‘Twixt A Rock And A Vortex

A chilly late December walk in the park and there’s Vinnie on a lakeside bench, staring at the geese and looking morose. “Hi, Vinnie, why so down on such a bright day?”

“Hi, Sy. I guess you ain’t heard. Frankie’s got the ‘rona.”

Frankie??!? The guys got the constitution of an ox. I don’t think he’s ever been sick in his life.”

“Probably not. Remember when that bug going around last January had everyone coughing for a week? Passed him right by. This time’s different. Three days after he showed a fever, bang, he’s in the hospital.”

“Wow. How’s Emma?”

“She had it first — a week of headaches and coughing. She’s OK now but worried sick. Hospital won’t let her in to see him, of course, which is a good thing I suppose so she can stay home with the kids and their schoolwork.”

“Bummer. We knew it was coming but…”

“Yeah. Makes a difference when it’s someone you know. Hey, do me a favor — throw some science at me, get my mind off this for a while.”

“That’s a big assignment, considering. Let’s see … patient, pandemic … Ah! E pluribus unum and back again.”

“Come again?”

“One of the gaps that stand between Physics and being an exact science.”

“I thought Physics was exact.”

“Good to fifteen decimal places in a few special experiments, but hardly exact. There’s many a slip ‘twixt theory and practice. One of the slips is the gap between kinematic physics, about how separate objects interact, and continuum physics, where you’re looking at one big thing.”

“This is sounding like that Loschmidt guy again.”

“It’s related but bigger. Newton worked on both sides of this one. On the kinematics side there’s billiard balls and planets and such. Assuming no frictional energy loss, Newton’s Three Laws and his Law of Gravity let us calculate exact predictions for their future trajectories … unless you’ve got more than three objects in play. It’s mathematically impossible to write exact predictions for four or more objects unless they start in one of a few special configurations. Newton didn’t do atoms, no surprise, but his work led to Schrödinger’s equation for an exact description of single electron, single nucleus systems. Anything more complicated, all we can do is approximate.”

“Computers. They do a lot with computers.”

“True, but that’s still approximating. Time‑step by time‑step and you never know what might sneak in or out between steps.”

“What’s ‘continuum‘ about then? Q on Star trek?”

“Hardly, we’re talking predictability here. Q’s thing is unpredictability. A physics continuum is a solid or fluid with no relevant internal structure, just an unbroken mass from one edge to the other. Newton showed how to analyze a continuum’s smooth churning by considering the forces that act on an imaginary isolated packet of stuff at various points in there. He basically invented the idea of viscosity as a way to account for friction between a fluid and the walls of the pipe it’s flowing through.”

“Smooth churning, eh? I see a problem.”

“What’s that?”

“The eddies and whirlpools I see when I row — not smooth.”

“Good point. In fact, that’s the point I was getting to. We can use extensions of Newton’s technique to handle a single well‑behaved whirlpool, but in real life big whirlpools throw off smaller ones and they spawn eddies and mini‑vortices and so on, all the way down to atom level. That turns out to be another intractable calculation, just as impossible as the many‑body particle mechanics problem.”

“Ah‑hah! That’s the gap! Newton just did the simple stuff at both ends, stayed away from the middle where things get complicated.”

“Exactly. To his credit, though, he pointed the way for the rest of us.”

“So how can you handle the middle?”

“The same thing that quantum mechanics does — use statistics. That’s if the math expressions are average‑able which sometimes they’re not, and if statistical numbers are good enough for why you’re doing the calculation. Not good enough for weather prediction, for instance — climate is about averages but weather needs specifics.”

“Yeah, like it’s just started to snow which I wasn’t expecting. I’m heading home. See ya, Sy.”

“See ya, Vinnie. … Frankie. … Geez.

~~ Rich Olcott

The Buck Rolls On, We Hope

<knock, knock> “Door’s open. Come in but maintain social distance.”

“Hiya, Sy. Here’s your pizza, still hot and everything but no pineapple.”

“Thanks, Eddie. Just put it on the credenza. There’s a twenty there waiting for you. Put the balance on my tab.”

“Whoa, I recognize this bill. It’s the one that Vinnie won off me at the after‑hours dice game last month before all this started. See, I initialed it down here on the corner ’cause Vinnie usually don’t do that well. How’d you get it from him?”

“I didn’t get it from Vinnie, I got it from Al when I sold him a batch of old astronomy magazines. Vinnie must have finally paid off his tab at Al’s coffee shop.”

“Funny how that one bill just went in a circle. Financed some risky business, paid off a loan, bought stuff, and here I get it again so I can buy stuff to make more pizza. That’s a lotta work for one piece of paper.”

