What Are the Paradoxes of the Light Clock Problem in Special Relativity?

[bloby]

The OP's language in presenting the paradox is similar to that found here. I'm guessing the OP has read this... others might do so as well to get a sense of the proposed problem.

#28 Has anyone read it?

I read the beginning of this document. My point of view:

"my long paper on Special Relativity" ...didn't read it, will not read it.
"this diagram creates a false visualization" ...maybe.
"A light clock works by emitting a light ray. This ray reflects from a mirror opposite the clock and returns.
One round-trip of the light is a tick of the clock." ...No the light clock is the emitter/reciver + the mirror.
"The diagram is meant to be a visualization of what a distant observer would see." ...No, the clock could fit
in your pocket, it would even be better if it does. It is meant to (visualize)describe the clock of a moving
observer, that is the clock he has pulled out from his pocket, seen by you who see this clock moving with velocity
v.
"The diagram must be from the point of view of a distant observer, since a local observer would not see the clock
moving." ...? Why?

 

[PeterDonis]

So please, given primed distances and times, taking the observer to be at A, a beam moving from A to B, and a signal is sent from B to A at the time of reflection (T'=1s), how would you solve? You know the distance between A and B is h' (1.12). What is the observed time for event B in the unprimed frame?

You still haven't answered the questions I asked in post #78. Is the motion of the light clock perpendicular to the direction the light pulse goes, or parallel to it? Are there two light pulses, or only one? Nobody can "solve" anything until we know what the scenario is.

 

[Dale]

Altergnostic, one further brief explanation why your premise is false. If an object moves along some arbitrary path r(t) in an inertial frame and at some time t emits a light signal towards the observer at the origin who receives the signal at a time T, then the observer can write:
r(t)^2=c^2 (T-t)^2
Which is one equation in one unknown and has a single root where T>t. So we can always obtain t given T, and so the delay from T to t is immaterial.

 

[altergnostic]

Since you agree that the manner of transmitting the data to the observer does not affect the operation of the clock, then logically you must conclude that the operation of the clock can be analyzed without considering the manner of transmission. If X doesn't affect Y then Y can be analyzed without considering X.

This refutes your premise.

EDIT: it is, of course, possible to take the results of the analysis of the clock and then use that to analyze the reception of signals by the observer. But it is not necessary in the clock analysis, and the clock analysis must be done before the signal analysis.

I'm not sure I follow your reasoning here. What is it that you call clock analysis and signal analysis precisely? If I understand you correctly, isn't the clock analysis simply the primed diagram? And if so, don't you agree that that the observer in the unprimed frame won't agree with that analysis?

Anyway, the maner of transmission diesn't affect the operation of the clock itself, it alters how the clock will seem to operate from a moving frame. It is just an apparent effect.

 

[altergnostic]

...Did your calculation test your claim that it (in fact the Lorentz transformation set) doesn't work?

That is similar to an accelerated electron, which is much more complex than the simple light ray problem that we are discussing - except if we approximate it with Newtonian mechanics. Light is much easier to calculate, as we can simply use c instead of a to-be-solved V. It only distracts from the topic as long as your topic problem is not solved.

The observer has nothing to with it, as explained by me and others. Instead you can have detectors, rulers and clocks everywhere you like. If I correctly recall, this whole topic came from your claim that it is impossible to detect (x,y,z,t) of light in transit, because if we could (but we can, as I showed), this would according to you cause a self-contradiction of SR. We all agree that there is no problem if there only is a single detector, so why would you discuss that?? Did you try to show that self-contradiction with a calculation based on my detailed scenario, and if so, did PeterDonis' answer suffice to show that it is no problem?

Yes, it showed that the Lorentz transformations are not useful here, which is not a big deal since it is already accepted that you can't use gamma to transform a body moving at the speed of light wrt an observer.

I don't see how replacing the beam with a slower moving bullet is so complex at all. It is the same analysis, with different primed time values, nothing more.

And Peter's analysis is incorrect because he used the speed of the ship while looking for numbers for the beam. By applying a spatial rotation on the observer's frame, the x-axis should be aligned with h'. The main problem here us that everyone is simply assuming that the beam will be observed to move at c from the moving frame's point of view, but Peter is the only
who actually tried to analyse the setup I proposed.

The basic contradiction with your proposed scenario is that you are using several observers to track the beam while I'm discussing how would the beem look like for a single observer. And as you are assuming that it would seem to move at c and refuse to follow my analysis I proposed you replace the beam with a slower projectile to see how that alters the analysis (because it doesn't), but you refused to look into it also.

 

[altergnostic]

You still haven't answered the questions I asked in post #78. Is the motion of the light clock perpendicular to the direction the light pulse goes, or parallel to it? Are there two light pulses, or only one? Nobody can "solve" anything until we know what the scenario is.

I thought this was already clear from the diagram but I'm glad to clarify. The clock moves perpendicular to the direction of the light pulse (the line drawn from one mirror to another in the rest frame). There's a light pulse going from A to B, which we have been calling the beam and is the pulse that makes the clock tick by moving between the mirrors, and there's another pulse from point B to the origin to communicate the time of the reflection event to the observer, since he can't see the beam in this setup. (harrylin has been proposing scattering so that the observer can see the beam also, so you can replace the signal with the scattered light coming from B, but it works just the same).
And since we are looking for numbers for the beam we really need the spatial rotation so that the observer's x-axis is in line with AB.

