Measuring Gravitational Redshifts in Galaxy Clusters
  Wojtak {\it et al} have stacked 7,800 clusters from the SDSS survey in
redshift space. They find a small net blue-shift for the cluster galaxies
relative to the brightest cluster galaxies, which agrees quite well with the
gravitational redshift from GR. Zhao {\it et al.} have pointed out that, in
addition to the gravitational redshift, one would expect to see transverse
Doppler (TD) redshifts, and that these two effects are generally of the same
order. Here we show that there are other corrections that are also of the same
order of magnitude. The fact that we observe galaxies on our past light cone
results in a bias such that more of the galaxies observed are moving away from
us in the frame of the cluster than are moving towards us. This causes the
observed average redshift to be $\langle \delta z \rangle = -\langle \Phi
\rangle + \langle \beta^2 \rangle / 2 + \langle \beta_x^2 \rangle$, with
$\beta_x$ is the line of sight velocity. That is if we average over galaxies
with equal weight. If the galaxies in each cluster are weighted by their
fluence, or equivalently if we do not resolve the moving sources, and make an
average of the mean redshift giving equal weight per photon, the observed
redshift is then opposite to the usual transverse Doppler effect. In the WHH
experiment, the weighting is a step-function because of the flux-limit for
inclusion in the spectroscopic sample and the result is different again, and
depends on the details of the luminosity function and the SEDs of the galaxies.
Including these effects substantially modifies the blue-shift profile. We show
that in-fall and out-flow have very small effect over the relevant range of
impact parameters but become important on larger scales.
