A Geometrical View of Time Dilation and the Twin Paradox - Comments

[Orodruin]

Orodruin submitted a new PF Insights post

A Geometrical View of Time Dilation and the Twin Paradox

Continue reading the Original PF Insights Post.

 

[RJLiberator]

Excellent write up. Well-written! Thanks.

 

[Dale]

Thanks Orodruin. This is definitely worth referencing.

The geometrical view does help clear up a lot of the confusing aspects of SR, but it seems to be a big leap for new students. I wonder which is more difficult for most students: understanding the geometrical view or understanding the relativity of simultaneity.

 

[Greg Bernhardt]

Facebook comment: "This is Gold"

 

[Orodruin]

The geometrical view does help clear up a lot of the confusing aspects of SR, but it seems to be a big leap for new students.

I find the geometrical comparison to Euclidean space helps students and it is how I start the presentation of the subject in my SR class, by introducing Lorentz transforms as the hyperbolic rotations leaving the Minkowski line element invariant. Admittedly, these are students who are in their fourth year of university and they may be more ready for it. I had the idea of writing this Insight three days ago when I was giving the corresponding lecture, although I have tried to make it slightly more accessible than the level I present it on in class.

 

[SlowThinker]

Perhaps you're going a bit fast near the end.

Just applying the modified Pythagorean theorem it becomes clear that the proper times along the world lines are given by
$$\tau_1=2\tau,\ \ \ \ \tau_2=2\tau\sqrt{1-\frac{v^2}{c^2}}\ \ \ \ \ \implies\ \ \ \ \ \frac{\tau_1}{\tau_2}=\gamma$$
and therefore less proper time will pass for the twin staying behind, i.e., the red curve.

It seems the twin staying behind should have more proper time, since $\frac{\tau_1}{\tau_2}=\gamma\geq1$.

 

[Orodruin]

Perhaps you're going a bit fast near the end.

It seems the twin staying behind should have more proper time, since $\frac{\tau_1}{\tau_2}=\gamma\geq1$.

Youre right of course, fixed.

 

[Smattering]

Orodruin submitted a new PF Insights post

This is indeed a very good read. Thanks a lot for the effort.

 

[greswd]

What do you think of this derivation of the space-time diagram from the traveling twin's perspective?

As described in this paper: http://arxiv.org/abs/gr-qc/0104077v2

 

[greswd]

What do you think of this derivation of the space-time diagram from the traveling twin's perspective?

As described in this paper: http://arxiv.org/pdf/gr-qc/0104077v2.pdf

I think many people are interested in the traveling twin's perspective.

This diagram is bizarre because both twins are in the SAME FRAME for a finite duration of time, this is not the case in the staying twin's frame.

And despite both being in the same frame, the traveling twin sees the other twin undergo time compression (the opposite of time dilation).

 

[Orodruin]

I think many people are interested in the traveling twin's perspective.

This diagram is bizarre because both twins are in the SAME FRAME for a finite duration of time, this is not the case in the staying twin's frame.

And despite both being in the same frame, the traveling twin sees the other twin undergo time compression (the opposite of time dilation).

This is full of misconceptions. First of all, all objects exist in all frames. Frames are just different ways of assigning coordinates to events. You can use any viable frame to describe any situation as long as you do it properly. Some frames are better suited for the analysis, but that does not imply a special status. The "radar coordinates" introduced in the paper is a non-minkowski coordinate system. You should be as little surprised about having different proper time per radar time as you should be that changing a polar coordinate results in different displacements for different distances from the origin in normal Euclidean space.

Apart from that, I do not see anything particularly new or of additional pedagogical value here. Just the added complexity of not using a minkowski coordinate chart.

 

[greswd]

This is full of misconceptions. First of all, all objects exist in all frames. Frames are just different ways of assigning coordinates to events. You can use any viable frame to describe any situation as long as you do it properly. Some frames are better suited for the analysis, but that does not imply a special status. The "radar coordinates" introduced in the paper is a non-minkowski coordinate system. You should be as little surprised about having different proper time per radar time as you should be that changing a polar coordinate results in different displacements for different distances from the origin in normal Euclidean space.