 

[altergnostic]

Altergnostic, one further brief explanation why your premise is false. If an object moves along some arbitrary path r(t) in an inertial frame and at some time t emits a light signal towards the observer at the origin who receives the signal at a time T, then the observer can write:
r(t)^2=c^2 (T-t)^2
Which is one equation in one unknown and has a single root where T>t. So we can always obtain t given T, and so the delay from T to t is immaterial.

I agree. We are given t'B=1s and we are looking for T. Place AB=h'=1.12 as your arbitrary direction, find T with a given t'B=1s. That's the time the observer will see the object reach B, and the "object" is the beam. What is the apparent velocity?

 

[PeterDonis]

The clock moves perpendicular to the direction of the light pulse (the line drawn from one mirror to another in the rest frame).

Ok, good.

There's a light pulse going from A to B, which we have been calling the beam and is the pulse that makes the clock tick by moving between the mirrors, and there's another pulse from point B to the origin to communicate the time of the reflection event to the observer, since he can't see the beam in this setup.

Ok, this was the part I hadn't understood. Now it's clear. See below for revised analysis.

And since we are looking for numbers for the beam we really need the spatial rotation so that the observer's x-axis is in line with AB.

It doesn't matter whether it's the x or the y axis; I had picked the x-axis for the direction of motion of the light clock as a whole since that's a common convention in SR problems. To humor you I'll re-do my analysis with the light pulses moving in the x-direction and the light clock as a whole moving in the y direction relative to the observer.

We have two frames, the "observer" frame (the frame in which the observer is at rest), which will be the unprimed frame, and the "clock" frame (the frame in which the clock is at rest), which will be the primed frame. The relative velocity between the two frames is v = 0.5; the light clock is moving in the positive y-direction in the observer frame (and therefore the observer is moving in the negative y-direction in the clock frame). The gamma factor associated with this v is 1.16 (approximately). So the transformation equations are:

Unprimed to Primed Frame

t' = 1.16 ( t - 0.5 y )
x' = x
y' = 1.16 ( y - 0.5 t )

Primed to Unrimed Frame

t = 1.16 ( t' + 0.5 y' )
x = x'
y = 1.16 ( y' + 0.5 t' )

We have six events of interest (two pairs of events occur at the same point in spacetime and so have identical coordinates, in either frame):

D0 - The light clock source emits a light pulse towards the mirror.

A0 - The observer is co-located with the light clock source at the instant that the pulse is emitted. Thus, events A0 and D0 happen at identical points in spacetime. This point is taken to be the common origin of both frames (moving the origin elsewhere would just add a bunch of constant offsets in all the formulas, making the math more complicated without changing any of the results).

B1a - The light pulse reflects off the mirror.

B1b - A light signal is emitted by the mirror back towards the observer, carrying the information that the light pulse has struck the mirror. Events B1a and B1b happen at identical points in spacetime.

A2 - The observer receives the light signal emitted from event B1b.

D2 - The light clock detector (which is co-located with the source) receives the light pulse that was reflected off the mirror.

We know that the spatial distance between the light clock source/detector and the mirror, in the clock frame, is 1. This, combined with the information that events A0/D0 are at the origin, fixes the following coordinates (primes on the event labels denote coordinates in the primed frame):

A0 = A0' = D0 = D0' = (0, 0, 0)

B1a' = B1b' = (1, 1, 0)

D2' = (2, 0, 0)

Simple application of the transformation equations above gives the unprimed coordinates of B1a/B1b and D2:

B1a = B1b = (1.16, 1, 0.58)

D2 = (2.32, 0, 1.16)

All of this is the same as I posted previously, just with the x and y coordinates switched, since you prefer to have the x-axis oriented in the direction the light pulse travels.

It only remains to calculate the coordinates of event A2. It is easiest to do this in the unprimed frame, since the observer is at rest at the spatial origin in this frame. Therefore, a light pulse emitted towards the observer from event B1b has to travel from spatial point (1, 0.58) to spatial point (0, 0). A light pulse's worldline must have a zero spacetime interval, so the elapsed time in the unprimed frame must satisfy the equation:

\Delta t^2 - \Delta x^2 - \Delta y^2 = 0

or

\Delta t = \sqrt{ \Delta x^2 + \Delta y^2 } = \sqrt{ 1 + (0.58)^2 } = 1.16

Which of course is just the gamma factor. We could have seen this directly by realizing that the time elapsed in the unprimed frame from event A0 to event B1a must be the same as the time elapsed in the unprimed frame from event B1b to event A2; but I wanted to calculate it explicitly to show how everything fits together. [Edit: In other words, I wanted to show that we don't have to *assume* that the light signal travels at c, which is what "zero spacetime interval" means. We can *prove* that it must, by comparing the result we get from the direct method I just gave, with the result we get from the interval calculation I just gave, and seeing that they are the same.]

So we have the unprimed coordinates for event A2:

A2 = (2.32, 0, 0)

Again, a simple calculation using the above transformation formula gives:

A2' = (2.68, 0, -1.34)

What is this telling us? Well, the observer is moving in the negative y-direction in the primed (clock) frame, so the y-coordinate of event A2 is negative in this frame. The time in this frame is *larger* than that in the unprimed frame because it takes extra time for the light signal to catch up to the observer since the observer is moving away from it. We also expect this from the relativity of simultaneity: events A2 and D2 are simultaneous in the unprimed frame (the reason why should be obvious from the discussion I gave above), so they won't be simultaneous in the primed frame; the event that is in the opposite direction from the relative motion (A2 in this case) will occur later in the primed frame.

