Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion

TL;DR

This paper introduces a geometric mapping principle that extends Einstein's equivalence principle to spacetimes with curvature and torsion, leading to new equations of motion and a quantum path integral formulation.

Contribution

It presents a nonholonomic mapping principle that derives laws in curved and torsioned spacetimes from flat spacetime laws, extending the equivalence principle.

Findings

Derives a new action principle for particles in torsioned spacetimes.

Shows trajectories are autoparallel due to torsion force.

Provides a generalized path integral transformation for quantum spaces.

Abstract

I explain the geometric basis for the recently-discovered nonholonomic mapping principle which permits deriving laws of nature in spacetimes with curvature and torsion from those in flat spacetime, thus replacing and extending Einstein's equivalence principle. As an important consequence, it yields a new action principle for determining the equation of motion of a free spinless point particle in such spacetimes. Surprisingly, this equation contains a torsion force, although the action involves only the metric. This force makes trajectories autoparallel rather than geodesic. Its geometric origin is the closure failure of parallelograms in the presence of torsion, A simple generalization of the mapping principle transforms path integrals from flat spacetimes to those with curvature and torsion, thus playing the role of a quantum equivalence principle, applicable at present only to spaces with gradient torsion.