Another train and platform

[JVNY]

Here is another train and platform example, offered to examine ghwellsjr's view equating what you see with what is real.

Assumptions:

Train has proper length 100.
Platform has proper length 60.
Train and platform are in relative inertial motion at 0.8c.
There are two observers on the platform, Far (on the farther end of the platform from the train's perspective), and Near (on the nearer end of the platform from the train's perspective).
Near and Far have watches, synchronized in the platform frame.
Near's watch is an LED watch that flashes a signal with a time stamp toward Far as Near's watch records elapsed time.
Simultaneously in the platform frame, the front of the train is at the far end of the platform, and the rear of the train is at the near end of the platform. Hooks at the front and rear catch Far and Near, like mailbags on the old mail trains. In the platform frame, both watches read 60 at this time.

What happens according to Far?

One theory: Far is hooked aboard the train and joins the train's reference frame when the far end of the platform and the front end of the train are aligned; Far's watch reads 60; but the platform is length contracted so that the near end of the platform is well ahead of the rear end of the train. For Far, the platform length contracts instantaneously toward him, so that Near contracts from being 60 away to being only 36 away. Near's watch runs backwards (the opposite of clock advancement that occurs in the usual acceleration examples, because the usual examples refer to planets or stars ahead of the accelerating object, not behind it as here).

Stella inhabits an accelerated frame, and clocks in such a frame can . . . run slower or even backwards! The details depend not only on their motions, but also on their positions relative to Stella. . . Light from events below the plane can never reach you at all, and this prompts the plane to be called a "horizon". In fact, although you cannot know what is happening below this plane, it turns out that you can infer time below it is going backwards.

See http://math.ucr.edu/home/baez/physics/Relativity/SR/movingClocks.html​

Then, Near's watch runs slowly (time dilation), so that by the time Near has aligned with the rear of the train and she is hooked aboard her watch is back up to 60. Near is younger than Far, as one would expect (Near's clock ran backwards, then ran more slowly than Far's).

This theory seems to imply that Near's watch will send three sets of inconsistent time stamp signals to Far. First, just as Far's watch reads 60, Far will only have received Near's watch's flash time stamped 0 (because Far is 60 away). Thus signals time stamped 1-60 are still on their way toward Far. Second, Near's watch runs backwards, so the time stamps of each successive backward tick of the watch is sent toward Far. Third, Near's watch runs forward again, sending new successively forward time stamped signals. One could argue that there is a Rindler horizon when Far accelerates, preventing him from seeing Near's watch signals (as suggested in the quotation above). However, once he stops accelerating the horizon disappears, so any signals that occurred behind the horizon would be able to catch up to him. See, for example:

Of course, this is not the same as a black hole’s event horizon in two very important respects. Firstly, it’s always possible to stop the spaceship accelerating, so this horizon’s persistence is a matter of choice, not physical law.

http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html​

A second theory: it is absurd to think that Far receives these multiple and conflicting sets of time stamped signals. Far receives the signal time stamped "0" just as he is hooked and joins the train, and thereafter he receives only forward ticking signals from Near. There is simply a gap in the set of time stamped signals that Far sees. This gap explains why Near ages less than Far (what you see is what you get). ghwellsjr, is this how you would analyze this example?

 

[PeterDonis]

What happens according to Far?

What does "according to Far" mean? As you've set up the scenario, at the instant $t = 60$ in the platform frame, both Near and Far experience a sudden discontinuous change in their motion, from being at rest relative to the platform, to moving at 0.8c relative to the platform. So to say "what happens according to Far", if that's going to mean anything more than what Far directly observes (for example, to assign a "time according to Far" to events spatially separated from Far, such as events on Near's worldline, which is what you have to do if you want to assign a "time dilation" to Near, relative to Far), you have to decide what kind of coordinate chart you are going to use to represent "what happens according to Far".

If, OTOH, you are willing to limit "what happens according to Far" to what he directly observes, then it's easy. See further comments below.

One theory: Far is hooked aboard the train and joins the train's reference frame

Observers don't "join" reference frames. A reference frame (in the sense you are using the term--a better term would be "coordinate chart") is an arbitrary convention. Far does not have to use a coordinate chart in which he is always at rest; and if he is not inertial forever (and he isn't--he is non-inertial at least at the instant that his motion changes discontinuously, as above), there is no unique way to construct a coordinate chart in which he is always at rest. That means there is no unique way for Far to assign Near a distance from him, or a time relative to him. So the question you're asking, in the sense you're asking it, isn't well-defined; it depends on how Far chooses a coordinate chart, and there is no unique way for him to do so.

I should note that even good sources of information about physics are often sloppy in their language on this point; for example, the Usenet Physics FAQ entry which you quote:

Stella inhabits an accelerated frame, and clocks in such a frame can . . . run slower or even backwards!

The "accelerated frame" being referred to here is not uniquely picked out by the physics, and Stella "inhabits" it only if she chooses to use such a chart. It can be useful to do so, as long as you recognize the limitations of doing so.

A second theory: it is absurd to think that Far receives these multiple and conflicting sets of time stamped signals.

Of course it is. What signals Far receives, and in what order, is a direct observable and must be the same regardless of how you calculate it or what coordinate chart you use. Your next sentence is quite correct, but incomplete:

Far receives the signal time stamped "0" just as he is hooked and joins the train, and thereafter he receives only forward ticking signals from Near.

He receives "forward ticking signals from Near" *all* the time, before he is hooked as well as after. However, this...

There is simply a gap in the set of time stamped signals that Far sees.

...is not correct; there is no "gap". All that changes is the Doppler shift that Far observes Near's signals to have (see below for a more detailed description). Also, this...

This gap explains why Near ages less than Far (what you see is what you get).

...is not correct as it stands either; there is no invariant way to say how much Near ages relative to Far, because they are spatially separated. However, there is a (frame-dependent) sort of "differential aging" going on; here's how that works. We look at everything in terms of the signals Far sees arriving from Near.

When Far's clock reads 60, he receives the signal from Near time stamped 0. At this instant, he is hooked by the train, so the Doppler shift of Near's signals instantaneously changes: before being hooked, there was no shift; after being hooked, there is a Doppler redshift. The Doppler factor is 3, so the signals from Near now appear to Far slowed down by a factor of 3; i.e., for every 3 ticks of Far's clock, he receives signals from Near whose time stamps differ by 1 tick.

The Doppler redshift of Near's signals continues until Far receives the signal time stamped 60 from Near. Since the Doppler shift factor is 3, Far will receive this signal when his clock reads 240. At this point, the Doppler shift of Near's signals goes back to zero (because when Near's clock reads 60, he is hooked by the train). So where before, Near's signals were 60 ticks behind Far's clock (Far received Near's signal time stamped 0 at tick 60 of his clock), now they are 180 ticks behind Far's clock (Far receives Near's signal time stamped 60 at tick 240 of his clock).

How does Far interpret this? It depends on how he chooses to interpret it. For example, he could say that Near's light signals fell behind by 120 ticks during the period where they had a Doppler shift--they went from being 60 ticks behind Far's clock to being 180 ticks behind Far's clock. He could interpret this as Near "aging less" during this period.

But Near was moving away from Far during this period at 0.8c; that's shown by the Doppler redshift observed in his light signals. Moving away at 0.8c for 180 ticks of Far's clock means that, at the end of the Doppler shift period, Near was 144 units further away from Far. So purely based on the increase in distance, Far would expect Near's light signals to be 144 ticks further behind; yet they only fell behind by 120 ticks. That would seem to indicate that Near aged *more*, not less, than Far, during the Doppler shift period!

All this is really showing, of course, is that, as I said above, there is no invariant way to specify the "relative aging" of Near and Far in this scenario; it depends on the coordinate chart you use (and in the reasoning I gave above, I implicitly switched coordinate charts without saying so, which is how I came up with two apparently contradictory conclusions about how much Near aged). But you *can* specify what Far directly observes; I did it above. And those direct observations are invariant; you can calculate them using any coordinate chart you like and you will get the same answer. In fact, you don't even need to use a coordinate chart to calculate them; I didn't use one above. I only used the observed Doppler shift, which is easy to compute from the specified relative velocity of 0.8c. All of which supports, I think, ghwellsjr's position equating what you see with what is real.

 

[JVNY]

What does "according to Far" mean?

It is intentionally ambiguous so that we can then consider the two approaches: in one I try to be very conventional, and in the other I try to think like ghwellsjr (focusing on what Far sees).

. . . So to say "what happens according to Far", if that's going to mean anything more than what Far directly observes (for example, to assign a "time according to Far" to events spatially separated from Far, such as events on Near's worldline, which is what you have to do if you want to assign a "time dilation" to Near, relative to Far), you have to decide what kind of coordinate chart you are going to use to represent "what happens according to Far".

If, OTOH, you are willing to limit "what happens according to Far" to what he directly observes, then it's easy.

The first approach does mean to say more than what Far observes, so it does try to bring in something more, using the term reference frame rather than coordinate chart, though.

Observers don't "join" reference frames. A reference frame (in the sense you are using the term--a better term would be "coordinate chart") is an arbitrary convention.

I think that the conventional view is that observers do join (or change or switch) reference frames, so I put it that way in the first approach. People regularly use that language when explaining the twin paradox, for example:

“Prime observes those clocks from DIFFERENT frames of reference on the way out and on the way back” http://www.phys.vt.edu/~jhs/faq/twins.html

“At some point he turns around, thus switching reference frames again, and when he gets back home he now is back in reference frame of the Earth.” http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/971109a.html

“The acceleration causes the traveling twin to change from one constant velocity reference frame to another” http://www.csupomona.edu/~ajm/materials/twinparadox.html​

That said, I don't think that ghwellsjr would agree that this description is correct.

What signals Far receives, and in what order, is a direct observable and must be the same regardless of how you calculate it or what coordinate chart you use. . . He receives "forward ticking signals from Near" *all* the time, before he is hooked as well as after. However . . . there is no "gap". . . And those direct observations are invariant; you can calculate them using any coordinate chart you like and you will get the same answer. In fact, you don't even need to use a coordinate chart to calculate them; I didn't use one above. I only used the observed Doppler shift, which is easy to compute from the specified relative velocity of 0.8c. All of which supports, I think, ghwellsjr's position equating what you see with what is real.

Let me see whether I understand this. There are no missing time stamped signals. For example, Far does not receive signals from Near's watch showing -2, -1, 0, then 12, 13, 14. Rather, there is just a delay in the time that Far observes between receiving some of the signals (a gap in a different sense, a delay). Before accelerating, the signals arrive at a constant rate as measured by Far's proper watch -- the same rate that Far's watch ticks. But then, after Far's acceleration, there is a longer time gap between the receipt of some of the signals (as measured by Far's watch). But later yet, Far begins to receive the signals from Near at the same rate as Far's watch ticks (because they are both at rest with respect to each other at opposite ends of the train). Is that right?

