# Another train and platform

1. Jan 24, 2014

### 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?

2. Jan 24, 2014

### Staff: Mentor

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.

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:

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.

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:

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

...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...

...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.

3. Jan 24, 2014

### JVNY

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).

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.

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 [Broken]​

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

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.

Last edited by a moderator: May 6, 2017
4. Jan 24, 2014

### Staff: Mentor

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).

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

Correct.

Correct.

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.

Yes.

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.

Yes.

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.

5. Jan 24, 2014

### ghwellsjr

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.

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?

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6. Jan 24, 2014

### bahamagreen

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

7. Jan 24, 2014

### ghwellsjr

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.

8. Jan 24, 2014

### 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?

9. Jan 24, 2014

### ghwellsjr

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.

10. Jan 24, 2014

### Staff: Mentor

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).

11. Jan 24, 2014

### ghwellsjr

Here's a reference to it by DaleSpam:

12. Jan 25, 2014

### pervect

Staff Emeritus
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

13. Jan 25, 2014

### 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.

14. Jan 25, 2014

### Staff: Mentor

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.

15. Jan 25, 2014

### Staff: Mentor

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.

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.

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.

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?

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.

16. Jan 25, 2014

### ghwellsjr

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

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.

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.

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.

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.

17. Jan 25, 2014

### 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?

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.

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).

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).

18. Jan 25, 2014

### pervect

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

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?

19. Jan 25, 2014

### Staff: Mentor

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.

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".

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.

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.

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

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

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.

20. Jan 26, 2014

### 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.

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