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

Hysteresis Everywhere

“We’ve known each other for a long time, ain’t we, Sy?”

“That we have, Vinnie.”

“So I get suspicious when we’ve specific been talking about a magnetic field making something else magnetic and you keep using general words like ‘driver‘ and ‘deviation‘. You playing games?”

“You caught me. The hysteresis idea spreads a lot farther than magnetism. It addresses an entire dimension Newton was too busy to think about — time.”

“Wait a minute. Newton was all about velocity and acceleration and both of them are something‑per‑time. It’s right there in the units. Twice for acceleration.”

“True, but each is really about brief time intervals. Say you’re riding a roller‑coaster. Your velocity and acceleration change second‑by‑second as forces come at you. Every force changes your net acceleration immediately, not ten minutes from now. Hysteresis is about change that happens because of a cause some time in the past. Newton didn’t tackle time‑offset problems, I suppose mostly because the effects weren’t detectable with the technology of his time.”

“They had magnets.”

“Permanent ones, not electromagnets they could control and measure the effects of. Electromagnetic hysteresis generates effects that Newton couldn’t have known about. Fahrenheit didn’t invent temperature measurement until two years before Newton died, so science hadn’t yet discovered temperature‑dependent hysteresis effects. The microscope had been around for a half‑century or so but in Newton’s day people were still arguing about whether cells were a necessary part of a living organism. Newton’s world didn’t have an inkling of cellular biophysics, much less biophysical hysteresis. At human scale, country‑level economic data if it existed at all was a military secret — not a good environment for studying cases of economic hysteresis.”

“So what you’re saying is that Newton couldn’t have tackled those even if he’d wanted to. Got it. But that’s a pretty broad list of situations. How can you say they’re all hystereseseses, … loopy things?”

“They’ve all got a set of characteristics that you can fit into similar mathematical models. They’re all about some statistical summary of a complex system. The system is under the influence of some outside driver, could be a physical force or something more abstract. The driver can work in either of two opposing directions, and the system can respond to the driver to change in either of two opposing ways. Oh, and a crucial characteristic is that the system has a buffer of some sort that saves a memory of what the driver did and serves it up some time later.”

“Wait, lemme see if I can match those pieces to my magnetic nail. OK, the driver is the outside magnetic field, that’s easy, the system is the magnetic iron atoms, and the summary is the nail’s field. The driver can point north‑to‑south or south‑to‑north and the atoms can, too. Ah, and the memory is the domains ’cause the big ones hold onto the direction the field pointed last. How’d I do?”

“Perfect.”

“Goody for me. So why are those guys on the radio saying the economy is hysterical, ‘scuse, has hysteresis? What’s which part?”

“Economies are complex beasts, with a lot of separate but interacting hysteresis loops. These guys, what were they discussing at the time?”

“Unemployment, if I remember right. They said the job market is sticky, whatever that means.”

“Good example. Here’s our basic hysteresis loop with some relabeling. Running across we’ve got our driver, the velocity of money, which claims to measure all the buying and selling. Up‑and‑down we’ve got total employment. The red dot is the initial equilibrium, some intermediate level where there’s just enough cash flowing around that some but not all people have jobs. Then a new industry, say cellphones, comes in. Suddenly there’s people making cellphones, selling cellphones, repairing cellphones –“

“I get the idea. More activity, money flows faster, more jobs and people are happy. OK, then the pandemic comes along, money slows down, jobs cut back and around we go. But where’s the stickiness?”

“In people’s heads. If they get into Depression thinking, everyone holds onto cash even if there’s a wonderful new cellphone out there. People have to start thinking that conditions will improve before conditions can improve. That’s the delay factor.”

“Hysterical, all right.”

~~ Rich Olcott

The Hysterical Penguin

“Sy, you said that hysteresis researchers filled in two of Newton’s Physics gaps. OK, I get that he couldn’t do atomic stuff ’cause atoms hadn’t been discovered yet. What’s the other one?”

Proposition XI, Problem VI
from Book I of Newton’s Principia

“Non‑linearity.”

“You’re gonna have to explain that.”

“It’s a math thing. I know you don’t go for equations, so here’s a picture to get you started on how Newton solved problems. Look at all familiar?”

