Thought Provoker
Posts: 530
Joined: April 2007
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Hi Keiths,
Thank you for your response. I think it helped me with my presentation, a lot.
The difference between you and Zachriel is that Zachriel approaches his discussions like he would a military maneuver. He is careful not to over-extend.
You may yet win the battle of convincing people I made a semantic mistake in my use of the term "General Relativity" but it may cost you the war in that you end up demonstrating that my efforts are honest and rational.
Page 422 of The Road to Reality is part of Chapter 18 which is titled "Minkowskian geometry".
Starting at the top of Page 420...
"In passing from [Euclidean geometry] to [Minkowskian geometry], there are also changes that relate to inequalities. The most dramatic of these contains the essence of the so-called 'clock paradox' (or 'twin paradox') of special relativity. ... if we accept that the passage of time, as registered by a moving clock, is really a kind of 'arc length' measured along a world line, then the phenomenon is not more puzzling than the path along which this distance is measured. Both are measured by the same formula, namely [integral of ds], but in the Euclidian case, the straight path represents the minimizing of the measured distance between two fixed end-points, whereas in the Minkowski case, it turns out that the straight, i.e. inertial, path represents the maximizing of the measured time between two fixed end events (see also 17.9)." [emphasis Penrose's]
Penrose goes on to explain how in the shortest distance between two points in Euclidean geometry is a straight line and how that is not true for Minkowskian geometry.
This is why I say the traveling twin took a short cut.
Page 421 has some pictures explaining all of this. Penrose also explains why this is NOT due to accelerations and is purely a geometry problem.
Continuing on page 422...
"It is frequently argued that it would be necessary to pass to Einstein's general relativity in order to handle acceleration, but this is completely wrong. The answer for the clock times is obtained using the formula [integral of ds] (with ds>0) in both theories. The astronaut is allowed to accelerate in special relativity, just as in general relativity."
Penrose was trying to explain a concept using terms people understand. I was focused on Penrose's concept. If you want to claim victory over semantics, be my guest.
As for how to "fix" your chart. Simply re-label the chart to read...
| Quote | Traveler's clock as seen by Homebody
Traveler's clock
Homebody's clock |
You and Lasky are handwaving a preferential choice of a frame of reference. What is the basis of this choice? Acceleration?
I suggest that is why Penrose was trying to explain that worrying about acceleration was "completely wrong".
The Twin Paradox is a geometry problem.
And it doesn't matter whether you want to mouth the words "Special Relativity" or "General Relativity" in the process of figuring out the arc lengths.
The traveling twin took a short cut.
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