There is a general misunderstanding about Nature’s speed limit. It is often asserted that nothing can travel faster than the speed of light in a vacuum. This is incorrect. There are several easily measurable things that can move much faster than the speed of light. One class of such things is shadows, more precisely, the edge of a shadow.
The idea that a shadow can move faster than the light that produces it can be a stumbling block even for some folks who should know better, for example, PhD’s in physics. The proof only requires simple trigonometry.
First, let me clarify the claim. Imagine a straight wall standing perpendicular to the ground (90° angle). Sunlight shines from one side of the wall casting a shadow onto the ground; the angle that the light hits the ground will be smaller when the sun is lower in the sky. We’ll call that angle “θ”. In the present example, θ is less than 90° and more than 0°. The claim is that lowering (decreasing the height of) the wall at speeds far slower than the speed of light can cause the velocity of the shadow to move as fast as one likes: 10 times the speed of light, 100 times the speed of light, all while the wall moves slower than the speed of light. It’s easiest to imagine the wall being lowered some set distance and then stopping, causing the shadow to shorten some set distance and stop. In fact, if you lower the wall “just right”, the edge of the shadow can arrive at its destination prior to moving from its starting point. On the other hand, raising the wall will never cause the shadow to move faster than the speed of light. Those are the claims.
The speed of light will be called “c”. The speed of the wall being lowered is kc, where k is always less than one. This assures that the speed of the wall is always less than c in the statements that follow. Some people will find the proof tiresome, others will be hungry for it. I’ll get to the proof. It’s not that tricky, but first the result.
For a wall being lowered at velocity kc, the velocity (v) of the shadow’s edge along the ground for any given θ is:
Equation 1) v = kc*cos(θ)/(sin(θ) – k)
That’s it.
sin(θ) and cos(θ) will always be between zero and one in this example; so will k, as defined above. There are only two variables in this equation, the angle at which the light hits the ground (θ), and the velocity of the wall (kc; k is really the variable here, because c, the speed of light, is a constant). The height of the wall and how far it moves play no role.
Assuming for the moment that I know what I’m doing and the equation is true, all the claims are implicit in equation 1. The key is the denominator: sin(θ)-k. Because sin(θ) has to be between zero and one, and k has to be between zero and one, for any θ, the wall can be lowered at a velocity kc where k is as close to the value of sin(θ) as you like. That is to say, sin(θ)-k can be as small as you would like it to be. In fact, it could be equal to zero. (Zero in the denominator! Oh no! I’ll return to that singularity later.) For now the point is that v can be made to be as large as you like by bringing k closer and closer to sin(θ). All of this while k < 1, and the wall is moving slower than the speed of light (kc). For example, using equation 1, if θ is 20° and k is 0.30 (so the wall is being lowered at 30% the speed of light), the velocity of the shadow on the ground is more than 6 times the speed of light. For the same θ and k = 0.33, the shadow moves at more than 25 times the speed of light (sin(θ) ≈ 0.342 when θ = 20°).
Enough for now. I’ll return with the singularity and how the shadow can reach its destination before leaving its origin, all without breaking any law of physics. Then the proof that equation 1 is true.
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