Demote Indigo (5/24/13)
I've been thinking about the rainbow, and it's not clear to me that indigo should have its own band. It should be demoted, much like Pluto was demoted from being a planet. Don't get me wrong, I'm not saying indigo isn't a color in the rainbow, or that The Indigo Girls shouldn't have a band. But teal is also a color in the rainbow, and no one is lobbying for teal to have its own band. Indigo is pretty much just dark blue, it's neither special nor distinct when you look at a rainbow.* I'm pretty sure that indigo is only a band because the people who decided these things liked the number 7. In addition to being a lame band, removing indigo gives a rational symmetry to the rainbow. With 6 bands, the rainbow is in alignment with introductory art classes, where we learn of three primary colors (red, yellow, blue) and three complementary colors (green, violet, orange). Remove indigo from the rainbow, badda bing, badda boom, Nature, art and physics are one. I know that ROY G BIV is an iconic acronym, so in order to remember the new system I recommend the expression, "ROY G BIV, but I am demoted."
I'm just here to help.
While I'm on the topic of colors and rainbows, I would also like to advocate using "violet" only to refer to the short wavelengths in the rainbow, and to reserve "purple" for combinations of red and blue. Admittedly, this probably only bothers me. Still, it's my blog. At least I think this is a blog. In any case, although our eyes cannot distinguish purple from violet, they are fundamentally different in terms of energy spectrum and would behave differently, say, when passed through a prism. Even a trained eye cannot discern the difference between combinations of wavelengths of light and a single wavelength. This is in contrast to the trained ear which can discern the different notes within a chord.
The similarity between purple and violet fascinated 19th century metaphysicists. They were hot on analogies and statements like, "As above, so below." Here's how it came up: the entire spectrum of visible light (the light in the rainbow) goes from wavelengths of about 390 nanometers (nm) (violet) to about 700 nm (red). Thus, the entire visible spectrum is slightly smaller than a doubling of frequencies/wavelengths. In musical terms, this would be just shy of an octave. To a metaphysicist, this meant that just after violet, another red was coming, perhaps a pastel, I don't know, but another red. It made sense of why violet could be mimicked by red and blue. In the metaphysical theory, violet is a transitional "mix" of red and blue, except the red involved is the next octave above the one we see. I always liked this idea, although it makes little physical or physiological sense. The facts that visible light covers a frequency range slightly smaller than an octave and that violet looks like blue plus red are seductive. However, as I used to continually remind graduate students, "A lot of things make sense, but only a few things are true." It's why we do experiments rather than see who wins a debate about what ought to be.
*I am compelled here to assure the disgruntled reader that I am fully aware that the rainbow represents a continuous spectrum of wavelengths and the bands we see are subjective in nature, dependent on the mechanistic vagaries of the pigments in our eyes and our cognitive architecture. I'm not addressing this at the moment because it is obvious and boring, unless you didn't know that, in which case it's pretty fascinating. Either way, we perceive bands, so I get to talk about them.
The Speed of Darkness (5/27/13)
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.
The Speed of Darkness: Singularities (5/28/13)
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.
The Speed of Darkness: Resolution (5/29/13)
I don't know. 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.