Tag: Inertial frame

Tropical beach with palm trees next to icy polar region with glaciers.

The Big Water Pump

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment

“Springtime! Could you make me a lilac latte, Cal?”

“Maybe if I left out the coffee, Cathleen. Lilac’s too delicate to stand up to coffee’s punch. How about a cold brew of light roast? I just made a batch. Plenty of caffeine in there, not too much intensity and you can imagine the flowery part.”

“I’ll have that, and a lemon scone, please.”

“Here you go, fresh from the filter. Hey, you sure lit a fire under Sy. He’s done a whole string of posts about Coriolois Effects.”

“Tsk, Cal, the scientist’s name was Coriolis. I’m not surprised there’s been multiple posts — the same pseudo‑force shows up in many ways.”

“Pseudo‑force?”

<looks around> “Good, Sy’s not here. He’d talk our ears off about inertial frames. My quick answer from a planet scientist perspective is that real forces are the ones that make things happen in systems where everything’s moving in straight lines at a steady pace.”

“Like on a pool table?”

“Mm-hm. You can generally make good predictions on systems like that, which is how pool sharks make their money. But if part of the system is accelerating in some way, maybe it’s rotating, you’ve got two choices for predicting how the system will behave. The hard way is to calculate each individual component’s motion in a single coordinate system using just the real forces. The easier way is to group components that have a common acceleration. Pick a convenient group to serve as your base subsystem. Define another subsystem for the components that all have the same acceleration relative to the first subsystem and so on. Then you pretend a pseudo‑force drives the interactions between your subsystems.”

“Like Earth and our Moon make a subsystem ’cause they orbit the Sun together and you said rotation’s a kind of acceleration. The pseudo‑force is centrifugal, fighting against the Sun’s gravity to keep Earth’s subsystem in orbit!”

“I love it when that kind of connection‑making happens in my classroom. Thank you, Cal.”

“You’re welcome. So your subsystems are what Sy calls frames?”

“Pretty much. Skipping some technical caveats, that’s the idea. When I think about atmosphere dynamics, I could try to calculate the planet’s whole atmosphere as an incredibly messy collection of atoms. I prefer to think of the Earth as a subsystem hosting some number of air mass subsystems, all embedded in the Universe system. The Universe enforces straight‑line inertia and the Earth adds rotational acceleration but the air masses are constrained to the planet’s spherical geometry. The Coriolis pseudo‑force summarizes all three effects. The calculation’s still messy, but it’s a lot more manageable. And then there’s water.”

“Water?”

“The piston that drives the climate. Water molecules are small so they move easily through the atmosphere. The important thing is, they’re good at transporting heat energy.”

“How’s that? They’d be the same temperature as everything else.”

“Temperature doesn’t always measure energy. Water molecules like to hold onto other water molecules. It takes energy to get them apart. When they get back together, the energy’s released so it’s like the freed‑up molecules store heat energy. In solid water, every molecule is locked into position. Melting a given mass of water amounts to breaking those locks. The liquid mass at freezing temperature contains more energy than the ice did. When liquid water evaporates, the gas contains even more energy, because the molecules can roam even more freely. Visualize a bucket of water someplace warm.”

“A Hawai’ian beach.”

“That bucketful absorbs heat energy as it evaporates, cooling the Pacific Ocean. Winds sweep up the gas and carry it north to the Arctic where it freezes. In the process it warms the ice cap by giving up its liquid‑to‑gas heat and also its solid‑to‑liquid heat. Water’s two active phase transitions make it a far more efficient heat transporter than dry air alone.”

“One bucket’s teeny in the ocean, though.”

“Multiply that by gazillions. We have gigatons of surface water. The evaporate/freeze/melt process cycles as the icecaps degrade, continuously acting to moderate Earth’s temperature differences. If Earth were dry, the gradient would be far steeper. Thermal gradients drive air movement. A dry Earth’s extreme temperature discrepancies would generate permanent gale‑force winds towards the poles.”

~ Rich Olcott

Atmospheric Jiu-jitsu

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment

A gorgeous early Spring day for a walk by the lake — blue sky, air just the right side of crisp, trees showing their young green leaves, geese goosily paddling around. As I pass the park bench I hear a familiar voice. “Hello, Mr Moire.”

“Morning, Walt. It’s been a while. What do your people want to know about now?”

“We’ve been reading your series of posts about the Coriolis Effect. You have masses of air pushing each other around, you have pendulums twisting about, and you have objects flying weird orbits around latitudes instead of the planet’s center. Which is the real Effect?”

“All three.”

“They’re so different. How can all three be right?”

“Knowing something of your interests, I can think of another Coriolis application will help clarify the connection. How do you steer an old‑fashioned artillery shell?”

“You don’t, you aim it.” <His eyes are looking inward.> “Those old howitzers, you traverse to the target’s coordinates, set the elevation for the distance and your munitions and whatever your barrel’s still good for, load ‘er up, let ‘er rip and wait for the forward observer to tell you how to adjust.”

