Gin And The Art of Quantum Mechanics

On May 30, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, Uncategorized2 Comments

“Fancy a card game, Johnny?”
“Sure, Jennie, deal me in. Wot’re we playin’?”
“Gin rummy sound good?”

Great idea, and it fits right in with our current Entanglement theme. The aspect of Entanglement that so bothered Einstein, “spooky action at a distance,” can be just as spooky close-up. Check out this magic example — go ahead, it’s a fun trick to figure out.

Spooky, hey? And it all has to do with cards being two-dimensional. I know, as objects they’ve got three dimensions same as anyone (four, if you count time), but functionally they have only two dimensions — rank and suit.

When you’re looking at a gin rummy hand you need to consider each dimension separately. The queens in this hand form a set — three cards of the same rank. So do the three nines. In the suit dimension, the 4-5-6-7 run is a sequence of ranks all in the same suit.

A physicist might say that evaluating a gin rummy hand is a separable problem, because you can consider each dimension on its own. <Hmm … three queens, that’s a set, and three nines, another set. The rest are hearts. Hey, the hearts are in sequence, woo-hoo!>

“Gin!”

If you chart the hand, the run and sets and their separated dimensions show up clearly even if you don’t know cards.

A standard strategy for working a complex physics problem is to look for a way to split one kind of motion out from what else is going on. If the whole shebang is moving in the z-direction, you can address the z-positions, z-velocities and z-forces as an isolated sub-problem and treat the x and y stuff separately. Then, if everything is rotating in the xy plane you may be able to separate the angular motion from the in-and-out (radial) motion.

But sometimes things don’t break out so readily. One nasty example would be several massive stars flying toward each other at odd angles as they all dive into a black hole. Each of the stars is moving in the black hole’s weirdly twisted space, but it’s also tugged at by every other star. An astrophysicist would call the problem non-separable and probably try simulating it in a computer instead of setting up a series of ugly calculus problems.

The card trick video uses a little sleight-of-eye to fake a non-separable situation. Here’s the chart, with green dots for the original set of cards and purple dots for the final hand after “I’ve removed the card you thought of.” The kings are different, and so are the queens and jacks. As you see, the reason the trick works is that the performer removed all the cards from the original hand.

The goal of the illusion is to confuse you by muddling ranks with suits. What had been a king of diamonds in the first position became a king of spades, whereas the other king became a queen. You were left with an entangled perception of each card’s two dimensions.

In quantum mechanics that kind of entanglement crops up any time you’ve got two particles with a common history. It’s built into the math — the two particles evolve together and the model gives you no way to tell which is which.

Suppose for instance that an electron pair has zero net spin (spin direction is a dimension in QM like suit is a dimension in cards). If the electron going to the left is spinning clockwise, the other one must be spinning counterclockwise. Or the clockwise one may be the one going to the right — we just can’t tell from the math which is which until we test one of them. The single test settles the matter for both.

Einstein didn’t like that ambiguity. His intuition told him that QM’s statistics only summarize deeper happenings. Bohr opposed that idea, holding that QM tells us all we can know about a system and that it’s nonsense to even speak of properties that cannot be measured. Einstein called the deeper phenomena “elements of reality” though they’re currently referred to as “hidden variables.” Bohr won the battle but maybe not the war — Einstein had such good intuition.

~~ Rich Olcott

Oh, what an entangled wave we weave

On May 23, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

“Here’s the poly bag wiff our meals, Johnny. ‘S got two boxes innit, but no labels which is which.”
“I ordered the mutton pasty, Jennie, anna fish’n’chips for you.”
“You c’n have this box, Johnny. I’ll take the other one t’ my place to watch telly.”
…
<ring>
” ‘Ullo, Jennie? This is Johnny. The box over ‘ere ‘as the fish. You’ve got mine!”

In a sense their supper order arrived in an entangled state. Our friends knew what was in both boxes together, but they didn’t know what was in either box separately. Kind of a Schrödinger’s Cat situation — they had to treat each box as 50% baked pasty and 50% fried codfish.

But as soon as Johnny opened one box, he knew what was in the other one even though it was somewhere else. Jennie could have been in the next room or the next town or the next planet — Johnnie would have known, instantly, which box had his meal no matter how far away that other box was.

By the way, Jennie was free to open her box on the way home but that’d make no difference to Johnnie — the box at his place would have stayed a mystery to him until either he opened it or he talked to her.

Information transfer at infinite speed? Of course not, because neither hungry person knows what’s in either box until they open one or until they exchange information. Even Skype operates at light-speed (or slower).

