There’s Always An Angle

“No, Moire, when I said the glasses get dark or light depending I was talking about those glasses that just block out shiny, like from windows across the street when the Sun hits ’em just wrong.”

“I got this, Sy. That’s about polarized light, Feder, and polarized sunglasses. Sy and me, we talked about that when we were thinkin’ Star Trek weapons.”

“You guys talk about everything, Vinnie.”

“Pretty much. Anyhow, it goes back to how electrons make light. Electrons got charge and that makes an electric field around them, right? When you jiggle an electron up and down the field jiggles and sooner or later that’ll make some other electron jiggle like maybe in your eye and you see that as light. How’m I doing, Sy?”

“You’re on a roll. Keep it going.”

“Okay, so the electron doesn’t have to jiggle only up and down, it can do side‑to‑side if it feels like it or anything in between and the field goes along with all of that. When you got a lot of electrons doing that together, different‑angle waves go out and that poor second electron gets shoved all around the compass, right?”

“Hey, don’t all those jiggles just cancel each other out?”

“Nah, ’cause their timing’s off. They’re not in sync or nothing so the jiggles push in every direction random‑like.”

“How about lasers? I thought their waves all marched in sync.”

“They’re in sync strong‑and‑weak, but I guess whether they’re up‑and‑down in sync depends on the technology, right, Sy?”

“Right, Vinnie. Simple diode laser beams usually aren’t polarized, but special-purpose lasers may be designed with polarization in the package. Of course, any beam can be polarized if it’s bounced off something at just the right angle.”

“What’s the angle got to do with it, Moire?”

“I bet I know. Sy. Is that bounce angle connected to the prism stuff?”

“Nice shot, Vinnie. Carry on.”

“Ok, Feder, follow me ’cause this is a little complicated. Sy, can I borrow your whiteboard?”

“Sure.”

“Thanks. All right, this thick green wiggle is a regular light ray’s electric field, coming in at a low angle and jiggling in all directions. It hits a window or something, that’s the black line, and some of it gets reflected, that’s the red wiggle, and some gets through but not as much which is why the second green line is skinny. The fast‑slow marks are about wave speeds but it’s why the skinny wiggle runs at that weird angle. We good?”

“Mostly, I guess, but where does the polarization come in?”

“I’m gettin’ there. That’s what the dots are about. I’m gonna pretend that all those different polarization directions boil down to either up‑and‑down, that’s the wiggles, and side to side, that’s the dots. Think of the dots as wiggle coming out and going back in cross‑ways to the up‑and‑down. It’s OK to do that, right, Sy?”

“Done in the best families, Vinnie. Charge on.”

“So anyway, the up‑and‑down field can sink into the window glass and mess around with the atoms in there. They pass some of the energy down through the glass but the rest of it gets gets thrown back out like I show it.”

“But there’s no dots going down.”

“Ah-HAH! The side‑to‑side field doesn’t sink into the glass at all ’cause the atoms ain’t set up right for that. That side‑to‑side energy bounces back out and hits you in the eyes which is why you use those polarizing sunglasses.”

“But how do those glasses work is what I asked to begin with.”

“That’s all I got, Sy, your turn.”

“Nice job, Vinnie. How they cut the glare, Mr Feder, is by blocking only Vinnie’s side‑to‑side waves. Glare is mostly polarized light reflected off of horizontal surfaces like water and roadway. Block that and you’re happy. How they work is by selective absorption. The lenses are made of long, skinny molecules stretched out in parallel and doped with iodine molecules. Iodine’s a big, mushy atom with lots of loosely-held electrons, able to absorb many frequencies but only some polarizations. If a light wave passes by jiggling in the wrong direction, its energy gets slurped. No more glare.”

~~ Rich Olcott

Dark Glasses

My office door THUMPs as Richard Feder barrels in. Vinnie’s half out of his chair with his fists balled up but he settles back down when he sees who it is. “Moire, I gotta question.”

“Afternoon, Mr Feder. What brings you to the 12th floor of the Acme Building?”

“My dentist’s up here. They gave me these really dark glasses for when they aimed a bright light in my mouth to harden something in there so I wondered why’re they so dark an’ what about those glasses that can’t make up their minds?”

“Well, Mr Feder, as usual you’ve asked a jumbled question. Let’s see. The answers all boil down to what light is made of and what the glasses are made of.”

“I thought it was photon particles, Sy. The light, I mean.”

“It is, Vinnie, but photons only act like particles when they’re emitted and when they’re absorbed. In between, they act like waves. Dark glasses are all about photons as waves. The simplest case is the plain dark glasses.”

