Length Contraction Paradox Scenario

[bmbuncher]

Hi! I have been pondering a scenario involving a paradox with length contraction. I brought it up with my physics professor, and I somewhat understand what is supposed to happen, but I'm still somewhat confused, so I was wondering if you could help me figure out what is going on.

In this scenario, a meter stick is traveling at relativistic speeds above the flat ground. On the ground, there is a setup where two laser tripwires (consisting of a laser emitter and receiver), each located one meter apart, are lying in the path of the meter stick. When both lasers are tripped simultaneously (as in the receivers simultaneously detect that there is no light entering it), an electrical signal is transmitted that turns on a light. So, from what I understand, when the meter stick is traveling extremely fast, it experiences length contraction (say, it is traveling fast enough to contract to 0.9 meters), and so it will never obstruct the path of both lasers simultaneously; therefore, in the frame of reference where the lasers are still and the meter stick is moving, no electronic pulse will be created, and so the light will not turn on. However, in the frame of reference that the lasers are moving and the meter stick is still, the distance between the lasers will contract (say, to 0.9 meters), so both lasers will be obstructed simultaneously, and so the light will turn on.

When I posed this to the professor, he said that the observation in both frames of reference should be analogous because causality must be preserved; therefore, the light will either turn on in both scenarios, or will not remain in both scenarios. He said that, while the length contraction will occur in both scenarios, the time dilation caused by special relativity should "balance this out" so that the same result occurs regardless of the frame of reference. However, I am still somewhat confused. In this scenario, how does time dilation cause the light to behave similarly in both frames of reference? Can anyone explain what is going on in this scenario that prevents causality from being violated? Thank you!

 

[phyzguy]

This is the well-known "Pole in the Barn Paradox". If you google search on this phrase you will find many explanations for how it is resolved by correctly considering the relativity of simultaneity. Here is one of them.

 

[PeterDonis]

When both lasers are tripped simultaneously (as in the receivers simultaneously detect that there is no light entering it), an electrical signal is transmitted that turns on a light.

This is one element of your scenario which is not really present in the standard "pole and barn" paradox. To help resolve your scenario, in addition to all the good info you will get by considering the pole and barn paradox, you might want to think carefully about exactly how the setup you describe here, that turns on a light if both lasers are tripped "simultaneously", will be realized physically. (Hint: "simultaneously" is frame-dependent; which frame is meant? And how do you physically detect "simultaneity" in that frame?)

 

[bmbuncher]

So if I'm understanding this correctly, simultaneous events are not necessarily simultaneous because the Lorentz transformations cause different locations experience different effects (I know that is somewhat unscientific, but I haven't spent a huge amount of time investigating the mathematics behind special relativity). Therefore, when both lasers would be covered because of the length contraction of the distance between the lasers, the event still may not occur because of a new requirement for simultaneity due to time dilation. Is this correct? Thank you!

 

[PeterDonis]

simultaneous events are not necessarily simultaneous because the Lorentz transformations cause different locations experience different effects

The correct statement is: pairs of events that are simultaneous in one frame are not simultaneous in other frames.

Therefore, when both lasers would be covered because of the length contraction of the distance between the lasers, the event still may not occur because of a new requirement for simultaneity due to time dilation. Is this correct?

Not really.

Let me repeat the suggestion I made in my previous post: consider the frame in which the laser triggers are at rest. You have said that when the two lasers are triggered "simultaneously", an electrical signal is sent that turns on a light. How, specifically, do you propose to do this? How is the signal sent? From where to where? And since there must actually be at least two signals (one from each laser), how do you determine that both are sent "simultaneously"? Think carefully about how you would actually design the apparatus to do all this.

 

[bmbuncher]

I see. My thought is that a receiver would trigger when the light is interrupted like a normal laser tripwire would activate. Both of these would feed into an AND gate, which would send an electrical pulse when both were active simultaneously; however, I do not know enough about electronics and logic gates to really understand what is going on in this scenario.

 

[PeterDonis]

My thought is that a receiver would trigger when the light is interrupted like a normal laser tripwire would activate.

Okay, but the light interruption is happening at two spatially separated locations, one meter apart. How can that trigger a single receiver? Wouldn't you need two of them?

Both of these would feed into an AND gate, which would send an electrical pulse when both were active simultaneously

Where is the AND gate located in space? How does the information about the light interruption get to where the AND gate is? How does the gate detect when both lights are interrupted "simultaneously"?

