Category: Uncategorized

A Shot Through The Dark

On By rolcottIn Cosmology, General: Fun Stuff, UncategorizedLeave a comment

<THUNK!!> “Oh, dear. Is this the same elevator that you and Vinnie got trapped in, Sy?”

“Afraid so, Cathleen, but at least we had lights. This looks like a power outage, not a stuck door mechanism. Calling the building super probably won’t help. Hope you’re okay being stuck in the dark.”

“I’m an astronomer, Sy. A dark night’s my best thing. Remember the time we got locked with no light in my Mom’s closet?”

<chuckle> “Mm-hm. It was our pretend spaceship to Mars. We had no idea that closet had a catch we couldn’t reach. We were stuck there until your Mom came home. <sigh> We’ll have to wait ’til power comes back.”

<FZzzzzttPOP!!> … <then a voice like molten silver> “Oh, there you are, Sy! I’ve been looking all over for you. Who’s this?”

“Been a while, Anne. This is Cathleen. Cathleen, meet Anne. Anne’s an … explorer.”

“Ooo, where do you explore? For that matter, how did you get in here, and why is your dress (is it satin?) glowing like that?”

“Yes, it is satin, at the moment. It figures out whatever I need and makes that happen. It’s glowing because we’re in the dark.”

“I suspect your dress saved you when you met anti‑Anne.”

“Auntie Anne?”

“No, Cathleen, anti‑Anne, another me in the anti‑Universe. You might be right, Sy. It would have held anti‑Earth’s anti‑atoms away long enough for me to escape annihilation. Maybe I should explain.”

“I wish you would.”

“Wellll, I’ve got this super‑power for jumping across spacetime. Sy helped me calibrate my jumps and we even worked out how I can change size and use entropy to navigate between probabilities. So I explore everywhere and everywhen and that’s how I got into this elevator.” <brief fizzing sound> “Don’t worry, power will be back on soon but we’ve got time for Sy to explain my most recent experience.”

“Ah‑boy, now what?”

“Well, it seemed like a fun thing to do — go back to the earliest time I could, maybe even watch the Big Bang. I did some reading so I had an idea of what to expect as I dove down the time axis — gas clouds collapsing with glittering bursts of star formation, stars collecting into galaxies, galaxies streaming by like granular gas — so beautiful, especially because I can tweak my time rate and watch it all in motion!”

“And did you see all that?”

“Oh, yes, but then I hit a wall I couldn’t get past and I don’t understand why.”

“What were things like just before you hit the wall?”

“This was just beyond when I saw the very first stars turning on. There were vague clouds glowing here and there but basically the Universe became pitch black, no light at all for a while until the background started to glow with a very deep red just before I was blocked.”

“Ah. Cathleen, this is more your bailiwick than mine. Anne, Cathleen teaches Astronomy and Cosmology.”

“Just as a check, Anne, do you know exactly how far into the past you got?”

“Sorry, no. My time sense is pretty well calibrated for hours‑to‑centuries but this was billions of years. You probably know when I was better than I do.”

“On the evidence, I’d say you got 99.98% of the way back to your goal, nearly to the beginning of the Dark Age.”

“Dark Age? I’ve been there — 10th‑century Earth, bad times for everyone unless you were at the top of the heap but you wouldn’t stay there long. But I was too far out in space to see Earth. I couldn’t even pick out the Milky Way.”

“No, this was the Universe’s Dark Age, a couple hundred million years between when atoms formed and stars formed. Nothing could make new light. The Dark Age started at Big Bang plus 370 000 years when temperature cooled to 4000 K. The dark red you saw everywhere was atoms emitting blackbody radiation at 4000 K. Just 0.01% further into the past, the Universe was a billion‑degree quark plasma where not even atoms could survive. No wonder your dress wouldn’t let you enter.”

<THUNK!!> “Oh, good, power’s back on. We have light again!”

