Non-relativistic Geometry and the Equivalence Principle

TL;DR

This paper presents a geometric formulation of the equivalence principle in non-relativistic physics, linking it to symmetry principles and showing its implications for classical and quantum systems.

Contribution

It introduces a symmetry-based approach to non-relativistic gravity, connecting the equivalence principle to time-reparameterization symmetry and clarifying wave-function transformations.

Findings

The Newtonian potential can be replaced by a curved spatial metric.

The equivalence principle is tied to a remnant of relativistic time-reparameterization symmetry.

Quantum wave-functions transform under frame changes in a specific way.

Abstract

We describe a geometric and symmetry-based formulation of the equivalence principle in non-relativistic physics. It applies both on the classical and quantum levels and states that the Newtonian potential can be eliminated in favor of a curved and time-dependent spatial metric. It is this requirement that forces the gravitational mass to be equal to the inertial mass. We identify the symmetry responsible for the equivalence principle as the remnant of time-reparameterization symmetry of the relativistic theory. We also clarify the transformation properties of the Schroedinger wave-function under arbitrary changes of frame.