Length Contraction Paradox Scenario

[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.