Thermodynamic Property of Schwarzschild Black Hole in Non-Commutative Gauge Theory of Gravity

Abstract

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.

On Modified First Law of Black hole Thermodynamics in The Non-Commutative Gauge Theory

“…In our previous works [24,25], we used the tetrad formalism and both the star * −product and the SW map to construct a non-commutative gauge theory for a static metric with spherical symmetric. One can use a perturbation form for the SW map to describe the deformed tetrad fields êa µ (x, Θ) as a development in the power of Θ up to the second-order, which can be obtained by following the same approach in Ref.…”

“…where the commutative entropy of SBH S = πr 2 h . In our previous work [24], we use the usual first law of black hole thermodynamics in the NC spacetime, which is defined as:…”

“…Our aim is to investigate the NC correction as a thermodynamically variable in the first law of BH thermodynamics, which is motivated by our previous work [24], where we found that in the phase transition of the NC SBH in presence of the pressure the NC parameter Θ play the same role as a thermodynamical variable.…”

“…In the same context as in Ref. [24], we study the influence of the pressure on phase transition and the stability condition, but in this work, we use the modified first law of the BH thermodynamics. Let's considered the pressure term in the relation (34), so we get:…”

“…The black hole thermodynamics play an important role in modern physics, and provide a real connection between gravity and quantum mechanics [12][13][14]. Recently there has been a lot of interest in modified thermodynamics proprieties of the black hole by introducing the non-commutativity [15][16][17][18][19][20][21][22][23][24].…”