Math Poetry

Eddie serves a good pizza. I amble over to the gelato stand for a chaser. “Evening, Jeremy. You’re looking a little distraught.”

“I am, Mr Moire. Just don’t ask me to quantify it! Math is getting me down. Why do they shove so much of it at us? You don’t put much math into your posts and they make sense mostly.”

“Thanks for the mostly. … Do you enjoy poetry?”

“Once I read some poems I liked. Except in English class. They spend too much time classifying genre and rhyme scheme instead of just looking at what the poet wrote. All that gets in the way.”

“Interesting. What is it that you like about poetry?”

“Mmm, part of it is how it can imply things without really saying them, part of it is how compact a really good one is. I like when they cram the maximum impact into the fewest possible words — take out one word and the whole thing falls apart. That’s awesome when it works.”

“Well, how does it work?”

“Oh, there’s lots of techniques. Metaphor’s a biggie — making one thing stand for something else. Word choice, too — an unexpected word or one with several meanings. Sometimes it’s a challenge finding the word that has just the right rhythm and message.”

“Ah, you write, too. When you compose something, do you use English or Navajo?”

“Whichever fits my thought better. Each language is better at some things, worse at others. A couple of times I’ve used both together even though only rez kids would understand the mix.”

“Makes sense. You realize, of course, that we’ve got a metaphor going here.”

“We do? What standing for what?”

“Science and Poetry. I’ve often said that Physics is poetry with numbers. Math is as much a language as English and Navajo. It has its own written and spoken forms just like they do and people do poetry with it. Like them, it’s precise in some domains and completely unable to handle others. Leaning math is like learning a very old language that’s had time to acquire new words and concepts. No wonder learning it is a struggle.”

“Poetry in math? That’s a stretch, Mr Moire.”

“Prettiest example I can think of quickly is rhyming between the circular and hyperbolic trigonometric systems. The circular system’s based on the sine and cosine. The tangent and such are all built from them.”

“We had those in class — I’ll remember ‘opposite over hypotenuse‘ forever and I got confused by all the formulas — but why do you call them circular and what’s ‘hyperbolic‘ about?”

“Here, let me use Ole Reliable to show you some pictures. I’m sure you recognize the wavy sine and cosine graphs in the circular system. The hyperbolic system is also based on two functions, ‘hyperbolic sine‘ and ‘hyperbolic cosine,’ known in the trade as ‘sinh‘ and ‘cosh.’ They don’t look very similar to the other set, do they?”

“Sure don’t.”

“But for every circular function and formula there’s a hyperbolic partner. Now watch what happens when we combine a sine and cosine. I’ll do it two ways, a simple sum and the Pythagorean sum.”

“Pythagorean?”

“Remember his a2+b2=c2? The orange curve comes from that, see in the legend underneath?”

“Oh, like a right triangle’s hypotenuse. But the orange curve is just a flat straight line.”

“True, as we’ve known since Euler’s day. Are you familiar with polar coordinates?”

“A little. There’s a center, one coordinate is distance from the center, and the other coordinate is the angle you’ve rotated something, right?”

“Good enough. Here’s what the same two combinations look like in polar coordinates..”

“Wow. Two circles. I never would have guessed that.”

“Mm-hm. Check the orange circle, the one that was just a level straight line on the simple graph. It’s centered on the origin. That tells us the sum of the squares is invariant, doesn’t change with the angle.”

“Do the hyperbolic thingies make hyperbolas when you add them that way?”

“Not really, just up-curving lines. The plots for their differences are interesting though. For these guys the Pythagorean difference is invariant. Einstein’s relativity is based on that property.”

“Pretty, like you say.”

~~ Rich Olcott

Imagine A Skyrocket Inside A Black Hole

Vinnie’s never been a patient man. “We’re still waiting, Sy. What’s the time-cause-effect thing got to do with black holes and information?”

“You’ve got most of the pieces, Vinnie. Put ’em together yourself.”

“Geez, I gotta think? Lessee, what do I know about black holes? Way down inside there’s a huge mass in a teeny singularity space. Gravity’s so intense that relativity theory and quantum mechanics both give up. That can’t be it. Maybe the disk and jets? No, ’cause some holes don’t have them, I think. Gotta be the Event Horizon which is where stuff can’t get out from. How’m I doing, Sy?”

“You’re on the right track. Keep going.”

“Okay, so we just talked about how mass scrambles spacetime, tilts the time axis down to point towards where mass is so axes stop being perpendicular and if you’re near a mass then time moves you even closer to it unless you push away and that’s how gravity works. That’s part of it, right?”

“As rain. So mass and gravity affect time, then what?”

