  The observed flat rotation curves of galaxies require either the presence of
dark matter in Newtonian gravitational potentials or a significant modification
to the theory of gravity at galactic scales. Detecting relativistic Doppler
shifts and gravitational effects in the rotation curves offers a tool for
distinguishing between predictions of gravity theories that modify the inertia
of particles and those that modify the field equations. These higher-order
effects also allow us in principle, to test whether dark matter particles obey
the equivalence principle. We calculate here the magnitudes of the relativistic
Doppler and gravitational shifts expected in realistic models of galaxies in a
general metric theory of gravity. We identify a number of observable quantities
that measure independently the special- and general-relativistic effects in
each galaxy and suggest that both effects might be detected in a statistical
sense by combining appropriately the rotation curves of a large number of
galaxies.
