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Talk:Counterexamples to Relativity (view source)
Revision as of 04:07, August 1, 2010
, 04:07, August 1, 2010→Curl of the gravitaional field: "path length" = "local time"
:: Simeon, your mathematical work is rigorous and correct. However, the twin paradox example is interesting to study here. I am aware that the twin paradox is solved by the non-inertial turn-around of the ship when it is going back home. However, in this solution, it is still noted that there is an age difference between the twins. [http://mentock.home.mindspring.com/twins.htm Wikipedia affirms this] and so do [http://mentock.home.mindspring.com/twins.htm other sites]. Such an age difference in twins shows that there is some sort of path dependence. I understand that traveling at near-c speeds in space is not the same thing as moving from point A to B in a gravitational field, but the concept does seem to be a bit similar. Could you maybe explain this for us a bit? Thanks. [[User:PhyllisS|PhyllisS]] 00:52, 31 July 2010 (EDT)
:: Simeon, your mathematical work is rigorous and correct. However, the twin paradox example is interesting to study here. I am aware that the twin paradox is solved by the non-inertial turn-around of the ship when it is going back home. However, in this solution, it is still noted that there is an age difference between the twins. [http://mentock.home.mindspring.com/twins.htm Wikipedia affirms this] and so do [http://mentock.home.mindspring.com/twins.htm other sites]. Such an age difference in twins shows that there is some sort of path dependence. I understand that traveling at near-c speeds in space is not the same thing as moving from point A to B in a gravitational field, but the concept does seem to be a bit similar. Could you maybe explain this for us a bit? Thanks. [[User:PhyllisS|PhyllisS]] 00:52, 31 July 2010 (EDT)
OK, I think I understand. I assumed that the "conservative field" / "curl is zero" stuff referred to the gravitational force field. If it refers to the passage of time, that's different. It isn't true that "all physical parameters are path independent". An extremely important one that isn't path independent is the arc length of the path or arc. You can draw a short straight line from A to B, or a long loopy line that starts at A, wanders around, and eventually gets to B. Why is this relevant to the twin paradox? Because, in relativity, an observer's own elapsed time ("local time") is really just the arc length of his "world line" in Minkowski space. Minkowski was an extremely smart guy, by the way. The twin that stays home takes a direct route from point A (their birth) to point B (the moment they compare ages and see that one has gray hair and wrinkled skin.) The other twin takes a very roundabout route, getting in a rocket and going to Alpha Centauri and back. Their path lengths are their local times, which are different. (Why is the length of the roundabout path actually shorter, so that that twin ages less? Because, in Minkowski space, using the "timelike convention" that all the best people use :-), motion in space subtracts from the elapsed time. That's just the way it works.)
Now I assume that there is no dispute about the facts of relativistic time dilation. In addition to being predicted exactly by the Lorentz transform, it has been observed in practice in cosmic ray muon decays, as well as countless observations in particle accelerators. The "twin paradox" is just an extreme consequence of this. It has of course never been observed in that form, just as we don't know whether Schrodinger's cat is alive.
The "twin paradox" is a consequence of special relativity, not general, and hence does not relate to gravity. I hate to be the umpteenth person to tell you that general relativity is too hard to explain, but it's kind of true. I barely understand the most rudimentary basics. (When Eddington made his comment about only 3 people in the world who understand gen. rel., I wasn't the third! :-) But I can say that you don't need to worry about general relativity to understand the "twin paradox". You can finesse the Minkowski-space curvature of the path during the turnaround at Alpha Centauri, and just say that the twin went there and came back. So was something physically different, that the twins could observe? You bet. The "younger" twin will remember having experienced 6 months of horrendous acceleration in the ionic-drive rocket, followed by a year of horrendous turnaround, and another 6 months of horrendous deceleration at the end. She will have soft, smooth skin, but at a great cost. :-)
Sorry to be so long-winded. In quick summary, the thing that's different about the paths is their length, and that is exactly the local elapsed time. [[User:Simeon|Simeon]] 00:07, 1 August 2010 (EDT)