Spacetime diagram - Twin paradox

PAllen

Well, there is a question of: "what is meant by Pam's perspective?". When Pam looks out of the rocket portal, takes measurements, etc. coordinates are not involved. Coordinates are an artifact of a mathematical model of reality. Given the philosophy that Einstein called general covariance and is now called diffeomorphism invariance, you have the result that it doesn't matter what coordinates you use to answer any possible physical definition of "Pam's perspective". Given this hard and fast mathematical foundation, why not answer any question about Pam's perspective using the simplest possible coordinates? These would be, any convenient inertial frame. Similarly in GR, nobody worries about having coordinates in which some world line is the time axis; you just pick some computationally convenient coordinates and compute what any observer sees or measures.

A.T.

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

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

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.

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.

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).

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A.T.

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

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

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

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.

In the interactive version there are check boxes to switch on the light paths
http://www.adamtoons.de/physics/twins.swf
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 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

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.

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

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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PeterDonis

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

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

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.

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