Testing gravity theories via transverse Doppler and gravitational redshifts in galaxy clusters

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 6\sigma^2/c and 3\sigma^2/2c in order to fit the observed velocity dispersion \sigma self-consistently. This result is from the Virial Theorem for a spherical isotropic cluster, and is insensitive to the theory of gravity. In any case, a gravitational redshift signal cannot directly distinguish between the Einsteinian and f(R) gravity theories, because the mass of the cluster dark halo must be treated as an unknown fitting parameter, whose value must vary according to the theory adopted, otherwise the system would be in equilibrium in one gravity theory and out of equilibrium in another.