  There is growing interest in testing alternative gravity theories using the
subtle gravitational redshifts in clusters of galaxies. However, current models
all neglect a transverse Doppler redshift of similar magnitude, and some models
are not self-consistent. An equilibrium model would fix the gravitational and
transverse Doppler velocity shifts to be about 6sigma^2/c and 3sigma^2/2c in
order to fit the observed velocity dispersion sigma self-consistently. This
result comes from the Virial Theorem for a spherical isotropic cluster, and is
insensitive to the theory of gravity. A gravitational redshift signal also does
not directly distinguish between the Einsteinian and f(R) gravity theories,
because each theory requires different dark halo mass function to keep the
clusters in equilibrium. When this constraint is imposed, the gravitational
redshift has no sensitivity to theory. Indeed our N-body simulations show that
the halo mass function differs in f(R), and that the transverse Doppler effect
is stronger than analytically predicted due to non-equilibrium.