“Yeah, Moire, simple’s good.”

“They’re black because they’ve been doped with black chemicals. If your glasses are actually made of glass, the manufacturer probably dumped iron and sulfur into the melt. When heated those elements combine to form black iron sulfide particles spread throughout the mass. If the glasses are plastic, the manufacture mixed black dye into the formula. Either way, the more dopant added, the blacker the product and the fewer waves make it through the lens.”

“Great, Sy, but how come the black? I remember that Sun-spectrum poster that Al had up in his shop once. Lotsa sharp dark lines that Cathleen said were from different elements absorbing little slices of that rainbow background. But there were plenty of colors left over to make white.”

“Impressive memory, Vinnie. That was what, three years ago? Anyhow, those absorption lines come from separated atoms floating in the hot gas of the solar atmosphere. Quantum mechanics says that an isolated atom has a characteristic set of electron configurations, each with its own energy level. Say an incoming photon meets a gas atom. If the photon’s energy just matches the difference between the atom’s current configuration and some other configuration, suddenly the atom’s in the new configuration and no more photon. It has to match just right or no absorption. Those sharp lines come from that selectivity, OK?”

“So how do you get total black from selective atoms?”

“You don’t. You get black from less‑selective molecules and larger structures. Atoms right next to each other bring entanglement into the action — which electron is where on which atom? Many more configurations, many more differences between energy level pairs, many more lines that can overlap to make broad absorption bands. Suppose you’ve got some glass or plastic doped to have a single band sucking up everything between orange and green. Shine white light into it. Only red light and blue light come through. We see that as purple, a color that’s not even in the spectrum. Make that band even broader like it is with metals and rocks and iron sulfide; nothing gets through.”

“Then how do they do those glasses that get dark or light depending? The factory can’t put chemicals in but take ’em out temporary‑like when you walk inside.”

“Good point. In fact, the glass composition stays the same, sort of. The factory puts in chemicals that change their structure depending on the light level. If you dope optical glass with silver chloride crystallites, for instance, UV light can energize a chloride’s electron up to where it can leave the chloride and be captured by a silver ion. Do that with enough silver ions in the crystallite and you have a tiny piece of silver metal. Enough pieces and the glass looks gray, at least until heat energy joggles things back to the silver chloride ground state. For plastic lenses they use a subtler strategy — large‑ish molecules with spread‑out electron structures. UV light energizes an electron to another level and the molecule twitches to an opaque alternate form that relaxes when heat shakes it down.”

“Heat, huh? No wonder mine don’t work so good on the beach.”

~~ Rich Olcott

Things That Won’t Work

Vinnie gets a far-away look in his eye. I wait. “Ya know, Sy, there oughtta be a way.”

“A way to what?”

“I ain’t giving up on this faster-than-light communication stuff. I know Einstein said it couldn’t happen because it’d flip cause and effect and he didn’t like that, but that feels too much like philosophy books I’ve read that boil down to, ‘This thing can’t be true because I don’t want it to be.’ Maybe there’s something we ain’t thought of yet.”

“Lots of people have played with that challenge for decades. Do you have any fresh ideas?”

“A couple possibles. Lessee if I’ve got this straight. We’ve got two separate message channels going — one that works instant-like for information between entangled quantum thingies, and one for everything else that’s stuck at lightspeed or less. Suppose I’ve got two entangled pizzas— nah, we’re really talking quantum stuff like electrons and photons so I’ll just say particles. Suppose I’ve got two entangled particles that are some ugly mix of red and green but we know when they’re de-linked they’ll be opposite. I send one to you the regular way but they’re still linked. I look at the one I still got and it’s red, say. The same moment, yours instantly went green but you don’t know that yet until you look or you get status information from me through the not‑instant channel. So the problem is getting information to leak between the two channels, right?”

“That’s about the size of it.”

“OK, try this one. How about I use a magnetic field or something to force mine to red? And maybe a set time later I make it green to confirm I’m in control and it’s a real signal.”

“Sorry, as soon as you manipulate properties in part of an entangled system you break the entanglement and the other part is free to do whatever it wants to. Next?”

“Uhh … time synchronization. How about you and me set a certain time for me to look at mine? You can watch yours and when it flips or not you’ll know.”

“All that does is move the manipulation to the other end of the setup. Me looking at my particle resets yours to whatever color mine isn’t and that breaks the entanglement. Next?”