“No correction for the Earth turning west‑to‑east beneath the shell’s trajectory?”

“No need. Inside a max 10‑mile range, the artillery, target and shell all share the same initial eastward vector. Windage and temperature inversions are more of a problem than Coriolis forces. That’s where judgement, feedback and reload speed come in.”

“Now stand that up against a cruise missile.”

“Very different situation. With cannons, all the propulsion happens at the start. That’s why they call it ballistic. Cruise missiles have an extended boost phase, maybe more than one, so they can do in‑flight steering. On the other hand, range is hundreds of miles or more so you do need to figure in relative easting.”

“And the easting correction takes power, right?”

“Of course.”

“From the missile’s point of view, that power goes to counteract the Coriolis force pushing it off‑course. You don’t see it from the ground but the missile does. Clearer now?”

“Give me a minute.” <sketches on his notepad> “Okay, counteracting attempt to deflect course — got it. Hmm, a pendulum’s even simpler, because it’s not trying to keep in sync with the Earth’s rotation. No forces in play crosswise to the swing plane so it can maintain orientation relative to the Universe. To museum visitors it looks like something’s twisting, but it’s us doing the moving. The air masses, though … forces are in play with that one.”

“It’s always important to keep track of who’s doing what to whom. That system has four distinct frames of reference: the Earth, a moving air mass, the air mass it collides with, and the Universe.”

“The Universe?”

“Sets the stage for Newton’s First Law, about conservation of linear momentum. Say there’s an air mass hovering over Dallas, latitude 30° north. Relative to the Earth it’s stationary, but relative to the Sun and the rest of the Universe it has an eastward vector clocking 1450 km/hour. Now suppose that mass moves north relative to the Earth.”

“But there’s already an airmass taking up space there, say in Manitoba. There’ll be a collision, northbound momentum against Manitoban inertia.”

“Here’s where Coriolis gets into the game. Manitoba may have zero motion relative to Earth, but Manitoba and its air mass are also moving eastward relative to the Universe. Manitoba’s speed is slower than Dallas’ but it’s not zero. Manitoba’s momentum deflects the Dallas mass into an even more easterly vector.”

“You’re saying that Coriolis plays jiu‑jitsu with the atmosphere.”

“I wouldn’t have come up with that interpretation, but it’s reasonable.”

“What about the weird orbits?”

“Not really orbits, more like equilibrium bands. The concept comes from the theoretical notion that every latitude along a meridian has a natural equilibrium speed where air pressure balances other forces. The bands would be parallel circles around the globe except for geography and transient disturbances. Dallas’ 1450‑km/h number was an example. If you exceed your local natural speed, centrifugal force moves you towards the Equator; if you’re a slowpoke, you’re shoved towards the nearest pole. Real weather’s more complicated.”

“Everything’s always more complicated.”

~ Rich Olcott

The Polar Expression

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, Uncategorized2 Comments

“Good afternoon, Mr … Moire, yes?”

“The same. Can I help you?”

“Yes. I am Tomas Frashko. I am new to this University. I could not help overhearing—”

“The whole neighborhood couldn’t help overhearing.”

“Mmm, yes. My sympathy. But I have some questions, if you have a moment.”

“My coffee mug’s not empty yet. Please sit down. I’ll help if I can.”

“Thank you. I have often seen the Coriolis Effect explained as an atmospheric effect — northbound air with high‑speed low‑latitude momentum deflected eastward by slower‑moving air already at higher latitudes. The last part of your recent post goes to some trouble to avoid that explanation. Why is that?”

“Because the Effect doesn’t only play with the atmosphere. It drives gyre currents in the oceans and probably the magma flows deep inside Earth’s mantle.”

“So fluids, not just air. But it is still a matter of fluid with a velocity in one direction being diverted by fluid with a different velocity. Also, these cases are planet‑scale effects operating over large distances. Surely systems at small scale do not experience a measurable amount of Coriolis force.”

“But they do. Museum Foucault pendulums swing on a scale measured in meters. There’s dozens of them on display all over the world, they act just as Coriolis’ ideas predicted, and the host institutions go to a great deal of trouble to ensure the steady swinging isn’t disturbed by rushing air.”

“Ah, yes. I have seen the pendulum exhibit in our museum in the city where I grew up. A hypnotic thing, swinging back and forth on its wire, each swing a little closer to knocking down a pin … finally! Then slowly turning direction to knock down another one. The museum docent said the plane of the pendulum’s swing pivots to demonstrate Earth’s rotation, but then she mentioned that the full circle takes more than a day to complete. She couldn’t explain why.”

“If it were swinging from a point above the North or South Pole it would be a one-day completion, 15 arcseconds per second.. Scientists tried mounting one at the South Pole and that’s exactly what they determined. The poles are the only points on Earth’s surface where the the pendulum’s inertial frame matches Earth’s so it looks like the Earth is simply turning beneath the pendulum. On the other hand, along the Equator the Coriolis force doesn’t affect a pendulum’s motion at all.”

“Not at all?”

