The Speed of Darkness: Singularities (Part 2 of 3)

If you’ve stumbled here in your search for the Speed of Darkness Series, I encourage you to first read Part 1 of this series, The Speed of Darkness.

So, shadows traveling faster than the speed of light are one thing, singularities in equations are quite another.

First, I should put some specifics on this situation for clarity. Let’s say the wall is 20 feet tall. That’s pretty specific. And let’s say the sun is relatively low in the sky, so that the angle that light hits the ground after just missing the top of the wall is 20°. Also specific. Under these conditions, the length of the shadow cast by the wall is about 55 feet. This comes from the equation, for any height of the wall (Y), and an incident angle of light (θ), the length of the shadow (L) is:

Equation 2) L = Y*cot(θ)

For Y = 20 feet and θ = 20°, L ≈ 55 feet. In case you’re playing at home, the cotangent (cot) is just cosine/sine. That is to say, cot(θ) = cos(θ)/sin(θ). Frankly, if you’re still giving me the benefit of the doubt about equation 1 at this point, you should really trust me about equation 2. It’s a far easier slog. I’m getting to the problematic singularity, I really am.

We’ll lower the height of the wall from 20 feet to 10 feet quickly, but much slower than the speed of light. In that case, the shadow on the ground will shorten from a length of 55 feet to 27.5 feet. Sensible. Sometimes things should be sensible, and here is such a thing. As an aside, it is worth noting that as the wall is lowered, the angle (θ) of the light hitting the ground remains constant. That’s because we’re assuming the sun is far away, which it is, and so the “rays” of light hitting the wall are essentially parallel to each other.

So equation 1 describes the velocity of the shadow’s edge as it shortens from 55 to 27.5 feet in length as a function of the speed of the wall being lowered (kc). That equation demonstrates that as k is raised in value to be closer and closer to the value of sin(θ), the speed of the shadow gets faster and faster. When k = sin(θ), we have an equation with zero in the denominator. It is difficult to express how big a problem this is potentially.

In mathematics, you can never, never, ever divide by zero. It’s not just a big number, it is indecipherable mathematically. It means that something is occurring that is not mathematically valid. It is a singularity. If you let me divide by zero, I can prove 2 = 1. I’m not exaggerating. I can prove 2 = 1 if you let me divide by zero just once. That’s how big a deal it is. If you’re curious how that is done, I can show you later, just ask (it’s also a pretty easy google).

Having a singularity in an equation that is supposed to describe a physical problem, like our beloved equation 1, is often a clear sign that the equation is incorrect. Physicists generally frown on singularities, although they have been embraced relatively recently via black holes and big bangs… but we’re just dealing with a wall that gets shorter.

Part of why I love equation 1 and this entire problem is because the singularity in this case has a perfectly good physical interpretation. The shadow shortens from 55 to 27.5 feet. It goes faster and faster as k approaches sin(θ). What happens when k = sin(θ)? When the wall is lowered at a speed of kc, where k = sin(θ), light appears from 55 to 27.5 feet everywhere instantly. Instantly. At that exact speed of the wall lowering, the idea of the velocity of the shadow makes no sense. It has no value. “Instantly” is no more a speed than “infinity” is a number. So even at the singularity of equation 1, the physical situation is aptly described. Come on! That is lovely. Math is beautiful. Even the Qualmish smile.

As long as we stay slower than the speed of light, nothing prevents k from becoming larger than sin(θ). In that case the sign of the velocity of the shadow becomes negative; it reverses direction. <tilting head> If equation 1 is true truly truly, this must also have a physical interpretation. I’ve already indicated what that interpretation is above. I’ll elaborate next on how that allows the shadow to arrive at its destination prior to leaving its starting point. I love this. Don’t you? I do.

Equation 1 is correct, you’ll see, and it reveals what Nature’s speed limit really means. This problem is NOT an exception to the physical laws governing the velocity of things. Exceptions DO NOT prove rules. At their best, apparent exceptions explore the rule and its appropriate applications. This is what equation 1 is doing for us.