Apart from that, I do not see anything particularly new or of additional pedagogical value here. Just the added complexity of not using a minkowski coordinate chart.

By "same frame", I mean same inertial reference frame. Both being at rest w.r.t. each other.

I'm not sure what you mean by "special status", I don't think I mentioned that any frame had a special status.

The "time compression" may be viewed as a form of gravitational blueshift. This and the "same frame" effect are described as being due to "shock like scale discontinuity".

Speaking of additional pedagogical value, I was just curious about the traveling twin's perspective. I think many people have never seen such a diagram before too.

 

[Nugatory]

This diagram is bizarre because both twins are in the SAME FRAME for a finite duration of time

By "same frame", I mean same inertial reference frame. Both being at rest w.r.t. each other.

Whether the twins are at rest relative to one another has nothing to do with whether they are in the same frame. Both twins are always in all frames all the time.

 

[PAllen]

This is full of misconceptions. First of all, all objects exist in all frames. Frames are just different ways of assigning coordinates to events. You can use any viable frame to describe any situation as long as you do it properly. Some frames are better suited for the analysis, but that does not imply a special status. The "radar coordinates" introduced in the paper is a non-minkowski coordinate system. You should be as little surprised about having different proper time per radar time as you should be that changing a polar coordinate results in different displacements for different distances from the origin in normal Euclidean space.

Apart from that, I do not see anything particularly new or of additional pedagogical value here. Just the added complexity of not using a minkowski coordinate chart.

Note that radar coordinates are exactly Minkowski coordinates if the defining world line is eternally inertial. They are asymptotically the same as Fermi-Normal coordinates near the defining world line. They have the specific feature that as long as the defining world line is inertial for all proper time before some event, and also inertial for all proper time after some event, no matter how complex the motion in between, and however 'long' such complex motion occurs, they give a complete coordinate chart of Minkowski space (unlike Fermi-Normal coordinates). Thus, they can be seen as a generalization of Minkowski coordinates to describe the 'experience' of a non-inertial observer, having several nice properties (including also they are built from more achievable measurements than positing very long rulers).

All the same, one must never attach 'more reality' to a description in radar coordinates than in some other coordinates.

 

[Dale]

What do you think of this derivation of the space-time diagram from the traveling twin's perspective?

As described in this paper: http://arxiv.org/abs/gr-qc/0104077v2

I like it. It is one of my favorites.

That said, I wouldn't worry too much about details like the part of the trip where the distance is constant. It is a non-inertial frame so distance and speed don't have their ordinary meanings anyway. The point is mostly that the traveling twin's "perspective" is not symmetrical to the home twin.

 

[Orodruin]

I like it. It is one of my favorites.

That said, I wouldn't worry too much about details like the part of the trip where the distance is constant. It is a non-inertial frame so distance and speed don't have their ordinary meanings anyway. The point is mostly that the traveling twin's "perspective" is not symmetrical to the home twin.

But exactly here lies the problem. I am not arguing that the coordinates are abad idea, I just question the pedagogical value to people who are just learning relativity. I do not see the point of introducing non-inertial coordinate systems on top of the struggles they already have.

 

[Dale]

But exactly here lies the problem. I am not arguing that the coordinates are abad idea, I just question the pedagogical value to people who are just learning relativity. I do not see the point of introducing non-inertial coordinate systems on top of the struggles they already have.

I agree. Pedagogically, for a novice I would encourage them to stick with inertial frames. Then if they persist use the paper in the spirit of "your question does have an answer, here it is, come back to it when you are ready".

 

[the_emi_guy]

Seems to me that SR cannot be applied to a problem that involves three inertial frames and only two
observers. Isn't an observer moving from one inertial frame to another is out of the scope of SR?
Seems that any solution would require a footnote indicating that
the presumably infinite acceleration experienced by the "travelling" twin is assumed to have no implication.

On the other hand, allowing three observers, one in each frame, is a valid SR problem, and its solution is a trivial
application of time dilation and removes all of the complexity of the two observer problem.