“Whoa, looks like something toward the end of my Geometry class.”

“Exactly. Newton was trained as a geometer and he was good at it. His general strategy was to translate a physical system to a geometrical structure and then work out its properties as a series of geometric proofs. The good news was that he proved a lot of things that started us on the way to quantitative science. The bad news was that his proofs were hard to extend to situations where the geometry wasn’t so easy.”

“That’s easy?”

“For Newton, maybe it was. Who knows? Anyway, the toolkit they gave you in Geometry class was what Newton had to work with — logic, straight lines and some special curves like ellipses and parabolas whose properties had been studied since Euclid, all on a flat plane. Nearly everything depended on finding proportionalities between different distances or areas — this line is twice that one but equal to a third, that sort of thing. Proportionality like that is built into equations like here+(velocity×time)=there. See how distance traveled is proportional to time? The equation plots as a straight line, which is why it’s called a linear equation.”

“So what’s non‑linear look like — all wiggle‑waggle?”

“Not necessarily. Things can vary smoothly along curves that aren’t those classical ones. Newton’s methods are blocked on those but Leibniz’s algebra‑based calculus isn’t. That’s why it won out with people who needed answers. What’s important here is that Newton’s lines can’t describe everything. Mmm… where does a straight line end?”

“Either at a T or never. Same thing for a parabola. Hey, ellipses don’t really end, either.”

“Mm-hm. Newton’s lines either stop abruptly or they continue forever. They don’t grow or peter out exponentially like things in real life do. Suppose something’s velocity changes, for instance.”

“That’s acceleration. I like accelerating.”

“So true, I’ve experienced your driving. But even you don’t accelerate at a constant rate. You go heavy or light or maybe brake, whatever, and our speed goes up or down depending. The only way Newton’s geometry can handle variable acceleration is to break it into mostly‑constant pieces and work one piece at a time. Come to think of it, that may be where he got the idea for his fluxions method for calculus. Fortunately for him, some things like planets and artillery shells move pretty close to what his methods predict. Unfortunately, things like disease epidemics and economies don’t, which is why people are interested in non‑linearity.”

“So what do these hysteresis guys do about it?”

“Mostly algebraic calculus or computer approximations. But there wasn’t just one group of hysteresis guys, there was a bunch of groups, each looking at different phenomena where history makes a difference. Each group had their own method of attack.”

“Like your elephant thing with Anne, lots of notions about entropy.”

Typical hysteresis loop
Red — initial evolution
Blue — subsequent changes

“How’d you find out about that?”

You wrote those posts, Sy, about three years ago.”

“Oh, that’s right. Talk about history. Anyway, it took decades for the ecologists, epidemiologists, civil engineers and several kinds of physicist to realize that they all have systems that behave similarly when driven by a stressor. Starting at some neutral situation, the system evolves in the driver’s direction to some maximum deviation where increased stress has no further effect. When the stress is relieved, the system may stick temporarily at the strained position. When it does evolve away from there, maybe a reverse driver is needed to force a return to the starting situation. In fact, if the forward and reverse drivers are applied repeatedly the system may never get back to the initial unstressed position.”

“Like that iron nail. Not magnetic, then magnetic, then reversed.”

~~ Rich Olcott

A Beetled Brow

Vinnie’s brow was wrinkling so hard I could hear it over the phone. “Boltzmann, Boltzmann, where’d I hear that name before? … Got it! That’s one of those constants, ain’t it, Sy? Molecules or temperature or something?”

“The second one, Vinnie. Avagadro was the molecule counter. Good memory. Come to think of it, both Boltzmann and Avagadro bridged gaps that Loschmidt worked on.”

“Loschmidt’s thing was the paradox, right, between Newton saying events can back up and thermodynamics saying no, they can’t. You said Boltzmann’s Statistical Mechanics solved that, but I’m still not clear how.”

“Let me think of an example. … Ah, you’ve got those rose bushes in front of your place. I’ll bet you’ve also put up a Japanese beetle trap to protect them.”

“Absolutely. Those bugs would demolish my flowers. The trap’s lure draws them away to my back yard. Most of them stay there ’cause they fall into the trap’s bag and can’t get out.”

