This is the best explanation I have ever seen about this. I knew it was due to a change in reference frame, but having seen this particular explanation I feel for the first that I can explain the resolution to someone else.
I don’t think so. This handy calc ulator says that travel to Tau Ceti (11.9 ly) would take 5Y2M under a constant 1g acceleration and braking at the end. On earth 13Y9M would pass.
I have no way of knowing if the site is correct, but from what I’ve read in other places, “only” 1 g is enough to produce amazing results [YMM indeed V]. Now to power such a ship is another thing…
You will still get a twin paradox if you consider 1g acceleration/deceleration instead of infinite acceleration at the beginning, middle, and end of the trip, yes.
And you can do special relativity just fine in non-inertial reference frames. You just need calculus to do it, instead of mere algebra.
I’ve been trying to visualise the practical problems with this question, but the maths is defeating me.
1/ Cosmic inflation is causing many of the galaxies which are visible in the sky to recede away from us at speeds faster than light. The closest galaxies that are receding faster than c are at something like z=1.4, which means the light from those galaxies has been travelling for eight billion years or so in order to get here. We can call this point x.
2/ If we want to travel to a galaxy which is receding from Earth at a speed faster than light, we don’t have to go all the way to point x; because of expansion we only need to travel to a much closer than that. Call this point y; by the time we get there (billions of years in the future) the universe will have expanded even more, so point y is much closer than point x. But congratulations; you would now be both standing still and receding at light speed away from Earth.
3/ But the twin paradox doesn’t really kick in untill the two twins come together again - if the distant twin turns round at point y and tries to get back to Earth, the Earth will be receding even faster so it will take billions more years to get back. Good luck trying to work out when the distant twin ever gets back to Earth - it may be never.
Because the universe is expanding, it is possible for a galaxy to be so far away that, no matter how fast you go away from Earth, you will never catch up to it. For the same reason, if you
are able to get someplace, you may not be able to get back—note that it is possible, here on Earth, to see objects that are really, really far away.
Thinking about it again, a traveller should be able to get back to Earth from point y, since it is just beyond the sphere where recession exceeds the speed of light. Light could reach Earth from point x in the past, so it should be able to get back to us from point y.
But the round trip will be more than 16 billion years.
I am not following your reasoning. Galaxies have been observed that are now 30 billion light years away. How are you going to get back there, especially when there is a cosmological event horizon only 16 or so billion light years away?
Well, since point ‘y’ is only just over the lightspeed barrier, we should be able to get back from there in the same way that light reached us from point ‘x’ much earlier. The theory is that a traveller would travel from the volume of space that is expanding away from us at ‘c’ and immediately travel into the space which is not quite expanding away from Earth at ‘c’, so ultimately you can get back here. Of course by that time the Sun will be a white dwarf.
Unless the acceleration of the expansion of the universe increases and everything flies apart anyway.
Think of it this way: even if the point you are trying to get to is reachable (i.e., not outside a event horizon), by the time you get there your original point will be much farther away, because space is expanding. So far away that you can never make it back. The round trip is not symmetrical.
I know what you mean, but in this case, there are two different points;
1/ point ‘x’, which is currently the closest location that is receding from Earth at ‘c’,
2/ point ‘y’, which is the point which would be the closest location that would be receding from Earth at ‘c’ when we eventually got there (assuming we could travel at the speed of light ourselves). This is much closer.
All this is subject to the uncertainties of simultaneity, which are considerable at these speeds and scales.
We could not get back from point ‘x’, but we could get back from point ‘y’.
Your items (1) and (2) are unclear to me. If we stay put on earth, the (proper) distance today to the point at which the recession velocity is c is x, and if we ask again in a billion years, the answer would be y with y > x. This seems the opposite of your description.
Yes, you are correct, except that the travel time to ‘y’ would probably be more than 8 billion years. By which time ‘x’ would be many, many more billions of light years away. But it should still be possible to travel from ‘y’ to Earth 8 billion years from now, or at least the white dwarf where the Sun used to be.
Ah, I see where the confusion lies. Yes, the distance to point ‘x’ now is smaller than the distance to where point ‘y’ will be 8 billion years from now, when we get there; but 8 billion years from now point ‘x’ will have moved much further away (because of expansion).
Moved without actually moving, of course.
Distances in an expanding universe are hard.
Thanks. I should have just read your posts more carefully, as you were clear before.