I do not know enough about electronics and logic gates to really understand what is going on in this scenario.

You don't need to; I'm not trying to get into the details of logic gates. An AND gate for combining the two signals is fine, once both signals are at the same spatial location; the question is how to get to that point. I'm trying to get you to think specifically about how each piece of the apparatus is laid out in space and how the signals required to make it work have to travel in space and time.

 

[bmbuncher]

I'm sorry, it would be two receivers. Each of these receivers would send an electrical signal into the AND gate at a separate location when the individual laser is covered. When both are covered, an electrical pulse would leave the AND gate and turn on the light. The AND gate and wires leading from the lasers are still in relation to the lasers. The AND gate would be created using electronic components, as detailed in the link below. Does that help at all?

http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/and.html

 

[PeterDonis]

I'm sorry, it would be two receivers. Each of these receivers would send an electrical signal into the AND gate at a separate location when the individual laser is covered.

Where is the separate location? How much time will the signals from each receiver take to travel from the receivers to the separate location?

And, once you've got all this clear in the frame in which the receivers (and, presumably, the AND gate as well) are at rest, transform to the frame in which the moving meter stick is at rest. What do things look like there? How does the transformation affect the time it takes for each signal to reach the AND gate, now that both receivers and the gate are moving at a relativistic speed?

 

[bmbuncher]

Oh, I see! It never occurred to me that the electronics would be affected, but that does make sense. I'll do some research on Lorentz contractions to figure out how to apply the equations (I've never used them before), and if it still doesn't make sense, I'll ask my questions here. Thank you!

 

[PeterDonis]

It never occurred to me that the electronics would be affected, but that does make sense.

The electronics are affected, but that's not really the key point. The key point is the signals that have to travel from the receivers to the AND gate.

 

[Bill_K]

The electronics are affected, but that's not really the key point. The key point is the signals that have to travel from the receivers to the AND gate.

Peter, I dissent. I think you've misunderstood the problem. It's just Pole in the Barn.

 

[PeterDonis]

Peter, I dissent. I think you've misunderstood the problem. It's just Pole in the Barn.

I agree the Pole and Barn scenario captures most of it; but in the Pole and Barn scenario, the key question is how the pole can fit in the barn even though, in the pole's rest frame, the barn is shorter. Here the key question is how the two signals from the laser triggers can *fail* to light up the light even though, in the meter stick's rest frame, the distance between the triggers is shorter than the stick, so in that frame, both triggers are triggered simultaneously. To resolve that question, at least if you want to actually fully understand *how* things work in the meter stick's rest frame, just considering relativity of simultaneity is not enough; you also have to consider how the signals from the triggers actually travel to a common location, which is something that isn't necessary in the standard Pole and Barn scenario.

To put it another way, the standard Pole and Barn paradox can be resolved just by pointing out relativity of simultaneity; but this "paradox" additionally requires you to think about how "simultaneity", in any frame, is actually *tested for*, physically.

 

[bmbuncher]

I see, that makes sense. I think Peter is right, in that the main question here is not how the meter stick will behave when experiencing length contraction; rather, it is how to resolve the appearance that the light will be on or off depending on which reference frame is used, which, with my limited knowledge of causality and how that affects the scenario, should be impossible. Is that what you're saying?

 

[nearlynothing]

Say you can determine to infinite accuracy if the meter stick blocks the lights simultaneously.
Still, this refers to simultaneity on one frame of reference, when you go to the rest frame of the meter stick, you can see the lights are blocked simultaneously "in the rest frame of the meter stick", but things are arranged so that the light turns on only when the meter stick blocks both lights simultaneously on the rest frame of the detectors.

 

[PeterDonis]

it is how to resolve the appearance that the light will be on or off depending on which reference frame is used, which, with my limited knowledge of causality and how that affects the scenario, should be impossible. Is that what you're saying?

I'm saying that the light must be either on or off in all reference frames, yes. (In fact I telegraphed the answer in my response to Bill_K; the light stays off.) The light being on or off is a direct observable, and direct observables must be frame-independent. The question is how to explain why the light stays off from the viewpoint of the meter stick frame, even though in that frame, both lasers are triggered simultaneously.

 

[PeterDonis]

things are arranged so that the light turns on only when the meter stick blocks both lights simultaneously on the rest frame of the detectors.

And *how* are things arranged so they work that way? That's the question.

