Category: Cosmology

Now And Then And There

On By rolcottIn Cosmology, James Webb Space Telescope, Physics: Light and Relativity, UncategorizedLeave a comment

Still at our table in Al’s otherwise empty coffee shop. We’re leading up to how Physics scrambled ‘Now‘ when a bell dings behind the counter. Al dashes over there. Meanwhile, Cathleen scribbles on a paper napkin with her colored pencils. She adds two red lines just as Al comes back with a plate of scones. “Here, Sy, if you’re going to talk Minkowski space this might be useful.”

“Hah, you’re right, Cathleen, this is perfect. Thanks, Al, I’ll have a strawberry one. Mmm, I love ’em fresh like this. OK, guys, take a look at Cathleen’s graphy artwork.”

“So? It’s the tile floor here.”

“Not even close, Mr Feder. Check the labels. The up‑and‑down label is ‘Time’ with later as higher. The diagram covers the period we’ve been sitting here. ‘Now‘ moves up, ‘Here’ goes side‑to‑side. ‘Table‘ and ‘Oven‘, different points in space, are two parallel lines. They’re lines because they both exist during this time period. They’re vertical because neither one moves from its relative spatial position. Okay?”

“Go on, Moire.”
  ”Makes sense to me, Sy.”

“Good. ‘Bell‘ marks an event, a specific point in spacetime. In this case it’s the moment when we here at the table heard the bell. I said ‘spacetime‘ because we’re treating space and time as a combined thing. Okay?”

“Go on, Moire.”
  ”Makes sense to me, Sy.”

“So then Al went to the oven and came back to the table. He traveled a distance, took some time to do that. Distance divided by time equals velocity. ‘Table‘ has zero velocity and its line is vertical. Al’s line would tilt down more if he went faster, okay?”

“Mmmm, got it, Sy.”
  ”Cute how you draw the come-back label backwards, lady. Go on, Moire.”

“I do my best, Mr Feder.”

“Fine, you’ve got the basic ideas. Now imagine all around us there’s graph paper like this — except there’s no paper and it’s a 4‑dimensional grid to account for motion in three spatial dimensions while time proceeds. Al left and returned to the same space point so his spacetime interval is just the time difference. If two events differ in time AND place there’s special arithmetic for calculating the interval.”

“So where’s that get us, Moire?”

“It got 18th and 19th Century Physics very far, indeed. Newton and everyone after him made great progress using math based on a nice stable rectangular space grid crossed with an orderly time line. Then Lorentz and Poincaré and Einstein came along.”

“Who’s Poincaré?”

“The foremost mathematician of nineteenth Century France. A mine safety engineer most days and a wide‑ranging thinker the rest of the time — did bleeding‑edge work in many branches of physics and math, even invented a few branches of his own. He put Lorentz’s relativity work on a firm mathematical footing, set the spacetime and gravity stage for Minkowsky and Einstein. All that and a long list of academic and governmental appointments but somehow he found the time to have four kids.”

Simultaneity Animation by “Acdx”
licensed under Creative Commons Attribution-Share Alike 4.0 International.

“A ball of fire, huh? So what’d he do to Newton’s jungle gym?”

“Turned its steel rod framework into jello. Remember how Cathleen’s Minkowski diagram connected slope with velocity? Einstein showed how Lorentz’s relativity factor sets a speed limit for our Universe. On the diagram, that’d be a minimum slope. Going vertical is okay, that’s standing still in space. Going horizontal isn’t, because that’d be instantaneous travel. This animation tells the ‘Now‘ story better than words can.”

“Whah?”
  ”Whah?”

“We’re looking down on three space travelers and three events. Speeds below lightspeed are within the gray hourglass shape. The white line perpendicular to each traveler’s time line is their personal ‘Now‘. The travelers go at different velocities relative to us so their slopes and ‘Now‘ lines are different. From our point of view, time goes straight up. One traveler is sitting still relative to us so its timeline is marked ‘v=0‘ and parallels ours. We and the v=0 traveler see events A, B and C happening simultaneously. The other travelers don’t agree. ‘Simultaneous‘ is an illusion.”

~~ Rich Olcott

Hyperbolic plane image by Murdock Grewar,
licensed under Creative Commons Attribution-Share Alike 4.0 International.

The Bottom of Time

On By rolcottIn Astronomy: Techniques, Cosmology, James Webb Space Telescope, Uncategorized2 Comments

“Cathleen, one of my Astronomy magazines had an article, claimed that James Webb Space Telescope can see back to the Big Bang. That doesn’t seem right, right?”

“You’re right, Al, it’s not quite right. By our present state of knowledge JWST‘s infrared perspective goes back only 98% of the way to the Bang. Not quite the Bottom of Time, but close.”

“Whaddaya mean, ‘Bottom of Time‘? I’ve heard people talking about how weird it musta been before the Big Bang. And how can JWST see back in time anyway? Telescopes look across space, not time.”

“So many questions, Mr Feder, and some hiding behind others. That’s his usual mode, Cathleen. Care to tag-team?”