“Maybe something with a bunch of particles all entangled together? How about—”

“Nup, can’t base a strategy on that. Like everything else quantum, entanglement is statistical. There’s no guarantee that even in our two‑particle system I’ll see green if you see red — the odds are high but not 100%. There’s a proven theorem that says if two particles are ‘maximally entangled,’ adding a third to the system reduces the odds that any two will coordinate their behaviors. A bunch of particles would be even less stable. It’s called the monogamy theorem, care to guess why?”

“Physics fun with metaphors again, cute, but I can see this is a good one. You got anything?”

“Not having to do with entanglement, but I have been playing with a different idea, sort of a blank‑sky approach.”

“You mean blue‑sky.”

“Uh-uh, blank. Think about a sky made of dark matter. Dark matter’s subject to gravity but so far as we know it has absolutely no interaction with electromagnetism of any kind — doesn’t play with electrons, light waves, nothing. Einstein based part of his relativity work on Maxwell’s electromagnetism equations. In fact, that’s where the idea came from that ‘c‘ was the speed limit for the Universe. It was a good idea and there’s a huge amount of evidence that he was right. Everything in our Standard Model except the photon is subject to the Lorentz factor. Both light and gravity acting on normal matter travel at c‑speed. Well, maybe the value of c has something to do with how quarks work. Dark matter doesn’t have quarks. What if dark matter has a different speed limit, maybe a lot higher than c or even no limit at all? Maybe we could exploit that property somehow. How about a dark‑matter telegraph?”

“I’m thinking of my Grampa’s recipe for rabbit stew. ‘First you gotta catch your rabbit,’ he used to say,”

~~ Rich Olcott

The Pizza Connection

“Wait a minute, Sy. If Einstein’s logic proves we can’t have faster‑than‑light communication, what about all the entanglement hype I see in my science magazines?”

“Hype’s the right word, Vinnie. Entanglement’s a real effect, but it doesn’t play well as a communication channel.”

“OK, why not?”

“Let’s set the stage. We’re still in our personal spaceships and we’ve just ordered pizza from Eddie. The entanglement relationship is independent of time and distance so I’m going to skip over how fast we’re going and pretend that Eddie’s using transporter delivery technology, ok?”

“Fine with me,”

“Good. You order your usual double pepperoni with extra cheese, I ask for Italian sausage. Two pizza boxes suddenly appear on our respective mess tables. No reflection on Eddie, but suppose he has a history of getting orders crossed. The quantum formalism says because our orders were filled at the same time and in a single operation, the two boxes are entangled — we don’t know which is which. Before we open the boxes, each of us has a 50:50 shot of getting the right order. It’s like we’ve got a pair of Schrödinger pizzas, half one order and half the other until we look, right?”

“Won’t happen, Eddie’s a pro.”

“True, but stay with me here. I open my box and immediately I know which pizza you received, no matter how far away your ship is from mine. Is that instantaneous communication between us?”

“Of course not, I’m not gonna know which pizza either of us got until I open my own box. Then I’ll know what my meal’s gonna be and I’ll know what you’re having, too. Actually, I’m probably gonna know first because I get hungry sooner than you.”

“Good point. Anyway, entanglement doesn’t transmit human‑scale information. The only communication between us in our spaceships is still limited by Einstein’s rules. But this is a good setup for us to dig a little deeper into the quantum stuff. You rightly rejected the Schrödinger pizza idea because pizza’s human‑scale. One of those boxes definitely holds your pizza or else it definitely holds mine. There’s no in‑between mixtures with human‑scale pizzas. Suppose Eddie sent quantum‑scale nanopizzas, though. Now things get more interesting.”

“Eddie doesn’t mess up orders.”

<sigh> “Even Eddie can’t keep things straight if he sends out a pair of quantum‑scale pizzas. What’s inside a specific entangled box is called a local property. John Stewart Bell proved some statistical criteria for whether a quantum system’s properties are local or are somehow shared among the entangled objects. Scientists have applied his tests to everything from entangled photons up to little squares of diamond. They’ve tracked quantum properties from spin states to vibration modes. A lot of work went into plugging loopholes in Bell’s criteria.”

“What’d they find?”

“The results keep coming up non-local. Our quantum pizzas truly do not have separate characteristics hiding inside their boxes unless Eddie marked a box to destroy the symmetry. All the objects in an entanglement share all the applicable quantum property values until one object gets measured. Instantly, all the entangled objects snap into specific individual property values, like which box holds which pizza. They stop being entangled, too. That happens no matter how far apart they are. Those experimental results absolutely rule out the local‑property idea which was the most appealing version of the ‘underlying reality‘ that Einstein and Bohr argued over.”

“Wait, I can’t tell you anything faster than light, but these quantum thingies automatically do that instant‑like?”