“Glad it works so well for you. OK, Newton would look at individual beetles. He’d see right off that they fly mostly in straight lines. He’d measure the force of the wind and write down an equation for how the wind affects a beetle’s flight path. If the wind suddenly blew in the opposite direction, that’d be like the clock running backwards. His same equation would predict the beetle’s new flight path under the changed conditions. You with me?”

“Yeah, no problem.”

“Boltzmann would look at the whole swarm. He’d start by evaluating the average point‑to‑point beetle flight, which he’d call ‘mean free path.’ He’d probably focus on the flight speed and in‑the‑air time fraction. With those, if you tell him how many beetles you’ve got he could generate predictions like inter‑beetle separation and how long it’d take an incoming batch of beetles to cross your yard. However, predicting where a specific beetle will land next? Can’t do that.”

“Who cares about one beetle?”

“Well, another beetle might. …
Just thought of a way that Statistical Mechanics could actually be useful in this application. Once Boltzmann has his numbers for an untreated area, you could put in a series of checkpoints with different lures. Then he could develop efficiency parameters just by watching the beetle flying patterns. No need to empty traps. Anyhow, you get the idea.”

Japanese Beetle, photo by David Cappaert, Bugwood.org
under Creative Commons BY 3.0

“Hey, I feel good emptying that trap, I’m like standing up for my roses. Anyway, so how does Avagadro play into this?”

“Indirectly and he was half a century earlier. In 1805 Gay‑Lussac showed that if you keep the pressure and temperature constant, it tales two volumes of hydrogen to react with one volume of oxygen to produce one volume of water vapor. Better, the whole‑number‑ratio rule seemed to hold generally. Avagadro concluded that the only way Gay‑Lussac’s rule could be general is if at any temperature and pressure, equal volumes of every kind of gas held the same number of molecules. He didn’t know what that number was, though.”

“HAW! Avagadro’s number wasn’t a number yet.”

“Yeah, it took a while to figure out. Then in 1865, Loschmidt and a couple of others started asking, “How big is a gas molecule?” Some gases can be compressed to the liquid state. The liquids have a definite volume, so the scientists knew molecules couldn’t be infinitely small. Loschmidt put numbers to it. Visualize a huge box of beetles flying around, bumping into each other. Each beetle, or molecule, ‘occupies’ a cylinder one beetle wide and the length of its mean free path between collisions. So you’ve got three volumes — the beetles, the total of all the cylinders, and the much larger box. Loschmidt used ratios between the volumes, plus density data, to conclude that air molecules are about a nanometer wide. Good within a factor of three. As a side result he calculated the number of gas molecules per unit volume at any temperature and pressure. That’s now called Loschmidt’s Number. If you know the molecular weight of the gas, then arithmetic gives you Avagadro’s number.”

“Thinking about a big box of flying, rose‑eating beetles creeps me out.”

  • Thanks to Oriole Hart for the story‑line suggestion.

~~ Rich Olcott

Abstract Horses

It was a young man’s knock, eager and a bit less hesitant than his first visit.

“C’mon in, Jeremy, the door’s open.”

“Hi, Mr Moire, it’s me, Jerem…  How did ..?  Never mind.  Ready for my black hole questions?”

“I’ll do what I can, Jeremy, but mind you, even the cosmologists are still having a hard time understanding them.  What’s your first question?”

“I read where nothing can escape a black hole, not even light, but Hawking radiation does come out because of virtual particles and what’s that about?”

“That’s a very lumpy question.  Let’s unwrap it one layer at a time.  What’s a particle?”

“A little teeny bit of something that floats in the air and you don’t want to breathe it because it can give you cancer or something.”

“That, too, but we’re talking physics here.  The physics notion of a particle came from Newton.  He invented it on the way to his Law of Gravity and calculating the Moon’s orbit around the Earth.  He realized that he didn’t need to know what the Moon is made of or what color it is.  Same thing for the Earth — he didn’t need to account for the Earth’s temperature or the length of its day.  He didn’t even need to worry about whether either body was spherical.  His results showed he could make valid predictions by pretending that the Earth and the Moon were simply massive points floating in space.”

Accio abstractify!  So that’s what a physics particle is?”

