"Forced Conclusion" in Train-Embankment Experiment

[Ian432]

In the train-embankment thought experiment described by Einstein in his "public" version of STR, he says the following:
“Hence the observer will see the beam of light emitted from B [the front of the train] earlier than he will see that emitted from A [the rear of the train]. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B [front of train] took place earlier than the lightning flash A [rear of train].”
[ REF: https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Part_I ]

I take issue with the phrase "must ... come to the conclusion." I actually think it very unlikely that the passenger would jump to that conclusion. Here is my reasoning:

The only way to "see" a difference in the arrival-time of these two light beams would be to detect it using sophisticated machinery. For example, assuming a 20-meter long train car traveling at 120 kph, the difference in arrival time would be around 200 trillionths of a second (if my calculations are correct). Those with access to such sophisticated machinery would already be quite aware of the hazards in jumping to any conclusion about the "real" time-origin of the two beams, especially knowing all the debates in physics that have been raging about exactly such problems. Thus, the passenger, having "observed" the phenomena on a measuring-device, would probably come up with any number of theories to explain the findings, and would withhold judgment until more definitive evidence was uncovered. Note also that anyone able to measure such beam arrival-times would probably also be able to detect a red- or blue-shift in the frequency of such light beams. Would it not be true that, in Einstein's scenario, there would be a blue-shift in the "B" (front) beam and a red-shift in the "A" (rear) beam? If so, the sophisticated passenger would have stronger (but not definitive) evidence that s/he was not standing still relative to A and B, but was actually in an inertial frame moving toward B and away from A. This color-shift example casts even more doubt on the claim that the passenger would be forced to make a specific conclusion from the observed phenomena.

 

[jbriggs444]

It's a thought experiment. All the instruments are perfect and trustworthy and all the assumptions that go into the scenario are guaranteed to hold.

 

[PeterDonis]

assuming a 20-meter long train car traveling at 120 kph, the difference in arrival time would be around 200 trillionths of a second (if my calculations are correct).

I get a difference of $7.4 \times 10^{-15}$ seconds, which is about five orders of magnitude smaller. The difference will be $\gamma v \Delta x / c^2$, where $\gamma = 1 / \sqrt{1 - v^2 / c^2}$. I get $v = 33.3$ (remember that the units need to be m/s, not kph), $\gamma = 6.22 \times 10^{-15}$, and $\Delta x = 20$ by hypothesis.

Those with access to such sophisticated machinery would already be quite aware of the hazards in jumping to any conclusion about the "real" time-origin of the two beams, especially knowing all the debates in physics that have been raging about exactly such problems.

What debates are you referring to?

Would it not be true that, in Einstein's scenario, there would be a blue-shift in the "B" (front) beam and a red-shift in the "A" (rear) beam?

Compared to the frequencies observed by the observer on the embankment, yes.

If so, the sophisticated passenger would have stronger (but not definitive) evidence that s/he was not standing still relative to A and B, but was actually in an inertial frame moving toward B and away from A.

This is true if "A" and "B" refer to the sources of the light beams (and if we also assume that the "natural" frequencies of both beams are the same, but that's a minor point). But the sources of the light beams are objects, not events. The conclusion Einstein is talking about is about the events of the two lightning strikes--in other words, specific points in spacetime, not objects. The fact that the passenger can (correctly) conclude that the inertial frame in which he is at rest is moving relative to the sources of the light beams he sees does not change his conclusion about the timing of the events of the lightning strikes, because one cannot be "at rest" or "moving" relative to an event (as opposed to an object).

Here's another way to look at it: the lightning flash at A happens at a particular instant at A; and the lightning flash at B happens at a particular instant at B. At the instant the flash from A is emitted, A is co-located with the rear of the train; and at the instant the flash from B is emitted, B is co-located with the front of the train. The fact that A and B move, relative to the rear and front of the train (respectively), after the flashes are emitted is irrelevant to determining the times of the flashes relative to the passenger, because the speed of the light coming from each flash is independent of the motion of A and B relative to the passenger (that is the key physical fact that was not recognized in pre-relativity physics). And the passenger, by hypothesis, is equidistant from the front and rear of the train, therefore he is also equidistant from the events at which the flashes were emitted. That is why the passenger must conclude that the lightning flashes happened at different times in his frame.

