How much more do you want to know?
You have the wall lowering at exactly k = sin(θ) so the light appears instantly on the ground from the 55 foot mark to the 27.5 foot “destination”. If you increase k further, the denominator will become negative, suggesting the velocity of the shadow will now reverse.
How can that be? Well, if you move past the singularity (instantly) in this scenario, which you do when you make k > sin(θ), now the light will appear at the 27.5 foot mark before the shadow’s edge moves at all from the 55 foot mark. From there, the sliver of light at the 27.5 foot mark will expand backwards toward the 55 foot shadow’s edge, generating a velocity of the shadow that now has the opposite direction; the shadow’s edge is now moving from the 27.5 foot “destination” towards the 55 foot starting place. After this, the faster you lower the wall with k > sin(θ), the slower the shadow moves from 27.5 to 55 feet. It’s all in equation 1.
The reason all these strange behaviors of shadows are acceptable physically is that a shadow has no mass. The speed limit of light is only for things with mass (also for information, save that for another time). If you think of what is moving when a shadow moves, there is no mass moving in the direction of the shadow. It is a timing issue, that is to say, the timing of when the light hits the ground. Since the distance to the ground is shorter when the wall is lowered, you can get light to hit the ground at the destination before light can reach the ground at the original shadow’s edge. That is how equation 1 is derived and why it breaks no laws. Nothing with mass is moving faster than light.
Think of the wave that people generate at ball games by raising their arms as the person next to them raises theirs. Let’s turn those people into pistons shooting up. The wave can be created by timing the pistons to shoot up at particular moments. There need be no communication between the pistons at all. You just need a timer that tells each piston when to shoot up. If they all shoot up at the same time, there will be no wave, but if you time the pistons so that each one rises a very short time after the previous one, you generate a wave of pistons rising.
In principle, that wave can be as fast as you want it to be… yes, faster than the speed of light. No mass or information is moving in the direction of the wave. The pistons have mass, but the wave does not; it is perfectly permissible for the wave to travel faster than the speed of light. This is true as long as the pistons are just controlled by timers. If the behavior of one piston is initiated by the previous piston (a bit more like the human version), NOW there is communication required for the pistons to rise, and it is impossible for the wave to travel faster than light.
If you ponder the shadow situation with the wall, you’ll find that it is akin to the timing of pistons, except it is the timing of light hitting the ground. Timing, not communication. Timing, not mass. That is how equation 1 is derived.
If anyone wants to see the derivation, let me know from this site by email. I’ll provide it. I’ll make a link if enough people want it. I originally published it in Quantum (1996) volume 7(2), “Shady Computations” pages 34-36.
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