“Yup, just something that has mass and location and maybe a velocity.  That’s all you need to know to do motion calculations, unless the distance between the objects is comparable to their sizes, or they’ve got an electrical charge, or they move near lightspeed, or they’re so small that quantum effects come into play.  All other properties are irrelevant.”

“So that’s why he said that the Moon was attracted to Earth like the apple that fell on his head was — in his mind they were both just particles.”

“You got it, except that apple probably didn’t exist.”

“Whatever.  But what about virtual particles?  Do they have anything to do with VR goggles and like that?”

“Very little.  The Laws of Physics are optional inside a computer-controlled ‘reality.’  Virtual people can fly, flow of virtual time is arbitrary, virtual electrical forces can be made weaker or stronger than virtual gravity, whatever the programmers decide will further the narrative.  But virtual particles are much stranger than that.”

“Aw, they can’t be stranger than Minecraft.  Have you seen those zombie and skeleton horses?”Horses

“Yeah, actually, I have.  My niece plays Minecraft.  But at least those horses hang around.  Virtual particles are now you might see them, now you probably don’t.  They’re part of why quantum mechanics gave Einstein the willies.”

“Quantum mechanics comes into it?  Cool!  But what was Einstein’s problem?  Didn’t he invent quantum theory in the first place?”

“Oh, he was definitely one of the early leaders, along with Bohr, Heisenberg, Schrödinger and that lot.  But he was uncomfortable with how the community interpreted Schrödinger’s wave equation.  His row with Bohr was particularly intense, and there’s reason to believe that Bohr never properly understood the point that Einstein was trying to make.”

“Sounds like me and my Dad.  So what was Einstein’s point?”

“Basically, it’s that the quantum equations are about particles in Newton’s sense.  They lead to extremely accurate predictions of experimental results, but there’s a lot of abstraction on the way to those concrete results.  In the same way that Newton reduced Earth and Moon to mathematical objects, physicists reduced electrons and atomic nuclei to mathematical objects.”

“So they leave out stuff like what the Earth and Moon are made of.  Kinda.”

“Exactly.  Bohr’s interpretation was that quantum equations are statistical, that they give averages and relative probabilities –”

“– Like Schrödinger’s cat being alive AND dead –”

“– right, and Einstein’s question was, ‘Averages of what?‘  He felt that quantum theory’s statistical waves summarize underlying goings-on like ocean waves summarize what water molecules do.  Maybe quantum theory’s underlying layer is more particles.”

“Are those the virtual particles?”

“We’re almost there, but I’ve got an appointment.  Bye.”

“Sure.  Uhh… bye.”

~~ Rich Olcott

The question Newton couldn’t answer

250 years ago, when people were getting used to the idea that the planets circle the Sun and not the other way around, they wondered how that worked.  Isaac Newton said, “I can explain it with my Laws of Motion and my Law of Gravity.”

The first Law of Motion is that an object will move in a straight line unless acted upon by a force.  If you’re holding a ball by a string and swing the ball in a circle, the reason the ball doesn’t fly away is that the string is exerting a force on the ball.  Using Newton’s Laws, if you know the mass of the ball and the length of the string, you can calculate how fast the ball moves along that circle.

Newton said that the Solar System works the same way.  Between the Sun and each planet there’s an attractive force which he called gravity.  If you can determine three points in a planet’s orbit, you can use the Laws of Motion and the Law of Gravity to calculate the planet’s speed at any time, how close it gets to the Sun, even how much the planet weighs.

Astronomers said, “This is wonderful!  We can calculate the whole Solar System this way, but… we don’t see any strings.  How does gravity work?”

Newton was an honest man.  His response was, “I don’t know how gravity works.  But I can calculate it and that should be good enough.”

And that was good enough for 250 years until Albert Einstein produced his Theories of Relativity.  This graphic shows one model of Einstein’s model of “the fabric of space.”  According to the theory, light (the yellow threads) travels at 186,000 miles per second everywhere in the Universe.

Fabric of Space 4a

As we’ve seen, the theory also says that space is curved and compressed near a massive object.  Accordingly, the model’s threads are drawn together near the dark circle, which could represent a planet or a star or a black hole.  If you were standing next to a black hole (but not too close). you’d feel fine because all your atoms and the air you breathe would shrink to the same scale.  You’d just notice through your telescope that planetary orbits and other things in the Universe appear larger than you expect.FoS wave

This video shows how a massive object’s space compression affects a passing light wave.  The brown dot and the blue dot both travel at 186,000 miles per second, but “miles are shorter near a black hole.”  The wave’s forward motion is deflected around the object because the blue dot’s miles are longer than the miles traveled by the brown dot.

