
#19
Feb1413, 02:58 PM

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#20
Feb1413, 03:24 PM

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(You can construct any number of coordinate systems in which Pam has constant spatial coordinate position, but none of these constitute a global frame for Pam in the sense that there are global inertial frames). 



#21
Feb1413, 04:08 PM

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It looks like I need to get busy on the Twin Paradox FAQ that I have been meaning to draft for this forum. 



#22
Feb1413, 05:03 PM

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Personally I like to use spacepropertime diagrams to visualize the twins, because you see the age difference directly. In this interactive version there is both types of diagrams, and the three inertial frames of the quickturnaround version:
http://www.adamtoons.de/physics/twins.swf As has been mentioned: If you want to have just one rest frame of noninertial twin, you have to smooth the acceleration to avoid discontinuities. In the simplest case the acceleration is constant, and you have a constant gravity in the restframe of the noninertial twin. In spacepropertime diagrams gravity looks something like this: From: http://www.physics.ucla.edu/demoweb/...spacetime.html So the twin frames would look something like this: Keep in mind that both worldlines are supposed to have the same length in each diagram, because everything advances at the same rate in spacepropertime. Also note that this is equivalent with throwing up an object (red) from the surface of the Earth (green). 



#23
Feb1413, 05:16 PM

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 Jim's worldline is straight  Pam's worldline is curved 



#24
Feb1413, 05:20 PM

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A.T.,
We've been over how this is not a valid coordinate chart of Minkowski space because one event in Minkowski space appears in two places on this chart. It could conceivably be treated as a chart of a completely different manifold. In terms of utility, how do you draw light paths on this? It seems all light paths are on top of each other along the bottom of the diagram. Not very instructive. Also not instructive is intersections of paths on this chart do not represent coincident events in the real world. I think this highly nonstandard tool will only serve to confuse. 



#25
Feb1413, 05:22 PM

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Edit: I should have noted, as PAllen did, that your diagram is not a coordinate chart; the coordinate transformation I was talking about would be done from a standard inertial chart to a noninertial chart in which Pam's worldline was the "time" axis. 



#26
Feb1413, 05:28 PM

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Let me remind to look at my post #8, where I describe carefully how the diagram looks for Pam at rest in a typical coordinate chart. This description is based on FermiNormal coordinates for Pam.




#27
Feb1413, 05:44 PM

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http://www.adamtoons.de/physics/twins.swf 



#28
Feb1413, 06:00 PM

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Can you give a simple answer to how they appear? It seems they can only be as I supposed because they have no proper time. [Edit: Ok, I see you have to press the animate button. But that just adds more confusion: you draw light paths as horizontal lines from where they originate, so based on proper time of whatever world line emitted them. It's clear as mud where they end; they don't end on the line for the other twin (in general). I don't see the logic of where they end. (Oh, I figured that out). Additional point: such diagrams are completely uninterpretable except in the presence of the Minkowski diagram, because a twin situation or a situation where two rockets never meet at all could be indistinguishable. ] 



#29
Feb1413, 06:21 PM

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I don't think light signals are useful or needed in the spacepropertime diagram of the twins. In the Minkowski diagram they are used as an indirect way to show the differential aging. But that is already shown directly in the spacepropertime diagram.




#30
Feb1413, 06:26 PM

P: 97

Also, when she switches on her gravity machine (her rocket motors) she will see Jims clock speed up due to the difference in gravitational potential. So a more correct diagram would be like this  In answer to Peter's question, his view is that of a mathematicuian. All the correct numbers can be obained from one static reference frame, and one doesn't have to look at the situation from any other point of view. But some of us want a bit more than numbers. We want to be able to visualize the situation from different points of view to enable us to understand it a bit more deeply. Of course, there are limits to what our imagining can achieve  I still like the Bohr atom with its circular and elliptical orbits, and electrons jumping from one orbit to the other! Mike NB. I didn't count the year dots on Jims turnaround line. NB2. The two lines of simultaneity shouldn't meet at 4 on Pams time line  than would assume that turnaround is accomplished in zero time. 



#31
Feb1413, 07:23 PM

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#32
Feb1413, 10:40 PM

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The direct metric method, or picking any inertial reference frame method, or doppler analysis all work fine for a W shaped trajectory, but these lines of simultaneity break down. It simply means there are limitations to that simultaneity convention for more complex noninertial motion. You can pick a different simultaneity convention, getting a different type of diagram in which Pam is 'at rest', that does work for this case. One example is radar coordinates. 



#33
Feb1413, 11:56 PM

P: 97

Thanks, PAllen, for mentioning the shoot out to the left. I hadn't realised that that happened, but its obvious when you consider Lorentz contraction and decreasing relative velocity.




#34
Feb1513, 03:39 AM

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But I agree that they should be used together with Minkowski diagrams, because both have their weak points:  Spacepropertime diagrams don't show meetings directly as intersections of world lines  Minkowski diagrams don't show popertime directly as a length 



#35
Feb1513, 04:09 AM

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In the meantime, I decided to redraw the diagrams from the link that Jaumzaum provided specifically to combine the signals for both Jim and Pam in each drawing. Here is the first one for the Inertial Reference Frame (IRF) of Jim (shown in bluePam is in black): Next is the last diagram shown in the link which is the IRF in which Pam is at rest during the return part of the trip: I don't know why they only showed the messages going from Pam to Jim. It's just as easy to show Jim's messages going to Pam. And here is a diagram they didn't show which is the IRF in which Pam is at rest during the first part of the trip when she is traveling away: Please note that each drawing illustrates exactly the same information. You can follow any message being sent by either twin, noting the year it was sent, and track how it was received by the other twin in which year it was received. On the next post, I will show another aborted attempt to marry portions of these last two diagrams together in which Pam is always at rest and then I will show a successful way to depict a noninertial diagram in which Pam is always at rest and it also correctly describes the paths of the messages. 



#36
Feb1513, 04:09 AM

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Jaumzaum's link shows an aborted attempt to combine the last two diagrams from the last post. Here I will show a better way to do this but they still have problems and they cannot show the paths of the message for both twins on the same diagram. First is the combined diagram in which Pam is always at rest and in which she is receiving the messages from Jim. Note that everything she sees is accurate:
Next is the combined diagram in which Pam is always at rest and in which she is sending messages to Jim. Although it correctly shows when she sent the messages, she cannot tell the path they take to Jim. This final noninertial drawing in which Pam is always at rest correctly shows the timings for both Pam and Jim in terms of when they send and receive all the messages: Note that Pam could always use a radar method to determine how far away Jim was and this diagram takes advantage of that information. 


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