The main thing I am trying to do here is to focus just on what Far sees, because that is what ghwellsjr argues is real. I think that using concepts like moving between frames and the distance between Near and Far length contracting at acceleration is all contrary to ghwellsjr's analysis. These things are not real in that analysis.

 

[PeterDonis]

I think that the conventional view is that observers do join (or change or switch) reference frames, so I put it that way in the first approach. People regularly use that language when explaining the twin paradox

You have to bear a couple of things in mind when reading these descriptions.

First, they are descriptions in English, not math; they are not describing how the answers are actually calculated, they are only trying to fit the answers that have already been calculated into some sort of intuitive framework. In other words, they are not describing the actual theory; they are describing a way of trying to interpret what the theory says that makes some sort of intuitive sense.

Second, the language is sloppy in the way I described in my last post. Reference frames are a convention; they are not given by the physics. A more careful description would make clear the distinction between switching reference frames, which is just switching the conventions you use to describe the physics, and switching states of motion, which is an actual physical change (you have to fire rockets or otherwise experience proper acceleration).

That said, I don't think that ghwellsjr would agree that this description is correct.

I wouldn't say it's incorrect, just sloppy. See above.

There are no missing time stamped signals.

Correct.

For example, Far does not receive signals from Near's watch showing -2, -1, 0, then 12, 13, 14.

Correct.

Rather, there is just a delay in the time that Far observes between receiving some of the signals (a gap in a different sense, a delay).

Sort of. If you think of Near as sending discrete signals, for example as sending a signal each time his clock ticks to a new integer value, then there is a gap between every pair of successive signals; what varies is the length of the gap as measured by Far's clock.

Before accelerating, the signals arrive at a constant rate as measured by Far's proper watch -- the same rate that Far's watch ticks.

Yes.

But then, after Far's acceleration, there is a longer time gap between the receipt of some of the signals (as measured by Far's watch).

Not some of them; all of them, until the Doppler shift goes back to zero. But this is a "gap" in the sense I gave above: each successive signal emitted by Near at a tick of his clock arrives 3 ticks later by Far's clock, instead of 1 tick later.

But later yet, Far begins to receive the signals from Near at the same rate as Far's watch ticks (because they are both at rest with respect to each other at opposite ends of the train).

Yes.

I think that using concepts like moving between frames and the distance between Near and Far length contracting at acceleration is all contrary to ghwellsjr's analysis. These things are not real in that analysis.

I would say they are conventions, because they are based on picking a coordinate chart, which is a convention. Whether that counts as "real" or not is, IMO, a question about language, not physics. The important contrast, IMO, is between conventions and invariants; invariants are things like what Doppler shift Far observes in Near's signals.

 

[ghwellsjr]

Here is another train and platform example, offered to examine ghwellsjr's view equating what you see with what is real.

Assumptions:

Train has proper length 100.
Platform has proper length 60.
Train and platform are in relative inertial motion at 0.8c.
There are two observers on the platform, Far (on the farther end of the platform from the train's perspective), and Near (on the nearer end of the platform from the train's perspective).
Near and Far have watches, synchronized in the platform frame.
Near's watch is an LED watch that flashes a signal with a time stamp toward Far as Near's watch records elapsed time.
Simultaneously in the platform frame, the front of the train is at the far end of the platform, and the rear of the train is at the near end of the platform. Hooks at the front and rear catch Far and Near, like mailbags on the old mail trains. In the platform frame, both watches read 60 at this time.

What happens according to Far?

First off, thanks PeterDonis for providing all the textual explanation so that I can focus on some diagrams.

Here's a spacetime diagram showing the Inertial Reference Frame (IRF) of the platform, actually the two observers with their watches that start off at either end of the platform. I'm defining the speed of light to be 1 foot per nanosecond. Far is in green and Near is in blue. The locomotive of the train is shown in black and the caboose is in red. After the hooks pick up the two observers with their watches, I don't bother to show the train or the platform as they would just clutter up the diagrams:

You can see that prior to the hookups, green Far sees the blue Near watch synchronized but reading 60 nsec earlier since they are separated by 60 feet. At the moment of hookup, green Far's watch is at 60 nsec and he is seeing blue Near's watch at 0 nsec and immediately starts to see the blue Near watch ticking at 1/3 of his own rate. This continues for 180 more nsecs or until the green Far watch reaches 240 nsecs and the near Blue watch appears to be at 60 nsecs which is when the blue Near observer appears to get hooked up. From this point on, the two watches tick at the same rate but there is a 180 nsec difference between them.

Now you're getting ready to do some frame jumping so let's first take a look at the rest frame of the train:

I want you to confirm that all the signals that were sent from blue Near to green Far are depicted exactly the same in this diagram as in the first one. This diagram was created simply by taking the coordinates of all the dots (events) in the first diagram and Lorentz Transforming them at a speed of 0.8c. Both frames contain identical information. If we have one, we can get to the other.

One theory: Far is hooked aboard the train and joins the train's reference frame when the far end of the platform and the front end of the train are aligned; Far's watch reads 60; but the platform is length contracted so that the near end of the platform is well ahead of the rear end of the train. For Far, the platform length contracts instantaneously toward him, so that Near contracts from being 60 away to being only 36 away. Near's watch runs backwards (the opposite of clock advancement that occurs in the usual acceleration examples, because the usual examples refer to planets or stars ahead of the accelerating object, not behind it as here).

Stella inhabits an accelerated frame, and clocks in such a frame can . . . run slower or even backwards! The details depend not only on their motions, but also on their positions relative to Stella. . . Light from events below the plane can never reach you at all, and this prompts the plane to be called a "horizon". In fact, although you cannot know what is happening below this plane, it turns out that you can infer time below it is going backwards.

See http://math.ucr.edu/home/baez/physics/Relativity/SR/movingClocks.html​

Then, Near's watch runs slowly (time dilation), so that by the time Near has aligned with the rear of the train and she is hooked aboard her watch is back up to 60. Near is younger than Far, as one would expect (Near's clock ran backwards, then ran more slowly than Far's).

This theory seems to imply that Near's watch will send three sets of inconsistent time stamp signals to Far. First, just as Far's watch reads 60, Far will only have received Near's watch's flash time stamped 0 (because Far is 60 away). Thus signals time stamped 1-60 are still on their way toward Far. Second, Near's watch runs backwards, so the time stamps of each successive backward tick of the watch is sent toward Far. Third, Near's watch runs forward again, sending new successively forward time stamped signals. One could argue that there is a Rindler horizon when Far accelerates, preventing him from seeing Near's watch signals (as suggested in the quotation above). However, once he stops accelerating the horizon disappears, so any signals that occurred behind the horizon would be able to catch up to him. See, for example:

Of course, this is not the same as a black hole’s event horizon in two very important respects. Firstly, it’s always possible to stop the spaceship accelerating, so this horizon’s persistence is a matter of choice, not physical law.

http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html​

A second theory: it is absurd to think that Far receives these multiple and conflicting sets of time stamped signals. Far receives the signal time stamped "0" just as he is hooked and joins the train, and thereafter he receives only forward ticking signals from Near. There is simply a gap in the set of time stamped signals that Far sees. This gap explains why Near ages less than Far (what you see is what you get). ghwellsjr, is this how you would analyze this example?

The example is 100% completely analyzed in the first diagram. There is no new information or knowledge or insight to be gained by looking at other diagrams. We do it just for fun. I have already transformed to a second IRF. But you want to see a non-inertial diagram for the green Far observer and his watch.

OK, my favorite way to do this is to have green Far send out radar signals which bounce off blue Near along with the time on blue Near's watch. I show a representative sampling of these signals using the first diagram:

Green Far collects all this data and then uses it to construct his non-inertial rest frame. He does this by dividing the difference between each radar sent and received times by two and assuming that the signals propagated in the same amount of time both ways (Einstein's second postulate) to derive a distance that light would travel in that amount of time. He takes the average of those two radar times and assumes that the distance applies at that time. He plots the observed times on blue Near's watch as a function of time and distance and gets this diagram:

Notice how all the same signals that appear in the first two diagrams and when they were sent by blue Near and received by green Far and propagated at c are identical in this diagram, in other words, no new information, no new insight, no new knowledge. It's just another arbitrary way to present the data.

Can you draw a non-inertial diagram of the type that you described as the first theory that also preserves all the actual data sent and received by both parties and maintains the speed of light at c? I challenge you to try it?

 

[bahamagreen]

Hoping Far and Near are Born ridgid about their t=60...

 

[ghwellsjr]

Hoping Far and Near are Born ridgid about their t=60...

We treat them as point particles and don't worry about how they survive acceleration.

But if you're going to worry about that kind of detail, you also have to worry about those hooks, and how they are fastened to the train. It can go on and on.

 

[JVNY]

Thanks George. I look forward to analyzing the diagrams.

With conceptual revisions courtesy of PeterDonis and you, I think that I understand the "what you see" approach, and it generally makes sense to me (leaving aside any disagreement over the use of the word "real"). Although it is interesting that the last diagram shows a maximum distance of 120 between Near and Far (rather than 100, the rest length of the train), presumably as a result of the radar convention.

For your question, no I don't think that it is possible to diagram the first theory to preserve the data -- particularly given that the first theory seems to require multiple inconsistent sets of signals. However, based on PeterDonis's comments I suspect that I may have created a straw man in theory one. I was trying to describe a typical reference frame-centric (or coordinate chart-centric) approach, but I may not have done a good job at it.

Did you ask the question to suggest that theory one is a poor summary of the typical approach?

Or did you ask the question to suggest that it is a reasonable summary of the typical approach, and to suggest that the typical approach is wrong because one cannot diagram the example properly using it?

 

[ghwellsjr]

Thanks George. I look forward to analyzing the diagrams.

With conceptual revisions courtesy of PeterDonis and you, I think that I understand the "what you see" approach, and it generally makes sense to me (leaving aside any disagreement over the use of the word "real"). Although it is interesting that the last diagram shows a maximum distance of 120 between Near and Far (rather than 100, the rest length of the train), presumably as a result of the radar convention.

For your question, no I don't think that it is possible to diagram the first theory to preserve the data -- particularly given that the first theory seems to require multiple inconsistent sets of signals. However, based on PeterDonis's comments I suspect that I may have created a straw man in theory one. I was trying to describe a typical reference frame-centric (or coordinate chart-centric) approach, but I may not have done a good job at it.