When Einstein presented his General Theory of Relativity in 1916, his calculations led him to predict that this effect would cause a star’s apparent position to be altered by the Sun’s gravitational field. Fabric of Space 4b

An observer at the bottom of this diagram can pinpoint the position of star #1 by following its light ray back to the star’s location.  Star #2, however, is so situated that its light ray is bent by our massive object.  To the observer, star #2’s apparent position is shifted away from its true position.

In 1919, English physicist-astronomer Arthur Eddington led an expedition to the South Atlantic to test Einstein’s prediction.  Why the South Atlantic?  To observe the total eclipse of the sun that would occur there.  With the Sun’s light blocked by the Moon, Eddington would be able to photograph the constellation Taurus behind the Sun.

Sure enough, in Eddington’s photographs the stars closest to the Sun were shifted in their apparent position relative to those further way.  Furthermore, the sizes of the shifts were almost embarrassingly close to Einstein’s predicted values.

Eddington presented his photographs to a scientific conference in Cambridge and thus produced the first public confirmation of Einstein’s theory of gravity.

Wait, how does an object bending a light ray connect with that object’s pull on another mass?  Another piece of Einstein’s theory says that if a light ray and a freely falling mass both start from the same point in spacetime, both will follow the same path through space.  American physicist John Archibald Wheeler said, “Mass bends space, and bent space tells mass how to move.”

 

~~ Rich Olcott

Gargh, His Heirs, and the AAAD Problem

Gargh the thinkerGargh, proto-humanity’s foremost physicist 2.5 million years ago, opened a practical investigation into how motion works.  “I throw rock, hit food beast, beast fall down yes.  Beast stay down no.  Need better rock.”  For the next couple million years, we put quite a lot of effort into making better rocks and better ways to throw them.  Less effort went into understanding throwing.

There seemed to be two kinds of motion.  The easier kind to understand was direct contact — “I push rock, rock move yes.  Rock stop move when rock hit thing that move no.”  The harder kind was when there wasn’t direct contact — “I throw rock up, rock hit thing no but come back down.  Why that?

Gargh was the first but hardly the last physicist to puzzle over the Action-At-A-Distance problem (a.k.a. “AAAD”).  Intuition tells us that between pusher and pushee there must be a concrete linkage to convey the push-force.  To some extent, the history of physics can be read as a succession of solutions to the question, “What linkage induces this apparent case of AAAD?”

Most of humanity was perfectly content with AAAD in the form of magic of various sorts.  To make something happen you had to wish really hard and/or depend on the good will of some (generally capricious) elemental being.

aristotle 1Aristotle wasn’t satisfied with anything so unsystematic.  He was just full of theories, many of which got in each other’s way.  One theory was that things want to go where they’re comfortable  because of what they’re made of — stones, for instance, are made of earth so naturally they try to get back home and that’s why we see them fall downwards (no concrete linkage, so it’s still AAAD).

Unfortunately, that theory didn’t account for why a thrown rock doesn’t just fall straight down but instead goes mostly in the direction it’s thrown.  Aristotle (or one of his followers) tied that back to one of his other theories, “Nature hates a vacuum.”  As the rock flies along, it pushes the air aside (direct contact) and leaves a vacuum behind it. More air rushes in to fill the vacuum and pushes the rock ahead (more direct contact).

We got a better (though still AAAD) explanation in the 17th Century when physicists invented the notions of gravity and inertia.Newton 204

Newton made a ground-breaking claim in his Principia.  He proposed that the Solar System is held together by a mysterious AAAD force he called gravity.  When critics asked how gravity worked he shrugged, “I do not form hypotheses” (though he did form hypotheses for light and other phenomena).

