Virial Yang And Yin

On By rolcottIn Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

“But Mr Moire, how does the Virial Equation even work?”

“Sometimes it doesn’t, Jeremy. There’s an ‘if’ buried deep in the derivation. It only works for a system in equilibrium. Sometimes people use the equation as a test for equilibrium.”

“Sorry, what does that mean?”

“Let’s take your problem galaxy cluster as an example. Suppose the galaxies are all alone in the Universe and far apart even by astronomical standards. Gravity’s going to pull them together. Galaxy i and galaxy j are separated by distance Rij. The potential energy in that interaction is Vij = G·mi·mj / Rij. The R‘s are very large numbers in this picture so the V attractions are very small. The Virial is the average of all the V’s so our starting Virial is nearly zero.”

“Nearly but not quite zero, I get that. Wait, if the potential energy starts near zero when things are far apart, and a falling‑in object gives up potential energy, then whatever potential energy it still has must go negative.”

“It does. The total energy doesn’t change when potential energy converts to kinetic energy so yes, we say potential energy decreases even though the negative number’s magnitude gets larger. It’d be less confusing if we measured potential energy going positive from an everything-all-together situation. However, it makes other things in Physics much simpler if we simply write (change in potential energy)+(change in kinetic energy)=0 so that’s the convention.”

“The distances do eventually get smaller, though.”

“Sure, and as the objects move closer they gain momentum and kinetic energy. Gaining momentum is gaining kinetic energy. You’re used to writing kinetic energy as T=m·v²/2, but momentum is p=m·v so it’s just as correct to write T=p²/2m. The two are different ways of expressing the same quantity. When a system is in equilibrium, individual objects may be gaining or losing potential energy, but the total potential energy across the system has reached its minimum. For a system held together by gravity or electrostatic forces, that’s when the Virial is twice the average kinetic energy. As an equation, V+2T=0.”

“So what you’re saying is, one galaxy might fall so far into the gravity well that its potential energy goes more negative than –2T. But if the cluster’s in equilibrium, galaxy‑galaxy interactions during the fall‑in process speed up other galaxies just enough to make up the difference. On the flip side, if a galaxy’s already in deep, other galaxies will give up a little T to pull it outward to a less negative V.”

“Well stated.”

“But why 2? Why not 3½ or some other number?”

“The 2 comes from the kinetic energy expression’s ½. The multiplier could change depending on how the potential energy varies with distance. For both gravity and electrostatic interactions the potential energy varies the same way and 2 is fine the way it is. In a system with a different rule, say Hooke’s Law for springs and rubber bands, the 2 gets multiplied by something other than unity.”

“All that’s nice and I see how the Virial Equation lets astronomers calculate cluster‑average masses or distances from velocity measurements. I suppose if you also have the masses and distances you can test whether or not a collection of galaxies is in equilibrium. What else can we do with it?”

“People analyze collections of stars the same way, but Professor Hanneken’s a physicist, not an astronomer. He wouldn’t have used class time on the Virial if it weren’t good for a broad list of phenomena in and outside of astronomy. Quantum mechanics, for instance. I’ll give you an important example — the Sun.”

“One star, all by itself? Pretty trivial to take its average.”

“Not averaging the Sun as an object, averaging its plasma contents — hydrogen nuclei and their electrons, buffeted by intense heat all the way down to the nuclear reactions that run near the Sun’s core. It’s gravitational potential energy versus kinetic energy all over again, but at the atomic level this time. The Virial Theorem still holds, even though turbulence and electromagnetic effects generate a complicated situation.”

“I’m glad he didn’t assign that as a homework problem.”

“The semester’s not over yet.”

~~ Rich Olcott

A Virial Homework Problem

On By rolcottIn Astronomy: Techniques, Physics: Entropy and Thermodynamics, Uncategorized1 Comment

“Uh, Mr Moire? Would you mind if we used Old Reliable to do the calculations on this problem about the galaxy cluster’s Virial?”

Data extracted and re-scaled from Fig 2 of Smith (1936), The Mass of the Virgo Cluster

“Mm, only if you direct the computation, Jeremy. I want to be able to face Professor Hanneken with a clear conscience if your name ever comes up in the conversation. Where do we start?”

“With the data he printed here on the other side of the problem sheet. Old Reliable can scan it in, right?”

“Certainly. What are the columns?”

