a new minimum to this effective potential
witch give us multiples stable circular orbits. In (a) when $\Theta$ increases
the maximum peak of the curve decrease and shifted a little off the event
horizon. We need to note that the divergence around the event horizon is a
consequence of the non-commutative geometry, which play the role of barrier to
prevent particles at the high energy from falling into the event horizon. From
Fig. 2 (b), we can see that the increase of mass shifted the effective
potential off the event horizon and increases the depth of the potential well
in the NC space-times. Like we see in (c) the effective potential depends on
the energy of the test particle in the NC space-time (LABEL:eqt3.73), then the
increase of the energy applied the decrease in the level of the effective
potential and increase in the depth of the potential well. While for a
particle with weak energy $E\ll 1$, the new minimum of the effective potential
disappear then this particle falls into the event horizon. As another new note
from (d), we should mention that in the NC space-times it always exists a
minimum of the effective potential near the event horizon whatever the value
of the orbital momentum, and when we increase $l$ the depth of the potential
well decrease and shifted to the event horizon, and the other extremes of the
effective potential has been restored when we get $l_{crt}>2\sqrt{3}m$.

In this scenario the NC geometry plays the role of the potential well near the
event horizon when all matter absorbed by the black-hole is compressed into
this region before entering the event horizon, this leads to the formation of
an accretion disk with high-density and high-temperature around this black
hole and becomes very bright, this is so-called "Black Hole Accretion Disk
Theory" see Ref.BHAD1 ; BHAD2 ; BHAD3 ; BHAD4  which is also known in
astronomy as "Quasar", Ref. quasar1 ; quasar2 .

This new minimum showing in the behaviors of the effective potential in Fig.
2, can be found in other theories like a Reissner–Nordström charged black hole
RN1 ; RN2  or in the theory of non-singularity black hole hayw2 ; hayw3
,..etc, while this theory has a problem where this minimum is located inside
the event horizon which can not be interpreted as a stable circular orbit. In
our work, the non-commutativity shifted this new minimum to the outside of the
event horizon, which gives us a possibility to see it as a stable circular
orbit near the event horizon, we see all of that in the following section.

###  3.1 Stability condition

In which follows, we treat the circular orbits and the stability condition in
the NC space-time, to see how this deformed geometry affects this class of
orbits. For that, take the case of circular orbits ($\dot{r}=0$), the
corresponding effective potential must satisfy:

$V_{eff}(r,\Theta)=V^{2}(r,\Theta)-E^{2}=0$ (59)

Now, we can find the extreme of the non-commutative effective potential given
by the relation (LABEL:eqt3.73), to get the stable and unstable orbit, we need
to solve this equation:

$\frac{dV_{eff}}{dr}=0$ (60)

In NC space-time, the minimum value of $V_{eff}$ appears as soon as
$l\geqslant 0$, (if $l=0$, in this case, the NC parameter play the role of
angular momentum), this corresponds to the Newtonian case but the existence of
the maximum value of $V_{eff}$ requires a condition on the angular momentum
$l$, which is given by $l_{crt}>2\sqrt{3}m$, this corresponds to the
relativistic case in commutative space-time. It is shown that the
gravitational field gauge theory in NC Schwarzschild geometry using Seiberg-
Witten maps is equivalent to the Newtonian case and the relativistic case in
commutative Schwarzschild geometry.

Table 1: Some numerical values of unstable circular orbit $r_{uns}$ and the multiple stable circular orbit $r_{sta}$ in the commutative and the NC case 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
---|---|---|---|---|---|---
$r_{sta}(internal)$ |  | 2.16349 | 2.21421 | 2.25862 | 2.29837 | 2.33435
$r_{uns}$ | 3.83278 | 3.83684 | 3.8419 | 3.84894 | 3.85791 | 3.86876
$r_{sta}(external)$ | 13.8072 | 13.8074 | 13.8076 | 13.8078 | 13.8081 | 13.8086

This table show some numerical solution of equation (60), where it represents
the variation of unstable and multiple stable circular orbits as a function of
NC parameter $\Theta$, where the three type of circular orbit increase with
the increase of $\Theta$. This behavior can be seen in Fig. 3.

(a)(b)(c)(d)(e)(f)

Figure 3: The behaviors of the radius of circular orbit for a particle in the
NC space-time. Unstable and multiple stable circular orbit as function of
$\Theta$ and for fixed $l=4.2$, $E=0.998$, $m=1$ in (a), (b) and (c), and as
function of $l$ and for fixed $\Theta=0.2$, $E=0.998$, $m=1$ in (d), (e) and
(f).

