The Motion of Massive Test Particles in the Dark Matter with an a₀/r² Energy Density

The motion of massive test particles in dark matter is studied. It is shown that if the energy density of the dark matter making up a galactic halo has a large $r$ behavior of $1/r^2$, then contrary to intuition the motion of these test particles are not govern by Newtonian gravity, but rather by the equations of geodesic motion from Einstein's theory of general relativity. Moreover, the rotational velocity curves of orbiting massive test particles in this energy density do not approach a constant value at large $r$ but will instead always increase with the radius of the orbit $r_c$.