Tag: Inertial frame

Hyperbolas But Not Hyperbole

On By rolcottIn Astronomy: Techniques, Physics: Light and Relativity, Uncategorized2 Comments

“Minus? Where did that come from?”

<Gentle reader — If that question looks unfamiliar, please read the preceding post before this one.>

Jim’s still at the Open Mic. “A clever application of hyperbolic geometry.” Now several of Jeremy’s groupies are looking upset. “OK, I’ll step back a bit. Jeremy, suppose your telescope captures a side view of a 1000‑meter spaceship but it’s moving at 99% of lightspeed relative to you. The Lorentz factor for that velocity is 7.09. What will its length look like to you?”

“Lorentz contracts lengths so the ship’s kilometer appears to be shorter by that 7.09 factor so from here it’d look about … 140 meters long.”

“Nice, How about the clocks on that spaceship?”

“I’d see their seconds appear to lengthen by that same 7.09 factor.”

“So if I multiplied the space contraction by the time dilation to get a spacetime hypervolume—”

“You’d get what you would have gotten with the spaceship standing still. The contraction and dilation factors cancel out.”

“How about if the spaceship went even faster, say 99.999% of lightspeed?”

“The Lorentz factor gets bigger but the arithmetic for contraction and dilation still cancels. The hypervolume you defined is always gonna be just the product of the ship’s rest length and rest clock rate.”

His groupies go “Oooo.”

One of the groupies pipes up. “Wait, the product of x and y is a constant — that’s a hyperbola!”

“Bingo. Do you remember any other equations associated with hyperbolas?”

“Umm… Yes, x2–y2 equals a constant. That’s the same shape as the other one, of course, just rotated down so it cuts the x-axis vertically.”

Jeremy goes “Oooo.”

Jim draws hyperbolas and a circle on the whiteboard. That sets thoughts popping out all through the crowd. Maybe‑an‑Art‑major blurts into the general rumble. “Oh, ‘plus‘ locks x and y inside the constant so you get a circle boundary, but ‘minus‘ lets x get as big as it wants so long as y lags behind!”

Another conversation – “Wait, can xy=constant and x2–y2=constant both be right?”
  ”Sure, they’re different constants. Both equations are true where the red and blue lines cross.”

A physics student gets quizzical. “Jim, was this Minkowski’s idea, or Einstein’s?”

“That’s a darned good question, Paul. Minkowski was sole author of the paper that introduced spacetime and defined the interval, but he published it a year after Einstein’s 1905 Special Relativity paper highlighted the Lorentz transformations. I haven’t researched the history, but my money would be on Einstein intuitively connecting constant hypervolumes to hyperbolic geometry. He’d probably check his ideas with his mentor Minkowski, who was on the same trail but graciously framed his detailed write‑up to be in support of Einstein’s work.”

One of the astronomy students sniffs. “Wait, different observers see the same s2=(ct)2–d2 interval between two events? I suppose there’s algebra to prove that.”

“There is.”

“That’s all very nice in a geometric sort of way, but what does s2 mean and why should we care whether or not it’s constant?”

“Fair questions, Vera. Mmm … you probably care that intervals set limits on what astronomers see. Here’s a Minkowski map of the Universe. We’re in the center because naturally. Time runs upwards, space runs outwards and if you can imagine that as a hypersphere, go for it. Light can’t get to us from the gray areas. The red lines, they’re really a hypercone, mark where s2=0.”

From the back of the room — “A zero interval?”

“Sure. A zero interval means that the distance between two events exactly equals lightspeed times light’s travel time between those events. Which means if you’re surfing a lightwave between two events, you’re on an interval with zero measure. Let’s label Vera’s telescope session tonight as event A and her target event is B. If the A–B interval’s ct difference is greater then its d difference then she can see B — if the event is in our past but not beyond the Cosmic Microwave Background. But if a Dominion fleet battle is approaching us through subspace from that black dot, we’ll have no possible warning before they’re on us.”

Everyone goes “Oooo.”

~~ Rich Olcott

Thinking in Spacetime

On By rolcottIn Cosmology, Mathematics: Dimensions, UncategorizedLeave a comment

The Open Mic session in Al’s coffee shop is still going string. The crowd’s still muttering after Jeremy stuck a pin in Big Mike’s “coincidence” balloon when Jim steps up. Jim’s an Astrophysics post‑doc now so we quiet down expectantly. “Nice try, Mike. Here’s another mind expander to play with. <stepping over to the whiteboard> Folks, I give you … a hypotenuse. ‘That’s just a line,’ you say. Ah, yes, but it’s part of some right triangles like … these. Say three different observers are surveying the line from different locations. Alice finds her distance to point A is 300 meters and her distance to point B is 400. Applying Pythagoras’ Theorem, she figures the A–B distance as 500 meters. We good so far?”

