Particle Motion and Perturbed Dynamical System in Warped Product Spacetimes

TL;DR

This study analyzes particle motion and dynamical systems in five-dimensional warped spacetimes, focusing on geodesic behavior, perturbations, and the effects of brane displacement on the system's energy and critical points.

Contribution

The paper introduces a dynamical systems approach to analyze geodesic motion in warped five-dimensional spacetimes, considering perturbations and brane displacements, which is a novel application in higher-dimensional gravity.

Findings

Energy function depends on warp factor and measures brane displacement perturbation.

Geodesic motion is affected by metric perturbations, altering critical points.

Perturbations influence null and timelike geodesic behavior near the brane.

Abstract

In this paper we have used the dynamical systems analysis to study the dynamics of a five-dimensional universe in the form of a warped product spacetime with a spacelike dynamic extra dimension. We have decomposed the geodesic equations to get the motion along the extra dimension and have studied the associated dynamical system when the cross-diagonal element of the Einstein tensor vanishes, and also when it is non-vanishing. In the first case, introducing the concept of an energy function along the phase path in terms of the extra-dimensional coordinate, we have examined how the energy function depends on the warp factor. The energy function has been used as a measure of the amount of perturbation caused by a brane displacement. Geometrically the effect of brane displacement is manifested in terms of a coordinate translation along the extra dimension, thereby producing a change in the geodesic motion along the extra dimension in the region close to the brane. Then we studied the geodesic motion under a conventional metric perturbation in the form of homothetic motion and conformal motion and examined the nature of critical points for a Mashhoon-Wesson-type metric. Finally we investigated the motion for null and timelike geodesics under the condition when the cross-diagonal element of the Einstein tensor is non-vanishing and examined the effects of perturbation on the critical points of the dynamical system.