Geodesic congruences in warped spacetimes

TL;DR

This paper investigates the behavior of timelike geodesic congruences in various warped five-dimensional spacetimes, deriving analytical and numerical results to understand how branes and extra dimensions influence geodesic evolution.

Contribution

It provides analytical expressions for ESR parameters in warped geometries with branes and explores their evolution through numerical solutions, extending understanding of geodesic flows in such spacetimes.

Findings

Analytical formulas for expansion scalar in Randall--Sundrum geometry.

Effects of thin branes on geodesic expansion evolution.

Numerical analysis of ESR evolution with initial conditions and curvature.

Abstract

In this article, we explore the kinematics of timelike geodesic congruences in warped five dimensional bulk spacetimes, with and without thick or thin branes. Beginning with geodesic flows in the Randall--Sundrum AdS (Anti de Sitter) geometry without and with branes we find analytical expressions for the expansion scalar and comment on the effects of including thin branes on its evolution. Later, we move on to congruences in more general warped bulk geometries with a cosmological thick brane and a time-dependent extra dimensional scale. Using analytical expressions for the velocity field, we interpret the expansion, shear and rotation (ESR) along the flows, as functions of the extra dimensional coordinate. The evolution of a cross-sectional area orthogonal to the congruence, as seen from a local observer's point of view, is also shown graphically. Finally, the Raychaudhuri and geodesic equations in backgrounds with a thick brane are solved numerically in order to figure out the role of initial conditions (prescribed on the ESR) and spacetime curvature on the evolution of the ESR.