Minimal Length Effects on Keplerian Scattering and Gravitational Lensing

Abstract

We study the impact of a minimal length, implied by generalized uncertainty principles and quantum gravity models, on unbounded (scattering) trajectories in the Kepler problem. The analysis is based on the precession of the Hamilton vector, which serves as a sensitive probe of orbital perturbations. Within the framework of the deformed Heisenberg algebra, we derive the correction to the trajectory arising from minimal length effects. It is shown that these quantum-gravitational corrections lead to a reduction in the scattering angle. In particular, for massless particles such as photons, the quantization of space results in a weakening of the gravitational lensing effect. Using available experimental data from the observation of the Einstein ring, we estimate the deformation parameter and the corresponding minimal length for the electron and Mercury. These findings highlight potential observational signatures of minimal length scenarios in high-energy astrophysics and gravitational optics.

Figures (2)

• Figure 1: Scattering trajectory near a central force center $S$. The impact parameter is denoted by $b$, and the scattering angle is $\theta$. The change in scattering angle due to perturbation is $\Delta\theta$, while $\Delta\varphi$ represents the deviation of the angle between the major axis of the hyperbola and the asymptote from its unperturbed value $\varphi_0$.
• Figure 2: Formation of an Einstein ring when the source $S$, the lens $L$, and the observer $O$ are perfectly aligned. The dashed ellipse represents the Einstein ring. The angle $\alpha_E$ denotes the Einstein angle. The angle $\theta$ is the deflection angle caused by the lens. The angle $\phi$ is measured between the original (unlensed) direction of the light and the line connecting the source to the lens.