 

[Ibix]

Einstein is describing the results of an experiment in a hypothetical universe where the speed of light is constant and independent of the motion of the source, and where this is known to the experimenters. The conclusion of simultaneity (or lack thereof) is correct in such a universe, independent of any argument over how the real universe behaves.

Einstein then goes on to show that then-outstanding problems in physics can be resolved if this is how our universe behaves. Not so hypothetical after all.

You are correct that the experiment isn't practical to do, but that doesn't really matter. One can always scale up the train and run it in free space instead of on the Earth. Whether there's a Doppler shift depends on details that we didn't discuss. Again we can vary the experiment slightly. Use four identical lamps, one mounted at the front and one at the rear of the train, and one at the front and one at the rear of the platform. You can arrange a mechanism so that the lamps flash as they pass each other. Then both the train and platform observers will receive one shifted and one non-shifted pulse from each end of the train and can verify that both "front" emission events were simultaneous and both "rear" emission events were simultaneous.

 

[Ian432]

Please allow me to rephrase my original question with greater clarity, and in line with your comments: the creation of the light-beams (due to the lightning strikes at A and B) are events in space-time. The instant after those events occur, each light-beam travels toward the position of the passenger. When the light-beams arrive, the passenger detects that the front beam (from the event at B) is blue-shifted relative to the rear beam (from the event at A), and that the front beam arrives before the rear beam. The passenger is free to speculate that the speed of both beams is constant and equal, in an "absolute" sense. The passenger is also free to use the clue of the color-shift differences to conclude that the two events in space-time may actually have happened simultaneously, even though the information about the event at B arrived at the passenger's location before the information about of the event at A.

The passenger may further overcome the perplexing situation of realizing that the information about the event at B arrived "more quickly" than the information about the event at A, even though the carriers of that information (the light-beams) must be moving at exactly the same constant speed. Simply put, the passenger concludes that s/he is moving toward the information that is arriving from the event at B, and away from the information that is arriving from the event at A.

 

[jbriggs444]

The passenger is free to speculate that the speed of both beams is constant and equal, in an "absolute" sense.

The passenger has no room for speculation. It is a given of the problem that the speed of both beams as measured in his rest frame is equal. The givens of a thought experiment are not debatable. They are facts of the matter and must be accepted.

The passenger is also free to use the clue of the color-shift differences to conclude that the two events in space-time may actually have happened simultaneously, even though the information about the event at B arrived at the passenger's location before the information about of the event at A.

The frequency of the received light will depend on the frequency generated at the emitter and the velocity of the emitter relative to the receiver. It can not depend on any "absolute" state of motion of the receiver. That is a given of the problem. It is also completely irrelevant to the thought experiment.

The passenger may further overcome the perplexing situation of realizing that the information about the event at B arrived "more quickly" than the information about the event at A, even though the carriers of that information (the light-beams) must be moving at exactly the same constant speed.

The information is carried on the leading edge of the light pulse. It moves at the speed of the light pulse.

Simply put, the passenger concludes that s/he is moving toward the information that is arriving from the event at B, and away from the information that is arriving from the event at A.

It is a given of the problem that no such conclusion can be reached: The laws of physics are identical regardless of one's choice of inertial reference frame.

 

[Ian432]

The frequency of the received light [...] is also completely irrelevant to the thought experiment.

There is nothing in the thought experiment that says that the frequency of the light is irrelevant to the thought experiment. Moreover, this difference in frequency is entirely relevant to my questions about the thought experiment, and it is those questions to which you are, in fact, responding.

The information is carried on the leading edge of the light pulse. It moves at the speed of the light pulse.

Your statement does not overcome the fact that the information, like the light pulses, arrive(s) at different times to the passenger.

The passenger has no room for speculation.

