Vertical stability of circular orbits in relativistic razor-thin disks

TL;DR

This paper establishes a vertical stability criterion for circular orbits in static, axially symmetric thin disks within general relativity, linking stability to the strong energy condition and introducing an approximate third integral for near-equatorial orbits.

Contribution

It provides the first analytical vertical stability criterion for geodesics crossing thin disk planes and introduces an approximate third integral for slightly inclined orbits.

Findings

Strong energy condition ensures vertical stability.

Adiabatic invariance yields an approximate third integral.

Results applicable to static spacetimes and potentially rotating disks.

Abstract

During the last few decades, there has been a growing interest in exact solutions of Einstein equations describing razor-thin disks. Despite the progress in the area, the analytical study of geodesic motion crossing the disk plane in these systems is not yet so developed. In the present work, we propose a definite vertical stability criterion for circular equatorial timelike geodesics in static, axially symmetric thin disks, possibly surrounded by other structures preserving axial symmetry. It turns out that the strong energy condition for the disk stress-energy content is sufficient for vertical stability of these orbits. Moreover, adiabatic invariance of the vertical action variable gives us an approximate third integral of motion for oblique orbits which deviate slightly from the equatorial plane. Such new approximate third integral certainly points to a better understanding of the analytical properties of these orbits. The results presented here, derived for static spacetimes, may be a starting point to study the motion around rotating, stationary razor-thin disks. Our results also allow us to conjecture that the strong energy condition should be sufficient to assure transversal stability of periodic orbits for any singular timelike hypersurface, provided it is invariant under the geodesic flow.