“Annoying, isn’t it? But it’s a sparse form of messaging. My quantum pizza box can tell yours only two things, ‘I’ve been opened‘ and ‘I hold Italian sausage pizza.’ They’re one‑time messages at the quantum level and you as an observer can’t hear either one. Quantum theoreticians call the interaction ‘wave function collapse‘ but Einstein called it ‘spooky action at a distance.’ He hated even that limited amount of instantaneous communication because it goes directly against the first principle of Special Relativity. Relativity has been vigorously tested for over a century. It’s stood up to everything they’ve thrown at it — except for this little mouse nibbling at its base.”

~~ Rich Olcott

Speed Limit

“Wait, Sy, there’s something funny about that Lorentz factor. I’m riding my satellite and you’re in your spaceship to Mars and we compare notes and get different times and lengths and masses and all so we have to use the Lorentz factor to correct numbers between us. Which velocity do we use, yours or mine?”

“Good question, Vinnie. We use the difference between our two frames. We can subtract either velocity from the other one and replace v with that number. Strictly speaking, we’d subtract velocity components perpendicular to the vector between us. If I were to try to land on your satellite I’d have to expend fuel and energy to change my frame’s velocity to yours. When we matched frames the velocity difference would be zero, the Lorentz factor would be 1.0 and I’d see your solar array as a perfect 10×10‑meter square. Our clocks would tick in sync, too.”

“OK, now there’s another thing. That Lorentz formula compares our subtracted speeds to lightspeed c. What do we subtract to get c?”

“Deep question. That’s one of Einstein’s big insights. Suppose from my Mars‑bound spaceship I send out one light pulse toward Mars and another one in the reverse direction, and you’re watching from your satellite. No matter how fast my ship is traveling, Einstein said that you’d see both pulses, forward and backward, traveling at the same speed, c.”

“Wait, shouldn’t that be that your speed gets added to one pulse and subtracted from the other one?”

“Ejected mass works that way, but light has no mass. It measures its speed relative to space itself. What you subtract from c is zero. Everywhere.”

“OK, that’s deep. <pause> But another ‘nother thing—”

“For a guy who doesn’t like equations, you’re really getting into this one.”

“Yeah, as I get up to speed it grows on me. HAW!”

“Nice one, you got me. What’s the ‘nother thing?”

“I remembered how velocity is speed and direction but we’ve been mixing them together. If my satellite’s headed east and your spaceship’s headed west, one of us is minus to the other, right? We’re gonna figure opposite v‑numbers. How’s that work out?”

“You’re right. Makes no difference to the Lorentz factor because the square of a negative difference is the same as the square of its positive twin. You bring up an important point, though — the factor applies to both of us. From my frame, your clock is running slow. From your frame, mine’s the slow one. Einstein’s logic says we’re both right.”

“So we both show the same wrong time, no problem.”

“Nope, you see my clock running slow relative to your clock. I see exactly the reverse. But it gets worse. How about getting your pizza before you order it?”

“Eddie’s good, he ain’t that good. How do you propose to make that happen?”

“Well, I don’t, but follow me here. <working numbers on Old Reliable> Suppose we’re both in spaceships. I’m loafing along at 0.75c relative to Eddie’s pizza place on Earth and your ship is doing 3c. Also, suppose that we can transmit messages and mass much faster than lightspeed.”

“Like those Star Trek transporters and subspace radios.”

“Right. OK, at noon on my personal clock you tell me you’ve ordered pizza so I get one, too. Eddie slaps both our pizzas into his transporter 10 minutes later. The math works out that according to my clock you get your pizza 8.9 minutes before you put in your order. You like that?”

“Gimme a sec … nah, I don’t think so. If I read that formula right with v1 being you and v2 being me, if you run that formula for what I’d see with my velocity on the bottom, that’s a square root of a minus which can’t be right.”

“Yup, the calculation gives an imaginary number, 4.4i minutes, whatever that means. So between us we have two results that are just nonsense — I see effect before cause and you see a ridiculous time. To avoid that sort of thing, Einstein set his speed limit for light, gravity and information.”

“I’m willing to keep under it if you are.”

“Deal.”

~~ Rich Olcott

The Relativity Factor

“Sy, it’s nice that Einstein agreed with Rayleigh’s wave theory stuff but why’d you even drag him in? I thought the faster‑than‑light thing was settled.”

“Vinnie, faster‑than‑light wasn’t even an issue until Einstein came along. Science had known lightspeed was fast but not infinite since Rømer measured it in Newton’s day. ‘Pretty fast,’ they said, but Newtonian mechanics is perfectly happy with any speed you like. Then along came Einstein.”