It is a given of the problem that no such conclusion can be reached

Your statements would lead one to conclude that this is a thought experiment that precludes thought itself within the experiment. By which one could only conclude that thinking is forbidden within the world of relativity physics, and that therefore the only valid frames in which relativity physics can even be considered are those in which thinking cannot exist. Yet in reality, I think, and indeed many other people think (including you). Therefore, since thinking cannot exist within a relativity frame, relativity itself cannot exist for many people (including you). Translation: I would hope for a more thoughtful reply. To my mind, you have dismissed and sidestepped, but not actually or thoughtfully answered, my core questions about the thought experiment.

The fact remains that the passenger could easily detect a difference in the frequency of the light beams arriving from the two different events. If allowed even a small modicum of thought, the passenger could conclude that the two events occurred simultaneously but that the information from event B arrived before the information from event A, and that the passenger is moving toward the incoming information from event B and away from the incoming information from event A.

 

[Dale]

When the light-beams arrive, the passenger detects that the front beam (from the event at B) is blue-shifted relative to the rear beam (from the event at A),

The Doppler shift is incidental. Other variations of the scenario could eliminate it without changing the conclusion regarding simultaneity, like what Ibix described.

In addition to being incidental, it is also irrelevant. Light of any frequency travels at c. The observed Doppler shift does not change the speed of propagation.

The passenger is free to speculate that the speed of both beams is constant and equal,

It is a given. The passenger is required to reason based on this assumption. The question is what conclusion is required based on this assumption. Indeed, with this as a given the passenger must conclude that the flashes were not simultaneous.

The passenger is also free to use the clue of the color-shift differences to conclude that the two events in space-time may actually have happened simultaneously,

No, he isn't. The given is that light of any color or any source travels at c, so observing the Doppler shift cannot change the conclusion.

 

[Ian432]

Givens: The light moves at light speed. The passenger receives the information about event B before receiving the information about event A.
Challenge: I propose that the passenger could perceive the color of the light beams, and that the color is relevant, because it provides a clue that could help explain, to the passenger, the reason for the sequence in the arrival-times of the light beams.

You say that my challenge is irrelevant, but the color difference in the light beams is entirely consistent with all the terms of the thought experiment, and therefore it is entirely relevant.

 

[jbriggs444]

Givens: The light moves at light speed. The passenger receives the information about event B before receiving the information about event A.
Challenge: I propose that the passenger could perceive the color of the light beams, and that the color is relevant, because it provides a clue that could help explain, to the passenger, the reason for the sequence in the arrival-times of the light beams.

You say that my challenge is irrelevant, but the color difference in the light beams is entirely consistent with all the terms of the thought experiment, and therefore it is entirely relevant.

Being consistent with the experiment and being relevant to the experiment are two different things. The light being being emitted by a ruby laser attached to the train is consistent with the experiment. The light being emitted by a strobe mounted on the embankment is consistent with the experiment. But both are irrelevant. The important thing is that a pulse of light was emitted at the time and place of the lightning strike.

 

[Ian432]

Use four identical lamps, one mounted at the front and one at the rear of the train, and one at the front and one at the rear of the platform. You can arrange a mechanism so that the lamps flash as they pass each other. Then both the train and platform observers will receive one shifted and one non-shifted pulse from each end of the train and can verify that both "front" emission events were simultaneous and both "rear" emission events were simultaneous.

You mean: both "front" emission events were simultaneous with one another and both "rear" emission events were simultaneous with one another. This is definitely an interesting scenario, but aside from muddying the water, I do not see how they definitively put to rest the passenger's conjecture as to whether the front and rear events were simultaneous with one another.

 

[Ian432]

Being consistent with the experiment and being relevant to the experiment are two different things.

Dale and jbriggs444: Let me use an analogy. By your arguments, the fact that Maxwell's demon used up energy while busily opening and shutting a door between two containers of gas was "consistent with but irrelevant to" Maxwell's original thought experiment, because it did not lead to Maxwell's conclusion. Yet this challenge--that the demon must use energy to open and shut that door--allowed the second law of thermodynamics to be maintained despite Maxwell's proposed demon.

Do you think it would have been valid for Maxwell to have done as you are proposing to do--to dismiss a constructive challenge to his thought experiment by using the term "irrelevant," without responding to the actual challenge itself?

 

[peety]

More simply, it is a given that the passenger is at rest in his/her frame and therefore cannot conclude that s/he is 'moving towards the information that is arriving...'