“You’re on, Sy. Well, Mr Feder. The ‘look back in time‘ part comes from light not traveling infinitely fast. We’ve known that for three centuries, ever since Rømer—”

“Roamer?”

“Ole Rømer, a Danish scientist who lived in Newson’s time. Everyone knew that Jupiter’s innermost large moon Io had a dependably regular orbit, circling Jupiter every 49½ hours like clockwork. Rømer was an astronomer when he wasn’t tutoring the French King’s son or being Copenhagen’s equivalent of Public Safety Commissioner. He watched Io closely, kept notes on exactly when she ducked behind Jupiter and when she reappeared on the other side. His observed timings weren’t quite regular, generally off by a few minutes. Funny thing was, the irregularities correlated with the Earth‑Jupiter distance — up to 3½ minutes earlier than expected when Earth in its orbit was closest to Jupiter, similarly late when they were far apart. There was a lot of argument about how that could be, but Rømer, Huygens, even Newton, all agreed that the best explanation was that we only see Io’s passage events after light has taken its time to travel from there to here.”

“Seems reasonable. Why should people argue about that?”

“The major sticking point was the speed that Huygens calculated from Rømer’s data. We now know it’s 186000 miles or 300000 kilometers or one lightsecond per second. Different ways of stating the same quantity. Huygens came up with a somewhat smaller number but still. The establishment pundits had been okay with light transmission being instantaneous. Given definite numbers, though, they had trouble accepting the idea that anything physical could go that fast.”

“Tag, my turn. Flip that distance per time ratio upside down — for every additional lightsecond of distance we’re looking at events happening one second farther into the past. That’s the key to JWST‘s view into the long‑ago. Al, you got that JWST‘s infrared capabilities will beat Hubble‘s vis‑UV ones for distance. Unless there’s something seriously wrong with Einstein’s assumption that lightspeed’s an absolute constant throughout spacetime, we expect JWST to give us visibility to the oldest free photons in the Universe, just 379000 years upward from the Big Bang.”

“Wait, I heard weaseling there. Free photons? Like you gotta pay for the others?”

“Ha, ha, Mr Feder. During those first 379000‑or‑so years, we think the Universe was so hot and so dense that no photon’s wave had much of a chance to spread out before it encountered a charged something and got absorbed. At last, things cooled down enough for atoms to form and stay in one piece. Atoms are neutral. Quantum rules restrict their interaction to only photons that have certain wavelengths. The rest of the photons, and there’s a huge number of them, were free to roam the expanding Universe until they happen to find a suitable absorber. Maybe someone’s eye or if we’re lucky, a sensor on JWST or some other telescope.”

Thanks for this to George Derenburger

“What about before the 300‑and‑something thousand years? Like, before Year Zero? Musta been weird, huh?”

“Well, there’s a problem with that question. You’re assuming there was a Year Minus‑One, but that’s just not the case.”

“Why not? Arithmetic works that way.”

“But the Universe doesn’t. Stephen Hawking came up with a good way to think about it. What on Earth is south of the South Pole?”

“Eeayahh … nope. Can’t get any further south than that.”

“Well, there you are, so to speak. Time’s bottom is Year Zero and you can’t get any further down than that. We think.”

~~ 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

The Gelato Model

On By rolcottIn Cosmology, Mathematics, Physics: Newton and Gravity, UncategorizedLeave a comment

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

“Glad you like it, Anne.”

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

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

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

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

“S’what I said.”

“Measured in dollars or trayfuls?”

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

“Batches all the same size?”

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

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

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

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

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

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

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

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

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

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

“You know we want to know, Sy.”

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

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

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

“So what about my gelato?”

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

~~ Rich Olcott

Symmetrical Eavesdropping

On By rolcottIn Cosmology, Mathematics, Physics: Fundamentals, UncategorizedLeave a comment

“Wait, Sy, you’ve made this explanation way more complicated than it has to be. All I asked about was the horrible whirling I’d gotten myself into. The three angular coordinates part would have done for that, but you dragged in degrees of freedom and deep symmetry and even dropped in that bit about ‘if measurable motion is defined.’ Why bother with all that and how can you have unmeasurable motion?”

“Curiosity caught the cat, didn’t it? Let’s head down to Eddie’s and I’ll treat you to a gelato. Your usual scoop of mint, of course, but I recommend combining it with a scoop of ginger to ease your queasy.”

“You’re a hard man to turn down, Sy. Lead on.”

<walking the hall to the elevators> “Have you ever baked a cake, Anne?”

“Hasn’t everyone? My specialty is Crazy Cake — flour, sugar, oil, vinegar, baking soda and a few other things but no eggs.”

“Sounds interesting. Well, consider the path from fixings to cake. You’ve collected the ingredients. Is it a cake yet?”

“Of course not.”

“Ok, you’ve stirred everything together and poured the batter into the pan. Is it a cake yet?”

