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#199
Jan2413, 04:50 PM

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#200
Jan2413, 05:06 PM

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However you misinterpreted my use of the word "experienced." Webster's dictionary gives two or three definitions for the use of that term. You chose the wrong definition where I thought the context made the definition I was applying very clear. I'm not insisting anyone embrace that sense of specialness. Of course forum members can consider that observation or dismiss itwhatever their preference. 


#201
Jan2413, 05:06 PM

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#202
Jan2413, 05:14 PM

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#203
Jan2413, 05:18 PM

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#204
Jan2413, 05:29 PM

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A different question is what is 'experienced' as a simultaneity surface. You can approach this mathematically or physically. I prefer physically, and note that there is a well defined 'frame' for an observer when/where different physically reasonable simultaneity definitions agree (to some desired precision). Thus, agreement to some precision between radar simultaneity and Born rigid ruler simultaneity defines the size of a physically meaningful frame for an observer. The longer since your last significant (to desired precision) deviation from inertial, the larger the spatial extent of your physically meaningful simultaneity slice. Mathematically, along a world line, you can use whatever spacelike surfaces you want as simultaneity slice, for a region of spacetime in which they don't intersect. If you want to cover a region where one choice has intersections, choose a different set of surfaces that don't intersect there. 


#205
Jan2413, 07:38 PM

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Bear with me a little more here. I'm trying to get my head into your concept and use of Lorentz frames and simultaneous spaces.
Focus just on the stayathome twin. Is he in an inertial frame? Does he exist in a continuous sequence of simultaneous spaces, each one parallel to his X1 axis? 


#206
Jan2413, 08:04 PM

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Also, there is no empirically discernable sense in which he exists in one simultaneity convention and not in another. If he exists, then he exists regardless of which simultaneity convention you adopt. 


#207
Jan2513, 06:38 AM

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... Now I transform to the IRF in which the traveling twin is stationary during the outbound portion of his trip: And the transform to the IRF in which the traveling twin is stationary during the inbound portion of his trip: Now how do you get information from these last two IRF's to draw the extra lines in the first IRF and to come to the conclusions that you do? 


#208
Jan2513, 01:58 PM

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I had originally simply tried to give the discussion a larger context by expanding the picture by adding frame coordinates for the outgoing and return trip frames. Of course subsequent comments (beginning with an observation by Vandam) led to the addition of a second Red guy in the rest frame with events A and B and the intersections of the blue X1 and X1' axes with that 2nd Red guy's worldline. My picture of the blue guy jumping frames (to use a phrase from Rindler's text book) then generated a series of pushbacks from others. But, to answer your question I've displayed my construction, sketch b) below, to show how my additional features would be added to your sketch a). I just wanted to show the coordinates associated with your frames (Rindler makes a distinction between "frames" and "frame coordinates"). So, I began by establishing the X4 and X4' coordinates along the direction of the inertial worldlines. Then, I used 45degree green lines to represent photon worldlines. The simultaneous spaces for the two coordinate systems are then established by adding in the X1 and X1' axes. These X1 and X1' axes are of course placed such that the photon worldlines bisect the angles between the X4X1 pairs. The 2nd Red guy worldline was just added in to illustrate the interesting feature mentioned by Vandam. [edit: Note the slight discrepancy in my sketch b). The green photon worldline is not exactly 45degrees. I hope this does not distract from illustrating the basic concepts. 


#209
Jan2513, 03:56 PM

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Now the rest frame for the traveler during the outbound: You should know that I adjusted the coordinate time of event B in the first frame so that the event appears simultaneous with the common origin of all the frames. That is another way of forcing event B to be simultaneous with your X_{1} axis. And the rest frame for the traveler during the inbound: Again, I adjusted the coordinate time of event A in the first frame so that the event appears simultaneous with the traveling twin's turnaround event. That is another way to force event A to be simultaneous with your X'_{1} axis. Now I think your interesting observation was that in the traveler's simultaneous space, event A occurs after event B. However, I think you can see that following your definition of simultaneous space, you really should say that the interesting observation is that in the traveler's simultaneous space, event A occurs both before and after event B. And now that I've pointed that out, you can easily go back to your original sketch or mark up mine to show this interesting observation. 