A couple of Jeremy’s groupies look doubtful. Maybe‑an‑Art‑Major shyly raises a hand. “The formula they taught us is a2+b2=c2. And aren’t the x and y supposed to go horizontal and vertical?”

“Whoa, nice questions and important points. In a minute I’m going to use c for the speed of light. It’s confusing to use the same letter for two different purposes. Also, we have to pay them extra for double duty. Anyhow, I’m using d for distance here instead of c, OK? To your next point — Alice, Bob and Carl each have their own horizontal and vertical orientations, but the A–B line doesn’t care who’s looking at it. One of our fundamental principles is that the laws of Physics don’t depend on the observer’s frame of reference. In this situation that means that all three observers should measure the same length. The Pythagorean formula works for all of them, so long as we’re working on a flat plane and no-one’s doing relativistic stuff, OK?”

Tentative nods from the audience.

“Right, so much for flat pictures. Let’s up our game by a dimension. Here’s that same A–B line but it’s in a 3D box. <Maybe‑an‑Art‑Major snorts at Jim’s amateur attempt at perspective.> Fortunately, the Pythagoras formula extends quite nicely to three dimensions. It was fun figuring out why.”

Jeremy yells out. “What about time? Time’s a dimension.”

“For sure, but time’s not a length. You can’t add measurements unless they all have the same units.”

“You could fix that by multiplying time by c. Kilometers per second, times seconds, is a length.” His groupies go “Oooo.”

“Thanks for the bridge to spacetime where we have four coordinates — x, y, z and ct. That makes a big difference because now A and B each have both a where and a when — traveling between them is traveling in space and time. Computationally there’s two paths to follow from here. One is to stick with Pythagoras. Think of a 4D hypercube with our A–B line running between opposite vertices. We’re used to calculating area as x×y and volume as x×y×z so no surprise, the hypercube’s hypervolume is x×y×z×(ct). The square of the A–B line’s length would be b2=(ct)2+d2. Pythagoras would be happy with all of that but Einstein wasn’t. That’s where Alice and Bob and Carl come in again.”

“What do they have to do with it?”

“Carl’s sitting steady here on good green Earth, red‑shifted Alice is flying away at high speed and blue‑shifted Bob is flashing toward us. Because of Lorentz contractions and dilations, they all measure different A–B lengths and durations. Each observer would report a different value for b2. That violates the invariance principle. We need a ruggedized metric able to stand up to that sort of punishment. Einstein’s math professor Hermann Minkowski came up with a good one. First, a little nomenclature. Minkowski was OK with using the word ‘point‘ for a location in xyz space but he used ‘event‘ when time was one of the coordinates.”

“Makes sense, I put events on my calendar.”

“Good strategy. Minkowski’s next step quantified the separation between two events by defining a new metric he called the ‘interval.’ Its formula is very similar to Pythagoras’ formula, with one small change: s2=(ct)2–d2. Alice, Bob and Carl see different distances but they all see the same interval.”

“Minus? Where did that come from?”

~~ Rich Olcott

Speed Limit

On By rolcottIn Physics: Light and Relativity, UncategorizedLeave a comment

“Wait, Sy, there’s something funny about that Lorentz factor. I’m riding my satellite and you’re in your spaceship to Mars and we compare notes and get different times and lengths and masses and all so we have to use the Lorentz factor to correct numbers between us. Which velocity do we use, yours or mine?”

“Good question, Vinnie. We use the difference between our two frames. We can subtract either velocity from the other one and replace v with that number. Strictly speaking, we’d subtract velocity components perpendicular to the vector between us. If I were to try to land on your satellite I’d have to expend fuel and energy to change my frame’s velocity to yours. When we matched frames the velocity difference would be zero, the Lorentz factor would be 1.0 and I’d see your solar array as a perfect 10×10‑meter square. Our clocks would tick in sync, too.”

“OK, now there’s another thing. That Lorentz formula compares our subtracted speeds to lightspeed c. What do we subtract to get c?”