But that’s not quite quantum entanglement, because there’s definite content (meat pie or batter-fried cod) in each box. In the quantum world, neither box holds something definite until at least one box is opened. At that point, ambiguity flees from both boxes in an act of global correlation.

There’s strong experimental evidence that entangled particles literally don’t know which way is up until one of them is observed. The paired particle instantaneously gets that message no matter how far away it is.

Niels Bohr’s Principle of Complementarity is involved here. He held that because it’s impossible to measure both wave and particle properties at the same time, a quantized entity acts as a wave globally and only becomes local when it stops somewhere.

Here’s how extreme the wave/particle global/local thing can get. Consider some nebula a million light-years away. A million years ago an electron wobbled in the nebular cloud, generating a spherical electromagnetic wave that expanded at light-speed throughout the Universe.

cats-eye nebula — The Cat’s Eye Nebula (NGC 6543)
courtesy of NASA’s Hubble Space Telescope

Last night you got a glimpse of the nebula when that lightwave encountered a retinal cell in your eye. Instantly, all of the wave’s energy, acting as a photon, energized a single electron in your retina. That particular lightwave ceased to be active elsewhere in your eye or anywhere else on that million-light-year spherical shell.

Surely there was at least one other being, on Earth or somewhere else, that was looking towards the nebula when that wave passed by. They wouldn’t have seen your photon nor could you have seen any of theirs. Somehow your wave’s entire spherical shell, all 10¹² square lightyears of it, instantaneously “knew” that your eye’s electron had extracted the wave’s energy.

But that directly contradicts a bedrock of Einstein’s Special Theory of Relativity. His fundamental assumption was that nothing (energy, matter or information) can go faster than the speed of light in vacuum. STR says it’s impossible for two distant points on that spherical wave to communicate in the way that quantum theory demands they must.

Want some irony? Back in 1906, Einstein himself “invented” the photon in one of his four “Annus mirabilis” papers. (The word “photon” didn’t come into use for another decade, but Einstein demonstrated the need for it.) Building on Planck’s work, Einstein showed that light must be emitted and absorbed as quantized packets of energy.

It must have taken a lot of courage to write that paper, because Maxwell’s wave theory of light had been firmly established for forty years prior and there’s a lot of evidence for it. Bottom line, though, is that Einstein is responsible for both sides of the wave/particle global/local puzzle that has bedeviled Physics for a century.

~~ Rich Olcott

Think globally, act locally. Electrons do.

On May 16, 2016November 30, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

“Watcha, Johnnie, you sure ‘at particle’s inna box?”
“O’course ’tis, Jennie! Why wouldn’t it be?”
“Me Mam sez particles can tunnel outta boxes ’cause they’re waves.”
“Can’t be both, Jessie.”

Maybe it can.

Nobel-winning (1965) physicist Richard Feynman said the double-slit experiment (diagrammed here) embodies the “central mystery” of Quantum Mechanics.

When the bottom slit is covered the display screen shows just what you’d expect — a bright area opposite the top slit.

When both slits are open, the screen shows a banded pattern you see with waves. Where a peak in a top-slit wave meets a peak in the bottom-slit wave, the screen shines brightly. Where a peak meets a trough the two waves cancel and the screen is dark. Overall there’s a series of stripes. So electrons are waves, right?

But wait. If we throttle the beam current way down, the display shows individual speckles where each electron hits. So the electrons are particles, right?

Now for the spooky part. If both slits are open to a throttled beam those singleton speckles don’t cluster behind the slits as you’d expect particles to do. A speckle may appear anywhere on the screen, even in an apparently blocked-off region. What’s more, when you send out many electrons one-by-one their individual hits cluster exactly where the bright stripes were when the beam was running full-on.

It’s as though each electron becomes a wave that goes through both slits, interferes with itself, and then goes back to being a particle!

By the way, this experiment isn’t a freak observation. It’s been repeated with the same results many times, not just with electrons but also with light (photons), atoms, and even massive molecules like buckyballs (fullerene spheres that contain 60 carbon atoms). In each case, the results indicate that the whatevers have a dual character — as a localized particle AND as a wave that reacts to the global environment.

Physicists have been arguing the “Which is it?” question ever since Louis-Victor-Pierre-Raymond, the 7th Duc de Broglie, raised it in his 1924 PhD Thesis (for which he received a Nobel Prize in 1929 — not bad for a beginner). He showed that any moving “particle” comes along with a “wave” whose peak-to-peak wavelength is inversely proportional to the particle’s mass times its velocity. The longer the wavelength, the less well you know where the thing is.

