Earliest stages of the non-equilibrium in axially symmetric,
  self-gravitating, dissipative fluids
  We report a study on axially and reflection symmetric dissipative fluids,
just after its departure from hydrostatic and thermal equilibrium, at the
smallest time scale at which the first signs of dynamic evolution appear. Such
a time scale is smaller than the thermal relaxation time, the thermal
adjustment time and the hydrostatic time. It is obtained that the onset of
non--equilibrium will critically depend on a single function directly related
to the time derivative of the vorticity. Among all fluid variables (at the time
scale under consideration), only the tetrad component of the anisotropic tensor
in the subspace orthogonal to the four--velocity and the Killing vector of
axial symmetry, shows signs of dynamic evolution. Also, the first step toward a
dissipative regime begins with a non--vanishing time derivative of the heat
flux component along the meridional direction. The magnetic part of the Weyl
tensor vanishes (not so its time derivative), indicating that the emission of
gravitational radiation will occur at later times. Finally, the decreasing of
the effective inertial mass density, associated to thermal effects, is clearly
illustrated.