“Speed cop, was he?”

“Funny, Vinnie. No, Einstein showed that the Universe enforces the lightspeed limit. It’s central to how the Universe works. Come to think of it, the crucial equation had been around for two decades, but it took Einstein to recognize its significance.”

“Ah, geez, equations again.”

“Just this one and it’s simple. It’s all about comparing v for velocity which is how fast something’s going, to c the speed of light. Nothing mystical about the arithmetic — if you’re going half the speed of light, the factor works out to 1.16. Ninety‑nine percent of c gives you 7.09. Tack on another 9 and you’re up to 22.37 and so on.”

“You got those numbers memorized?”

“Mm-hm, they come in handy sometimes.”

“Handy how? What earthly use is it? Nothing around here goes near that fast.”

“Do you like your GPS? It’d be useless if the Lorentz factor weren’t included in the calculations. The satellites that send us their sync signals have an orbit about 84 000 kilometers wide. They run that circle once a sidereal day, just shy of 86 400 seconds. That works out to 3 kilometers per second and a Lorentz factor of 1.000 005.”

“Yeah, so? That’s pretty close to 1.0.”

“It’s off by 5 parts per million. Five parts per million of Earth’s 25 000-mile circumference is an eighth of a mile. Would you be happy if your GPS directed you to somewhere a block away from your address?”

“Depends on why I’m going there, but I get your point. So where else does this factor come into play?”

“Practically anywhere that involves a precision measurement of length or duration. It’s at the core of Einstein’s Special Relativity work. He thought about observing a distant moving object. It’s carrying a clock and a ruler pointed along the direction of motion. The observer would see ticks of the clock get further apart by the Lorentz factor, that’s time dilation. Meanwhile, they’d see the ruler shrink by the factor’s inverse, that’s space compression.”

“What’s this ‘distant observer‘ business?”

“It’s less to do with distance than with inertial frames. If you’re riding one inertial frame with a GPS satellite, you and your clock stay nicely synchronized with the satellite’s signals. You’d measure its 1×1‑meter solar array as a perfect square. Suppose I’m riding a spaceship that’s coasting to Mars. I measure everything relative to my own inertial frame which is different from yours. With my telescope I’d measure your satellite’s solar array as a rectangle, not a square. The side perpendicular to the satellite’s orbit would register the expected 1 meter high, but the side pointing along the orbit would be shorter, 1 meter divided by the Lorentz factor for our velocity difference. Also, our clocks would drift apart by that Lorentz factor.”

“Wait, Sy, there’s something funny about that equation.”

“Oh? What’s funny?”

“What if somebody’s speed gets to c? That’d make the bottom part zero. They didn’t let us do that in school.”

“And they shouldn’t — the answer is infinity. Einstein spotted the same issue but to him it was a feature, not a bug. Take mass, for instance. When they meet Einstein’s famous E=mc² equation most people think of the nuclear energy coming from a stationary lump of uranium. Newton’s F=ma defined mass in terms of a body’s inertia — the greater the mass, the more force needed to achieve a certain amount of acceleration. Einstein recognized that his equation’s ‘E‘ should include energy of motion, the ½mv² kind. He had to adjust ‘m‘ to keep F=ma working properly. The adjustment was to replace inertial mass with ‘relativistic mass,’ calculated as inertial mass times the Lorentz factor. It’d take infinite force to accelerate any relativistic mass up to c. That’s why lightspeed’s the speed limit.”

~~ Rich Olcott

Three-speed Transmission

“Have I got this straight, Sy? You’re saying that prisms throw rainbows because light goes slower through glass than in air and that bends the beam, but every frequency lightwave bends a different amount. Also you’re saying all the bending happens when speeds switch at the glass face, not inside the glass. Am I right so far?”

“Perfect, Vinnie, but you skipped an important detail.”

“Which one?”

“Snell’s ‘index of refraction‘, the ratio of wave speed in vacuum to wave speed in the medium. The higher the frequency, the higher the speed in the medium so the index decreases towards 1.0. The definition lets us calculate wave speed in the medium from that frequency’s refraction index. For most materials the index is usually greater than 1.0, meaning that the speed inside the material is usually slower than in space.”

“Still using those ‘most‘ and ‘usually‘ weasel‑words.”

“Guilty as charged, because we’ve finally gotten to the ‘multiple speeds of light‘ thing. Which means I need more precise wording. The wave speed we’ve been talking about so far applies to a specific part of the wave, say the peak or trough. Those are wave phases, so I’m going to call that speed the ‘phase speed‘, OK?”