“Nope. Centrifugal force, a little bit, but not Coriolis force.”

“Does the one become the other?”

“Oh no, they’re quite different. Centrifugal force represents competition between dissimilarly rotating frames; Coriolis force represents their coupling. If you’re riding on a merry‑go‑round—”

“A what?”

“Mm, you’d probably call it a carousel.”

“Ah. Yes, go on.”

“If you’re riding on a carousel, your straight‑line inertia in the fairgrounds frame tries to drive you forward. To stay in position on the rotating carousel, you fight that outward inertial impetus by holding onto something. In the ride’s rotating frame, that looks like you’re exerting centripetal force to counterbalance a centrifugal force that the fairgrounds frame doesn’t see.”

“Yes, yes, but how does that differ from Coriolis force?”

Cycle time = 24 hours / Sin(latitude)

“Centrifugal force depends on an object’s distance from the center of rotation. Coriolis force doesn’t care about that. It rises with the sine of the angle between the object’s vector and the axis of rotation. On a sphere the relevant angle is the latitude. A northbound object, could be a pendulum bob, arrives at the North Pole traveling perpendicular to the Earth’s axis. Perpendicular angles have the maximum sine, 1.0. The Coriolis coupling is strongest there and that’s why a pendulum’s reference frame is locked to the Earth’s 24‑hour period. At the equator a northbound object moves parallel to the polar axis. Parallel lines have zero angle with zero sine so the Coriolis coupling’s zero. A pendulum’s plane of motion at the equator stays where it started, infinite precession completion time.”

“And in‑between?”

“In between. A pendulum’s cycle would run 27.7 hours in Helsinki, more than 60 hours at the Tropic of Cancer.”

“Small coupling, not much swerving.”

~ Rich Olcott

  • Thanks to Ric Werme for his thoughtful comments and suggestions.

Directional Reset

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, Uncategorized1 Comment

Professor of Astronomy Cathleen O’Meara barges into Cal’s Coffee Shop. “There you are, Sy Moire! You numbskull! You addlepate! You … nincompoop! “

We’ve known each other since we were kids but I’ve rarely seen her this angry. “What have I done this time, Cathleen? I apologize, but what for?”

“That last post you put up. One of the hardest things to get across to planet science students is the Coriolis Effect. You got it exactly backwards, you lummox! Confused the be-jeepers out of half my students and it’s going to take a whole class period to unwind it.”

All those exclamation points sting when they strike home. “It did feel funny. All the sources I checked said Coriolis skews travel to the right in the northern hemisphere but I worked hard for hours on that video and it clearly shows ‘left‘.”

<sniff> “Stupid waste of time, chump! That video doesn’t show Coriolis.” <she grabs one of Cal’s graph-paper napkins and starts sketching> “Your balloon or whatever isn’t traveling north along Earth’s surface. It’s going out into space. That dark line tracks the thing’s shadow, or it would if you had the Sun behind it instead of off to the side. It has nothing at all to do with the cloud stream at the top of the hurricane and by the way those winds in the picture are outward, not inward as you’d’ve known if you’d’ve thought about for even a moment, blockhead! Here, look at a sideways view.”

“You’re saying my balloon’s not following the surface, it’s vectored away from the surface parallel to the north‑south axis. Also that the shadow points that I plotted on Earth trend westward only because the Earth turns west‑to‑east underneath the balloon. … Okay, I can see that. Goes so high up I guess it can’t be a balloon, huh?”

“Don’t try to deflect the conversation, nitwit. Figure out what you got wrong and put up a correction post that gives a proper account of Coriolis. Sorry, Cal, I’ll need my coffee in a sippy‑cup. Gotta go revise my lesson plan, again.”

She grabs her caffeine to‑go, flings me a final “Dolt! ” and storms out the door trailing a cloud of grumbles.

Vinnie’s open-mouthed. “Geez, Sy, she does have a temper.”

“You know it, Vinnie. Fortunately she saves it up for deserving occasions but don’t ever get her started on politics. So let’s see, what part of what I posted did I get right?”

“Well, there’s the part about Helsinki’s rotation around the Earth runs fewer kilometers per hour than Quito’s. That’s just fact, can’t argue with it.”

“Yeah, Mr Moire, and there’s Conservation of Momentum.”

“Right, Jeremy.” Synapses connect in my head. “Got it! Vinnie, what’s the rule between speed and orbit size?”

“The closer the faster. The Moon’s a quarter‑million miles away, takes a month to go round the Earth; the ESS is 250 miles up, circles us every 90 minutes. If you’re in some orbit and wanna go lower, you gotta speed up. Took me an hour to convince Larry that’s the way it works. He was all about centrifugal force forcing you outward, but if you want to get deeper in the gravity well you need the extra speed to balance the extra gravity.”

“That’s the rule for space orbits, alright, but things work exactly the opposite for travel on the surface of a rotating sphere. Gravity pulls centerward with the same strength everywhere so gravity’s not what balances the centrifugal force.”

“What does?”