I just had to put numbers to de Broglie’s equation. With Newton in mind, I measured one of the apples in my kitchen. To scale everything, I assumed each object moved by one of its diameters per second. (OK, I cheated for the electron — modern physics says it’s just a point, so I used a not-really-valid classical calculation to get something to work with.)

“Particle”	Mass, kilograms	Diameter, meters	Wavelength, meters	Wavelength, diameters
Apple	0.2	0.07	7.1×10^-33	1.0×10^-31
Buckyball	1.2×10^-24	1.0×10^-9	0.083	8.3×10⁺⁷
Hydrogen atom	1.7×10^-27	1.0×10^-10	600	6.0×10⁺¹²
Electron	9.1×10^-31	3.0×10^-17	3.7×10⁺¹²	1.2×10⁺²⁹

That apple has a wave far smaller than any of its hydrogen atoms so I’ll have no trouble grabbing it for a bite. Anything tinier than a small virus is spread way out unless it’s moving pretty fast, as in a beam apparatus. For instance, an electron going at 1% of light-speed has a wavelength only a nanometer wide.

Different physicists have taken different positions on the “particle or wave?” question. Duc de Broglie claimed that both exist — particles are real and they travel where their waves tell them to. Bohr and Heisenberg went the opposite route, saying that the wave’s not real, it’s only a mathematical device for calculating relative probabilities for measuring this or that value. Furthermore, the particle doesn’t exist as such until a measurement determines its location or momentum. Einstein and Schrödinger liked particles. Feynman and Dirac just threw up their hands and calculated.

Which brings us to the other kind of quantum spookiness — “entanglement.” In fact, Einstein actually used the word spukhafte (German for “spooky”) in a discussion of the notion. He really didn’t like it and for good reason — entanglement rudely collides with his own Theory of Relativity. But that’s another story.

~~ Rich Olcott

Location, Location, Location

On May 9, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

“Hoy, Johnny, still got that particle inna box?”
“Sure do, Jessie.”
“So where’s hit in there?”
“Me Pap says hit’s spread-out like but hit’s mostly inna middle.”
“Why’s hit spread then?”
“The more I taps the box, the wider hit spreads. Sommat to do wiff energy.”

Newton would have answered Jessie’s question by saying, sort of, “Pick a point anywhere in the box. The probability that the particle is at that point is equal to the probability that it’s at any other point.”

Quantum physicists take a different approach. They start by saying, “We know there’s zero probability that the particle is anywhere outside of the box, so there must be zero probability that it’s exactly at any wall.”

Now for a trick that we’re actually quite used to. When you listen to an orchestra, you can usually pick out the notes being played by a particular instrument. Someone blessed/cursed with perfect pitch can tell when a note is just a leetle bit flat, say an A being played at 438 cycles instead of 440. You can create any sound by mixing together the right frequencies in the right proportion. That’s how an MP3 recorder does it.

QM solutions use that strategy the other way round. They calculate probabilities by adding together sets of symmetric elementary shapes, all of which are zero at certain places, like the box walls. For instance, on average Johnnie’s particle will be near the middle of his box, so we start a set with an orange mound of probability right there. That mound is like our base frequency — it has no nodes, no non-wall places where the probability is zero.

Then we add a first overtone, the one-node yellow shape that represents equal probability on either side of a plane of zero probability.

Two nodal planes at right angles give us the four-peaked green shape. Further steps up have more and more nodal planes (cyan then blue, and so on). The video shows the running total up to 46 nodes.

.
As we add more nodes, the cumulative shape gets smoother and broader. After a huge number of steps, the sum will look pretty much like Newton’s (except for right at the walls, of course).

So if the classical and QM boxes wind up looking the same, why go to all that trouble? Because those nodes don’t come for free.

Suppose you’re playing goalie in an inverse tennis game. There’s a player in each service box. Your job is to run the net line using your rackets to prevent either player from getting a ball into the opposing half-court. Basically, you want the ball’s locations to look like the single-node yellow shape up above. You’ll have to work hard to do that.

Now suppose they give you a second, crosswise net (the green shape). You’re going to have to work twice as hard. Now add a third net, and so on … each additional nodal plane is going to be harder (cost more energy) to keep empty. Not a problem if you have an infinite amount of energy.

Enter Planck and Einstein. They showed there’s a limit for small systems like atoms and molecules. Electrons dash about in atom- or molecule-shaped boxes, but the principle is the same. The total probability distribution is still the sum of bounded elementary shapes. However, you can’t use an infinite number of them. Rather, you start with the cheapest shapes (the fewest nodes) and build upward.

Tally two electrons for each shape you use. Why two? Because that’s the rule, no arguments.

