I have read and read again your reply, and while I was writing the following translation of my comments, I received another answer to your "Proposal", which as I told you I did send to many people. This time the answer came from one of the Italian physicists most competent in relativity. He wrote rather bluntly that: "It is not worthwhile to discuss once again the twin paradox, it would be just the same as debating Pythagora's theorem".
Well, I must admit that I think more or less the same, with one difference: that after a discussion such as the one you brought forward, one can say to have understood the question even better, and that he would then be able to teach it better. But let me say frankly that I believe that there is no hope to find anything new, or original, or interesting, and that I really do not understand why you say for instance that:
> With Part II of the attached, you will see the issues beginning to become sharper.
I do not see anything "sharper", just a simple "smoothing" of corners (exactly the same purpose I tried to achieve with my hyperbolic branch), which does not change the UNIQUE CORRECT interpretation of this problem. You add that:
> You give a lot of equations and very little explicit physics…
but let me say that to introduce one smoothing (as I shall do in the following), or even three smoothings as you did (or infinite ones, as I did before) DOES NOT CHANGE at all the mathematics of the situation: this is just simple plain mathematics, in some sense no physics at all (the physics is only in the "interpretation").
I am sorry to say that, since this seems to have been your favourite topic for many years, and you assert that:
> it is a very complex topic ... it is very sophisticated and even perverse in its subtlety - it goes well beyond SRT.
I do not believe that, and I am always surprised to see many good people get entagled in the same TRAP (I am surprised to see Mendel Sachs between the people you quote, the others are not so good; as I told you, I do not know too much literature, since in mathematics it is not necessary to read many different proofs of the same theorem, when you know one of them, and you have well understood it, this is enough, a truth does always remain a truth). Namely, not being able to completely understand the essence of relativity, which is the RELATIVITY OF SIMULTANEITY. What can be shown instead, once again, at the end of this discussion, is that Delbert Larson is correct (see point 1 in my web page concerning the Foundations of Physics) when he remarks that:
"If we try to come up with theoretical arguments to show how special relativity is wrong [or "incomplete", or whatever else you like], we will lose. SR has been studied and celebrated for generations now. If there was a theoretical flaw it would have been found long ago ... from a mathematical (and therefore theoretical) sense, special relativity is completely consistent and correct. Arguing that point merely shows a misunderstanding of the theory ... it takes a long time, generally, to find out why the opponent is wrong. But after days of thought one generally (90% or more) finds that the opponent is wrong".
Before going into details, let me remark one point. You speak of:
> the vagueness of "it's a GLOBAL effect"…
and of course you are right since the language is always a bit "vague". But it is not vague for people who is able to understand what kind of "semantic reference" that vague term was trying to suggest. As I shall proof immediately, the misunderstanding that you seem to share with other people is that you wish to "compare" different times relative to the "lives" of different observers, and this is not in principle (in general) possible at all. Let me be more precise: when one introduces two general (accelerated) observers A and B in Minkowski space-time (let us even suppose it 2-dimensional), which meet in an event E0, and then take an event say it E in the life for instance of B, it is sound to ask what is the proper time elapsed FOR B from E0 to E, but it has not meaning in general to ask the same for A. The point is that one has to define one "corresponding" event of E in the life of A, and that is not even possible (or uniquely defined) for accelerated observers. The case of inertial observers is of course more simple, but even in this case there are subtleties which one must take care of. This is why I talked of a "global effect", when the two observers meet TWICE, in E0 and in E1, then it is possible to compare the elapsed proper times for BOTH observers from E0 to E1, and this the twin effect; but there is not in principle possibility to compare times concerning only PARTS of the trip, as you wish to. Anyway, since you study a particularly simple case, then something more can be said, and I shall do it immediately.
Your computations in point II are indeed correct - there is no doubt about that, after all they are very simple - but the point is: what do they teach to us? What can we learn from them? As a matter of fact, their correctness notwithstanding, they can lead unexperienced people to misunderstandings.
Let me make things simpler, modifying the classical "three observers scenario" (see drawing N. 2 in the attached Figure I) with only one smoothing of the corner, as you suggested (see drawings N. 4 and N. 5 in Figure II). Now we have only one "travelling twin", let us call him B, from the event E0, when he meets A, with uniform speed v (w.r.t. A; v < 1, I shall always use geometrical unities) to the event E1, when he starts as you say "the 1st half of the turn around acceleration". This ends in the event E2, where we can introduce the 2nd half of the turn around acceleration, till the event E3, and from this event on then B travels once again with uniform speed -v (w.r.t. A). In the event E4 the two twins meet again. Well, you can correctly ask what is the (proper) time spent by B from E0 to E1, say 2500000 sec; and then from E1 to E2, say 24; from E2 to E3, once again 24; from E3 to E4, once again 2500000, for a total time of 5000048 sec. Then, if you want to do a "comparison", you must introduce CORRESPONDING events in the life of A, which are DIFFERENT events, except for E0 and E4. You can introduce indeed the event F1, which is defined as the only one event in the life of A which is SIMULTANEOUS WITH RESPECT TO A with the event E1, and so on, and you correctly remark, with your parameters choice (v = 0.8886, sqr(1-v^2) = 0.5), that the clock of A marks:
from E0 to F1, 5000000 sec
from F1 to F2, 30 sec
from F2 to F3, 30 sec
from F3 to E4, once again 5000000 sec,
for a total time of 10000060.