“Fine with me.”

“Good, because the second speed is different. Among his many important contributions, Lord Rayleigh pointed out that you can’t have a pulse that’s one pure frequency. A single‑frequency wave never starts and never ends. Do you remember the time I combined waves to draw a camel?”

“You did, mostly, but there was funny stuff at his nose and butt.”

“Because I only included about a hundred component waves. It’d take many more to kill those boundary zig‑zags. Any finite wave has the same issue. Rayleigh said that an individual wave has a phase speed, but any ‘peculiarity,’ like a pulse rise or fall, could only be created by a group of waves. The peculiarity could travel at a different speed from the component waves, like a pair of scissors where the cutting point moves faster than either blade.”

“Sounds like carrier wave and sidebands on my ham radio. But if different frequencies have different speeds they’d get all out of sync with each other. How does a photon stay in one piece?”

“The vacuum is non-dispersive — the photon’s component waves all travel at the same speed and stay together. If a medium absorbs some frequency, that makes it dispersive and that changes things.”

“Ah, that’s why you hedged about transparency.”

“Exactly. Throw in a few absorbing atoms, like cobalt that absorbs red or gold that absorbs blue, and you get interesting effects from your sideband components interacting. Skipping some math, the bottom line is simple and cute. The group speed’s equation is just like the phase speed’s except there’s a positive or negative correction term in the denominator.”

“Sy, I don’t like equations, remember? I suppose f is frequency in your correction term but what’s slope?”

“That’s a measure of how rapidly the index changes as the frequency changes. For most frequencies and most media, the slope is very slightly negative because the index slowly descends towards 1.0 at high energies. The vg fraction’s denominator stays just less than nf so the group goes slightly faster than the phase. Near an absorption line, though, things get sloppy. Waves that are just a little off the absorber’s favorite frequency can still interact with it. That changes their speed and the ‘corrected’ refraction index.”

“Gimme a sec … guess I’m OK with the positive slopes but there’s that yellow part where the slope is negative. Wouldn’t that make the fraction’s bottom smaller and the group speed higher?”

“Certainly. In fact, under the right conditions the denominator can be less than 1.0. That pushes the group speed above c — faster than light in vacuum, even though the component waves all run slower than vacuum lightspeed. It’s only the between‑component out‑of‑syncness relationship that scissors along beyond c.”

“You said there’s a third speed?”

“Signals. In a dispersive medium those sideband waves get chaotic and can’t carry information. Wave theory and Einstein agree — chaos may be able to travel faster than light, but information can’t.”

~~ Rich Olcott

Chasing Rainbows

“C’mon, Sy, Newton gets three cheers for tying numbers to the rainbow’s colors and all that, but what’s it got to do with that three speeds of light thing which is where we started this discussion?”

“Vinnie, they weren’t just numbers, they were angles. The puzzle was why each color was bent to a different degree when entering or leaving the prism. That was an inconvenient truth for Newton.”

“Inconvenient? There’s a loaded word.”

“Indeed. A little context — Newton was in a big brouhaha about whether light was particles or waves. Newton was a particle guy all the way, battling wave theory proponents like Euler and Descartes and their followers on the Continent. Even Hooke in London had a wave theory. Newton’s problem was that his beam deflections happened right at the prism’s air‑glass interfaces.”

“What difference does that … wait, you mean that there’s no bending inside the prism? Light inside still goes straight but in a different direction?”

“That’s it, exactly. The deflection angles are the same, whether the beam hits the prism near the short‑path tip or the long‑path base. No evidence of further deviation inside the prism unless it has bubbles — Newton had to discard or mask off some bad prisms. Explaining the no‑curvature behavior is difficult in a particle framework, easy in a wave framework.”

“Really? I don’t see why.”

Left: faster medium, right: slower medium
Credit: Ulflund, under Creative Commons 1.0

“Suppose light is particles, which by definition are local things affected only by local forces. The medium’s effects on a particle would happen in the bulk material rather than at the interface. The effect would accumulate as the particles travel further through the medium. The bend should be a curve. Unfortunately for Newton, that’s not what his observations showed.”

“OK, scratch particles. Why not scratch waves, too?”

“Waves have no problem with abrupt variation at an interface, They flip immediately to a new stable mode. For example. here’s an animation showing an abrupt speed change at the interface between a fast‑travel medium like air and a slow‑travel medium like glass or water. See how one end of each bar gets slowed down while the other end is still moving at speed? By the time the whole bar is inside, its path has slewed to the refraction angle.”