“Geometry. In space orbits, velocity and kinetic energy increase toward the core. On a sphere’s surface, the highest velocity is farthest away from the rotational axis, at the equator. Velocity falls off to zero at both poles. Every latitude has its characteristic velocity and kinetic energy. Suppose you’re loose on Earth’s northern hemisphere and moving east too fast for your latitude. You’ll drift southward, away from the axis, until you hit a latitude that matches your speed. Meanwhile, because you’re moving east the landscape will flow westward beneath you. The blend is the Coriolis Effect.”

“So if I’m slower than my latitude I drift north and Coriolis sends me east?”

“Cathleen would agree, Jeremy.”

~ Rich Olcott

When It’s Not The Same Frame – Never Mind

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment
  • Author‘s note — Please ignore everything below the separator line. It’s bogus. No excuses, it’s just wrong. I intend to embarrass Vinnie and Sy just as soon as I get my head straight. My apologies to every reader, especially teachers, that I’ve confused.

“Hey, Sy, I couldn’t help overhearing—”

<chuckle> “Of course not, Cal. Overhearing what?”

“When you said Quito goes round the world twice as fast as Helsinki. That can’t be true! Things would collide and we’d get all kinds of earthquakes and stuff.”

“Well, sure, Cal, if those two airports moved relative to each other. But they don’t, they’re stuck 10750 kilometers apart just like they’ve always been. I hated flying that route. Mountains to dodge at both ends, in between there’s bad weather a lot of the time and no place good to set down if something goes wrong. … Wait — different speeds — it’s frames again, ain’t it, Sy?”

“Exactly, Vinnie, even though it’s not black holes for a change. Relative to an inertial frame on the Earth’s surface, the Earth itself doesn’t move and neither does either city. Relative to a Sun‑centered frame, though, the Earth spins on its axis once every 24 hours. In the Sun’s frame, Quito on Earth’s 40‑thousand kilometer Equator does 1666 kilometers per hour. Helsinki’s at 60° North. Its circle around the spin axis is only 20 thousand kilometers so its linear speed is 833 kilometers per hour even though it does the same 15 degrees per hour that Quito does.”

“Hi, Mr Moire. Welcome back. I couldn’t help overhearing—”

<chuckle> “Of course not, Jeremy. Overhearing what?”

“You talking about places on Earth moving different speeds. We just studied about that in Dr O’Meara’s planet science class but it’s still loose in my head. It has to do with why storms go counterclockwise, right?”

“It has everything to do with that, except the counterclockwise storms are only in the northern hemisphere. Southern hemisphere storms rotate the other way.”

“I got this, Sy. Bring up that movie you got on Old Reliable, the one that shows the northern hemisphere. Yeah, that one. Jeremy, some guy in a balloon is the dark line on his way from Kansas to the North Pole to meet Santa. In his frame the earth is moving left‑to‑right relative to his northbound course. See how the red star’s moving?”

“Yeah, it’s moving towards sunrise so his movie’s got the rotation right. Why Kansas?”

“‘Cause he’s got a good long shot over flatlands before any mountains or big lakes get in the way, okay? So, the other section of Sy’s movie is like it was shot from a satellite in geostationary orbit. In its frame the Earth is standing still, but the balloon guy’s swerving to his left which is west. Also counterclockwise.”

“Mmm, okay. So you’re saying that in our earthbound frame we see northerly winds getting twisted to their left which is west but it’s really the Earth turning under the atmosphere and that’s why hurricanes turn the way they do.”

“There are other ways to analyze it, guys.”

“Like what, Sy?”

“Let’s get back to Quito and Helsinki. In the northern hemisphere the latitude lines make shorter circles as you go north so your distance traveled per day gets smaller.”

“Makes sense, yeah.”

“Right. Your balloon guy’s at rest somewhere in the Earth’s frame before he starts his trip so the satellite sees him traveling eastward at say 1200 kilometers per hour. The atmosphere around him is doing about the same. Suppose he suddenly moves a few hundred kilometers north where the atmosphere’s moving significantly slower but he still has his original eastward momentum. What happens?”

“He gets slowed down.”

“Why?”

“Drag from the slower air. He dumps some of his momentum to the air molecules.”

“Conservation of Momentum does apply, Vinnie. That’s an explanation I see a lot in the pop‑sci press, but I’m not happy with it. An astronaut in a shuttlecraft going point‑to‑point across the airless Moon would see the same between‑frames contrast.”

“Oh! Newton’s First Law says you can’t change momentum unless an external force acts on you. So that’s the Coriolis Force, Mr Moire?”

“It’s related, Jeremy. Gravity restricts planet‑bound travelers to surface motion. Geometry and the force of gravity give that westward push in the planet’s frame to northbound objects in the northern hemisphere. The balloon guy and the astronaut don’t observe the Coriolis Effect unless they look out the window.”

~ Rich Olcott

When It’s Not The Same Frame

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment
  • Author‘s note — Please ignore everything below the separator line. It’s bogus. No excuses, it’s just wrong. I intend to embarrass Vinnie and Sy just as soon as I get my head straight. My apologies to every reader, especially teachers, that I’ve confused.

