Minimal length discretization and properties of modified metric tensor and geodesics

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.