  The gravitational stability of a two-dimensional self-gravitating and
differentially rotating gaseous disk in the context of post-Newtonian
(hereafter PN) theory is studied. Using the perturbative method and applying
the second iterated equations of PN approximation, the relativistic version of
the dispersion relation for the propagation of small perturbations is found. We
obtain the PN version of Toomre's local stability criterion by utilizing this
PN dispersion relation. In other words, we find relativistic corrections to
Toomre's criterion in the first PN approximation. Two stability parameters
$\eta$ and $\mu$ related to gravity and pressure are introduced. We illustrate
how these parameters determine the stability of the Newtonian and PN systems.
Moreover, we show that, in general, the differentially rotating fluid disk is
more stable in the context of PN theory relative to the Newtonian one. Also, we
explicitly show that although the relativistic PN corrections destabilize
non-rotating systems, they have the stabilizing role in the rotating thin
disks. Finally, we apply the results to the relativistic disks around
hypermassive neutron stars (HMNSs), and find that although Newtonian
description predicts the occurrence of local fragmentations, PN theory remains
in agreement with the relevant simulations, and rules out the existence of
local fragmentations.