Once again, this is all straightforward analysis and I don't see a paradox anywhere; it just requires being careful about defining events and frames. I'll put any responses I have to other comments you've made (now that I know we are both talking about the same scenario) in a separate post.

Edit: I suppose I should add that it's easy to confirm that *all* of the light pulses travel at c, in both frames, from the coordinates that I gave above.

 

[PeterDonis]

AC=x't=0.5 lightseconds = distance traveled by the light clock in the x direction between reflection events T_A and T_B

This is not correct; what you are trying to calculate here is the unprimed coordinates of an event that I didn't list in my analysis:

D1 - The event at which the light clock source/detector is located at the instant the light pulse reflects off the mirror, in the unprimed frame.

The unprimed coordinates of this event are obvious from my analysis:

D1 = (1.16, 0, 0.58)

i.e., the same t and y coordinates as B1a/B1b, just x = 0 instead of x = 1. This means, of course, that the distance AC is 0.58, *not* 0.5, in the unprimed frame. And, of course, the velocity of the light clock source/detector, which is just the y-coordinate of D1 divided by the t-coordinate, is 0.5, as it should be.

The transformation equations give for the primed coordinates of event D1:

D1' = (1, 0, 0)

which is obviously what we expect given the primed coordinates of B1a/B1b.

AB = h' = 1.12
T_B = T'_B + h = 2.12s
V_AB = h'/T_B = .53c

These are incorrect as well, as you can see from my analysis:

- The distance AB, in the unprimed frame, is equal to the gamma factor, 1.16; I don't know where you got 1.12 from.

- The time T_B is the t-coordinate of event A2 in the unprimed frame, which is 2.32, i.e., twice the gamma factor; it is *not* obtained by adding T'_B and h, which makes no sense since you are adding quantities from different frames.

- The relative velocity V_AB, which as I understand it is supposed to be the velocity of the light clock as calculated by the observer, is, as I showed above, 0.5 no matter how you calculate it, as long as it's a valid calculation. Even with a correct value for h' and T'_B, dividing them to get a relative velocity is not valid, and I don't understand why you think it makes any sense.

 

[harrylin]

Yes, it showed that the Lorentz transformations are not useful here

Of course the LT or their equivalent are useful here; it's what the light clock example makes clear. You can use Lorentz contraction and clock synchronization to find time dilation with it.

, which is not a big deal since it is already accepted that you can't use gamma to transform a body moving at the speed of light wrt an observer.

There is no body moving at the speed of light in this example...

I don't see how replacing the beam with a slower moving bullet is so complex at all. It is the same analysis, with different primed time values, nothing more.

The velocity transformation equation for bullets is more complex than putting c, especially at an angle. See section 5 of http://www.fourmilab.ch/etexts/einstein/specrel/www/

And Peter's analysis is incorrect because he used the speed of the ship while looking for numbers for the beam. By applying a spatial rotation on the observer's frame By applying a spatial rotation on the observer's frame, the x-axis should be aligned with h'.

Apparently he has corrected it now. However, I missed why anyone would need a spatial rotation - S and S' are moving in parallel and the light ray reflects, it doesn't rotate.

The main problem here us that everyone is simply assuming that the beam will be observed to move at c from the moving frame's point of view, but Peter is the only
who actually tried to analyse the setup I proposed.

See again post #71. Is it needed? Your set-up is supposedly the standard one of textbooks, which everyone including myself analysed (but with pen and paper many years ago).

The basic contradiction with your proposed scenario is that you are using several observers to track the beam while I'm discussing how would the beem look like for a single observer. And as you are assuming that it would seem to move at c and refuse to follow my analysis I proposed you replace the beam with a slower projectile to see how that alters the analysis (because it doesn't), but you refused to look into it also.

Apparently you now mean with "observer", not a non-local observer of instruments but a light detector with a clock next to it (right?). This topic started as follow-up of the other thread with "You can't detect light at a distance. [..] You can't be aware of light moving in any direction other than straight into your eyes (or detectors). So how can a non-local observer see those light rays?" I think (but I did not see you acknowledge it) that that problem has now been solved. Correct? Also, as far as I can see all your questions here have been answered, which were:

- How can we correctly diagram undetected light like that? How does the emitter adjust the angle of emission?
- Conversely, how can light be emitted at an angle if it's speed is not affected by the motion of the emitter?
- Shouldn't we apply SR transforms primarily with light that is actually observed and than use that information to diagram the interior of the light clock?
- Isn't diagraming the light clock in the moving frame like that illegal? Aren't those vectors purely imaginary?

And so you moved on to a different issue, which is, if understand you correctly, that according to you the Lorentz transformation don't work. Once more: if more people need to search for the error in your calculations, just ask!

 

[altergnostic]

Peter, you made what I believe to be the same mistake I pointed out earlier. You again plugged in the speed of the light clock into gamma. I insist that you can't do it that way. We are not looking for numbers for the light clock, we are looking for numbers for the beam. My analysis is useless if you start assuming the beam going at c frim the beginning. As I said, the speed of the light clock only helps to find the opposite side AC. If you really want to humor me, place the x-axis aling with AB. Imagine the whole light clock as invisible and all you can see is the signal coming from B after a while or something. We actually don't even need the speed of the light clock at all, since we have km marks all iver the place and we know the primed time. Take the primed time for event B (1s) and the given distance AB (1.12).

And the relative velocity Vab is the number we are seeking: the observed speed of the beam.