 

[nearlynothing]

And *how* are things arranged so they work that way? That's the question.

I'm having problems understanding why the "how" you determine this simultaneity is the key question here.
When you think about the pole in the barn paradox, how do you then determine when both ends of the pole are inside the barn? what is the detection apparatus used there?
The detection is just a means to make it apparent to us that something is somewhere at a given time on a given reference frame, but wether you detect this or not doesn't change the physical reality.

 

[PeterDonis]

When you think about the pole in the barn paradox, how do you then determine when both ends of the pole are inside the barn? what is the detection apparatus used there?

There doesn't have to be one, because the positions of the pole don't cause anything else in the scenario to happen. In the scenario under discussion here, that's not the case: the positions of the ends of the meter stick directly cause the light to turn on (or not).

The detection is just a means to make it apparent to us that something is somewhere at a given time on a given reference frame, but wether you detect this or not doesn't change the physical reality.

It doesn't change the physical reality of the paths through spacetime of each end of the pole (or meter stick). But if the scenario is set up so the detection of the positions of the ends of the meter stick has other consequences (like whether or not a light turns on), then how the detection is done *does* make a difference for physical reality.

 

[PAllen]

I'm having problems understanding why the "how" you determine this simultaneity is the key question here.
When you think about the pole in the barn paradox, how do you then determine when both ends of the pole are inside the barn? what is the detection apparatus used there?
The detection is just a means to make it apparent to us that something is somewhere at a given time on a given reference frame, but wether you detect this or not doesn't change the physical reality.

Well, in my favorite formulation of the Barn Pole paradox, there is detection. In the barn frame, the doors close momentarily, enclosing the pole. The fact they are both closed, and the fact that they were timed to close at the same time per the Barn frame, is then what must explained in the pole frame. You then get into the question of how the doors were preprogrammed to close and open, specifically, how the clocks controlling them were synchronized.

This example is very similar, but the difference is a different mechanism for probing simultaneity. Instead of pre-programmed closing and opening of doors, you have communication of beam interruption to something that determines simultaneity. And the thing to model is how the beams are both interrupted in both frames, but if the simultaneity detector is set up to work in the laser beam/detector frame, it finds the interruptions non-simultaneous. Meanwhile, in the rod frame, the interruptions are simultaneous, but the moving detector (having been set up per the laser frame), determines otherwise. All facts are readily explained in both frames, but the details are interestingly related to how you propose to detect the simultaneity of the interruption.

 

[ghwellsjr]

Well, in my favorite formulation of the Barn Pole paradox, there is detection. In the barn frame, the doors close momentarily, enclosing the pole. The fact they are both closed, and the fact that they were timed to close at the same time per the Barn frame, is then what must explained in the pole frame. You then get into the question of how the doors were preprogrammed to close and open, specifically, how the clocks controlling them were synchronized.

You haven't described your favorite formulation in which there is detection. At least I can't see it in the description you just gave which I thought was the standard formulation of the Barn Pole paradox, not some variation. Can you please provide the details of what is detected, where the detector is and how it effects the outcome of the scenario?

 

[ghwellsjr]

Hi! I have been pondering a scenario involving a paradox with length contraction. I brought it up with my physics professor, and I somewhat understand what is supposed to happen, but I'm still somewhat confused, so I was wondering if you could help me figure out what is going on.

This is a very interesting "paradox" that you brought up with your physics professor. I'm curious: was it your own idea or did you read about it somewhere? If you read about it somewhere, where? It would be interesting to see exactly how it was stated.

In this scenario, a meter stick is traveling at relativistic speeds above the flat ground. On the ground, there is a setup where two laser tripwires (consisting of a laser emitter and receiver), each located one meter apart, are lying in the path of the meter stick. When both lasers are tripped simultaneously (as in the receivers simultaneously detect that there is no light entering it),...

You have clearly stated that there are two receivers located one meter apart, but...

...an electrical signal is transmitted that turns on a light.

..you have not clearly stated the details of this electrical signal. This is what Peter has been trying to get you to specify, so far, unsuccessfully. If there is just one electrical signal, as you imply, then it must travel from one of the receivers to the other receiver where the AND gate is located and where the light can be located. Is that what you had in mind?

I'm going to draw some spacetime diagrams to depict the various possibilities of your scenario to illustrate why Peter is so adamant about getting you to nail down exactly where you want that AND gate to be. That is what this problem is about, not what happens when you transform to the rest frame of the meter stick. That is a trivial application of the Lorentz Transformation process which will automatically resolve what happens in different frames.

