Length Contraction Paradox Scenario

[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.

 

[ghwellsjr]

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.

Of course, you can add delay in either signal line from a receiver to its input of the AND gate to change or correct, as you say, the setup to make it work like it would if you had put the AND gate at the midpoint between the two receivers, but how are you going to know how much delay to add? Tweak a pot until the light goes out?

The point is that if you put the AND gate at the midpoint between the receivers (and have equal length wires) then you are automatically following Special Relativity's definition of simultaneity and you're letting the setup define the simultaneity instead of assuming that it will come out right by some method of tweaking or correcting, as you call it.

How exactly would you correct the circuitry if you had the AND gate colocated with one of the receivers?

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

This statement violates special relativity.

Then how do you explain what I already asked you to look at in post #22?

Look at this diagram showing the rest frame of the apparatus. I have drawn in two black lines to show the gap between them where there is no simultaneity between the black and red receivers detecting the meter stick (between the blue and green worldlines).

Now here is the same scenario transformed to the frame in which the meter stick is at rest. I have drawn in two black lines to show the overlap where there is simultaneity between the black and red receivers detecting the meter stick.

If you still disagree, then I would ask you for a reputable reference that states that the lighting of a lamp is frame variant.

 

[PeterDonis]

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.

Yes, *if* there were a difference in travel times, you *could* correct it this way. But the OP specified (in response to questions from ghwellsjr and me) that in the scenario under discussion here, the light travel time from each receiver to the AND gate is the same, in the frame in which both receivers and the AND gate are at rest. So your comment, while correct in principle, is irrelevant to this discussion.

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.

If all you want is to make the correct prediction about whether the lamp lights up, yes, this is all you need. But the OP wanted more than that: he wanted an explanation of how it can be that the lamp does not light, even though, in the frame in which the meter stick is at rest, both laser receivers are covered simultaneously. That's what the extra discussion ghwellsjr and I have been having with the OP is about.

If you, personally, don't need all that extra discussion, good for you. But how about letting the OP tell us whether the extra discussion is helping him?

 

[PeterDonis]

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

Careful! This is not correct as you state it. I suspect what you meant to say is: the lighting of the lamp is independent of which frame we *analyze* the scenario in, i.e., of which simultaneity convention we adopt for purposes of *describing* the experiment. But it is certainly *not* independent of which frame the receivers are covered simultaneously in! The latter can be defined physically, independent of any choice of frame; in fact, that's part of what we are trying to get the OP to recognize.

If the receivers were covered simultaneously in the rest frame of the receivers and the AND gate, the lamp would light. Since they are not covered simultaneously in that frame--they are only covered simultaneously in the rest frame of the meter stick--the lamp does not light. So which frame the receivers are/are not covered simultaneously in certainly *does* make a difference to whether the lamp lights.

Note that we can describe the scenario in either frame--the rest frame of the receivers/gate or the rest frame of the meter stick--and our description does not have to include any statements like "the receivers are not covered simultaneously in the rest frame of the receivers/gate, but they are covered simultaneously in the rest frame of the meter stick". But those statements will have to be *implicit* in our description, if it is a correct description: that is, our description will be equivalent, physically, to one in which those statements are made explicitly.

 

[Orodruin]

If you still disagree, then I would ask you for a reputable reference that states that the lighting of a lamp is frame variant.

I never claimed this. In fact, it is exactly what your statement claimed, but Peter has already addressed this.

My statement was that if you know the frame in which the receivers need to be covered simultaneously for the lamp to light up, then you know everything about the device that you need to know in order to resolve the paradox. If you know which frame this is you may deduce from this the time difference between the coverings which will light the lamp in your own frame.

 

[ghwellsjr]

My statement was that if you know the frame in which the receivers need to be covered simultaneously for the lamp to light up, then you know everything about the device that you need to know in order to resolve the paradox.

The paradox, as stated by the OP was that in the rest frame of the receivers, they are not covered simultaneously and so the light will stay off whereas in the rest frame of the meter stick, the receivers are covered simultaneously and so the light will come on. How do you resolve this paradox?

