Geodesic equation in non-commutative gauge theory of gravity

TL;DR

This paper develops a non-commutative gauge theory of gravity for spherically symmetric metrics, revealing that non-commutativity can remove black hole singularities, alter horizons, and produce observable effects at large scales.

Contribution

It introduces a second-order non-commutative deformation of gravity for spherical metrics, analyzing its effects on black hole properties and planetary orbits, with potential implications for dark matter and high-energy physics.

Findings

Non-commutativity removes Schwarzschild singularity.

New stable orbits appear near the event horizon.

Estimated non-commutative parameter affects planetary motion.

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 $\Theta^{\mu\nu}$, 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.