In this paper, we constructed the deformed Schwarzschild black hole in the non-commutative (NC) gauge theory of gravity. Where the singularity at the origin is shifted by the non-commutativity to the finite radius r h = 2m and changed the spherical symmetry of the black hole to the ellipse symmetry (deformed the NC event horizon). The thermodynamics property of the NC black hole is analyzed. As a first step, we describe the Hawking temperature and the entropy of the NC Schwarzschild black hole, where the result shows a difference in the pole-equator temperature of the black hole, and give us a new scenario of the black hole evaporation. Our estimation for the NC parameter is close to the Planck scale Θ ≈ 2.257 × 10 −35 m. Then, the description of the ADM mass, the heat capacity, and the Gibbs free energy of the deformed black hole show the effect of the NC gauge theory on the thermodynamic stability and the phase transitions. Finally, we investigated the influence of the pressure of the black hole in the first law of thermodynamics on the stability and the phase transition of the Schwarzschild black hole in the NC spacetime, where the result shows that the NC parameter plays a similar role as the thermodynamical function and defined the critical point as the pressure in the modified first law of thermodynamics which means a second-order phase transition or a continuous one. The stability and the phase transition of the Schwarzschild black hole are affected by the non-commutativity.

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 Θ µν , 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 Θ are obtained. We have also specified the problem of Mercury's perihelion and used the experimental data to estimate the NC parameter Θ, then we show that Θ 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.

In this work, we construct a non-commutative gauge theory of the modified structure of the gravitational field using the Seiberg-Witten map and the general tetrad fields of Schwarzschild space-time, to show that the non-commutative geometry removes the singularity in the origin of the black hole, thus obtaining a non-singular Schwarzschild black hole. 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 the ordinary Schwarzschild spacetime. The noncommutative periastron advance of Mercury orbit are obtained, and with the available experimental data, we find a parameter of non-commutativity of order of 10−25 s.kg−1. This result shows that the new fundamental length √~ is of order 10−31 m.

In this paper, we investigated the thermodynamic properties of Schwarzschild black hole (SBH) in the non-commutative (NC) gauge theory of gravity. According to our previous work, we modify the first law of the black hole (BH) thermodynamics by the physical quantity (NC potential) A which is the conjugate to the NC parameter Θ, which leads to this expression d M = T d Ŝ + AdΘ. Our result shows that the NC SBH has a phase transition, and the non-commutativity affected this transition. And the NC potential A is effective only in the final stage of the BH evaporation, where it increases the Gibbs free energy at this stage, and the NC parameter in this study can represent the tension of the spacetime. Then the study of the pressure of the SBH in the modified first law of the BH thermodynamic shows a second-order phase transition, and the critical value of the thermodynamical variables are related to each value of Θ, and that leads to this parameter to play the same role as a thermodynamical variable.

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