 

[jbriggs444]

Dale and jbriggs444: Let me use an analogy. By your arguments, the fact that Maxwell's demon used up energy while busily opening and shutting a door between two containers of gas was "consistent with but irrelevant to" Maxwell's original thought experiment, because it did not lead to Maxwell's conclusion. Yet this challenge--that the demon must use energy to open and shut that door--allowed the second law of thermodynamics to be maintained despite Maxwell's proposed demon.

Do you think it would have been valid for Maxwell to have done as you are proposing to do--to dismiss a constructive challenge to his thought experiment by using the term "irrelevant," without responding to the actual challenge itself?

The challenge you pose, if correct, would contradict the givens of the experiment and prove the assumptions underlying relativity to be false. However, your challenge is not well thought out. It is not correct.

 

[Ian432]

However, your challenge is not well thought out. It is not correct.

I invite you, or others, to explain--without simply dismissing me--why my challenge is "not correct." So far, no one on this forum, in this thread, has responded in a way that seems consistent with the spirit with which Einstein's theory would have been received and discussed at the time it was proposed. Negating a serious question about a topic as "irrelevant" or "not correct" or being "not well thought out" (despite the abundance of evidence that, indeed, I have thought this out quite carefully and it is a robust argument, still waiting for an equally robust answer) is hardly educational.

To quote Einstein himself: "I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations." (Public STR)

 

[Ian432]

Perhaps this will clear the air. I believe all the previous posters have relied upon one assumption: that, in Einstein's thought experiment, it is a "given" that the time at which information about events are received by a receiver, requires the receiver actually to understand or conclude the timing and sequencing of the events themselves.

All I need is for you to quote from the STR itself, to show me where the above was actually proposed as a "given" by Einstein himself, in the STR.
Here is a source you can use:
https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Part_I

 

[Ian432]

Just to tie up some loose ends:

1. The challenge to Maxwell's demon was meaningful and relevant because it exposed something about the necessary nature of the demon itself. Maxwell could have said, "oh, but this is a special demon that doesn't really use up any energy," but that would have violated the laws of physics as they were at that time understood, even though those laws stood outside his thought-experiment.

2. My challenge to Einstein's passenger is meaningful and relevant because it exposes something about the necessary nature of that passenger, in ways that refer to laws of nature that lie outside the thought-experiment but are clearly relevant to it. In order for a passenger to "conclude" anything, the passenger must be able to reason logically--clearly, the passenger is not just an inanimate object. And in order for the passenger to perceive a difference in the arrival-times of the different light-beams, the passenger must have access to extremely sophisticated equipment (as PeterDonis has pointed out, the time difference is even smaller than I had originally thought). This is a requirement within the thought experiment itself. My stipulation about the color-shift in the light beams is as consistent with the requirements of original thought-experiment, as is the stipulation about Maxwell's demon actually using up energy when it opens and shuts the door.

3. The conclusion to the challenge to Maxwell's demon is that the second law of thermodynamics still holds. The conclusion to my challenge about Einstein's passenger would be that the passenger is NOT forced to conclude anything definitive about the actual timing of the events at A and B.

I have raised this challenge to those on this forum, and I am still waiting for a meaningful reply that not only supports, but also lives up to the spirit of thoroughness, thoughtfulness, civility, and intellectual integrity of, the original proponent of the experiment. You may recall that Einstein himself was at times the recipient of challenges to other of his thought experiments, and he treated his challengers with respect and never doubted either their intellectual capacity, or the seriousness of the challenges themselves.

 

[jbriggs444]

Perhaps this will clear the air. I believe all the previous posters have relied upon one assumption: that, in Einstein's thought experiment, it is a "given" that the time at which information about events are received by a receiver, requires the receiver actually to understand or conclude the timing and sequencing of the events themselves.

If you, given suitably accurate instruments, careful setup and an assurance that light always travels at a particular fixed speed were to receive a light signal that had followed a straight path and was emitted from something one light-second away, could you conclude that the time the light was emitted was one second prior to the time of its arrival?