It’s important to realize that QM does NOT say that two specific electrons occupy one shape. All the charge is spread out over all the shapes — we’re just keeping count.

When you run out of electrons the accumulated model shows everything we can know about the electronic configuration. You won’t know where any particular electron is, but you’ll know where some electron spends some time. For a chemist that’s the important thing — the peaks and nodes, the centers of negative and positive charge, are the most likely regions for chemical reactions to happen.

Johnnie’s energetic taps make his particle boldly go where no particle has gone before.

~~ Rich Olcott

Particles and Poetry

On May 2, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

“Hoy, Johnny, wotcher got inna box?”
“Hit’s a particle, Jessie.”
“Ooo, lovely for you. Umm… wot’s a particle then?”
“Me Pap says hit’s sommat you calc’late about wiffout knowin’ wot ’tis.”

Pap’s right. Newton was a particle guy all the way (he was a strong supporter of the idea that light is composed of particles). One of his most important insights was that he could simplify gravitational calculations if he replaced an object with an equally massive “particle” located at the object’s center of mass. Could be a planet, or a moon, or that apple — he could treat each of them as a “particle.” That worked fine for his purposes, because the distances between his object centers were vastly larger than the object sizes.

Fleas — “Great fleas have little fleas upon their backs to bite ’em / And little fleas have lesser fleas and so on infinitum.” ~~ Augustus De Morgan

It took Roche to work out what happens when the distances get small. Gravitational forces break the original “particles” into littler particles. And when two of the little ones approach closely enough they break up, and then those break up… You get the idea. Take the process far enough and you get Saturn’s Rings, for instance.

But the analysis can keep going. Consider one “particle” in Saturn’s A-ring. It’s probably about 3″ across, made of ice, and contains something like 10²⁴ particles that happen to be molecules of H₂O. Each molecule contains 3 nuclei (2 protons and one oxygen nucleus) and 10 electrons, all 13 of which merit “particle” status if you’re calculating molecules. They’re all held together by a blizzard of photons carrying the electromagnetic forces between them. The oxygen nucleus contains 16 nuclear particles, each of which contains 3 quarks. The quark structures would fly apart except for a host of gluons that pass back and forth transmitting the nuclear strong force. Hooboy, do we got particles.

“Particle” is a slippery word. For Newton’s purposes, if an object is small relative to its distance from other objects, that was all he needed to know to treat it as a particle.

One dictionary specifies “a small localized object which has identifiable physical or chemical properties such as volume or mass.” However, there are theoretical grounds to believe that the classic “particle of light,” the photon, has neither mass nor volume. Physicists have had long arguments trying to devise a good working definition. Nobelist (1999) Gerard ‘t Hooft ended one such discussion by saying, “A particle is fundamental when it’s useful to think of it as fundamental.”

It may seem a little strange for a physicist to argue for imprecision. In fact, ‘t Hooft was arguing for a broad, even poetic but still precise understanding of the word.

Poets use metaphor to help us understand the world. Part of their art is to pack as much meaning as they can into the minimum number of words. In the same way, scientists use mathematics to pack observed relationships into a simile called an equation — a brief bit of math may connect and illuminate many disparate phenomena.

Think of physics as metaphor, with numbers.

Newton’s Law of Gravity works for for galaxies roving through a cluster and for basketball-sized satellites orbiting Earth and for stars circling a black hole (if they don’t get too close). Maxwell’s Equations, just 30 symbols including parentheses and equal signs, give the speed of light and describe the operation of electric motors. The particle physicists’ Standard Model makes predictions that match experimental results to more than a dozen decimal places.

Good equations are so successful that Nobelist (1963) Eugene Wigner wrote an influential paper entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

We sometimes get into trouble by confusing metaphor with reality. Poetic metaphors can be carried too far — Hamlet’s lungs were not in fact filling with water from his “sea of troubles.”

Mathematical models can also be carried too far. Popular (and practitioner) discussion of quantum mechanics is rife with over-extended metaphors. QM calculations yield only statistical results — an average position, say, plus or minus so much. It’s an average, but of what? The “many worlds” hypothesis is an unnecessarily long jump. There are simpler, less extravagant ways to account for statistical uncertainty.

~~ Rich Olcott

Perturbed? You’re not the only one

On April 25, 2016November 28, 2025 By rolcottIn Physics: Fundamentals, Physics: Quantum Mechanics, UncategorizedLeave a comment

Dolls — Successive approximations
to a real girl, but still not there

It started with the Babylonians. The Greeks abhorred the notion. The Egyptians and Romans couldn’t have gotten along without it. Only 1600 years later did Newton gave final polishing to … The Method of Successive Approximations.

