Effects of Lens Motion and Uneven Magnification on Image Spectra
  Counter to intuition, the images of an extended galaxy lensed by a moving
galaxy cluster should have slightly different spectra in any metric gravity
theory. This is mainly for two reasons. One relies on the gravitational
potential of a moving lens being time-dependent (the $\text{Moving}$
$\text{Cluster}$ $\text{Effect}$, $\text{MCE}$). The other is due to uneven
magnification across the extended, rotating source (the $\text{Differential}$
$\text{Magnification}$ $\text{Effect}$, $\text{DME}$). The time delay between
the images can also cause their redshifts to differ because of cosmological
expansion. This Differential Expansion Effect is likely to be small. Using a
simple model, we derive these effects from first principles.
  One application would be to the Bullet Cluster, whose large tangential
velocity may be inconsistent with the $\Lambda CDM$ paradigm. This velocity can
be estimated with complicated hydrodynamic models. Uncertainties with such
models can be avoided using the MCE. We argue that the MCE should be observable
with ALMA.
  However, such measurements can be corrupted by the DME if typical spiral
galaxies are used as sources. Fortunately, we find that if detailed spectral
line profiles were available, then the DME and MCE could be distinguished. It
might also be feasible to calculate how much the DME should affect the mean
redshift of each image. Resolved observations of the source would be required
to do this accurately.
  The DME is of order the source angular size divided by the Einstein radius
times the redshift variation across the source. Thus, it mostly affects nearly
edge-on spiral galaxies in certain orientations. This suggests that observers
should reduce the DME by careful choice of target, a possibility we discuss in
some detail.
