I Why Is the Green Observer Always the Bent One in the Twin Paradox?

[stevendaryl]

I think I just had an "aha!" moment. Tell me if I'm on the right track. In descriptions of different observers' reference frames, people often say things like, "Pretend Alice is holding a ruler in her hand that she uses to measure distances relative to her and Bob is holding a ruler to measure distances relative to him." Well, I didn't realize that they ever let go of their rulers! I thought that even when changing direction, the green guy would keep his ruler in his hand. That's why, to me, the diagram shown in the FAQ...

View attachment 245184

...describes a completely symmetric situation, with the red guy arbitrarily chosen as the one at rest. If the green guy continues to hold on to his ruler, then from his point of view the situation would look like what I put in my original post...

Something to consider is the analogous question for Euclidean geometry. Suppose the two paths, the green path and the red path, instead of being paths in spacetime, are paths in space. Suppose that they are two different ways to get from one location, on the lower left, to another location, on the lower right. If two cars met in the lower left, and one car took the green path, and one car took the red path, then when they arrive at the lower right, they could compare distances as measured on the cars' odometers. The two odometer readings will be different. Is that a physical effect? Did the odometers work differently along the green and red paths? Or do the odometers just accurately reflect the difference in lengths of the two paths?

 

[SamRoss]

Did the odometers work differently along the green and red paths? Or do the odometers just accurately reflect the difference in lengths of the two paths?

Of course the odometers are simply measuring the distance along each path. I think my mistake was in taking too literally statements like "Alice's reference frame" and "Bob's reference frame". I thought Bob's reference frame was a frame that moved with him. Bob would always be at x=0 no matter what actions he took. Now I'm realizing that the reference frame might have been named after Bob because he happened to be moving along with it at first, but it will not follow him if he decides to change direction. This misunderstanding prevented me from viewing world lines as fixed. I thought the world line would change shape when looked at from the point of view of a different observer.

 

[jbriggs444]

Now I'm realizing that the reference frame might have been named after Bob because he happened to be moving along with it at first, but it will not follow him if he decides to change direction.

Nicely and accurately put!

Inertial reference frames are... inertial. They keep moving.

There is such a thing as an accelerating reference frame but it usually takes more than the trajectory of a single object to define one. There are rules involving continuity, not double-mapping events and such that must be followed. Simply stringing together a series of instantaneous tangent inertial frames does not always meet the requirements.

Measurements of time and space referenced against an accelerating frame are even weirder than those referenced against inertial frames.

 

[Ibix]

Simply stringing together a series of instantaneous tangent inertial frames does not always meet the requirements.

Indeed - see post #22 for one of the problems.

 

[SamRoss]

There is such a thing as an accelerating reference frame but it usually takes more than the trajectory of a single object to define one. There are rules involving continuity, not double-mapping events and such that must be followed. Simply stringing together a series of instantaneous tangent inertial frames does not always meet the requirements.

Measurements of time and space referenced against an accelerating frame are even weirder than those referenced against inertial frames.

Interesting. I would have thought you could string together an infinite number of inertial frames to get an accelerating one. Could you direct me to where I can read about measuring time and space in an accelerating reference frame and how this is different from stringing together inertial frames?

 

[Orodruin]

Interesting. I would have thought you could string together an infinite number of inertial frames to get an accelerating one.

This is the issue. There is no unique way of doing this and ”string together” is not a very well defined concept. I would suggest readibg up on curvilinear coordinates in general for Euclidean spaces before trying to look at non-inertial reference frames in relativity. One major issue is the relativity of simultaneity, which makes it non-trivial to define what ”string together” actually means.

 

[Dale]

Could you direct me to where I can read about measuring time and space in an accelerating reference frame and how this is different from stringing together inertial frames?

Here is one reference on the topic that I found recently:

https://arxiv.org/abs/gr-qc/0006095
Figure 1 identifies the main mathematical problem associated with naively stringing together reference frames.

 

[stevendaryl]

Interesting. I would have thought you could string together an infinite number of inertial frames to get an accelerating one. Could you direct me to where I can read about measuring time and space in an accelerating reference frame and how this is different from stringing together inertial frames?

Here's the problem: Suppose that you are at rest in one frame up until time $t=0$, then you accelerate quickly so that you are now at rest in a second frame. The first frame uses coordinates $(x,t)$ and the second frame uses coordinates $(x',t')$. How can you piece together these two inertial frames so that you have a noninertial frame? Since "frame" is a little ambiguous, let me switch the problem to that of coordinate systems. How can you piece together two inertial coordinate systems, $(x,t)$ and $(x',t')$ to come up with a noninertial coordinate system?

