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

1. Nov 12, 2015

### Orodruin

Staff Emeritus
2. Nov 12, 2015

### RJLiberator

Excellent write up. Well-written! Thanks.

3. Nov 12, 2015

### Staff: Mentor

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.

4. Nov 12, 2015

### Greg Bernhardt

5. Nov 12, 2015

### Orodruin

Staff Emeritus
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.

6. Nov 12, 2015

### SlowThinker

Perhaps you're going a bit fast near the end.
It seems the twin staying behind should have more proper time, since $\frac{\tau_1}{\tau_2}=\gamma\geq1$.

7. Nov 12, 2015

### Orodruin

Staff Emeritus
Youre right of course, fixed.

8. Nov 16, 2015

### Smattering

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

9. Apr 18, 2016

### greswd

Last edited by a moderator: Apr 18, 2016
10. Apr 18, 2016

### greswd

I think many people are interested in the travelling 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 travelling twin sees the other twin undergo time compression (the opposite of time dilation).

11. Apr 18, 2016

### Orodruin

Staff Emeritus
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.

12. Apr 18, 2016

### greswd

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 travelling twin's perspective. I think many people have never seen such a diagram before too.

13. Apr 18, 2016

### Staff: Mentor

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.

14. Apr 18, 2016

### PAllen

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.

15. Apr 18, 2016

### Staff: Mentor

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 travelling twin's "perspective" is not symmetrical to the home twin.

16. Apr 18, 2016

### Orodruin

Staff Emeritus
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.

17. Apr 19, 2016

### Staff: Mentor

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".

18. Apr 19, 2016

### 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.

19. Apr 19, 2016

### Orodruin

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

20. Apr 19, 2016

### 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?

21. Apr 19, 2016

### Orodruin

Staff Emeritus
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.

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

22. Apr 19, 2016

### the_emi_guy

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

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.

23. Apr 19, 2016

### PAllen

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.

24. Apr 19, 2016

### Orodruin

Staff Emeritus

Again, a very common misconception. What you fail to realise 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.

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

25. Apr 19, 2016

### 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.