“Ah, Einstein said that cause‑and‑effect runs parallel with time ’cause you can’t have an effect before what caused it. You’re saying that if gravity tilts time, it’ll tilt cause‑and‑effect?”

“So far as we know.”

“That’s a little weasel-ish.”

“Can’t help it. The time‑directed flow of causality is a basic assumption looking for counter‑examples. No‑one’s come up with a good one, though there’s a huge literature of dubious testimonials. Something called a ‘closed timelike curve‘ shows up in some solutions to Einstein’s equations for extreme conditions like near or inside a black hole. Not a practical concern at our present stage of technology — black holes are out of reach and the solutions depend on weird things like matter with negative mass. So anyhow, what happens to causality where gravity tilts time?”

“I see where you’re going. If time’s tilted toward the singularity inside a black hole, than so is cause‑and‑effect. Nothing in there can cause something to happen outside. Hey, bring up that OVR graphics app on Old Reliable, I’ll draw you a picture.”

“Sure.”

“See, way out in space here this circle’s a frame where time, that’s the red line, is perpendicular to the space dimensions, that’s the black line, but it’s way out in space so there’s no gravity and the black line ain’t pointing anywhere in particular. Red line goes from cause in the middle to effect out beyond somewhere. Then inside the black hole here’s a second frame. Its black line is pointing to where the mass is and time is tilted that way too and nothing’s getting away from there.”

“Great. Now add one more frame right on the border of your black hole. Make the black line still point toward the singularity but make the red line tangent to the circle.”

“Like this?”

“Perfect. Now why’d we put it there?”

“You’re saying that somewhere between cause-effect going wherever and cause-effect only going deeper into the black hole there’s a sweet spot where it doesn’t do either?”

“Exactly, and that somewhere is the Event Horizon. Suppose we’re in a mothership and you’re in our shuttlecraft in normal space. You fire off a skyrocket. Both spacecraft see sparks going in every direction. If you dive below an Event Horizon and fire another skyrocket, in your frame you’d see a normal starburst display. If we could check that from the mothership frame, we’d see all the sparks headed inward but we can’t because they’re all headed inward. All the sparkly effects take place closer in.”

“How about lighting a firework on the Horizon?”

“Good luck with that. Mathematically at least, the boundary is infinitely thin.”

“So bottom line, light’s trapped inside the black hole because time doesn’t let the photons have an effect further outward than they started. Do I have that right?”

“For sure. In fact, you can even think of the hole as an infinite number of concentric shells, each carrying a causality sign reading ‘Abandon hope, all ye who enter here‘. So what’s that say about information?”

“Hah, we’re finally there. Got it. Information can generate effects. If time can trap cause‑effect, then it can trap information, too.”

~~ Rich Olcott

Tilting at Black Holes

“What’s the cause-effect-time thing got to do with black holes and information?”

“We’re getting there, Al. What happens to spacetime near a black hole?”

“Everybody knows that, Sy, spacetime gets stretched and squeezed until there’s infinite time dilation at the Event Horizon.”

“As usual, Vinnie, what everybody knows isn’t quite what is. Yes, Schwarzschild’s famous solution includes that Event Horizon infinity but it’s an artifact of his coordinate system. Al, you know about coordinate systems?”

“I’m a star-watcher, Sy. Sure, I know about latitude and longitude, declination and right ascension, all that stuff no problem.”

“Good. Well, Einstein wrote his General Relativity equations using generalized coordinates, like x,y,z but with no requirement that they be straight lines or at right angles. Schwarzschild solved the equations for a non‑rotating sphere so naturally he used spherical coordinates — radius, latitude and longitude. Since then other people have solved the equations for more complicated cases using more complicated coordinate systems. Their solutions don’t have that infinity.”

“No infinity?”

“Not that one, anyhow. The singularity at the hole’s geometric center is a real thing, not an artifact. So’s a general Event Horizon, but it’s not quite where Schwarzschild said it should be and it doesn’t have quite the properties that everybody thinks they know it has. It’s still weird, though.”

“How so?”

“First thing you have to understand is that when you get close to a black hole, you don’t feel any different. Except for the spaghettification, of course.”

“It’s frames again, ain’t it?”

“With black holes it’s always frames, Vinnie. If you’re living in a distorted space you won’t notice it. Whirl a meter‑long sword around, you’d always see it as a meter long. A distant observer would see you and everything around you as being distorted right along with your space. They’ll see that sword shrink and grow as it passes through different parts of the distortion.”

“Weird.”

“We’re just getting started, Al. Time’s involved, too. <grabbing a paper napkin and sketching> Here’s three axes, just like x,y,z except one’s time, the G one points along a gravity field, and the third one is perpendicular to the other two. By the way, Al, great idea, getting paper napkins printed like graph paper.”