~ Rich Olcott

It’s All About The Coupling

On By rolcottIn UncategorizedLeave a comment

The game‘s over but there’s still pizza on the table so Eddie picks up the conversation. “So if gadolinoleum has even more unpaired electrons than iron, how come it’s not ferromagnetic like iron is?”

Vinnie’s tidying up the chips he just won. “I bet I know part of it, Eddie. Sy and me, we talked about magnetic domains some years ago. If I remember right, each iron atom in a chunk is a tiny little magnet, which I guess is the fault of its five unpaired electrons, but usually the atom magnets are pointing in all different directions so they all average out and the whole chunk doesn’t have a field. If you stroke the chunk with a magnet, that collects the little magnets into domains and the whole thing gets magnetic. How come gadomonium” <winks at Eddie, Eddie winks back> “doesn’t play the domain game, Susan?”

“It’s gadolinium, boys, please. As to the why, part’s at the atom level and part’s higher up. My lab neighbor Tammy schooled me on rare earth magnetism just last week. She does high‑temperature solid state chemistry with lanthanide‑containing materials. Anyway, she says it’s all about coupling.”

“I hope she told you more than that.”

The angular portion of the lowest-energy spherical harmonic modes
Credit: Inigo.quilez, under CCA SA 3.0 license

“She did. Say you’ve got a single gadolinium atom floating in space. Its environment is spherically symmetrical, no special direction to organize the wave‑orbitals hosting unpaired charges. Now turn on a magnetic field to tell the atom which way is up, call that the z‑axis. The atom’s wave‑orbital with zero angular momentum orients along z. Six more wave‑orbitals with non‑zero angular momentum spin one way or the other at various angles to the z‑axis. Those charges in motion build the atom’s personal magnetic field.”

“But we’re on Earth, not in space.”

“Bear with me. First, as a chemist I must say that most of the transition and lanthanide elements happily lose two electrons so in general we’re dealing with ions. Before you ask, Vinnie, that goes even for metals where the ions float in an electron sea. When Tammy said ‘coupling’ she was talking about how strongly one ion feels the neighboring fields. Iron and other ferromagnetic materials have a strong coupling, much stronger than the paramagnetics do.”

“Why’s the ferro- coupling so much stronger?”

“Two effects. You can read both of them right off the Periodic Table. Physical size, for one. Each row down in the table represents one electronic shell which takes up space. The atom or ion in any row is bigger than the ones above it. Yes, the heavy elements have more nuclear charge to pull electronic charge close, but shielding from their completed lower shells lets the outer charge cloud expand. Tammy told me that gadolinium’s ions are about 20% wider than iron’s.”

“Makes sense — you make the ions get further apart, they won’t connect so good. What’s the other effect?”

“It’s about how each orbital distributes its charge. There are tradeoffs between shell number, angular momentum and distance from the nucleus. Unpaired charge concentration in gadolinium’s high‑momentum 4f‑orbitals on the average stays inside of all its 3‑shell waves. The outermost charge shelters the unpaired waves inside it. That weakens magnetic coupling with unpaired charge in neighboring ions. Bottom line — gadolinium and its cousins are paramagnetic because they’re bigger and less sensitive than ferromagnetic iron is.”

“Then how come rare earth supermagnets the Chinese make are better than the cheapie ironic kinds we can make here?”

“The key is getting the right atoms into the right places in a crystalline solid. Neodymium magnets, for instance, have clusters of iron atoms around each lanthanide. The cluster arrangement aligns everyone’s z‑axes letting the unpaired charges gang up big‑time. You find materials like that mostly by luck and persistence. Tammy’s best samples are multi‑element oxides that arrange themselves in planar layers. Pick a component just 1% off the ideal size or cook your mixture with the wrong temperature sequence and the structure has completely different properties. Chinese scientists worked decades to perfect their recipes. USA chose to starve research in that area.”