Observers:
Observer1 (Earthbound observer) is on earth.
Observer2 (outbound observer) passes Earth traveling toward Alpha Centauri (and never comes back)
Observer3 (inbound observer) is traveling from Alpha Centauri and first passes outbound observer, then later passes earth.

Events:
Earth passes his time to outbound observer when they meet.
Outbound observer passes his time to inbound observer when they meet.

Analysis:
Outbound observer's point of view (exclusively): Earthbound clock is running slow (simple time dilation)
thus inbound observer's clock is set ahead of Earth's clock when outbound and inbound meet.

Now continuing along with outbound observer (*not* switching inertial frames). Earth's clock is still running slow, but the
inbound observer's clock is running even slower since this relative velocity is larger.
Thus, by the time that the outbound observer sees the inbound observer reach earth, he will see that the inbound observer's clock will have gone from being ahead of Earth's to being behind Earth's by simple application of time dilation applied to this larger relative velocity.

This can be worked using simple SR time dilation from any of the three inertial frames to obtain identical results.

 

[Orodruin]

Seems to me that SR cannot be applied to a problem that involves three inertial frames and only two
observers. Isn't an observer moving from one inertial frame to another is out of the scope of SR?

No. It is perfectly within the scope of SR. You do not need an actual observer to define an inertial frame in SR.

 

[the_emi_guy]

We have to allow the observers to have physical bodies, and the clocks to have actual mass, otherwise this is not a physics problem but a mathematical exercise.

More importantly, why would we take a problem that when formulated with three real human observers is trivial to analyze, and formulate it in a way that is very complicated (look at how many questions PF gets on this topic), and requires hypothetical massless observers?

 

[Orodruin]

We have to allow the observers to have physical bodies, and the clocks to have actual mass, otherwise this is not a physics problem but a mathematical exercise.

No, this is not true. You can use any actual observer to define a reference frame, but it is not necessary for the theory to make sense. Changes of inertial frames are just changes of how you assign coordinates. What you are claiming is essentially the equivalent of saying that you cannot use spherical coordinates unless you have an actual sphere, which is obviously absurd. It may not be the most convenient coordinate system unless you have some sort of spherical symmetry, but you certainly can use it.

A physical exercise is any exercise which makes experimentally verifiable predictions. You can use any coordinate system which might be convenient to make those predictions. Like many novices, you are putting way too much relevance into coordinates instead of focusing on the actual issue in the twin paradox problem - the geometry of space-time. You can use any coordinate system you wish, the geometry will not change.

More importantly, why would we take a problem that when formulated with three real human observers is trivial to analyze, and formulate it in a way that is very complicated (look at how many questions PF gets on this topic), and requires hypothetical massless observers?

There are no massless hypothetical observers in this example. Just different coordinate systems on the same geometrical object.

 

[the_emi_guy]

No, this is not true. You can use any actual observer to define a reference frame, but it is not necessary for the theory to make sense.

If we are not talking about actual masses traveling in these inertial frames then we are not talking about physics.

Changes of inertial frames are just changes of how you assign coordinates. What you are claiming is essentially the equivalent of saying that you cannot use spherical coordinates unless you have an actual sphere, which is obviously absurd. .

Nope, I can have an observer (real human observer) in an single inertial frame and change coordinates all day. While traveling through New York my position can be rendered in Cartesian coordinates, then when I cross the border into New Jersey we decide to render my position in some polar coordinates. Nothing about this change in coordinates implies a change in inertial frame. On the other hand, the "traveling" twin is undergoing acceleration bringing him into a different inertial frame. This is outside the scope of SR unless we assume a non-physical observer.

I am not just a "novice" making this up. I have seen 30 page dissertations on the subject of how to properly handle the acceleration problem inherent in the two observer twin paradox. Just go to Wikipedia under "twin paradox" and search for the word "acceleration".

My question is why bother? With three observers, each in his own unchanging inertial frame, we get the expected result with simple plug 'n chug application of time dilation applied twice to of the three observers, all comfortably within SR.

 

[PAllen]

If we are not talking about actual masses traveling in these inertial frames then we are not talking about physics.