 

[Dale]

I'm not sure I follow your reasoning here. What is it that you call clock analysis and signal analysis precisely?

The clock analysis is the analysis of the clock itself, how its mechanism functions in any reference frame. It is, by your own admission, unaffected by how signals about its operation are transmitted to any observer, and so the analysis of those signals is unnecessary for the analysis of the clock mechanism, contrary to your premise.

 

[Dale]

I agree. We are given t'B=1s and we are looking for T. Place AB=h'=1.12 as your arbitrary direction, find T with a given t'B=1s. That's the time the observer will see the object reach B, and the "object" is the beam. What is the apparent velocity?

Why are we looking for T? You already agreed that it cannot affect the operation of the clock in any way. It is irrelevant.

 

[altergnostic]

There is no body moving at the speed of light in this example...

But there is! The beam is our object of analysis here, and it is going at c in the primed frame. We are not seeing it directly, so we need the signal from event B to know its speed.

The velocity transformation equation for bullets is more complex than putting c, especially at an angle.

Do it like the train and embankment problem and simply ditch the angle. You are given the primed distance between AB and the primed time for event B, you don't need anything else.

Apparently you now mean with "observer", not a non-local observer of instruments but a light detector with a clock next to it (right?). This topic started as follow-up of the other thread with "You can't detect light at a distance. [..] You can't be aware of light moving in any direction other than straight into your eyes (or detectors). So how can a non-local observer see those light rays?" I think (but I did not see you acknowledge it) that that problem has now been solved. Correct?

I aknowledge that you found a setup that allows us to observe the beam, and you did it only by sending light from the to the detector, so I give you that. But that also proves what I have been saying, that you can't simply take local or primed numbers for the beam and use them as unprimed data. The beam has since acted like an object subject to relative velocities, it is just like a moving train that we see with light reflected from it. The light between the train and embankment acts the same as light between beam and detector, and it is this light that we directly observe that we know moves at c. It is this light that brings us the coordinates for the beam.

- How can we correctly diagram undetected light like that? How does the emitter adjust the angle of emission?

Answered: with a new setup. Current setup is incomplete.

- Conversely, how can light be emitted at an angle if it's speed is not affected by the motion of the emitter?

Actually this hasn't been answered at all, but I already concluded that light that is moving in any direction other than directly at us doesn't have to follow any constancy, since we can't be considered neither source nor observer in that case.

- Shouldn't we apply SR transforms primarily with light that is actually observed and than use that information to diagram the interior of the light clock?

Clearly the answer is yes, otherwise we wouldn't have even mentioned scattering or signaling.

- Isn't diagraming the light clock in the moving frame like that illegal? Aren't those vectors purely imaginary?

You showed it can be done with a rigorous setup, but I still contend the numbers are different from current diagrams.

And so you moved on to a different issue, which is, if understand you correctly, that according to you the Lorentz transformation don't work. Once more: if more people need to search for the error in your calculations, just ask!

That is a conclusion I came to only in recent posts (I actually learned a lot from your scattering setup) so it is not like I changed the subject, I only went where the discussion led me. You refuse to do the analysis with the set of givens I proposed, which are very realistic. And now you claim that substituting the beam with a prijectile is overly complex when I showed that it isn't, you don't even have to take any angle into account since I gave you the distance AB measured locally so you can put the x-axis directly in line with it, and I also gave primed times so you can calculate the speed.

 

[Dale]

But there is! The beam is our object of analysis here, and it is going at c in the primed frame. We are not seeing it directly, so we need the signal from event B to know its speed.

Besides the funny self-contradiction, it is one of the postulates of SR that it is going at c in all frames. We know its speed according to SR, and alternative theories that disagree with SR are not permitted here.

 

[harrylin]

OK, I think that we are getting somewhere:

[little mutual misunderstanding ...]
Do it like the train and embankment problem and simply ditch the angle. You are given the primed distance between AB and the primed time for event B, you don't need anything else.

A bullet doesn't have the same speed in both frames; I'll stick to the topic instead, which allows much better clarity.

[..] The light between the train and embankment acts the same as light between beam and detector, and it is this light that we directly observe that we know moves at c. It is this light that brings us the coordinates for the beam.

I already illustrated that it is not necessarily to use that light for the time data. So, if you mean that it is light or radio waves or other means that brings us the clock synchronisation, then I completely agree.

[..] Actually this hasn't been answered at all, but I already concluded that light that is moving in any direction other than directly at us doesn't have to follow any constancy, since we can't be considered neither source nor observer in that case.

In post #8 I gave you the answer (in the link). However, you did not comment on it.
[EDIT: In fact, my answer took care of the spatial rotation misconception]

[..]You showed it can be done with a rigorous setup, but I still contend the numbers are different from current diagrams.

Of course, we narrowed down the problem as stemming from a calculation error - either made by "all" textbooks and students, or made by you. My purpose was (and still is) to get rid of the bug, after which you can be an asset to this forum - and you will feel better.

[..]I actually learned a lot from your scattering setup

Thanks - such feedback is helpful to remain motivated with this voluntary work!

so it is not like I changed the subject, I only went where the discussion led me. [..]

OK. I will next look into your calculation example with the light beams, and give my analysis.

 

[altergnostic]

The clock analysis is the analysis of the clock itself, how its mechanism functions in any reference frame. It is, by your own admission, unaffected by how signals about its operation are transmitted to any observer, and so the analysis of those signals is unnecessary for the analysis of the clock mechanism, contrary to your premise.

