  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.
