George

2008-Apr-17, 02:15 AM

Ant analogy of the cosmos expansion.

[Revision: The analogy is contrary to what is held in modern cosmology regarding the expansion. The proper analogy would be one where the unfixed end of the rope is being pulled always at a constant velocity. So, as the ant walks toward the fixed end, it will be progressively advance into regions of the elastic rope that has less and less of a retractive velocity. Thus, the ant will always, eventually, reach the fixed end.

We have altered our dicussions to this revision, so it will be confusing if you join the fun at this point and don't realize we have changed to a constant pull rate (velocity) for the rubber rope.]

[Using the attachment.]

I can't see how the ant can reach its destination without a speed at least half that of the rope's speed where the ant starts.

Let the ant's velocity be 1 mps (meter per second), and the rope streching rate be set for 1 mps per meter. So, at P2, the rope velocity relative to P1 is 1 mps, and at P3 it is 2 mps.

Now, to simplify, let's not stretch the rope until after we give the ant the advantage of moving the first second from position P3. [Its goal is P1, of course.] The ant must stop after the first second and then give the next second to our rope stretchers (located off the page in some dark region. ;) ). What happens?

After the first second, our ant will first reach P2, but in the next second it will be pulled back to P3 since at P2 the rope stretch rate is 1 mps. Thus, the ant has accomplished nothing! This procedure could be done forever and we'll get the same results.

Note that the ant has not only the advantage of going first, which means it reaches a point that has less of a pull-back velocity, but also we are ignoring the fact that the pull-back rate will actually increase more than 1 mps as it gets further from P2 and gets closer to P3.

If we were to allow the ant more than 1.0 seconds for its travel time, then it will reach P1 eventually. If we, however, give the ant less time for its head start, then the ant looses ground.

I would expect the later case would be more of an accurate differential approach to the problem, so the ant would have to travel somewhat faster than 1/2 the pull-back rate relative to its destination.

I'm probably wrong, but why? :doh:

[Revision: The analogy is contrary to what is held in modern cosmology regarding the expansion. The proper analogy would be one where the unfixed end of the rope is being pulled always at a constant velocity. So, as the ant walks toward the fixed end, it will be progressively advance into regions of the elastic rope that has less and less of a retractive velocity. Thus, the ant will always, eventually, reach the fixed end.

We have altered our dicussions to this revision, so it will be confusing if you join the fun at this point and don't realize we have changed to a constant pull rate (velocity) for the rubber rope.]

[Using the attachment.]

I can't see how the ant can reach its destination without a speed at least half that of the rope's speed where the ant starts.

Let the ant's velocity be 1 mps (meter per second), and the rope streching rate be set for 1 mps per meter. So, at P2, the rope velocity relative to P1 is 1 mps, and at P3 it is 2 mps.

Now, to simplify, let's not stretch the rope until after we give the ant the advantage of moving the first second from position P3. [Its goal is P1, of course.] The ant must stop after the first second and then give the next second to our rope stretchers (located off the page in some dark region. ;) ). What happens?

After the first second, our ant will first reach P2, but in the next second it will be pulled back to P3 since at P2 the rope stretch rate is 1 mps. Thus, the ant has accomplished nothing! This procedure could be done forever and we'll get the same results.

Note that the ant has not only the advantage of going first, which means it reaches a point that has less of a pull-back velocity, but also we are ignoring the fact that the pull-back rate will actually increase more than 1 mps as it gets further from P2 and gets closer to P3.

If we were to allow the ant more than 1.0 seconds for its travel time, then it will reach P1 eventually. If we, however, give the ant less time for its head start, then the ant looses ground.

I would expect the later case would be more of an accurate differential approach to the problem, so the ant would have to travel somewhat faster than 1/2 the pull-back rate relative to its destination.

I'm probably wrong, but why? :doh: