The Gelato Model

“Eddie, this ginger gelato’s delicious — not too sweet and just the right amount of ginger bite.”

“Glad you like it, Anne.”

On the way down here, Sy was telling me about how so many things in the Universe run on the same mathematics if you look at them with the right coordinate system. Sy, how do you pick ‘the right coordinate system?”

“The same way you pick the right property to serve as a momentum in Newton’s Equation of Motion — physical intuition. You look for things that fit the system. Sometimes that puts you on the road to understanding, sometimes not. Eddie, you keep track of your gelato sales by flavor. How are they doing?”

“Pistachio’s always a good seller, Sy, but ginger has been coming on strong this year.”

“In motion terns, pistachio’s momentum is constant but ginger is gaining momentum, right?”

“S’what I said.”

“Measured in dollars or trayfuls?”

“In batches. I make it all in-house. I’m proud of that. Dollars, too, of course, but that’s just total for all flavors.”

“Batches all the same size?”

“Some are, some not, depending. If I had a bigger machine I could make more but I do what I can.”

“There you go, Anne, each gelato flavor is like a separate degree of freedom. Eddie’s tracked sales since he started so we can take that date as the origin. Measuring change along any degree in either batches or dollars we have perfectly respectable coordinates although the money view of the system is fuzzier. Velocity is batches per unit time, there’s even a speed limit, and ginger has accelerated. Sound familiar?”

“Sounds like you’re setting up a Physics model.”

“Call it gelato trend physics, but I don’t think I can push the analogy much further. The next step would be to define a useful momentum like Newton did with his Law of Motion.”

F=ma? That’s about acceleration, isn’t it?”

“Probably not in Newton’s mind. Back in his day they were arguing about which was conserved, energy or momentum. It was a sloppy argument because no‑one agreed on crisp definitions. People could use words like ‘quantity of motion‘ to refer to energy or momentum or even something else. Finally Newton defined momentum as ‘mass times velocity‘, but first he had to define ‘mass‘ as ‘quantity of matter‘ to distinguish it from weight which he showed is a force that’s indirectly related to mass.”

“So is it energy or momentum that’s conserved?”

“Both, once you’ve got good definitions of them. But my point is, our car culture has trained us to emphasize acceleration. Newton’s thinking centered on momentum and its changes. In modern terms he defined force as momentum change per unit time. I’m trying to think of a force‑momentum pair for Eddie’s gelato. That’s a problem because I can’t identify an analog for inertia.”

“Inertia? What’s that got to do with my gelato?”

“Not much, and that’s the problem. Inertia is resistance to force. Who can resist gelato? If it weren’t for inertia, the smallest touch would be enough to send an object at high speed off to forever. The Universe would be filled with dust because stars and planets would never get the chance to form. But here we are, which I consider a good thing. Where does inertia come from? Newton changed his mind a couple of times. To this day we only have maybe‑answers to that question.”

“You know we want to know, Sy.”

“Einstein’s favorite guess was Mach’s Principle. There’s about a dozen different versions of the basic idea but they boil down to matter interacting with the combined gravitational and electromagnetic fields generated by the entire rest of the Universe.”

“Wow. Wait, the stars are far away and the galaxies are much, much further away. Their fields would be so faint, how can they have any effect at all?”

“You’re right, Anne, field intensity per star does drop with distance squared. But the number of stars goes up with distance cubed. The two trends multiply together so the force trends grow linearly. It’s a big Universe and size matters.”

“So what about my gelato?”

“We’ll need more research, Eddie. Another scoop of ginger, Anne?”

~~ Rich Olcott

A Summertime Slice of π

So you think you’re standing still?  Let’s run some circles, all variations on the theme of 2πR…

Circles in circles
The Earth rotates on its axis,
as it and the moon revolve around their barycenter,
as the barycenter revolves around the Sun.
Not to scale, of course.

The Earth’s radius is 4,000 miles and it completes one rotation every 24 hours.  Its circumference at the Equator (2πR) is 25,000 miles, so if you’re reading this in Ecuador you’re doing 25000/24 = 1041 miles per hour.

I’m writing this in Denver, at 39.75oN, where the circumference perpendicular to the axis of rotation is only 19,200 miles.  Sitting here I’m circling the Earth at 800 miles per hour.  But that’s not all.

The Earth and the Moon both revolve around their common center of gravity (their barycenter).  The barycenter is inside the Earth, offset from its center by 2881 miles.  The center of the Earth runs a circle around the barycenter once every month (27.3 days), at a relatively piddly 27.6 miles per hour.  But that’s not all.

Circles in circles
One year of Denver’s distance from Sag A*,
the black hole at the center of the Milky Way.
The green curve is our orbit around the Sun.
The blue curve adds in the monthly wobble
as we revolve around the barycenter.
The red curve includes our daily rotation. (Should be 365 cycles but I used artistic license to cut that down to 36. Besides, the vertical scale’s all wrong. Sue me.)

Earth’s orbit is (nearly) a circle.  The orbit’s radius is 93 million miles so its circumference is 584 million miles.  If you ran that many miles in a year you’d have to hit a pace of 66,600 miles per hour (no rest stops).  But that’s not all.

The Sun’s not just standing still all alone in space.  It’s part of the Milky Way Galaxy, which rotates once per 230 million years.  The Sun is about 26,000 light-years (152.8 quadrillion miles) from the center of the galaxy, so in one cycle it travels some 960 quadrillion miles.  That’s a rate of 476,000 miles per hour.  But that’s not all.

The Milky Way is one of about 50 galaxies in the Local Group.  The galaxies move with respect to each other and the whole assembly undoubtedly rotates.  Unfortunately, the astronomers are just now devising technology that can measure all that motion.  Expect large numbers for the net speeds when they figure them out.  But that’s not all.

The entire Local Group is flying towards a point in the constellation Centaurus.  Our flight speed has been measured at about 1,430,000 miles per hour.  The astronomers think the flight is linear, but on a larger scale it may be part of yet another rotation.

Feeling a bit dizzy?  Have a frosty glass of iced tea with your delicious π and just let the Earth spin along.

~~ Rich Olcott

Gargh, His Heirs, and the AAAD Problem

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

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

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

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

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

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

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

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

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

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

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

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

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

 ~~ Rich Olcott