“Mm-hm. Everyone’s $20 better off now, all because the bill kept moving. Chalk it off to ‘the velocity of money.‘ If Vinnie didn’t spend that money the velocity’d be zero and none of the rest would have happened.”

“That sounds suspiciously like Physics, Sy.”

“Guilty as charged, Eddie. Just following along with what Isaac Newton started back when he was staying at his mother’s place, hiding out from the bubonic plague.”

Newton, after a day at the beach
while wearing an anti-viral mask

“What’s that got to do with money? Was Newton a banker?”

“Not quite, although the last 30 years of his life he headed up England’s Royal Mint. The core of his work during his Science years was all about change and rate of change. His Laws of Motion quantified what it takes to cause change. He developed his version of calculus to bridge between how fast change happens and how much change has happened.”

“Hey, that’s those graphs you showed me, with the wave on the top line and the slope underneath.”

“Bingo. Pandemics are a long way from the simple systems that Newton studied, but the important point is that to study his planets and pendulums he developed general strategies for tackling complex situations. He started with just a few basic concepts, like position and speed, and expanded on them.”

“Speed’s speed, what’s to expand?”

“Newton expanded the notion of speed to velocity, which also includes direction. From Newton’s point of view, the velocity of a planet in orbit is continuously changing even if its miles per hour is as steady as … a planet.”

“Who cares?”

“Newton did, because he wanted to know what makes the change happen. His starting point was if there’s any motion, it’s got to be at constant speed and in a straight line unless some force causes a velocity change. That’s where his notion of gravity came from — he invented the idea of ‘the force of gravity‘ to account for us not flying off the rotating Earth and the Earth not zooming away from the Sun. His methods set the model that physicists have followed ever since — if we see motion, we measure how fast it’s happening and then we look for the force or forces that can explain that.”

“Now I see where you’re going. That ‘velocity of money‘ thing is about how fast the paper changes hands, isn’t it? Wait, if Vinnie had put that twenty up on his wall as a trophy, then the chain would’ve been broken.”

“Right, or if Al had diverted it to buy, say, coffee beans. That’s why we say velocity of money and not speed, because the direction of flow counts.”

“Smelling more and more like Physics, Sy. Like, there’s astrophysics and biophysics and you’re coming up with econophysics.”

“Well, yeah, but I didn’t invent the term. It’s already out there, with textbooks and academic study groups and everything. It’s just interesting to use economics as a metaphor for physics and vice-versa. The fun is in seeing where the metaphors break down.”

“I see one already, Sy. Those forces — we all had different reasons to kick the bill along.”

“Good point. Now we figure out those forces.”

~~ Rich Olcott

Rhythm Method

A warm Summer day.  I’m under a shady tree by the lake, watching the geese and doing some math on Old Reliable.  Suddenly a text-message window opens up on its screen.  The header bar says 710-555-1701.  Old Reliable has never held a messaging app, that’s not what I use it for.  The whole thing doesn’t add up.  I type in, Hello?

Hello, Mr Moire.  Remember me?

Suddenly I do.  That sultry knowing stare, those pointed ears.  It’s been a yearHello, Ms Baird.  What can I do for you?

Another tip for you, Mr Moire.  One of my favorite star systems — the view as you approach it at near-lightspeed is so ... meaningful.  Your astronomers call it PSR J0337+1715.

So of course I head over to Al’s coffee shop after erasing everything but that astronomical designation.  As I hoped, Cathleen and a few of her astronomy students are on their mid-morning break.  Cathleen winces a little when she sees me coming.  “Now what, Sy?  You’re going to ask about blazars and neutrinos?”

I show her Old Reliable’s screen.  “Afraid not, Cathleen, I’ll have to save that for later.  I just got a message about this star system.  Recognize it?”

“Why, Sy, is that a clue or something?  And why is the lettering in orange?”

“Long story.  But what can you tell me about this star system?”

“Well, it’s probably one of the most compact multi-component systems we’re ever going to run across.  You know what compact objects are?”

“Sure.  When a star the size of our Sun exhausts most of its hydrogen fuel, gravity wins its battle against heat.  The star collapses down to a white dwarf, a Sun-full of mass packed into a planet-size body.  If the star’s a bit bigger it collapses even further, down to a neutron star just a few miles across.  The next step would be a black hole, but that’s not really a star, is it?”

“No, it’s not.  Jim, why not?”

“Because by definition a black hole doesn’t emit light.  A black hole’s accretion disk or polar jets might, but not the object itself.”