“My location’s between the Physics and Astronomy buildings, Sy. Gotta consider my clientele. Besides, I got a deal on the shipment. What’s the twirly around that third axis?”

“It’s a reminder that there’s a couple of space dimensions that aren’t in the picture. Now suppose the red ball is a shuttlecraft on an exploration mission. The blue lines are its frame. The thick vertical red line shows it’s not moving because there’s no spatial extent along G. <another paper napkin, more sketching> This second drawing is the mothership’s view from a comfortable distance of the shuttlecraft near a black hole.”

“You’ve got the time axis tilted. What’s that about?”

“Spacetime being distorted by the black hole. You’ve heard Vinnie and me talk about time dilation and space compression like they’re two different phenomena. Thing is, they’re two sides of the same coin. On this graph that shows up as time tilted to mix in with the BH direction.”

“How about those twirly directions?”

“Vinnie, you had to ask. In the simple case where everything’s holding still and you’re not too close to the black hole, those two aren’t much affected. If the big guy’s spinning or if the Event Horizon spans a significant amount of your sky, all four dimensions get stressed. Let’s keep things simple, okay?”

“Fine. So the time axis is tilted, so what?”

“We in the distant mothership see the shuttlecraft moving along pure tilted time. The shuttlecraft doesn’t. The dotted red lines mark its measurements in its blue‑line personal frame. Shuttlecraft clocks run slower than the mothership’s. Worse, it’s falling toward the black hole.”

“Can’t it get away?”

“Al, it’s a shuttlecraft. It can just accelerate to the left.”

“If it’s not too close, Vinnie. The accelerative force it needs is the product of both masses, divided by the distance squared. Sound familiar?”

“That’s Newton’s Law of Gravity. This is how gravity works?”

“General Relativity cut its teeth on describing that tilt.”

~~ Rich Olcott

Cause, Effect And Time

We’re still at Vinnie’s table by the door of Al’s coffee shop. “Long as we’re talking about black holes, Sy, I read in one of my astronomy magazines that an Event Horizon traps information the same way it traps light. I understand how gravity makes escape velocity for photons go beyond lightspeed, but how does that trap information?”

“Well, to start with, Al, you understand wrong. The whole idea of escape velocity applies to massive objects like rockets that feel the force of gravity. Going up they trade kinetic energy for potential energy; given enough kinetic energy they escape. Photons have zero mass — the only way gravity influences them is by bending the spacetime they fly through.”

“Does the bending also affect information or is that something else?”

Minkowski’s spacetime diagram…

“Fair question, but it’ll take some background to answer it. Good thing I’ve got Old Reliable and my graphics files along. Let’s start with this one. Vinnie’s seen a lot of spacetime graphs like this, Al, but I don’t think you have. Time runs upward, distance runs sideward, okay? Naming a specific time and location specifies an event, just like a calendar entry. Draw a line between two events; the slope is the speed you have to go to get from one to the other.”

“Just the distance, you’re not worrying about direction?”

“Good question. You’re thinking space is 3D and this picture shows only one space dimension. Einstein’s spacetime equations take account of all four dimensions mixing together, which is one reason they’re so hard to solve except in special cases. For where we’re going, distance will be enough, okay?”

“Not gonna argue.”

… compartmentalized by Einstein’s speed limit …

“Now we roll in Einstein’s speed limit. Relativity says that nothing can go faster than light. On a Minkowski diagram like this we draw the lightspeed slope at a 45″ angle. Any physical motion has a slope more vertical than that.”

“Huh?”

“See, Al, you’re going one second per second along time, right? If you’re not making much progress distance‑wise, you don’t do much on Sy’s sideways axis. You move mostly up.”

“Exactly, Vinnie. The bottom and top sections are called ‘timelike‘ because, well, they’re mostly like time.”

“Are the other two sections spacelike?”

“Absolutely. You can’t get from ‘Here & Now‘ to the ‘Too far to see‘ event without going faster than light. Einstein said that’s a no‑no. Suppose that event’s a nova, ‘Now‘ but far away. Astronomers will have to just wait until the nova’s light reaches them at ‘Here‘ but at a later ‘Now.’ Okay, Vinnie, here’s a graphic you haven’t seen yet.”

… and re-interpreted in terms of causality.

“Looks pretty much the same, except for that arrow. What’s cause and effect got to do with time?”

