Geodesic equation in non-commutative gauge theory of gravity*

Abstract

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.

The Bound of the Non-Commutative Parameter Based on Gravitational Measurements

“…As a first step, we obtain a correction to the periastron advance of mercury up to the second order in θ [21], where our results show that the NC parameter is close to the Planck scale θ phy ∼ 10 −31 m, and the reach of this application to other planets of our solar system shows that θ phy is of the same order and acts as a fundamental constant of the solar system. Then, we compute the correction to the light deflection, the red shift, and the time delay in the NC spacetime.…”

“…In our previous studies [21], we used the tetrad formalism and both the star product and the SW map [22] to construct the NC gauge theory for a static metric with spherical symmetry. A perturbation form for the SW map can also be used to describe the deformed tetrad fields êa µ as a development in the power of θ up to the second order, which can be obtained by following the same approach mentioned in Ref.…”

“…The deformed tetrad fields are calculated in Ref. [21]; thus, we follow the same step to compute the deformed metric components of SBH in the equatorial plane θ = π 2 ,…”

“…In our previous study [21], we derive the expression of the angle deviation after one revolution in the NC SBH metric ( 7)-( 10) using the perturbation form of the geodesic equation outlined in Ref. [24].…”

“…Recently, there has been a lot of research in determining a lower bound of the NC parameter and studies on quantum gravity effects, with several approaches [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Our aim is to estimate a lower bound on the NC parameter using the four classical experimental tests of GR inspired by the NC geometry based on the gauge theory of gravity and compared to the other results obtained by another approach of NC geometry [12,16].…”