Changing frames in the twin paradox?

[anuttarasammyak]

[mentors’ note: this thread has been spun off from a now closed thread on the twin paradox]
Ages of twins depend on in what frame of reference each of them has been.

 

[Nugatory]

I am afraid it is a false statement. Ages of twins depend on in what frame of reference each of them has been.

That is not right (and makes no sense - reference frames aren’t things that people, whether twins or not, are in or out of).

The age of the twin is what it is at any given point on that twin’s worldline and has nothing to do with any frame of reference. The thing that depends on our choice of reference frame is the age of the twin at the same time that something else somewhere else happens.

 

[anuttarasammyak]

So what is the right describtion of the twins condition instead of saying "One twin stays on Earth and the other twin is a pilot of go-return rocket" ?

 

[Nugatory]

So what is the right description of the twins condition instead of saying "One twin stays on Earth and the other twin is a pilot of go-return rocket" ?

That’s a good description. We could also say things like “one twin follows a straight-line path through spacetime, the other does not” or “one twin experiences proper acceleration and the other does not” or the like. But note that no frames are involved in any of these descriptions.

Reference frames only come in when we want to do some calculations, and then we pick some reference frame and don't change it. It may be the one in which the earth is at rest, it may be one in which the traveler is at rest on the outbound leg, it may be one in which the traveler is at rest on the return leg, or it may be one in in which neither is ever at rest (in practice we usually choose the frame in which the earth is at rest because the symmetry simplifies the calculations). We use this frame to assign coordinates to the three relevant events (departure, turnaround, reunion), then calculate the aging along each twin's worldlines - this will be the same no matter what frame we've chosen - and find that the traveler has aged less.

And there we are: no "changing frames", no "was in one frame, changed to another", no "in" a frame.

 

[vanhees71]

So what is the right describtion of the twins condition instead of saying "One twin stays on Earth and the other twin is a pilot of go-return rocket" ?

You just take their time-like trajectories and calculate their proper times between starting and meeting again.

 

[Dale]

Ages of twins depend on in what frame of reference each of them has been

As others have mentioned, there is no such thing as being "in" a reference frame. What people mean when they say "Object A is in frame B" is actually "Object A is at rest in frame B". In other words, what you want to do is specify the motion with respect to a given frame.

 

[phinds]

@anuttarasammyak this whole business of frame of reference seems to be confusing to some people before they get a handle on it. The terminology can also lead to misunderstanding.

For example, when @Nugatory said there is no "in a frame" there is a sense in which he is correct but it's fairly common usage to say, for example, "in frame A", BUT ... what you have to really interpret that to mean is "from the point of view of frame A".

So in that sense, everything and every place in the entire universe is "in" every imaginable frame of reference in the sense that you can describe their existence from the point of view of that frame**

A person can be motionless in one frame, moving at a constant velocity in another frame, and accelerating in yet another frame.

For example, a man is walking down the aisle of a spaceship that is accelerating as it passes the Earth. From the frame of reference in which his belt buckle is at rest, he is at rest. In the reference frame of the floor on which he is walking, he is moving with constant velocity. In the reference frame of a person on Earth, he is accelerating. So he is "in" all of those frames in that you can reference his motion to each frame.

What matters to a person's age is HIS/HER own world line (in his/her frame of reference). Everyone is always at rest in their own frame of reference and their rate of aging is their proper time, regardless of what the rest of the universe is doing relative to them. Different paths of motion through space-time can take differing amounts of proper time which is how the twins end up with different ages in the Twin Paradox.

** I should point out that, as I understand it, this not really true of receding galaxies and things in them, relative to us here on Earth, because of recession vs proper motion, but that's a whole 'nother can of worms.

 

[Nugatory]

For example, when @Nugatory said there is no "in a frame" there is a sense in which he is correct but it's fairly common usage to say, for example, "in frame A", BUT ... what you have to really interpret that to mean is "from the point of view of frame A".

Even that is not quite right because a frame is not something that can have a point of view - it's like saying "from the point of view of the quadratic formula", the concept just does not apply. "From the point of view of an observer at rest in that frame" makes more sense, but still encourages OP's misunderstanding.

Maybe best is to say that "in a frame" is a convenient natural language shorthand for "using the coordinates assigned by that frame" and applies to calculations made using those coordinates. "The frame of X" should be understood as "That frame in which X is at rest"; "X is in this frame" should be avoided altogether because it encourages OP's misunderstanding.

 

[anuttarasammyak]

As others have mentioned, there is no such thing as being "in" a reference frame. What people mean when they say "Object A is in frame B" is actually "Object A is at rest in frame B". In other words, what you want to do is specify the motion with respect to a given frame.

Thanks all. It is a good exercise of not only physics but English for me.

anuttarasammyak said:
Ages of twins depend on in what frame of reference each of them has been.

I try some small improvemets on this :

Ages of twins depend on in what frame of reference each of them has been at rest.

Ages of twins depend on what frames of reference have been their own.

 

[Dale]

Ages of twins depend on in what frame of reference each of them has been at rest.

I like this one a little more than the other one.

 

[Ibix]

I must say I wouldn't refer to frames at all, and would say "the ages of the twins depend on their velocity history".

I don't know what thread this was spun off from, though. Perhaps there was reason to have frames explicitly mentioned.

 

[PeroK]

Thanks all. It is a good exercise of not only physics but English for me.

anuttarasammyak said:
Ages of twins depend on in what frame of reference each of them has been.

I try some small improvemets on this :

Ages of twins depend on in what frame of reference each of them has been at rest.

Ages of twins depend on what frames of reference have been their own.

