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.
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