Vertical stability of circular orbits in relativistic razor-thin 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.

Stability aspects of relativistic thin magnetized disks

“…whereδ(z) stands for the covariant Dirac δ-function [29], R σ αγβ is the (smooth) Riemann curvature tensor for z = 0, and…”

“…Assuming the system to be symmetric under reflections z → −z, one can calculate σ, P r , and P ϕ for static axisymmetric spacetimes with metric (2), leading to [29] …”

“…Since (2) is static and axisymmetric, its geodesic equations will have two independent constants of motion: the total energy H and the angular momentum around the z direction L z . The (reduced) Hamiltonian for the geodesics of the metric (2) is given by [29] …”

“…Such * vpachecof@gmail.com † asaa@ime.unicamp.br parameters can be chosen to achieve some desired physical properties, with special emphasis, of course, to the stability of the solution. For the stability analysis of the disk, we consider, besides the usual generalized radial Rayleigh criteria [27,28], also the vertical stability (oblique orbits) [29]. We show that one indeed has a large continuous family of magnetized disks satisfying these stability criteria, which could be useful, in principle, for the study of the abundant astrophysical situations involving disks of matter and magnetic fields.…”

Curvature dependence of relativistic epicyclic frequencies in static, axially symmetric spacetimes

“…In particular, g µν,θ = 0 on the equatorial plane (defined by θ = π/2). We must stress that the formalism presented in this paper is valid for smooth metrics; the vertical stability criteria for circular orbits in razor-thin disks were analyzed in [6,7]. We adopt the conventions of [5].…”