“Deep question. That’s one of Einstein’s big insights. Suppose from my Mars‑bound spaceship I send out one light pulse toward Mars and another one in the reverse direction, and you’re watching from your satellite. No matter how fast my ship is traveling, Einstein said that you’d see both pulses, forward and backward, traveling at the same speed, c.”

“Wait, shouldn’t that be that your speed gets added to one pulse and subtracted from the other one?”

“Ejected mass works that way, but light has no mass. It measures its speed relative to space itself. What you subtract from c is zero. Everywhere.”

“OK, that’s deep. <pause> But another ‘nother thing—”

“For a guy who doesn’t like equations, you’re really getting into this one.”

“Yeah, as I get up to speed it grows on me. HAW!”

“Nice one, you got me. What’s the ‘nother thing?”

“I remembered how velocity is speed and direction but we’ve been mixing them together. If my satellite’s headed east and your spaceship’s headed west, one of us is minus to the other, right? We’re gonna figure opposite v‑numbers. How’s that work out?”

“You’re right. Makes no difference to the Lorentz factor because the square of a negative difference is the same as the square of its positive twin. You bring up an important point, though — the factor applies to both of us. From my frame, your clock is running slow. From your frame, mine’s the slow one. Einstein’s logic says we’re both right.”

“So we both show the same wrong time, no problem.”

“Nope, you see my clock running slow relative to your clock. I see exactly the reverse. But it gets worse. How about getting your pizza before you order it?”

“Eddie’s good, he ain’t that good. How do you propose to make that happen?”

“Well, I don’t, but follow me here. <working numbers on Old Reliable> Suppose we’re both in spaceships. I’m loafing along at 0.75c relative to Eddie’s pizza place on Earth and your ship is doing 3c. Also, suppose that we can transmit messages and mass much faster than lightspeed.”

“Like those Star Trek transporters and subspace radios.”

“Right. OK, at noon on my personal clock you tell me you’ve ordered pizza so I get one, too. Eddie slaps both our pizzas into his transporter 10 minutes later. The math works out that according to my clock you get your pizza 8.9 minutes before you put in your order. You like that?”

“Gimme a sec … nah, I don’t think so. If I read that formula right with v1 being you and v2 being me, if you run that formula for what I’d see with my velocity on the bottom, that’s a square root of a minus which can’t be right.”

“Yup, the calculation gives an imaginary number, 4.4i minutes, whatever that means. So between us we have two results that are just nonsense — I see effect before cause and you see a ridiculous time. To avoid that sort of thing, Einstein set his speed limit for light, gravity and information.”

“I’m willing to keep under it if you are.”

“Deal.”

~~ Rich Olcott

The Relativity Factor

On By rolcottIn Physics: Light and Relativity, UncategorizedLeave a comment

“Sy, it’s nice that Einstein agreed with Rayleigh’s wave theory stuff but why’d you even drag him in? I thought the faster‑than‑light thing was settled.”

“Vinnie, faster‑than‑light wasn’t even an issue until Einstein came along. Science had known lightspeed was fast but not infinite since Rømer measured it in Newton’s day. ‘Pretty fast,’ they said, but Newtonian mechanics is perfectly happy with any speed you like. Then along came Einstein.”

“Speed cop, was he?”

“Funny, Vinnie. No, Einstein showed that the Universe enforces the lightspeed limit. It’s central to how the Universe works. Come to think of it, the crucial equation had been around for two decades, but it took Einstein to recognize its significance.”

“Ah, geez, equations again.”

“Just this one and it’s simple. It’s all about comparing v for velocity which is how fast something’s going, to c the speed of light. Nothing mystical about the arithmetic — if you’re going half the speed of light, the factor works out to 1.16. Ninety‑nine percent of c gives you 7.09. Tack on another 9 and you’re up to 22.37 and so on.”

“You got those numbers memorized?”

“Mm-hm, they come in handy sometimes.”

“Handy how? What earthly use is it? Nothing around here goes near that fast.”

“Do you like your GPS? It’d be useless if the Lorentz factor weren’t included in the calculations. The satellites that send us their sync signals have an orbit about 84 000 kilometers wide. They run that circle once a sidereal day, just shy of 86 400 seconds. That works out to 3 kilometers per second and a Lorentz factor of 1.000 005.”

“Yeah, so? That’s pretty close to 1.0.”

“It’s off by 5 parts per million. Five parts per million of Earth’s 25 000-mile circumference is an eighth of a mile. Would you be happy if your GPS directed you to somewhere a block away from your address?”

