What’s that funnel about, really?

On July 18, 2016November 28, 2025 By rolcottIn Physics: Newton and Gravity, UncategorizedLeave a comment

If you’ve ever watched or read a space opera (oh yes, you have), you know about the gravity well that a spacecraft has to climb out of when leaving a planet. Every time I see the Museum’s gravity well model (photo below), I’m reminded of all the answers the guy gave to, “Johnny, what can you make of this?”

The model’s a great visitor-attracter with those “planets” whizzing around the “Sun,” but this one exhibit really represents several distinct concepts. For some of them it’s not quite the right shape.

The simplest concept is geometrical. “Down” is the direction you move when gravity’s pulling on you.

HS cone — Gravitational potential energy change
for small height differences

A gravity well model for that concept would be just a straight line between you and the neighborhood’s most intense gravity source.

You learned the second concept in high school physics class. Any object has gravitational potential energy that measures the amount of energy it would give up on falling. Your teacher probably showed you the equation GPE = m·g·h, where m is the mass of the object, h is its height above ground level, and g is a constant you may have determined in a lab experiment.

If the width of the gravity well model at a given height represents GPE at that level, the model is a simple straight-sided cone.

Newton energy cone — Gravitational potential energy change
for large height differences
The h indicates
an approximately linear range
where the HS equation could apply.

But of course it’s not that simple. Newton’s Law of Gravity says that the potential energy at any height r away from the planet’s center is proportional to 1/r.

Hmm… that looks different from the “proportional to h” equation. Which is right?

Both equations are valid, but over different distance scales. The HS teachers didn’t quite lie to you, but they didn’t give you the complete picture either. Your classroom was about 4000 miles (21,120,000 feet) from Earth’s center, whereas the usual experiments involve height differences of at most a dozen feet. Even the 20-foot drop from a second-story window is less than a millionth of the way down to Earth’s center.

Check my numbers:

Height h	1/(r+h) × 10⁸	Difference in 1/(r+h) × 10¹⁴
0	4.734,848,484	0
20	4.734,844,001	4.48
40	4.734,839,517	8.97
60	4.734,835,033	13.45
80	4.734,830,549	17.93
100	4.734,826,066	22.42

Sure enough, that’s a straight line (see the chart). Reminds me of how Newton’s Law of Gravity is valid except at very short distances. The HS Law of Gravity works fine for small spans but when the distances get big we have to use Newton’s equation.

We’re not done yet. That curvy funnel-shaped gravity well model could represent the force of gravity rather than its potential energy. Newton told us that the force goes as 1/r² so it decreases much more rapidly than the potential energy does as you get further away. The gravity force well has a correspondingly sharper curve to it than the gravity energy well.

Newton force cone — The *force* of gravity
or an embedding diagram

The funnel model could also represent the total energy required to get a real spacecraft off the surface and up into space. Depending on which sci-fi gimmickry is in play, the energy may come from a chemical or ion rocket, an electromagnetic railgun, or even a tractor beam from some mothership way up there.

No matter the technology, the theoretical energy requirement to get to a given height is the same. In practice, however, each technology is optimal for some situations but forbiddingly inefficient in others. Thus, each technology’s funnel has its own shape and that shape will change depending on the setting.

In modern physics, the funnel model could also represent Einstein’s theory of how a mass “bends” the space around it. (Take a look at this post, which is about how mass curves space by changing the local distance scale.) Cosmologists describe the resulting “shapes” with embedding diagrams that are essentially 2D pictures of 3D (or 4D) contour plots. The contours are closest together where space is most compressed, just as lines showing a steep hillside on a landscape contour map are close together.

The ED around a non-spinning object looks just like the force model picture above. No surprise — gravitational force is how we we perceive spatial curvature.