We can conclude from Fig. 3 that as the NC parameter $\Theta$ increases, all
the types of radius increase in (a), (b), and (c). Therefore unstable circular
orbital has a greater radius in NC spaces as the parameter increases, this
indicates a strong gravitational field. Where we can see as the angular
momentum $l$ increase, the unstable and internal stable circular orbit
decrease, while the external stable circular orbit increase in (d), (e), and
(f).

In astrophysics, the innermost stable circular orbit (ISCO) has a big
importance to describe the motion of a test body around a compact object.
Where this class of orbits can be obtained from the stability condition, which
is given by:

$\frac{d^{2}V_{eff}}{dr^{2}}\geqslant 0$ (61)

The numerical solution of these condition show that, in the commutative
Schwarzschild space with $l_{crit}$, we get $r^{C}_{min}\geqslant 6$ and for a
NC Schwarzschild space using the Seiberg-Witten maps withe parameter
$\Theta=0.2$ we get two conditions of stability orbits (see Fig.4) $r_{s}\ll
r^{NC}_{min}\leqslant 2.46729$ and $r^{NC}_{min}\geqslant 6.00772$, which
corresponding to the multiple innermost stable orbits.

(a)(b)(c)

Figure 4: The condition for stability of circular orbits for different
$\Theta$ and fixed other parameter: (a) $E=1$, $l_{crit}=2\sqrt{3}$, $m=1$.
(b) $E=1$, $l_{crit}=\sqrt{3}$, $m=0.5$. (c) $E=1$, $l_{crt}=6\sqrt{3}/14$,
$m=3/14$.

The Figure 4, show the behaviors of the composite conditions equations (60)
and (61) for fixed $E$ and different other parameters $l_{crit}$, $m$ and
$\Theta$, as we see in the commutative space $\Theta=0$ we have just one
condition for the innermost stable circular orbit, but the NC space increase
this condition of the innermost stable circular orbit and he add a new
condition for the stable circular orbit near the event horizon of the static
black hole. Another note can be seen from the graph when we decrease the mass
of the black hole we found that the NC effect increase and that suggests to
the NC term correction to be in the proportionality to $(\propto\frac{1}{m})$.

(a)(b)

Figure 5: The condition for the stability of circular orbits for fixed $E=1$ .
(a) different $m$, $l_{crit}=2\sqrt{3}m$ and fixed $\Theta=0.2$. (b) different
$\Theta$ and fixed $m=0.50$

As we see in Fig 5, the behaviors for the stability condition of circular
orbits as a function of the mass $m$ in (a) and as a function of the NC
parameter $\Theta$ in (b) is showing, then we see with the increase of the
mass the two stability condition in the NC space-times increase, and the same
note when we increase the NC parameter this two stability condition increase.
From this behavior in Fig. 5, we can see that, the NC parameter $\Theta$ plays
the same role as the mass $m$, and that may be can use to explain the dark
matter in this universe.

Table 2: Some numerical solution for the radius condition of innermost stable circular orbit with different parameter $\Theta$ and fixed $E=1,\quad l_{crit}=2\sqrt{3}m$, $m$. (a) $m=1$, (b) $m=0.5$, (c) $m=3/14$. $\Theta$ | 0 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30
---|---|---|---|---|---|---
$r_{(a)min}\geqslant$ | 6 | 6.00127 | 6.00286 | 6.00507 | 6.00792 | 6.01138
$r_{s}\ll r_{(a)min}\leqslant$ |  | 2.39118 | 2.48542 | 2.5655 | 2.63613 | 2.69974
$r_{(b)min}\geqslant$ | 3 | 3.00254 | 3.00569 | 3.01008 | 3.01566 | 3.02241
$r_{s}\ll r_{(b)min}\leqslant$ |  | 1.28275 | 1.34987 | 1.40569 | 1.45373 | 1.49587
$r_{(c)min}\geqslant$ | 1.28571 | 1.29157 | 1.29869 | 1.3083 | 1.32011 | 1.33377
$r_{s}\ll r_{(b)min}\leqslant$ |  | 0.616476 | 0.657125 | 0.688445 | 0.713273 | 0.733258

Like we see in table 2, some numerical solutions are obtained according to
conditions equations (60) and (61) as show in Fig. 4, which represent the
variation of innermost stable circular orbit radius as function of $\Theta$,
which is found to increase with increasing $\Theta$. Where we see, the NC
space predicted a new stable circular orbit near the event horizon, which was
unknown in the commutative space, and all of this is shown in Fig 6.

