Minimal Length Effects on Keplerian Scattering and Gravitational Lensing

TL;DR

This paper investigates how a minimal length scale, suggested by quantum gravity theories, affects Keplerian scattering and gravitational lensing, leading to measurable deviations in astrophysical observations.

Contribution

It derives the correction to scattering trajectories due to minimal length effects within deformed Heisenberg algebra frameworks, linking quantum gravity to observable phenomena.

Findings

Minimal length reduces scattering angles in Keplerian trajectories.

Quantum gravity effects weaken gravitational lensing, as seen in Einstein ring observations.

Estimated minimal length scales for electron and Mercury based on astrophysical data.

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.