- #176
bobc2
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ghwellsjr said:I followed your link and found another similar diagram to the one that I asked about in post #161.
I asked why the blue circles were placed where they wer
Again, I wonder about the placement of the blue circles, why are they placed differently this time?
I really don't understand why you put the blue circles in either place, they have no significance either way to the blue traveler. And what further doesn't make any sense to me is why you draw the blue lines intersecting with the black circles. Now I realize that the slope of the blue lines are indicating simultaneity for the blue traveler no matter where they are placed but you must have some reason to pick these particular lines.
Now you can easily see the symmetry of Time Dilation between the IRF's of the two twins. Or does that matter to what you are trying to convey?
Very good graphics, ghwellsjr. Your graphics make it very clear. Good job.
I was just trying to get across a concept of an observer's sequence of simultaneous spaces, and just happened to approach it this way. In both of the sketches I was simply indicating that the stay-at-home twin and the traveling twin each advances along his respective worldline (along the red and blue X4 axes). I wanted to illustrate the concept of the two different sequences of simultaneous spaces associated with each twin. In one case I put blue dots on the blue worldline to indicate where the black rest simultaneous spaces intersect with the blue worldline. In the other case I placed blue dots on the blue worldline simultaneous space sequence such that the blue simultaneous spaces would intersect with the black worldline. That’s all. It was just intended to help visualize the concept of simultaneous spaces. I wasn’t interested in numbers—just the concept.
The sketch below includes hyperbolic calibration curves to identify the ten year locations with respect to the rest frame origin. The sketch on the right shows the traveling twin’s two different inertial frames, one for the trip out and one for the return trip. I emphasize the these frames do not include the actual turnaround. The path length during turnaround is so short compared to the path length of the total trip out and back, that we would need a magnifying glass to see the curve. My interest was restricted to comparing the inertial trip out and the inertial trip back (after turnaround is complete).
I could overlay any inertial coordinates over the lower right sketch below to identify proper times for both rest system and any other inertial coordinates. I've given a short derivation of the hyperbolic curves. It begins with a sketch of red and blue guys moving at the same speed in opposite directions with respect to the black coordinates.
The sketch below gets messy, but it illustrates a couple of more details that one may or may not find interesting. Notice that the event A on the 2nd Red stay-at-home guy (displaced from the first red twin) is presented to the returning twin’s trip simultaneous space before it is presented to the outgoing twin’s simultaneous space. Notice that this does not in any way imply that the 2nd Red guy's time is flowing backwards for that Red guy sitting at rest in his own black inertail frame. It's just a feature of special relativity and is no more mysterious than the two twins having different ages after they reunite.
By the way, the blue dots on the traveling twin's worldline are placed with same proper time increments as the black worldline dots (one year intervals of proper time on both worldlines, in accordance with your preference). The hyperbolic calibration curves show the five year lapses.
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