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If a moving Galaxy/Cluster doesn't have radially outward motion component from us, we are unable to detect its redshift. The answer may look obvious: There's no wavelength squeezing/expanding ju-ju.

But, here's my problem: As universe expands, the gravitational grip becomes weaker because it's inversely proportional to square of the distance. So, technically, when we receive a light (which is actually from past), light has travelled from strong gravitational field to weak gravitational field (I am talking about whole universe here) which is subject to Gravitational Time Dilation. As frequency is inversely proportional to time, there must be redshift of all lights we receive.

Then, why are Galaxies/Clusters moving parallel to us not redshifted?

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I think, you mix two or even three kinds of redshift effects.

First, the classical Doppler shift due to which light emitted from a moving object is shifted to the red or blue. This is dependent on relative movement of sender and receiver only. For transverse motion we still get a redshift (see Wikipedia URL below), but it is very small for low speeds.

Second, there is the cosmological redshift which occurs in an expanding universe (or a blueshift in a contracting one). This is only dependent on the universe's sizes at the times of emitting and receiving the light and not affected by any kind of peculiar speeds, neither parallel nor radial. However, if a galaxy is very near to the Milky Way, it is not following the Hubble flow (the expansion of the universe) but rather stays gravitationally bound to it, so there is no cosmological redshift.

Third, the gravitational redshift which occurs if the photon is escaping a gravitational potential. This is only dependent on the source's mass (or whatever mass is creating a potential well in the emission point. If the universe is perfectly homogeneous, then there are no potential wells and no gravitational redshift. In reality it's not, but I guess, this is not the effect you are interested in.

I think, this table summarizes pretty much what I wrote so far: http://en.wikipedia.org/wiki/Redshift#Redshift_formulae

If this doesn't answer your question, please specify the conditions more precisely so that it is clear which types of redshift apply in which way.

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