If you know which frame this is you may deduce from this the time difference between the coverings which will light the lamp in your own frame.

I know which frame this is, it's the rest frame of the meter stick but I have no idea what my frame is supposed to be or what the time difference is you are referring to. Please make it clear when you resolve the paradox.

 

[ghwellsjr]

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.

Careful! This is not correct as you state it. I suspect what you meant to say is: the lighting of the lamp is independent of which frame we *analyze* the scenario in, i.e., of which simultaneity convention we adopt for purposes of *describing* the experiment. But it is certainly *not* independent of which frame the receivers are covered simultaneously in! The latter can be defined physically, independent of any choice of frame; in fact, that's part of what we are trying to get the OP to recognize.

If the receivers were covered simultaneously in the rest frame of the receivers and the AND gate, the lamp would light. Since they are not covered simultaneously in that frame--they are only covered simultaneously in the rest frame of the meter stick--the lamp does not light. So which frame the receivers are/are not covered simultaneously in certainly *does* make a difference to whether the lamp lights.

Note that we can describe the scenario in either frame--the rest frame of the receivers/gate or the rest frame of the meter stick--and our description does not have to include any statements like "the receivers are not covered simultaneously in the rest frame of the receivers/gate, but they are covered simultaneously in the rest frame of the meter stick". But those statements will have to be *implicit* in our description, if it is a correct description: that is, our description will be equivalent, physically, to one in which those statements are made explicitly.

I included the complete quote and its context.

Orodruin was arguing that the circuitry is irrelevant to the lighting of the light. I said that the relative lengths of the two wires connecting the receivers to the AND gate was the only thing that mattered. If those two lengths are the same, then the light will not come on even in the rest frame of the meter stick where the two receivers are covered simultaneously. And that is why I said, "The lighting of the lamp is independent of which frame the receivers are covered simultaneously in". Why is that wrong?

 

[Orodruin]

The paradox, as stated by the OP was that in the rest frame of the receivers, they are not covered simultaneously and so the light will stay off whereas in the rest frame of the meter stick, the receivers are covered simultaneously and so the light will come on. How do you resolve this paradox?

By knowing that the device is constructed so that it turns on only if the receivers are covered simultaneously in the receiver rest frame. Even if I am in the stick frame I can easily deduce which events will be simultaneous in the receiver frame. Of course, this will then imply the receivers being covered with a fixed time difference in my (ie the stick's) rest frame, but I can compute that from the given information without problems.

I know which frame this is, it's the rest frame of the meter stick but I have no idea what my frame is supposed to be or what the time difference is you are referring to. Please make it clear when you resolve the paradox.

What I stated above. If I know that the lamp will turn on if the receivers are covered simultaneously in their rest frame, it will turn on if the receivers are covered with a computable time difference in a different frame. To make it concrete, assume that the receivers are located in x = 0 and x = L in their rest frame and the stick is moving with velocity v. Assume that the stick covers the x=0 receiver at time t = 0, by the construction of the device, the lamp will then turn on if also the receiver at x = L is covered at t = 0.

Those events transform to the stick rest frame by Lorentz transformation to

Δt' = -vγΔx = -vγL

It follows that the statement of having the receivers covered simultaneously in their rest frame is equivalent to the statement that they are covered with a time difference -vγL in the stick rest frame. Thus the lamp will turn on if the furthest receiver is covered at t' = t0' - vγL and the closest at t' = t0'. Now obviously L is measured in the receiver rest frame, but this can easily be swapped for an expression in the stick rest frame by Lorentz contraction.

 

[Orodruin]

I included the complete quote and its context.

Orodruin was arguing that the circuitry is irrelevant to the lighting of the light. I said that the relative lengths of the two wires connecting the receivers to the AND gate was the only thing that mattered.

And all this boils down to is specifying the frame in which the events need to be simultaneous as this is all you need to know in order to know if the lamp turns on or not, which was my entire argument. I am not saying that it is wiring independent, I am saying that the wiring defines a frame and that all you need to know about the wiring is which frame this is.

