Estimating the strength of Lorentzian distribution in non-commutative geometry by solar system tests

Abstract: In this paper, we study four classical tests of Schwarzschild space-time with Lorentzian distribution in non-commutative geometry. We performed detailed calculations of the first-order corrections induced by the non-commutative parameter on planetary orbital precession, light deflection, radar wave delay, and gravitational redshift. The study showed that the impact of the non-commutative parameter on the time-like geodesics is significantly greater than its effect on the null geodesics. By using a series of precise experimental observations, the allowable range for the non-commutative parameter is ultimately constrained within $\Theta\leq0.067579~\mathrm{m}^{2}$, which is given by Mercury's orbital precession. This result aligns with the view that $\sqrt{\Theta}$ is of the order of the Planck length. Moreover, this constrained parameter range exceeds the Planck scale by a significant margin.