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Originally Posted by Cougar
Not much, if the "object" is a photon.
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Of course not much, but even such a tiny effect after billions of years of accumulation finally becomes visible.
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Originally Posted by Cougar
No, that's called gravitational interaction.
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Yes. It is gravitational interaction. Astrophysicists call this kind of gravitational interaction "dynamical friction" though since it has certain particular features differentiating it from other kinds of gavitational interaction. We are free to use a different name but since this one is already used in the literature then changing the name might confuse some people. I propose not to multiply names unnecessarily and leave the name already used by specialists.
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Originally Posted by Cougar
More likely, it will be absorbed and re-emitted by dust particles or atoms or molecules, which opens up a whole slew of possibilities for what happens.
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Of course but I'm not concerned with this effect here since I analyze here only the dynamical friction.
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Originally Posted by Cougar
But if a photon makes it through without hitting anything, then depending on the mass of the cloud, it will be gravitationally redshifted as it climbs out of the cloud's gravity well.
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It may be shown that this effect is negligible comparing to the gravitational friction. You may notice that the photon before climbing out of a well has to fall into it first, and it leaves only the first well in which it was generated as a subject of any concern, while the effects of dynamical friction accumulate during the whole lifetime of the photon, from its source to the detector. So even being exceedingly small they may be not negligible.
For the time being I'm not commenting on the nature of gravitational redshift since popular saying "climbing out of gravity well", despite that it does not represent contemporary physics of the curved spacetime, suffices here as a didactic tool.
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Originally Posted by Cougar
Physicists and cosmologists are very well aware of gravitational redshifting. I don't know why you think it needs another name. 
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I don't need another name for "gravitational redshift" and I'm surprised that you think that I do, as I never expressed such a need.
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Originally Posted by Cougar
According to calculations, for very distant and very massive objects, the gravitational redshift is nevertheless negligible when compared to the cosmological redshift due to Hubble expansion. It contributes less than 1% of the measured redshift.
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You got it just right, however on what observation you base the assumption that it is an expansion? I call it just "Hubble redshift" since I don't see any reason to call it "due to Hubble expansion". As my calculations indicate, it is only the redshift due to dynamical friction. It seems a simpler solution since it does not require to invent new things and the old dynamical friction suffices to explain this redshift.
The only problem might be that someone has to calculate this redshift instead of assuming that it is Doppler shift which might be easier but not necessarily reflect the reality. I'm puzzled why it is so difficult to find any trace of calculations of the redshift resulting from the dynamical friction of photons in the literature of the subject. It is the reason why I made these calculations myself and presented the results here for astronomers to tell me why they are not interested in them (as editors of scientific journals keep telling me that it is the case and that's why those results aren't worth publishing in a scientific journal despite being formally right according to the referees who evaluated them).