“The first one’s clear. The second column is the distance between the galaxy and the center of the cluster. Professor Hanneken said the published data was in degrees but he converted that to kiloparsecs to get past a complication of some sort. The third column is, umm, ‘the relative line‑of‑sight velocity.’ I understand the line‑of‑sight part, but the numbers don’t look relativistic.”

“You’re right, they’re much smaller than lightspeed’s 300,000 km/s. I’m sure the author was referring to each galaxy’s motion relative to the other ones. That’s what the Virial’s about, after all. I’ll bet John also subtracted the cluster’s average velocity from each of the measured values because we don’t care about how the galaxies move relative to us. Okay, we’ve scanned your data. What do we do next?”

“Chart it, please, in a scatter plot. That’s always the first thing I do.”

“Wise choice. Here you go. What do we learn from this?”

“On the whole it looks pretty flat. Both fast and slow speeds are spread across the whole cluster. If the whole cluster’s rotating we’d see faster galaxies near the center but we don’t. They’re all moving randomly so the Virial idea should apply, right?”

“Mm-hm. Does it bother you that we’re only looking at motion towards or away from us?”

“Uhh, I hadn’t thought about that. You’re right, galaxy movements across the sky would be way too slow for us to detect. I guess the slowest ones here could actually be moving as fast as the others but they’re going crosswise. How do we correct for that?”

“Won’t need much adjustment. The measured numbers probably skew low but the average should be correct within a factor of 2. What’s next?”

“Let’s do the kinetic energy piece T. That’d be the average of galaxy mass m times v²/2 for each galaxy. But we don’t know the masses. For that matter, the potential energy piece, V=G·M·m/R, also needs galaxy mass.”

“If you divide each piece by m you get specific energy, joules/kilogram of galaxy. That’s the same as (km/s)². Does that help?”

“Cool. So have Old Reliable calculate v²/2 for each galaxy, then take the average.”

“We get 208,448 J/kg, which is too many significant figures but never mind. Now what?”

“Twice T would be 416,896 which the Virial Theorem says equals the specific potential energy. That’d be Newton’s G times the cluster mass M divided by the average distance R. Wait, we don’t know M but we do know everything else so we can find M. And dividing that by the galaxy count would be average mass per galaxy. So take the average of all the R distances, times the 416,896 number, and divide that by G.”

“What units do you want G in?”

“Mmm… To cancel the units right we need J/kg times parsecs over … can we do solar masses? That’d be easier to think about than kilograms.”

“Old Reliable says G = 4.3×10-3 (J/kg)·pc/Mʘ. Also, the average R is … 890,751 parsecs. Calculating M=v²·R/G … says M is about 90 trillion solar masses. With 29 galaxies the average is around 3 trillion solar masses give or take a couple of factors of 2 or so.”

“But that’s a crazy number, Mr Moire. The Milky Way only has 100 billion stars.”

“Sometimes when the numbers are crazy, we’ve done something wrong. Sometimes the numbers tell us something. These numbers mutter ‘dark matter‘ but in the 1930s only Fritz Zwicky was listening.”

~~ Rich Olcott

  • Thanks again to Dr KaChun Yu for pointing out Sinclair Smith’s 1936 paper. Naturally, any errors in this post are my own.

Viral, Virial, What’s The Difference?

On By rolcottIn Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

A young man’s knock at my office door, eager yet a bit hesitant. “C’mon in, Jeremy, the door’s open.”

“Hi, Mr Moire. Got a minute?”

“It’s slow season, Jeremy. What can I do for you?”

“It’s my physics homework, sir. Professor Hanneken asked a question that I don’t understand.”

“John’s a bit of a joker but asking unsolvable questions isn’t usually one of his things. Well, except for that one about how long it would take to play Mahler’s Piano Quartet if you had only two musicians because of budget restrictions. What’s the question?”

“He wants us to use something called ‘the viral theorem‘ to deduce things about a certain galaxy cluster. I know what viral memes are but I don’t think I’ve ever heard of a theorem that spreads like a virus. I’ve done searches on my class notes and online textbook — nothing. So what is it and how am I supposed to use it?”

“Do you have the question with you?”

“Yessir, it’s #4 on this sheet.”

“Ah, just as I thought. Read it again. The word is ‘virial,’ not ‘viral.’ Big difference.”

“I suppose, but what’s a virial then?”

“We need some context. Imagine a cluster of free‑floating objects bound together by mutual forces of attraction. No central attractor, just lots of pairwise pulling, okay?”

“What kind of forces?”