~ Rich Olcott

Flipping An Edge Case

On By rolcottIn Chemistry, Physics: Quantum Mechanics, UncategorizedLeave a comment

“Why’s the Ag box look weird in your chart, Susan?”

“That’s silver, Eddie. It’s an edge case. The pure metal’s diamagnetic. If you alloy silver with even a small amount of iron, the mixture is paramagnetic. How that works isn’t my field. Sy, it’s your turn to bet and explain.”

I match Eddie’s bet (the hand’s not over). “It’s magnetism and angular momentum and how atoms work, and there are parts I can’t explain. Even Feynman couldn’t explain some of it. Vinnie, what do you remember about electromagnetic waves?”

“Electric part pushes electrons up and down, magnetic part twists ’em sideways.”

“Good enough, but as Newton said, action begets reaction. Two centuries ago, Ørsted discovered that electrons moving along a wire create a magnetic field. Moving charges always do that. The effect doesn’t even depend on wires — auroras, fusion reactor and solar plasmas display all sorts of magnetic phenomena.”

“You said it’s about how atoms work.”

“Yes, I did. Atoms don’t follow Newton’s rules because electrons aren’t bouncing balls like those school‑book pictures show. An electron’s only a particle when it hits something and stops; otherwise it’s a wave. The moving wave carries charge so it generates a magnetic field proportional to the wave’s momentum. With me?”

“Keep going.”

“That picture’s fine for a wave traveling through space, but in an atom all the charge waves circle the nucleus. Linear momentum in open space becomes angular momentum around the core. If every wave in an atom went in the same direction it’d look like an electron donut generating a good strong dipolar magnetic field coming up through the hole.”

“You said ‘if’.”

“Yes, because they don’t do that. I’m way over‑simplifying here but you can think of the waves pairing up, two single‑electron waves going in opposite directions.”

“If they do that, the magnetism cancels.”

“Mm‑hm. Paired‑up configurations are almost always the energy‑preferred ones. An external magnetic field has trouble penetrating those structures. They push the field away so we classify them as diamagnetic. The gray elements in Susan’s chart are almost exclusively in paired‑up configurations, whether as pure elements or in compounds.”

“Okay, so what about all those paramagnetic elements?”

The angular portion of the lowest-energy spherical harmonic modes
Credit: Inigo.quilez, under CCA SA 3.0 license

“Here’s where we get into atom structure. An atom’s electron cloud is described by spherical harmonic modes we call orbitals, with different energy levels and different amounts of angular momentum — more complex shapes have more momentum. Any orbital hosting an unpaired charge has uncanceled angular momentum. Two kinds of angular momentum, actually — orbital momentum and spin momentum.”

“Wait, how can a wave spin?”

“Hard to visualize, right? Experiments show that an electron carries a dipolar magnetic field just like a spinning charge nubbin would. That’s the part that Feynman couldn’t explain without math. A charge wave with spin and orbital angular momentum is charge in motion; it generates a magnetic field just like current through a wire does. The math makes good predictions but it’s not something that everyday experience prepares us for. Anyway, the green and yellow‑orange‑ish elements feature unpaired electrons in high‑momentum orbitals buried deep in the atom’s charge cloud.”

“So what?”

“So when an external magnetic field comes along, the atom’s unpaired electrons join the party. They orient their fields parallel to the external field, in effect allowing it to penetrate. That qualifies the atom as paramagnetic. More unpaired electrons means stronger interaction, which is why iron goes beyond paramagnetic to ferromagnetic.”

“How does iron have so many?”

“Iron’s halfway across its row of ten transition metals—”

“I know where you’re going with this, Sy. It’ll help to say that these elements tend to lose their outer electrons. Scandium over on the left ionizes to Sc3+ and has zero d‑electrons. Then you add one electron in a d orbital for each move to the right.”

“Thanks, Susan. Count ’em off, Vinnie. Five steps over to iron, five added d‑electrons, all unpaired. Gadolinium, down in the lanthanides, beats that with seven half‑filled f‑orbitals. That’s where the strength in rare earth magnets arises.”

