The principle of least action for test particles in a four-dimensional spacetime embedded in 5D

TL;DR

This paper derives the 4D equations of motion for test particles in a 5D spacetime using the principle of least action, revealing that deviations from geodesic motion are due to variable rest mass influenced by the extra dimension.

Contribution

It provides a direct derivation of 4D particle motion from the least action principle, linking mass variation to the extra dimension and clarifying the physical meaning of geometric quantities.

Findings

Deviations from 4D geodesic motion are due to variable rest mass.

Mass variation is induced by the scalar field and metric dependence on the extra coordinate.

The least action principle yields physically meaningful equations of motion.

Abstract

It is well known that, in the five-dimensional scenario of braneworld and space-time-mass theories, geodesic motion in 5D is observed to be non-geodesic in 4D. Usually, the discussion is purely geometric and based on the dimensional reduction of the geodesic equation in 5D, without any reference to the test particle whatsoever. In this work we obtain the equation of motion in 4D directly from the principle of least action. So our main thrust is not the geometry but the particle observed in 4D. A clear physical picture emerges from our work. Specifically, that the deviation from the geodesic motion in 4D is due to the variation of the rest mass of a particle, which is induced by the scalar field in the 5D metric and the explicit dependence of the spacetime metric on the extra coordinate. Thus, the principle of least action not only leads to the correct equations of motion, but also provides a concrete physical meaning for a number of algebraic quantities appearing in the geometrical reduction of the geodesic equation.