“That’s the thing, it doesn’t matter. Gravitational, electrostatic, rubber bands even. The only restriction is that the force between each pair of objects follows the same force‑distance rule. For rubber bands it’s mostly just 1/distance until you get near the elastic limit. For gravity and electrostatics the rule is that force runs as 1/distance2. Got that picture?”

<grin> “It’d be tricky rigging up those rubber bands to not get tangled. Anyhow, instead of planets around the Sun you want me to think of stars held in a cluster by each other’s gravity. Do they all have to be the same size or the same distance apart?”

“No, because of what happens next. You’re thinking right — a heavier star pulls harder than a lighter one, and two stars close together feel more mutual force than stars far apart. We account for that variation by taking an average. Multiply force times distance for every possible pairing, then divide the total by the number of objects. The averaged number is the Virial, symbol V.”

“Wait, force times distance. That’s the Physics definition of work, like pulling something up against gravity.”

“Exactly. Work is directed energy. What we’re talking about here is the amount of energy required to pull all those objects away from the center of mass or charge or whatever, out to their current positions. The Virial is the average energy per object. It’s average potential energy because it depends on position, not motion.”

“They’d release all that energy if they just fell together so why don’t … wait, they’re going all different directions so momentum won’t let them, right?”

“You’re on your way. Motion’s involved.”

“Umm … Kepler’s Law — the closer any two of them get, the faster they orbit each other … OH! Kinetic energy! When things fall, potential energy’s converted to kinetic energy. Is there an average kinetic energy that goes up to compensate for the Virial getting smaller?”

“Bingo. When you say ‘average kinetic energy‘ what quantity springs to mind?”

“Temperature. But that’s only for molecules.”

“No reason we can’t define a galactic analog. In fact, Eddington did that back in 1916 when he brought the notion from gas theory over to astrophysics. He even used ‘T‘ for the kinetic average, but remember, this T refers to kinetic energy of stars moving relative to each other, not the temperatures of the stars themselves. Anyhow, the Virial Theorem says that a system of objects is in gravitational or electrostatic equilibrium when the Virial is T/2. Clausius’ ground‑breaking 1870 proof for gases was so general that the theorem’s been used to study everything from sub‑atomic particles to galaxies and dark matter.”

<bigger grin> “With coverage like that, the Virial’s viral after all. Thanks, Mr Moire.”

~~ Rich Olcott

  • Thanks to Dr KaChun Yu for helpful pointers to the literature. Naturally, any errors in this post are my own.

A Fleeting Shadowed Sky

On By rolcottIn Astronomy: Planetary Science, Chemistry, UncategorizedLeave a comment

“Hey, Uncle Sy, I’ve got a what‑if for you.”

“What’s that, Teena?”

“Suppose we switched Earth’s air molecules with helium. No, wait, except for the oxygen molecules. I know we need them.”

“First off, a helium-oxygen atmosphere wouldn’t last very long, not on the geological time scale. That’s an unstable situation.”

“Why, would the helium burn up like I’ve seen hydrogen do?”

“No, helium doesn’t burn. Helium atoms are smug. They’re happy with exactly the electrons they have. They don’t give, take or share electrons with oxygen or anything else. No, the issue is that helium’s so light.”

“What difference does that make?”

“The oxygen and helium won’t stay mixed together.”

“The air’s oxygen and nitrogen molecules are all mixed together. They told us that in Science class.”

“That’s correct. But oxygen and nitrogen molecules weigh nearly the same. It would take eight balloon‑fulls of helium to match the weight of one balloon‑full of oxygens. Suppose you had a bunch of equal‑weighted marbles, say red ones and blue ones. Pretend you pour them into a big bucket and stir them around like an atmosphere does. Which color would wind up on top?”

“Both, they’d stay mixed together.”

“Uh-huh. Now replace the blue marbles with marble‑sized ping‑pong balls and stir well.”

“The heavy marbles slide to the bottom. The light balls need to be somewhere so they get bullied up to the top.”

“Exactly. That’s what the oxygen molecules would do — sink down toward the ground and shove the helium atoms up to the top of the atmosphere. Funny thing though — the shoving happens faster than the sinking.”

“Why’s that?”

“It’s the mass thing again. At any given temperature, helium atoms in a gas zip around four times faster than oxygen molecules do. Anyway, the helium atoms that arrive up top won’t stay there.”

“Where else would they go?”

“Anywhere else, basically. Have you heard the phrase, ‘escape velocity‘?”

“It has something to do with rockets, doesn’t it?”