Stay with me, we’ll get to The Chicken soon.

Suppose for some weird reason you wanted to know the square root of 2701. Any Babylonian could see immediately that 2701 is a bit less than 3600 = 60², so as a first approximation they’d guess ½(60 + (2701/60)) = 52.5. They’d do the multiplication to check: 52.5×52.5 = 2756.25.

Well, 52.5 is closer than 60 but not close enough. So they’d plug that number into the same formula to get the next successive approximation: ½(52.5 + 2701/52.5) = 51.97. Check it: 51.97×51.97 = 2700.88. That was probably good enough for government work in Babylonia, but if the boss wanted an even better estimate they could go around the loop again.

Scientists and engineers tackle a complex problem piecewise. Start by looking for a simple problem you know how to solve. Adjust that solution little by little to account for the ways in which the real system differs from the simple case. Successive Approximation is only one of many adjustment strategies invented over the centuries.

The most widely-used technique is called Perturbation Theory (which has nothing to do with the ways kids find to get on their parents’ nerves). The strategy is to find some single parameter, maybe a ratio of two masses or the relative strength of a particle-particle interaction. For a realistic solution, it’s important that the parameter’s value be small compared to other quantities in the problem.

Simplify the original problem by keeping that parameter in the equations but assume that it’s zero. When you’ve found a solution to that problem, you “perturb” the solution — you see what happens to the model when you allow the parameter to be non-zero.

There’s an old story, famous among physicists and engineers, about an association of farmers who wanted to design an optimum chicken-raising operation. Maybe with an optimal chicken house they could heat the place with the birds’ own body heat, things like that. They called in an engineering consultant. He looked around some running farms, took lots of measurements, and went away to compute. A couple of weeks later he came back, with slides. (I told you it’s an old story.) He started to walk the group though his logic, but he lost them when he opened his pitch with, “Assume a spherical chicken…”

Fat chick bank — Henrietta
*Fat Chicken Bank* by Becky Zee

Now, he may actually have been on the right track. It’s a known fact that many biological processes (digestion, metabolism, drug dosage, etc.) depend on an organism’s surface area. A chicken’s surface area could be key to calculating her heat production. But chickens (for example, our charming Henrietta) have a complicated shape with a poorly-defined surface area. The engineer’s approximation strategy must have been to estimate each bird as a sphere with a tweakable perturbation parameter reflecting how spherical they aren’t.

Then, of course, he’d have to apply a second adjustment for feathers, but I digress.

Now here’s the thing. In quantum mechanics there’s only a half-dozen generic systems with exact solutions qualifying them to be “simple” Perturbation Theory starters. Johnny’s beloved Particle In A Box (coming next week) is one of them. The others all depend in similar logic — the particle (there’s always only one of them) is confined to a region which contains places where the particle’s not allowed to be. (There’s one exception: the Free Particle has no boundaries and therefore is evenly smeared across the Universe.)

Virtually all other quantum-based results — multi-electron atoms, molecular structures, Feynman diagrams for sub-atomic physics, string theories, whatever — depend on Perturbation Theory. (The exceptions are topology and group-theory techniques that generally attempt to produce qualitative rather quantitative predictions.) They need those tweakable parameters.

In quantum-chemical calculations the perturbation parameters are generally reasonably small or at least controllable. That’s not true for many of the other areas. This issue is especially problematic for string theory. In many of its proposed problem solutions no-one knows whether a first-, second- or higher-level approximation even exists, much less whether it would produce reasonable predictions.

I find that perturbing.

~~ Rich Olcott

How rockets don’t work

On April 18, 2016November 28, 2025 By rolcottIn General: Fun Stuff, Physics: Fundamentals, Physics: Newton and GravityLeave a comment

I was only 10 years old but already had Space Fever thanks to Chesley Bonestell’s artwork in Collier’s and Life magazines. I eagerly joined the the movie theater ticket line to see George Pal’s Destination Moon. I loved the Woody Woodpecker cartoon (it’s 12 minutes into the YouTube video) that explained rockets to a public just getting used to jet planes. But the explanation’s wrong.

Go ahead, follow the link and watch the cartoon. I’ll wait here.

Pretty far-sighted for 1950, eh? And it’s amazing how much they got right, including how the driving force for the Space Race was international politics. But oh, the physics…

Yeah, they tacitly acknowledged Newton’s Third Law: For every action there is an equal and opposite reaction. The cartoon implies that the action is the pellets coming out of the barrel and the reaction is Woody getting knocked back. But that can’t be right: if it were true you wouldn’t get any kick when you fire a blank cartridge — but you do. Let’s take a close look at just what actions are in play.