Did you ask the question to suggest that theory one is a poor summary of the typical approach?

Or did you ask the question to suggest that it is a reasonable summary of the typical approach, and to suggest that the typical approach is wrong because one cannot diagram the example properly using it?

It is a good summary of a very popular approach but I think if people would understand the radar method it might become more popular, especially since it also works for inertial observers. I never saw the need to create a non-inertial frame in the first place, since it adds nothing. But when dealing with someone a long time ago, I discovered the radar method before I knew that it had a name and if we have to go non-inertial it seems like a much more satisfying way to go.

 

[PeterDonis]

if we have to go non-inertial it seems like a much more satisfying way to go.

I believe the Dolby & Gull paper that goes into detail about the radar method (can't find the link to it right now, it's been linked to in previous threads on similar topics) gives a proof that this method is the only method that can cover all of spacetime (more precisely, all of spacetime that is causally connected to the worldline of the observer who is "at rest" in the non-inertial frame) and assign a unique time to each event; any other method must assign multiple times to some events (because multiple "surfaces of simultaneity" cross at those events).

 

[ghwellsjr]

I believe the Dolby & Gull paper that goes into detail about the radar method (can't find the link to it right now, it's been linked to in previous threads on similar topics) gives a proof that this method is the only method that can cover all of spacetime (more precisely, all of spacetime that is causally connected to the worldline of the observer who is "at rest" in the non-inertial frame) and assign a unique time to each event; any other method must assign multiple times to some events (because multiple "surfaces of simultaneity" cross at those events).

Here's a reference to it by DaleSpam:

There is no single standard reference frame for non inertial observers. You are going to have to specify exactly how you plan to construct the non inertial reference frame.

My favorite method is the Dolby and Gull approach:
http://arxiv.org/abs/gr-qc/0104077

 

[pervect]

My $.02

It seems to me that one or more of $\Gamma^x{}_{tt}, \Gamma^y{}_{tt}, \Gamma^z{}_{tt}$ must be infinite at the point of instantaneous turnaround, because the proper acceleration there is infinite and the $\Gamma^{*}{}_{tt}$ can be shown to be equal to the proper acceleration along the worldline.

Therefore I don't t think there is _any_ set of coordinates involving an instantaneous turnaround that will be suitable to perform general relativity with , as one need twice differentiability of the metric to construct important elements of the theory like the Riemann.

So if one wants to use general relativity, coordinate systems based on infinitely fast turnarounds are in fact problematic. This doesn't strike me as an unreasonable or overly burdensome restriction on coordinate choices, infinite proper acclelerations are not really very "physical". One has a great deal of freedom of choice in determining coordinates, but it's important to have a manifold that is at least twice differentiable to fully apply the theory.

It would be interesting to see the resultant metric that the Dolby and Gull coordinate choice gives, and exactly how bad the singularities are - I suspect that every time in the diagram that the lines of simultaneity they draw in the paper make a sharp bend, the metric isn't twice differentiable, but I could be mistaken

 

[JVNY]

I hope that members who subscribe to the typical frame-centric method will weigh in. Here are some thoughts about how to apply that method.

The basic theory is that when Far hooks aboard the train he should stop analyzing his own radar signals and pay attention only to radar signals that have been sent by an observer on the train (because he is now on the train).

Assume that there is a rider on the train's front hook (Front). Front has been sending out radar signals to Near and receiving them, thus observing Near's distance. When Far hooks aboard the train, he finds himself seated with Front. Far is now in the train frame (using frame-centric language), with his watch reading 60. Front's watch reads some arbitrary time, say 100. Front and Far notice that their watches tick at the same rate.

They chat after receiving some of their returning radar signals and ask each other how far away Near was simultaneously with Far hooking aboard. Because they are at the same place at that time, they should agree on simultaneity then (at Far time 60, Front time 100). Front says Near was 36 away. Far says she was 60 away. That is odd, because each is in the same place at the same time at rest with respect to each other, describing how far away the same person was at that time. Yet they have two different answers. One must conclude that distance to a single object can differ for two observers at rest with respect to each other at the same place at the same time. We discussed a slightly different version of this at further length in the thread on synchronizing clocks on a rotating platform (WannabeNewton noting that simultaneity might depend on the observer's entire segment, not just a single point). Is this conclusion defensible?

Say that Far and Front then compare their series of radar rangings. They find that Front has a very smooth series of rangings showing Near with a straight world line in inertial motion in a single direction at 0.8c until she accelerates when she hooks aboard. This is consistent with the fact that Front and Near were in inertial movement as just described. The radar (and coordinate) distance between them grows smoothly from 0 (when Front is aligned with Near) to 100 (when Near hooks aboard). Far will show a kinked world line for Near in the final diagram in post 5, which includes the conclusions that (a) Near was as far away as 120 at one point, and (b) that there were three changes of direction by one or the other or both of Far and Near (considering the final diagram in terms of Near relative to Far, one change of direction to the left at Near time 0, then to the right at Near time 60, then to the left again at Near time 80).

But there is no reference frame in which Near is 120 away from Far at the time shown in the final chart. Moreover, a change of direction is something one actually feels (proper acceleration). Far and Near can radio each other, and each will advise that they only accelerated once, for a total of two changes of direction (not three). The final diagram is not just an arbitrary way to present data -- it generates a world line for Near that is inconsistent with the total number of actual (proper, or felt) accelerations.

So the frame-centric theorist says that the final diagram is the wrong way to describe the example. Instead, after changing frames you should use the radar signals of the observer in the frame you have joined, and everything will work out fine. When Far accelerated to 0.8c relative to Near, length contraction occurred. Near contracted from 60 away to 36 away, and thereafter Near traveled at 0.8c for a distance of 64 and then hooked on board. Near only accelerated once; she did not accelerate when moving from 60 apart to 36 apart -- that change in distance occurred because of length contraction, not any acceleration by her.

 

[PeterDonis]

every time in the diagram that the lines of simultaneity they draw in the paper make a sharp bend, the metric isn't twice differentiable

Shouldn't this just be "differentiable"? The connection coefficients are first derivatives of the metric.

Also, just on basic calculus grounds, a "sharp bend" in a curve means its derivative is not well-defined at the point of the bend, so the proper acceleration would not be well-defined, even if you didn't pick a coordinate chart (and hence didn't define connection coefficients); just trying to evaluate $d / d \tau$ at the bend point would be enough.

I'm not sure this entirely invalidates the Dolby & Gull method, because that method only requires round-trip light signals, which don't depend on any curves being differentiable, just continuous. But physically, of course, just continuity is not enough; the proper acceleration needs to be well-defined and finite. So each sharp corner really should be "rounded off" in a physically realistic model.

 

[PeterDonis]

I hope that members who subscribe to the typical frame-centric method will weigh in.

I don't really "subscribe to" the "frame-centric method", because it focuses on things that are frame-dependent instead of things that are invariant, which means it causes more confusion than it solves. But I do have a few comments.

One must conclude that distance to a single object can differ for two observers at rest with respect to each other at the same place at the same time.

No, one must conclude that if you use two different frames, you will get two different answers for the distance to a single object. Front's answer is obtained using Front's frame (the train frame); Far's answer is obtained using Far's original frame (in which he was at rest before he hooked onto the train). In other words, "distance" is frame-dependent. This is no mystery, but it is, as I noted above, a reason to be wary of the "frame-centric" method, since it can easily cause confusion if you try to attribute "reality" to frame-dependent things like distance.

But there is no reference frame in which Near is 120 away from Far at the time shown in the final chart.

Sure, there is; ghwellsjr explicitly showed it. It's just not one of the frames you considered, because you restricted yourself to inertial frames. That's why Far had to "change frames" when he changed his state of motion. But this restriction to inertial frames is arbitrary; the general notion of "frame" does not require a frame to be inertial. Non-inertial frames are often useful in physics; in fact, the "frame" you are in right now, at rest on the surface of the Earth, is a non-inertial frame, but you use it every day.

Moreover, a change of direction is something one actually feels (proper acceleration).

No, this is not correct; proper acceleration does not have to correspond to a "change of direction". It does in an inertial frame, but not necessarily in a non-inertial frame. You, at rest on the surface of the Earth, are experiencing proper acceleration all the time; are you always changing direction?

When Far accelerated to 0.8c relative to Near, length contraction occurred. Near contracted from 60 away to 36 away, and thereafter Near traveled at 0.8c for a distance of 64 and then hooked on board. Near only accelerated once; she did not accelerate when moving from 60 apart to 36 apart -- that change in distance occurred because of length contraction, not any acceleration by her.

While this is indeed the inertial "frame-centric" description of what occurred, notice that it requires Near to move instantaneously from 60 apart to 36 apart, without feeling any acceleration, because of "length contraction". In other words, this viewpoint requires "length contraction" to magically teleport objects instantaneously from one place to another. This is inconsistent on its face with causality.

 

[ghwellsjr]

I hope that members who subscribe to the typical frame-centric method will weigh in. Here are some thoughts about how to apply that method.

I'd like to weigh in even though I don't like the typical frame-centric method.

The basic theory is that when Far hooks aboard the train he should stop analyzing his own radar signals and pay attention only to radar signals that have been sent by an observer on the train (because he is now on the train).

If green Far is going to stop analyzing his own radar signals then he has to ignore everything after the one he sent at his time of -60 nsecs and as can be seen by this diagram from post #5:

According to that measurement, blue Near was 60 feet away and the Proper Time on both their watches was 0 nsec.

Assume that there is a rider on the train's front hook (Front). Front has been sending out radar signals to Near and receiving them, thus observing Near's distance. When Far hooks aboard the train, he finds himself seated with Front. Far is now in the train frame (using frame-centric language), with his watch reading 60. Front's watch reads some arbitrary time, say 100. Front and Far notice that their watches tick at the same rate.

They chat after receiving some of their returning radar signals and ask each other how far away Near was simultaneously with Far hooking aboard. Because they are at the same place at that time, they should agree on simultaneity then (at Far time 60, Front time 100). Front says Near was 36 away. Far says she was 60 away.

You're comparing apples with oranges. As I said before, the last measurement that green Far could have made was 60 nsecs earlier when blue Near was 60 feet away. It would be presumptuous of green Far to assume that blue Near was still 60 feet away 60 nsec later. How does he know that? Maybe blue Near got hooked by another train going the other way.