“I don’t want to get into the metaphysical weeds here. There’s a gazillion theories about time — the Universe is expanding and that drives time; entropy always increases and that drives time; time is an emergent property of the underlying structure of the Universe, whatever that means. From an atomic, molecular, mechanical physics point of view, time is the result of causes driving effects. Causes always come first. Your finger bleeds after you cut it, not before. Cause‑effect runs along the time axis. Einstein showed us that cause‑effect can’t travel any faster than lightspeed.”

“That’s a new one. How’d he figure that?”

“Objects move objects to make things happen. They can’t move faster than lightspeed because of the relativity factor.”

“What if the objects are already touching?”

“Your hand and that cup are both made of atoms and it’s really their electric fields that touch. Shifting fields are limited by lightspeed, too.”

“So you’re saying that cause-effect is timelike.”

“Got it in one. Einstein would say causality is not only timelike, but exactly along the time axis. That’s one big reason he was so uncomfortable about action at a distance — a cause ‘Here‘ having an effect ‘There‘ with zero time elapsed would be a horizontal line, pure spacelike, on Minkowski’s graph. Einstein invented the principle of entanglement as a counterexample, thinking it impossible. He’d probably be shocked and distressed to see that today we have experimental proof of entanglement.”

~~ Rich Olcott

Holes in A Hole?

Mid-afternoon coffee break time so I head over to Al’s coffee shop. Vinnie’s at his usual table by the door, fiddling with some spilled coffee on the table top. I notice he’s pulled some of it into a ring around a central blob. He looks at it for a moment. His mental gears whirl then he looks up at me. “Hey Sy! Can you have a black hole inside another black hole?”

“That’s an interesting question. Quick answer is, ‘No.’ Longer answer is, ‘Sort of, maybe, but not the way you’re thinking.’ You good with that, Vinnie?”

“You know me better than that, Sy. Pull up a chair and give.”

I wave at Al, who brings me a mug of my usual black mud. “Thanks, Al. You heard Vinnie’s question?”

“Everyone on campus did, Sy. Why the wishy-washy?”

“Depends on your definition of black hole.”

Sky-watcher Al is quick with a response. “It’s a star that collapsed denser than a neutron star.”

Vinne knows me and black holes better than that. “It’s someplace where gravity’s so strong that nothing can get out, not even light.”

“Both right, as far as they go, but neither goes deep enough for Vinnie’s question.”

“You got a better one, I suppose?”

“I do, Vinnie. My definitition is that a black hole is a region of spacetime with such intense gravitation that it wraps an Event Horizon around itself. Al’s collapsed star is one way to create one, but that probably doesn’t account for the Event Horizons around supermassive black holes lurking in galactic cores. Your ‘nothing escapes‘ doesn’t say anything about conditions inside.”

“Thought we couldn’t know what happens inside.”

“Mostly correct, which is why your question is as problematical as you knew it was. Best I can do is lay out possibilities, okay? First possibility is that the outer black hole forms around a pre-existing inner one.”

“Can they do that?”

“In principle. What makes a black hole is having enough mass gathered in close proximity. Suppose you have a black hole floating our there in space, call it Fred, and a neutron star comes sidling by. If the two bodies approach closely enough, the total amount of mass could be large enough to generate a second Event Horizon shell enclosing both of them. How long that’d last is another matter.”

“The outer shell’d go away?”

“No chance of that. Once the shell’s created, the mass is in there and the star is doomed … unless the star’s closest approach matches Fred’s ISCO. That’s Innermost Stable Circular Orbit, about three times Fred’s Event Horizon’s half-diameter if Fred’s not rotating. Then the two bodies might go into orbit around their common center of gravity.”

“How’s rotation come into this?”

“If the mass is spinning, then you’ve got a Kerr black hole, frame-dragging and an ISCO each along and against the spin direction. Oh, wait, I forgot about tidal effects.”

“Like spaghettification, right.”

“Like that but it could be worse. Depending on how tightly neutronium holds itself together, which we don’t know, that close approach might be inside the Roche limit. Fred’s gravity gradient might simply shred the star to grow the black hole’s accretion disk.”

“Grim. You said there’s other possibilities?”

“Sorta like the first one, but suppose the total mass comes from two existing black holes, like the collision that LIGO picked up accidentally back in 2014. Suppose each one is aimed just outside the other’s ISCO. Roche fragmentation wouldn’t happen, I think, because each body’s contents are protected inside its own personal Event Horizon. Uhh … darn, that scheme won’t work and neither will the other one.”

“Why not?”
 ”Why not?”

“Because the diameter of an Event Horizon is proportional to the enclosed mass. The outer horizon’s diameter for the case with two black holes would be exactly the sum of the diameters of the embedded holes. If they’re at ISCO distances apart they’re can’t be close enough to form the outer horizon. For the same reason, I don’t think a neutron star could get close enough, either.”