“Depends on why I’m going there, but I get your point. So where else does this factor come into play?”

“Practically anywhere that involves a precision measurement of length or duration. It’s at the core of Einstein’s Special Relativity work. He thought about observing a distant moving object. It’s carrying a clock and a ruler pointed along the direction of motion. The observer would see ticks of the clock get further apart by the Lorentz factor, that’s time dilation. Meanwhile, they’d see the ruler shrink by the factor’s inverse, that’s space compression.”

“What’s this ‘distant observer‘ business?”

“It’s less to do with distance than with inertial frames. If you’re riding one inertial frame with a GPS satellite, you and your clock stay nicely synchronized with the satellite’s signals. You’d measure its 1×1‑meter solar array as a perfect square. Suppose I’m riding a spaceship that’s coasting to Mars. I measure everything relative to my own inertial frame which is different from yours. With my telescope I’d measure your satellite’s solar array as a rectangle, not a square. The side perpendicular to the satellite’s orbit would register the expected 1 meter high, but the side pointing along the orbit would be shorter, 1 meter divided by the Lorentz factor for our velocity difference. Also, our clocks would drift apart by that Lorentz factor.”

“Wait, Sy, there’s something funny about that equation.”

“Oh? What’s funny?”

“What if somebody’s speed gets to c? That’d make the bottom part zero. They didn’t let us do that in school.”

“And they shouldn’t — the answer is infinity. Einstein spotted the same issue but to him it was a feature, not a bug. Take mass, for instance. When they meet Einstein’s famous E=mc² equation most people think of the nuclear energy coming from a stationary lump of uranium. Newton’s F=ma defined mass in terms of a body’s inertia — the greater the mass, the more force needed to achieve a certain amount of acceleration. Einstein recognized that his equation’s ‘E‘ should include energy of motion, the ½mv² kind. He had to adjust ‘m‘ to keep F=ma working properly. The adjustment was to replace inertial mass with ‘relativistic mass,’ calculated as inertial mass times the Lorentz factor. It’d take infinite force to accelerate any relativistic mass up to c. That’s why lightspeed’s the speed limit.”

~~ Rich Olcott

Chasing Rainbows

On By rolcottIn Physics: Light and Relativity, UncategorizedLeave a comment

“C’mon, Sy, Newton gets three cheers for tying numbers to the rainbow’s colors and all that, but what’s it got to do with that three speeds of light thing which is where we started this discussion?”

“Vinnie, they weren’t just numbers, they were angles. The puzzle was why each color was bent to a different degree when entering or leaving the prism. That was an inconvenient truth for Newton.”

“Inconvenient? There’s a loaded word.”

“Indeed. A little context — Newton was in a big brouhaha about whether light was particles or waves. Newton was a particle guy all the way, battling wave theory proponents like Euler and Descartes and their followers on the Continent. Even Hooke in London had a wave theory. Newton’s problem was that his beam deflections happened right at the prism’s air‑glass interfaces.”

“What difference does that … wait, you mean that there’s no bending inside the prism? Light inside still goes straight but in a different direction?”

“That’s it, exactly. The deflection angles are the same, whether the beam hits the prism near the short‑path tip or the long‑path base. No evidence of further deviation inside the prism unless it has bubbles — Newton had to discard or mask off some bad prisms. Explaining the no‑curvature behavior is difficult in a particle framework, easy in a wave framework.”

“Really? I don’t see why.”

Left: faster medium, right: slower medium
Credit: Ulflund, under Creative Commons 1.0

“Suppose light is particles, which by definition are local things affected only by local forces. The medium’s effects on a particle would happen in the bulk material rather than at the interface. The effect would accumulate as the particles travel further through the medium. The bend should be a curve. Unfortunately for Newton, that’s not what his observations showed.”

“OK, scratch particles. Why not scratch waves, too?”

“Waves have no problem with abrupt variation at an interface, They flip immediately to a new stable mode. For example. here’s an animation showing an abrupt speed change at the interface between a fast‑travel medium like air and a slow‑travel medium like glass or water. See how one end of each bar gets slowed down while the other end is still moving at speed? By the time the whole bar is inside, its path has slewed to the refraction angle.”

“Like a car sliding on ice when a rear wheel sticks for an moment, eh Sy?”

“That was not a fun ride, Vinnie.”

“I enjoyed it. Whatever, I get how going air‑to‑glass or vice‑versa can change a beam’s direction. But if everything’s going through the same angle, how do rainbows happen?”