A lot of introductory material on SR focuses on "observer" and over-emphasises the observer's rest frame, IMO. One of my first insights when I was learning SR was that this is not actually the way physics has ever been done. An example is the heliocentric model of the solar system, which does not require us to leave the surface of the Earth and establish ourselves in the Sun's rest frame. In other words, regardless of your state of motion, you are free to choose any reference frame in which to study a problem. Quite early on in my studies I dropped what I considered to be this "observer nonsense" and focused on the relationship between measurements in different reference frames as being at the heart of SR.

This is one reason I don't really understand why in the twin paradox it is so important to analyse things from the travelling twin's rest frame. The travelling twin can do things the easy way and use the most convenient inertial reference frame. It's a bit like not allowing yourself to use the heliocentric model of the solar system while being stuck on the surface of the spinning Earth.

To get to my point. I think you are over-emphasising the role of an object's rest frame in determining its proper time between events. Of course, you can do it that way if you want. But, it's much easier just to describe your scenario in a suitable inertial reference frame and use that make the calculations. You don't need that frame to be the rest frame of any particular protagonist in your scenario!

That said, it's critical, of course, that overall the travelling twin has non-inertial motion of some description.

A good exercise, in fact, is to analyse the twin paradox from an arbitary inertial reference frame and confirm that the results of the differential ageing are indeed invariant (as they must be). Note that each twin's rest frame is not, per se, critical to the calculation.

 

[Ibix]

This is one reason I don't really understand why in the twin paradox it is so important to analyse things from the travelling twin's rest frame.

I think it's an artefact of how it's taught. The paradox arises in a failed attempt to build a non-inertial frame, and "...so what's the right way to do it then?" is not an unreasonable followup question from a student. But there's quite a lot more maths in working with non-inertial frames and more than one "right way to do it", so the answer can only be "we'll come to that later".

It would probably be better not to frame the thing as a paradox in the first place. Perhaps get them to construct the frames themselves, lead them to the regions not covered and multiply covered, and ask what would go wrong if one were so foolish as to use a naive time dilation calculation in this case. Then you could show qualitatively how to construct a valid non-inertial frame (use curved or piecewise lines instead of straight ones) and explain that this has a lot of consequences for the maths of vectors and promise to discuss it in the first lecture or two of the GR course.

I think @Orodruin has argued something like this case before.

 

[PeroK]

I think it's an artefact of how it's taught ...

There are two points. The first is how SR is taught; and, the second is how each of us ultimately understands the theory. My point is quite fundamental. It started with my questioning the idea of observers "disgreeing". This is a quite a powerful idea that causes a lot of confusion and consternation. But, we don't employ this language in classical physics. "One observers measures the car as moving to the left, while the other observer measures it moving to the right, so the observer's disagree about the direction the car is moving!"

I feel like I cleared a lot of clutter away when I was learning SR and was able to get down to brass tacks, as they say.

It would probably be better not to frame the thing as a paradox in the first place.

Ideally, I would only introduce the twin "thing" once the student has enough knowledge to see through any issues. But, the twins are out there with all their clutter and obfuscation, whether I like it or not.

 

[anuttarasammyak]

We are familiar with diargtrams like what I picked up from https://www.feynmanlectures.caltech.edu/I_15.html
for Galilean and Lorentz transformations. We learn the concept of coordinate systems or Frame of Reference.

However, in some GR discussion, e.g., twin 'paradox' discussion, is FR concept not suitable ? Reading the comments this question comes to my mind.

 

[Dale]

However, in some GR discussion, e.g., twin 'paradox' discussion, is FR concept not suitable ?

Not suitable for what?

 

[anuttarasammyak]

Well, I found some critics above about I referred FR for twin “paradox “.

 

[PeterDonis]

Well, I found some critics above about I referred FR for twin “paradox “.

If you mean posts in this thread, nobody was saying you can't use frames of reference. Of course you can. But you need to do so correctly. Posts in this thread have corrected wrong things that you said about frames of reference. That is not the same as saying you can't use them at all.

 

[Ibix]

Reading the comments this question comes to my mind.

You can use frames to discuss the twin paradox, but there are a couple of points worth considering.

First, if all you are doing with the frames is having somebody be at rest in them then they are an unnecessary complication. Saying "A is at rest in system S and B is at rest in system S', where S and S' are inertial reference frames and S' is moving at velocity v with respect to S" is a very long winded way of saying "A and B are moving inertially with relative velocity v". If you are going to do some analysis with the frames (e.g. introduce S'' with velocity -v and explain the frame change simultaneity issues that lead to the "paradox") then you need to introduce them, but there's none of that in what you said.

The other point is the pedagogical one that @PeroK and I were discussing, that there might be better ways to teach the issues with simultaneity than calling a bad analysis of a scenario a paradox and then dumping it in students' laps and saying "here, it's broken, fix it". As PeroK points out, though, the "paradox" is out in the wild and we have to live with it, as well as dreaming of a better world.

 

[Orodruin]

It would probably be better not to frame the thing as a paradox in the first place. Perhaps get them to construct the frames themselves, lead them to the regions not covered and multiply covered, and ask what would go wrong if one were so foolish as to use a naive time dilation calculation in this case. Then you could show qualitatively how to construct a valid non-inertial frame (use curved or piecewise lines instead of straight ones) and explain that this has a lot of consequences for the maths of vectors and promise to discuss it in the first lecture or two of the GR course.

I think @Orodruin has argued something like this case before.

I usually leave out any discussion on ”paradoxes” until very late in the course. I discuss most things geometrically so there is essentially always also a Euclidean equivalent.

However, you do need to address the ”paradoxes” because you can safely bet that each and every student in the class will already be familiar with it at a cursory level at least. Then it is your job as lecturer to set the record straight so they don’t keep misconceptions such as ”it is accceleration that causes the difference” or ”the twins will actually disagree”.