“No hole in a hole, huh?”

“I’m afraid not.”

~~ Rich Olcott

  • Thanks to Alex and Xander, who asked the question.

Footprints in The Glasses

I think he sometimes lies in wait for me like a cheetah crouching to ambush prey. No, more like a frog. Today I’m on my daily walk when suddenly — “Hey Moire, I got questions!”

Yeah, more like a frog. “Morning, Mr Feder. Out early today, aren’t you?”

“It’s gonna be hot today so I figured you’d walk the park early. I like it down here by the lake.”

Yup, definitely a frog. “Well, what can I do for you?”

“I’m wearing these new glasses, okay?”

“I can see that. Very … stylish.”

“So I read what you wrote about how light slows down when it goes through stuff and I wonder, does the light slow down enough going through these glasses that I might not see a bus in time? And how does stuff slow down light anyway?”

<drawing Old Reliable from its holster> “That first question is quantitative so let’s gather the numbers. The speed of light in vacuum is about 186 000 miles per second, that’s 300 megameters per second or 300 millimeters per nanosecond. Metric system conversions are kinda fun, aren’t they?”

“Hang on — megameters per second is meters per microsecond, take it down another thousand top and bottom…. I guess that’s okay.”

“Old Reliable doesn’t lie. Alright, your eyeglass lenses look like they’re a couple of millimeters thick. I’ll call it three millimeters to make the numbers pretty. If your lenses were vacuum space a short light pulse would pass through in 0.01 nanosecond, okay?”

“Not that thick, but go on.”

“The slow‑down factor is technically called the refractive index. Old Reliable says that eyeglass refractive indexes typically run about 1.5 so with the slow‑down our light pulse would take 0.015 nanosecond instead of 0.01. Is that enough increase to affect your rection time significantly? Let’s see … Says here that a typical nerve impulse travels at about 50 meters per second. Keeping the numbers pretty I’ll guess that between your eye and the vision centers in the back of your brain is about 2 inches or 5 centimeters. You good with that?”

“Not that short, but anything for pretty numbers. Go on.”

“Five centimeters is 0.05 meters, at 50 meters per second comes to 0.001 second. Slowing down that pulse lengthens your reaction time from 0.001 second to 0.001 000 000 015 second. Not enough of a difference to worry about.”

“But how come it slows at all seeing as I’ve heard it’s mostly empty space between the atoms?”

“There’s empty and there’s empty. You’re thinking of little solar‑system atoms, aren’t you, with particle electrons orbiting the nucleus and what space is left is vacuum. We’ve known for a century that it’s not that way. The electrons aren’t particles, they’re fuzzy blobs, and the volume around them is chock full of lumpy electric field. The incoming lightwave, really an electromagnetic wave, doesn’t see one electron here and another one way over there and free passage in between. Nope, it interacts with the whole field and that’s where the slow‑down happens.”

“Lots of quantum jumps and like that, huh?”

“No quantum jumps unless your glasses are tinted. Mmm… You ever run along the seashore?”

“I’m from Jersey. Of course I have.”

 Time periodicity at a point,
 space periodicity at a moment.

“Visualize running across hard sand and suddenly you hit a patch of soft sand. You keep your feet oscillating up and down at the same rate, but you make less progress along the beach. Your footprints get closer together, right?”

“Sometimes I fall down. So?”

“Something similar happens with a lightwave. It repeats in time like your foot going up and down and it repeats in space like your footprints in the sand. The wave’s energy changes with repeat time. When light passes through an electric field like the one inside clear, colorless glass, the field doesn’t absorb energy — no change in repeat time. What does happen is the field squeezes the peak‑to‑peak distance. The wave acts like your footprints getting closer together. Less distance divided by the same time means lower speed. The wave slows down inside the glass.”

“Does light ever fall down?”

“Only if its energy quantum matches an absorber’s gap.”

~~ Rich Olcott

Stars Are REALLY Warm-hearted

“I don’t understand, profesora. The Sun’s fuel is hydrogen. The books say when the Sun runs out of fuel it will eject much hydrogen and collapse to a white dwarf. So it didn’t run out of fuel, yes?”

“That’s an excellent question, Maria. Your simple sketch of layered zones is adequate for a stable star like our Sun is now. When things go unstable we need to pay more attention to dynamic details like mass, pressure and diffusion. The numbers matter.”

“I had that the fusion zone is 30% up from the center, and the top of the radiation zone is at 70%.”

“Yes, but percentages of a straight line don’t really give us a feel for the volumes and masses. Volumes grow as the radius cubed. The Sun’s core, the part inside your 30% radius, holds (30%)3 which is less than 3% of the Sun’s volume. The convection shell on the outside is also 30% thick, but that zone accounts for ⅔ of the star’s volume.”