“So unpaired electrons from iron flip alloyed silver paramagnetic?”

“Vinnie wins this pot.”

~ Rich Olcott

Was Ramses Pharaoh-magnetic?

On By rolcottIn Chemistry, UncategorizedLeave a comment

Kareem puts in another couple of chips. “Hold your horses, Cal. The conversation‘s just getting interesting.”

Vinnie raises him a few chips. “Hey, Mr Geology. Just how rare are these lanthanide rare earths? And if they’re metals, how come they’re called earths?”

Source

“Not that rare.” <pulls up an image on his phone> “Here’s a quick abundance chart for the lanthanides and a few other elements averaged over all of Earth’s continental crust. Cerium’s more abundant than copper and 350 times more common than lead. Of course, that’s an average. Lanthanide concentrations in economically viable ores are much higher, just like with copper, lead, tin and other important non‑ferrous metals.”

“Funny zig-zag pattern there.”

“Good catch, Cal. Even‑number elements are generally more abundant than their odd‑numbered neighbors. That’s the Oddo-Harkins Rule in action—”

“ODDo-Harkins, haw!”

“You’re—” <Susan’s catches Vinnie’s frown and quickly drops few chips onto the pile> “Sorry, Vinnie. You’re not the first person to flag that pun. Two meteorite chemists named Giuseppe Oddo and William Harkins developed the rule a century ago. We’re pretty sure the pattern has to do with how stars fuse even‑numbered alpha particles to build up the elements heavier than hydrogen and helium. As to why the rare earths are called earths, back when Chemistry was just splitting away from alchemy, an ‘earth‘ was any crumbly mineral. Anybody heard of diatomaceous earth?”

Cal perks up. “Yeah, I got a bag of that dust in my garden shed to kill off slugs.”

“Mm‑hm. Powdery, mostly silica with some clay and iron oxide. The original ‘earth’ definition eventually morphed to denote minerals that dissolve in acid” <grin> “which diatomaceous earth doesn’t do. A few favorable Scandinavian mines gave the Swedish chemists lanthanide‑enriched ores to work on. Strictly speaking, in metallic form the lanthanides are rare earth metals, not rare earths, but people get sloppy.”

Eddie pitches in some chips. “So they’re <snort> chemical odd‑ities. Why would anyone but a chemist care about them?”

<sigh> “Magnetism.” <shows her laptop’s screen> “Here’s a chart that highlights the elements that are most magnetically active. The lanthanides are that colored strip below the main table. Chemically they’d all fit into that box with the red circle. They’re—”

“Wait, there’s more than one kind of magnetism?”

“Oh, yes. The distinction’s about how an element or material interacts with an external magnetic field. Most elements are at least weakly paramagnetic, which means they’re pulled into the field; diamagnets push away from it. Diamagnetic reaction is generally far weaker. Manganese is the strongest paramagnet, about 70 times stronger per atom than the strongest diamagnet, bismuth. Then there’s iron, cobalt and nickel — they do ferromagnetism, which means their atoms interact so strongly with the field that they get their neighbors to join in and make a permanent magnet.”

Schematic of a Gouy Balance

“How does anyone find out whether the field’s pulling or pushing?”

“Good question, Cal (you owe the pot, by the way). Basically, the idea is to somehow weigh a sample both with and without a surrounding field. Tammy’s lab down the hall from me has a nice Gouy Balance setup which is one way to make that measurement. The balance stands on a counter over a hole that leads down to a hollow glass tube that guards against air currents. There’s also a big powerful permanent magnet down there, mounted on a hinged arrangement. Your sample hangs on a piece of fishline hooked to the balance pan. Take a weight reading, swing the magnet into position just below the sample, read the weight again, do some arithmetic and you’re done. A higher weight reading when the field’s in place means your sample’s paramagnetic, less weight means it’s diamagnetic.”

