Stability of perturbed geodesics in $nD$ axisymmetric spacetimes

TL;DR

This paper investigates the stability of perturbed circular geodesics in higher-dimensional axisymmetric spacetimes, analyzing how extra dimensions influence oscillation frequencies and stability of test particles in disk models.

Contribution

It extends stability analysis of geodesics to five and six-dimensional spacetimes, including Randall-Sundrum models, revealing conditions for stability in higher dimensions.

Findings

Stable geodesics are possible in 6D systems for radial perturbations.

In 5D systems, stability occurs only in specific cases for radial directions.

Vertical stability is achieved in all cases for both 5D and 6D spacetimes.

Abstract

The effect of self-gravity of a disk matter is evaluated by the simplest modes of oscillation frequencies for perturbed circular geodesics. It is plotted the radial profiles of free oscillations of an equatorial circular geodesic perturbed within the orbital plane or in the vertical direction. The calculation is carried out to geodesics of an axisymmetric $n$-dimensional spacetime. The profiles are computed by examples of disks embeded in five-dimensional or six-dimensional spacetime, where it is studied the motion of free test particles for three axisymmetric cases: (i) the Newtonian limit of a general proposed $5D$ and $6D$ axisymmetric spacetime; (ii) a simple Randall-Sundrum $5D$ spacetime; (iii) general $5D$ and $6D$ Randall-Sundrum spacetime. The equation of motion of such particles is derived and the stability study is computed for both horizontal and vertical directions, to see how extra dimensions could affect the system. In particular, we investigate a disk constructed from Schwarzschild and Chazy-Curzon solutions with a simple extension for extra dimensions in the case (i), and by solving vacuum Einstein field equations for a kind of Randall-Sundrum-Weyl metric in cases (ii) and (iii). We find that it is possible to compute a range of possible solutions where such perturbed geodesics are stable. Basicaly, the stable solutions appear, for the radial direction, in special cases when the system has $5D$ and in all cases when the system has $6D$; and, for the axial direction, in all cases when the system has both $5D$ or $6D$.