“Like a car sliding on ice when a rear wheel sticks for an moment, eh Sy?”

“That was not a fun ride, Vinnie.”

“I enjoyed it. Whatever, I get how going air‑to‑glass or vice‑versa can change a beam’s direction. But if everything’s going through the same angle, how do rainbows happen?”

“Everything doesn’t go through the same angle. Frequencies make a difference. Go back to the video and keep your eye on one bar as it sweeps up the interface. See how the sweep’s speed controls the deflection angle?”

“Yeah, if the sweep went slower the beam would get a chance to bend further. Faster sweeps would bend it less. But what could change the sweep speed?

“Two things. One, change the medium to one with a different transmission speed. Two, change the wave itself so it has a different speed. According to Snell’s Law, the important parameter for a pair of media is their ratio of fast‑speed divided by slow‑speed. If the fast medium is a vacuum that ratio is the slow medium’s index of refraction. The greater the index, the greater the bend.”

“Changing the medium doesn’t apply. I got one prism, it’s got one index, but I still get a whole rainbow.”

“Right, rainbows are about how one prism treats a bunch of waves with different time and space frequencies.”

“Space frequency?”

“If you measure a wave in meters it’s cycles per meter.”

“Wavelength upside down. Got it.”

“Whether you figure in frequencies or intervals, the wave speed works out the same.”

“Speed of light, finally.”

“Point is, when a wave goes through any medium, its time frequency doesn’t change but its space frequency does. Interaction with local charge shortens the wavelength. Short‑wavelength blue waves are held back more than long‑wavelength red ones. The different angles make your rainbow. The hold‑back is why refraction indices are usually greater than one.”

“Usually?”

~~ Rich Olcott

Through A Prism Brightly

Familiar footsteps outside my office. “C’mon in, Vinnie, the door’s open.”

“Hi, Sy, gotta minute?”

“Sure, Vinnie, business is slow. What’s up?”

“Business is slow for me, too. I was looking over some of your old posts—”

“That slow, eh?”

“You know it. Anyway, I’m hung up on that video where light’s got two different speeds.”

“Three, really.”

“That’s even worse. What’s the story?”

“Well, first thing, it depends on where the light is. If you’re out in the vacuum, far away from atoms, they’re all the same, c. Simple.”

“Matter messes things up, then.”

“Of course. Our familiar kind of matter, anyway, made of charges like quarks and electrons. Light’s whole job is to interact with charges. When it does, things happen.”

“Sure — photon bangs into a rock, it stops.”

“It’s not that simple. Remember the wave-particle craziness? Light’s a particle at either end of its trip but in between it’s a wave. The wave could reflect off the rock or diffract around it. Interstellar infra-red astronomy depends upon IR scooting around dust particles so we can see the stars behind the dust clouds. What gets interesting is when the light encounters a mostly transparent medium.”

“I get suspicious when you emphasize ‘mostly.’ Mostly how?”

“Transparent means no absorption. The only thing that’s completely transparent is empty space. Anything made of normal matter can’t be completely transparent, because every kind of atom absorbs certain frequencies.”

“Glass is transparent.”

“To visible light, but even that depends on the glass. Ever notice how cheap drinking glasses have a greenish tint when you look down at the rim? Some light absorption, just not very much. Even pure silica glass is opaque beyond the near ultraviolet. … Okay, bear with me on this. Why do you suppose Newton made such a fuss about prisms?”

“Because he saw it made a rainbow in sunlight and thought that was pretty?”

“Nothing so mild. We’re talking Newton here. No, it had to do with one of his famous ‘I’m right and everyone else is wrong‘ battles. Aristotle said that sunlight is pure white‑color, and that objects emit various kinds of darkness to overcome the white and produce colors for us. That was academic gospel for 2000 years until Newton decided it was wrong. He went to war with Aristotle using prisms as his primary weapons.”

“So that’s why he invented them?”

“No, no, they’d been around for millennia, ever since humans discovered that prismatic quartz crystals in a beam of sunlight throw rainbows. Newton’s innovation was to use multiple prisms arrayed with lenses and mirrors. His most direct attack on Aristotle used two prisms. He aimed the beam coming out of the first prism onto a reversed second prism. Except for some red and violet fringes at the edges, the light coming out of the second prism matched the original sunlight beam. That proved colors are in the light, not in Aristotle’s darknesses.”

“Newton won. End of story.”