~~ Rich Olcott

Throwing a Summertime curve

On June 27, 2016November 28, 2025 By rolcottIn Cosmology, Mathematics: Dimensions, Physics: Black Holes and Relativity, UncategorizedLeave a comment

All cats are gray in the dark, and all lines are straight in one-dimensional space. Sure, you can look at a garden hose and see curves (and kinks, dammit), but a short-sighted snail crawling along on it knows only forward and backward. Without some 2D notion of sideways, the poor thing has no way to sense or cope with curvature.

Up here in 3D-land we can readily see the hose’s curved path through all three dimensions. We can also see that the snail’s shell has two distinct curvatures in 3D-space — the tube has an oval cross-section and also spirals perpendicular to that.

But Einstein said that our 3D-space itself can have curvature. Does mass somehow bend space through some extra dimension? Can a gravity well be a funnel to … somewhere else?

No and no. Mathematicians have come up with a dozen technically different kinds of curvature to fit different situations. Most have to do with extrinsic non-straightness, apparent only from a higher dimension. That’s us looking at the hose in 3D.

Einstein’s work centered on intrinsic curvature, dependent only upon properties that can be measured within an object’s “natural” set of dimensions.

On a surface, for instance, you could draw a triangle using three straight lines. If the figure’s interior angles sum up to exactly 180°, you’ve got a flat plane, zero intrinsic curvature. On a sphere (“straight line” = “arc from a great circle”) or the outside rim of a doughnut, the sum is greater than 180° and the curvature is positive.

If there’s zero curvature and positive curvature, there’s gotta be negative curvature, right? Right — you’ll get less-than-180° triangles on a Pringles chip or on the inside rim of a doughnut.

Some surfaces don’t have intersecting straight lines, but you can still classify their curvature by using a different criterion. Visualize our snail gliding along the biggest “circle” he/she/it (with snails it’s complicated) can get to while tethered by a thread pinned to a point on the surface. Divide the circle’s circumference by the length of the thread. If the ratio’s equal to 2π then the snail’s on flat ground. If the ratio is bigger than 2π, the critter’s on a saddle surface (negative curvature). If it’s smaller, then he/she/it has found positive curvature.

In a sense, we’re comparing the length of a periphery and a measure of what’s inside it. That’s the sense in which Einsteinian space is curved — there are regions in which the area inside a circle (or the volume inside a sphere) is greater than or less than what would be expected from the size of its boundary.

Here’s an example. The upper panel’s dotted grid represents a simple flat space being traversed by a “disk.” See how the disk’s location has no effect on its size or shape. As a result, dividing its circumference by its radius always gives you 2π.

In the bottom panel I’ve transformed^* the picture to represent space in the neighborhood of a black hole (the gray circle is its Event Horizon) as seen from a distance. Close-up, every row of dots would appear straight. However, from afar the disk’s apparent size and shape depend on where it is relative to the BH.

By the way, the disk is NOT “falling” into the BH. This is about the shape of space itself — there’s no gravitational attraction or distortion by tidal spaghettification.

Visually, the disk appears to ooze down one of those famous 3D parabolic funnels. But it doesn’t — all of this activity takes place within the BH’s equatorial plane, a completely 2D place. The equations generate that visual effect by distorting space and changing the local distance scale near our massive object. This particular distortion generates positive curvature — at 90% through the video, the disk’s C/r ratio is about 2% less than 2π.

As I tell Museum visitors, “miles are shorter near a black hole.”

~~ Rich Olcott

^* – If you’re interested, here are the technical details. A Schwarzchild BH, distances as multiples of the EH radius. The disk (diameter 2.0) is depicted at successive time-free points in the BH equatorial plane. The calculation uses Flamm’s paraboloid to convert each grid point’s local (r,φ) coordinates to (w,φ) to represent the spatial configuration as seen from r>>w.

Gargh, His Heirs, and the AAAD Problem

On June 13, 2016November 28, 2025 By rolcottIn Physics: Fundamentals, Physics: Newton and Gravity, UncategorizedLeave a comment

Gargh, 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 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 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.

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 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

Is there stuff behind the stats?

On June 6, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

It would have been awesome to watch Dragon Princes in battle (from a safe hiding place), but I’d almost rather have witnessed “The Tussles in Brussels,” the two most prominent confrontations between Albert Einstein and Niels Bohr.