“Hey, Sy, I couldn’t help overhearing—”

<chuckle> “Of course not, Cal. Overhearing what?”

“When you said Quito goes round the world twice as fast as Helsinki. That can’t be true! Things would collide and we’d get all kinds of earthquakes and stuff.”

“Well, sure, Cal, if those two airports moved relative to each other. But they don’t, they’re stuck 10750 kilometers apart just like they’ve always been. I hated flying that route. Mountains to dodge at both ends, in between there’s bad weather a lot of the time and no place good to set down if something goes wrong. … Wait — different speeds — it’s frames again, ain’t it, Sy?”

“Exactly, Vinnie, even though it’s not black holes for a change. Relative to an inertial frame on the Earth’s surface, the Earth itself doesn’t move and neither does either city. Relative to a Sun‑centered frame, though, the Earth spins on its axis once every 24 hours. In the Sun’s frame, Quito on Earth’s 40‑thousand kilometer Equator does 1666 kilometers per hour. Helsinki’s at 60° North. Its circle around the spin axis is only 20 thousand kilometers so its linear speed is 833 kilometers per hour even though it does the same 15 degrees per hour that Quito does.”

“Hi, Mr Moire. Welcome back. I couldn’t help overhearing—”

<chuckle> “Of course not, Jeremy. Overhearing what?”

“You talking about places on Earth moving different speeds. We just studied about that in Dr O’Meara’s planet science class but it’s still loose in my head. It has to do with why storms go counterclockwise, right?”

“It has everything to do with that, except the counterclockwise storms are only in the northern hemisphere. Southern hemisphere storms rotate the other way.”

“I got this, Sy. Bring up that movie you got on Old Reliable, the one that shows the northern hemisphere. Yeah, that one. Jeremy, some guy in a balloon is the dark line on his way from Kansas to the North Pole to meet Santa. In his frame the earth is moving left‑to‑right relative to his northbound course. See how the red star’s moving?”

“Yeah, it’s moving towards sunrise so his movie’s got the rotation right. Why Kansas?”

“‘Cause he’s got a good long shot over flatlands before any mountains or big lakes get in the way, okay? So, the other section of Sy’s movie is like it was shot from a satellite in geostationary orbit. In its frame the Earth is standing still, but the balloon guy’s swerving to his left which is west. Also counterclockwise.”

“Mmm, okay. So you’re saying that in our earthbound frame we see northerly winds getting twisted to their left which is west but it’s really the Earth turning under the atmosphere and that’s why hurricanes turn the way they do.”

“There are other ways to analyze it, guys.”

“Like what, Sy?”

“Let’s get back to Quito and Helsinki. In the northern hemisphere the latitude lines make shorter circles as you go north so your distance traveled per day gets smaller.”

“Makes sense, yeah.”

“Right. Your balloon guy’s at rest somewhere in the Earth’s frame before he starts his trip so the satellite sees him traveling eastward at say 1200 kilometers per hour. The atmosphere around him is doing about the same. Suppose he suddenly moves a few hundred kilometers north where the atmosphere’s moving significantly slower but he still has his original eastward momentum. What happens?”

“He gets slowed down.”

“Why?”

“Drag from the slower air. He dumps some of his momentum to the air molecules.”

“Conservation of Momentum does apply, Vinnie. That’s an explanation I see a lot in the pop‑sci press, but I’m not happy with it. An astronaut in a shuttlecraft going point‑to‑point across the airless Moon would see the same between‑frames contrast.”

“Oh! Newton’s First Law says you can’t change momentum unless an external force acts on you. So that’s the Coriolis Force, Mr Moire?”

“It’s related, Jeremy. Gravity restricts planet‑bound travelers to surface motion. Geometry and the force of gravity give that westward push in the planet’s frame to northbound objects in the northern hemisphere. The balloon guy and the astronaut don’t observe the Coriolis Effect unless they look out the window.”

~ Rich Olcott

A No-Charge Transaction

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

“I ain’t done yet, Sy. I got another reason for Dark Matter being made of faster‑then‑light tachyons.”

“I’m still listening, Vinnie.”

“Dark Matter gotta be electrically neutral, right, otherwise it’d do stuff with light and that doesn’t happen. I say tachyons gotta be neutral.”

“Why so?”

“Stands to reason. Suppose tachyons started off as charged particles. The electric force pushes and pulls on charges hugely stronger than gravity pulls—”

“1036 times stronger at any given distance.”

“Yeah, so right off the bat charged tachyons either pair up real quick or they fly away from the slower‑than‑light bradyon neighborhood leaving only neutral tachyons behind for us bradyon slowpokes to look at.”

“But we’ve got un‑neutral bradyon matter all around us — electrons trapped in Earth’s Van Allen Belt and Jupiter’s radiation belts, for example, and positive and negative plasma ions in the solar wind. Couldn’t your neutral tachyons get ionized?”

