This is an interesting question! Suppose that
the object we are considering is spherical, and
first let’s consider that the light source is
coming from the center of the object. If we
assume that the medium of the object does not
interfere with the light as it passes through the
object, then we would not observe any red shifting
via the Doppler Effect. We would observe some
sort of gravitational red shift.
However, if the light source were at the
surface of the object, then we would see some type
of red shifting. This is because the [front]
surface is moving away from us due to its
contraction, and hence it would be red shifted via
the Doppler Effect.
Interestingly, if the object were transparent to
its own light source, then you would observe a
blue-shift of light from the other side.
Ultimately, the red shifting is likely to be
negligible because the object would need to
contract/implode at an immense rate in order to
have sufficient velocity for the Doppler Effect.
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