“But not ⅔ of the mass, I think. The core is the most dense, yes?”

“Truly. The core is <chuckle> at the core of the matter. It’s obviously under compression from all the mass above it, but there’s a subtler and more important reason. The Sun’s internal temperatures are so high that everything acts like an ideal gas, even near the center. Once you’re beneath the convection zone, the only transport mechanism is diffusion influenced by gravity. Helium nuclei weigh four times what hydrogen nuclei do. Helium and heavier things tend to sink toward the center, hydrogen tends to float upward. What effect does that have on the core’s composition?”

“The core is heavy with much helium, not as much hydrogen.”

“Good. Now, what’s next above the core?”

“The fusion zo– Oh! The place where there’s enough hydrogen to do the fusing.”

“If the temperature and pressure are right. That turns out to be a delicate balance. Too much heat makes that region expand, average distance between atoms increases and that slows down the fusion reaction. Too much pressure slows diffusion which then slows the reaction by hindering hydrogen’s entry and helium’s exit. Too little heat or too little pressure do the opposite. Now you know why the fusion zone is so narrow in our diagrams, only about 10% of a radius.”

“No fusion in the other layers?”

“Less than 1% of the total. So we’ve got nearly all the heat in the star coming from hydrogen‑to‑helium fusion in this diffusion‑controlled gaseous reaction zone buried deep in the star.”

“Ah! Now I see. It is wrong to say the star dies because it runs out of fuel. There is still much hydrogen in the upper zones, but the diffusion doesn’t let enough enter the fusion zone. That is why the fire goes out. What happens then?”

“It mostly depends on the star’s mass. Really big ones have a sequence of deeper, hotter fusion layers in their core, forming heavier and heavier atoms all the way down to iron. Each layer is diffusion‑limited, of course, and the whole thing is like a stack of Jenga blocks supported by heat coming from below. If reaction in any layer overruns its fuel delivery then it stops producing heat. The whole stack collapses violently to form a neutron star or a black hole. Nearby infalling atoms collide and radiate in an exponential heat‑up. But the stars are many millions of kilometers across. The outermost layers don’t have time to fall all the way in. Their imploding gases slam into gases exploding from the collapse zone — BLOOEY! — there’s a nova spewing hydrogen and stardust across the Universe.”

“That is how our Sun will die?”

“No, it’s too small for such violence so it’s fated for a gentler old age. Five billion years from now its core will be mostly carbon and oxygen. Fuel delivery won’t be able to sustain further fusion reactions. The radiation and convection layers will simply settle inward, releasing enough gravitational potential energy to start hydrogen fusion in an expanding cool red shell outside the core.”

“Hee-hee — no lo va la nova, profesora, the nova doesn’t go.

  • Thanks to Victoria, who asked the question.

~~ Rich Olcott

Layer Upon Layer

“Excuse me, profesora, you wanted me to come to your office?”

“Yes, Maria. Come in, please. I wanted to have a chat with you before you give your class presentation tomorrow.”

“I am a little nervous about it.”

“I thought you might be. I wanted to help with that. I’ll start by saying that your English language skills have gotten much better than you give yourself credit for. Better yet, you’ll be speaking before friends who want you to succeed. I’m sure you’ll do fine. I think if we go over your material together you’ll be more confident. Come open your laptop on my desk where we can both see it. Now bring up your first slide.”

“Yes, profesora. Already you know that the title of my presentation is ‘The Structure of The Sun.’ I only have one slide, this one, that shows a slice of a star like our Sun.”

“How did the star get that way?”

“It condensed from a galactic gas cloud that was mostly hydrogen. I plan to talk about that with waving of the hands because a good picture of it needs to be in motion and I don’t know how to do that yet.”

“Fair enough, just don’t skip over it. Beginnings are important. Now talk me through your diagram.”

“It starts in the middle ¿see the fusion zone? where protons, that’s hydrogen atoms without their electrons, are squeezed together to release energy and make alpha particles, that’s helium atoms without their electrons. The protons have the same charge so they push each other away, but they are beneath many kilometers of mass that push them together. Also, the temperature is very hot, tens of millions of degrees. Hot atoms move fast, so when the protons are pushed together it happens with enough force and speed .. sorry, I need a word, superar?”

“Overcome.”

“Thank you. The protons are pushed together with enough force and speed to overcome the charge barrier. The actual reactions are complicated. At the end there is an alpha particle, four times heavier than a proton, and there is much more energy than the overcoming used up. The fusion zone makes heat and the heavy alpha particles fall down into the ash zone. The heat must go somewhere. Already the center is hotter so the new heat goes upward into the radiation zone.”