“Why does that Ag box look weird in your table, sort of half‑brown and half‑gray?”

“That’s silver, Eddie. It’s an edge case. The pure metal’s diamagnetic but alloy a sample with even a small fraction of some ferromagnetic atoms and you’ve made it paramagnetic. Magnetism’s one test that people in the silver trade use to check if a coin or bar is pure. How that works isn’t my field. Sy, it’s your turn to bet and explain.”

~ Rich Olcott

That Lump in The Table

On By rolcottIn Astronomy: Planetary Science, Chemistry, UncategorizedLeave a comment

The Acme Building Science and Pizza Society is back in session. It’s Cal’s turn to deal the cards and the topic. “This TV guy was talking about rare earths that China’s got a lock on and it’s gonna mess up our economy, but he didn’t say what they are or why we should care about them. What’s goin’ on?”

Vinnie passes but Susan tosses a chip into the pot. “The rare earths are oxides of the lanthanide elements—”

“Wait, they’re from the planet that the Strange New Worlds engineering prof is from?”

“Put in a chip, Vinnie, you know the rules.” <He does.> “No, they have nothing to do with Pelia or her home planet. She’s a Lanthanite, these elements are lanthanides. Although these days we’re supposed to call them lanthanoids because ‑ides are ionic compounds like oxides.”

It’s not Kareem’s turn yet but he chuckles and flips in a chip. “Funny. The geology community settled on meteoroids as rocks floating in space, meteors when they flash through the sky, and meteorites when they hit the ground. I don’t think there’s such a thing as a meteoride. Sorry, Susan, go on.”

“As a matter of fact, Kareem, I once did a high‑rated downhill mountain bike path in Arizona called the Meteoride. Once. Didn’t wipe out but I admit I used my brakes a whole lot. Where was I? Oh, yes, the lanthanides. They’re a set of fourteen near‑identical twins, chemistry so similar that it took decades of heroic effort by 19th‑century Swedish chemists in the long, cold Swedish nights to separate and identify them.”

Benfey‘s “Periodic Snail” Image by DePiep under CC BY-SA 3.0

“Similar how?”

“They all act like aluminum.” <pulls laptop from her purse, points to two stickers on its lid> “You’ve all at least heard of the Periodic Table, right? Back in the mid-1800s, the chemists had isolated dozens of chemical elements, enough that they could start classifying them. They didn’t know what atoms were yet but they had developed ways to measure average atomic weights. Some theorists played with the idea of arranging elements with similar chemistries according to their atomic weights. Mendeleev did the best job, even predicting three elements to fill empty slots in his tabulation. These guys in the lime green row and the pale pink bulge were his biggest puzzlement.”

“Why’s that? They’re all spread out nice.”

“Because like I said, Vinnie, they all have pretty much the same chemistry. Aluminum’s a soft silvery metal, oxidizes readily to a 3+ ion and stays there. Same for almost all the lanthanides. Worse yet, all their atoms are nearly the same size, less than 8% difference from the largest to the smallest.”

“Why’s that make a difference?”

“Because they can all fit into the same crystalline structure. Nineteenth‑century chemistry’s primary technique for isolating a metallic element was to dissolve a likely‑looking ore, purify the solution, add an organic acid or something to make crystalline salts, burn away the organics, add more acid to dissolve the ash, purify the solution and re‑crystallize most it. Do that again and again until you have a provably pure product. All the lanthanide ions have the same charge and nearly the same size so the wrong ions could maliciously infiltrate your crystals. It took a lot of ingenious purification steps to isolate each element. There were many false claims.”

Kareem contributes another chip. “Mm‑hm, because geology doesn’t use chemically pure materials to create its ores. Four billion years ago when our planet was coated with molten magma, the asteroids striking Earth in the Late Heavy Bombardment brought megatons of stone‑making lithophile elements. The lanthanides are lithophiles so random mixtures of them tended to concentrate within lithic silicate and phosphate blobs that later cooled to form rocky ores. Industry‑scale operations can tease lanthanides out of ores but the processes use fierce chemicals and require close control of temperature and acidity. Tricky procedures that the Chinese spent billions and decades to get right. For the Chinese, those processes are precious national security assets.”