Maybe it’s the pellets plus the gases behind it pushing forward and the gun pushing backward? Sort of, but where do the gases come from? Right, the exploding charge next to your cheek in the receiver. Those gases move equally in all directions. Some of them push pellets down the barrel. Some of them push on the back end of the receiver which pushes the gun stock which mashes your shoulder. But there’s bunches of molecules that uselessly collide with the receiver’s walls.

Action and reaction balance out just fine but only when you consider the gases moving outward from the center of the BANG. For instance, if left and right didn’t balance perfectly the piece would crash into your ear or swing around and flatten your nose or the back of your head.

Both shotguns and conventional rockets get their propulsive energy from chemical combustion. The reason gun parts have to be strong is all those hot molecules dashing in every direction other than down and up the barrel. A chemical rocket casing has to be strong for the same reason.

Chemical combustion is just not an efficient use of propellant mass. Just look at this NASA image of a SpaceX Falcon 9 during a DSCOVR launch — huge side-flare from molecules that make no contribution to forward thrust:
Wouldn’t it be nice if we had a way to put all our propulsion energy into moving the vehicle forward?

There’s good news and not-so-good news. People are working on a few other options, all of which depend on forces we know how to steer: electric and magnetic. Unfortunately, each of them has drawbacks.

Unlike rockets, ion thrusters use an electric or magnetic field to accelerate ions (duh!) away from the vehicle. It’s a much more efficient process because there’s little off-axis action/reaction — all the propellant heads out the nozzle (action) and all the push-back force (reaction) acts directly on the vehicle.

But… ions resist being crowded together so you can’t blast huge quantities out the nozzle like you’d need to for a launch from Earth. Up in space, though, ion thrusters are perfect for satellite attitude adjustment and similar low-power tasks. The Dawn mission to Vesta and Ceres used an ion thruster to boost the spacecraft continuously from Earth to target. It’d be impractical to build a chemical-powered system to do that.

Rather than send out atoms one by one, a rail-gun drive could use high-power magnetic fields to accelerate lumps of iron down a track and away. Iron goes one way, vessel goes the other. Might work in the Asteroid Belt where lumps of iron are there by the billions, but on the other hand I’d rather not be a Belter tooling along in my mining tug only to be hit amidships by someone’s cast-off reaction mass.

And then there’s the Q-thruster and EmDrive. I hope to eventually include enough physics background in this blog that we can discuss the controversies and prospects for new-physics drives based on space warps and such. You can check out Dr Harold White’s video for some of that. It’d be sooo cool if they work.

~~ Rich Olcott

Another slice of π, wrapped up in a Black Hole crust

On April 11, 2016November 28, 2025 By rolcottIn Physics: Black Holes and Relativity, Physics: Fundamentals, UncategorizedLeave a comment

Last week a museum visitor wondered, “What’s the volume of a black hole?” A question easier asked than answered.

Let’s look at black hole (“BH”) anatomy. If you’ve seen Interstellar, you saw those wonderful images of “Gargantua,” the enormous BH that plays an essential role in the plot. (If you haven’t seen the movie, do that. It is so cool.)

A BH isn’t just a blank spot in the Universe, it’s attractively ornamented by the effects of its gravity on the light passing by:

Gargantua 2c — Gargantua,
adapted from Dr Kip Thorne’s book, *The Science of “Interstellar”*

Working from the outside inward, the first decoration is a background starfield warped as though the stars beyond had moved over so they could see us past Gargantua. That’s because of gravitational lensing, the phenomenon first observed by Sir Arthur Eddington and the initial confirmation of Einstein’s Theory of General Relativity.

No star moved, of course. Each warped star’s light comes to us from an altered angle, its lightwaves bent on passing through the spatial compression Gargantua imposes on its neighborhood. (“Miles are shorter near a BH” — see Gravitational Waves Are Something Else for a diagrammatic explanation.)

Moving inward we come to the Accretion Disc, a ring of doomed particles destined to fall inward forever unless they’re jostled to smithereens or spat out along one of the BH’s two polar jets (not shown). The Disc is hot, thanks to all the jostling. Like any hot object it emits light.

Above and below the Disc we see two arcs that are actually images of the Accretion Disc, sent our way by more gravitational lensing. Very close to a BH there’s a region where passing light beams are bent so much that their photons go into orbit. The disc’s a bit further out than that so its lightwaves are only bent 90^o over (arc A) and under (arc B) before they come to us.

By the way, those arcs don’t only face in our direction. Fly 360^o around Gargantua’s equator and those arcs will follow you all the way. It’s as though the BH were embedded in a sphere of lensed Disclight.