Inertia is also AAAD.  Those 17th Century savants showed that inertial forces push mass towards the Equator of a rotating object.  An object that’s completely independent of the rest of the Universe has no way to “know” that it’s rotating so it ought to be a perfect sphere.  In fact, the Sun and each of its planets are wider at the equator than you’d expect from their polar diameters.  That non-sphere-ness says they must have some AAAD interaction with the rest of the Universe.  A similar argument applies to linear motion; the general case is called Mach’s Principle.
JCMaxwell

The ancients knew of the mysterious AAAD agents electricity and its fraternal twin, magnetism.  However, in the 19th Century James Clerk Maxwell devised a work-around.  Just as Newton “invented” gravity, Maxwell “invented” the electromagnetic field.  This invisible field isn’t a material object.  However, waves in the field transmit electromagnetic forces everywhere in the Universe.  Not AAAD, sort of.

It wasn’t long before someone said, “Hey, we can calculate gravity that way, too.”  That’s why we now speak of a planet’s gravitational field and gravitational waves.

But the fields still felt like AAAD because they’re not concrete.  Some modern physicists stand that objection on its head.  Concrete objects, they say, are made of atoms which themselves are nothing more than persistent fluctuations in the electromagnetic and gravitational fields.  By that logic, the fields are what’s fundamental — all motion is by direct contact.einstein-tongue edged

Einstein moved resolutely in both directions.  He negated gravity’s AAAD-ness by identifying mass-contorted space as the missing linkage.  On the other hand, he “invented” quantum entanglement, the ultimate spooky AAAD.

 ~~ Rich Olcott

Sir Isaac, The Atom And The Whirlpool

Newton and atomNewton definitely didn’t see that one coming.  He has an excuse, though.  No-one in in the 17th Century even realized that electricity is a thing, much less that the electrostatic force follows the same inverse-square law that gravity does. So there’s no way poor Isaac would have come up with quantum mechanics.

Lemme ‘splain.  Suppose you have a mathematical model that’s good at predicting some things, like exactly where Jupiter will be next week.  But if the model predicts an infinite value under some circumstances, that tells you it’s time to look for a new model for those particular circumstances.

For example, Newton’s Law of Gravity says that the force between two objects is proportional to 1/r2, where r is the distance between their centers of mass.  The Law does a marvelous job with stars and satellites but does the infinity thing when r approaches zero.  In prior posts I’ve described some physics models that supercede Newton’s gravity law at close distances.

Electrical forces are same song second verse with a coda.  They follow the 1/r2 law, so they also have those infinity singularities.  According to the force law, an electron (the ultimate “particle” of negative charge) that approaches another electron would feel a repulsion that rises to infinity.  The coda is that as an electron approaches a positive atomic nucleus it would feel an attraction that rises to infinity.  Nature abhors infinities, so something else, some new physics, must come into play.

I put that word “particle” in quotes because common as the electron-is-a-particle notion is, it leads us astray.  We tend to think of the electron as this teeny little billiard-ballish thing, but it’s not like that at all.  It’s also not a wave, although it sometimes acts like one.  “Wavicle” is just  a weasel-word.  It’s far better to think of the electron as just a little traveling parcel of energy.  Photons, too, and all those other denizens of the sub-atomic zoo.

An electron can’t crumble or leak mass or deform to merge the way that sizable objects can.  What it does is smear. Quantum mechanics is all about the smear.  Much more about that in later posts.


 

Newton in whirlpoolIf Newton loved anything (and that question has been discussed at length), he loved an argument.  His battle with Liebniz is legendary.  He even fought with Descartes, who was a decade dead when Newton entered Cambridge.

Descartes had grabbed “Nature abhors a vacuum” from Aristotle and never let it go.  He insisted that the Universe must be filled with some sort of water-like fluid.  He know the planets went round the Sun despite the fluid getting in the way, so he reasoned they moved as they did because of the fluid.

Surely you once played with toy boats in the bathtub.  You may have noticed that when you pulled your arm quickly through the water little whirlpools followed your arm.  If a whirlpool encountered a very small boat, the boat might get caught in it and move in the same direction.  Descartes held that the Solar System worked like that, with the Sun as your arm and the planets caught in Sun-stirred vortices within that watery fluid.

Newton knew that couldn’t be right.  The planets don’t run behind the Sun, they share the same plane.  Furthermore, comets orbit in from all directions.  Crucially, Descartes’ theory conflicted with his own and that settled the matter for Newton.  Much of Principia‘s “Book II” is about motions of and through fluid media.  He laid out there what a trajectory would look like under a variety of conditions.  As you’d expect, none of the paths do what planets, moons and comets do.

