Stability of general relativistic static thick disks: the isotropic Schwarzschild thick disk

We study the stability of general relativistic static thick disks, as an application we consider the thick disk generated by applying the ``displace, cut, fill and reflect" method, usually known as the image method, to the Schwarzschild metric in isotropic coordinates. The isotropic Schwarzschild thick disk obtained from this method is the simplest model to describe, in the context of General Relativity, real thick galaxies. The stability under a general first order perturbation of the disk is investigated. The first order perturbation, when applying to the conservation equations, leads to a set of differential equations that are, in general, not self-consistent. This opens the possibility of performing various kinds of perturbations to transform the resulting system of equations into a self-consistent system. We perform a complete classification of the perturbations as well as the stability analysis for all the relevant physical perturbations. We found that, in general, the isotropic Schwarzschild thick disk is stable under these kinds of perturbations.