Geodesic equation in non-commutative gauge theory of gravity

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 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.

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

“…This result is close to the one obtained in our previous work [24] and close to the result obtained in Ref. [11], and confirm that the NC propriety of the spacetime appears close to the Planck scale.…”

“…7 show the behavior of the black hole mass as a function of the horizon radius. In the commutative case Θ = 0, the mass has linear comportment with the horizon radius M = r h /2 and vanishes at the origin, while in the NC spacetime the mass vanishes before the origin in the r 0 h (in this case the NC parameter Θ play the role of the mass as we found in our previous work [24]), and this radius increase with the increase of Θ. Where in the region r h < r 0 h the mass of the black hole become negative, this behavior is responsible for the negative behavior of the temperature in Fig.…”

“…If we pot Θ → 0, we recover the commutative case (2.1), and in the equatorial plane θ = π 2 the diagonal form of the metric is obtain as in [24].…”

“…For the non-commutative correction to this metric, it has been obtained in Ref. [24], using the Seiberg-Witten map and the Moyal product. The deformed tetrad fields êa µ (x, Θ) can be described as a development in the power of Θ up to the second-order, which can be obtained by following the same approach in Ref.…”

“…The non-zero components of the NC tetrad fields êa µ are calculated in Ref. [24]. Then, using the definition (2.10), to obtain the non-zero components of the non-commutative metric ĝµν up to the second-order of Θ:…”