This is a nonsensical claim, and the rest of your claims follow from it. So far as I know, there is no reputable modern theoretical physicist who would agree with the claim that acceleration and accelerated observers are outside the scope of SR.

 

[Orodruin]

If we are not talking about actual masses traveling in these inertial frames then we are not talking about physics.

Wrong again. Please read my previous post.

Nope, I can have an observer (real human observer) in an single inertial frame and change coordinates all day. While traveling through New York my position can be rendered in Cartesian coordinates, then when I cross the border into New Jersey we decide to render my position in some polar coordinates. Nothing about this change in coordinates implies a change in inertial frame. On the other hand, the "traveling" twin is undergoing acceleration bringing him into a different inertial frame. This is outside the scope of SR unless we assume a non-physical observer.

Again, a very common misconception. What you fail to realize is that the changes from Cartesian to polar coordinates is just the same as applied to space as a Lorentz transformation is to space-time (not completely true, a Lorentz transformation is the equivalent of a change between different Cartesian coordinate systems in space). You can easily use a curvilinear coordinate system in SR as well and there are many examples. Unless you accept that SR is not a theory about observers but a theory about the geometry of space-time, you will be very hard pressed to try to learn GR.

My question is why bother? With three observers, each in his own unchanging inertial frame, we get the expected result with simple plug 'n chug application of time dilation applied twice to of the three observers, all comfortably within SR.

Everything in this thread is SR. That SR cannot deal with acceleration is a myth.

 

[the_emi_guy]

So acceleration (and presumably gravity by equivalence property) is withing the scope of SR, I'll have to take your word for this. Still, I'll repeat from earlier:

Why bother with all of this complexity when formulating the problem with three observers (first proposed by Lord Halsbury) gets the expected result with simple plug 'n chug application of time dilation applied twice to of the three observers.

 

[PAllen]

So acceleration (and presumably gravity by equivalence property) is withing the scope of SR, I'll have to take your word for this. Still, I'll repeat from earlier:

No, per modern understanding, gravity involves curved spacetime, while SR involves flat spacetime. The equivalence principle is local,and is analogous to the statement that a small region of sphere is well approximated by a plane. Tidal gravity effects distinguish gravity from acceleration, but they become very small in a small region.

Why bother with all of this complexity when formulating the problem with three observers (first proposed by Lord Halsbury) gets the expected result with simple plug 'n chug application of time dilation applied twice to of the three observers.

What if there are only two observers? How is it useful, pedogogically, to say we must introduce a third to explain what happens? Especially, if the spacetime geometry viewpoint is very simple and requires only one frame (not necessarily one in which any observer is at rest).

 

[PeterDonis]

I can have an observer (real human observer) in an single inertial frame

You are misusing the term "inertial frame". The correct way of stating what you are trying to state here is: I can have an observer in the same inertial state of motion... The observer's state of motion is the physical thing. The "inertial frame" is a mathematical construct that we use to help us describe what is going on. And, as others have remarked, the same observer's worldline, with the same state of motion, can be described using as many different inertial frames--or non-inertial frames, for that matter--as we like.

the "traveling" twin is undergoing acceleration bringing him into a different inertial frame.

No. The traveling twin undergoes acceleration which changes his state of motion. (Even this terminology has limitations, but it will do for this discussion.) But there is no need to switch inertial frames to describe this; you can describe it all perfectly well using a single inertial frame.

So acceleration (and presumably gravity by equivalence property)

You are misstating this as well. "Gravity" is an ambiguous term. The correct statement, which has already been given in this thread, is that SR deals with any scenario in which spacetime is flat. GR deals with scenarios in which spacetime can be curved.

 

[the_emi_guy]

What if there are only two observers? How is it useful, pedogogically, to say we must introduce a third to explain what happens? Especially, if the spacetime geometry viewpoint is very simple and requires only one frame (not necessarily one in which any observer is at rest).

Thanks for asking this, it gets right to my point.