This is in an experimental setup that is free from smoke, mirrors or other gimmicks. You get to set it up yourself. Nobody is trying to pull the wool over your eyes.

 

[Ian432]

That's not a well-thought-out question or experimental setup. For starters, since you propose that I am within the experiment, I can tell you that I could not possibly set up such an experiment nor validate it.

If you have a point to make, please make it, and please also make it relevant to Einstein's thought-experiment and my challenge to it.

 

[jbriggs444]

In the interest of staying polite, I'm out.

 

[Ibix]

I don't understand what you think you are proving. The Doppler shift can tell you something about the velocity of the source with respect to the receiver, but so what? You could use lights moving at 0.9c in different directions that just happen to be at the ends of the platform when the ends of the train pass by. Then you know that the lights were fast moving. So what?

 

[Ibix]

...and your point about precise instrumentation is irrelevant. Just make the train longer and/or faster - the Andromeda paradox is the typical example.

 

[pervect]

Please allow me to rephrase my original question with greater clarity, and in line with your comments: the creation of the light-beams (due to the lightning strikes at A and B) are events in space-time. The instant after those events occur, each light-beam travels toward the position of the passenger. When the light-beams arrive, the passenger detects that the front beam (from the event at B) is blue-shifted relative to the rear beam (from the event at A), and that the front beam arrives before the rear beam. The passenger is free to speculate that the speed of both beams is constant and equal, in an "absolute" sense.

You're getting close to the right track, but perhaps a bit off, I think. Einstein has assumed (in deriving special relativity) that the speed of light is constant for all observers. You call this "speculation", a nicer name for it is "hypothesis". The issue at hand is to derive the logical consequences of this hypotesis ("speculation", in your words), to see the ramifications. At some point we may start to try and further address experimental issues as to how to actually carry out such an experiment, but at this point we are merely trying to apply logic to determine the logical consequence of our hypothesis.

So when Einstein is talking about the "forced conclusion", he is talking about the conclusion being forced in the context of special relativity, which has previously _assumed_ that the speed of light is constant for all observers. Basically, he's assuming you've been following what he's been saying in the previous chapters of his book (or perhaps you are quoting from a different source than http://www.bartleby.com/173/9.html in which case it may not be a book), and not just opening the book at the chapter in question and taking his remakrs out of context.

If one makes the assumption that the speed of light is constant for all observers, one comes to the conclusion that the lighning strikes that were simultaneous for the embankment observer are NOT simultaneous for the train observer.

The passenger is also free to use the clue of the color-shift differences to conclude that the two events in space-time may actually have happened simultaneously, even though the information about the event at B arrived at the passenger's location before the information about of the event at A.

The passenger is free to make other hypothesis other than special relativity, and to explore the logical consequences. This isn't, however, what Einstein is doing.

What Einstein is saying is that if one makes the logical assumptions that special relativity makes, simultaneity is NOT absolute, but it depends on the observer. A sub-point of this is that if one makes the assumption that simultaneity is absolute and independent of the observer, one cannot do special relativity properly, as one has made inconsistent logical assumptions. In fact, making the _assumption_ that simultaneity is absolute is a frequent barrier to students learning special relativity, they make the assumption without even thinking about it, and because this assumption is logically inconsistent with the postulate of special relativity, they derive various false conclusions, which I won't get into. So the intended message is one of caution - one must be careful not to make the seemingly natural assumption that simultaneity is "absolute" (i.e. independent of the observer) if one wishes to understand special relativity.

When deciding what assumptions are consistent with experiment, one obviously needs more than just a single thought experiment, one needs to look at the actual experimental results from a wide variety of experiments. This is outside the context of the current discussion, but has been done in other threads.

If you're interested in the later topic, it would probably be best to start a different thread. "Einstein's train" illustrates an important logical consequence of the assumptions of special relativity, logical consequences which are perfectly self-consistent but unexpected, and which are recognized as being notoriously confusing to students and readers. See for instance "The challenge of changing deeply held student beliefs about the relativity of simultaneity", https://www.aapt.org/doorway/TGRU/articles/Vokos-Simultaneity.pdf.

 

[PeterDonis]

There is nothing in the thought experiment that says that the frequency of the light is irrelevant to the thought experiment.