My premise has nothing to do with the operation of the clock, it has to do with apparent effects. Like doppler and such things.

Why are we looking for T? You already agreed that it cannot affect the operation of the clock in any way. It is irrelevant.

Because T is the time shown in the unprimed frame when the observer receives the signal from B, which is different from the primed time, and which is the time that determines the perceived speed of the beam from the observer at the origin. It is the time the beam seems to take to cross the distance AB. It is far from irrelevant.

Besides the funny self-contradiction, it is one of the postulates of SR that it is going at c in all frames. We know its speed according to SR, and alternative theories that disagree with SR are not permitted here.

There's no contradiction whatsoever. Observed/detected light is going c in all frames. Einstein never once thought about how light would look like at a distance since it seemed ridiculous to do so, since we can't have any knowledge of undetected light. He and everyone else always addressed this issue by placing another observer in the path of light, but that says nothing about the observed behavior of light receding from the observer (I really believe that taking harrylin setup of scattering light from the beam's path plus my suggestion that we take this scattered light and bring it to one single point - the observer - is a novelty).
How do you measure the speed of the train? You take the distance it covers over the time it takes to cover it. This velocity is not the same as marked on the trains own speedometer, for instance. You are on the train and it marks .5c on the speedometer. You cross 1 light second, from your perspective, in 2 seconds.
An observer at the embankment would see the train cross that distance in 2 seconds + the time it takes light to reach back to him from that distance (which would be the time marked on the observers watch at the moment he sees the train at that distance). The speed will seem to be slower relative to the speed as seen on the onboard speedometer!

I have a diagram on this from a discussion I had a few months ago:

But I'm afraid this will divert us from the topic, maybe this needs a thread itself.

 

[PeterDonis]

Apparently he has corrected it now.

My previous analysis was correct, it just didn't include the extra light signal from the mirror back to the observer. Altergnostic apparently still thinks I made a mistake (which I didn't); I'll address his issue in a separate post.

However, I missed why anyone would need a spatial rotation - S and S' are moving in parallel and the light ray reflects, it doesn't rotate.

You are correct that it doesn't matter how the x and y axes are oriented; either choice leads to the same physics. I switched it in my revised analysis only because I didn't see the point of getting into a protracted wrangle about it when I can do the analysis with altergnostic's preferred axis orientation and still show that there is no problem.

 

[altergnostic]

A bullet doesn't have the same speed in both frames; I'll stick to the topic instead, which allows much better clarity.

This is precisely my point! The same is true for the beam!

I already illustrated that it is not necessarily to use that light for the time data. So, if you mean that it is light or radio waves or other means that brings us the clock synchronisation, then I completely agree.

That's almost what I mean. I take that the sync has been done previously (at the origin) and the light actually brings time information, in this particular case. We have, at the origin, the same time marked both on the observer's watch as the light clock's time. The light clock departs. The observer will see the next second on the light clock at the moment he sees the signal, but the time marked on his watch when the signal reaches him is no t'B=1s, it is t'B + the time it takes light to cross BA.

In post #8 I gave you the answer (in the link). However, you did not comment on it.
[EDIT: In fact, my answer took care of the spatial rotation misconception]

I guess I forgot about that post. But there's no disagreement here.

Of course, we narrowed down the problem as stemming from a calculation error - either made by "all" textbooks and students, or made by you. My purpose was (and still is) to get rid of the bug, after which you can be an asset to this forum - and you will feel better.

You are welcome! That was ingenious indeed.

Thanks - such feedback is helpful to remain motivated with this voluntary work!

I started this post certain that the observer couldn't possibly trace the beam (and from current diagrams he really can't - you are the only one who made this possible so far in all the discussions I have participated, I really respect you for this). My last diagram was only possible because of this insight, I am only afraid that my conclusions are so far off current accepted diagrams that I won't get any credit for this. I don't care if I am wrong, really, I just want a really rigorous analysis, not just more of the same assumption that the beam must be traveling at the speed of light for this particular observer because it always travels at c for all observers - but I insist this is not your common observer, it is a non-observer, if you like. He is not observing the beam directly. The speed of light must apply to light that reaches him, since that is the light he detects going at c directly. The beam's velocity has to be calculated from that, just like you said above, the bullet has a different velocity in each frame, and so must this beam. I think this is even more consistent with such experiments than the opposite assumption that the beam must seem to move at c in this situation. Than we would have to explain why should the beam act differently from the bullet, since the setup is fundamentally the same.

OK. I will next look into your calculation example with the light beams, and give my analysis.

Thanks!

 

[Dale]

My premise has nothing to do with the operation of the clock,

You have explicitly stated multiple times that the standard analysis of the operation of a light clock is incorrect or incomplete. Do you now agree that the standard analysis is both a correct and complete analysis of the operation of the clock? If not, then please state exactly your objection to the standard analysis of the operation of the clock which you agree has nothing to do with any signals sent to any observers.

Note, the standard analysis only purports to be an analysis of the operation of the clock and not an analysis of the subsequent transmission of the results to any observer. Such considerations are irrelevant to the operation, as you have already agreed.

Observed/detected light is going c in all frames. Einstein never once thought about how light would look like at a distance since it seemed ridiculous to do so, since we can't have any knowledge of undetected light.

You are factually wrong on this count. The postulate of SR is that all light pulses travel at c, regardless of whether they are detected or not. Please read the postulates and note that they do not mention anything about detection. The postulate is that light travels at c, end of story, no distinction whatsoever between light going towards or away from any observer or about the detectability of the light. Get your facts straight.