“Not by a long shot. Aristotle had the strength of tradition behind him. A lot of Continental academics and churchmen had built their careers around his works. Newton’s earlier battles had won him many enemies and some grudging respect but few effective allies. Worse, Newton published his experiments and observations in a treatise which he wrote in English instead of the conventional scholarly Latin. Typical Newtonian belligerence, probably. The French academicians reacted by simply rejecting his claims out of hand. It took a generational turnover before his thinking was widely accepted.”

“Where do speeds come into this?”

“Through another experiment in Newton’s Optics treatise. If he used a card with a hole in it to isolate, say, green light in the space between the two prisms, the light beam coming from the second prism was the matching green. No evidence of any other colors. That was an important observation on its own, but Newton’s real genius move was to measure the diffraction angles. Every color had its own angle. No matter the conditions, any particular light color was always bent by the same number of degrees. Newton had put numbers to colors. That laid the groundwork for all of spectroscopic science.”

“And that ties to speed how?”

~~ Rich Olcott

‘Twixt A Rock And A Vortex

A chilly late December walk in the park and there’s Vinnie on a lakeside bench, staring at the geese and looking morose. “Hi, Vinnie, why so down on such a bright day?”

“Hi, Sy. I guess you ain’t heard. Frankie’s got the ‘rona.”

Frankie??!? The guys got the constitution of an ox. I don’t think he’s ever been sick in his life.”

“Probably not. Remember when that bug going around last January had everyone coughing for a week? Passed him right by. This time’s different. Three days after he showed a fever, bang, he’s in the hospital.”

“Wow. How’s Emma?”

“She had it first — a week of headaches and coughing. She’s OK now but worried sick. Hospital won’t let her in to see him, of course, which is a good thing I suppose so she can stay home with the kids and their schoolwork.”

“Bummer. We knew it was coming but…”

“Yeah. Makes a difference when it’s someone you know. Hey, do me a favor — throw some science at me, get my mind off this for a while.”

“That’s a big assignment, considering. Let’s see … patient, pandemic … Ah! E pluribus unum and back again.”

“Come again?”

“One of the gaps that stand between Physics and being an exact science.”

“I thought Physics was exact.”

“Good to fifteen decimal places in a few special experiments, but hardly exact. There’s many a slip ‘twixt theory and practice. One of the slips is the gap between kinematic physics, about how separate objects interact, and continuum physics, where you’re looking at one big thing.”

“This is sounding like that Loschmidt guy again.”

“It’s related but bigger. Newton worked on both sides of this one. On the kinematics side there’s billiard balls and planets and such. Assuming no frictional energy loss, Newton’s Three Laws and his Law of Gravity let us calculate exact predictions for their future trajectories … unless you’ve got more than three objects in play. It’s mathematically impossible to write exact predictions for four or more objects unless they start in one of a few special configurations. Newton didn’t do atoms, no surprise, but his work led to Schrödinger’s equation for an exact description of single electron, single nucleus systems. Anything more complicated, all we can do is approximate.”

“Computers. They do a lot with computers.”

“True, but that’s still approximating. Time‑step by time‑step and you never know what might sneak in or out between steps.”

“What’s ‘continuum‘ about then? Q on Star trek?”

“Hardly, we’re talking predictability here. Q’s thing is unpredictability. A physics continuum is a solid or fluid with no relevant internal structure, just an unbroken mass from one edge to the other. Newton showed how to analyze a continuum’s smooth churning by considering the forces that act on an imaginary isolated packet of stuff at various points in there. He basically invented the idea of viscosity as a way to account for friction between a fluid and the walls of the pipe it’s flowing through.”

“Smooth churning, eh? I see a problem.”

“What’s that?”

“The eddies and whirlpools I see when I row — not smooth.”

“Good point. In fact, that’s the point I was getting to. We can use extensions of Newton’s technique to handle a single well‑behaved whirlpool, but in real life big whirlpools throw off smaller ones and they spawn eddies and mini‑vortices and so on, all the way down to atom level. That turns out to be another intractable calculation, just as impossible as the many‑body particle mechanics problem.”

“Ah‑hah! That’s the gap! Newton just did the simple stuff at both ends, stayed away from the middle where things get complicated.”

“Exactly. To his credit, though, he pointed the way for the rest of us.”

“So how can you handle the middle?”

“The same thing that quantum mechanics does — use statistics. That’s if the math expressions are average‑able which sometimes they’re not, and if statistical numbers are good enough for why you’re doing the calculation. Not good enough for weather prediction, for instance — climate is about averages but weather needs specifics.”

“Yeah, like it’s just started to snow which I wasn’t expecting. I’m heading home. See ya, Sy.”

“See ya, Vinnie. … Frankie. … Geez.

~~ Rich Olcott