So, from what I understand, when the meter stick is traveling extremely fast, it experiences length contraction (say, it is traveling fast enough to contract to 0.9 meters), and so it will never obstruct the path of both lasers simultaneously; therefore, in the frame of reference where the lasers are still and the meter stick is moving, no electronic pulse will be created, and so the light will not turn on.

The first spacetime diagram is for the rest frame of the lasers without any electrical signals. We'll call this the apparatus rest frame. The units for time are light-meters, the amount of time it takes for light to travel one meter, which is also about 3.3 nanoseconds:

The two receivers are shown in blue and green, one meter apart and the meter stick, moving to the right at a speed of 43.589% of the speed of light is Length Contracted to 0.9 meters. You can see this clearly at the Time Coordinate of zero. The endpoints of all lengths must be specified at the same Coordinate Time, in other words, horizontally on the diagram. The front end of the meter stick is shown in black and the rear end is shown in red.

Clearly, the two receivers do not detect the presence of the meter stick at the same time. In fact there is a gap of about 0.2 light-meters between the blue receiver detecting the red rear end of the meter stick and the green receiver detecting the black front end of the meter stick. So your mission is to decide where to put that AND gate so that it will not detect a meter stick traveling at relativistic speed across the two receivers.

However, in the frame of reference that the lasers are moving and the meter stick is still, the distance between the lasers will contract (say, to 0.9 meters), so both lasers will be obstructed simultaneously, and so the light will turn on.

Here is a diagram for the rest frame of the meter stick:

You are correct that the distance between the laser (and receivers) is contracted to 0.9 meters and that there is a time interval when both lasers are obstructed simultaneously but...

When I posed this to the professor, he said that the observation in both frames of reference should be analogous because causality must be preserved; therefore, the light will either turn on in both scenarios, or will not remain in both scenarios. He said that, while the length contraction will occur in both scenarios, the time dilation caused by special relativity should "balance this out" so that the same result occurs regardless of the frame of reference. However, I am still somewhat confused. In this scenario, how does time dilation cause the light to behave similarly in both frames of reference? Can anyone explain what is going on in this scenario that prevents causality from being violated? Thank you!

...as your professor said, if the light does not turn on in one frame, it won't turn on in any other frame and we already decided that in the apparatus frame the light shouldn't come on so if you can arrange things so it doesn't come on in the apparatus frame, then it won't come on in the meter stick frame.

However, your professor is not correct about Time Dilation having anything to do with what happens in this scenario. But we'll leave the explanation until after you finish specifying how the light is hooked up to the receivers.

In the mean time, we can investigate the possibility of having just one electrical signal. Let's suppose you decide to put the AND gate at the location of the green receiver. Then the blue receiver has to send the electrical signal to the green receiver where the AND gate is located. For the sake of simplicity, we will assume that the electrical signal travels at the speed of light.

Now let's see what happens:

Clearly, this isn't going to work because the last part of the detection from the blue receiver overlaps the first part of the detection from the green receiver by about 0.8 light-meters.

So let's try it the other way around. Let's send the electrical signal from the green receiver to the blue receiver and put the AND gate at the blue receiver:

This does have the desirable effect of not detecting an overlap but will it work if the meter stick is traveling in the other direction? We don't want to have an experiment that works differently in two different directions:

Well this has the same problem as the first method so I think we can only conclude that we're going to need two electrical signals and put the AND gate somewhere other than at one of the receivers. That's your problem: tell us where to put the AND gate and make sure it works with the meter stick going either way.

 

[PAllen]

You haven't described your favorite formulation in which there is detection. At least I can't see it in the description you just gave which I thought was the standard formulation of the Barn Pole paradox, not some variation. Can you please provide the details of what is detected, where the detector is and how it effects the outcome of the scenario?

It seems obvious to me from the description. The detection mechanism is simply that the doors are programmed to close at the same time per the barn frame, based on clocks synchronized with the Einstein convention; this time being chosen that (per this barn frame) the pole is completely inside the barn. The successful closing and opening of doors such that no collisions occur is the simultaneity detection mechanism. The point I was answering is that this form has no simultaneity detection. To me it does. Maybe you prefer to call it a simultaneity assertion mechanism rather than a simultaneity detection mechanism.