#210
Jan2513, 04:57 PM

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#211
Jan2513, 07:53 PM

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#212
Jan2513, 08:09 PM

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#213
Jan2513, 08:32 PM

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Quote by bobc2
Quote by bobc2 I think that this thread is basically misdirected and is missing the crucial point. Which is the inherent problem with charting accelerated systems in Minkowski space. So I think that the problem is not with CMRF's and adopting their simultaneity but the fact that a system based such a series of frames is incorrectly charted in such a diagram. I submit that, within certain limitations of acceleration magnitude and spatial extent , a chart correctly constructed with the synchronization of a sequence of CMRFs would be perfectly tractable throughout an extended domain. That it would produce smooth continuity from a conventional inertial system through acceleration to a final inertial state without overlap or gap over an large spatial area. With no temporal ambiguities and complete agreement on events with Earth and all other inertial frames.That the overlap that occurs in the outward region in a Minkowski chart is neither inherent in nor an accurate representation of such a system but is purely an artifact of Minkowski graphing. There is one resulting condition of such an implementation; the synchronous coordinate time generated would not have a uniform rate throughout the system. Specifically,,if we assume the traveler location as the point of synchronization for the frame , then the coordinate time on clocks running back toward Earth would be slowed down by increasing degrees relative to the proper rate of a natural traveler clock there. Comparably the coordinate time outward from the traveler would have increasing rates. Although the time rate would slow down towards Earth it would still proceed forward out to a distance dependent on the acceleration rate. Beyond that it would actually increment backwards. At low accelerations this would occur at very large distance which would decrease as acceleration increased. Finally at maximal, instant acceleration, the extent of continuity would reduce to the single traveler point. With overlap increasing toward earth as clocks were set back to a previous reading and an increasing gap outward as the coordinate time was suddenly set forward. Compare with the Minkowski diagram. In this the traveler frame is portrayed as rotating clockwise, Resulting in temporal displacement. Into the Earth frame future back at Earth and into the Earth frame past outward from the traveler. With the resulting anomalies where the outward traveler frame intersects and overlaps itself , while the inward region jumps forward I think the actuality is the opposite. The Earth line of equal time is rotated counterclockwise. Into the traveler coordinate past at Earth and into the coordinate future outward from the traveler. The difference is that the minkowski diagram produces a literally impossible picture which could not have frame agreement with inertial observers. I.e. NO inertial observer could be colocated with a traveler with a post turnaround clock reading and an Earth clock with a preturnaround Earth time reading at point A And no physical system could meet and overlap itself. The second picture has a coordinate time discontinuity outward but no overlap , while the overlap actually occurs towards Earth but has no temporal implications whatsoever and is clearly simply a coordinate issue. As far as that goes , while we prefer that coordinates are smoothly continuous this is not really a serious matter. it can be accomodated with a little relabeling,perhaps PS for post synchronization attached to the redundent times readings toward earth A little calculational stitching yes? Forgive me if I have run on. I wanted to keep it as simple as possible but it may have gotten away from me ;0 So i think the root of the problem is that the direct Minkowki graphing implements an implicit assumption of actual simultaneity within a frame at equal time readings I also think I can pinpoint how this is implemented and why it produces the incorrect results but this is too long already. 


