Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

TL;DR

This paper reformulates the uncertainty principle as a geometric constraint on spacetime geodesics in general relativity, revealing a minimum length scale and a geodesic exclusion zone near singularities.

Contribution

It introduces a covariant geometric uncertainty principle (GeUP) that constrains geodesics and predicts a minimum length scale in spacetime, linking quantum uncertainty with classical geometry.

Findings

Confirmed a minimum length in spacetime geodesics.

Identified a geodesic exclusion zone near the Schwarzschild singularity.

Showed perturbations consistent with quantum gravity theories.

Abstract

The classical uncertainty principle inequalities were imposed over the general relativity geodesic equation as a mathematical constraint. In this way, the uncertainty principle was reformulated in terms of proper space-time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This reformulation confirmed the need for a minimum length of space-time line element in the geodesic, which depended on a Lorentz-covariant geodesic-derived scalar. In agreement with quantum gravity theories, GeUP imposed a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone was found around the singularity where uncertainty in space-time diverged to infinity.