On the stability of general relativistic geometric thin disks

TL;DR

This paper investigates the stability of various general relativistic thin disk models under different perturbations, finding stability in Schwarzschild-based disks with radial pressure and instability in Chazy-Curzon and Zipoy-Voorhees models.

Contribution

It provides a comprehensive stability analysis of relativistic thin disks generated by different metrics under first-order perturbations, highlighting conditions for stability and instability.

Findings

Schwarzschild thin disks with radial pressure are stable.

Perturbations in Chazy-Curzon and Zipoy-Voorhees disks lead to instability.

Radial perturbations can promote ring formation in stable disks.

Abstract

The stability of general relativistic thin disks is investigated under a general first order perturbation of the energy momentum tensor. In particular, we consider temporal, radial and azimuthal "test matter" perturbations of the quantities involved on the plane $z=0$. We study the thin disks generated by applying the "displace, cut and reflect" method, usually known as the image method, to the Schwarzschild metric in isotropic coordinates and to the Chazy-Curzon metric and the Zipoy-Voorhees metric ($\gamma$-metric) in Weyl coordinates. In the case of the isotropic Schwarzschild thin disk, where a radial pressure is present to support the gravitational attraction, the disk is stable and the perturbation favors the formation of rings. Also, we found the expected result that the thin disk models generated by the Chazy-Curzon and Zipoy-Voorhees metric with only azimuthal pressure are not stable under a general first order perturbation