“Well, them, too. The general idea is that once you reach a certain threshold speed relative to a planet or something, you’re going too fast for its gravity to pull you back down. There’s a formula for calculating the speed. The fun thing is, the speed depends on the mass of what you’re escaping from and your distance from the object’s center, but it doesn’t depend on your own mass. It applies to everything from rockets to gas molecules.”

“And we were just talking about helium being zippy. Is it zippy enough to escape Earth?”

Chart by Cmglee, licensed under
CCA-SA 3.0 Unported

“Good thinking! That’s exactly where I was going. The answer is, ‘Maybe.’ It depends on temperature. Warm molecules are zippy, cold molecules not so much. At the same temperature, light molecules are zippier than heavy ones. There’s a chart that shows thresholds for different molecules escaping from different planets. Earth could hold onto its helium atoms, but only if our atmosphere were more than a hundred degrees colder than it is. Warm as we are, bye‑bye helium.”

“How long would that take?”

“That’s a complicated question with lots of ‘It depends’ in the answer. Probably the most important has to do with water.”

“I didn’t say anything about water, just helium and oxygen.”

“I know, but much of Earth’s weather is driven by water vaporizing or condensing or just carrying heat from place to place. Water‑powered hurricanes and even big thunderstorms stir up the atmosphere enough to swoosh helium up to bye‑bye territory. On the other hand, suppose our helium‑Earth is dry. The atmosphere’s layers would be mostly stable, light atoms would be slow to rise. We’d have a very odd‑looking sky.”

“No clouds.”

“Pretty much. But it wouldn’t be blue, either.”

“Would it be pink? I like pink.”

“Sorry, sweetie, it’d be dark dark blue, some lighter near the horizon. Light going past an atomic or molecular particle can scatter from its temporarily distorted electron cloud. Nitrogen and oxygen molecules distort more easily than helium atoms do. Earth skies are blue thanks to sunlight scattered by oxygen and nitrogen. Helium skies wouldn’t have much of that.”

~ Rich Olcott

  • Thanks again to Xander, who asked a really good helium question.

A Blast from The Past

On By rolcottIn General: Fun Stuff, UncategorizedLeave a comment

Back at the beginning of the Plague Era when things were (mostly) shut down, I started posting daily memes to reflect my shelter‑in‑place state of mind. Here’s the first one

The initial day‑count reflected my expectation that the lock-down would last less than a month. Hah!

I gave up on numbers for a while

April rolled around and I went topical

Remember the Great Toilet Paper Shortage?

I’ll never know how many readers got this one, but I like it

This is a Physics/Astronomy/Cosmology blog so I posted this to stay on‑topic…

This was meant to be satire, but I saw posts from folks actually doing it…

Yes, cabin fever is a thing

It was a time for sudden insights

We all got used to e-meetings

and smart speakers, if only for the conversations

Soon I had to go for higher numbers

Staying at home had its bright side (unless you’re invested in Big Oil)

And its bad‑omen side

The seasons passed and the day count increased…

Ending this retrospective with one of my favorites

~~ Rich Olcott

In Which Tone of Voice?

On By rolcottIn Physics: Waves, UncategorizedLeave a comment

“Oh. OH! Wow, Uncle Sy, this changes everything!”

“Which ‘this,’ Teena?”

“The resonator thing. Our music teacher, Ms Searcy, has been going on about us singing too much in our heads. I thought she’s been saying we’re thinking about the music too much and should just let the singing happen.”

“Sounds very Zen.”

“I’m too young to know Zen stuff. But now I think maybe she’s saying we’re using those head resonators you just told me about — singing with our sinuses and nasal cavities instead of somewhere else. She’s never been clear on what we can do about that.”

“You’re probably right. No music teacher since Harold Hill would try to get away with ‘think the music.’ I’m sure Ms Searcy’s complaint is about head tones as opposed to chest tones. If so, she’s got a good excuse for being unclear.”

“How can I have chest tones when you said vibrations come from my voice box and that’s above my chest?”

“Sound wave energy is about molecules colliding against any neighboring molecules. Up, down left, right — none of those matter. When air from your lungs makes your vocal cords buzz against each other, much of that buzzing goes up through your throat and head resonators. However, some of the buzz energy travels into your chest cavity. That’s your biggest resonator and it’s where your lowest tones come from.”

“I guess Ms Searcy wants us to send even more buzzing there, but how do we do that?”

“That’s a hard question. People have been interested in it since they started teaching singing and oratory to other people. We learned one part of the answer in medieval times when we began studying anatomy up close and personal. For instance, your voice box, which I really ought to call your larynx now we’re getting into detail, relates to about a dozen different muscles.”

