Consequences of Minimal Length Discretization on Line Element, Metric Tensor and Geodesic Equation

TL;DR

This paper explores how minimal length uncertainty from the generalized uncertainty principle (GUP) affects the line element, metric tensor, and geodesic equations in general relativity, revealing potential quantum geometric effects.

Contribution

It introduces a method to incorporate GUP into classical gravitational equations, leading to modified line elements and geodesic equations that include higher-order kinematic effects.

Findings

Modified line element and geodesic equations incorporating GUP effects

Emergence of acceleration, jerk, and snap in particle motion

Potential explanations for accelerating expansion phenomena

Abstract

When minimal length uncertainty emerging from generalized uncertainty principle (GUP) is thoughtfully implemented, it is of great interest to consider its impacts on {\it "gravitational} Einstein field equations (gEFE) and to try to find out whether consequential modifications in metric manifesting properties of quantum geometry due to quantum gravity. GUP takes into account the gravitational impacts on the noncommutation relations of length (distance) and momentum operators or time and energy operators, etc. On the other hand, gEFE relates {\it classical geometry or general relativity gravity} to the energy-momentum tensors, i.e. proposing quantum equations of state. Despite the technical difficulties, we confront GUP to the metric tensor so that the line element and the geodesic equation in flat and curved space are accordingly modified. The latter apparently encompasses acceleration, jerk, and snap (jounce) of a particle in the {\it "quasi-quantized"} gravitational field. Finite higher-orders of acceleration apparently manifest phenomena such as accelerating expansion and transitions between different radii of curvature, etc.