“Everything doesn’t go through the same angle. Frequencies make a difference. Go back to the video and keep your eye on one bar as it sweeps up the interface. See how the sweep’s speed controls the deflection angle?”

“Yeah, if the sweep went slower the beam would get a chance to bend further. Faster sweeps would bend it less. But what could change the sweep speed?

“Two things. One, change the medium to one with a different transmission speed. Two, change the wave itself so it has a different speed. According to Snell’s Law, the important parameter for a pair of media is their ratio of fast‑speed divided by slow‑speed. If the fast medium is a vacuum that ratio is the slow medium’s index of refraction. The greater the index, the greater the bend.”

“Changing the medium doesn’t apply. I got one prism, it’s got one index, but I still get a whole rainbow.”

“Right, rainbows are about how one prism treats a bunch of waves with different time and space frequencies.”

“Space frequency?”

“If you measure a wave in meters it’s cycles per meter.”

“Wavelength upside down. Got it.”

“Whether you figure in frequencies or intervals, the wave speed works out the same.”

“Speed of light, finally.”

“Point is, when a wave goes through any medium, its time frequency doesn’t change but its space frequency does. Interaction with local charge shortens the wavelength. Short‑wavelength blue waves are held back more than long‑wavelength red ones. The different angles make your rainbow. The hold‑back is why refraction indices are usually greater than one.”

“Usually?”

~~ Rich Olcott

Several Big Sloshes

On By rolcottIn Astronomy: Planetary Science, UncategorizedLeave a comment

“I call distraction, Sy. You were going to explain how come the Moon’s drifting away from us but you got us into radians and stuff. What’s that got to do with the Moon flying away by dragging a big wave around the Earth?”

“It’s not dragging a localized bulge of water like you’re thinking, Vinnie, nothing like that wave on Miller’s Planet. For that matter, the Miller’s Planet wave had a sharply‑rising front which also doesn’t look like the textbook tidal bulge.”

“There’s a textbook on this stuff?”

“Many, Al. Heavy-duty people have spent a lot of time on tides. Think about all the military and commercial navies that depend on boats being able to leave port and dock on schedule.”

“And not run aground <heh heh>”

“Well, yes, Vinnie. Anyhow, like a lot of pre‑computer Physics, that work was based on a simplified ideal system — a moon orbiting a smooth planet with a world‑covering ocean. Water’s drawn horizontally towards the sub‑lunar points making an egg‑shaped surface and everything’s neat.”

“Probably nothing like real life.”

“Of course. Here’s a video I built from satellite altimetry data. The grey dot is roughly the point underneath the Moon as that day progressed. The red‑to‑blue height scale’s in meters.. Not as neat as theory, is it?”

“Wow, that’s a mess. Looks like the Moon’s pulling water along the Canada‑Alaska coast okay, and the western Pacific starts to get a dome going. But the water never catches up before the Moon’s gone.”

“Hey, Vinnie, look how the tides just go round and round New Zealand. And what’s that, Hudson Bay, it’s a pinwheel.”

“Yeah, and in between Africa and Madagascar it’s completely out of phase from what it oughta be.”

“What you’re looking at is slosh. Once again, reality overwhelms a pretty theory. Each basin has its own preferred set of oscillations. None of them match up with the Moon. But the other thing — “

“Tiny numbers. Everything’s like less than a couple meters, not not a big bulge at all.”

“Bingo, Vinnie. Against Earth’s 6.4‑million‑meter radius, those small chaotic sloshes don’t make for effective energy transfer driving the Moon away from us. That theory’s toast.”

“So what’s doing it?”

“There’s two theories that I know of, and they’re probably both right. The first one is Earth tides — that bump you think of as traveling around the planet, but the bump is rock instead of water.”

“That can’t be a big effect. Rocks don’t bend.”

“On a planetary scale they’re not as solid as you think, Vinnie. The rock crust is brittle and really thin, less than half a percent of Earth’s radius. It floats on molten outer mantle which has the fluidity of tapioca pudding. When that structure gives under stress the crust layer cracks. The seismologists and GPS techs have measured surface motion all over the world. When they analyze the maps, the lunar component accounts for up to a meter of coordinated vertical daily movement. Figure the whole Earth is continually being squeezed and pulled to that extent and you’ve got a lot of energy being expended every 24 hours.”

“How about the other theory?”