So if you're going to compare apples with apples, then you would have to use the last signal that Front has available to him at the time that green Far joined him as can be seen in Front's rest frame:

And that would be 20 feet away when the Proper Time on blue Near's watch was 0. It's true that if Front continues to monitor radar signals, then later on (after their chat) he can determine that blue Near was 36 feet away at the assigned time corresponding to when green Far joined him but you can't compare that to any measurement that green Far made because you have disallowed all his radar measurements that overlap his acceleration.

And if you're going to assert that green Near should switch to Front's radar measurements when he becomes colocated with Front, why shouldn't he have also used another observer's radar measurements that he was colocated with on the platform but now remains on the platform? He could eventually get that information from him.

That is odd, because each is in the same place at the same time at rest with respect to each other, describing how far away the same person was at that time. Yet they have two different answers. One must conclude that distance to a single object can differ for two observers at rest with respect to each other at the same place at the same time. We discussed a slightly different version of this at further length in the thread on synchronizing clocks on a rotating platform (WannabeNewton noting that simultaneity might depend on the observer's entire segment, not just a single point). Is this conclusion defensible?

As I have pointed out, there aren't two different answers, you have discredited one of them. All of blue Far's radar signals that overlapped his acceleration would include everything from the Coordinate Time of 0 to 160 nsecs on this diagram:

Everyone agrees with the blue Near's worldline prior to the watch's Proper Time of 0 nsec and after 80 nsecs but the question is how to connect those two points.

Say that Far and Front then compare their series of radar rangings. They find that Front has a very smooth series of rangings showing Near with a straight world line in inertial motion in a single direction at 0.8c until she accelerates when she hooks aboard. This is consistent with the fact that Front and Near were in inertial movement as just described. The radar (and coordinate) distance between them grows smoothly from 0 (when Front is aligned with Near) to 100 (when Near hooks aboard). Far will show a kinked world line for Near in the final diagram in post 5, which includes the conclusions that (a) Near was as far away as 120 at one point, and (b) that there were three changes of direction by one or the other or both of Far and Near (considering the final diagram in terms of Near relative to Far, one change of direction to the left at Near time 0, then to the right at Near time 60, then to the left again at Near time 80).

But there is no reference frame in which Near is 120 away from Far at the time shown in the final chart. Moreover, a change of direction is something one actually feels (proper acceleration). Far and Near can radio each other, and each will advise that they only accelerated once, for a total of two changes of direction (not three). The final diagram is not just an arbitrary way to present data -- it generates a world line for Near that is inconsistent with the total number of actual (proper, or felt) accelerations.

So the frame-centric theorist says that the final diagram is the wrong way to describe the example. Instead, after changing frames you should use the radar signals of the observer in the frame you have joined, and everything will work out fine. When Far accelerated to 0.8c relative to Near, length contraction occurred. Near contracted from 60 away to 36 away, and thereafter Near traveled at 0.8c for a distance of 64 and then hooked on board. Near only accelerated once; she did not accelerate when moving from 60 apart to 36 apart -- that change in distance occurred because of length contraction, not any acceleration by her.

Why don't you download my last diagram and draw in your method for connecting them? It would be a lot easier to visualize than a textual description.

 

[JVNY]

Thanks, PeterDonis. One question, then a few responses.

The question is: how can the final diagram accurately represent the world lines of Far and Near when it shows three changes of direction rather than the two that actually occurred?

You, at rest on the surface of the Earth, are experiencing proper acceleration all the time; are you always changing direction?

This is an SR problem, so gravity does not exist in it and one cannot refer to any feeling owing to gravity. The diagram shows three changes of direction; in SR, that must mean three felt instances of acceleration. But there were only two.

one must conclude that if you use two different frames, you will get two different answers for the distance to a single object. Front's answer is obtained using Front's frame (the train frame); Far's answer is obtained using Far's original frame (in which he was at rest before he hooked onto the train). In other words, "distance" is frame-dependent.

Far did not obtain his answer using his original frame. He obtained his answer by using a signal emitted in his original frame but received after accelerating and joining Front's frame. The radar method involves two frames for him. Only Front's measurement is based on a signal emitted and received in a single frame.

To focus on the simultaneity issue, make one more change to the example so that we can determine simultaneity for Far entirely in a single frame. Say that Far hooks onto the train 100 ahead of the train rear, but not at the front. The train has proper length 136, so Far hooks on 36 behind the front. Now, immediately upon his hooking onto the train, lightning bolts strike simultaneously in the train frame at the front of the train (36 ahead of Far) and at the near end of the platform (36 behind Far). The lightning flashes arrive simultaneously for Far and his hook companion. They can later measure the locations of the burn marks and determine that one bolt struck the train and Near 36 behind them; the other bolt struck the train 36 ahead of them; and both bolts struck simultaneously in the train frame at train time 100. Far and his hook companion are at rest with respect to each other at the same point at the same time and agree on the simultaneity of distant events and the distance between them, using the very standard Einstein lightning method.

That is because Far has joined the train frame, is in inertial motion just like his hook companion, and bases all determinations on events that occur while he is in that frame. The reason they disagreed before is that the radar method used an event that occurred while Far was in one frame (his emitting a signal when he was in the platform frame) and another event that occurred while he was in a different frame (his receiving the reflected signal when he was in the train frame).

While this is indeed the inertial "frame-centric" description of what occurred, notice that it requires Near to move instantaneously from 60 apart to 36 apart, without feeling any acceleration, because of "length contraction". In other words, this viewpoint requires "length contraction" to magically teleport objects instantaneously from one place to another. This is inconsistent on its face with causality.

Correct, but that is because the example uses instantaneous acceleration, which is probably not physically possible. If the acceleration occurs over a very small but positive time, then there is no teleportation. Near's relative distance changes at greater than the speed of light, but the literature seems to accept that as not contradicting SR (I think, although I am not sure, because no information can be conveyed faster than the speed of light even if length contraction occurs).

 

[pervect]

Shouldn't this just be "differentiable"? The connection coefficients are first derivatives of the metric.

In general you really need the Riemann, which means being able to differentiate the Christoffel symbols.

I'm not sure this entirely invalidates the Dolby & Gull method, because that method only requires round-trip light signals, which don't depend on any curves being differentiable, just continuous. But physically, of course, just continuity is not enough; the proper acceleration needs to be well-defined and finite. So each sharp corner really should be "rounded off" in a physically realistic model.

I wouldn't say the method is "invalid" at all. And I don't think there is any method that does better for the case of infinite acceleration.

But I think that the case of infinite proper acceleration is rather pathological, I believe some posters have convinced themselves that it's not so bad, and I wanted to post a bit on the downside.

The acceleration related issues don't arise until one tries to do more sophisticated things, like calculate geodesics (requires first order derivatives / Christoffel symbols) or the Riemann tensor (requires second order). These calculations may be more sophisticated, but I would say that an attempt at a frame of reference and/or coordinate system where you can't calculate force-free motion has some issues.

Another possible and unrelated limitation of the Doby & Gull method that I can think of is the issue of multiple radar returns. It's possible to have multiple images of a target, for example by gravitational lensing. I'm not sure how that's handled by the method - one could maybe take the first radar return always, but can one in that situation still have a 1:1 mapping between coordinates and events?

 

[PeterDonis]

how can the final diagram accurately represent the world lines of Far and Near when it shows three changes of direction rather than the two that actually occurred?

You continue to miss the fundamental point: "changes of direction" are frame-dependent. There is no invariant meaning to "changes of direction". It depends on the coordinate chart you adopt. So your use of the term "actually occurred" is not correct here.

This is an SR problem

Non-inertial frames are perfectly valid in SR, and proper acceleration does not correspond to "changes of direction" in a non-inertial frame. If you don't like my example with gravity, consider this one: you are on the inside surface of a space station that is rotating rapidly enough so that you can "stand" on the inside surface just like you would stand on the ground on Earth; i.e., you feel a 1 g acceleration. There is no gravity anywhere: spacetime is flat. But you are feeling a constant proper acceleration, yet you can describe physics perfectly well using a non-inertial frame in which you are always at rest and therefore do not "change direction".

Far did not obtain his answer using his original frame. He obtained his answer by using a signal emitted in his original frame but received after accelerating and joining Front's frame.

No, he obtained his answer by assigning coordinates to events. Just receiving light signals is not enough to assign coordinates to events, *unless* you are using the radar method that ghwellsjr is using. But you were explictly *not* using that method, so you can't rely on light signals to assign coordinates.

The radar method involves two frames for him.

No, the radar method involves a single, non-inertial frame in which Far is always at rest. If you're "changing frames", you're not using the radar method. And in any case, I thought the whole point was to *not* use the radar method, but instead to use the "frame-centric" method.

The reason they disagreed before is that the radar method used an event that occurred while Far was in one frame (his emitting a signal when he was in the platform frame) and another event that occurred while he was in a different frame (his receiving the reflected signal when he was in the train frame).

This is not correct; see my comments above about the radar method.

Correct, but that is because the example uses instantaneous acceleration, which is probably not physically possible. If the acceleration occurs over a very small but positive time, then there is no teleportation.

No, just motion faster than the speed of light, as you admit in your very next sentence:

Near's relative distance changes at greater than the speed of light

Which, if you insist on treating "distance" as something physically real, is a problem.

the literature seems to accept that as not contradicting SR

The literature does not treat "distance" as something physically real; it's a frame-dependent thing, and if you switch frames rapidly enough, the frame-dependent distance can change faster than the speed of light. That's OK because the "distance" is not physically real; it's just a coordinate.

Just to be clear, I'm not saying the "frame-centric" method is invalid; I'm just saying it has limitations, and you have to understand the limitations. The reason I pointed out that Near's relative distance changes faster than light is that you had earlier pointed out that "changes of direction" in the radar method (ghwellsjr's final diagram) don't match up with instances of proper acceleration. If that's a valid objection to the "radar method", then my objection about faster-than-light changes of distance is an equally valid objection to the "frame-centric" method. Conversely, if we accept that "distance" can change faster than light in the frame-centric method because "distance" isn't physically real, we have to also accept that "changes of direction" can fail to match up with proper acceleration in the radar method, because "changes of direction" aren't physically real.

The bottom line is that there is *no* way to set up a "frame", coordinate chart, or any other way of describing physics in relativity that satisfies *all* of our intuitions. It just isn't possible, because reality doesn't satisfy all of our intuitions. So any method you use is going to include things that seem "wrong" based on our intuitions. It's just a question of which intuitions you want to try to preserve, and which ones you're OK with throwing away. Different choices will lead to different methods of describing physics. I prefer the "radar method" here because it focuses on invariants, not frame-dependent quantities, and invariants generalize much better to more complicated cases. But that doesn't mean my preferred method satisfies all of our intuitions; it doesn't. No method can.

