  We obtain the Maxwell-J\"uttner distribution function at first order in the
post-Newtonian approximation within the framework of general relativity. Taking
into account the aforesaid distribution function, we compute the particle
four-flow and energy-momentum tensor. We focus on the search of static
solutions for the gravitational potentials with spherical symmetry. In doing
so, we obtain the density, pressure and gravitational potential energy profiles
in terms of dimensionless radial coordinate by solving the aforesaid equations
numerically. In particular, we find the parametric profile for the equation of
state $p/\rho$ in terms of the dimensionless radial coordinate. Due to its
physical relevance, we also find the galaxy rotation curves using the
post-Newtonian approximation. We join two different kinds of static solutions
in order to account for the linear regime near the center and the typical
flatten behavior at large radii as well.

General Relativity and Quantum Cosmology;Astrophysics of Galaxies;High Energy Physics - Theory