 

[altergnostic]

My previous analysis was correct, it just didn't include the extra light signal from the mirror back to the observer. Altergnostic apparently still thinks I made a mistake (which I didn't); I'll address his issue in a separate post.
You are correct that it doesn't matter how the x and y axes are oriented; either choice leads to the same physics. I switched it in my revised analysis only because I didn't see the point of getting into a protracted wrangle about it when I can do the analysis with altergnostic's preferred axis orientation and still show that there is no problem.

Sorry Peter, but I have to insist that you reconsider this carefully.
I insist that you have to align the x-axis with AB and completely ditch the speed of the light clock in the former x axis, since it is actually causing a distraction here.
Please note that the observer at the origin doesn't even need to know that speed to calculate anything:
He is given the distance AB
He knows the primed times, since he has synchronized his watch with the light clock back at the origin. He knows that the reflection at B occurs at t'=1s.
He receives the signal event from B at T=t'+ the time it took for the light signal to cross BA and reach him - that is the time he actually sees the signal, and is the time he observes the beam to reach the point B.

From the given distances and the observed times he can only observe that the beam crossed AB in T seconds, and then subtract the time it takes light to cross BA to get the primed time - which will be 1s again.

This seems all extremely consistent, and in accordance with what we would expect to have by replacing the beam with a projectile. If a projectile hits B at t'=2s, the observer will observe it crossing AB at T=t' + the time it took light to cross BA, and then subtract the time it takes light to cross BA to get the primed time - which will be 2s again.

By this method, the primed velocity will make the primed distance smaller than AB (length contraction), and the calculated distance will become y', which again is fully consistent with the setup. Like so:
Observed speed of the beam = AB/T = 1.12/2.12 =0.528c
Transformed time of event B = t'= 1s

We know that light travels at c locally from many many experiments, so to get the distance traveled by light in the primed frame, we just have to figure out the distance light crossed in 1 second (the primed time) = 1lightsecond = y'

Do you see my reasoning? You may not agree with it for some reason, but it is very consistent, and all the numbers add up. I can't see any error on it, if you can, please do point it out, since I am not here to defend any preconceived opinion, I'm just going where logic and math has taken me and sharing this with you for open analysis.

And just as a footnote, I don't think this contradicts any postulates nor disproves or changes SR in any way. It is simply a setup that hasn't been thought of yet, as far as I know (and I have to share credit for this with harrylin, even if he disagrees with my analysis). Observed light still travels at c. It also travels at c relative to the source. It is undetected light, or indirectly observed light, that I conclude doesn't need to follow the constant, but this is not your ordinary light. It is not the light that enters any known equation as c. This is light moving at a distance. Not a single light equation takes c to be light at a distance, everytime you have c in an equation, that c is referring strictly to light that reaches the observer, which is the light you do transforms with.
That's the importance of the signal arriving from B: it is with this light that we observe the beam, and it is with this light that we must do transforms, just like in any other setup.
To give the speed of light both for this incoming light and for the beam, as seen in the unprimed frame, is similar to believing that the speed of the man walking inside a moving train is the same for both primed (the train) and unprimed (embankment) frames.
If you still think the beam will be observed to go at c from A to B as observed from the observer at the origin A, than you have to explain why wouldn't the speed of a projectile (in place of the the beam) be the same in the primed and unprimed frame as well.

Please consider this carefully and think about this with no prejudice (I mean no harm to SR )

 

[PeterDonis]

Peter, you made what I believe to be the same mistake I pointed out earlier. You again plugged in the speed of the light clock into gamma.

It isn't a mistake (if there's any mistake involved here it's yours, in not understanding how relative motion is modeled in SR).

We are not looking for numbers for the light clock, we are looking for numbers for the beam.

Yes, but in order to get "numbers for the beam", you have to model the motion of the light clock as well. Otherwise you can't enforce the constraint that the beam has to be co-located with particular parts of the light clock at particular events (the source/detector at events D0 and D2, and the mirror at event B1a). Without that constraint in place any analysis of the beam's motion is meaningless.

My analysis is useless if you start assuming the beam going at c frim the beginning.

I didn't *assume* that the beam was moving at c; I *proved* that it moves at c, by enforcing the constraint I referred to above. Please read my analysis again, carefully. What I did was compute the locations of the light clock's source/detector at events D0 and D2, and compute the location of the mirror at event B1a, using only facts about the light clock's motion relative to the observer. I then showed that the spacetime interval between events D0 and B1a, and between events B1a and D2, is of zero length; this proves that the light pulse travels at c between those pairs of events. I did this in both frames to confirm that the interval is invariant (as it should be).

I also, once I understood that there was a second light signal involved (from event B1b to A2), did the same kind of computation for that light signal: compute the location of the observer at event A2 (we already know the mirror's location at event B1b because event B1b is co-located with event B1a), and confirm that the spacetime interval between events B1b and A2 is of zero length, again in both frames.

If you really want to humor me, place the x-axis aling with AB.

Sure, by all means let's do the analysis as many ways as you like; the answer will remain the same.

We now have the x-axis oriented in such a way that the event coordinates in the unprimed frame are as follows:

A0 = D0 = (0, 0, 0)

B1a = B1b = (t1, x1, 0)

A2 = (t2, 0, 0)

D2 = (t2, x2, y2)

Our objective is to find t1, x1, t2, x2, and y2, and then compute the spacetime intervals (A0 to B1a), (B1b to A2), and (B1b to D2), and verify that they are all zero.

