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

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.