# No way to catch up with galaxies currently receding at >c?

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In summary, a passenger in a rocket traveling to a far-away galaxy with a constant proper-acceleration of 1g for 50 proper-years experiences 50 years of proper-acceleration, followed by inertial motion from that point forwards. At the moment when the 1g proper-acceration ceases and the rocket resumes inertial motion, the passenger imputes a current relative velocity (Vnow) between the far-away galaxy and the rocket of 2x. Earthlings impute a current relative velocity (Vnow) between the far-away galaxy and Earth of also 2x. The rocket passenger imputes

TL;DR Summary
Help me do this math problem that assumes a traveler with constant proper acceleration of 1g. What happens to perceived distance of far galaxies currently receding at >c?
Consider a far-away galaxy that is considered to be currently receding from Earth at 2x the speed of light. (With this 2x c recession velocity, we are speaking of the Vnow, the imputed relative velocity of the galaxy compared to Earth now, not the velocity of the galaxy relative to Earth at the time of the emission of the light that is reaching us now from that galaxy).

Then consider a passenger in a rocket who travels in the direction of this far-away galaxy with a constant proper-acceleration of 1g for 50 proper-years. At the end of these 50 proper-years (i.e. 50 years of proper-acceleration of 1g as experienced by the passenger on the rocket), imagine that the rocket ceases its acceleration, but does not decelerate (i.e. the rocket coasts with inertial motion from that point forwards).

At the moment when the 1g proper-acceration ceases and the rocket resumes inertial motion:
1. What amount of time does the rocket passenger perceive to have elapsed? (Should be 50 years as given from the scenario above, no?)
2. What distance from Earth does the rocket passenger perceive that he/she has traveled? (Should be a smidgen under 50 light-years, no?)
3. What amount of time do Earthlings perceive to have elapsed by the time they receive signals from the rocket that indicate to Earth that the rocket has ceased its acceleration? (Should be something like 15 billion years, no?)
4. What distance from Earth do Earthlings perceive the rocket to be by the time they receive signals from the rocket that indicate to Earth that the rocket has ceased its acceleration? (Should be something like a smidgen under 15 billion light-years, no?)
5. What distance from the far-away galaxy does the rocket traveler perceive him/herself to be?
6. What distance from the far-away galaxy do Earthlings perceve the rocket traveler to be?
7. What current relative velocity (Vnow) between the far-away galaxy and the rocket does the rocket passenger impute?
8. What current relative velocity (Vnow) between the far-away galaxy and the rocket do Earthlings impute?
9. What current relative velocity (Vnow) between the far-away galaxy and Earth does the rocket passenger impute?
10. What current relative velocity (Vnow) between the far-away galaxy and the Earth do Earthlings impute?
11. For how much additional proper-time would the rocket passenger have to inertially coast before the rocket passenger would have no ability to ever travel back to Earth even with arbitrarily-large and sustained proper-acceleration? (due to the expansion rate of the universe and accelerating recession velocities even in the absence of Newtonian forces)?
12. For how much additional Earth-time would the rocket passenger have to inertially coast before the rocket passenger would have no ability to ever travel back to Earth even with arbitrarily-large and sustained proper-acceleration? (due to the expansion rate of the universe and accelerating recession velocities even in the absence of Newtonian forces)?

Doesn't "dark energy" or the scale-factor expansion of the universe change the whole ball game regarding whether "faster than light" travel is possible, at least in terms of recessional velocities?

Hear me out: I understand that if we disregarded scale-factor expansion, then a rocket could have an arbitrarily-high amount of proper acceleration away from Earth, and yet...if that rocket were sending light pulses back to Earth spaced apart 1 second each in the rocket's reference frame, those light pulses would arrive back at Earth with larger and larger time gaps in between...first 2 seconds, then 3 seconds...eventually 10 quadrillion years between received light pulses (if they could be detected from the lower energy and red shift, and of course assuming that Earth still existed). And yet, if someone on Earth waited an arbitrarily long time, one would still receive every light pulse from that rocket eventually. In other words, observers on Earth would never witness the rocket appearing to travel faster than light because that would entail one of the light pulses being the last light pulse that Earth would ever receive.

However, when we introduce scale-factor expansion of distances in the universe, and we get a rocket sufficiently far enough away from Earth, then we really could have, from the rocket's point of view, a last light pulse that would ever make it back to Earth, and likewise from Earth's point of view, a last light pulse that they would ever receive, even if they waited arbitrarily long for the next one. Is that right? If so, then how would it be inaccurate to say that the rocket is receding from Earth at a faster-than-light velocity? Keep in mind that the Earth will never witness the rocket doing this because, by definition, once the rocket is receding faster than that, Earth will be past its "event horizon." So, it is both true to say that an observer will never observer anything in the universe traveling as fast, or faster, than the speed of light, but also that objects from their own proper point of view can indeed recede from other objects faster than the speed of light, and thus leave other objects behind, behind an event horizon, with the help of the universe's scale-factor expansion. (Of course, this does NOT mean that objects can approach each other at faster-than-light speeds. The only reason they can recede from each other at faster-than-light speeds is because at some point they get enough "help" from the universe's scale-factor expansion).

Is this correct?

In general relativity the relevant laws of physics are that light moves as measured locally at an invariant speed ##c## and that massive objects move locally at speeds less than ##c##.

The concept of measuring the speed of a distant object is fundamentally ambiguous in curved spacetime. Your example of the expanding universe is a case in point. It's valid to say that a distant galaxy is receding at some speed greater than ##c##; and,. equally valid to say that the space between Earth and the galaxy is expanding at greater than ##c## and the galaxy itself has a local velocity of less than ##c##.

So special relativity says that relative distances locally can't be changing at a rate greater than c? Does this mean that, in a fundamental sense, recessional vs. approaching velocities have nothing to do with it? Does this leave open the possibility, if there were just the right spacetime geometry somewhere, or if there were scale-factor contraction in the universe in general (which is, of course, not the universe we appear to inhabit), then far away objects could in effect be approaching us at a rate greater than c (as long as they were "locally" not reaching c)? Even with the scale-factor expansion of the universe that we observe in our universe, are there any areas of particularly warped spacetime geometry that would allow far-away objects to be locally traveling less than c but "globally" traveling greater than c towards some other object?

So special relativity says that relative distances locally can't be changing at a rate greater than c? Does this mean that, in a fundamental sense, recessional vs. approaching velocities have nothing to do with it? Does this leave open the possibility, if there were just the right spacetime geometry somewhere, or if there were scale-factor contraction in the universe in general (which is, of course, not the universe we appear to inhabit), then far away objects could in effect be approaching us at a rate greater than c (as long as they were "locally" not reaching c)? Even with the scale-factor expansion of the universe that we observe in our universe, are there any areas of particularly warped spacetime geometry that would allow far-away objects to be locally traveling less than c but "globally" traveling greater than c towards some other object?
If the density of the universe were greater than a certain critical density, then the expansion would eventually stop and the universe would begin to contract. This would be expansion in reverse and objects far enough away would have a superluminal contraction velocity towards us. It appears, however, that the universe has less than the critical density and that the influence of dark energy will cause an ever accelerating expansion. There will be no big crunch, in other words.

You can postulate unusual spacetime geometries - see the Alcubierre drive, for example - but whether anything so exotic may be achieved is another matter.