The first thing to note is that the time coordinates are unchanged from my previous analysis; i.e., we still have t1 = 1.16 and t2 = 2.32.

The second thing to note is that x1 must be given by the sum of the squares of the x and y coordinates for events B1a/B1b in my previous analysis (since it's the same spatial distance, we've just rotated the axes to put that distance all along one axis); i.e., we must have

x_1 = \sqrt{ 1 + ( 0.58 )^2 } = 1.16

The third thing to note is that the sum of the squares of x2 and y2 must equal the square of the y coordinate of event D2 in my previous analysis (again, the same spatial distance from the origin, just with the axes rotated); i.e., we must have

x_2^2 + y_2^2 = ( 1.16 )^2 = x_1^2

This tells us something very important: the distances AB, BD, and AD in your diagram are all *equal*. (We could have seen this directly by noting that the light clock, which moves half as fast as the light pulse, covers distance AD in the same time as the light pulse covers distance AB + BD; therefore AB + BD = 2AD, which combined with AB = BD gives AB = BD = AD.) This means that the angle between lines AB and AD is 60 degrees (since it's an angle of an equilateral triangle), so we must have

\frac{y_2}{x_2} = tan (60 deg) = \sqrt{3}

Substituting this into the equation above gives

x_2^2 + 3 x_2^2 = x_1^2

or

2 x_2 = x_1

Now we can confirm that the spacetime intervals I mentioned are zero. First, we have t1 = x1, so the interval (A0 to B1a) is obviously zero. Similarly, we have t2 - t1 = t1 = x1, so the interval (B1b to A2) is obviously zero. Finally, the interval (B1a to D2) is given by

(t_2 - t_1)^2 - (x_2 - x_1)^2 - (y_2 - 0)^2 = ( 1.16 )^2 - ( x_2^2 + y_2^2 ) - x_1^2 + 2 x_2 x_1 = ( 1.16 )^2 - ( 1.16 )^2 - x_1^2 + x_1^2 = 0

As I said, the answer remains the same.

Take the primed time for event B (1s) and the given distance AB (1.12).

These numbers are from different frames; trying to mix them in formulas is going to give you meaningless answers.

And the relative velocity Vab is the number we are seeking: the observed speed of the beam.

Oh, so by Vab you meant the speed of light, not the speed of the clock? Fine, but your calculation of it still makes no sense; once again, you can't mix numbers from different frames and expect to get meaningful answers. See above.

 

[PeterDonis]

I have a diagram on this from a discussion I had a few months ago

The second part of your diagram (the "TRAIN FRAME" part) is incorrect; in the train frame the blinker is motionless, so all of the space coordinates should be marked as zero.

But I'm afraid this will divert us from the topic, maybe this needs a thread itself.

Before starting one, I strongly advise you to review a good basic SR textbook, such as Taylor & Wheeler.

 

[altergnostic]

The second part of your diagram (the "TRAIN FRAME" part) is incorrect; in the train frame the blinker is motionless, so all of the space coordinates should be marked as zero.



Before starting one, I strongly advise you to review a good basic SR textbook, such as Taylor & Wheeler.

That is the train frame measuring its velocity against the ground, just like you do in your car. It is the speed marked on its apeedometer.

 

[PeterDonis]

Sorry Peter, but I have to insist that you reconsider this carefully.
I insist that you have to align the x-axis with AB and completely ditch the speed of the light clock in the former x axis, since it is actually causing a distraction here.

Done; see my previous post (actually the last but one from this one). The answer remains the same.

He is given the distance AB
He knows the primed times, since he has synchronized his watch with the light clock back at the origin.

But he can't use the primed times along with unprimed distances to calculate anything meaningful. To correctly calculate a speed you must use the distance and time from the same frame. The "distance AB" that he is given is in the unprimed frame. That distance is *not* perpendicular to the relative motion of the light clock and the observer in the unprimed frame (it is in the *primed* frame, but it is *not* in the unprimed frame--perhaps this is one of the points where you are confused). So the observer *can't* combine it with a time in the primed frame to get a meaningful answer, because of length contraction.

From the given distances and the observed times he can only observe that the beam crossed AB in T seconds, and then subtract the time it takes light to cross BA to get the primed time - which will be 1s again.

Nope, this mixes numbers from different frames again.

This seems all extremely consistent, and in accordance with what we would expect to have by replacing the beam with a projectile.

The velocity of an object moving at less than c *does* change when you change frames. The velocity of light does not. Please review a good basic relativity textbook, paying particular attention to the formula for relativistic velocity addition; you will note that that formula always gives c for the velocity of a light beam.

Do you see my reasoning? You may not agree with it for some reason, but it is very consistent, and all the numbers add up. I can't see any error on it, if you can, please do point it out, since I am not here to defend any preconceived opinion, I'm just going where logic and math has taken me and sharing this with you for open analysis.

Read my latest analysis again, carefully; also read my comments above, carefully. Your error is that you are mixing numbers from different frames and expecting them to give you meaningful answers. Also, you may be mistakenly assuming that the distance AB is the same in both frames; it's not, for the reason I gave above.

And just as a footnote, I don't think this contradicts any postulates nor disproves or changes SR in any way.

Then you don't understand SR very well.

Observed light still travels at c. It also travels at c relative to the source. It is undetected light, or indirectly observed light, that I conclude doesn't need to follow the constant, but this is not your ordinary light.