Then, even if you do not do that, you lead your reader to ask himself (as Dingle asked himself): well, A is older than B at reunion in the event E4, this is a fact, but we must admit a second fact too, that the greater difference of times is coming up from E0 to F1, and from F3 to E4, the "pieces of travel" in which the two twins are perfectly "symmetric" in a relativistic sense. How is this then possible, since for B it should happen the same? (that is to say, B should see the same time dilation happen to A; you write: "if we have A and B at rest in different inertial frames, then whatever A observes about B, B observes about A. If A observes the time dilation/length contraction factor for B to be 0.675823, then B observes the time dilation/length contraction factor for A to be 0.675823 etc.", which is of course quite correct).
The simple answer to this "paradox" (I repeat that you did not explicitely state it!; you just asked to "cut and paste Part II and fill in your answers and send me the result", BUT THIS WOULD HAVE BEEN TOO MUCH OBVIOUS, your computations ARE correct!!) is that the previous "decomposition" of the travel of A IS NOT the "correct" decomposition FROM THE POINT OF VIEW OF B! From the point of view of B, in fact, the event E1 of his own life is simultaneous to an event G1 in the life of A which is QUITE DIFFERENT from F1 (see drawing 5),and so on (with the obvious exception of the events: G2, which is equal to F2; and once again E4, which is common to both observers). This means that, FROM THE POINT OF VIEW OF B, when he travels from E0 to E1, spending as you have said 2500000 sec, A travels from E0 to G1 (AND NOT F1!), spending 1250000 sec, as it MUST BE for symmetry reasons. Then B goes from E1 to E3, spending only 48 sec of his time, but in this "short" ACCELERATION PERIOD his twin A goes from G1 to G3, spending 7500060 sec!! Then B goes from E3 to E4, spending once again 2500000 sec, and A goes from G3 to E4, spending 1250000 sec.
Summing up, from the point of view of B, the most of the time which A has spent in his life - the very reason for his (A) becoming asymmetrically older with respect to him (B) - it has been spent exactly in that short acceleration period.
Conclusion: from the point of view of A, whose "state of motion" is always the same "with respect to the space-time" (this means with respect to ANY inertial reference frame: it would be possible to see all this "scenario" from the point of view of another inertial observer X, which sees A always moving of a uniform motion, and B accelerated), B is always getting younger, in all parts of his travel, by relativistic time dilation. From the point of view of B, A is getting younger TOO by the same relativistic time dilation (and exactly in the same quantitative proportion as before), in his two (long enough) uniform motion periods, but then A IS LOOSING ALL HIS YOUTH in the short acceleration period of B, from E1 to E3. In this part of the travel, the "time" of B almost stops down, while the time of A runs away...
If I must point out again but in other words where is "hidden" the "misunderstanding" in your argument, it is in the expression:
> Ticks Accumulated Between the Two Named Events.
The "two named events", for instance E1 and E2 (in my set-up), BELONG TO THE LIFE OF B, and NOT to the life of A, so in some sense you are not allowed to talk of the time accumulated by A "between these two events". You must CHANGE these events, either introducing F1 and F2 (point of view of A, A which "sees" B), OR G1 and G2 (point of view of B), and then you get quite different results!
Once again, one must be very precise in talking, otherwise he makes "mistakes" (no common sense in relativity, this is the point). Perhaps I learned from this discussion the following lesson, that it could be useful to underline to students, when teaching this argument. When one thinks of two inertial not parallel observers in the 2-dimensional Minkowski space-time (see drawing 1), they meet just once, say in E0, and there exist TWO functions, the first, H, or more precisely H(A,B), which uniquely associate to an event in the life of A an event in the life of B (you used before H^-1, namely you introduced: F1=H^-1(E1), and so on), and another one K, that is to say K = H(B,A), which, in similar (quite symmetric) way, associates to an event in the life of B an event in the life of A. BUT THE POINT IS THAT H AND K ARE NOT ONE THE INVERSE OF THE OTHER, that is to say, H^-1(E1) DOES NOT COINCIDE with K(E1), and that is all.
You suggested to be:
> rigorously precise and quantitative - speaking a minimum of prose and a maximum of physics…
and this is exactly what I have done, now and before, the real problem is that too often physicists do physics hiding just in their "prose" the most important things. No prose, only mathematics, I agree.
I sincerely hope that you will find the previous analysis correct and conclusive, since we are not discussing something personal of the kind "I am right, you are wrong" (it is not me to be right, but only the truth, which is not "mine"; and this truth is the same that I learned from ALL good physics teachers and textbooks, always the same), and that you will consider this old debated question settled once for all. Dingle was irremediably wrong, and one must fight SR, of course if he wishes to, with other more effective (experimental) "weapons"...
Once again best wishes from yours most sincerely
P.S. I added more drawings to the attached Figures I and II, suggesting how one could go on with this study, tackling more "exercises". Doing so, one could learn better how the asymmetric ageing does depend from DIFFERENT LENGTHS of space-time paths (this is what one refer generically to as "nature of the space-time", "geometry of the space-time", and so on). At last, I inserted a drawing (N. 8) of the situation which I believe is the most interesting one from a physical (namely experimental) pont of view, the muon experiment, which IS indeed different from the twin case. The Hafele-Keating experiment (drawing N. 7), which I do not know well in the details, is indeed similar to a twin case, but in that experimental situation one has to get into account possible gravitational effects, and more than all Sagnac effect. I do not know if the theoretical analysis of this experiment has been carried on quite properly, and if its final data really confirm SR or not (but I remember what Hafele himself admitted about the "value"of his findings: "Most people (including myself) would be reluctant to agree that the time gained by any one of these clocks is indicative of anything" … "The difference between theory and measurement is disturbing" - see point N. 12 in the quoted web page). This would be an analysis truly worth to be done...