  It has long been recognized that the finite speed of light can affect the
observed time of an event. For example, as a source moves radially toward or
away from an observer, the path length and therefore the light travel time to
the observer decreases or increases, causing the event to appear earlier or
later than otherwise expected, respectively. This light travel time effect
(LTTE) has been applied to transits and eclipses for a variety of purposes,
including studies of eclipse timing variations (ETVs) and transit timing
variations (TTVs) that reveal the presence of additional bodies in the system.
Here we highlight another non-relativistic effect on eclipse or transit times
arising from the finite speed of light---caused by an asymmetry in the
transverse velocity of the two eclipsing objects, relative to the observer.
This asymmetry can be due to a non-unity mass ratio or to the presence of
external barycentric motion. Although usually constant, this barycentric and
asymmetric transverse velocities (BATV) effect can vary between sequential
eclipses if either the path length between the two objects or the barycentric
transverse velocity varies in time. We discuss this BATV effect and estimate
its magnitude for both time-dependent and time-independent cases. For the
time-dependent cases, we consider binaries that experience a change in orbital
inclination, eccentric systems with and without apsidal motion, and
hierarchical triple systems. We also consider the time-independent case which,
by affecting the primary and secondary eclipses differently, can influence the
inferred system parameters, such as the orbital eccentricity.