Cal’s getting impatient. “Hey, guys, are we playing cards or what?”

~ Rich Olcott

A Cosmological Horse Race

On By rolcottIn Astronomy: Multi-messenger, Cosmology, Physics: Einstein, Physics: Electromagnetism, UncategorizedLeave a comment

A crisp Fall day, perfect for a brisk walk around the park. I see why the geese are huddled at the center of the lake — Mr Feder, not their best friend, is on patrol again. Then he spots me. “Hey, Moire, I gotta question!”

“Of course you do, Mr Feder. What is it?”

“Some guy on TV said Einstein proved gravity goes at the speed of light and if the Sun suddenly went away it’d take eight minutes before we went flying off into space. Did Einstein really say that? Why’d he say that? Was the TV guy right? And what would us flying across space feel like?”

“I’ll say this, Mr Feder, you’re true to form. Let’s see… Einstein didn’t quite prove it, the TV fellow was right, and we’d notice being cold and in the dark well before we’d notice we’d left orbit. As to why, that’s a longer story. Walk along with me.”

“Okay, but not too fast. What’s not quite about Einstein’s proving?”

“Physicists like proofs that use dependable mathematical methods to get from experimentally-tested principles, like conservation of energy, to some result they can trust. We’ve been that way since Galileo used experiments to overturn Aristotle’s pure‑thought methodology. When Einstein linked gravity to light the linkage was more like poetry. Beautiful poetry, though.”

“What’s so beautiful about something like that?”

“All the rhymes, Mr Feder, all the rhymes. Both gravity and light get less intense with the square of the distance. Gravity and light have the same kinds of symmetries—”

“What the heck does that mean?”

“If an object or system has symmetry, you can execute certain operations on it yet make no apparent difference. Rotate a square by 90° and it looks just the same. Gravity and light both have spherical symmetry. At a given distance from a source, the field intensity’s the same no matter what direction you are from the source. Because of other symmetries they both obey conservation of momentum and conservation of energy. In the late 1890s researchers found Lorentz symmetry in Maxwell’s equations governing light’s behavior.”

“You’re gonna have to explain that Lorentz thing.”

“Lorentz symmetry has to do with phenomena an observer sees near an object when their speed relative to the object approaches some threshold. Einstein’s Special Relativity theory predicted that gravity would also have Lorentz symmetry. Observations showed he was right.”

“So they both do Lorentz stuff. That makes them the same?”

“Oh, no, completely different physics but they share the same underlying structure. Maxwell’s equations say that light’s threshold is lightspeed.”

“Gravity does lightspeed, too, I suppose.”

“There were arguments about that. Einstein said beauty demands that both use the same threshold. Other people said, ‘Prove it.’ The strongest argument in his favor at the time was rough, indirect, complicated, and had to do with fine details of Earth’s orbit around the Sun. Half a century later pulsar timing data gave us an improved measurement, still indirect and complicated. This one showed gravity’s threshold to be with 0.2% of lightspeed.”

“Anything direct like I could understand it?”

“How about a straight‑up horse race? In 2017, the LIGO facility picked up a gravitational signal that came in at the same time that optical and gamma ray observatories recorded pulses from the same source, a colliding pair of neutron stars in a galaxy 130 million lightyears away. A long track, right?”

“Waves, not horses, but how far apart were the signals?”

“Close enough that the measured speed of gravity is within 10–15 of the speed of light.”

“A photo-finish.”

