  We show that the fast moving component of the "bullet cluster" (1E0657-56)
can induce potentially resolvable redshift differences between multiply-lensed
images of background galaxies. The moving cluster effect can be expressed as
the scalar product of the lensing deflection angle with the tangential velocity
of the mass components, and it is maximal for clusters colliding in the plane
of the sky with velocities boosted by their mutual gravity. The bullet cluster
is likely to be the best candidate for the first measurement of this effect due
to the large collision velocity and because the lensing deflection and the
cluster fields can be calculated in advance. We derive the deflection field
using multiply-lensed background galaxies detected with the Hubble Space
Telescope. The velocity field is modeled using self-consistent
N-body/hydrodynamical simulations constrained by the observed X-ray and
gravitational lensing features of this system. We predict that the
triply-lensed images of systems "G" and "H" straddling the critical curve of
the bullet component will show the largest frequency shifts up to ~0.5 km/sec.
This is within the range of the Atacama Large Millimeter/sub-millimeter Array
(ALMA) for molecular emission, and is near the resolution limit of the new
generation high-throughput optical-IR spectrographs. A detection of this effect
measures the tangential motion of the subclusters directly, thereby clarifying
the tension with LCDM, which is inferred from gas motion less directly. This
method may be extended to smaller redshift differences using the Ly-alpha
forest towards QSOs lensed by more typical clusters of galaxies. More
generally, the tangential component of the peculiar velocities of clusters
derived by our method complements the radial component determined by the
kinematic SZ effect, providing a full 3-dimensional description of velocities.