[rest of post snipped]

This is all confused. All light travels at c in SR, in any inertial frame. If you read my latest analysis, you will see that the light beam traveling from B1a to D2 travels at c, even though it is not "observed" (both events are "at a distance" from the observer).

 

[PeterDonis]

That is the train frame measuring its velocity against the ground, just like you do in your car. It is the speed marked on its apeedometer.

But the ground is moving in the opposite direction, and the blinker is not moving with it. A correct "TRAIN FRAME" diagram would have the blinker on the right, motionless, and points on ground moving to the left. Then you would have to calculate *which* points on the ground would be at which spatial coordinates in the train frame at which times.

A better tool for this job, IMO, would be a proper spacetime diagram, since that directly shows both space and time relationships and allows you to draw actual worldlines so you can see how they are related. If you aren't familiar with spacetime diagrams, I would recommend learning them; they really help to remove confusion for scenarios like this one.

 

[altergnostic]

But the ground is moving in the opposite direction, and the blinker is not moving with it. A correct "TRAIN FRAME" diagram would have the blinker on the right, motionless, and points on ground moving to the left. Then you would have to calculate *which* points on the ground would be at which spatial coordinates in the train frame at which times.

A better tool for this job, IMO, would be a proper spacetime diagram, since that directly shows both space and time relationships and allows you to draw actual worldlines so you can see how they are related. If you aren't familiar with spacetime diagrams, I would recommend learning them; they really help to remove confusion for scenarios like this one.

Agreed. But still it is not hard to see, from the bottom diagram, that the speed of the ground measured from the train (the speed on its speedometer, to simplify things) will not be the same speed as that observed by the man at the origin. This actually compromises gamma a bit, since it is hard to decide which velocity to plug into it.

But please, let's leave this be for now, since we will surely divert from the topic. If you feel this deserves attention and want to discuss it further, I will gladly start a thread on this, but let's please not digress here, since this thread is already dense enough as it is. Deal?

 

[Dale]

This is precisely my point! The same is true for the beam!

No, it isn't. The invariance of the speed of light is a postulate of SR. If that is precisely your point then your point is WRONG.

 

[PeterDonis]

Agreed. But still it is not hard to see, from the bottom diagram, that the speed of the ground measured from the train (the speed on its speedometer, to simplify things) will not be the same speed as that observed by the man at the origin.

You're going to have to explain how this is "not hard to see", because I don't see it. The only thing that changes about the velocity from frame to frame is the sign: in the ground frame, the train moves at +v, and in the train frame, the ground moves at -v.

I suspect you are confused because you think length contraction affects the velocity from frame to frame. It doesn't, because length contraction and time dilation *both* come into play, and their effects cancel when applied to the relative velocity. If you don't see how that works, read through my analysis again, carefully; for example, look at how the coordinates of events A2 and D2 transform between the primed and the unprimed frame, and note that the time and space coordinates change in concert to keep the relative velocity the same.

[Edit: To expand on this a little more, events D0 and D2 both lie on the worldline of the light clock source/detector, which is equivalent to the "train"; and events A0 and A2 both lie on the worldline of the observer, which is equivalent to the "ground". Events A0 and D0 are co-located at the origin, so their coordinates drop out of the analysis, and we can use the coordinates of event D2 in the unprimed frame, and A2 in the primed frame, to compute the relative velocity. In the unprimed frame, event D2 has coordinates (t2, x2, y2), and the relative velocity is given by sqrt(x2^2 + y2^2) / t2, which works out to 0.5. In the primed frame, event A2 has coordinates (t2', x2', y2'), and the relative velocity is given by sqrt(x'2^2 + y2'^2) / t2', which works out to 0.5. Then we just have to remember to set the sign appropriately, based on the sign of (x2, y2) or (x2', y2'), which gives v = +0.5 in the unprimed frame and v = -0.5 in the primed frame.]

But please, let's leave this be for now, since we will surely divert from the topic. If you feel this deserves attention and want to discuss it further, I will gladly start a thread on this, but let's please not digress here, since this thread is already dense enough as it is. Deal?

I don't see how starting another thread will help. This is an absolutely fundamental point that underlies the "problem" you posed in the OP, and all of my analysis, and indeed all of SR. If you think it's wrong, you basically think SR is wrong; and if you can't back your claim up cogently (which you haven't so far) and you won't abandon it, any further discussion of your claim is basically out of bounds here on PF, since we don't discuss personal speculations that contradict SR. Moving the discussion to another thread won't change that.

 

[phyti]

On the left, the drawing above the Ux axis is a space-time description of the 1-dimensional motion of observer A while monitoring his light clock.
The events (small circles) involve 2-dimensional motion of light in an arbitrary plane perpendicular to the Ux axis, labelled as Up. The Ut axis is only labelled in terms of t and gamma for the general case. Applying the drawing to the specific case v=.5c, gamma =1.155, d=1 ls, and t=1. Coordinate notation is (x,p,t).
Event sequence is:
U1(0,0,0) emit photon,
U2(.578,1.00,.578) reflect,
U3(1.16,0,1.16) detect.
On the right A's motion has been removed, and time dilation (.866) applied.
Event sequence is:
A1(0,0,0)
A2(0,1,1)
A3(0,0,2)
A is coincident with events 1 and 3, therefore values for event 2 are calculated based on additional information (symmetry and constant c).
U is coincident with event 1, therefore values for events 2 and 3 are calculated based on additional information (some form of light interaction between devices or observer A).