From Newton’s point of view, the only use for Book II was to demolish Descartes.  For us in later generations, though, he’d invented the science of hydrodynamics.

Which was a good thing so long as you don’t go too far upstream towards the center of the whirlpool.  As you might expect (or I wouldn’t even be writing this section), Book II is littered with 1/rn formulas that go BLOOIE when the distances get short.  What happens near the center?  That’s where the new physics of turbulence kicks in.

~~ Rich Olcott

Squeezing past Newton’s infinity

One of the most powerful moments in musical theater — Philip Quast Quastin his Les Miz role of Inspector Javert, praising the stars for the steadfastness and reverence for law that they signify for him.  The performance is well worth a listen.

Javert’s certitude came from Newton’s sublimely reliable mechanics — the notion that every star’s and planet’s motion is controlled by a single law, F~(1/r2).  The law says that the attractive force between any pair of bodies is inversely proportional to the square of the distance between their centers.  But as Javert’s steel-clad resolve hid a fatal spark of mercy towards Jean Valjean, so Newton’s clockworks hold catastrophe at their axles.

Newton’s gravity law has a problem.  As the distance approaches zero, the predicted force approaches infinity.  The law demands that nearby objects accelerate relentlessly at each other to collide with infinite force, after which their combined mass attracts other objects.  In time, everything must collapse in a reverse of The Big Bang.

Victor Hugo wrote Les Misérables about 180 years after Newton published his Principia.  A decade before Hugo’s book, Professeur Édouard Roche (pronounced rōsh) solved at least part of Newton’s problem.

Roche realized that Newton had made an important but crucial simplification.  Early in the Principia, he’d proven that for many purposes you can treat an entire object as though all of its mass were concentrated at a single point (the “center of mass”).  But in real gravity problems every particle of one object exerts an attraction for every particle of the other.

That distinction makes no difference when the two objects are far apart.  However, when they’re close together there are actually two opposing forces in play:

  • gravity, which preferentially affects the closest particles, and
  • tension, which maintains the integrity of each structure.

contact_binary_1
Binary star pair demonstrating Roche lobes, image courtesy of Cronodon.com

Roche noted that the gravity fields of any pair of objects must overlap.  There will always be a point on the line between them where a particle will be tugged equally in either direction.  If two bodies are close and one or both are fluid (gases and plasmas are fluid in this sense), the tension force is a weak competitor.  The partner with the less intense gravity field will lose material across that bridge to the other partner. Binary star systems often evolve by draining rather than collision.

Now suppose both bodies are solid.  Tension’s game is much stronger.  Nonetheless, as they approach each other gravity will eventually start ripping chunks off of one or both objects.  The only question is the size of the chunks — friable materials like ices will probably yield small flakes, as opposed to larger lumps made from silicates and other rocky materials.  Roche described the final stage of the process, where the less-massive body shatters completely.  The famous rings of Saturn and the less famous rings of Neptune, Uranus and Jupiter all appear to have been formed by this mechanism.

Roche was even able to calculate how close the bodies need to be for that final stage to occur. The threshold, now called the Roche Limit, depends on the size and mass of each body. You can get more detail here.

Klingon3And then there’s spaghettification.  That’s a non-relativistic tidal phenomenon that occurs near an extremely dense body like a neutron star or a black hole.  Because these objects pack an enormous amount of mass into a very small volume, the force of gravity at a close-in point is significantly greater than the force just a little bit further out. Any object, say a Klingon Warbird that ignored peril markings on a space map (Klingons view warnings as personal challenges), would find itself stretched like a noodle between high gravity on the side near the black hole and lower gravity on the opposite side.  (In this cartoon, notice how the stretching doesn’t care which way the pin-wheeling ship is pointed.)

Nature abhors singularities.  Where a mathematical model like Newton’s gravity law predicts an infinity, Nature generally says, “You forgot something.”  Newton assumed that objects collide as coherent units.  Real bodies drain, crumble, or deform to slide together.  Look to the apparent singularities to find new physics.

~~ Rich Olcott