The Dynamics of Test Particles and Pointlike Gyroscopes in the Brane World and Other 5D Models

TL;DR

This paper investigates the motion of test particles and gyroscopes in 5D brane world models, deriving gauge-invariant equations, analyzing confinement forces, and exploring spin dynamics with cosmological implications.

Contribution

It provides a covariant framework for particle and gyroscope dynamics in 5D models, including confinement forces and spin evolution, with applications to cosmology.

Findings

Derived gauge-invariant equations of motion for particles and gyroscopes.

Identified the form of non-gravitational forces confining particles to a 4D brane.

Found cosmological variations in the spin of confined gyroscopes.

Abstract

We study the dynamics of test particles and pointlike gyroscopes in 5D manifolds like those used in the Randall-Sundrum brane world and non-compact Kaluza-Klein models. Our analysis is based on a covariant foliation of the manifold using 3+1 dimensional spacetime slices orthogonal to the extra dimension, and is hence similar to the ADM 3+1 split in ordinary general relativity. We derive gauge invariant equations of motion for freely-falling test particles in the 5D and 4D affine parameterizations and contrast these results with previous work concerning the so-called ``fifth force''. Motivated by the conjectured localization of matter fields on a 3-brane, we derive the form of the classical non-gravitational force required to confine particles to a 4D hypersurface and show that the resulting trajectories are geometrically identical to the spacetime geodesics of Einstein's theory. We then discuss the issue of determining the 5D dynamics of a torque-free spinning body in the point-dipole approximation, and then perform a covariant (3+1)+1 decomposition of the relevant formulae (i.e. the 5D Fermi-Walker transport equation) for the cases of freely-falling and hypersurface-confined point gyroscopes. In both cases, the 4D spin tensor is seen to be subject to an anomalous torque. We solve the spin equations for a gyroscope confined to a single spacetime section in a simple 5D cosmological model and observe a cosmological variation of the magnitude and orientation of the 4D spin.