“And it’s called that because…?”

“Because atom motion is so, mm, frantic?”

“Good word.”

“… So frantic that there’s no moving in the same direction together, no convection like when steam rises over boiling water. Heat can only travel by convection, conduction or radiation. If there is no convection, moving heat must go neighbor‑to‑neighbor by conduction which is collision or by radiation which is photons jumping between atoms again and again until they escape. I have read that one photon’s energy can take 10000 years to cross the radiation zone.”

“So how is the next zone different?”

“It is much higher up from the center, nearly ¾ of the way to the surface. The pressure is 100 times less than in the fusion zone. The atoms have more room to move around together and form winds to carry the heat up by convection. But they can’t only go up, they have to come down, too, and that’s why my drawing has loops.”

“Is there a name for the loops?”

“Oh, yes, they are called Bénard cells and they’re very much like what I see looking into a pot of water just before it boils.”

“What’s the orange above the convection zone?”

“That’s the part of the Sun that we see, the photosphere that emits light in a continuous spectrum. The Fraunhofer lines, the dark lines in the astronomer’s spectrum, are the shadows of atoms high in in the photosphere that absorb only certain colors. I was surprised to learn how narrow the photosphere is, not even 0.02% of the Sun’s radius. Anyway, that’s my presentation, but now I have a question. The Sun’s fuel is hydrogen. The books say when the Sun runs out of fuel it will eject much of its hydrogen mass and collapse to a white dwarf. So it didn’t run out of fuel, yes?”

~~ Rich Olcott

A Match Game

<chirp chirp> “Lab C-324, Susan Kim speaking.”

<hoarsely> “Hi Susan, it’s Sy. Fair warning. The at‑home test I just ran says I’ve got Covid. I’ve had all four shots but it looks like some new variant dodged in anyway. We had coffee together at Al’s yesterday so I wanted to warn you. Better stock up on cough medicine and such.”

“Ooh. Thanks, Sy, sorry to hear that. If it’s any consolation, you’re not alone. About half the lab’s empty today because of Covid. I’m just waiting for this last extraction to complete and then I’m outta here myself. There’s chicken soup going in the slow‑cooker at home.”

“Ah, yes, a Jewish mother’s universal remedy.”

“Korean mothers, too, Sy, except we use more garlic. Chicken soup’s a standard all over the world — soothing, easy on the stomach and loaded with protein.”

“While you’re in wait mode, maybe you could explain something to me.”

“I can try. What is it?”

“How do these tests work? I swabbed my nose, swirled the yuck with the liquid in the little vial and put three drops into the ‘sample port‘ window. In the next few minutes fluid crept across the display window next to the port and I saw dark bars at the T and C markers. What’s that all about?”

“Miracles of modern immunochemistry, Sy, stuff we wouldn’t have been able to execute fifty years ago. What do you know about antibodies?”

“Not much. I’ve read a little about immunology but I always get the antibodies confused with the antigens and then my understanding goes south.”

“Ignore the ‘anti‘ parts — an antigen is usually a part of something from outside that generates an immune response. As part of the response, cells in your body build antibodies, targeted proteins that stick to specific antigens. Each unique antibody is produced by just a few of your cells. When you’re under a disease attack, your antibodies that match the attacker’s antigens lock onto the attacker to signal your defender cells what needs chewing up. About half‑a‑dozen Nobel Prizes went to researchers who figured out how to get a lab‑grown cell to react to a given antigen and then how to clone enough copies of that cell to make industrial quantities of the corresponding antibody. You follow?”

“So far, so good.”

“One more layer of detail. All antibodies are medium-sized proteins with the same structure like a letter Y. There’s a unique targeting bit at the end of each upper arm. An antigen can be anything — a fragment of protein or carbohydrate, a fatty acid, even some minerals.”

“Wait. If a protein can be an antigen, does that mean that an antibody can be an antigen, too?”

“Indeed, that’s the key for your test kit’s operation. The case holds a strip of porous plastic like filter paper that’s been treated with two narrow colorless stripes and a dot. The T stripe contains immobilized antibody for some fragment of the virus. The C stripe contains immobilized antibody antibody.”

“Hold on — an antibody that targets another antibody like maybe the bottom of the Y?”

“Exactly. That’s the control indicator. The dot holds virus antibodies that can move and they’re linked to tiny particles of gold. Each gold particle is way too small to see, but a bunch of them gathered together looks red‑brown. Okay, you put a few drops of yuckified liquid on top of the dot and the mixture migrates along the porous material. You tell me what happens.”