“Mm-hm.  Sy, your ‘object’ is actually three compact objects orbiting  around each other.  There’s a neutron star with a white dwarf going around it, and another white dwarf swinging around the pair of them.  Vivian, does that sound familiar?”

“That’s a three-body system, like the Moon going around the Earth and both going around the Sun.  Mmm, except really both white dwarfs would go around the neutron star because it’s heaviest and we can calculate the motion like we do the Solar System.”

“Not quite.  We can treat the Sun as motionless because it has 99% of the mass.  J0337+1715’s neutron star doesn’t dominate its system as much as the Sun does ours.  That outermost dwarf has 20% of its system’s mass.  Phil, what does that suggest to you?”

“It’d be like Pluto and Charon.  Charon’s got 10% of their combined mass and so Pluto and Charon both orbit a point 10% of the way out from Pluto.  From Earth we see Pluto wobbling side to side around that point.  So the neutron star must wobble around the point 20% outward towards the heavy dwarf.  Hey, star-wobble is how we find exoplanets.  Is that what this is about, Mr Moire?  Did someone measure its red-shift behavior?”PSR J0337+1715Cathleen saves me from answering.  “Not quite.  The study Sy’s chasing is actually a cute variation on red-shift measurements.  That ‘PSR‘ designation means the neutron star is a pulsar.  Those things emit electromagnetic radiation pulses with astounding precision, generally regular within a few dozen nanoseconds.  If we receive slowed-down pulses then the object’s going away; sped-up and it’s approaching, just like with red-shifting.  The researchers  derived orbital parameters for all three bodies from the between-pulse durations.  The heavy dwarf is 200 times further out than the light one, for instance.  Not an easy experiment, but it yielded an important result.”

My ears perk up.  “Which was…?”

“The gravitational force between the pulsar and each dwarf was within six parts per million of what Newton’s Laws prescribe.  That observation rules out whole classes of theories that tried to explain galaxies and galaxy clusters without invoking dark matter.”

Cool, huh?


~~ Rich Olcott

Gravity and other fictitious forces

In this post I wrote, “gravitational force is how we we perceive spatial curvature.”
Here’s another claim — “Gravity is like centrifugal force, because they’re both fictitious.”   Outrageous, right?  I mean, I can feel gravity pulling down on me now.  How can it be fictional?

Fictitious triangle
A fictitious triangle

“Fictitious,” not “fictional,” and there’s a difference.  “Fictional” doesn’t exist, but a fictitious force is one that, to put it non-technically, depends on how you look at it.

Newton started it, of course.  From our 21st Century perspective, it’s hard to recognize the ground-breaking impact of his equation F=a.  Actually, it’s less a discovery than a set of definitions.  Its only term that can be measured directly is a, the acceleration, which Newton defined as any change from rest or constant-speed straight-line motion.  For instance, car buffs know that if a vehicle covers a one-mile half-mile (see comments) track in 60 seconds from a standing start, then its final speed is 60 mph (“zero to sixty in sixty”).  Furthermore, we can calculate that it achieved a sustained acceleration of 1.47 ft/sec2.

Both F and m, force and mass, were essentially invented by Newton and they’re defined in terms of each other.  Short of counting atoms (which Newton didn’t know about), the only routes to measuring a mass boil down to

  • compare it to another mass (for instance, in a two-pan balance), or
  • quantify how its motion is influenced by a known amount of force.

Conversely, we evaluate a force by comparing it to a known force or by measuring its effect on a known mass.

Once the F=a. equation was on the table, whenever a physicist noticed an acceleration they were duty-bound to look for the corresponding force.  An arrow leaps from the bow?  Force stored as tension in the bowstring.  A lodestone deflects a compass needle?  Magnetic force.  Objects accelerate as they fall?  Newton identified that force, called it “gravity,” and showed how to calculate it and how to apply it to planets as well as apples.  It was Newton who pointed out that weight is a measure of gravity’s force on a given mass.

Incidentally, to this day the least accurately known physical constant is Newton’s G, the Universal Gravitational Constant in his equation F=G·m1·m2/r2.  We can “weigh” planets with respect to each other and to the Sun, but without an independently-determined accurate mass for some body in the Solar System we can only estimate G.  We’ll have a better value when we can see how much rocket fuel it takes to push an asteroid around.

CoasterBut there are other accelerations that aren’t so easily accounted for.  Ever ride in a car going around a curve and find yourself almost flung out of your seat?  This little guy wasn’t wearing his seat belt and look what happened.  The car accelerated because changing direction is an acceleration due to a lateral force.  But the guy followed Newton’s First Law and just kept going in a straight line.  Did he accelerate?