 

[JVNY]

Lots of great comments from all. I disagree with many of them (for example, you are changing direction when rotating, and you can determine this without question using a Foucault pendulum), but rather than covering the comments I have allotted my time to trying to diagram the frame-centric view of Front, as ghwellsjr suggested.

Start with the first diagram, which is the platform frame without any accelerations. Assumptions are as before, except that the arbitrary time on train Front's watch is 60 when Far hooks onto the train (and when Far's watch also reads 60). Light flashes from Near appear pretty closely spaced together. Then view the second diagram, which is the train frame without any accelerations. The light flashes from Near appear more spread apart. Finally, consider the third diagram, which is what happens according to Far, although not including all of the light flashes for reasons to be discussed. I don't know whether you can call this Far's "reference frame," because I don't know whether you can have a one observer reference frame (or instead must distinguish observers from frames).

Green Far and blue Near are vertical lines, 60 apart, until platform time 60. Then, Far hooks onto black Front of the train. This assumes instantaneous acceleration. Blue Near distance contracts from 60 to 36 away from Far. Near's proper time as Near presents to Far retrogresses from 60 to 12. Red Rear of train distance expands from 60 to 100 away from Far. Rear's proper time as Rear presents to Far retrogresses from 140 to 60.

There is a clear discontinuity in the Near and Rear worldlines. There is also a discontinuity in Rear and Near's clocks. In the discontinuous case you don't have to worry about clocks running backwards; Near's clock, for example, simply switches from 60 to 12. You can make the worldlines continuous by using acceleration over a short period of time. But that might create retrogressing time stamped flashes from Near's clock, which we have discussed is not sensible (Far does not receive any backward signals in the inertial platform diagram). There would still have to be a discontinuous clock retrogression.

Thereafter, Far, Front and Rear have vertical worldlines and synchronized clocks. Near is in inertial motion toward the left relative to them, and her clock is running slow to them. At train and Far time 140, Near hooks aboard the train and has a vertical worldline.

Now, how do we draw in the light flashes? Clearly the flash from Near at her time 0 struck Far just as Far hooked aboard, so I show it in this diagram. From a frame-centric view, the Near flash time stamped Tnear=12 is simultaneous (along the dashed line) with Far being on the train and in the train frame, so I can confidently draw in that flash.

What about the flashes stamped 1-11? From this drawing, they occurred before Far hooked aboard. My earlier suggestion was that they might disappear; there might be a gap in flashes between 0 and 12; my thought was that they occurred before Far joined, so maybe they don't exist for Far. But PeterDonis pointed out that they strike Far in the inertial platform frame, so they must strike him in his own frame. The conclusion: if an event occurred in the frame before Far joined the frame, then the event is in the frame for him when he arrives.

So I should draw these lines in. However, I don't know how. Do they stay continuous the whole way starting from Near's vertical blue world line at x=0? Or are they discontinuous, like the Near and Rear worldlines, disappearing from below the dashed line of simultaneity at a certain point and then reappearing above the dashed line?

If instead the acceleration takes a period of time, are the lines of light flashes 1-11 continuous? But then would they be curved?

Next, what about the flashes 13-60? Under the frame-centric view, they occurred while Far was in the platform frame. They existed; they were "real"; they were on their way toward him. When Far joins the train frame, however, these flashes occur above the dashed line of simultaneity; they occur in his future. So one alternative I suggested before is that Far will receive duplicate sets of these time stamped flashes. In this alternative, the flashes occurred for Far thus he must receive them; they must exist in his future already, and thus be drawn in above him along Near's worldline. But Near's clock is only at 12 simultaneously with him in his new frame, so Near's clock will tick 13 to 60 again and send a duplicate set of signals time stamped 13-60. I would have to draw two sets of light arrows 13-60. This is unappealing.

Alternatively, Far might conclude that the flashes stamped 13-60 "disappear" when he hooks onto the train. They existed and were real for him before stepping onto the train. But they have not happened yet in the train frame. By changing frames he causes something that did exist for him into something that has not yet happened for him. This makes sense under the relativity of simultaneity. The order of events in one frame can differ from the order of the same events in another frame. However, it is unappealing (even if only intuitively) for a frame-centric view, because that view implies "real" existence and location of distant events, and it is unappealing to have "real" events exist and then not exist for the same person (and then exist again if he jumps off the train straightaway). Perhaps this is a problem. Or perhaps it is not a problem for the frame-centric view at all, but merely shows a failure to fully internalize the relativity of simultaneity.

Perhaps, however, there is a compromise view. Only the flashes that Far receives show what is real. But the flashes up to Tnear=0 come from one (platform) frame, and those from Tnear=1 onward come from the second (train) frame. Far might be able to use only one way signals to determine distance of events. He may not need to use two way radar signals. For example, if lightning strikes at his time 60 just as he is aboard the train, and the lightning strikes are at 36 behind him and 36 ahead of him, he can determine that Near was 36 behind him when the bolts struck, using only the one way signals from the lighting strikes.

 

[PeterDonis]

you are changing direction when rotating, and you can determine this without question using a Foucault pendulum

And if you use a Foucault pendulum or some similar apparatus (something which detects rotation) as your definition of "changing direction", then you are *not* changing direction when you accelerate in a straight line. But your earlier comments indicated that you do regard that as "changing direction", so your own definitions are not mutually consistent. The root problem, as I've said before, is that you are trying to attribute an absolute reality to things that are frame-dependent. That won't work.

I don't know whether you can call this Far's "reference frame," because I don't know whether you can have a one observer reference frame (or instead must distinguish observers from frames).

If we're going to go into that level of detail about terminology, then we might as well go all the way. Here is what I understand to be the correct standard technical terminology (which, unfortunately, is often not used correctly even in some technical literature).

* A "reference frame", strictly speaking, is a set of four orthonormal basis vectors at a single event. (An "event" is a single point in spacetime; events are defined by what happens at them, for example, "Far hooks onto the train" is an event.) One of these vectors is timelike; it defines what state of motion is momentarily at rest, according to the reference frame, at the given event. The other three vectors are spacelike, and define a set of spatial axes at the given event that are momentarily at rest in the reference frame.

* An "observer" is a worldline in spacetime, i.e., a curve which is everywhere timelike. At any event on this worldline, there is a 4-velocity, which is a unit timelike vector that is tangent to the worldline. We can use the 4-velocity vector at an event on the worldline to define a reference frame at that event; we just need to pick a set of 3 orthonormal spacelike vectors that are momentarily at rest with respect to the observer's 4-velocity at the chosen event. Such a frame is often called the "rest frame" (or more precisely, the "momentarily comoving frame") of the given observer at the given event.

* A "coordinate chart" is a one-to-one mapping of 4-tuples of numbers to events in some region of spacetime. Often when the term "reference frame" is used, what is really meant is "coordinate chart". The diagrams ghwellsjr has been posting are diagrams of a region of spacetime using different coordinate charts. You also appear to be using the term "reference frame" to mean "coordinate chart", at least implicitly; in what follows I'll assume that "coordinate chart" is what you're really talking about.

The important thing to note is that many aspects of the above are arbitrary; they are conventions that can be chosen in various ways, not "real" physical things. Observers are not conventional, since worldlines are invariant geometric objects. Reference frames and coordinate charts can be chosen arbitrarily; often there are some features of the physics that make particular choices useful (for example, we may want to choose a particular observer's rest frame at a particular event, and there may be aspects of the physics that pick out particular spatial directions and therefore make it useful to choose the spatial vectors of a reference frame in particular ways), but we can always describe the physics using any reference frame or coordinate chart we like--as long as the frame or chart is valid. See below.

Near's proper time as Near presents to Far retrogresses from 60 to 12.

No, this is not correct. Proper time is monotonic; it can't "retrogress", because it's the actual reading on Near's clock, which always increases. What you mean is that the time coordinate assigned to the same event on Near's worldline is 60 in one coordinate chart, and 12 in a different coordinate chart. Note that they must be *different* charts, because a single coordinate chart must be a one-to-one mapping of coordinate values to event; a chart that assigns two different values of the time coordinate to the same event is not a valid chart.

Red Rear of train distance expands from 60 to 100 away from Far.

This is also misstated: the correct statement is that the distance between two particular events is 60 in one chart and 100 in a different chart. Once again, a single chart can't do this: it can't assign two different space coordinates to the same event.

Rear's proper time as Rear presents to Far retrogresses from 140 to 60.

Same comment as above.

There is also a discontinuity in Rear and Near's clocks.

No, there isn't. The physical clocks advance normally and continuously. What you mean is that there is a discontinuity in the *description* of Rear and Near's clocks, because you are switching coordinate charts. See above.

Note also that the same comment applies if you "smooth out" the sharp bends in the worldlines by using finite acceleration for a short period of time; you still need to switch coordinate charts if you want to talk about two different time or space coordinates being assigned to the same event.

Now, how do we draw in the light flashes?

I won't comment on most of this since I think you have already taken a wrong turning in the above, and I would recommend re-thinking your entire approach based on my comments above. However, I do want to comment on a couple of things:

if an event occurred in the frame before Far joined the frame, then the event is in the frame for him when he arrives.

This makes no sense. Events are invariant parts of spacetime; they're either there or they're not. If they're there, they're there for any coordinate chart that covers the region of spacetime containing them. You can't make an event appear or disappear by changing charts, unless you perversely define a chart to arbitrarily exclude a region of spacetime--but then it will be easy to show that the chart has "edges" that don't correspond to anything physical.

Do they stay continuous the whole way

Worldlines are always continuous, whether they are worldlines of observers (timelike curves) or of light rays (null curves). Any valid coordinate chart will represent worldlines as continuous curves; if you find yourself wondering whether a worldline is continuous or not, it means you're not using a valid chart (or you're trying to switch charts in midstream, so to speak).

By changing frames he causes something that did exist for him into something that has not yet happened for him.

Here you are using "frame" in a different sense: basically it means a coordinate chart, *plus* an additional premise that events having a time coordinate less than or equal to a certain value in that chart are "real", while events having a greater time coordinate are "not yet real". Changing charts then amounts to changing what is "real", which, as you note, is not appealing. However, the problem is easily avoided by not adopting the additional premise about what is "real", which is superfluous; you don't need it to make any physical predictions.

the flashes up to Tnear=0 come from one (platform) frame, and those from Tnear=1 onward come from the second (train) frame.

No, this is not correct. Events and worldlines don't "come from" a coordinate chart; they are invariant geometric objects in spacetime. Coordinate charts are just *descriptions* of regions of spacetime using numbers.

if lightning strikes at his time 60 just as he is aboard the train, and the lightning strikes are at 36 behind him and 36 ahead of him, he can determine that Near was 36 behind him when the bolts struck, using only the one way signals from the lighting strikes.