 

[ghwellsjr]

It seems obvious to me from the description. The detection mechanism is simply that the doors are programmed to close at the same time per the barn frame, based on clocks synchronized with the Einstein convention; this time being chosen that (per this barn frame) the pole is completely inside the barn. The successful closing and opening of doors such that no collisions occur is the simultaneity detection mechanism. The point I was answering is that this form has no simultaneity detection. To me it does. Maybe you prefer to call it a simultaneity assertion mechanism rather than a simultaneity detection mechanism.

In bmbuncher's scenario, there are two physical detectors that control the light action to take place in response to the presence of the meter stick at any time and with any number of meter sticks (or sticks of any length) traveling at any speed and in either direction.

Although it is possible for a barn/pole scenario to be devised in such a manner that the doors close and open simultaneously (in the barn frame) in response to the detected presence of a pole traveling at a random speed (in either direction) at a random time, it is not possible for the pole to survive under such terms.

The only way that the barn/pole scenario can actually happen so that everything can survive is if, as you said, it is preprogrammed with no detectors. That is, we need to have separate clock controlling mechanisms located at each door and with the pole. We could put in detectors that validate that the doors actually did close simultaneously and that the pole was actually inside the barn but they cannot control or initiate the simultaneous closing and opening of the doors.

That is what makes the difference between the barn/pole paradox and bmbuncher's scenario and why Peter made this important distinction. But beyond that, bmbuncher's scenario is incomplete and he needs to fill in the details and that is what is important here.

 

[bmbuncher]

The AND gate, which is located 10 meters from the centerpoint between the lasers, is connected to each of the laser receivers. One wire begins at each receiver, which then experiences a right angle turn towards the centerpoint of the lasers, and once they reach the center, they make another right angle towards the AND gate. Each wire is a total of 10.5 meters long (0.5 meters to the centerpoint of the lasers, 10 meters to the AND gate). The AND gate consists of two resistors (detailed in the first diagram of this page: http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/trangate.html), and the output is connected to a light bulb one meter from the gate. The light bulb circuit is completed by another one meter of wire that travels to the +6V input (shown in the diagram) on the AND gate This yields a total of 23 meters of wire, divided into: 10.5 meters in each wire that leads from the laser receivers to the AND gate, 1.0 meter that leads from the AND gate output to the light bulb, and 1.0 meter that leads from the light bulb to the battery (which powers the light bulb) and back to the AND gate.

I'm sorry about the miscommunications, does this answer enough to be able to analyze the question?

 

[PeterDonis]

The AND gate, which is located 10 meters from the centerpoint between the lasers

Why not put it *at* the center point between the lasers? The extra 10 meters just complicates things, as you'll see below when I leave it out. I'll also assume that the light bulb itself is co-located with the AND gate; the extra 1 meter you have from the gate to the light bulb also just complicates things.

is connected to each of the laser receivers. One wire begins at each receiver

To make things simpler, I'm going to assume that "wire" actually means "fiber optic cable with negligible effect on the speed of light within it", so what we actually have is effectively a light signal propagated from each laser receiver to the center point between them.

All the rest of the specifications are irrelevant to the question under discussion. Remember I said I wasn't asking about details of how AND gates work.

Now, with the apparatus as above, let's first look at things from the frame in which the apparatus is at rest and the meter stick is moving. When the front end of the meter stick passes the first laser receiver (call that receiver A), the receiver sends a signal to the gate. The signal travels 0.5 meters at the speed of light, so it takes 0.5 meters of time (we're using units where the speed of light is 1, so we're measuring time in meters; in conventional units, 1 meter of time = 3.3 nanoseconds, so the light takes 1.65 nanoseconds to travel from receiver A to the gate). So if we call the time when the front end of meter stick passes receiver A time t = 0, the signal reaches the gate at t = 0.5.

When the back end of the meter stick passes receiver A, the signal stops. As ghwellsjr pointed out, the meter stick is traveling at about v = 0.436, since it is length contracted to 0.9 meters. So the back end of the meter stick will travel 0.9 meters and reach receiver A at t = 0.9 / .436 = 2.06 meters. Since it takes 0.5 meters for the stoppage of the signal to travel to the gate, the gate sees the signal stop at t = 2.56.

When the front end of the meter stick passes the second receiver (call that receiver B), it starts sending a signal to the gate. This will happen at time t = 1 / .436, or about t = 2.29. It will take 0.5 meters for this signal to reach the gate, so the gate will see the second signal start at t = 2.79. Since this is after the first signal shuts off at t = 2.56, the gate never sees both signals at the same time, so it never lights the light bulb.