“What’re the other parts?”

“The easiest part was kinesiology — figuring out what action each muscle supports. The hardest part was teaching a student how to feel and control the right muscles to make their voice do exactly what they want.”

“How can that be hard? I can flex my arm and leg muscles any time I want.”

“Can you make your larynx move up and down in your throat?”

“Easy.”

“That motion depends on a chain of muscles running between the back of your tongue and the top of your chest. One way to send buzzes downward is to activate the muscles that pull the larynx in that direction. Shorter distance makes for more efficient buzz transmission. It can also help to pull your head back a little. That tenses those pulled‑down tissues and improves transmission even more.”

<slightly deeper voice> “Like this?”

“That’s the idea. Learning to do that without having to pay attention to it is part of vocal training. Now, can you spread your toes out?”

“That’s weird. I can on my right foot but not my left.”

“Very common. Each foot has little tiny muscles, called intrinsic muscles, buried deep inside. Some people can control the whole set, some not so much. Your larynx has two pairs of intrinsic muscles that govern how your vocal cords work together. Some voice teachers claim the intrinsic muscles inside the larynx are the key to proper voice technique. Unfortunately, you can’t see them or get a feel for controlling them other then ‘keep trying things until you get it right and remember what you did.’ That’s Ms Searcy’s strategy. It’d be much harder with helium.”

“That’s right! Our voices with the balloons were all head tones. How come?”

“The speed of sound.”

“That’s a sideways answer, Uncle Sy.”

“Okay, this is more direct. We’ve said resonance was about waves whose wavelength just fit across a cavity. Picture two waves, the same number of waves per second, but the wave in helium travels about 3½ times faster than the one in air. The helium wave stretches about 3½ times farther between peaks. Whatever peaks per second your vocal cords make, in helium your chest cavity is 3½ times too small to resonate. Your head cavities, though, can resonate to overtones of those frequencies.”

“Squeaky overtones.”

~ Rich Olcott

When Sounds Rebound

On By rolcottIn Physics: Waves, UncategorizedLeave a comment

<chirp chirp> “Moire here.”

“Hi, Uncle Sy.”

“Hi, Teena, How was the birthday party?”

“Pretty fun. We had balloons but Mom wouldn’t let us fly them into the sky.”

“Because animals might eat them when they come down, I suppose.”

“Yeah, that’s what she said. What we did was we untied the knotty part so we could breathe in the helium and sing squeaky Happy Birthdays. It didn’t make much difference to my voice but Brian’s came out weird ’cause his voice is breaking anyway. It was almost a yodel.”

“I thought you didn’t like Brian any more.”

“That was last month, Uncle Sy. He ‘poligized on accounta he’s still learning social skills. We had fun playing with the sounds.”

“Sure you did. Do you remember the slide whistle I gave you once?”

“That was a long time ago. It was fun until Brian bent the slider and it didn’t work any more.”

“No surprise. How’d you make it give a high note?”

“I’d push the slider all the way in. Slider-out made a low note, but Brian could make a high note even there if he blew really hard.”

“Well, he would. It all has to do with resonant cavities.”

“I don’t have any cavities! Mom makes sure I brush and floss every day. And what’s resonant?”

“You don’t have dental cavities but there are other kinds. ‘Cavity‘ is another word for ‘hole‘ and some of them are important. You breathe through two nasal cavities that join up to be your posterior nasal cavity and that connects to sinus cavities in your skull and down to your voice box and lungs. Your mouth cavity resonates with all those cavities and vibrations from your voice box to make your speaking sounds.”

“That’s twice you used that ‘resonate‘ word I don’t know.”

“Break it down. ‘–son–‘ means ‘sound,’ like ‘sonic.’ Then—”

“‘Re–‘ means ‘again‘ like ‘rebuild‘ so resonate is sounding again like an echo.”

“Right. Except a resonant cavity is picky about what sounds it works with. Here, I’m sending you a video. Did you get it?”

“Yeah, I see a couple of wiggly lines. Wait, the blue line stays the same size but the orange line gets littler until it’s all gone and then the picture goes yellow and starts over.”

“What happens over on the right‑hand side?”

“The blue line bounces back from zero but the orange line waves all over the place. Is that how sound works?”