How does he know how far away the lightning strikes are?

 

[JVNY]

And if you use a Foucault pendulum or some similar apparatus (something which detects rotation) as your definition of "changing direction", then you are *not* changing direction when you accelerate in a straight line. But your earlier comments indicated that you do regard that as "changing direction", so your own definitions are not mutually consistent. The root problem, as I've said before, is that you are trying to attribute an absolute reality to things that are frame-dependent. That won't work.

Both are changing directions, just differently. One is a straight line change in a single spatial direction, the other a series of changes in two spatial dimensions. So they are not inconsistent, just different. In SR there is an identity between changing direction and felt acceleration, however you characterize it. As to the reality of acceleration and the resulting length contraction, consider Rindler on the twin paradox and the resulting contraction of the distant point toward the accelerating twin:

B was accelerated out of his rest frame at P, at Q, and once again at P. These accelerations are recorded on B's accelerometer and he can therefore be under no illusion that it was he who remained at rest . . . [H]e has transferred himself to a frame in which the distance between P and Q is halved (length contraction), and this halving is real to him in every way.

Wolfgang Rindler, Introduction to Special Relativity (2nd ed. 1991), pp. 30-31.​

The acceleration is absolutely real, and recorded, and distinguishes the traveler from the stay at home twin without any illusion.

If we're going to go into that level of detail about terminology, then we might as well go all the way. Here is what I understand to be the correct standard technical terminology (which, unfortunately, is often not used correctly even in some technical literature). . .

I think that in many cases I am using reference frame, but I will carry on and you can decide.

No, this is not correct. Proper time is monotonic; it can't "retrogress", because it's the actual reading on Near's clock, which always increases. What you mean is that the time coordinate assigned to the same event on Near's worldline is 60 in one coordinate chart, and 12 in a different coordinate chart. Note that they must be *different* charts, because a single coordinate chart must be a one-to-one mapping of coordinate values to event; a chart that assigns two different values of the time coordinate to the same event is not a valid chart.

No, this is not what I mean. I should emphasize and explain an important part of the statement: "Near's proper time as Near presents to Far retrogresses from 60 to 12." In other words, Near's watch reads 60 (or "presents" as 60) in the platform frame when Far's clock reads (or "presents") 60 in the platform frame. Per Taylor and Wheeler in Spacetime Physics, assume a lattice of recording clocks that stretches along the x-axis to include Near and Far, at rest with respect to them and with clocks synchronized to their watches. The lattice records that Near's watch presents in the frame reading 60 at the same time that Far's watch does.

Literally assume an LED watch that is so close to the nearest recording clock that one can ignore signal delay between the LEDs shining a given time and the recording clock recording that time. So the LED watch shines "60" and the recording clock in the given frame records the image "60." That watch presents at 60 in the frame. It is not related to any coordinates. It is the time on Near's watch in the frame.

Then, Far hooks onto the train. The train also has a lattice of recording clocks, at rest with respect to the train with its clocks synchronized to the watches of Front and Rear. When Far hooks onto the train, the lattice of recording clocks records the following: Far's watch presents at 60, Front's watch presents at 60, Rear's watch presents at 60, and Near's watch presents at 12. So I am not creating any chart with a time coordinate of 12. The only chart you might be able to identify is the chart covering the train and Far, in which the time is 60. Near's watch, however, presents at 12. The lattice of recording clocks in the train frame records that Near's watch LEDs shine 12, and that is what I am reporting. There is no chart with any 12 on it.

This is also misstated: the correct statement is that the distance between two particular events is 60 in one chart and 100 in a different chart. Once again, a single chart can't do this: it can't assign two different space coordinates to the same event.

I am not sure whether this is a disagreement over semantics or not. Far is in one reference frame or chart (at rest with respect to the platform), where the distance is 60, then he is in another reference frame or chart (at rest with respect to the train), where the distance is 100. Call them two charts or two reference frames as you would like. I suspect that this is just another disagreement over what one calls "real." Following Rindler's quotation, first the distance is 60, then it is 100; both distances are real in every way in the frame-centric theory. Whether you agree with that theory or not is a separate issue -- I am just asserting that this is the correct way to describe the frame-shifting view.

The physical clocks advance normally and continuously. What you mean is that there is a discontinuity in the *description* of Rear and Near's clocks, because you are switching coordinate charts. See above.

Again, I am not here creating any coordinate chart. I am stating how the watches present, which is the same as how they would be recorded by a lattice of recording clocks that is at rest with respect to Far. A lattice at rest with respect to Far records Near's watch to read 60 at one moment (just before Far hooks aboard the train). A lattice at rest with respect to Far records Near's watch to read 12 in a subsequent moment. These are statements of recordings, not descriptions. I have not drawn any chart with a time 12 in it.

What the lattice or lattices record in between 60 and 12 is questionable. Remember that I question whether this frame-shifting approach works well. ghwellsjr suggested that my summary is an accurate enough description of the conventional theory, and that applying the theory to the example discredits the theory. So don't push against an open door. If I am doing a fair job of describing the conventional frame-shifting approach, then it seems that there are strong objections to that approach (which should be fine for you, because you do not subscribe to it).

This makes no sense. Events are invariant parts of spacetime; they're either there or they're not.

Yes, given that flashes 1-11 do occur (they are events in spacetime that occur), I have to draw them in the third diagram, which I have acknowledged. You are preaching to the converted here -- I acknowledged that I was wrong in initially thinking that they might disappear and create the kind of gap that you corrected me on in an earlier post.

If they're there, they're there for any coordinate chart that covers the region of spacetime containing them. You can't make an event appear or disappear by changing charts, unless you perversely define a chart to arbitrarily exclude a region of spacetime--but then it will be easy to show that the chart has "edges" that don't correspond to anything physical.
Worldlines are always continuous, whether they are worldlines of observers (timelike curves) or of light rays (null curves). Any valid coordinate chart will represent worldlines as continuous curves; if you find yourself wondering whether a worldline is continuous or not, it means you're not using a valid chart (or you're trying to switch charts in midstream, so to speak).

You are exactly right -- in order to diagram the frame-shifting approach I think that you have to "switch charts in midstream." That is why the third diagram titled "Front" is so different below the dashed line of simultaneity as above it. I am trying to accurately diagram the frame-centric view. That requires switching frames, which I think subscribers take more literally than you think they do. You think that it is a sloppy description. I think that they mean it literally. Witness the Rindler quotation: "B was accelerated out of his rest frame . . . These accelerations are recorded . . . he can therefore be under no illusion that . . . he . . . remained at rest . . . [H]e has transferred himself to a frame in which the distance between P and Q is halved (length contraction), and this halving is real to him in every way." I may or may not agree with the analysis, but I take the author's words seriously. He writes that B transferred himself to another frame, and that in that frame the distance is halved for real in every way. So I diagrammed such frame-shifting literally, with a shift between a very real distance of 60 from Far to Near just before hooking on (below the dashed line) to a very real distance of only 36 just after hooking on (above the dashed line).

How does he know how far away the lightning strikes are?

Just as in Einstein's example. He later takes a ruler and measures the distance to the char marks that the bolts made. He will measure a distance of 36 back to the char mark from the rear bolt.

I did my best to draw the frame-shifting diagram as ghwellsjr asked. He does a lot of work drawing diagrams for others. And I tried to do it in a way that is most fair to the frame-shifting theory, that makes it as good as it can be consistent with how people describe the theory. Maybe the diagram is invalid because of something I did wrong in interpreting the theory and then drawing the diagram. Or maybe it is invalid because the frame-shifting theory itself is simply invalid. We can't be sure until we have a diagram that accurately reflects the frame-shifting theory that we can all evaluate.

 

[Dale]

assume a lattice of recording clocks that stretches along the x-axis to include Near and Far, at rest with respect to them and with clocks synchronized to their watches.

...

The train also has a lattice of recording clocks, at rest with respect to the train with its clocks synchronized to the watches of Front and Rear.

These lattices are a physical implementation of two different coordinate charts, as Peter Donis mentioned earlier.

So, it is indeed incorrect to say that the proper time retrogresses, even with your formulation. It does not retrogress in either chart. If $\tau$ is the proper time and if $t$ is the platform lattice time and if $t'$ is the train lattice time then $d\tau/dt>0$ and $d\tau/dt'>0$. So the proper time does not retrogress.

 

[PeterDonis]

One is a straight line change in a single spatial direction

So if I'm moving in the positive x direction, and I accelerate in the positive x direction, I'm "changing direction" by this definition? I can't say this is inconsistent, exactly, but it does seem counterintuitive.

Anyway, this still doesn't address the point that "changing direction" is frame-dependent.

In SR there is an identity between changing direction and felt acceleration, however you characterize it.

No, that's not correct. What is going on here is that you are *defining* "changing direction" to mean "feeling acceleration". That's fine as a definition of terms as you use them, but you can't help yourself to it as a statement about physics that everyone has to agree with.

As to the reality of acceleration and the resulting length contraction, consider Rindler on the twin paradox and the resulting contraction of the distant point toward the accelerating twin:

B was accelerated out of his rest frame at P, at Q, and once again at P. These accelerations are recorded on B's accelerometer and he can therefore be under no illusion that it was he who remained at rest . . . [H]e has transferred himself to a frame in which the distance between P and Q is halved (length contraction), and this halving is real to him in every way.

Wolfgang Rindler, Introduction to Special Relativity (2nd ed. 1991), pp. 30-31.​

This is a valid argument for acceleration being real, because it's directly observable. It is not a valid argument for length contraction being real, because the change in the distance between P and Q is not directly observable; it's only there because of a change in coordinate charts (or frame fields--see further comments below). B can change charts (or frame fields) if he wants to when he accelerates, but there is nothing in the physics that forces him to; he can make correct physical predictions using any chart (or frame field) he likes.

The acceleration is absolutely real, and recorded, and distinguishes the traveler from the stay at home twin without any illusion.

For that particular way of setting up the scenario, yes. But there are other possible ways of setting up the scenario where nobody ever accelerates at all, so the full explanation of the twin paradox can't just be "acceleration makes the difference". But that's probably too much of a digression for this thread.

Per Taylor and Wheeler in Spacetime Physics, assume a lattice of recording clocks that stretches along the x-axis to include Near and Far, at rest with respect to them and with clocks synchronized to their watches.