From these two tables (1, 2), we can conclude that the NC space increases the
radius of the stable circular orbits and add a possibility of multiple stable
circular orbits near the event horizon of a static black hole.

$-6$$-4$$-2$$0$$2$$4$$6$$-6$$-4$$-2$$0$$2$$4$$6$ Figure 6: The position of the
innermost stable circular orbit with $E=1$, $m=1$, $h=1$,
$l_{crit}=2\sqrt{3}$, the circle with solid line represents ISCO for the
Schwarzschild black hole (black disk in center) in commutative case
$\Theta=0$. Dashed line represent the NC event horizon, dot lines represent
the new ISCO in internal region (near the event horizon), dot-dashed lines
represents ISCO in external region for the Schwarzschild black hole in the NC
case $\Theta=0.3$.

###  3.2 Orbital motion

In order to obtain the analytic formula for the periastron advance we need to
obtain the equation of motion (56) as a function of $\phi$, for that we need
to use the angular momentum equation (54), to write $r=r(\phi)$:

$\frac{dr}{d\tau}=\frac{dr}{d\phi}\frac{d\phi}{d\tau}=\frac{l}{\tilde{g}_{\phi\phi}(r,\Theta)}\frac{dr}{d\phi}$
(62)

we pot this equation into the equation (56), we get:

$\left(\frac{dr}{d\phi}\right)^{2}=-\frac{\tilde{g}_{\phi\phi}^{2}(r,\Theta)}{l^{2}}V_{eff}(r,\Theta)$
(63)

using the relations (56 and LABEL:eqt3.73), with case of a massive particle
$h=m_{0}^{2}$. We define a new variable $u=\frac{1}{r}$, and after some
algebra we can get this finale equation:

$\displaystyle\left(\frac{du}{d\phi}\right)^{2}$
$\displaystyle=\frac{(E^{2}-m_{0}^{2}c^{2})}{l^{2}}+\frac{2mm_{0}^{2}c^{2}}{l^{2}}u-u^{2}+2mu^{3}-\Theta^{2}\left\\{-u^{4}(1-2mu)\left(\frac{5}{8}-\frac{3}{8}\sqrt{1-2mu}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{16}mu\left(-17+\frac{5}{\sqrt{1-2mu}}\right)+\frac{m^{2}u^{2}}{(1-2mu)^{\frac{3}{2}}}\right)-\frac{2u^{2}}{l^{2}}\left(E^{2}+(-1+2mu)(m_{0}^{2}c^{2}+l^{2}u^{2})\right)\right.$
$\displaystyle\left.\times\left(\frac{5}{8}-\frac{3}{8}\sqrt{1-2mu}+\frac{1}{16}mu\left(-17+\frac{5}{\sqrt{1-2mu}}\right)+\frac{m^{2}u^{2}}{(1-2mu)^{\frac{3}{2}}}\right)\right.$
$\displaystyle\left.+\left(\frac{E^{2}mu^{3}(64u^{2}m^{2}+mu(-49+13\sqrt{1-2mu})+2(13-3\sqrt{1-2mu}))}{16l^{2}(1-2mu)^{2}}\right)\right.$
$\displaystyle\left.+\frac{mu^{3}(m_{0}^{2}c^{2}+l^{2}u^{2})(12u^{2}m^{2}+mu(-14+\sqrt{1-2mu})-(5+\sqrt{1-2mu}))}{8l^{2}(1-2mu)}\right\\}+\mathcal{O}(\Theta^{4})$
(64)

We use the effect that $mu\ll 1$, and we rewrite the above equation in the
linear form and we stop to the 3 order in $u$, so we find:

$\displaystyle\left(\frac{du}{d\phi}\right)^{2}$
$\displaystyle=\frac{(E^{2}-m_{0}^{2}c^{2})}{l^{2}}+\frac{2mm_{0}^{2}c^{2}}{l^{2}}u-u^{2}+2mu^{3}+\frac{\Theta^{2}}{2l^{2}}\left\\{(E^{2}-m_{0}^{2}c^{2})u^{2}+m(5m_{0}^{2}c^{2}-4E^{2})u^{3}\right\\}$
(65)

Derive the above equation with respect to $\phi$, then it yields:

$\displaystyle\frac{d^{2}u}{d\phi^{2}}+u=\frac{mm_{0}^{2}c^{2}}{l^{2}}+3mu^{2}+\frac{\Theta^{2}}{2l^{2}}\left\\{(E^{2}-m_{0}^{2}c^{2})u+\frac{3m}{2}(5m_{0}^{2}c^{2}-4E^{2})u^{2}\right\\}$
(66)

The above equation is the non-commutative geodesic equation.

