Minimal length discretization and properties of modified metric tensor and geodesics

TL;DR

This paper explores how minimal length discretization, incorporating gravitational effects into non-commutation relations, significantly alters spacetime geometry, leading to modified metric tensors and geodesic equations with higher-order derivatives.

Contribution

It introduces a novel approach to minimal length discretization that modifies the metric tensor and geodesic equations by including gravitational impacts and higher-order derivatives.

Findings

Modified metric tensor incorporates higher-order derivatives.

Altered geodesic equations reflect quantum-gravitational effects.

Properties of the new spacetime geometry are analyzed.

Abstract

We argue that the minimal length discretization generalizing the Heisenberg uncertainty principle, in which the gravitational impacts on the non--commutation relations are thoughtfully taken into account, radically modifies the spacetime geometry. The resulting metric tensor and geodesic equation combine the general relativity terms with additional terms depending on higher--order derivatives. Suggesting solutions for the modified geodesics, for instance, isn't a trivial task. We discuss on the properties of the resulting metric tensor, line element, and geodesic equation.