“There’s no direct evidence, so far as I know, but it seems reasonable on physical grounds. We’ve got two gyrations going on here, right? The Moon is on a 29½‑day orbit while the Earth rotates about thirty times faster. But the two motions use different frames. The Earth’s spin axis runs through the geometric center of the planet and tilts 23° from its orbit axis. Meanwhile the whole Earth‑Moon system rotates about its barycenter, their common center of gravity, which stays inside the Earth about ¾ of the way moonward from the Earth’s middle. That rotation is about 5° away from Earth’s orbit’s axis. Imagine a molten blob near the barycenter, happily following the Moon in the Earth-Moon frame, but the rest of the planet is saying, ‘No, no, you’re supposed to be moving hundreds of miles an hour in a different direction!‘ If the blob’s the least bit lighter or heavier than its neighbor blobs, inertial forces expend energy to kick it out of there.”

“So we got two ways to transfer energy steady-like.”

“I think so.”

~~ Rich Olcott

Here’s a Different Angle

On By rolcottIn Astronomy: Planetary Science, Physics: Fundamentals, UncategorizedLeave a comment

“OK, Sy, so there’s a bulge on the Moon’s side of the Earth and the Earth rotates but the bulge doesn’t and that makes the Moon’s orbit just a little bigger and you’ve figured out that the energy it took to lift the Moon raised Earth’s temperature by a gazillionth of a degree, I got all that, but you still haven’t told Al and me how the lifting works.”

“You wouldn’t accept it if I just said, ‘The Moon lifts itself by its bootstraps,’ would you?”

“Not for a minute.”

“And you don’t like equations. <sigh> OK, Al, pass over some of those paper napkins.”

“Aw geez, Sy.”

“You guys asked the question and this’ll take diagrams, Al. Ante up. … Thanks. OK, remember the time Cathleen and I caught Vinnie here at Al’s shop playing with a top?”

“Yeah, and he was spraying paper wads all over the place.”

“I wasn’t either, Al, it was the top sending them out with centri–…, some force I can never remember whether it’s centrifugal or centripetal.”

“Centrifugal, Vinnie, –fugal– like fugitive, outward‑escaping force. It’s one of those ‘depends on how you look at it‘ fictitious forces. From where you were sitting, the wads looked like they were flying outward perpendicular to the top’s circle. From a wad’s point of view, it flew in a straight line tangent to the circle. It’s like we have two languages, Room and Rotor. They describe the same phenomena but from different perspectives.”

“Hey, it’s frames again, ain’t it?”

“Newton’s inertial frames? Sort‑of but not quite. Newton’s First Law only holds in the Room frame — no acceleration, motion is measured by distance, objects at rest stay put. Any other object moves in a straight line unless its momentum is changed by a force. You can tackle a problem by considering momentum and force components along separate X and Y axes. Both X and Y components work the same way — push twice as hard in either direction, get twice the acceleration in that direction. Nice rules that the Rotor frame doesn’t play by.”

“I guess not. The middle’s the only place an object can stay put, right?”

“Exactly, Al. Everything else looks like it’s affected by weird, constantly‑varying forces that’re hard to describe in X‑Y terms.”

“So that breaks Newton’s physics?”

“Of course not. We just have to adapt his F=m·a equation (sorry, Vinnie!) to Rotor conditions. For small movements we wind up with two equations. In the strict radial direction it’s still F=m·a where m is mass like we know it, a is acceleration outward or inward, and F is centrifugal or centripetal, depending. Easy. Perpendicular to ‘radial‘ we’ve got ‘angular.’ Things look different there because in that direction motion’s measured by angle but Newton’s Laws are all about distances — speed is distance per time, acceleration is speed change per time and so forth.”

“So what do you do?”

“Use arc length. Distance along an arc is proportional to the angle, and it’s also proportional to the radius of the arc, so just multiply them together.”

“What, like a 45° bend around a 2-foot radius takes 90 feet? That’s just wrong!”

“No question, Al. You have to measure the angle in the right units. Remember the formula for a circle’s circumference?”

“Sure, it’s 2πr.”

“Which tells you that a full turn’s length is 2π times the radius. We can bridge from angle to arc length using rotational units so that a full turn, 360°, is 2π units. We’ll call that unit a radian. Half a circle is π radians. Your 45° angle in radians is π/4 or about ¾ of a radian. You’d need about (¾)×(2) or 1½ feet of whatever to get 45° along that 2-foot arc. Make sense?”