There's nothing that says it's relevant, either. If you think it is, it's up to you to demonstrate why. Just saying so isn't enough.

The passenger is also free to use the clue of the color-shift differences to conclude that the two events in space-time may actually have happened simultaneously

No, he isn't, unless he can show that the detected frequency of the light beams is related to their speed. But it isn't; the statement that the speed of light is independent of the motion of the source is one of the conditions of the problem, and that fact, as I pointed out before, makes the detected frequencies of the light flashes irrelevant to determining the times at which they happened.

Of course, you could explore the consequences of an alternate condition, that the speed of light is not independent of the motion of the source; but then, as pervect pointed out, you wouldn't be discussing Einstein's train thought experiment, but a different one. (You also wouldn't be discussing SR.)

 

[Ian432]

I am sorry I ever mentioned the "supposition" about the speed of light in the experiment and I fully agree that it is a constant. I fully accede that point. But this does not get at the core of my challenge.

My challenge is about the logical consistency of what I think of as Einstein's argument that the passenger on the train "must come to the conclusion" about the sequencing of events. In this thread, the basic question continually arises as to whether Einstein's statement that the passenger "must come to the conclusion" is a core assumption (a "given") of the thought-experiment, as most of the posters have suggested, or whether it is a conclusion that Einstein himself is drawing as a consequence of the thought experiment (as I have suggested).

If, as has been suggested by other posters, the passenger's conclusion is a "given," then it makes the thought-experiment much less interesting and less like the world as we know it. Einstein would be proposing a world in which scientists would be forced to make definitive conclusions about the timing of events based upon two pieces of evidence (the time of receiving the light-beams, and the apparent distance to the source of the light-beams), while being forced to ignore a third piece of compelling evidence (the relative velocity of the recording devices in relation to the light-beams, which, if allowed to be entered as evidence, would provide circumstantial evidence about why the "front" light beam arrived first).

 

[Ibix]

the relative velocity of the recording devices in relation to the light-beams

The relative velocity of the light beams measured by the recording device is c by definition! That's the whole point. Einstein just works out the implications of that definition.

 

[Dale]

Givens: The light moves at light speed. The passenger receives the information about event B before receiving the information about event A.

You forgot one important given. The distance between the passenger and event A is equal to the distance between the passenger and event B.

With that additional given the conclusion logically follows irrespective of any Doppler shift.

 

[Ian432]

There's nothing that says it's relevant, either. If you think it is, it's up to you to demonstrate why. Just saying so isn't enough.

I have done so. To recap: it's relevant because of the point you made earlier--the time differential of the light beam arrivals is extremely small. The sophistication required to measure that differential would also allow sophistication sufficient to measure the light-shift in each of the beams. This is no different than pointing out that Maxwell's demon uses energy to open and shut the door.

the detected frequencies of the light flashes irrelevant to determining the times at which they happened.

My understanding is that the different frequencies of the light beams would suggest to the passenger that there are differences in the relative velocities of the light-sources and the passenger relative to one another, and that the passenger might in fact be moving toward one light-beam and away from the other light-beam. It would not lead the passenger to conclude anything definitive about the actual timing of the events that gave rise to the beams, but it would allow the passenger at least to question his own perceived timing of origin. The passenger would therefore not be forced to "conclude" anything about the timing.

 

[Ian432]

The relative velocity of the light beams measured by the recording device is c by definition! That's the whole point. Einstein just works out the implications of that definition.

You forgot one important given. The distance between the passenger and event A is equal to the distance between the passenger and event B.
With that additional given the conclusion logically follows irrespective of any Doppler shift.

OK, thanks, these are good responses. I will think about these for a while and come back to this thread later.

 

[Ian432]

Sorry, but even after reviewing these most recent posts, the challenge still persists.

1. To Ibix's point: I was not challenging the relative velocity of the light-beams themselves in relation to the recording device. I was only pointing out the relative velocity of the recording device relative to the origin-loci (space coordinates) of events A and B--a difference in velocity that accounts for the color-shift differences in those light-beams, as perceived by the passenger. I fully accede the point that the speed of light is a constant. But this does not eliminate the perturbing additional variable of the color-shift difference in the light beams. All of this is fully in line with the original stipulations of the original thought-experiment.