“Sort of. The air molecules in a sound wave don’t go up‑and‑down, they go to‑and‑fro. The wiggly lines are a graph of where the energy is. Where the molecules bunch together they bang into each other more often than in the in‑between places. In open air the energy pushes along even though the molecules stay pretty much where they are. In a resonant cavity, sounds are trapped like the blue line if they have just the right wavelength. A cavity’s longest trappable wavelength is its lowest note, called its fundamental. The energy in the fundamental is sustained as long as new energy’s coming in.”

“What happens to the orange line?”

“It doesn’t get a chance to build up. The energy in those waves spreads out until the wave just isn’t any more.”

“Aww, poor wavey. Wait, what about when Brian blows extra‑hard and gets that high note?”

“He gives the commotion inside the whistle enough energy to excite a second trapped wave with twice the number of crowded places. That’s called an overtone of the fundamental. Sometimes you want to do that, sometimes you don’t.”

“Brian always did on the whistle.”

“Well, he would. The resonant cavity thing also explains why Brian’s voice breaks and how talking works. Your windpipe is a resonating cavity. Brian’s windpipe has grown just large enough that sometimes it resonates in his new fundamental and sometimes switches to some other fundamental or overtone. Talking depends on tuning the resonances inside your mouth cavity. Try saying ‘ooo‑eee‘ while holding your lips steady in ‘ooo‘ position. On ‘eee‘ your tongue rises up to squeeze out the low‑pitched long waves in ‘ooo‘, right?”

“oooeeeoooeeeooo“

  • Thanks to Xander, whose question inspired this story arc.

~ Rich Olcott

Why Physics Is Complex

On By rolcottIn Mathematics, Physics: Light and Relativity, Physics: Quantum Mechanics, UncategorizedLeave a comment

“I guess I’m not surprised, Sy.”

“At what, Vinnie?”

“That quantum uses these imaginary numbers — sorry, you’d prefer we call them i‑numbers.”

“Makes no difference to me, Vinnie. Descartes’ pejorative term has been around for three centuries so that’s what the literature uses. It’s just that most people pick up the basic idea more quickly without the woo baggage that the real/imaginary nomenclature carries along. So, yes, it’s true that both i‑numbers and quantum mechanics appear mystical, but really quantum mechanics is the weird one. And relativity.”

“Wait, relativity too? That’s hard to imagine, HAW!”

“Were you in the room for Jim’s Open Mic session where he talked about Minkowski’s geometry?”

“Nope, missed that.”

“Ah, okay. Do you remember the formula for the diagonal of a rectangle?”

“That’d be the hypotenuse formula, c²=a²+b². Told you I was good at Geometry.”

“Let’s use ‘d‘ for distance, because we’re going to need ‘c‘ for the speed of light. While we’re at it, let’s replace your ‘a and ‘b‘ with ‘x‘ and ‘y,’ okay?”

“Sure, why not?”

<casting image onto office monitor> “So the formula for the body diagonal of this box is…”

“Umm … That blue line across the bottom’s still √(x²+y²) and it’s part of another right triangle. d‘s gotta be the square root of x²+y²+z².”

“Great. Now for a fourth dimension, time, so call it ‘t.’ Say we’re going for light’s path between A at one moment and B some time t later.”

“Easy. Square root of x²+y²+z²+t².”

“That’s almost a good answer.”

“Almost?”

“The x, y and z are distance but t is a duration. The units are different so you can’t just add the numbers together. It’d be like adding apples to bicycles.”

“Distance is time times speed, so we multiply time by lightspeed to make distance traveled. The formula’s x²+y²+z²+(ct)². Better?”

“In Euclid’s or Newton’s world that’d be just fine. Not so much in our Universe where Einstein’s General Relativity sets the rules. Einstein or Minkowski, no‑one knows which one, realized that time is fundamentally perpendicular to space so it works by i‑numbers. You need to multiply t by ic.”

“But i²=–1 so that makes the formula x²+y²+z²–(ct)².”

“Which is Minkowski’s ‘interval‘ between an event at A and another event at B. Can’t do relativity work without using intervals and complex numbers.”

“Well that’s nice but we started talking about quantum. Where do your i‑numbers come into play there?”

“It goes back to the wave equation— no, I know you hate equations. Visualize an ocean wave and think about describing its surface curvature.”

Adapted from “The Great Wave off Kanagawa”
by Katsushika Hokusai, in public domain

“Curvature?”

“How abruptly the slope changes. If the surface is flat the slope is zero everywhere and the curvature is zero. Up near the peak the slope changes drastically within a short distance and we say the surface is highly curved. With me?”

“So far.”

