  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.

General Relativity and Quantum Cosmology