Which gets us to the next layer of weirdness. Astrophysicists believe that most BHs rotate, though maybe not as fast as Gargantua’s edge-of-instability rate. Einstein’s GR equations predict a phenomenon called frame dragging — rapidly spinning massive objects must tug local space along for the ride. The deformed region is a shell called the Ergosphere.

Frame dragging is why the two arcs are asymmetrical and don’t match up. We see space as even more compressed on the right-hand side where Gargantua is spinning away from us. Because the effect is strongest at the equator, the shell should really be called the Ergospheroid, but what can you do?

Inside the Ergosphere we find the defining characteristic of a BH, its Event Horizon, the innermost bright ring around the central blackness in the diagram. Barely outside the EH there may or may not be a Firewall, a “seething maelstrom of particles” that some physicists suggest must exist to neutralize the BH Information Paradox. Last I heard, theoreticians are still fighting that battle.

The EH forms a nearly spherical boundary where gravity becomes so intense that the escape velocity exceeds the speed of light. No light or matter or information can break out. At the EH, the geometry of spacetime becomes so twisted that the direction of time is In. Inside the EH and outside of the movies it’s impossible for us to know what goes on.

Finally, the mathematical models say that at the center of the EH there’s a point, the Singularity, where spacetime’s curvature and gravity’s strength must be Infinite. As we’ve seen elsewhere, Infinity in a calculation is Nature’s was of saying, “You’ve got it wrong, make a better model.”

So we’re finally down to the volume question. We could simply measure the EH’s external diameter d and plug that into V=(πd³)/6. Unfortunately, that forthright approach misses all the spatial twisting and compression — it’s a long way in to the Singularity. Include those effects and you’ve probably got another Infinity.

Gargantua’s surface area is finite, but its volume may not be.

~~ Rich Olcott

Michelson, Morley and LIGO

On April 4, 2016November 29, 2025 By rolcottIn Astronomy: Techniques, Gravitational astronomy, Physics: Electromagnetism, Physics: Light and Relativity, UncategorizedLeave a comment

Two teams of scientists, 128 years apart. The first team, two men, got a negative result that shattered a long-standing theory. The second team, a thousand strong, got a positive result that provided final confirmation of another long-standing theory. Both teams used instruments based on the same physical phenomenon. Each team’s innovations created whole new fields of science and technology.

Their common experimental strategy sounds simple enough — compare two beams of light that had traveled along different paths

Light (preferably nice pure laser light, but Albert Michelson didn’t have a laser when he invented interferometry in 1887) comes in from the source at left and strikes the “beam splitter” — typically, a partially-silvered mirror that reflects half the light and lets the rest through. One beam goes up the y-arm to a mirror that reflects it back down through the half-silvered mirror to the detector. The other beam goes on its own round-trip journey in the x-direction. The detector (Michelson’s eye or a photocell or a fancy-dancy research-quality CCD) registers activity if the waves in the two beams are in step when they hit it. On the other hand, if the waves cancel then there’s only darkness.

Getting the two waves in step requires careful adjustment of the x- and y-mirrors, because the waves are small. The yellow sodium light Michelson used has a peak-to-peak wavelength of 589 nanometers. If he twitched one mirror 0.0003 millimeter away from optimal position the valleys of one wave would cancel the peaks of the other.

So much for principles. The specifics of each team’s device relate to the theory being tested. Michelson was confronting the æther theory, the proposition that if light is a wave then there must be some substance, the æther, that vibrates to carry the wave. We see sunlight and starlight, so the æther must pervade the transparent Universe. The Earth must be plowing through the æther as it circles the Sun. Furthermore, we must move either with or across or against the æther as we and the Earth rotate about its axis. If we’re moving against the æther then lightwave peaks must appear closer together (shorter wavelengths) than if we’re moving with it.

Michelson designed his device to test that chain of logic. His optical apparatus was all firmly bolted to a 4′-square block of stone resting on a wooden ring floating on a pool of mercury. The whole thing could be put into slow rotation to enable comparison of the x– and y-arms at each point of the compass.

Interferometer 3 — Suppose the æther theory is correct. Michelson should see lightwaves cancel at some orientations.

According to the æther theory, Michelson and his co-worker Edward Morley should have seen alternating light and dark as he rotated his device. But that’s not what happened. Instead, he saw no significant variation in the optical behavior around the full 360^o rotation, whether at noon or at 6:00 PM.

Cross off the æther theory.