“Nice pun, Mr Feder. We’re about 8½ light-minutes away from the Sun so we’re also 8½ gravity-minutes from the Sun. As the TV announcer said, if the Sun were to suddenly dematerialize then Earth would lose the Sun’s orbital attraction 8½ minutes later. We as individuals wouldn’t go floating off into space, though. Earth’s gravity would still hold us close as the whole darkened, cooling planet leaves orbit and heads outward.”

“I like it better staying close to home.”

~ 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

Got To Be Good-lookin’ ‘Cause He’s So Hard To See

On By rolcottIn Cosmology, Crazy Theories, UncategorizedLeave a comment

I’ll be sorry when Acme Building’s management swaps out our old‑style door locks for electronic ones. Vinnie has such fun lock‑picking his way past my office door in the morning. “Morning, Vinnie.”

“Morning, Sy. Hey, I got a new Crazy Theory for you. Nobody knows what Dark Matter is, right?”

“Right. All we know is that it has about five times as much mass as normal matter so it participates in gravitational interactions. Some of it seems to gather in spherical halos around galaxies and some of it seems to collect in spikes near their centers. Cosmologists are arguing about whether or not Dark Matter is particles, much less how they’d be quantized. And we call it Dark because it absolutely doesn’t care about electromagnetism.”

“That’s what I thought. I remember you said if Dark Matter did play with light waves at all it’d block our view of the CMB. So yeah, absolute. Good.”

“I gather your theory is about Dark Matter.”

“Mm-hm. I thought of a way that all that mass could be hiding in plain sight except we can’t see it.”

“Alright, I’m listening.”

“Tachyons.”

“Come again?”

“Tachyons — particles that fly around faster than light. I read an article about ’em. Some people say they can’t exist but hear me out, okay? The reason they’re not supposed to exist is ’cause it would take an infinite amount of energy to boost something up past lightspeed. I got that, but suppose they were born above lightspeed, back when the Big Bang singularity had energy packed so tight the Physics laws we know don’t apply. A lot of particles got flung out below lightspeed, but maybe even more got flung out above it.”

“What does this have to do with dark matter?”

“I’m gettin’ there. The thing with tachyons is, the article said it’d take infinite energy to slow one down to lightspeed. A tachyon rock hits a slow rock, it don’t stop ’cause the slow rock don’t have the juice for that. The collision may take a little energy from the tachyon rock but that just changes its trajectory.”

“Mmm, those tachyon rocks can’t be a thing. The — what can I call it? slow matter?”

“The article called ’em bradyons.”

“Thanks. We know that 92% of all … bradyonic atoms in the Universe are hydrogens. Rocks are made of silicon, oxygen and other atoms that are even heavier. Everything heavier than hydrogen and maybe some helium was created by nuclear reactions inside a star. Tachyonic atoms zooming beyond lightspeed couldn’t gather together to form a star or even join one. No significant tachyonic fusion, no tachyonic rocks.”

“Okay, they all stay tachy‑hydrogen, still not a problem. The point is, there could be a lot of them and they could add up to a lot of mass. So the next thing I asked is, where would tachyons hang out? Gotta be around galaxies, but being tachyons going super‑lightspeed they can’t just hang, they orbit around the centers. They’d spend the most time where they go slowest which is where they’re farthest away ’cause that’s how orbits work. But they’d be thickest close in ’cause of gravity but that’s where they go fastest.”

“Cute, so you’re predicting galaxies with halos of tachyons, plus spikes of them at each center. That just happens to be the dark matter distribution the astronomers find.”

“It gets better, Sy. I’m not so sure of this because math, but it feels right. I don’t think tachyons can do electromagnetism things.”

“Why not?”

“No blue glow — you know, that blue glow in nuclear reactors when electrons go through the cooling water faster than light?”

“Cherenkov radiation, happens when fast electrons polarize the water. The polarizing slows light in water relative to a vacuum.”

“Right, but tachyons in space travel through vacuum. They ought to polarize the vacuum like what fast electrons do to water. Electromagnetic tachyons orbiting galaxies ought to make a blue glow but there isn’t one, so tachyons don’t do electromagnetism things and that makes them Dark Matter.”