“Gimme a sec … OK, I’m with you.”

“Great. So if angular distance is radius times angle, then angular momentum which is mass times distance per time becomes mass times radius times angle per time.”

“”Hold on, Sy … so if I double the mass I double the momentum just like always, but if something’s spinning I could also double the angular momentum by doubling the radius or spinning it twice as fast?”

“Couldn’t have put it better myself, Vinnie.”

~~ Rich Olcott

Breaking Up? Not So Hard

On By rolcottIn Physics: Black Holes and Relativity, UncategorizedLeave a comment

<transcript of smartphone dictation by Sy Moire, hard‑boiled physicist>
Day 173 of self‑isolation….
Perfect weather for a brisk solitary walk, taking the park route….
There’s the geese. No sign of Mr Feder, just as well….

Still thinking about Ms Baird and her plan for generating electric power from a black hole named Lonesome….
Can just hear Vinnie if I ever told him about this which I can’t….
“Hey, Sy, nothin’ gets out of a black hole except gravity, but she’s using Lonesome‘s magnetic field to generate electricity which is electromagnetic. How’s that happen?”
Good question….

Hhmph, that’s one angry squirrel….
Ah, a couple of crows pecking the ground under its tree. Maybe they’re too close to its acorn stash….

We know a black hole’s only measurable properties are its mass, charge and spin….
And maybe its temperature, thanks to Stephen Hawking….
Its charge is static — hah! cute pun — wouldn’t support continuous electrical generation….
The Event Horizon hides everything inside — we can’t tell if charge moves around in there or even if it’s matter or anti‑matter or something else….
The no‑hair theorem says there’s no landmarks or anything sticking out of the Event Horizon so how do we know the thing’s even spinning?

Ah, we know a black hole’s external structures — the jets, the Ergosphere belt and the accretion disk — rotate because we see red- and blue-shifted radiation from them….
The Ergosphere rotates in lockstep with Lonesome‘s contents because of gravitational frame-dragging….
Probably the disk and the jets do, too, but that’s only a strong maybe….
But why should the Ergosphere’s rotation generate a magnetic field?

How about Newt Barnes’ double‑wheel idea — a belt of charged light‑weight particles inside a belt of opposite‑charged heavy particles all embedded in the Ergosphere and orbiting at the black hole’s spin rate….
Could such a thing exist? Can simple particle collisions really split the charges apart like that?….

OK, fun problem for strolling mental arithmetic. Astronomical “dust” particles are about the size of smoke particles and those are about a micrometer across which is 10‑6 meter so the volume’s about (10‑6)3=10‑18 cubic meter and the density’s sorta close to water at 1 gram per cubic centimeter or a thousand kilograms per cubic meter so the particle mass is about 10‑18×103=10‑15 kilogram. If a that‑size particle collided with something and released just enough kinetic energy to knock off an electron, how fast was it going?

Ionization energy for a hydrogen atom is 13 electronvolts, so let’s go for a collision energy of at least 10 eV. Good old kinetic energy formula is E=½mv² but that’s got to be in joules if we want a speed in meters per second so 10 eV is, lemme think, about 2×10‑18 joules/particle. So v² is 2×2×10‑18/10‑15 which is 4×10‑3 or 40×10‑4, square root of 40 is about 6, so v is about 6×10‑2 or 0.06 meters per second. How’s that compare with typical speeds near Lonesome?

Ms Baird said that Lonesome‘s mass is 1.5 Solar masses and it’s isolated from external gravity and electromagnetic fields. So anything near it is in orbit and we can use the circular orbit formula v²=GM/r….
Dang, don’t remember values for G or M. Have to cheat and look up the Sun’s GM product on Old Reliable….
Ah-hah, 1.3×1020 meters³/second so Lonesome‘s is also near 1020….
A solar‑mass black hole’s half‑diameter is about 3 kilometers so Lonesome‘s would be about 5×103 meters. Say we’re orbiting at twice that so r‘s around 104 meters. Put it together we get v2=1020/104=1016 so v=108 meters/sec….
Everything’s going a billion times faster than 10 eV….
So yeah, no problem getting charged dust particles out there next to Lonesome….

Just look at the color in that tree…
Weird when you think about it. The really good color is summertime chlorophyll green when the trees are soaking up sunlight and turning CO2 into oxygen for us but people get excited about dying leaves that are red or yellow…

Well, now. Lonesome‘s Event Horizon is the no-going-back point on the way to its central singularity which we call infinity because its physics are beyond anything we know. I’ve just closed out another decade of my life, another Event Horizon on my own one‑way path to a singularity…

Hey! Mr Feder! Come ask me a question to get me out of this mood.

