This is a question I've been mulling over for a while and I'm hoping someone here can point me in the right direction. Sorry if it's a bit of a novice question. For the record, I don't fully know GR, but don't let that stop you from using it in the answer.
Since the universe is expanding - that is, the spacetime metric is expanding by way of a near-exponentially increasing scale factor - we can say that the distance between any two non-bound objects is increasing over time. Herein lays my dilemma; if there were two objects separated by a large distance that had no relative velocities initially, after a long time, the effects of expansion would cause them to have large apparent velocities away from each other. Given that there hasn't been any acceleration to cause these velocities, are there still relativistic effects in play? That is, is there time dilation between the two frames?
Furthermore, given long enough time, the rate of increasing distance between the two objects could place them outside of their visible horizon (ie they are travelling away from each other at superluminal velocities). Since there was still no acceleration to achieve this feat, what can one say about the relativistic effects in this case?
At first I thought this was an easy question. I thought of course there would be relativistic effects and when the objects go superluminal, the visible horizon is there to ensure there can never be causal contact and thus preserve physics. But then I thought what if spacetime stopped expanding abruptly (seems crazy but as far as I know, nothing makes this completely impossible)? Since there was no initial relative velocities, wouldn't the two objects return to being in the same inertial frame? And seeing as none of them experienced any sort of acceleration, how then could we describe their two final states? By which I mean, if we were to assume there were relativistic effects during transit, how would we overcome such simple paradoxes like the twin paradox, or other relevant ones?
At this point, I'm stumped. I even attended a lecture by Miguel Alcubierre since he would have had to consider these types of effects in his design... No help. Equations are great to illustrate a point, but I'm really going to need a conceptual answer as well to fully understand this.