“Probably not much. Remember, tachyon particles whiz past each other too fast to collect into a star and do fusion stuff so there’s nobody to generate tachyonic super‑high‑energy radiation that makes tachyon ions. No ionized winds either. If a neutral tachyon collides with even a high-energy bradyon, the tachyon carries so much kinetic energy that the bradyon takes the damage rather than ionize the tachyon. Dark Matter and neutral tachyons both don’t do electromagnetic stuff so Dark Matter’s made of tachyons.”

“Ingenious, but you missed something way back in your initial assumptions.”

“Which assumption? Show me.”

“You assumed that tachyon mass works the same way that bradyon mass does. The math says it doesn’t.” <grabbing scratch paper for scribbling> “Whoa, don’t panic, just two simple equations. The first relates an object’s total energy E to its rest mass m and its momentum p and lightspeed c.”

E² = (mc²)² + (pc)²

“I recognize the mc² part, that’s from Einstein’s Equation, but what’s the second piece and why square everything again?”

“The keyword is rest mass.”

“Geez, it’s frames again?”

“Mm‑hm. The (mc²)² term is about mass‑energy strictly within the object’s own inertial frame where its momentum is zero. Einstein’s famous E=mc² covers that special case. The (pc)² term is about the object’s kinetic energy relative to some other‑frame observer with relative momentum p. When kinetic energy is comparable to rest‑mass energy you’re in relativity territory and can’t just add the two together. The sum‑of‑squares form makes the arithmetic work when two observers compare notes. Can I go on?”

“I’m still waitin’ to hear about tachyons.”

“Almost there. If we start with that equation, expand momentum as mass times velocity and re‑arrange a little, you get this formula

E = mc² / √(1 – v²/c²)

The numerator is rest‑mass energy. The v²/c² measures relative kinetic energy. The Lorentz factor down in the denominator accounts for that. See, when velocity is zero the factor is 1.0 and you’ve got Einstein’s special case.”

“Give me a minute. … Okay. But when the velocity gets up to lightspeed the E number gets weird.”

“Which is why c is the upper threshold for bradyons. As the velocity relative to an observer approaches c, the Lorentz factor approaches zero, the fraction goes to infinity and so does the object’s energy that the observer measures.”

“Okay, here’s where the tachyons come in ’cause their v is bigger than c. … Wait, now the equation’s got the square root of a negative number. You can’t do that! What does that even mean?”

“It’s legal, when you’re careful, but interpretation gets tricky. A tachyon’s Lorentz factor contains √(–1) which makes it an imaginary number. However, we know that the calculated energy has to be a real number. That can only be true if the tachyon’s mass is also an imaginary number, because i/i=1.”

“What makes imaginary energy worse than imaginary mass?”

“Because energy’s always conserved. Real energy stays that way. Imaginary mass makes no sense in Newton’s physics but in quantum theory imaginary mass is simply unstable like a pencil balanced on its point. The least little jiggle and the tachyon shatters into real particles with real kinetic energy to burn. Tachyons disintegrating may have powered the Universe’s cosmic inflation right after the Big Bang — but they’re all gone now.”

“Another lovely theory shot down.”

~ Rich Olcott

Squaring The Circle

On By rolcottIn Cosmology, UncategorizedLeave a comment

Vinnie gives me the eye. “That crazy theory of yours is SO bogus, Sy, and there’s a coupla things you said we ain’t heard before.”

“What’s wrong with my Mach’s Principle of Time?”

“If the rest of the Universe is squirting one thing forward along Time, then everything’s squirting everything forward. No push‑back in the other direction. You might as well say that everything’s running away from the Big Bang.”

“That’s probably a better explanation. What are the couple of things?”

“One of them was, ‘geodesic,‘ as in ‘motion along a geodesic.‘ What’s a geodesic?”

“The shortest path between two points.”

“That’s a straight line, Mr Moire. First day in Geometry class.”

“True in Euclid’s era, Jeremy, but things have moved on since then. These days the phrase ‘shortest path’ defines ‘straight line’ rather than the other way around. Furthermore, the choice depends on how you define ‘shortest’. In Minkowski’s spacetime, for instance, do you mean ‘least distance’ or ‘least interval’?”

“How are those different?”

“The word ‘distance’ is a space‑only measurement. Minkowski plotted space in x,y,z terms just like Newton would have if he could’ve brought himself to use René Descartes’ cartesian coordinates. You know Euclid’s a²+b²=c² so you should have no problem calculating 3D distance as d=√(x²+y²+z²).”

“That makes sense. So what’s ‘interval’ about then?”

“Time has entered the picture. In Minkowski’s framework you handle two ‘events’ that may be at different locations and different times by using what he called the ‘interval,’ s. It measures the path between events as
s=√[(x²+y²+z²)–(ct)²]. Usually we avoid the square root sign and work with s².”

“That minus sign looks weird. Where’d it come from?”