Now the idea is to re-do the above analysis in the frame in which the meter stick is at rest and the apparatus is moving at v = - .436. Assuming that we place the spatial origin of the apparatus' rest frame at the AND gate, halfway between the receivers, we can assign the following coordinates to the events of interest in the apparatus rest frame:

Event #1: Front end of meter stick passes receiver A; (t, x) = (0, - 0.5).

Event #2: Signal from event #1 reaches AND gate; (t, x) = (0.5, 0).

Event #3: Back end of meter stick passes receiver A; (t, x) = (2.06, - 0.5).

Event #4: Signal from event #3 reaches AND gate; (t, x) = (2.56, 0).

Event #5: Front end of meter stick passes receiver B; (t, x) = (2.29, 0.5).

Event #6: Signal from event #5 reaches AND gate; (t, x) = (2.79, 0).

Since event #4 happens before event #6, the bulb does not light. Of course, in this frame, event #3 also happens before event #5, so it's easy to confuse that pair of events with #4 and #6 and think that the #3, #5 pair is what determines whether the bulb lights. But that won't work; see below.

Now it's just a matter of using the Lorentz transformation with v = 0.436 to find the coordinates (t', x') of all 6 of the above events in the meter stick rest frame. You should find that event #4 still happens before event #6 in that frame, so the bulb still does not light. But you should also find that event #3 happens *after* event #5 in the meter stick rest frame, which is what was confusing you: the two ends of the meter stick do cover both receivers simultaneously in this frame. But as the above analysis shows, that's not what determines whether the bulb lights; what determines that is when the signals reach the AND gate, i.e., events #4 and #6, *not* events #3 and #5.

 

[ghwellsjr]

The AND gate, which is located 10 meters from the centerpoint between the lasers, is connected to each of the laser receivers. One wire begins at each receiver, which then experiences a right angle turn towards the centerpoint of the lasers, and once they reach the center, they make another right angle towards the AND gate. Each wire is a total of 10.5 meters long (0.5 meters to the centerpoint of the lasers, 10 meters to the AND gate). The AND gate consists of two resistors (detailed in the first diagram of this page: http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/trangate.html), and the output is connected to a light bulb one meter from the gate. The light bulb circuit is completed by another one meter of wire that travels to the +6V input (shown in the diagram) on the AND gate This yields a total of 23 meters of wire, divided into: 10.5 meters in each wire that leads from the laser receivers to the AND gate, 1.0 meter that leads from the AND gate output to the light bulb, and 1.0 meter that leads from the light bulb to the battery (which powers the light bulb) and back to the AND gate.

I'm sorry about the miscommunications, does this answer enough to be able to analyze the question?

Yes, it does, excellent. In fact, it's more than enough. The important part is the 0.5 meter wiring from each receiver to the centerpoint between the lasers, at least as far as making a spacetime diagram goes because those are limited to in-line scenarios. All that the extra wiring to the AND gate and then to the light bulb does is add more delay from the time of detection to the time of display but it won't effect the decision of whether or not the light comes on or for how long.

In fact, you could have eliminated all that extra wiring and put the AND gate and light bulb right at the centerpoint between the receivers. So let's work on that problem. Here is a spacetime diagram with a grey worldline added to represent the location of the AND gate and light bulb:

Here's what I'd like you to do if you are willing: copy this image and paste it in your favorite drawing program and add in some of the electrical signals (traveling at the speed of light) to represent the beginning, midpoint, and ending of the reception of the detection of the meter stick similarly to what I did in post #22 except that the signals will terminate at the grey line. Then upload it and describe why the light will not come on.

If you can do that, I will upload another diagram transformed to the rest frame of the meter stick and you can repeat the process to see how the light bulb will still not come on. Can you do that? If you don't want to, I'll do it for you but I think it will be more meaningful to you if you do it yourself.

 

[ghwellsjr]

Here is a spacetime diagram based on Peter's coordinates in case you prefer to use his to add the signals in:

 

[ghwellsjr]

...does this answer enough to be able to analyze the question?

As I said in post #27, yes, it does, but now the question is, are you analyzing the question or are you waiting for someone else to do it?

 

[PeterDonis]

now the question is, are you analyzing the question or are you waiting for someone else to do it?