If those two lengths are the same, then the light will not come on even in the rest frame of the meter stick where the two receivers are covered simultaneously. And that is why I said, "The lighting of the lamp is independent of which frame the receivers are covered simultaneously in". Why is that wrong?

Because with that wording, you are stating that it does not matter to the lighting of the lamp which frame the events are simultaneous in, which is false as I can have a situation where the lamps are covered simultaneously in one frame but not in another. Thus, the specification of which frame the events need to be simultaneous in is crucial and the lighting is not independent of this.

 

[ghwellsjr]

The paradox, as stated by the OP was that in the rest frame of the receivers, they are not covered simultaneously and so the light will stay off whereas in the rest frame of the meter stick, the receivers are covered simultaneously and so the light will come on. How do you resolve this paradox?

By knowing that the device is constructed so that it turns on only if the receivers are covered simultaneously in the receiver rest frame. Even if I am in the stick frame I can easily deduce which events will be simultaneous in the receiver frame.

Sure you can, but that has no bearing on which events are simultaneous in the stick frame or why the light behaves the same way in all frames. Look, everyone, including the OP agrees that whatever the resolution is in one frame, it has to be the same in all other frames. But he doesn't understand why.

Of course, this will then imply the receivers being covered with a fixed time difference in my (ie the stick's) rest frame, but I can compute that from the given information without problems.

I know which frame this is, it's the rest frame of the meter stick but I have no idea what my frame is supposed to be or what the time difference is you are referring to. Please make it clear when you resolve the paradox.

What I stated above. If I know that the lamp will turn on if the receivers are covered simultaneously in their rest frame, it will turn on if the receivers are covered with a computable time difference in a different frame. To make it concrete, assume that the receivers are located in x = 0 and x = L in their rest frame and the stick is moving with velocity v. Assume that the stick covers the x=0 receiver at time t = 0, by the construction of the device, the lamp will then turn on if also the receiver at x = L is covered at t = 0.

The OP specified all the values of your variables:

L = 1 (meter)
1/γ = 0.9

Therefore:

γ = 1.11111
v = 0.43589c

Your two events are x=0, t=0 and x=1, t=0 and here is my previously drawn spacetime diagram with those two events marked as black dots:

Those events transform to the stick rest frame by Lorentz transformation to

Δt' = -vγΔx = -vγL

This evaluates to:

Δt' = -vγL = -0.43589 * 1.111111 * 1 = -0.48432

However, you didn't do the calculation for Δx' which evaluates to 1.111111.

So here is the diagram transformed to the stick rest frame with the same two events marked as black dots:

It follows that the statement of having the receivers covered simultaneously in their rest frame is equivalent to the statement that they are covered with a time difference -vγL in the stick rest frame. Thus the lamp will turn on if the furthest receiver is covered at t' = t0' - vγL and the closest at t' = t0'. Now obviously L is measured in the receiver rest frame, but this can easily be swapped for an expression in the stick rest frame by Lorentz contraction.

I still have no idea how transforming those two events determines a time difference that has any bearing on the subject. Maybe you could copy the spacetime diagram and mark it up to show what you are saying or at least provide specific details that will help explain what you are concluding. You need to explain what t0' is and which receiver is the furthest one (what color in my diagram). Also, I don't know why you made the comment about L in your last sentence.

 

[ghwellsjr]

I included the complete quote and its context.

Orodruin was arguing that the circuitry is irrelevant to the lighting of the light. I said that the relative lengths of the two wires connecting the receivers to the AND gate was the only thing that mattered.

And all this boils down to is specifying the frame in which the events need to be simultaneous as this is all you need to know in order to know if the lamp turns on or not, which was my entire argument. I am not saying that it is wiring independent, I am saying that the wiring defines a frame and that all you need to know about the wiring is which frame this is.

Here's what you said:

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.

If you are now changing your mind and agreeing that the wiring, which is the analysis of the signal, is important, then you are moving in the right direction. However, I still don't see any analysis of the signals in the wires on your part.