“When Minkowski was designing his spacetime, he needed a time scale that could be combined with the x,y,z lengths but was perpendicular to each of them. Multiplying time by lightspeed c gave a length, but it wasn’t perpendicular. He could get that if he multiplied by i=√(–1) to get cti as a partner for x,y,z. Fortunately, that forced the minus sign into the sum‑of‑squares
(x²+y²+z²)–(ct)² formula.”

Vinnie’s getting impatient. “What is an actual geodesic, who cares about them, and what do these equations have to do with anything?”

“A geodesic is a path in spacetime. Light always travels along a geodesic. The modern version of Newton’s First Law says that any object not subject to an outside force travels along a geodesic. By definition the geodesic is the shortest path, but you can’t select which path from A to B is the shortest unless you can measure or calculate them. There’s math to tell us how to do that. Time’s a given in a Newtonian Universe, not a coordinate, so geodesics are distance‑only. We calculate d along paths that Euclid would recognize as straight lines. That’s why the First Law is usually stated in terms of straight lines.”

“So the lines can go all curvy?”

“Depends, Vinnie. When you’re piloting an over‑water flight, you fly a steady bearing, right?”

“Whenever ATC and the weather lets me. It’s the shortest route.”

“So according to your instruments you’re flying a straight line. But if someone were tracking you from the ISS they’d say you’re flying along a Great Circle, the intersection of Earth’s surface with some planar surface. You prefer Great Circles because they’re shortest‑distance routes. That makes them geodesics for travel on a planetary surface. Each Circle’s a curve when viewed from off the surface.”

“Back to that minus sign, Mr Moire. Why was it fortunate?”

“It’s at the heart of Relativity Theory. The expression links space and time in opposite senses. It’s why space compression always comes along with time dilation.”

“Oh, like at an Event Horizon. Wait, can’t that s²=(x²+y²+z²)–(ct)² arithmetic come out zero or even negative? What would those even mean?”

“The theory covers all three possibilities. If the sum is zero, then the distance between the two events exactly matches the time it would take light to travel between them. If the sum is positive the way I’ve written it then we say the geodesic is ‘spacelike’ because the distance exceeds light’s travel time. If it’s negative we’ve got a ‘timelike’ geodesic; A could signal B with time to spare.”

~ Rich Olcott

The Time Is Out of Joint

On By rolcottIn Cosmology, Crazy Theories, UncategorizedLeave a comment

Vinnie galumphs over to our table. “Hi, guys. Hey, Sy, I just read your Confluence post. I thought that we gave up on things happening simultaneous because of Einstein and relativity but I guess that wasn’t the reason.”

“Oh, things do happen simultaneously, no‑one claims they don’t, it’s just that it’s impossible for two widely‑separated observers to have evidence that two widely‑separated events happened simultaneously. That’s a very different proposition.”

“Ah, that makes me feel better. The ‘nothing is simultaneous‘ idea was making me itchy ’cause I know for sure that a good juggler lets go with one hand just as they’re catching with the other. How’s Einstein involved then?”

“Lightspeed’s a known constant. Knowing distance and lightspeed lets you calculate between‑event time, right? The key to simultaneity was understanding why lightspeed is a constant. We’d known lightspeed wasn’t infinite within the Solar System since Rømer’s time, but people doubted his number applied everywhere. Maxwell’s theory of electromagnetism derived lightspeed from the properties of space itself so it’s universal. Only in Newton’s Universe was it possible for two distant observers to agree that two also‑distant events were simultaneous.”

“Why was Newton’s Universe special?

“Space held still and didn’t bend. Astronomers A and B had a stable baseline between them. After measuring the baseline with light they could measure the angles each observed between the events. Some trigonometry let them send each other congratulatory messages on seeing a simultaneous pair of incidents. After Einstein’s work, they knew better.”

“It’s frames again, ain’t it?”

“Of course, Vinnie. A‘s frame is almost certainly moving relative to B‘s frame. Motion puts the Lorentz relativity factor into the game, making each astronomer’s clock run faster than the other’s. Worse, each astronomer sees that the other’s yardsticks are too short.”

Jeremy gives me a confused look.

“Space compression goes along with time dilation, Jeremy. Professor Hanneken will explain it all when your class gets to that unit. Bottom line, things can happen simultaneously in Einstein’s Universe, but no‑one can agree on which things.”

“Wait, if every frame has its own time‑rate, how can two spaceships rendezvous for an operation?”

“Good question, Jeremy. Einstein had an answer but complications hide under the covers. He suggested that A start a timer when sending a light pulse to a mirror at B. A waits for the reflection. B starts a timer when they see A‘s pulse. A measures the pulse’s round‑trip time. Each creates a clock that advances one tick for half of the round-trip time. B sets their clock back by one tick. That done, they agree to meet some number of ticks later.”

“Hmm… That should work, but you said there are complications.”

“There are always complications. For instance, suppose B is slingshotting around a black hole so that pulse and reflection travel different pathlengths. Or suppose one frame is rotating edge‑on to the other. In practice the ships would re‑sync repeatedly while approaching the rendezvous point.”