Author’s note — Yes, ambient radiation in Lonesome‘s immediate vicinity probably would account for far more ionization than physical impact, but this was a nice exercise in estimation and playing with exponents and applied physical principles.

~~ Rich Olcott

The Solid Gold Bath Towel

On By rolcottIn Physics: Money, UncategorizedLeave a comment

“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

Supply, Demand And Friction

On By rolcottIn Physics: Money, UncategorizedLeave a comment

<chirp, chirp> “Moire here. Open for business on a reduced schedule.”

“Hiya, Sy. It’s Eddie, taking orders for tonight’s pizza deliveries. My 6:15 wave is full-up, can I schedule you for 6:45? Whaddaya like tonight?”

“Yeah, a little later’s OK, Eddie. Mmmm, I think a stromboli this time. Rolled up like that, it ought to stay hot longer.”

“Good idea. Hey, I been thinking about that ‘velocity of money‘ thing and the forces that change where it goes. Isn’t that just another name for ‘supply and demand’? Bad weather messes up the wheat crop, I gotta pay more for pizza flour, that kinda thing.”

“That’s one of the oldest theories in economics, the idea that low supply increases prices and conversely. Economists often use two hyperbolas to describe the trade-off. Unfortunately, the idea’s only sorta true and only for certain markets. Oh, and it’s only sorta related to how fast money flows through the economy.”

“C’mon, Sy, you’re talking to a professional here. I watch my costs pretty close. Supply-demand tells my story — a bad tomato harvest drives my red sauce price through the roof.”

“No question it works for some products where there’s many independent buyers, many independent sellers, everyone has the same information, and a few other technical only-ifs. It’s what they call a perfect market. How many different companies do you buy flour from?”

“Three or four in town here. I switch around. Keeps ’em on their toes and holds their price down.”

“Competition’s a good thing, right? No buyer pays more than they absolutely must and no seller takes less than their competition does. Negative feedback all over the place. If one vendor figures out an advantage and can make money selling the same stuff for a lower price, everyone else copies them and the market price settles into a new lower equilibrium and there’s no advantage any more.”

“Yeah, that’s the way it works for flour.”

“And a few other commodities like grains and metals and West Texas crude. Economic theorists love the perfect-market model because it sets prices so nicely. Physicists love ideal cases, too — frictionless pulleys on infinitely sharp pivots, that kind of thing, where you can ignore the practical details. Most markets have lots of practical considerations that gum up the works.”

“Devil’s in the details, huh?”

“Sure. I seem to recall you’ve got a favorite sausage supplier.”

“Yeah, my brother-in-law Joey. OK, he’s family, but he does good work — fresh meat ground exactly the way I want it, got a good nose for spices, dependable delivery, what’s not to like?”

“Is he more expensive?””

“A little, a little, but it’s worth it.”

“So hereabouts there’s an imperfect market for sausage. The economists might tally Joey’s extra profit from that premium price to an accounting column labeled ‘Goodwill.’ A physicist would have another name for it.”

“Goodwill. Joey’d like that. So what would the physicist call it?”

“Real mechanical systems are never perfectly energy‑efficient. Energy is always lost to friction. In my money‑physics framework money’s lost to friction. It’s the reason you pay a premium above what would be perfect‑market price for sausage. Nothing wrong with that so long as you know you’re doing it and why. Most real markets are loaded with friction of various sorts. Think of market regulators as mechanics, running around with oil cans as they reduce inefficiency and friction.”

“What other frictions … lemme think. Monopoly, for sure — some big chain takes over my market, drives me the rest of the way out of business and then they can charge whatever they want. Umm … collusion, either direction. Advertising, maybe, but that’s mostly legal.”

“You got the idea. So, how is business?”

“Are you kidding? Way off. I had to lay off people, now it’s just me baking and delivering.”

“Would you buy flour these days at the usual price?”

“Nah. At the rate things are going what I got will last me for a l‑o‑o‑n‑g time. I got no place to put any more.”

“Your customers aren’t buying, you’re not buying, money’s not changing hands. The velocity of money’s so low that supply-demand isn’t capable of setting price. That’s deflation, not market friction.”

“Either way, it hurts.”

~~ Rich Olcott