Null Geodesics in Five Dimensional Manifolds

TL;DR

This paper investigates null geodesics in five-dimensional warped spaces, revealing how 4D particles can appear massive and experience forces due to extra-dimensional effects, with implications for observable physics.

Contribution

It provides a detailed analysis of 5D null geodesics, their relation to 4D physics, and explores the effects of extra dimensions on particle properties and motions.

Findings

Massless 5D particles can appear massive in 4D.

Null geodesics can induce a fifth force unless reparameterized.

Variable cosmological constant affects particle rest mass and time dilation.

Abstract

We analyze a class of 5D non-compact warped-product spaces characterized by metrics that depend on the extra coordinate via a conformal factor. Our model is closely related to the so-called canonical coordinate gauge of Mashhoon et al. We confirm that if the 5D manifold in our model is Ricci-flat, then there is an induced cosmological constant in the 4D sub-manifold. We derive the general form of the 5D Killing vectors and relate them to the 4D Killing vectors of the embedded spacetime. We then study the 5D null geodesic paths and show that the 4D part of the motion can be timelike -- that is, massless particles in 5D can be massive in 4D. We find that if the null trajectories are affinely parameterized in 5D, then the particle is subject to an anomalous acceleration or fifth force. However, this force may be removed by reparameterization, which brings the correct definition of the proper time into question. Physical properties of the geodesics -- such as rest mass variations induced by a variable cosmological ``constant'', constants of the motion and 5D time-dilation effects -- are discussed and are shown to be open to experimental or observational investigation.