  The non-equilibrium contributions to the post-Newtonian hydrodynamic
equations are determined from a relaxation-time model of the post-Newtonian
Boltzmann equation. The Chapman-Enskog method is used to calculate the
non-equilibrium distribution function. The components of the energy-momentum
tensor are found from the knowledge of the non-equilibrium and the
post-Newtonian equilibrium Maxwell-J\"uttner distribution functions. The
linearized field equations for the mass, momentum and internal energy densities
coupled with the three Poisson equations of the post-Newtonian approximation
are investigated by considering a plane wave representation of the fields. The
constitutive equations for the viscous stress and heat flux vector are obtained
and it is shown that the transport coefficients of shear viscosity and heat
conductivity do depend on the Newtonian gravitational potential.
