Suppose we place two galaxies 100 Mpc apart, with zero initial velocity with respect to each other. In other words, they are static with respect to each other initially, with a negligible gravitational attraction, say. They are static is in the physical distance space, not necessarily in the comoving coordinate system.
Now, as the universe expands according to FRW metric, will the distance between the two galaxies increase with time as $d => a(t)d$?
If no, then is the expansion of universe really an expansion of space, or is it just that all matter are moving away from each other due to some initial explosion, like the big bang? Will the light emitted by one galaxy be redshifted to the other galaxy, as there is no relative velocity in the physical distance space, and hence no doppler effect?
If yes, then isn't it a violation of conservation of momentum? Two bodies initially with zero relative velocity and zero interaction suddenly starts to move away from each other? In that case, if the relative motion synchronises with the hubble expansion again with them receding away from each other, then what is the nature of the force/energy which is supplying the potential well to "glue" particles to the expanding comoving coordinate system?
Best Answer
Short answer: the premise of your question is flawed, and therefore your actual questions don't really have answers beyond "that's not how it works."
Longer answer:
The issue with the premise of your question is that you cannot compare the velocities of two galaxies separated by 100 Mpc. The velocity is a quantity that is defined locally (formally, the velocity is a vector in the tangent space at the position of the object). On the other hand, 100 Mpc is a large enough distance that spacetime curvature effects due to the expansion are not negligible (we cannot pretend that the two galaxies live in the same tangent space). So the velocities cannot be compared directly. As an analogy, if two ants on different lines of longitude on Earth pointed their arms directly "North", it doesn't mean anything to ask whether the ants are pointing their arms in the "same" direction on the surface of the sphere. The definition of "North" depends on the tangent space of the sphere you find yourself in, and can't be compared between tangent spaces.
What you can say, is that the distance between galaxies increases with time as the Universe expands, and that there is a gravitational redshift that occurs as light travels from one galaxy to another.
Note that gravitational redshift is like the Doppler effect if you picture the galaxies as having a relative velocity, but it's not really the same thing (since you can't directly compare velocities over such large distances). You can give different physical interpretations to where this redshift comes from (personally I like saying that the expansion of space stretches the photon's wavelength as it travels from one galaxy to the next), but a mathematical derivation of this redshift effect that everyone will agree on is to project the tangent vector of the light's path into the observers' tangent spaces (attached to the locations of each galaxy), and using this projection show that each observer will assign a different frequency to the light.
A related issue with the way your question is framed is that the Big Bang did not happen at a single point; it happened everywhere in space. It's just that space itself was very small, in the sense that the distances between points were small compared to the distances we observe today, or perhaps the distances were actually zero.
Essentially, the issue is that you are trying to apply intuition from your experience with uncurved spaces, to a curved spacetime, and finding (correctly) that your intuition leads to contradictions. What this is telling you is that your intuition is wrong, and the correct explanations require knowledge of how curved spacetimes work. This is a very common issue for people learning GR. On one level, an answer to your questions is for me just say "trust me, the math works" (which is sort of what I'm doing). On another level, if you really want to understand what's going on, the full story is too long for me to fit it into an answer on this site, so I recommend (a) taking a course if that is an option for you, (b) working through a book (the ones by Schutz and Hartle are aimed at beginners), and/or (c) watching some videos (such as the ones by Susskind).