If those two lengths are the same, then the light will not come on even in the rest frame of the meter stick where the two receivers are covered simultaneously. And that is why I said, "The lighting of the lamp is independent of which frame the receivers are covered simultaneously in". Why is that wrong?

Because with that wording, you are stating that it does not matter to the lighting of the lamp which frame the events are simultaneous in, which is false as I can have a situation where the lamps are covered simultaneously in one frame but not in another.

Yes, I am stating that it does not matter to the lighting of the lamp which frame the events are simultaneous in. Once the wiring is in place, even though both receivers are covered simultaneously in the stick frame, the light still does not come on. That's the issue that you seem unconcerned about.

Thus, the specification of which frame the events need to be simultaneous in is crucial and the lighting is not independent of this.

No, the wiring is crucial to getting the light to come on as specified by the OP. That's what Peter and I finally got the OP to address. He correctly understands why the light does not come on in the rest frame of the receivers where they never simultaneously detect the stick but he has not shown that he understands why the light also does not come on in the rest frame of the stick where the receivers do detect the stick simultaneously, and neither have you.

 

[PeterDonis]

I said that the relative lengths of the two wires connecting the receivers to the AND gate was the only thing that mattered.

You should have said "lengths in the rest frame of the receivers and the AND gate", since length is frame-dependent. But I understand that that's what you meant.

If those two lengths are the same, then the light will not come on even in the rest frame of the meter stick where the two receivers are covered simultaneously.

The light doesn't come on "in a frame". It either comes on or it doesn't. In the scenario under discussion, it doesn't, because the two receivers are *not* covered simultaneously in the rest frame of the receivers and the AND gate. But *if* the scenario were different, and the two receivers *were* covered simultaneously in that frame, the light *would* come on.

And that is why I said, "The lighting of the lamp is independent of which frame the receivers are covered simultaneously in". Why is that wrong?

Because, taking the statement as it stands, it appears to be saying that even in a different scenario, where the receivers were covered simultaneously in the receiver/AND gate rest frame, the light would not come on, which is false, as I noted above. That's how it came across to me, and it's evidently how it came across to Orodruin too.

I understand that wasn't what you meant to say, but that's why one has to be very careful when describing these things in English instead of math. In the mathematical description, or the spacetime diagrams you have drawn, there's no possibility of ambiguity.

 

[PeterDonis]

I am saying that the wiring defines a frame and that all you need to know about the wiring is which frame this is.

It's not just the wiring that defines a frame; it's the receivers plus the wiring plus the AND gate, all being at rest relative to one another, plus the relative distances being what they are (and the stipulation that there are no extra electronic corrections present).

Also, saying that all these things "define a frame" still leaves the question of *how* they "define a frame", i.e., what that means, physically, and how it appears in a different frame. This may be obvious to you, but the question is whether it's obvious to the OP, who still hasn't responded to this whole portion of the thread.

 

[pervect]

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.

We could say, abstractly, that in "physical reality", frame transformations use the kinematics of the Lorentz transform, and not the kinematics of the Gallilean transform. At this point, we can then say that the problem is a simple application of the Lorentz transform and go through the math.

But I don't think this is a very satisfying answer. I also think, from experience that it wouldn't be effective, and this would become obvious by the number of confused and/or argumentative posts in response.

So rather than go this route, one of us is asking people to describe the setup in more detail, so we can give a less abstract answer. I think this is a reasonable approach.