Michelson’s strategy depended on light waves getting out of step if something happened to the beams as they traveled through the apparatus. Alternatively, the beams could charge along just fine but something could happen to the apparatus itself. That’s how the LIGO team rolled.

Interferometer 2 — Suppose Einstein’s GR theory is correct. Gravitational wave stretching and compression should change the relative lengths of the two arms.

Einstein’s theory of General Relativity predicts that space itself is squeezed and stretched by mass. Miles get shorter near a black hole. Furthermore, if the mass configuration changes, waves of compressive and expansive forces will travel outward at the speed of light. If such a wave were to encounter a suitable interferometer in the right orientation (near-parallel to one arm, near-perpendicular to the other), that would alter the phase relationship between the two beams.

The trick was in the word “suitable.” The expected percentage-wise length change was so small that eLIGO needed 4-kilometer arms to see movement a tiny fraction of a proton’s width. Furthermore, the LIGO designers flipped the classical detection logic. Instead of looking for a darkened beam, they set the beams to cancel at the detector and looked for even a trace of light.

eLIGO saw the light, and confirmed Einstein’s theory.

~~ Rich Olcott

Gravitational Waves Are Something Else

On March 28, 2016November 29, 2025 By rolcottIn Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Fundamentals, UncategorizedLeave a comment

If you’re reading this post, you’ve undoubtedly seen at least one diagram like the above — a black hole or a planet or a bowling ball makes a dent in a rubber sheet and that’s supposed to explain Gravity. But it doesn’t, and neither does this spirally screen-grab from Brian Greene’s presentation on Stephen Colbert’s Late Show:

<Blush> I have to admit that the graphic I used a couple of weeks ago is just as bad.

Gravitational waves don’t make things go up and down like ocean waves, and they’re definitely not like that planet on a trampoline — after all, there’s nothing “below” to pull things downward so there can’t be a dent. And gravitational waves don’t do spirals, much.

Of all the wave varieties we’re familiar with, gravitational waves are most similar to (NOT identical with!!) sound waves. A sound wave consists of cycles of compression and expansion like you see in this graphic. Those dots could be particles in a gas (classic “sound waves”) or in a liquid (sonar) or neighboring atoms in a solid (a xylophone or marimba).

Contrary to rumor, there can be sound in space, sort of. Any sizable volume of “empty” space contains at least a few atoms and dust particles. A nova or similar sudden event can sweep particles together and give rise to successive waves that spread as those local collections bang into particles further away. That kind of activity is invoked in some theories of spiral galaxy structure and the fine details of Saturn’s rings.

In a gravitational wave, space itself is compressed and stretched. A particle caught in a gravitational wave doesn’t get pushed back and forth. Instead, it shrinks and expands in place. If you encounter a gravitational wave, you and all your calibrated measurement gear (yardsticks, digital rangers, that slide rule you’re so proud of) shrink and expand together. You’d only notice the experience if you happened to be comparing two extremely precise laser rangers set perpendicular to each other (LIGO!). One would briefly register a slight change compared to the other one.

Light always travels at 186,000 miles per second but in a compressed region of space those miles are shorter. Einstein noticed that implication of his Theory of General Relativity and in 1916 predicted that the path of starlight would be bent when it passed close to a heavy object like the Sun. The graphic shows a wave front passing through a static gravitational structure. Two points on the front each progress at one graph-paper increment per step. But the increments don’t match so the front as a whole changes direction. Sure enough, three years after Einstein’s prediction, Eddington observed just that effect while watching a total solar eclipse in the South Atlantic.

Unlike the Sun’s steady field, a gravitational wave is dynamic. Gravitational waves are generated by changes in a mass configuration. The wave’s compression and stretching forces spread out through space.

Here’s a simulation of the gravitational forces generated by two black holes orbiting into a collision. The contours show the net force felt at each point in the region around the pair.
We’re being dynamic here, so the simulation has to include the fact that changes in the mass configuration aren’t felt everywhere instantaneously. Einstein showed that space transmits gravitational waves at the speed of light, so I used a scaled “speed of light” in the calculation. You can see how each of the new features expands outward at a steady rate.

Even near the violent end, the massive objects move much more slowly than light speed. The variation in their nearby field quickly smooths out to an oval and then a circle about the central point, which is why the calculated gravity field generates no spiral like the ones in the pretty pictures.

Oh, and those “gravity well” pictures? They’re not showing gravitational fields, they’re really gravitational potential energy diagrams, showing how hard it’d be to get away from somewhere. In the top video, for example, the satellite orbits the planet because it doesn’t have enough kinetic energy to get out of the well. The more massive the attractor, the tighter it curves space around itself and the deeper the well.

~~ Rich Olcott

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