Many folks who visit PF have learned about time dilation between observers who are in relative motion but want to get a handle on how perfectly good clocks can wind up sitting side-by-side but out of sync due to some motion that occurred. These folks are better served by showing them how simple application of time dilation to all three inertial frames (see post #18) shows this in a trivial and intuitive manner (and I do mean "inertial frames" here, with each observer at rest in his frame).

As a follow on we could consider "what if there were only two observers?".
Well, what did the second observer do? Did he turn around instantaneously experiencing infinite acceleration (a non-physical physics problem)? What if he accelerates and decelerates gradually over time? What if he slingshots himself around another planet experiencing weightlessness the whole way? It may very well turn out that all of these produce the same result because spacetime does not get curved and SR still applies (or maybe not). Volumes have been written about all these various scenarios.

 

[PeterDonis]

These folks are better served by showing them how simple application of time dilation to all three inertial frames (see post #18) shows this in a trivial and intuitive manner (and I do mean "inertial frames" here, with each observer at rest in his frame).

Whether this is "trivial and intuitive" is debatable (see below), but it seems to me that these folks would be served better still by giving them the most general method of all: spacetime geometry and lengths of curves. What we have here is a triangle in spacetime: one side is the stay-at-home twin's worldline, and the other two sides are the traveling twin's outbound and inbound worldlines. Then all we need is the Minkowski spacetime analogue to the Pythagorean theorem--of which the time dilation equation you are implicitly using in your method is just a special case.

As a follow on we could consider "what if there were only two observers?"

That's not a follow-on; it's how the original scenario was posed. When you formulate it in terms of three observers, you are, on the face of it, talking about a different scenario. You then have to explain how the result you derive with three observers gives the right answer for the case of only two, which was the actual scenario posed. This is certainly doable, but I question whether it is "trivial and intuitive" to a person who is struggling with time dilation and differential aging.

what did the second observer do? Did he turn around instantaneously experiencing infinite acceleration (a non-physical physics problem)? What if he accelerates and decelerates gradually over time? What if he slingshots himself around another planet experiencing weightlessness the whole way? It may very well turn out that all of these produce the same result because spacetime does not get curved and SR still applies (or maybe not).

The first case is the one that is equivalent to your analysis--although I would prefer to phrase it in terms of taking the limit as the turnaround time (the time elapsed on the traveling twin's clock during the period when he is accelerating to turn around) goes to zero. The key point is that, if the answer we are interested in is elapsed proper time for the two twins, we don't really care about the details of how the traveling twin turns around, as long as the time elapsed on his clock while doing so is negligible compared to the time elapsed on his clock during the inbound and outbound legs.

The second case, accelerating and decelerating gradually over time, is not equivalent to your analysis. And the simplest way to state the reason why is by using spacetime geometry and worldlines: the worldline of the traveling twin is different in this case than it is in the idealized case you are analyzing (which is equivalent to the idealized case of instantaneous turnaround). So you would not expect the traveling twin's elapsed time to be the same. It will still be shorter than the stay-at-home twin's elapsed time when they meet up again, but by a different amount (which will depend on the details of how the twin accelerates and decelerates gradually). The spacetime geometry method adapts easily to this case; it just amounts to evaluating arc length along a curve instead of a pair of straight lines.

The two cases above can both be formulated in flat spacetime. The third, by contrast, requires curved spacetime, because in flat spacetime there is no way for the traveling twin to turn around without feeling acceleration, whereas in this third "slingshot" case, the traveling twin is weightless, feeling no acceleration, the whole way. The spacetime geometry method carries over to this case with no problem at all, since, as I noted, it is completely general.

 

[the_emi_guy]

Peter, thanks for the thoughtful response, I appreciate it.

I think that there is unrecognized value in seeing that an outbound (non-returning) observer sees Earth clock running slow, later synchronizes his "ahead" clock with an inbound clock, then watches the inbound clock go from being ahead to being behind Earth do to the greater inbound vs. outbound relative velocity. Note that this clock skew is relative to outbound's local clock, Earth's clock ended up being behind, and inbound's clock even more behind.

Doing this in all three observer rest frames and getting the same clock skew I think is compelling to a beginner, and the concept of proper time was not even needed or required.