“Good. Now, visualize the wave moving past you at some convenient speed. Does it make sense that the slope change per unit time is proportional to the curvature?”

“The pointier the wave segment, the faster its slope has to change. Yeah, makes sense.”

“Which is what the classical wave equation says — ‘time‑change is proportional to space‑change’. The quantum wave equation is fundamental to QM and has exactly the same form, except there’s an i in the proportionality constant and that changes how the waves work.” <casting a video> “The equation’s general solution has a complex exponential factor eix. At any point its value is a single complex number with two components. From the x‑direction, the circle looks like a sine wave. From the i‑direction it also looks like a sine wave, but out of phase with the x‑wave, okay?”

A traveling wave packet.
Image by “Xcodexif” under CCS-A 4.0 license

“Out of phase?”

“When one wave peaks, the other’s at zero and vice‑versa. The point is, rotation’s built into the quantum waves because of that i‑component.” <another video> “Here’s a lovely example — that black dot emits a photon that twists and releases the electromagnetic field as it moves along.”

~ Rich Olcott

…With Imaginary Answers

On By rolcottIn Mathematics, UncategorizedLeave a comment

“I’ve got another solution to Larry’s challenge for you, Vinnie.”

“Will I like this one any better, Sy?”

“Probably not but here it is. The a‑line has unit length, right?”

“That’s what the picture says.”

“And the b‑line also has unit length, along the i‑axis.”

“Might as well call it that.”

“And we’ve agreed it’s a right triangle because all three vertices touch the semicircle. A right triangle with two one‑unit sides makes it isosceles, right, so how big is the ac angle?”

“Isosceles right triangle … 45°. So’s the bc angle even though it don’t look like that in the picture.”

“Perfect. Now let me split the triangle in two by drawing in this vertical line.”

“That’s its altitude. I told you I remember Geometry.”

“So you did. Describe the ac triangle.”

“Mm, it’s a right triangle because altitudes work that way. We said the ac angle is 45° which means the top angle is also 45°.”

“What about length ca?”

“Gimme a sec, that’s buried deep. … If the hypotenuse is 1 unit long then each side is √½ units.”

“Vinnie, I’m impressed. So what’s the area of the right half of the semicircle?”

“That’s a quarter‑circle with radius ca which is √½ so the area would be ¼πca² which comes to … π/8.’

“Now play the same game with the b‑side.”

“Aw, geez, I see where you’re going. Everything’s the same except cb has an i in it which is gonna get squared. The area’s ¼πcb² which is … ¼π(i√½)² which is –π/8. Add ’em both together and the total area comes to zero again. … Wait!”

“Yes?”

“You said the i‑axis is perpendicular to everything else.” <sketching on the back of the card> “That’s not really a semicircle, it’s two arcs in different planes that happen to have the same center and match up at one point. The c‑line’s bent. Bo‑o‑o‑gus!”

“I said you wouldn’t like it. But I didn’t quite say ‘perpendicular to everything else‘ because it’s not. The problem is that the puzzle depends on a misleadingly ambiguous diagram. The only perpendicular to i that I worked with was the a‑line. You’ve also got it perpendicular to part of the c‑line.”

“Sy, if these i-number things are ambiguous like that, why do you physics and math guys spend so much time with them? And do they really call them i‑numbers?”

“Ya got me. No, they’re actually called ‘complex numbers,’ because they are complex. The problem is with what we call their components.”

“Components plural?”

Figure by “HB” under CCSA 4.0 license

“Mm‑hm. Mathematicians put numbers in different categories. There’s the natural numbers — one, two, three on up. Extend that through zero on down and you have the integers. Roll in the integer/integer fractions and you’ve got the rational numbers. Throw in all the irrationals like pi and √2 and you’re got the full number line. They call that category the reals.”

“That’s everything.”

“That’s everything along the real number axis. Complex numbers have two components, one along the real number line, the other along the perpendicular pure i‑number line. A complex number is one number but it has both real and i components.”

“If it’s not real, it’s imaginary, HAW!”

“More correct than you think. That’s exactly what the ‘i‘ stands for. If Descarte had only called the imaginaries something more respectable they wouldn’t have that mysterious woo‑quality and people wouldn’t have as much trouble figuring them out. Complex numbers aren’t mysterious or ambiguous, they’re just different from the numbers you’re used to. The ‘why’ is because they’re useful.”

“What good’s a number with two components?”