Ok, this clarifies how you are using the term "frame". Taylor and Wheeler do sometimes use the term "reference frame" to refer to such a lattice, yes. A more correct terminology, however, would be that it is a "frame field"--a one-to-one mapping of reference frames to events (note that the lattice T&W describe includes rulers as well as clocks: the rulers and their spatial orientations comprise the spatial vectors of the frame assigned to each event by the frame field), plus a clock synchronization convention (T&W are assuming the standard Einstein clock synchronization convention, but there are other possible conventions as well).

(Also, as DaleSpam pointed out, defining a lattice this way is equivalent to defining a coordinate chart, so you are using coordinate charts whether you realize it or not.)

Near's watch reads 60 (or "presents" as 60) in the platform frame when Far's clock reads (or "presents") 60 in the platform frame.

...

Near's watch, however, presents at 12 in the train frame.

You didn't add an important qualifier here; I've added it in bold. Strictly speaking, it should be "in the train frame field"; see my comments above. I'll use "frame" in what follows, but I'm using it to mean "frame field" as I've defined it above.

The more important point here is that you are talking about two *different* events on Near's worldline. One, at which Near's watch reads 60, is the event which is simultaneous with Far hooking on to the train in the platform frame. The other, at which Near's watch reads 12, is the event which is simultaneous with Far hooking on to the train in the train frame. These are two different events on Near's worldline, which is why Near's watch can have two different readings. Near's watch can't have two different readings at the same event.

This also, btw, illustrates why it's a mistake to call length contraction "real". The "length" that supposedly gets contracted when Far gets hooked on to the train is not a single length--that is, it's not a single spacelike curve. What is really happening is that the "distance" to Near in the platform frame is the distance between the "hook event"--the event at which Far hooks on to the train--and the event on Near's worldline at which his watch reads 60. The "distance" to Near in the train frame is the distance between the hook event and the event on Near's worldline at which his watch reads 12. Since those are two different events on Near's worldline, the two "distances" are measured along different spacelike curves, which is how they can be different (the distance along a single spacelike curve can't change; it's an invariant). So saying that "length contraction" is "really happening" is misleading: it gives the impression that some physical length is actually changing, when all that is changing is which spacelike curve is being used to give the "distance" to Near.

Far is in one reference frame or chart (at rest with respect to the platform), where the distance is 60, then he is in another reference frame or chart (at rest with respect to the train), where the distance is 100.

You clarified how you're using "reference frame"; see my comments above. However, there's also an issue with the word "in" here. Observers are not "in" reference frames (or frame fields, or coordinate charts, if it comes to that). Observers can *choose* to use particular frames to make it easier for them to describe and predict events. But an observer does not have to change frames when he changes his state of motion. He can if he chooses, but he's not required to; he can describe all the physics and predict all events using whatever frame he likes. He does not have to be at rest in the frame he is using, which is what is implied by the claim that he is "in" one frame, then he changes his state of motion and is "in" another frame. These are all arbitrary choices of how to describe events; they are not required by the physics.

I suspect that this is just another disagreement over what one calls "real."

To the extent that "real" is a term about physics, not philosophy, I think it's a mistake to call anything "real" that is frame-dependent. That's why I object to calling length contraction "real"; it's frame-dependent. Proper acceleration, on the other hand, is a direct observable, so it's an invariant, so I have no problem with calling it "real".

I am just asserting that this is the correct way to describe the frame-shifting view.

I agree that Rindler's description is probably as good a brief illustration as any of the frame-centric view. If that's all you're using it for, fine. But you appeared to me to be claiming that Rindler's description is a valid argument for length contraction being "real", which I object to for the reasons given above.

What the lattice or lattices record in between 60 and 12 is questionable.

No, it isn't. You know Near's worldline in one lattice, and you know at which events on that worldline his watch records 12 and 60. Since all of your lattices are inertial, transforming between them is simple; so if you know what Near's worldline looks like, and where the "12" and "60" events are, in one lattice, you know those things for every lattice.

Non-inertial frames bring in other issues, because the assumptions that go into constructing the lattice as T&W describe require the lattice to be inertial.

Remember that I question whether this frame-shifting approach works well.

It works fine as a method of calculating. The problems arise only if you try to claim that frame-dependent things like length contraction are "real". However, all of the calculating machinery of the frame-centric approach works just fine even if you don't make such claims. So it depends on whether you view frames as just calculating machinery or whether you insist that they are telling you what's "real".

in order to diagram the frame-shifting approach I think that you have to "switch charts in midstream."

It depends on how you try to draw the diagram. You can draw a diagram using one chart, and then draw the "grid lines" of the other chart (i.e., the worldlines of the lattice points and the simultaneity surfaces of the other chart) on the diagram. The other chart's grid lines will not be vertical/horizontal if you do it this way, but you can still represent their relationships.

What you can't do is draw a single diagram in which the "grid lines" of both charts are vertical/horizontal; nor can you draw a diagram where one section has the first chart's grid lines vertical/horizontal and another section has the second chart's grid lines vertical/horizontal. "Mixing charts" this way will produce an inconsistent diagram, where a single point on the diagram has to correspond to distinct events in the actual spacetime. That doesn't work, and I suspect the issues you are having are because you are trying to draw this kind of diagram that doesn't work. But I'm not sure the "frame-shifting approach" *requires* you to draw the diagram that way; certainly if there is going to be any single diagram that represents that approach, it can't be drawn that way if it's going to be consistent.

 

[Dale]

Ok, this clarifies how you are using the term "frame". Taylor and Wheeler do sometimes use the term "reference frame" to refer to such a lattice, yes. A more correct terminology, however, would be that it is a "frame field"--a one-to-one mapping of reference frames to events (note that the lattice T&W describe includes rulers as well as clocks: the rulers and their spatial orientations comprise the spatial vectors of the frame assigned to each event by the frame field), plus a clock synchronization convention (T&W are assuming the standard Einstein clock synchronization convention, but there are other possible conventions as well).

(Also, as DaleSpam pointed out, defining a lattice this way is equivalent to defining a coordinate chart, so you are using coordinate charts whether you realize it or not.)

Just to explain my thoughts, the reason that I called it a physical implementation of a coordinate chart was because he stipulated that the clocks were synchronized. A lattice of rods and clocks would be a physical realization of a frame field, but as soon as you specify that the clocks are synchronized you have introduced an additional structure that is not part of a frame field, but which is part of a coordinate chart.

 

[PeterDonis]

The reason that I called it a physical implementation of a coordinate chart was because he stipulated that the clocks were synchronized. A lattice of rods and clocks would be a physical realization of a frame field, but as soon as you specify that the clocks are synchronized you have introduced an additional structure that is not part of a frame field, but which is part of a coordinate chart.

Yes, agreed; I was getting at the same point when I said that the "lattice" specification is a frame field plus a clock synchronization convention. A frame field by itself doesn't include a synchronization convention, as you say; you have to add it as additional structure.

 

[JVNY]

So if I'm moving in the positive x direction, and I accelerate in the positive x direction, I'm "changing direction" by this definition? I can't say this is inconsistent, exactly, but it does seem counterintuitive. . . This is a valid argument for acceleration being real, because it's directly observable.

You are quite right -- I forgot to include change of speed. Properly corrected, I should say that there is an identity in SR of felt acceleration and either change of speed or change of direction. How would you reconcile for an SR student the example (which has two felt accelerations) and ghwellsjr's final diagram, which has three instances of change in the worldline?

Ok, this clarifies how you are using the term "frame". Taylor and Wheeler do sometimes use the term "reference frame" to refer to such a lattice, yes. A more correct terminology, however, would be that it is a "frame field"--a one-to-one mapping of reference frames to events (note that the lattice T&W describe includes rulers as well as clocks: the rulers and their spatial orientations comprise the spatial vectors of the frame assigned to each event by the frame field), plus a clock synchronization convention (T&W are assuming the standard Einstein clock synchronization convention, but there are other possible conventions as well).
(Also, as DaleSpam pointed out, defining a lattice this way is equivalent to defining a coordinate chart, so you are using coordinate charts whether you realize it or not.)
You didn't add an important qualifier here; I've added it in bold. Strictly speaking, it should be "in the train frame field"; see my comments above. I'll use "frame" in what follows, but I'm using it to mean "frame field" as I've defined it above.

OK, frame field makes sense to me; and it seems reasonable to say that the lattice is a coordinate chart.

The more important point here is that you are talking about two *different* events on Near's worldline. One, at which Near's watch reads 60, is the event which is simultaneous with Far hooking on to the train in the platform frame. The other, at which Near's watch reads 12, is the event which is simultaneous with Far hooking on to the train in the train frame. These are two different events on Near's worldline, which is why Near's watch can have two different readings. Near's watch can't have two different readings at the same event.

Agreed. When Near's watch ticks off a unit of elapsed proper time by flashing "12," that flashing of "12" is not itself a matter of coordinates. It is simply an event in spacetime, a flash like a flash of a firecracker going off or a lightning bolt. Where and when it occurs in any given lattice is a coordinate matter for that lattice. So, strictly applying the frame-shifting theory means that some events actually go backwards for the accelerating observer. Far is in one frame field at the same time at which the event "Near's watch flashes 60" occurs (a distant, simultaneous event). Then, Far transfers himself to another frame field and is in that field at the same time at which the event "Near's watch flashes 12" occurs (a distant, simultaneous event). The frame shifting theory does not say that proper time retrogresses. Only that events that occur in one order (proper time for Near always advances, 1, 2, 3, ...) can occur in a different order, even backwards, for an accelerating observer because that observer defines what happens by claiming as his own the series of distant simultaneous events in each frame field that he serially occupies (Near's watch flashes 60 then 12 if instantaneous acceleration, or perhaps 60, 59, . . . if continuous acceleration). Again, this may not be good physics, but I think that this is what the frame-shifting theory means.

You clarified how you're using "reference frame"; see my comments above. However, there's also an issue with the word "in" here. Observers are not "in" reference frames (or frame fields, or coordinate charts, if it comes to that). Observers can *choose* to use particular frames to make it easier for them to describe and predict events. But an observer does not have to change frames when he changes his state of motion. He can if he chooses, but he's not required to; he can describe all the physics and predict all events using whatever frame he likes. He does not have to be at rest in the frame he is using, which is what is implied by the claim that he is "in" one frame, then he changes his state of motion and is "in" another frame. These are all arbitrary choices of how to describe events; they are not required by the physics.

Perhaps "in" is the wrong word. But this is how the frame-centric approach uses it. There is a lattice, the observer is at rest with respect to the lattice, so you could say that he is in the lattice, or on the lattice, or anything like that. He can always go back and check the records of the recording clocks, see that there are two successive recordings with him at rest with respect to the recording clocks (thus "in" or "on" that lattice), and he will find one of those recordings with the event "Far is located next to nearest recording clock at lattice location x at time t" being simultaneous with "Near's watch flashes # at that lattice's nearest recording clock location x2 at the same time t".