(a)(b)

Figure 7: Time-like geodesic for a test particle $h=1$ around a non-
commutative Schwarzschild Black-Hole, with different values of $\Theta$ and
fixed other parameters in the plan $\theta=\frac{\pi}{2}$: (a)
$M=\frac{3}{14}$, $l=1.586$, $E=0.993$. (b) $M=\frac{3}{14}$, $l=0.915$,
$E=0.975$.

In figure 7, we plot the geodesic equation (65) for a massive particle around
a NC Schwarzschild Black-Hole, for different values of $l$ and $E$ with a
fixed black hole mass to $m=3/14$. Like we see in (a) and (b), the non-
commutativity of the spacetime decreases the major semi-axis of the particle
orbit, and he stays stable. This clarifies that the NC effects are responsible
for strengthening the strength of the gravitational field.

###  3.3 Periastron advance of Mercury orbit

Let’s write this equation into the perturbation form of the Keplerian
trajectory equation:

$\frac{d^{2}u}{d\phi^{2}}+u=\frac{m}{\tilde{l}^{2}}+\frac{g(u)}{\tilde{l}^{2}}$
(67)

where $\tilde{l}=\frac{l}{m_{0}c}$, and
$g(u)=3mu^{2}+\frac{\Theta^{2}}{2l^{2}}\left\\{(E^{2}-m_{0}^{2}c^{2})u+\frac{3m}{2}(5m_{0}^{2}c^{2}-4E^{2})u^{2}\right\\}$.
We follow the same steps in the Ref. adkins . The angle deviation after one
revolution it’s given by:

$\Delta\phi=\frac{\pi g_{1}}{\tilde{l}^{2}}$ (68)

where $g_{1}=\frac{dg(u)}{du}\mid_{u=\frac{1}{b}}$, and the distance $b$ is
defined by $b=m\alpha(1-e^{2})$, with $\alpha$, $e$ denote the major semi-axis
and the eccentricity of the movement. Using the relation (68), the angle
deviation in the NC space:

$\Delta\phi=\frac{6\pi
GM}{c^{2}\alpha(1-e^{2})}+\pi\Theta^{2}\left[\frac{(E^{2}_{0}/c^{2}-m^{2}_{0}c^{2})}{2m\alpha(1-e^{2})}+\frac{6(m^{2}_{0}c^{2}-E^{2}_{0}/c^{2})}{\alpha^{2}(1-e^{2})^{2}}+\frac{3m^{2}_{0}c^{2}}{2\alpha^{2}(1-e^{2})^{2}}\right]$
(69)

So, we found a result that is a little bit close with what was found in ulh1
where are used just the star product. But in our work, we are using the
Seiberg-Witten map. By using the relativistic relation of dispersion, we can
find:

$\Delta\phi=\frac{6\pi
GM}{c^{2}\alpha(1-e^{2})}+\pi\Theta^{2}\left[\frac{m^{2}_{0}v^{2}c^{2}}{2GM\alpha(1-e^{2})}-\frac{6m^{2}_{0}v^{2}}{\alpha^{2}(1-e^{2})^{2}}+\frac{3m^{2}_{0}c^{2}}{2\alpha^{2}(1-e^{2})^{2}}\right]$
(70)

It is clear that the first term represents the predictions of the well-known
general relativity and a correction which is given according to the NC
parameter.