“It’s two for the price of one. The real number line is good for displaying a single variable, but suppose one quantity links two variables, x and y. You can plot them using Descartes’ real‑valued planar coordinates. Or you could use complex numbers, z=x+iy and plot them on the complex plane. Interesting things happen in the complex plane. Look at the difference between ex and eix.”

“One grows, the other cycles.”

“Quantum mechanics depends on that cycling.”

~ Rich Olcott

A Complex Question

On By rolcottIn Mathematics, UncategorizedLeave a comment

A familiar footstep in the hall outside my office. “Door’s open, Vinnie, c’mon in.”

“Hi, Sy. Brought you something.” <lays a card on my desk> “So the question is, what’s the area inside the semicircle?”

<thoughtful pause> “Zero.”

“Dang, that’s both you and Larry say it’s zero. I don’t see it. ‘Splain, okay?”

“Larry gave you this?’

“Yeah. Can’t be zero area, it’s right there all spread out. How do you figure zero?”

“Start with the diagonal lines. The ‘a‘ line has a length of one unit, right?”

“The picture says so, but what’s i about on line ‘b‘?”

“That’s a special number, defined to be the solution x of the equation x²+1=0. Taking square roots, x=i and i²+1=0. I know you hate equations, but bear with me.”

<grumble> “Okay, go ahead.”

“If we had the radius r, the area would be ½πr². Line c is the diameter so the radius is r=c/2. We can get c by solving the triangle. We know it’s a right triangle because all three vertices touch a semicircle, right?”

“Geometry I remember; it was algebra gave me fits.”

“Then you remember good old Pythagoras and his a²+b²=c² formula, which we can use here because it’s a right triangle. Plugging in the known lengths we get c²=1²+i²=i²+1. Look familiar?”

“Yeahhh, i²+1=0 from the definition so c² is zero. What if b was 3?”

“We’d have c²=1²+(3i)²=1+9i²=1–9=–8. If both a and b are 3, then c²=3²+(3i)²=9+9i²=9–9 and we’re back to c²=0. It’s just arithmetic, but enhanced with i and its special rules.”

“Wait. If c²=0 then c=0 and c‘s the diameter and if that’s zero then so’s the radius and the area. The whole thing shrinks to a point but it can’t ’cause it’s right there spread out in the picture. Is this sort of thing why they told us x²–1=0 has a solution but x²+1=0 doesn’t?”

“It’s closely related, because i is involved. Incidentally, they lied to you in that algebra class. They showed you that famous quadratic equation ax²+bx+c=0, right, along with the magic b²–4ac formula. If that formula is positive you’ve got two roots, they said, but if it’s negative there’s no root.”

“Sounds familiar. Where’s the lie?”

“The first part’s true: with x²–1=0 you’ve got a=1,b=0,c=–1 with two roots x=1 and x=–1. The lie is in the second part: with x²+1=0 you’ve got a=1,b=0,c=+1 with two roots x=i and x=–i. All of those are perfectly good solutions.”

“But i‘s not a number! I can’t have i apples and I can’t write a check for i dollars. Why even play with it?”

“The i‑numbers are numbers, just not counting numbers. They’ve been part of mainstream math ever since the early Renaissance. Algebra was almost a blood sport at the time. Academics in Italy used to issue public challenges to solve problems that they’d already solved secretly. There were duels to settle questions of priority. Bombelli systematized i‑number arithmetic while working out one limited class of cubic equations. Cardano compiled solutions from multiple authors, including Bombelli, into his book Ars Magna. Then he systematically added his own solutions to a catalog of cubic and quartic equations. The Old Guard refused to believe in either i‑numbers or negative numbers, but Cardano showed you can’t escape them if you’re doing serious algebra.”

“Not even negative numbers?”

“Nope. The negatives weren’t widely accepted until the Venetian trading houses forced the issue — negatives are implicit in the merchants’ newly‑invented double‑entry bookkeeping. i‑numbers weren’t considered respectable for hundreds of years.”

Complex function centered at (1,i)
Adapted from image by Averse, under CC 2.0 license

“Then what happened?”

“Several advances. Euler connected i with trigonometry. Add Descarte‘s coordinate system, Fourier‘s wave analysis and Leibniz‘ notational innovations. For example, plotting an i‑number helix in 3D gives you these sine waves. When I think ‘i‑axis‘ I think ‘perpendicular to other coordinates‘.”

“Like Las Vegas.”

“Similar.”

Cardano image by R. Cooper
from Wellcome Trust
under the CCA International license
  • Thanks to Lloyd Boyer who raised several good points.

~~ Rich Olcott