Jacobi stability of the vacuum in the static spherically symmetric brane world models

TL;DR

This paper investigates the stability of vacuum solutions in brane world models using Jacobi stability analysis, revealing that stability depends on specific parameters and that unstable solutions exhibit chaotic behavior.

Contribution

It applies the Kosambi-Cartan-Chern (KCC) Jacobi stability theory to brane world vacuum solutions, providing new insights into their stability properties.

Findings

Jacobi stability is limited to a small parameter range.

Unstable trajectories exhibit chaotic behavior.

Jacobi analysis constrains physical properties of brane vacuum.

Abstract

We analyze the stability of the structure equations of the vacuum in the brane world models, by using both the linear (Lyapunov) stability analysis, and the Jacobi stability analysis, the Kosambi-Cartan-Chern (KCC) theory. In the brane world models the four dimensional effective Einstein equations acquire extra terms, called dark radiation and dark pressure, respectively, which arise from the embedding of the 3-brane in the bulk. Generally, the spherically symmetric vacuum solutions of the brane gravitational field equations, have properties quite distinct as compared to the standard black hole solutions of general relativity. We close the structure equations by assuming a simple linear equation of state for the dark pressure. In this case the vacuum is Jacobi stable only for a small range of values of the proportionality constant relating the dark pressure and the dark radiation. The unstable trajectories on the brane behave chaotically, in the sense that after a finite radial distance it would be impossible to distinguish the trajectories that were very near each other at an initial point. Hence the Jacobi stability analysis offers a powerful method for constraining the physical properties of the vacuum on the brane.