He doesn't have to wait; between us we've already done the analysis.

 

[ghwellsjr]

He doesn't have to wait; between us we've already done the analysis.

Not quite. You told him what to do in post #26 and what he should find but no one has yet posted the transformations to the rest frame of the meter stick:

Now it's just a matter of using the Lorentz transformation with v = 0.436 to find the coordinates (t', x') of all 6 of the above events in the meter stick rest frame. You should find that event #4 still happens before event #6 in that frame, so the bulb still does not light. But you should also find that event #3 happens *after* event #5 in the meter stick rest frame, which is what was confusing you: the two ends of the meter stick do cover both receivers simultaneously in this frame. But as the above analysis shows, that's not what determines whether the bulb lights; what determines that is when the signals reach the AND gate, i.e., events #4 and #6, *not* events #3 and #5.

 

[Orodruin]

The analysis of the signal really is unnecessary and I see no point in complicating life with it. It really just is the pole-in-barn paradox. The lighting of the lamp can be assumed to occur if both receivers are covered simultaneously in a given frame. If this frame is the rest frame of the receivers, the lamp will not light. An observer in the rest frame of the pole will not think the lamp should light up simply because (s)he can compute that the receivers have never been simultaneously covered in the receiver frame, which was the criterion for lighting the lamp. Dealing with signals is just going into detail about how such a device would be constructed, but adds nothing in terms of physics.

 

[ghwellsjr]

The analysis of the signal really is unnecessary and I see no point in complicating life with it.

Then you missed the point of the problem.

It really just is the pole-in-barn paradox.

No it's not. This problem was ambiguously stated. Didn't you read PeterDonis's posts or my post #22?

The lighting of the lamp can be assumed to occur if both receivers are covered simultaneously in a given frame.

That is so untrue. What is important is the relative lengths of the two wires (or fiber optic cables) coming from the two receivers to the AND gate. The lighting of the lamp is independent of which frame the receivers are covered simultaneously in.

If this frame is the rest frame of the receivers, the lamp will not light.

Different frames will not change what actually happens. All frames will analyze the exact same result. Several posters, including myself and Peter, have noted that in the rest frame of the meter stick, the receivers simultaneously detect the meter stick, and yet the light does not light. The question is, why?

An observer in the rest frame of the pole will not think the lamp should light up simply because (s)he can compute that the receivers have never been simultaneously covered in the receiver frame, which was the criterion for lighting the lamp.

The light and all the wiring between it and the receivers and the AND gate don't know or care what an observer thinks or analyzes.

Dealing with signals is just going into detail about how such a device would be constructed, but adds nothing in terms of physics.

Of course the construction is part of the physics and needs to be part of the analysis if the scenario is going to work as intended.

If you would read the OP, you would see what the problem is:

The OP stated that there were two receivers one meter apart and an electrical signal connecting them to a light. Obviously, this is impossible. There need to be two electrical signals connecting them through an AND gate (or equivalent) to the light. The details of how these are constructed alone determines whether the light will behave in the intended manner, not some analysis based on simultaneity according to a given frame. After prodding, the OP did fill in the missing details and posted a complete description of how the wiring could be constructed so that the scenario would work as desired. Now all that is left is the analysis for both frames.

Remember the OP's questions:

In this scenario, how does time dilation cause the light to behave similarly in both frames of reference? Can anyone explain what is going on in this scenario that prevents causality from being violated?

Those questions have not been answered and I was hoping that with enough help the OP could answer them himself and that's why I asked him if he was working on it in post #29 and I'm still waiting for his answer.

 

[PeterDonis]

Not quite. You told him what to do in post #26 and what he should find but no one has yet posted the transformations to the rest frame of the meter stick:

Ah, yes, good point.

 

[Orodruin]

That is so untrue. What is important is the relative lengths of the two wires (or fiber optic cables) coming from the two receivers to the AND gate. The lighting of the lamp is independent of which frame the receivers are covered simultaneously in.

Any difference in travel times (again in the frame where the receivers should be simultaneously activated to light the lamp) from the receivers to the lamps can be corrected by electronics. I maintain that gritting down into signalling is not necessary as long as we state in which frame the receivers should be simultaneously covered in order for the lamp to light up. Obviously we are going to have to make the correct setup happen, but this can be done.

The lighting of the lamp is independent of which frame the receivers are covered simultaneously in.

This statement violates special relativity.