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Spacetime diagram - Twin paradox

by jaumzaum
Tags: diagram, paradox, spacetime, twin
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A.T.
#19
Feb14-13, 02:58 PM
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Quote Quote by PAllen View Post
why not answer any question about Pam's perspective using the simplest possible coordinates?
What people are seeking is not the mathematically simplest way to get a numerical result. They seek a conceptual understanding of how the situation is not symmetrical, and why both twin's rest frames predict that Pam will age less.
PAllen
#20
Feb14-13, 03:24 PM
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Quote Quote by A.T. View Post
What people are seeking is not the mathematically simplest way to get a numerical result. They seek a conceptual understanding of how the situation is not symmetrical, and why both twin's rest frames predict that Pam will age less.
To which, numerous times, the answer has been given that just as there is no such thing as global frames in GR, there is no such thing as a global non-inertial frame in SR. Thus to the extent you are looking for a rest frame for Pam that is analogous to stay at home rest frame, the only correct answer is that there is no such thing.

(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).
PeterDonis
#21
Feb14-13, 04:08 PM
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Quote Quote by A.T. View Post
They seek a conceptual understanding of how the situation is not symmetrical
It's not symmetrical because Pam feels a force and Jim does not. I've said that a number of times in other twin paradox threads, but I see that I didn't emphasize it in this one; there have been so many threads on this topic lately that I've lost track. So now consider it emphasized here.

Quote Quote by A.T. View Post
and why both twin's rest frames predict that Pam will age less.
This is harder because there isn't a unique way to construct "Pam's rest frame". My point was that you don't have to to derive the result that Pam ages less. If someone absolutely insists on having *some* representation of "Pam's rest frame", then we'll have to pick one. This has been discussed in other recent threads as well.

It looks like I need to get busy on the Twin Paradox FAQ that I have been meaning to draft for this forum.
A.T.
#22
Feb14-13, 05:03 PM
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Personally I like to use space-propertime 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 quick-turnaround version:

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

As has been mentioned: If you want to have just one rest frame of non-inertial 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 rest-frame of the non-inertial twin.

In space-propertime 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 space-propertime.

Also note that this is equivalent with throwing up an object (red) from the surface of the Earth (green).
Attached Thumbnails
twins_hs.png  
A.T.
#23
Feb14-13, 05:16 PM
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Quote Quote by PeterDonis View Post
It's not symmetrical because Pam feels a force and Jim does not.
Yes, and that difference in the frame-invariant proper-acceleration means that in both diagrams:
- Jim's worldline is straight
- Pam's worldline is curved
PAllen
#24
Feb14-13, 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 non-standard tool will only serve to confuse.
PeterDonis
#25
Feb14-13, 05:22 PM
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Quote Quote by A.T. View Post
Yes, and that difference in the frame-invariant proper-acceleration means that in both diagrams:
- Jim's worldline is straight
- Pam's worldline is curved
If you draw the diagrams the way you did, yes. But you can always do a coordinate transformation that makes Pam's worldline look straight and Jim's look curved. It will be harder to read off the physics from such a diagram (which is part of the point of trying to explain to people that non-inertial frames are not quite what they imagine), but you can certainly draw one.

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 non-inertial chart in which Pam's worldline was the "time" axis.
PAllen
#26
Feb14-13, 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 Fermi-Normal coordinates for Pam.
A.T.
#27
Feb14-13, 05:44 PM
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Quote Quote by PAllen View Post
In terms of utility, how do you draw light paths on this?
In the interactive version there are check boxes to switch on the light paths
http://www.adamtoons.de/physics/twins.swf
Quote Quote by PAllen View Post
I think this highly non-standard tool will only server to confuse.
I think both diagrams can be used in parallel. It's not like the standard tool (Minkowski diagram) is very successful at providing an intuitive visualization of this (from what I have experienced).
Quote Quote by PAllen View Post
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 Fermi-Normal coordinates for Pam.
I would love to see such a diagram, even if it is only an approximate sketch. But the big problem of the "typical coordinate chart" is of course that you don't see the age difference directly in the diagram.
PAllen
#28
Feb14-13, 06:00 PM
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Quote Quote by A.T. View Post
I think both diagrams can be used in parallel. It's not like the standard tool (Minkowski diagram) is very successful at providing an intuitive visualization of this (from what I have experienced).

