Confinement and focusing of geodesics in warped spacetimes

We have explored certain characteristic features of test particle trajectories in five dimensional, warped bulk geometries with a single thick brane. After a brief introduction on such spacetimes, we have first discussed timelike geodesic motion. The geodesic equations, which reduce to a first order autonomous dynamical system, have been solved using analytical and numerical methods. We have demonstrated how a growing (decaying) warp factor leads to oscillatory (runaway) trajectories, suggesting confinement (deconfinement). Further, we have pointed out differences that arise when we have a cosmological brane and/or a time-dependent extra dimension. Next, we have moved on to the kinematics of geodesic congruences. The evolution of the kinematical variables (expansion, rotation and shear) along geodesic flows have been obtained using analytical and numerical approaches, with particular emphasis on the required conditions and occurence of geodesic focusing