This is one of those “depends on how you look at it” cases.  From a frame of reference locked to the car (arrows), he was accelerated outwards by a centrifugal force that wasn’t countered by centripetal force from his seat belt.  However, from an earthbound frame of reference he flew in a straight line and experienced no force at all.

Side forceSuppose you’re investigating an object’s motion that appears to arise from a new force you’d like to dub “heterofugal.”  If you can find a different frame of reference (one not attached to the object) or otherwise explain the motion without invoking the “new force,” then heterofugalism is a fictitious force.

Centrifugal and centripetal forces are fictitious.  The  “force” “accelerating” one plane towards another as they both fly to the North Pole in this tale is actually geometrical and thus also fictitious   So is gravity.

In this post you’ll find a demonstration of gravity’s effect on the space around it.  Just as a sphere’s meridians give the effect of a fictitious lateral force as they draw together near its poles, the compressive curvature of space near a mass gives the effect of a force drawing other masses inward.

~~ Rich Olcott

The direction Newton avoided facing

Reading Newton’s Philosophiæ Naturalis Principia Mathematica is less challenging than listening to Vogon poetry.  You just have to get your head working like a 17th Century genius who had just invented Calculus and who would have deep-fried his right arm in rancid skunk oil before he’d admit to using any of his rival Liebniz’ math notations or techniques.

Newton II-II ellipseNewton was essentially a geometer. These illustrations (from Book 1 of the Principia) will give you an idea of his style.  He’d set himself a problem then solve it by constructing sometimes elaborate diagrams by which he could prove that certain components were equal or in strict proportion.

Newton XII-VII hyperbolaFor instance, in the first diagram (Proposition II, Theorem II), we see an initial glimpse of his technique of successive approximation.  He defines a sequence of triangles which as they proliferate get closer and closer to the curve he wants to characterize.

The lines and trig functions escalate in the second diagram (Prop XII, Problem VII), where he calculates the force  on a body traveling along a hyperbola.

Newton XLIV-XIV precessionThe third diagram is particularly relevant to the point I’ll finally get to when I get around to it.  In Prop XLIV, Theorem XIV he demonstrates something weird.  Suppose two objects A and B are orbiting around attractive center C, but B is moving twice as fast as A.  If C exerts an additional force on B that is inversely dependent on the cube of the B-C distance, then A‘s orbit will be a perfect circle (yawn) but B‘s will be an ellipse that rotates around C, even though no external force pushes it laterally.

In modern-day math we’d write the additional force as F∼(1/rBC3), but Newton verbalized it as “in a triplicate ratio of their common altitudes inversely.”  See what I mean about Vogon poetry?

Now, about that point I was going to get to.  It’s C, in the center of that circle.  If the force is proportional to 1/r3, what happens when r approaches zero?  BLOOIE, the force becomes infinite.

In the previous post we used geometry to understand the optical singularity at the center of the Christmas ball.  I said there that my modeling project showed me a deeper reason for a BLOOIE.  That reason showed up partway through the calculation for the angle between the axis and the ring of reflected  light.  A certain ratio came out to be (1-x)/2x, where x is proportional to the distance between the LED and the ball’s center.  Same problem: as the LED approaches the center, x approaches zero and BLOOIE.  (No problem when x is one, because the ratio is 0/2 which is zero which is OK.)

Singularities happen when the formula for something goes to infinity.

Now, Newton recognized that his central-force (1/rn)-type equations covered gravity and magnetism and even the inward force on the rim of a rotating wheel.  It’s surprising that he didn’t seem too worried about BLOOIE.

I think he had two excuses.  First, he was limited by his graphical methodology.  In most of his constructions, when a certain distance goes to zero there’s a general catastrophe — rectangles and triangles collapse to lines or even points, radii whirl aimlessly without a vertex to aim at…  His lovely derivations devolve into meaninglessness.  Further advances would depend on the  algebraic approach to Calculus taken by the detested Liebniz.

Second (here’s the hook for this post’s title), Newton was looking outward, not inward.  He was considering the orbits of planets and other sizable objects.  r is always the distance between object centers.  For sizable objects you don’t have to worry about r=0 because “center-to-center equals zero” never occurs.  If the Moon (radius 1080 miles) were to drop down to touch the Earth (radius 3960 miles), their centers would still be 5000 miles apart.  No BLOOIE.

Actually, there would be CRUMBLE instead of BLOOIE because a different physical model would apply — but that’s a tale for another post.

The moral of the story is this.  Mathematical models don’t care about infinities, but Nature does.  Any conditions where the math predicts an infinite value (for instance, where a denominator can become zero) are prime territory for new models that make better predictions.

~~ Rich Olcott