What about this Hubble volume thought experiment is unphysical?

Question

I've been considering a situation with Hubble volumes that I'm having a difficult time resolving. For simplicity, assume the expansion of space is constant, rather than accelerating or decelerating.

Consider two small objects, A and B. A is completely inertial. B is a small distance away from A (A meter, a kilometer, a few light-minutes, whatever), and is inertial with respect to A. B can be powered, though, and only does so to maintain its exact distance and relative position to A - purely as a way to account for the expansion of space between A and B over time.

You start off a very far distance from A and B, but inertial with respect to them, not accounting for the expansion of space - so all relative motion between you and A/B comes from that. A and B are precisely positioned so A is inside of your Hubble volume, and B is outside of it. Before A can recede out of your Hubble volume, though, you quickly accelerate to a velocity sufficiently close to light speed to keep A within range and, eventually, to arrive at it.

After your very long journey, you arrive at A and decelerate so you're inertial with respect to it again. B's been actively maintaining a constant distance from A, so it should be nearby too.

That's the part that's giving me trouble - once B's outside of my Hubble volume, it should never again be able to enter it, but I'm having a difficult time finding what about my situation is unphysical.

I considered that length contraction in the direction of travel may play a role in bringing B back into my Hubble volume, but I don't think that's the case - no matter how much I contract length, I'll still never out-distance a photon I emit at the start of my journey, and a photon I emit at the start of my journey can never reach B since its distance from B only ever increases.

I also considered that the extreme time dilation during my journey would mean that by the time I got to A, the expansion of space would have necessarily pushed B out of A's own Hubble volume. I no longer think that's the case, but just in case it is, that's why I stipulated earlier that B is powered and can maintain its distance to A.

I've been going back and forth on whether B is reachable, but at this point I'm pretty firmly in the "not reachable" camp. Why though? What incorrect assumption did I make, or what did I overlook? Is it something with general relativity during the periods of acceleration and deceleration at the endpoints of the journey? Or does B remain reachable for some reason I haven't been able to figure out?

Thanks for any help.

Answer 1

I think I may finally have come up with the answer on my own. I came to the conclusion that two objects not in each others' respective Hubble Volumes, but with Hubble Volumes that overlap even a little bit, can still reach each other in theory, with careful coordination.

Consider objects B and C, separated by nearly a full Hubble Diameter - so their own Hubble Volumes only have a small region of intersection, which is where we'll say object A is. Since B and C are separated by nearly a Hubble Diameter, the recessional velocity between them should be nearly 2c. If both B and C accelerate toward A to a speed sufficiently close to the speed of light to ensure they eventually reach A, though, then they'll meet there in a few quadrillion years, time dilation accounted for.

From A's reference frame, this doesn't seem to present any contradictions: It measures B and C as both moving toward it at just-barely sub-light speed, and it so it measures the distance between B and C to be decreasing at very nearly twice the speed of light - which is allowable since neither B nor C are going FTL to accomplish that. So the distance between B and C decreases, rather than increases.

So my conclusion is any two objects with an overlapping Hubble Volume can still eventually meet and interact, in theory - but once their Hubble Volumes become entirely separate, they're separated forever.

I'm still shaky on some of the more minor details - for example, I don't really know what things would look like from the reference frame of B or C, either during acceleration, constant velocity, or deceleration. I'm thinking B and C wouldn't even see each other until they were very nearly at A. But as a whole I'm finally satisfied with this answer.