2. To Dale's point about distances, if we call the passenger's position "P," then the distance P-A and P-B are equal only at the time of the "actual" events A and B. By the time the light-beams hit the passenger, P-B is smaller than P-A. This is by definition.

To both of your points: Given that light has a constant speed, I do understand why a reasonable passenger would arise at the conclusion that event B happened before event A, if the passenger only understood that he was "equidistant" from both events, and that a beam of light from B reached him before a beam of light from A. But if the passenger was also given information about the significant color-shift differences in the light beams, he would be almost compelled to conjecture about his movement relative to the points of origin of the light-beams between the time that the events occurred and the time that the light-beams arrived at his detector, and to speculate about whether his own movement was what accounted for the difference in arrival-time of the light-beams.

One question that no one has raised yet is: "can the interpretation of light-frequencies reasonably be considered as part of the experience within the inertial frame of the passenger"? However, I believe the answer is "yes." If we already accept that the passenger knows (1) when the light hits him and (2) when it hits him, we can certainly accept that he also knows (3) the frequency of such light.

 

[PeterDonis]

I have done so.

No, you haven't. You've explained why the frequency shift would be measurable. You have not explained why the frequency shift would affect the speed of the light beams, or how any such effect could possibly be consistent with the assumptions that Einstein made in constructing the thought experiment.

This is no different than pointing out that Maxwell's demon uses energy to open and shut the door.

It is quite different. The point in the Maxwell's Demon case is that the demon's entropy (not energy) must be included in the analysis. But there is no analogous point here.

My understanding is that the different frequencies of the light beams would suggest to the passenger that there are differences in the relative velocities of the light-sources and the passenger relative to one another

Of course. Nobody is disputing that. But that is not sufficient to support the claim you are making.

the passenger might in fact be moving toward one light-beam and away from the other light-beam

No. Motion relative to the light beam is not the same as motion relative to the source of the light beam. In fact, if the speed of the light beam is constant for all observers, regardless of the motion of the source, then the concept of "motion relative to the light beam" does not even make sense, since any such "motion" would have to change the speed of the light beam, and that is impossible by hypothesis.

 

[Nugatory]

2. To Dale's point about distances, if we call the passenger's position "P," then the distance P-A and P-B are equal only at the time of the "actual" events A and B. By the time the light-beams hit the passenger, P-B is smaller than P-A. This is by definition.

The distance the light travels is the distance between where it is emitted (the position of A or B at the moment of emission) and where it is received (the position of the passenger's eyes at the moment of reception). The movement of A or B before and after the moment of emission is irrelevant to the calculation of that distance, as is the movement the passenger before and after the moment of reception. Once you have that distance, you divide it the speed of light to get the time that it took the light to cover that distance.

Einstein does assume (and discusses the assumption earlier) that if the light took x seconds to cover the distance, and reached the passenger's eyes at time T, then it was emitted at time T-x.

 

[PeterDonis]

I was only pointing out the relative velocity of the recording device relative to the origin-loci (space coordinates) of events A and B

There is no such thing. An object can have a velocity relative to another object, but it cannot have a velocity relative to a space coordinate, because "a space coordinate" is frame-dependent. When each light flash is emitted, two objects are spatially co-located: the point on the embankment where the lightning strikes (this is what you are considering the "source" of the light beam) and the corresponding end of the train (front for B, rear for A). Each of these are objects--one is moving relative to the train passenger and one is not. But neither of them are a "space coordinate".

if we call the passenger's position "P," then the distance P-A and P-B are equal only at the time of the "actual" events A and B. By the time the light-beams hit the passenger, P-B is smaller than P-A. This is by definition.

This is true, but it does not affect the calculation of the time it takes the light beam to travel. Once again, the relevant speed is the speed of the light beam relative to the receiver, not the speed of the light source relative to the receiver. The latter cannot possibly affect the time it takes the light to travel. Only the former can, and you have already admitted that the speed of the light beam is $c$ relative to the receiver.

he would be almost compelled to conjecture about his movement relative to the points of origin of the light-beams between the time that the events occurred and the time that the light-beams arrived at his detector, and to speculate about whether his own movement was what accounted for the difference in arrival-time of the light-beams.