It depends on how you try to draw the diagram. . . What you can't do is draw a single diagram in which the "grid lines" of both charts are vertical/horizontal; nor can you draw a diagram where one section has the first chart's grid lines vertical/horizontal and another section has the second chart's grid lines vertical/horizontal. "Mixing charts" this way will produce an inconsistent diagram, where a single point on the diagram has to correspond to distinct events in the actual spacetime. That doesn't work, and I suspect the issues you are having are because you are trying to draw this kind of diagram that doesn't work. But I'm not sure the "frame-shifting approach" *requires* you to draw the diagram that way; certainly if there is going to be any single diagram that represents that approach, it can't be drawn that way if it's going to be consistent.

Right, that seems to be a problem. How can a frame-shifting approach draw a diagram in which the accelerating observer goes from one set of grid lines (or one set of lattices) to another and keep everything properly overlapping. I am not sure how else to draw the diagram. Perhaps there is a way that I can't think of, or perhaps it just cannot be done (thus impugning the theory).

 

[PeterDonis]

How would you reconcile for an SR student the example (which has two felt accelerations) and ghwellsjr's final diagram, which has three instances of change in the worldline?

By emphasizing that both changes of speed and changes of direction are frame-dependent (because "speed" and "direction" themselves are frame-dependent). The correspondence between felt acceleration and change of speed/direction only holds for inertial frames; it does not hold for non-inertial frames.

When Near's watch ticks off a unit of elapsed proper time by flashing "12," that flashing of "12" is not itself a matter of coordinates. It is simply an event in spacetime, a flash like a flash of a firecracker going off or a lightning bolt. Where and when it occurs in any given lattice is a coordinate matter for that lattice.

All good.

So, strictly applying the frame-shifting theory means that some events actually go backwards for the accelerating observer.

No, it means that the accelerating frame constructed as you describe can only cover a limited region of spacetime. See below.

Only that events that occur in one order (proper time for Near always advances, 1, 2, 3, ...) can occur in a different order, even backwards, for an accelerating observer because that observer defines what happens by claiming as his own the series of distant simultaneous events in each frame field that he serially occupies (Near's watch flashes 60 then 12 if instantaneous acceleration, or perhaps 60, 59, . . . if continuous acceleration). Again, this may not be good physics, but I think that this is what the frame-shifting theory means.

I'm not sure about that. A key reason why I'm not sure is that you cite Taylor and Wheeler as an example of usage of the frame-shifting theory. Can you cite a specific passage where they say that the ordering of events can be reversed like this for an accelerating observer? The correct thing to say, and what I would expect a source like T&W to say, is that an "accelerating frame" constructed as you describe is only valid for a region of spacetime close enough to the accelerating observer's worldline that no "reversals" like this occur. The reason for that is that, if the event ordering far enough away reverses, there will be some event closer in (in this case, an event between Far and Near) which must be assigned multiple time coordinates in this "accelerating frame" (because multiple lines of simultaneity cross there). This makes the frame invalid at that event and any event further away, because a valid frame must assign a single time coordinate to every event in the region it covers, and any event where this uniqueness fails marks a boundary of the region of spacetime the frame can validly cover.

I should also note, as I've noted before, that the method you describe for constructing an accelerated observer's frame is not the only possible way of doing so; there are other methods that allow the entire spacetime (or at least the entire region of spacetime that is causally connected to the observer's worldline) to be covered. The diagram ghwellsjr drew is an example of such an accelerated frame ("coordinate chart" would be a better term).

Right, that seems to be a problem. How can a frame-shifting approach draw a diagram in which the accelerating observer goes from one set of grid lines (or one set of lattices) to another and keep everything properly overlapping.

The answer is, he can't except in a region of spacetime small enough that no grid lines cross (which equates to no lines of simultaneity crossing). Any region where the grid lines cross simply cannot be covered by an accelerating frame constructed this way. See above.

 

[JVNY]

I'm not sure about that. A key reason why I'm not sure is that you cite Taylor and Wheeler as an example of usage of the frame-shifting theory. Can you cite a specific passage where they say that the ordering of events can be reversed like this for an accelerating observer?

No, I don't know of any TW quotation like that. The frame-jumping explanations often occur using moving walkways as inertial reference frames, and jumping between walkways to accelerate. Thus, in explaining the twin paradox, some people describe the traveling twin as getting on a moving walkway outbound, then jumping on to a different moving walkway heading back in the opposite direction, then hopping off of that one when back home again. For example,

In the twin “paradox”, one twin, Bob, stays on earth, an inertial frame which we will call S. At
time zero, the other twin, Alice, gets on a moving walkway going with speed v = 4/5 c. This is also an inertial frame, which we call S′. She travels a distance 4 light years (as measured by Bob) and then jumps on to another moving walkway, moving in the opposite direction at the same speed, which carries her back to earth. This walkway is a third inertial frame, which we call S′′. . .

Bob stays in frame S. Alice is in frame S′ until she turns around, at which point she “jumps” into frame S′′. . .

We now determine the position and time of each of these events in each of the three frames. . .

[W]hen Alice jumps into frame S′′ she should adjust her watch by this amount for it to be synchronized with her new frame. Of course, she has not aged by this additional amount, just as, for example, ones age does not change by a hour when crossing into a new time zone.

young.physics.ucsc.edu/5I/twins.pdf​

And, with a focus on the frame-jumping view that changing frames changes the "now" of a distant event, the teaching of this view typically asks the following question:

If the traveling twin is asked "How old is your brother right now" what is the correct reply (a) just before she makes the jump, (b) just after she makes the jump

http://www.uAlberta.ca/~pogosyan/teaching/PHYS_458/FALL_2012/assignment1.pdf​

In these analyses, the distance and stay at home clock times "really" change with the jump to a different walkway/frame. They involve reversing direction, so the age of the stay at home twin "jumps" forward (his age, and his watch, move forward) when the traveler jumps frames. That is because the distant stay at home twin is ahead of the traveler. I have unfortunately lost references to ones in which the traveler does not change directions, but simply jumps to successively faster neighboring walkways going in the same direction. But if the traveling twin jumps to a faster outbound walkway while still traveling outbound, the answer to the question "how old is your brother now" after the jump would be younger than just before that jump.

Conceptually, think of it this way.

The lattice of recording clocks at rest with respect to the platform records a series of events: (a) Near's watch emitting time stamped signals 12-60 at near end of platform, (b) Far being on the far end of the platform as his watch emits time stamped signals 12-60, and (c) Far hooking onto the train at 60. Platform observers later download the images from the recording cameras and confirm that there are photos with Far on the platform at the same time that his watch and Near's watch emit signals 12-60.

The lattice of recording clocks at rest with respect to the train records a series of events: (a) Far hooking onto the train front at train time 60 and Far watch time 60, (b) thereafter Near's watch emitting time stamped signals 12-60, and (c) and Near hooking onto the train rear at train time 140, Near's watch time 60. Train observers later download the images from the recording cameras and confirm that there are photos with Far on the train at the same time that Near's watch emits signals 12-60.

Do you agree that this is how the lattices would work?

If so, there is no need to talk about retrogression. One simply states that Far was on the platform for the emission of signals 12-60 according to the platform lattice, and thereafter (as he defines before and after, because he is the one jumping frames) he was on the train for the emission of signals 12-60 according to the train lattice. This should not be controversial. It is just the relativity of simultaneity.

What might be controversial is how to interpret this. If Far thinks that distant events on an observer's line of simultaneity exist in his "now," he might have a problem. He would believe that the emissions of signals 12-60 occurred in his "now" when he was on the platform, and again in his "now" when he was on the train, and thus occurred "now" for him twice. But the same events cannot happen twice. The radar method does not suffer from this defect, because the time assigned to any distant event depends on the emission, reflection, and reception of a signal, which can only happen once for any event.

 

[PeterDonis]

Do you agree that this is how the lattices would work?

Yes, I don't think there is any disagreement over the calculational machinery. I think the issue is "interpretation"; see below.

What might be controversial is how to interpret this. If Far thinks that distant events on an observer's line of simultaneity exist in his "now," he might have a problem.

Right, but do the sources you linked to actually make that claim?

It seems to me that what's really going on here is bad pedagogy: trying to teach concepts like relativity of simultaneity while still attempting to preserve ordinary intuitions about concepts like "now". This leads to either an inconsistent interpretation--sometimes talking as if "now" is real and sometimes talking as if it's just a convention--or issues like the ones you raise, where it seems like single events happen "now" twice for an observer that changes his state of motion.

 

[JVNY]

It seems to me that what's really going on here is bad pedagogy: trying to teach concepts like relativity of simultaneity while still attempting to preserve ordinary intuitions about concepts like "now". This leads to either an inconsistent interpretation--sometimes talking as if "now" is real and sometimes talking as if it's just a convention--or issues like the ones you raise, where it seems like single events happen "now" twice for an observer that changes his state of motion.

I think that it is an issue of pedagogy. The citations do not give the answer, but the complete question posted on another site implies the answer. The complete question is here, http://www.physast.uga.edu/files/phys4102_fertig/HW%20Chapter%2015.pdf , and includes the following about the brother's age "right now" before and after the frame jump:

If the traveling twin is asked the question, "How old is your brother right now, and which of you is younger?", what is the correct reply (i) just before she makes the jump, (ii) just after she makes the jump? (Nothing dramatic happens to her brother during the split second between (i) and (ii) of course -- what does change radically is his sister's notion of what "right now" means.)​

The Dolby and Gull article points out similar issues at page 1257, using "Barbara" as the traveler:

Bohm goes on to say that ‘‘after the acceleration at E an event such as N is ascribed a smaller time coordinate than it had before!’’ . . .

Also, if Barbara’s hypersurfaces of simultaneity at a certain time depend so sensitively on her instantaneous velocity as these diagrams suggest, then she would be forced to conclude that the distant planets swept backwards and forwards in time whenever she went dancing!​

So Dolby and Gull draw the hypersurfaces of simultaneity using the radar method instead.

 

[phyti]

This drawing reduces the example to Greens perception of Reds motion.
The left part is essentially the same as drawn by George in post 5 using radar methods.
The transfer for each from the platform to the train occurs at t=0 instead of 60.
The original red path is retained (black dashed) for comparison of before and
after positions (magenta) of clock ticks.
The right part is refined to replace the instantaneous changes with
continuous paths for 40 time units, thus rounding all the corners.
When compared the all inertial version is a good first approximation.

https://www.physicsforums.com/attachments/66109