“You’re going to have to do better than that, Vinnie. Absence of evidence just might be evidence of absence. Maybe they’re not there to begin with.”

~ Rich Olcott

Properties of Space

On By rolcottIn Cosmology, Physics: Electromagnetism, Physics: Waves, UncategorizedLeave a comment

Vinnie gives me the side‑eye. “Wait, Sy. Back there you said Maxwell got the speed of light from the properties of space. What does any of that even mean?”

“Do you remember Newton’s equation for the force of gravity between two objects?”

“Of course not. Lessee… the force’d be bigger when either one gets bigger, and it’d get smaller when the distance between ’em gets bigger and there’s some constant number to make the units right, right?”

“Close enough, it’s the distance squared. The equation’s F=Gm1m2/r². The G is the constant you mentioned. It does more than turn mass‑units times mass‑units divided by length‑units‑squared into force‑units. It says how many force‑units. For one pair of objects at a certain distance, turn the G‑dial up and you get more force. Make sense?”

“Yeah, that looks right.”

“The value of G sets the force‑distance scale for how two objects attract each other everywhere in the Universe. That value is a property of space. So is the fact that the value is the same in all directions.”

“Huh! Never thought of it like a scale factor. Space has other properties like that?”

“Certainly. Coulomb’s Law for the electrostatic force between two charged objects has the same basic structure, FE=–(q1q2/r²)/CE. In any units you like you replace the q‘s with object charge amounts and r with the distance between them. For each set of change‑ and distance‑units there’s a well‑researched value of CE to convert your charge and distance numbers into force‑units. Under the covers, though, CE is a scale factor that controls the range of the electrostatic force. It’s the same everywhere in the Universe and it’s completely independent of Newton’s gravity scale factor.”

“Hey, what about ‘like charges repel, opposites attract’?”

“That’s what the minus sign’s in there for. If the q‘s have the same charge, the force is negative, that’s repulsion; opposite charges make for positive, attractive force.”

“If there’s a CE for electric there’s gotta be a CM for magnetic.”

“Sort of. The electrostatic force doesn’t care about direction. Magnetism does care so the equation’s more complicated. You’re right, though, there is a similar universal scale factor we might as well call CM.”

<chuckle> “Electric, magnetic, I don’t suppose we could mix those two somehow for an electromagnetic scale factor?”

<grin> “Did you read ahead in the book? Yes we can, and Maxwell’s equations showed us how. If you multiply the two C‘s together, you get one over the square of the speed of light. Re‑arranging a little, c=√(1/CECM), so c, the electromagnetic scale factor for velocity, is based on those space properties. Einstein showed that no material object can have a velocity greater than c.”

“I’ll take your word for the arithmetic, but how does that combination make for a speed limit?”

“There’s an easy answer you’re not going to like — it’s a speed because the units come out meters per second.”

“That’s a cheat. I don’t like it at all and it doesn’t account for the limit part. Explain it with Physics, no fancy equations.”

“Tough assignment. Okay, typical waves have a displacement force, like wind or something pushing up on an ocean wave, that works against a restoring force, such as gravity pulling down. Electromagnetic waves are different. The electric component supplies the up force, but the magnetic component twists sideways instead of restoring down. The wave travels as a helix. The CE and CM properties determine how tightly it spirals through space. That’s lightspeed.”

“And the limit part?”

“Einstein maintained that anything that happens must follow the same rules for all observers no matter how each is moving. The only way that can be true is if space is subject to the Lorentz contraction √[1-(v/vmax)²] for some universal maximum speed vmax. Maxwell’s electromagnetism equations showed that vmax is c. Okay?”

“I suppose.”

~ Rich Olcott

  • * Vinnie hates equations even with regular letters, Greek letters make it worse. Hence my using CE and CM instead of the conventional ε0 and μ0 notation. Sue me.

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