 

[epovo]

Okay. Let's call A the event when the front of the stick reaches the front detector, and B the event when the rear of the stick *clears* the rear detector.
We have a set-up in which if A happens before B in the ground frame a light will turn on and stay on, but not if A happens after B. There are a myriad ways in which you could set this up. The set-up does not care in which order A and B happen on the stick frame or on my grandmother's frame waving good-bye from the window of a rocket.
One possible (though maybe not very practical set-up) is to have two synchronized clocks on each detector which stop when they detect events A and B respectively. Afterwards (no rush) some electronic circuitry compares the readings and turns on the light. A copy of the design is mailed to the stick-riding observer, who also fits his stick with clocks at either end. If both accept that events can be differently ordered in different frames, none would raise an eyebrow

 

[ghwellsjr]

Okay. Let's call A the event when the front of the stick reaches the front detector, and B the event when the rear of the stick *clears* the rear detector.
We have a set-up in which if A happens before B in the ground frame a light will turn on and stay on, but not if A happens after B. There are a myriad ways in which you could set this up. The set-up does not care in which order A and B happen on the stick frame or on my grandmother's frame waving good-bye from the window of a rocket.

So what if there are a myriad of ways to set this up? They are not the topic of this thread. This thread is about why the specific wired circuit described by the OP in post #25 does not turn on the light in the stick frame even though the receivers are closer together than the length of the stick. The OP's professor told him it was because of Time Dilation. Is that the reason?

One possible (though maybe not very practical set-up) is to have two synchronized clocks on each detector which stop when they detect events A and B respectively. Afterwards (no rush) some electronic circuitry compares the readings and turns on the light. A copy of the design is mailed to the stick-riding observer, who also fits his stick with clocks at either end. If both accept that events can be differently ordered in different frames, none would raise an eyebrow

Although the two events that you described can be in different orders in the two frames under consideration, the OP specifically asked about how causality is not violated. You need to focus on the events related to causality that do happen in the same order in both frames to answer his question.

 

[ghwellsjr]

Okay. Let's call A the event when the front of the stick reaches the front detector, and B the event when the rear of the stick *clears* the rear detector.
We have a set-up in which if A happens before B in the ground frame a light will turn on and stay on, but not if A happens after B.

To help visualize what you are describing, I have put black dots (events) on a couple of my previous spacetime diagrams and marked them with labels "A" and "B". Here is the one for the ground frame, the rest frame of the apparatus including the two receivers depicted in blue and green:

Since the meter stick is Length Contracted to 0.9 meters, event A happens after event B and the gap between them makes sense that the light will not come on.

Here is the rest frame of the meter stick, the ends of which are depicted in red and black:

In this frame, the distance between the receivers is Length Contracted to 0.9 meters and event A happens before event B so there is an overlap of time when both receivers detect the meter stick but the light still will not come on. The OP wants to know why. In other words, how do the signals propagate from the two receivers to the inputs of the AND gate located midway between the two receivers in this frame so that they don't arrive simultaneously?

Since it appears that the OP has lost interest in his thread, maybe someone else (besides me or Peter) will mark up this spacetime diagram to show how the signals propagate from the receivers to the AND gate (depicted by the grey worldline):

Then here is the rest frame of the meter stick where the same propagations can be drawn in by someone to show how even though there is an overlap in time when they start out, there is gap when they arrive at the AND gate depicted by the grey worldline:

Any takers?

 

[ghwellsjr]

Any takers?

I think a week has been a long enough time for anyone interested in drawing in the signals as they propagate from the two laser receivers (in blue and green) to the AND gate (in gray) so I'll do it myself. Here is the rest frame of the apparatus showing the front end (in black) and the rear end (in red) of the Length Contracted meter stick as they are detected by the two receivers:

In this frame, since the meter stick is shorter than the distance between the receivers, there is a gap in time between the rear end of the stick leaving the rear detector (at B) and the front end of the meter stick arriving at the front receiver (at A) and this gap in time is maintained by the two signals as they propagate to the AND gate so that the light does not turn on.

Now we transform to the rest frame of the meter stick:

In this frame, the distance between the receivers is shorter than the meter stick so that there is an overlap in time between the front receiver arriving at the front of the meter stick (at A) and the rear receiver leaving the rear of the meter stick (at B) but because the AND gate is moving toward the rear receiver and away from the front receiver, the timing between the two signals changes from an overlap at the receivers to a gap at the AND gate and so the light does not come on.

Note that the professor's explanation that Time Dilation balances out the Length Contraction is wrong.