The Solid Gold Bath Towel

“C’mon, Sy, I heard weaseling there — ‘velocity‑based thinking‘ ain’t the same as velocity numbers.”

“Guilty as charged, Vinnie. The centuries-old ‘velocity of money‘ notion has been superceded for a half-century, but the theory’s still useful in the right circumstances. It’s like Newton’s Law of Gravity that way, except we’ve been drifting away from Newton for a full century.”

“What, gravity doesn’t work any more?”

“Sure it does, and most places the force is exactly what Newton said it should be — proportional to the mass divided by the distance. But it goes wrong when the mass‑to‑distance ratio gets huge, say close to a star or a black hole. That’s when we move up to Einstein’s theory. It includes Newton’s Law as a special case but it covers the high-ratio cases more exactly and accounts for more phenomena.”

“Just for grins, how about when the ratio is tiny?”

“We don’t know. Some cosmologists have suggested that’s what dark energy is about. Maybe when galaxies get really far apart, they’re not attracted to each other quite as much as Newton’s Law says.”

“I suppose the money theories have problems at high and low velocities?”

“That’s one pair of problems. Money velocity is proportional to nominal traffic divided by money supply. Suppose an average currency unit changes hands thousands of times a day. That says people don’t have confidence that money will buy as much tomorrow as it could today. They’ve got hyperinflation.”

“Ah, and at the low end it’d be like me putting Eddie’s autographed $20 in a frame on my wall. No spend, no traffic, zero velocity.”

“Right, but for the economy it’d be everyone putting all their money under their mattresses. Money that’s frozen in place doesn’t do anything except maybe make someone feel good. It’s like water in a stream, it has to be flowing to be useful in generating power.”

“Wait, you used a word back there, ‘nominal.’ What’s that about?”

“Good ears. It points up another important distinction between Physics and Economics. Suppose you’re engineering a mill at that stream and you measure water flow in cubic meters per second. Kinetic energy is mass times velocity squared and power is energy per unit time. If you know water’s density in kilograms per cubic meter you can calculate the stream’s available water power. Density is key to finding mass from volume when volume’s easy to measure, or volume from easily‑measured mass.”

“OK, so what’s that got to do with ‘nominal‘?”

“In economic situations, money is easy to measure — it’s just the price paid — but value is a puzzle. In fact, people say that understanding the linkage between price and value is the central problem of Economics. There’s a huge number of theories out there, with good counter-examples for every one of them. For example, consider the solid gold bath towel.”

“What a stupid idea. Thing like that couldn’t dry you off in the desert.”

“True, but it’s made out of a rare material and some people think rarity makes value. In the right setting it’d be beautiful and there are certainly people who think beauty makes value. A lot of person‑time would be required to create it and some people think labor input is what makes value. The people who think utility makes value would give that towel very low marks. Of course, if you’ve already got plenty of bath towels you’re not about to buy another one so you don’t care.”

“So how do they decide what its price should be?”

“Depends on where you are. Many countries use a supply‑demand auction system that measures value by what people are willing to pay. Planned‑economy countries set prices by government edict. Other countries use a mixed system where the government sets prices for certain commodities like bread and fuel but everything else is subject to haggling. Whatever system’s in use, ‘nominal‘ traffic is the total of all transaction prices and that’s supposed to measure value.”

“Velocity’s supposed to be money supply divided into value flow but we can’t use value so we fake it with money flow?”

“You got it. Then the government tries to manage the money supply so velocity’s in a sweet spot.”

“Sounds rickety.”

“Yup.”

~~ Rich Olcott

On Gravity, Charge And Geese

A beautiful April day, far too nice to be inside working.  I’m on a brisk walk toward the lake when I hear puffing behind me.  “Hey, Moire, I got questions!”

“Of course you do, Mr Feder.  Ask away while we hike over to watch the geese.”

“Sure, but slow down , will ya?  I been reading this guy’s blog and he says some things I wanna check on.”

I know better but I ask anyhow.  “Like what?”

“Like maybe the planets have different electrical charges  so if we sent an astronaut they’d get killed by a ginormous lightning flash.”

“That’s unlikely for so many reasons, Mr Feder.  First, it’d be almost impossible for the Solar System to get built that way.  Next, it couldn’t stay that way if it had been.  Third, we know it’s not that way now.”

“One at a time.”

“OK.  We’re pretty sure that the Solar System started as a kink in a whirling cloud of galactic dust.  Gravity spanning the kink pulled that cloud into a swirling disk, then the swirls condensed to form planets.  Suppose dust particles in one of those swirls, for whatever reason, all had the same unbalanced electrical charge.”

“Right, and they came together because of gravity like you say.”

I pull Old Reliable from its holster.  “Think about just two particles, attracted to each other by gravity but repelled by their static charge.  Let’s see which force would win.  Typical interstellar dust particles run about 100 nanometers across.  We’re thinking planets so our particles are silicate.  Old Reliable says they’d weigh about 2×1018 kg each, so the force of gravity pulling them together would be …  oh, wait, that’d depend on how far apart they are.  But so would the electrostatic force, so let’s keep going.  How much charge do you want to put on each particle?”

“The minimum, one electron’s worth.”

“Loading the dice for gravity, aren’t you?  Only one extra electron per, umm, 22 million silicon atoms.    OK, one electron it is …  Take a look at Old Reliable’s calculation.gravity vs electrostatic calculation Those two electrons push their dust grains apart almost a quintillion times more strongly than gravity pulls them together.  And the distance makes no difference — close together or far apart, push wins.  You can’t use gravity to build a planet from charged particles.”

“Wait, Moire, couldn’t something else push those guys together — magnetic fields, say, or a shock wave?”

“Sure, which is why I said almost impossible.  Now for the second reason the astronaut won’t get lightning-shocked — the solar wind.  It’s been with us since the Sun lit up and it’s loaded with both positive- and negative-charged particles.  Suppose Venus, for instance, had been dealt more than its share of electrons back in the day.  Its net-negative charge would attract the wind’s protons and alpha particles to neutralize the charge imbalance.  By the same physics, a net-positive planet would attract electrons.  After a billion years of that, no problem.”

“All right, what’s the third reason?”

“Simple.  We’ve already sent out orbiters to all the planets.  Descent vehicles have made physical contact with many of them.  No lightning flashes, no fried electronics.  Blows my mind that our Cassini mission to Saturn did seven years of science there after a six-year flight, and everything worked perfectly with no side-trips to the shop.  Our astronauts can skip worrying about high-voltage landings.”

“Hey, I just noticed something.  Those F formulas look the same.”  He picks up a stick and starts scribbling on the dirt in front of us.  “You could add them up like F=(Gm1m2+k0q1q2)/r2.  See how the two pieces can trade off if you take away some mass but add back some charge?  How do we know we’ve got the mass-mass pull right and not mixed in with some charge-charge push?”

Geese and ducks“Good question.  If protons were more positive than electrons, electrostatic repulsion would always be proportional to mass.  We couldn’t separate that force from gravity.  Physicists have separately measured electron and proton charge.  They’re equal (except for sign) to 10 decimal places.  Unfortunately, we’d need another 25 digits of accuracy before we could test your hypothesis.”

“Aw, look, the geese got babies.”

“The small ones are ducks, Mr Feder.”

~~ Rich Olcott

What’s that funnel about, really?

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.DMNS gravity well

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

rh lineSure 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