“Wait, what’s in that liquid?”

“It’s standard pH-buffered saline, keeps the proteins healthy.”

“Hmm. Alright, the dot’s gold‑labeled virus antibody grabs virus in my yuck and swims downstream. The T stripe’s virus antibody snags the virus‑antigen combination particles and I see red‑brown there. Or not, if there’s no virus. Meanwhile, the creeping liquid sweeps other gold‑labeled antibodies, virus‑bound or not, until they hit the C stripe and turn it red‑brown if things are working right. Uhhh, how much gold are we talking about?”

“Colloidal gold particles are typically balls maybe 50 nanometers across. Stripe area’s about 1 mm2, times 50 nanometers, density 19.32 kg/m3, gold’s $55 per gram today … about 5 microcents worth.”

~~ Rich Olcott

In vacuo veritas?

“Let’s see if my notes are complete, Mr Moire. We’re crossed off two possible Universe finales — falling into a Big Crunch or expanding forever while making new matter between the galaxies to keep itself in a steady state. Or the Universe might expand to some critical density and then stay there but we mostly ruled that out because a twitch would push it to either crunching or expanding forever. Einstein’s Cosmological Constant might or might not be dark energy but either way, Friedmann’s equation predicts that the Universe will expand exponentially. Is that all the ways we could end?”

“Of course not, Jeremy. The far distant future’s like anything we humans don’t know much about, we make lots of guesses. Vacuum energy, for instance.”

“Anything to do with getting my roommate off the couch when it’s their turn to do the floors?”

“Very funny, but no. The notion of ‘vacuum‘ has a history. Aristotle said it’s empty space and that’s nothing and you can’t talk about nothing, but of course that’s exactly what he was doing. It wasn’t until Newton’s day that we developed dependable technologies for producing and investigating ‘nothing.’ Turns out that a good vacuum’s hard to find and even outer space is a lot busier than you might think.”

“Yeah, Jim in the Physics lab says he’s working with Ultra‑High Vacuum, a millionth of a millionth of an atmosphere, and the molecules left in the apparatus still cause problems.”

“Wonder how many molecules that is. Time for Old Reliable. <muttering> Avagadro’s Number, 22.4 liters, 10-12 atmospheres … Wow, there’s nearly 30 billion molecules per liter in his rig, a couple hundred times more if he chills it. <scrolling> This Wikipedia article says the solar wind runs only ten thousand protons per liter; interstellar medium’s about a tenth of that. But all those are physical vacuums. Theoretical vacuums are completely empty except they’re sort‑of not.”

“How could they be empty but not? Is that a Schrödinger joke?”

“No, but it does point up how the word has acquired multiple technical meanings. Newton’s concept of a vacuum was basically equivalent to Aristotle’s — simply a space with no matter in it. Two centuries later, Maxwell pointed the way to electric and magnetic fields which meant we needed to define a new vacuum with no such fields. Einstein added his proviso about the speed of light in a vacuum but that was okay. Then along came quantum and strings and several new kinds of vacuum.”

“Why would we need new definitions? Nothing’s nothing, isn’t it?”

“Not necessarily in theory, and that’s the point. For instance, you might use a Maxwell‑inspired theory to think about how a certain charged object behaves in a certain electromagnetic field. You can’t isolate the field’s effects unless you can add it to a theoretical space containing no objects or electromagnetic fields. Make sense?”

“And that’s a Maxwell vacuum? Seems reasonable. Then what?”

“Quantum theories go in the other direction. They start by assuming that Maxwellian vacuums can’t exist, that space itself continually produces virtual particles from their associated fields.”

“Um, conservation of mass?”

“Valid question. This may feel like dodging, but there’s math and experiment to back it up. What’s really conserved, we think, is mass‑energy. Particles, anti‑particles and energy fluctuations averaging to zero over finite time intervals. A dab of energy concentrated to create a particle’s mass? No problem, because that particle will be annihilated and release its energy equivalent almost immediately. To replace the Maxwellian vacuum, quantum theorists co‑opted the word to refer to a system’s lowest possible quantum state or maybe the lowest possible set of states, depending on which kind of calculation is underway. The cosmology people picked up that notion and that’s when the doom‑saying started.”

“Awright, now we’re getting somewhere. What’s their vacuum like?”

“From what I’ve seen, a tall stack of ‘if‘s and hand‑waving. The idea is that our Universe may not be in the lowest possible quantum state and if so, sometime in the next 188 billion years we could suddenly drop from false to true vacuum, in which case everything goes haywire. I’m not convinced that the Universe even has a quantum state. Don’t panic.”

~~ Rich Olcott