The Tussles would be the Fifth (1927) and Seventh (1933) Solvay Conferences. Each conference was to center on a particular Quantum Mechanics application (“Electrons and Photons” and “The Atomic Nucleus,” respectively). However, the Einstein-Bohr discussions went right to the fundamentals — exactly what does a QM calculation tell us?

Einstein’s strength was in his physical intuition. By all accounts he was a good mathematician but not a great one. However, he was very good indeed at identifying important problems and guiding excellent mathematicians as he and they attacked those problems together.

Like Newton, Einstein was a particle guy. He based his famous thought experiments on what his intuition told him about how particles would behave in a given situation. That intuition and that orientation led him to paradoxes such as entanglement, the EPR Paradox, and the instantaneously collapsing spherical lightwave we discussed earlier. Einstein was convinced that the particles QM workers think about (photons, electrons, etc.) must in fact be manifestations of some deeper, more fine-grained reality.

Bohr was six years younger than Einstein. Both Bohr and Einstein had attained Directorship of an Institute at age 35, but Bohr’s has his name on it. He started out as a particle guy — his first splash was a trio of papers that treated the hydrogen atom like a one-planet solar system. But that model ran into serious difficulties for many-electron atoms so Bohr switched his allegiance from particles to Schrödinger’s wave theory. Solve a Schrödinger equation and you can calculate statistics like average value and estimated spread around the average for a given property (position, momentum, spin, etc).

Here’s where Ludwig Wittgenstein may have come into the picture. Wittgenstein is famous for his telegraphically opaque writing style and for the fact that he spent much of his later life disagreeing with his earlier writings. His 1921 book, Tractatus Logico-Philosophicus (in German despite the Latin title) was a primary impetus to the Logical Positivist school of philosophy. I’m stripping out much detail here, but the book’s long-lasting impact on QM may have come from its Proposition 7: “Whereof one cannot speak, thereof one must be silent.“

I suspect that Bohr was deeply influenced by the LP movement, which was all the rage in the mid-1920s while he was developing the Copenhagen Interpretation of QM.

An enormous literature, including quite a lot of twaddle, has grown up around the question, “Once you’ve derived the Schrödinger wave function for a given system, how do you interpret what you have?” Bohr’s Copenhagen Interpretation was that the function can only describe relative probabilities for the results of a measurement. It might tell you, for instance, that there’s a 50% chance that a particle will show up between here and here but only a 5% chance of finding it beyond there.

Following Logical Positivism all the way to the bank, Bohr denounced as nonsensical or even dangerously misleading any attachment of further meaning to a QM result. He went so far as to deny the very existence of a particle prior to a measurement that detects it. That’s serious Proposition 7 there.

I’ve read several accounts of the Solvay Conference debates between Einstein and Bohr. All of them agree that the conversation was inconclusive but decisive. Einstein steadfastly maintained that QM could not be a complete description of reality whilst Bohr refused to even consider anything other than inscrutable randomness beneath the statistics. The audience consensus went to Bohr.

None of the accounts, even the very complete one that I found in George Musser’s book Spooky Action at A Distance, provide a satisfactory explanation for why Bohr’s interpretation dominates today. Einstein described multiple situations where QM’s logic appeared to contradict itself or firmly established experimental results. However, at each challenge Bohr deflected the argument from Einstein’s central point to argue a subsidiary issue such as whether Einstein was denying the Heisenberg Uncertainty Principle.

Albert still stood at the end of the bouts, but Niels got the spectators’ decision on points. Did the ref make the difference?

~~ 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

Hard Science Ain't Hard

Tag: Newton

What’s that funnel about, really?

Throwing a Summertime curve

Gargh, His Heirs, and the AAAD Problem

Is there stuff behind the stats?

Think globally, act locally. Electrons do.

Location, Location, Location

Particles and Poetry

Perturbed? You’re not the only one

How rockets don’t work

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