#214
Jan2513, 09:12 PM

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#215
Jan2613, 01:56 AM

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Against an inertial chart (which covers the complete manifold), any valid alternative chart can be drawn, for whatever region of spacetime such a chart covers. Do you disagree with any of this? [This brings up and option I have discussed on other threads, but didn't want to further complicate this thread: It is perfectly routine to use different, overlapping coordinate charts on a manifold. You just specify the mapping that identifies the same events for the overlapping region(s). Of course, this approach gives up on the basically meaningless question of what is 'the' simultaneity map between the stay at home world line and the traveling world line, from 'the point of view' of the traveler. It leaves you with: In patch 1, there is a partial mapping; in patch two there is another partial mapping that isn't and has no need to consistent with the other patch for the overlapping region.] 


#216
Jan2613, 12:50 PM

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We often will talk about an observer being at rest in a particular Inertial Reference Frame (IRF) and we usually mean that his clock is synchronized to the coordinate time and it's in this sense that when we talk about the classic Twin Paradox, we assume that both of their clocks read zero when the one twin departs. And we can assume that prior to that time, both clocks and the coordinate time were all in sync with negative times on them. So now we consider what happens after the twins are reunited. In this particular scenario, the time on the traveling twin's clock will read 10 years when the StayAtHome (SAH) twin's clock reads 13 years and also when the coordinate time is 13 years. So do the twins share the same simultaneous space? I would say yes, because as I said before, simultaneity is defined for the coordinate system, not for any particular observers. But since you asked the question, you probably are using a different definition of simultaneous space that is defined for observers and not for coordinate systems and because their clocks have different times on them, maybe you'll say no. What do you say? But while we're on the subject, let's think about another issue: consider the question of the simultaneity between the traveling twin's turnaround event and the SAH twin. In the first IRF, this happens for the SAH twin when his clock reads 6.5 years (assuming zero at the start). But if we look at the next IRF that I drew, it happens at around 4.9 years and for the last IRF, it happens at around 9.1 years. So we see that the issue of simultaneity is IRF dependent. However, if we ask a different question, namely when will the SAH twin see the traveling twin turn around, we can get the answer in the following way: Look again at the first diagram from post #209: Note that at the moment of turnaround, the traveling twin is about 4.1 lightyears away from the SAH twin. Therefore, we conclude that it will take 4.1 years for the light from the turnaround event to reach the SAH twin and since his clock read 6.5 years at the moment of the turnaround event, he will see his twin turn around when his own clock reads 10.6 years. I have drawn in the blue signal going from the turnaround event to the SAH twin to illustrate this: But here's what I consider to be the interesting observation. We can do the same thing for the other two IRFs and we get the same answer even though the IRFdependent values are different. Let's look again at the second IRF diagram: We see here that the turnaround event occurs at a coordinate time of 5 years but the Proper Time on the SAH twin's clock is about 3.9 years and the traveling twin is closer than before, only about 3.2 lightyears away. (You have to count the red dots to determine what the Proper Time is on the SAH twin's clock.) But it doesn't take just 3.2 years for the image of the turnaround event to reach the SAH twin because he is moving away from it. We have to follow the path of light along a 45 degree angle to see where it intersects with the SAH twin. (Unfortunately, I didn't draw these diagram with the two axes having exactly the same scale so you have to pay attention to the grid lines when you define what 45 degrees means.) And here is the diagram showing the blue signal path for the second IRF. Again, you have to count the red dots to see that the SAH twin sees the turnaround event when his own clock reaches 10.6 years: Now let's look again at the third IRF diagram: Now the turnaround event occurs at a coordinate time of 11.8 years and the Proper Time on the SAH twin's clock is at about 9.1 years. And just like in the second IRF, the distance between the SAH twin and the traveling twin is only about 3.2 lightyears away but it doesn't take 3.2 years for the SAH twin to see the traveling twin turn around because he is traveling towards him. In fact, it takes only about 1.5 years and once again, in this IRF, the SAH twin sees the turnaround event when his own clock reaches about 10.6 years. Here's a diagram showing the blue signal path for this IRF: Don't you agree that this is an interesting observation? No matter what IRF we use, it doesn't change the observations that the observers make. 


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