Vinnie erupts. “HAW! Successive approximation again!”

“Indeed. If we could extend the method to more than two participants we’d have a true Universal Coordinated Time.”

“Don’t we have that, Mr Moire? The Big Bang happened 14 billion years ago. Couldn’t we measure time from that?”

“Sort of. Last I looked the number was 13.787 plus‑or‑minus 20 million years. Too much slop for an instantaneous fleet‑level rendezvous like the final battle scene in StarTrek:Picard. But you’ve brought up an interesting question for a Crazy Theories seminar. One of Cosmology’s deepest unsolved questions is, ‘How does inertia work?’ Do you remember Newton’s First Law?”

<closes eyes> “In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.“

“Right. In other words, every object resists change to its current steady motion along a geodesic. Why is that? There’s no coherent, well‑founded, well‑tested theory. Einstein liked Mach’s Principle, which says inertia exists because every object is attracted through space to all the mass in the Universe. Suppose there’s a Mach’s Principle for Time, saying that objects squirt up the Time axis because they’re repelled by all the mass in the Universe.”

Vinnie hoots, “Bo-o-o-ohh-GUS!”

~ Rich Olcott

Look, Look Again, Then Think

On By rolcottIn Astronomy, UncategorizedLeave a comment

Cathleen and I are sharing scones and memories when Vinnie trundles up to our table. “Glad I got you two together. I just ran across a couple news items and I need some explanations.”

“Astronomy AND Physics in the same news items? Do tell.”

“They’re only one paragraph each and read like someone wrote ’em before their morning coffee. They’re both about that big black hole they’ve been taking pictures of.”

“The one in our galaxy or the M87* supermassive black hole in the Messier‑87 galaxy?”

“The second one, Cathleen. This item says it shot out a jet traveling faster than light.”

<sigh> “Pop‑sci journalism at its worst, right, Sy? I know the work that’s based on and the academic reports don’t say that. Good observations leading to less flamboyant conclusions.”

“Maybe it was supposed to be a bigger article but the editors cut it down badly. That happens. I’m sure it’s not really a superluminal jet—”

“Superluminal’s faster‑than‑light, right?”

“Right, Vinnie. Sorry to get technical. Anyway, it’s an illusion.”

“Ah geez, it’ll be frames again, right?” <eyes suddenly open wide> “Wait, I got it! I betcha it’s about the time difference. Take a blob in that jet, it’s flying out at near lightspeed. Time dilation happens when relativity’s in the game, me and Sy talked about that, so blob‑frame seconds look like they take longer than ours do. We see the blob cramming a lightsecond of distance traveled into less than one of our seconds and that’s superluminal. Am I right, Sy?”

“Right answer to a different question, I’m afraid. You’re straight on the time dilation but it doesn’t apply to this situation. Something happening within the blob’s frame, maybe a star blowing up or something weird metabolizing in there, Special Relativity’s time distortion hijinks would show us that action taking place in slow motion. But this superluminal blob claim hinges on how the blob’s whole frame moves relative to ours. That motion isn’t superluminal but it can look that way if conditions are right. As I understand it, the M87* jet qualifies. Your bailiwick rather than mine, Cathleen.”

“Actually it is a frames thing, Vinnie, but timeframes, not spacetime. Those blobs move too slowly in our sky to watch in real time. We take snapshot A and then maybe a few years later we take snapshot B and compare. Speed is the ratio of distance to time. We need the A‑B distance in 3‑D space to compare to the known time between snapshots. But we can’t see the blob’s trajectory in 3‑D. All we can capture is its 2‑D arc C‑B across an imaginary spherical shell we call the sky. If the M87* jet were perpendicular to our line of sight the C‑B image on the sky‑sphere would match the 3‑D path. Multiply the image’s angle in radians by the distance to M87* and we’re done.”

“We’re not done?”

“Nope. This jet points only 20° away from our direct line of sight. I’ll spare you the trigonometry and just say that distance A‑B is about 3 times longer than C‑B.”

“So we measure C‑B, triple the angle and multiply by the M87* distance. No problem.”

“Problem. That tripling is what makes the blob’s A‑B journey appear to go faster than light. Three times 0.4c equals 1.2c. But you missed something important. Your arithmetic assumed you could use a simple ‘M87* distance’. Not in this case, because the blob moves towards us at close to lightspeed. Visualize two concentric sky‑spheres. The outer one’s radius runs from us to the blob’s location at A‑time. The inner sphere’s radius runs to the blob’s location at B‑time. The B‑sphere is our reference frame. The light we saw at A‑time had to travel from the outer sphere to the inner one before we could register the C‑B image.”

“Can’t be very far.”

“We’re talking years at lightspeed, so lightyears, so significant. A properly illusion‑free A‑B travel calculation must include the A‑C travel time in the denominator of the distance/time ratio. The true kilometers per second come out well below lightspeed. Oh, and relativity’s not involved.”

“Dang, Cathleen, it was such a cool illusion.”

~ Rich Olcott