.
I don't know about browser issue or whatever, but nothing happens when I check those boxes. I tried refreshing, checking the different 'whose frame' buttons on top, but no light paths appeared.

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 un-interpretable 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. ]
A.T.
#29
Feb14-13, 06:21 PM
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I don't think light signals are useful or needed in the space-propertime 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 space-propertime diagram.
Mike Holland
#30
Feb14-13, 06:26 PM
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Quote Quote by ghwellsjr View Post
If you've ignored the paths that light takes to travel, then you haven't addressed jaumzaum's issue, have you? But while we're waiting for your diagram that does that, maybe you could answer Peter's questions:
Jaunzaum's second diagram is incorect. It ignores the length contraction that will make Pam see a much shorter distance between the start and end of her journey while she is traveling. Only as they match speeds at the midpoint of the journey does this distance expand again to match the distance seen by Jim. So Pam sees Jim shoot out further out at turnaround, as has been mentioned before (by Peter, I think).

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.
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TwinParadox3.jpg  
PeterDonis
#31
Feb14-13, 07:23 PM
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Quote Quote by Mike Holland View Post
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.
I should clarify that I am sympathetic to this desire; I just think it's important to understand the limitations of using non-inertial frames. It's particularly important when you start moving on to curved spacetime and GR, where there are *no* global inertial frames, so *all* ways of describing a scenario have limitations.
PAllen
#32
Feb14-13, 10:40 PM
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Quote Quote by Mike Holland View Post
Jaunzaum's second diagram is incorect. It ignores the length contraction that will make Pam see a much shorter distance between the start and end of her journey while she is traveling. Only as they match speeds at the midpoint of the journey does this distance expand again to match the distance seen by Jim. So Pam sees Jim shoot out further out at turnaround, as has been mentioned before (by Peter, I think).

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.
Thanks for the drawing. You will see that this is exactly what I described in my post #8. An example of what Peter is referring to about limitations of the approach is try to do this for a sideways W shaped trajectory for Pam (where the center peak of the W does not reach all the way back to the stay at home world line). You find that you cannot construct coordinates of this type at all because the lines of simultaneity for this convention intersect, causing the the home world line to by multiply labeled: for range of events on it, each is given 2 time coordinates, which is inadmissible.

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 non-inertial 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.
Mike Holland
#33
Feb14-13, 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.
A.T.
#34
Feb15-13, 03:39 AM
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Quote Quote by PAllen View Post
Additional point: such diagrams are completely un-interpretable 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. ]
That is not quite true. Since the Euclidean length of the worldlines in the space-propertime corresponds to coordinate time, you can tell that objects meet: When they arrive at the same space coordinate, after the same Euclidean path distance.

But I agree that they should be used together with Minkowski diagrams, because both have their weak points:
- Space-propertime diagrams don't show meetings directly as intersections of world lines
- Minkowski diagrams don't show poper-time directly as a length
ghwellsjr
#35
Feb15-13, 04:09 AM
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Quote Quote by Mike Holland View Post
Quote Quote by ghwellsjr View Post
If you've ignored the paths that light takes to travel, then you haven't addressed jaumzaum's issue, have you? But while we're waiting for your diagram that does that, maybe you could answer Peter's questions:
Jaunzaum's second diagram is incorect. It ignores the length contraction that will make Pam see a much shorter distance between the start and end of her journey while she is traveling. Only as they match speeds at the midpoint of the journey does this distance expand again to match the distance seen by Jim. So Pam sees Jim shoot out further out at turnaround, as has been mentioned before (by Peter, I think).
Could you be a little more specific with which diagram you are referring to? I couldn't find one that was incorrect.

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 blue--Pam 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 non-inertial diagram in which Pam is always at rest and it also correctly describes the paths of the messages.
Attached Thumbnails
Married Frames 1.PNG   Married Frames 2.PNG   Married Frames 3.PNG  
ghwellsjr
#36
Feb15-13, 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 non-inertial 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.
Attached Thumbnails
Married Frames 4.PNG   Married Frames 5.PNG   Married Frames 6.PNG  


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