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I am currently learning quantum mechanics using Griffiths. In the appendix, he goes to talk about EPR and Bell's inequality, and that experimental verification of Bell's inequality rejects the "local hidden variable" theory. This means if Quantum Mechanics is correct, the collapse of the wave function is instantaneous, and not subject to locality.

He then tried to argue that many things in fact travel faster than the speed of light but we don't have to worry about them,

"if a bug flies across the beam of a movie project, the speed of its shadow is proportional to the distance to the screen; in principle, that distance can be as large as you like, and hence the shadow can travel at arbitrarily high velocity. However, the shadow does not carry any energy, nor can it transmit any information from one point on the screen to another".

I get how the shadow is not carrying any energy, but why can't it transmit information? For example, suppose Bob is on one side of the screen and encodes his message on the contour of the bug then sends it to the projector, then the bug flies across the project at a very close distance to the light source. Then can this message be transferred to Alice on the other side of the screen faster than the speed of light?

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Part 1:

In the scenario you describe, let's imagine the fastest possible bug, which travels at nearly the speed of light.

Alice: A

Bob: B

Projector Alice: PA

Projector Bob: PB

That bug has to travel from B to PB, then PB to PA, then the bug's shadow has to travel from PA to A since there are still light rays en route after the bug is in position--A can only "see the shadow" after the light en route finishes propagating. So the total time it takes for A to receive the message from B will be the time it takes for light to travel from B to PB to PA to A. Which is roundabout and hence a longer distance than the direct B to A (by the triangle inequality).

So the message will not be transferred to Alice faster than the speed of light.

Part 2:

In Griffith's description, the bug does NOT come from B in the first place, only PB, so obviously does not carry information from B.

Imagine at t=0 the bug is at PB and at t=dt the bug is at PA, and it takes time P for the shadow to be projected. Then the shadow reaches B (from PB) at t=P and reaches A (from PA) at t=P+dt, so indeed there is a time difference of only dt between A seeing the shadow and B seeing the shadow, and dt can be quite short compared to the A-B distance over the speed of light. So there is a sense in which the shadow "travels faster than light" (two shadow events separated by a distance longer than $c * \Delta t$) which is different from your scenario, but in this scenario, no information is being carried.

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    $\begingroup$ Great response. Addressed my confusion perfectly. $\endgroup$
    – Davidson Cheng
    Dec 7, 2022 at 7:25
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    $\begingroup$ Just adding to Part 1, the distance between PB and PA can be arbitrarily small. But the time it takes for light to travel from B to PB and PA to A (ignoring PB to PA for now) would have to be greater than the time it takes to travel from A to B, because of the triangle inequality. Please correct me if I am wrong. $\endgroup$
    – Davidson Cheng
    Dec 7, 2022 at 7:29
  • $\begingroup$ Exactly, well said. The triangle inequality :) $\endgroup$
    – Alwin
    Dec 7, 2022 at 8:26

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