The obvious thing would be to use the first coordinate system for events prior to $t=0$, and use the second coordinate system for events after $t=0$. Here's the problem with that. In the drawing below, I've drawn a representation of a region of spacetime. I've used the black lines to indicate the t-axis and the x-axis. I've used blue lines to indicate the t'-axis and the x'-axis. If you're wondering why the t' and x' axes seem tilted, that's just the Lorentz transformations. The t' axis is the line satisfying the equation $x' = 0$, which in terms of the original coordinates $(x,t)$ is the equation $\gamma (x - v t) = 0$ or $t = x/v$. So the t'-axis is a line with slope $1/v$ when plotted in terms of $x,t$. The x' axis is the line satisfying $t'=0$, which in terms of $(x,t)$ is the line $\gamma (t - \frac{vx}{c^2}) = 0$ or $t = \frac{vx}{c^2}$. So the x' axis is a line with slope $\frac{v}{c^2}$.

So our obvious way of combining coordinate systems is to use $(x,t)$ for events prior to the change of frames, and to use $(x',t')$ for events after the change. However, if you look at the chart, I've divided spacetime into numbered regions. Our rule works fine for regions 1, 3,4, 5, 7, and 8. However, there are two regions that are problematic: Region 6 is doubly-covered. According to coordinate system $(x,t)$, it covers event prior to the change of frames, and therefore should be described using the coordinates $(x,t)$. But according to the coordinate system $(x',t')$, it covers events after the change of frames and so should be described using the coordinates $(x',t')$. Both coordinate systems claim this region.

We have the opposite problem for region 2. It is not covered at all. According to the coordinate system $(x,t)$, it covers events that take place after the frame change, and so should be described using coordinates $(x',t')$. According to coordinate system $(x,t)$, region 2 takes place before the frame change, so should be described using coordinates $(x,t)$. Neither coordinate system claims this region.

There is no obvious way to come up with a coordinate system for an accelerated observer that

1. Covers every point in spacetime.
2. Doesn't double-cover any points in spacetime.
3. Has the observer at rest at all times.

 

[PeroK]

Interesting. I would have thought you could string together an infinite number of inertial frames to get an accelerating one. Could you direct me to where I can read about measuring time and space in an accelerating reference frame and how this is different from stringing together inertial frames?

To add to the above, this issue came up in a different guise recently, in a paradox about length contraction:

https://www.physicsforums.com/threads/implication-of-length-contraction.966815/#post-6137722

 

[Ibix]

Could you direct me to where I can read about measuring time and space in an accelerating reference frame and how this is different from stringing together inertial frames?

See post #22. It's just stringing two inertial frames together, but immediately you see that parts of spacetime (where the party was) get lost from the picture. Other parts get double-counted, in fact.

 

[SiennaTheGr8]

It's maybe worth emphasizing that constructing whole coordinate systems for an accelerating traveler is a different exercise from tracking the accelerating traveler's aging (elapsed proper time). For the latter, I believe it's always okay to divide the trip into infinitely many instantaneous inertial frames (for which the traveler is momentarily at rest) and integrate over the traveler's corresponding proper-time infinitesimals.

 

[stevendaryl]

It's maybe worth emphasizing that constructing whole coordinate systems for an accelerating traveler is a different exercise from tracking the accelerating traveler's aging (elapsed proper time). For the latter, I believe it's always okay to divide the trip into infinitely many instantaneous inertial frames (for which the traveler is momentarily at rest) and integrate over the traveler's corresponding proper-time infinitesimals.

Yes, that is true. What that construction does not do, however, is to give an unambiguous answer to the question: How old is this traveler compared with that traveler? (Unless they are in the same place and time.)

 

[Nugatory]

Yes, that is true. What that construction does not do, however, is to give an unambiguous answer to the question: How old is this traveler compared with that traveler? (Unless they are in the same place and time.)

Is there any construction that can give an unambiguous answer when the two travelers are not colocated? To remove the ambiguity we need an additional arbitrary choice of a simultaneity convention, do we not?

 

[PAllen]

I think debates about when, in the history of a general noninertial worldline, does the exra time of the inertial path correspond, are exactly as meaningful as asking:

Given a straight line and curved line between two points in a plane, which part of the curved line is the extra distance?

That is, it is nonsense. In both cases you can match up points on the two lines to get any answer you want.

 

[stevendaryl]

Is there any construction that can give an unambiguous answer when the two travelers are not colocated?

No. I don't think so.

 

[jbriggs444]

Is there any construction that can give an unambiguous answer when the two travelers are not colocated? To remove the ambiguity we need an additional arbitrary choice of a simultaneity convention, do we not?

Depends on what you are willing to consider as "arbitrary". For inertial travelers, the frame where their velocities are equal and opposite would be one obvious non-arbitrary choice.

 

[Ibix]

Depends on what you are willing to consider as "arbitrary". For inertial travelers, the frame where their velocities are equal and opposite would be one obvious non-arbitrary choice.

I think this is one source of confusion in this type of scenario. From the point of view of any given scenario there is always (or often, at least) a fairly narrow selection of sensible frames to use. One where a velocity is zero, or two velocities are equal, and suddenly some bit of maths is trivial. But from the point of view of the physical laws any choice is completely arbitrary.