No, he wouldn't, because the motion of the light sources relative to the receiver after the beams are emitted can't possibly affect the time the beams take to travel. Only the motion of the light beams themselves relative to the receiver can. See above.

 

[Dale]

2. To Dale's point about distances, if we call the passenger's position "P," then the distance P-A and P-B are equal only at the time of the "actual" events A and B. By the time the light-beams hit the passenger, P-B is smaller than P-A. This is by definition.

This doesn't make any sense. A and B are events, not objects. The distance at a different time is not even defined.

This is not very difficult to work out mathematically.

Given event A which occurs at time and location $(t_A,x_A)$ , then the second postulate says that light emitted from A will be received at all the events $(t,x)$ such that $(x-x_A)^2=c^2 (t-t_A)^2$. In a spacetime diagram this figure is called a lightcone. If you have an inertial observer, O, then their worldline can be written $x=v_O t+d_O$. With very little effort you can find the intersection of any given worldline and any given light cone, specifically if $d_O=0$ and R is the event of the intersection then $(t_R,x_R)=\left(\frac{c t_A+x_A}{c+v_O},\frac{v_O(c t_A+x_A)}{c+v_O}\right)$.

So, if you are given the second postulate and $\{t_R,x_A,v_O\}$ then you can uniquely solve for $t_A$. Since the solution is unique you are indeed "forced to conclude" the value of $t_A$

The first postulate says that all of these equations hold in every frame. By simply applying these equations for two emission events and two observers under the conditions given for the problem you determine that the two observers will disagree whether or not the emission events were simultaneous.

 

[Ian432]

No, he wouldn't, because the motion of the light sources relative to the receiver after the beams are emitted can't possibly affect the time the beams take to travel. Only the motion of the light beams themselves relative to the receiver can. See above.

1. By definition, the motion of the receiver relative to the light sources after the beams are emitted directly affects the sequence with which the beams hit the passenger (B first, A second). I think we agree on that.
2. I am not claiming anything about the time it took the beams to travel. I am making a very limited claim about what the passenger is forced to conclude, or not.

There is no such thing. An object can have a velocity relative to another object, but it cannot have a velocity relative to a space coordinate, because "a space coordinate" is frame-dependent. When each light flash is emitted, two objects are spatially co-located: the point on the embankment where the lightning strikes (this is what you are considering the "source" of the light beam) and the corresponding end of the train (front for B, rear for A). Each of these are objects--one is moving relative to the train passenger and one is not. But neither of them are a "space coordinate".

This is perplexing, to be sure, and it probably requires me to restate things. The passenger is not moving relative to the front or rear of the train. The passenger is moving relative to points B and A on the embankment, but the passenger does not perceive those points. Instead, the passenger perceives only photons arising from two events. Those two events occurred at the moment that the front and rear of the train were aligned with points B and A on the embankment. The photons that arose from event B arrive at the passenger's receiver before the photons that arise from event A, because the passenger is moving toward... the photons that were emitted from event B, and away from the photons that were emitted from event A. The passenger notes that the frequencies of the photons from B are higher than the frequencies of the photons from A. The passenger deduces that he could be moving toward the B photons and away from the A photons.

When I say that the passenger is moving "toward the photons," I mean "the passenger is moving with a velocity directly opposite to the velocity of the B photons, and directly in line with the velocity of the A photons." That should help clarify things.

The distance the light travels is the distance between where it is emitted (the position of A or B at the moment of emission) and where it is received (the position of the passenger's eyes at the moment of reception). The movement of A or B before and after the moment of emission is irrelevant to the calculation of that distance, as is the movement the passenger before and after the moment of reception. Once you have that distance, you divide it the speed of light to get the time that it took the light to cover that distance.

I believe the above reply to PeterDonis is sufficient but please let me know if not. My understanding is that the movement of the passenger in relation to points B and A on the embankment is actually critical to this entire discussion as it is the origin of the difference in arrival-time of the light beams at the passenger's receiver.