For a numerical application, we take the case of Mercury planet. We found that
the NC perihelion shift:

$\mid\delta\phi_{NC}\mid=\left(1.96689\times
10^{43}\right)\Theta^{2}Kg^{2}.s^{-2}$ (71)

And for the General Relativity predicts and the observed perihelion shift for
Mercury is given in benczik :

$\displaystyle\delta\phi_{obs}$ $\displaystyle=2\pi\left(7.98734\pm
0.00037\right)\times 10^{-8}rad/rev$ (72) $\displaystyle\delta\phi_{GR}$
$\displaystyle=2\pi\left(7.98742\right)\times 10^{-8}rad/rev$ (73)

We compered the NC correction to the observable data
($\mid\delta\phi_{NC}\mid\approx\delta\phi_{obs}$), we can estimate the value
of $\Theta$:

$\Theta\approx 1.597\times 10^{-25}s.Kg^{-1}$ (74)

or

$\sqrt{\hbar\Theta}\approx 1.029\times 10^{-29}m$ (75)

Now, we can define a lower bound for $\Theta$:

$\mid\delta\phi_{NC}\mid\quad\leq\quad\mid\delta\phi_{GR}-\delta\phi_{obs}\mid\approx
2\pi(1\times 10^{-12})rad/rev$ (76)

So we get:

$\Theta\leq 5.0553\times 10^{-28}s.kg^{-1}$ (77)

or

$\sqrt{\hbar\Theta}\leq 5.7876\times 10^{-31}m$ (78)

It’s clear that the NC parameter $\Theta$ is very small and is remarkable that
our result is very close to the obtained in Ref. rome1 ; mirza , who used the
classical mechanics in NC flat space, we need to note that our result has a
difference of the order $10^{-1}$ with the result in Ref. rome1 , this
difference exists because we use a curved space-time, and in the Ref. mirza
adds a new degree of freedom $\gamma$ and for the specific value of $\gamma$
he can get the same our result. This result leads us to the same conclusion as
Ref rome1 , the planetary system is very sensitive to the NC parameter, in
this way the NC parameter plays the role of a fundamental constant of the
system to describe the microstructure of the space-time in this region. So any
small change in $\Theta$ implies a sensible change to our system at a large
scale.

If we compare our result with the Planck length, then we find
$\sqrt{\hbar\Theta}>L_{P}$. The lower bound for the NC parameter is also have
a lower bound which is the Planck scale $L_{P}$:

$\sqrt{\hbar\Theta}\leq(3.5808\times 10^{4})L_{P}$ (79)

Use the natural units we can obtain the upper bound of the energy:

$3.39\times 10^{14}Gev\leq\frac{1}{\sqrt{\hbar\Theta}}$ (80)

and this upper bound have also a upper bound which was Planck energy $E_{P}$.

##  4 Conclusions

In this paper, we have investigated the geodesic motion of a test particle in
the NC Schwarzschild space-time. By using the Seiberg-Witten map and a general
form of the tetrad field for the Schwarzschild black hole we show that all the
non-zero components of the deformed metric $\tilde{g}_{\mu\nu}(r,\Theta)$
acquire a singularity in the NC correction term at the value $r=2m$ where we
don’t see in Ref. chai1 , and this singularity in the component
$\tilde{g}_{00}$ remove the singularity at the origin $r=0$ of the black hole,
this result emerged naturally from the NC structure of the spacetime itself,
then we get a non-singularity black hole. And we show the event horizon in the
NC is bigger than in the commutative case $r^{NC}_{H}>r^{C}_{H}$, so the
Schwarzschild radius plays the role as the radius of the compact object inside
the NC black hole.

The NC effective potential of the particles in the NC Schwarzschild spacetime
is calculated and through detailed analysis new stable circular orbits appear
near the event horizon. Therefore, the geodetic structure of this black hole
presents new types of motion next to the event horizon within stable orbits
that are not allowed by Schwarzschild spacetime. This difference around the
event horizon is a result of non-commutative geometry, which acts as a barrier
to prevent particles from falling into the event horizon. As we found that in
NC spacetime, the commutativity parameter plays the same role as the mass of
black hole, which can be used to explain dark matter .

Finally, we find that the NC space-time decreases the major semi-axis of the
particles orbit, this indicates that the effects of the non-commutativity
increase the strength of the gravitational field. Then we obtained the NC
periastron advance of Mercury orbit and with the experimental data, we get
that a $\Theta$ parameter of the order $10^{-25}s.Kg^{-1}$, gives observable
deviation in the perihelion shift of Mercury, and the lower bound to
$\sqrt{\hbar\Theta}$ show that the NC propriety appears before the Planck
length scale. The upper bound to the energy of the order of $10^{14}Gev$
confirms that the NC properties of space-time appear at the High Energy.

###### Acknowledgements.

This work is supported by project B00L02UN050120190001, Univ. Batna 1,
Algeria.

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