Can Shadows Move Faster Than Light?

A delightfully clear demonstration of a bug crawling up a window and casting a monstrous shadow across the universe is provided by David Griffiths on page 427 of his 2005 Introduction to Quantum Mechanics. This is touted as a demonstration of movement faster than light that carries no energy, cannot transmit a message, and cannot be causal
 
The author's first response (remember “intuition: useful, powerful, possibly wrong, potentially dangerous”) was that no shadow would move faster than light, that it could carry “energy”, can transmit a message, and can be causal. Careful geometry (2012-02-29) leads to the suggestion that, while message of the shadow takes distance/c seconds to arrive on the screen, multiple points on the screen may fall into shadow in less time than it takes light to travel from one of those points to the other. Afterward, those points can compare notes on when the shadow fell and decide how fast the “event” might have been traveling. As humans we have no trouble saying “those happened at the same time” or “everything happened at once.” So we should have no trouble imagining two events that happen so close together that one could not warn or notify the other. “It happened so fast, I couldn't put on the brakes” or “the wave came in so quick I couldn't shout to warn the other surfer.”. So the author was wrong to suggest (as written 2012-02-22) that “seeing the shadow or its edge moves faster then light is based on experimental and experiential fallacy.” Investigating why the edge may move faster is an interesting education.

If our creepy bug moves up at v, the image on the screen moves up at v' If the bug is 1 meter from the light source moving at 3m/sec (one hundred millionth the speed of light) and the screen is a 1x10^8 meters from the source, v is 3e-8c and v' is apparently 1c. Light takes longer to get to the screen the further the bug gets from the center line from projector to the screen. For now the screen is perpendicular to that center line. If we increase the screen distance to 2x10^8 meters, v' is apparently 2c. This will take some care to work out.

The bug starts at time 0 on the line from the projector to the screen, perpendicular to the screen's position. The bug is Bd from the point light source, the screen is Sd. The bug moves upward at v. The bug's position is measured as s above mid-line perpendicular to the screen and s=vt. Bd is considered tiny compared to Sd. At time 0, the screen has been lit. Light takes Sd/c to reach the screen, so screen position of the shadow s' = 0 occurs at t' = Sd/c. As the bug moves up, s=vt. At a given t, the shadow position will be s' = vtSd/Bd and the time for light to get to that point (or stop getting to that point) will be t+sqrt(Sd^2+s'^2)/c. The ds'/dt' with respect to t is the “speed” of the edge of the shadow. Numerically, calculating delta s'/delta t' makes it clear that if one is far enough away and the angle between the screen and the ray of light is not too great, that derivative will be higher than c.

The shadow can transmit information to a point on the screen, it can effectively transmit a lack of energy to trigger a reaction at that point on the screen, and it can be causal. That point on the screen cannot communicate with any other faster than the speed of light, so it may not be able to warn a nearby point that the shadow is coming if the shadow has also have fallen on that point before the message arrives. But there is no cause to abandon causality and no point in losing energy over the lack of energy in a shadow.

While the two turning points in the education of a physicist are Quantum Mechanics and Statistical Mechanics, we want to avoid over-learning and over-generalization too.

The Education Continues