“Actually, you sift the dry ingredients into the pan, then add the others separately, but I get your point. No, it’s not cake and it won’t be until it’s baked and I’ve topped it with my secret frosting. Some day, Sy, I’ll bake you one.”

<riding the elevator down to 2> “You’re a hard woman to turn down, Anne. I look forward to it. Anyhow, you see the essential difference between flour’s journey to cakehood and our elevator ride down to Eddie’s.”

“Mmm… OK, it’s the discrete versus continuous thing, isn’t it?”

“You’ve got it. Measuring progress along a discrete degree of freedom can be an iffy proposition.”

“How about just going with the recipe’s step number?”

“I’ll bet you use a spoon instead of a cup to get the right amount of baking soda. Is that a separate step from cup‑measuring the other dry ingredients? Sifting one batch or two? Those’d change the step‑number metric and the step-by-step equivalent of momentum. It’s not a trivial question, because Emmy Noether’s symmetry theorem applies only to continuous coordinates.”

“We’re back to her again? I thought—”

The elevator doors open at the second floor. We walk across to Eddie’s, where the tail‑end of the lunch crowd is dawdling over their pizzas. “Hiya folks. You’re a little late, I already shut my oven down.”

“Hi, Eddie, we’re just here for gelato. What’s your pleasure, Anne?”

“On Sy’s recommendation, Eddie, I’ll try a scoop of ginger along with my scoop of mint. Sy, about that symmetry theorem—”

“The same for me, Eddie.”

“Comin’ up. Just find a table, I’ll bring ’em over.”

We do that and he does that. “Here you go, folks, two gelati both the same, all symmetrical.”

“Eddie, you’ve been eavesdropping again!”

“Who, me? Never! Unless it’s somethin’ interesting. So symmetry ain’t just pretty like snowflakes? It’s got theorems?”

“Absolutely, Eddie. In many ways symmetry appears to be fundamental to how the Universe works. Or we think so, anyway. Here, Anne, have an extra bite of my ginger gelato. For one thing, Eddie, symmetry makes calculations a lot easier. If you know a particular system has the symmetry of a square, for instance, then you can get away with calculating only an eighth of it.”

“You mean a quarter, right, you turn a square four ways.”

“No, eight. It’s done with mirrors. Sy showed me.”

“I’m sure he did, Anne. But Sy, what if it’s not a perfect square? How about if one corner’s pulled out to a kite shape?”

“That’s called a broken symmetry, no surprise. Physicists and engineers handle systems like that with a toolkit of approximations that the mathematicians don’t like. Basically, the idea is to start with some nice neat symmetrical solution then add adjustments, called perturbations, to tweak the solution to something closer to reality. If the kite shape’s not too far away from squareness the adjusted solution can give you some insight onto how the actual thing works.”

“How about if it’s too far?”

“You go looking for a kite‑shaped solution.”

~~ Rich Olcott

Deep Symmetry

On By rolcottIn Cosmology, Mathematics, Physics: Fundamentals, UncategorizedLeave a comment

“Sy, I can understand mathematicians getting seriously into symmetry. They love patterns and I suppose they’ve even found patterns in the patterns.”

“They have, Anne. There’s a whole field called ‘Group Theory‘ devoted to classifying symmetries and then classifying the classifications. The split between discrete and continuous varieties is just the first step.”

“You say ‘symmetry‘ like it’s a thing rather than a quality.”

“Nice observation. In this context, it is. Something may be symmetrical, that’s a quality. Or it may be subject to a symmetry operation, say a reflection across its midline. Or it may be subject to a whole collection of operations that match the operations of some other object, say a square. In that case we say our object has the symmetry of a square. It turns out that there’s a limited number of discrete symmetries, few enough that they’ve been given names. Squares, for instance, have D4 symmetry. So do four-leaf clovers and the Washington Monument.”

“OK, the ‘4’ must be in there because you can turn it four times and each time it looks the same. What’s the ‘D‘ about?”

“Dihedral, two‑sided, like two appearances on either side of a reflection. That’s opposed to ‘C‘ which comes from ‘Cyclic’ like 1‑2‑3‑4‑1‑2‑3‑4. My lawn sprinkler has C4 symmetry, no mirrors, but add one mirror and bang! you’ve got eight mirrors and D4 symmetry.”

“Eight, not just four?”

“Eight. Two mirrors at 90° generate another one 45° between them. That’s the thing with symmetry operations, they combine and multiply. That’s also why there’s a limited number of symmetries. You think you’ve got a new one but when you work out all the relationships it turns out to be an old one looked at from a different angle. Cubes, for instance — who knew they have a three‑fold rotation axis along each body diagonal, but they do.”

“I guess symmetry can make physics calculations simpler because you only have to do one symmetric piece and then spread the results around. But other than that, why do the physicists care?”

“Actually they don’t care much about most of the discrete symmetries but they care a whole lot about the continuous kind. A century ago, a young German mathematician named Emmy Noether proved that within certain restrictions, every continuous symmetry comes along with a conserved quantity. That proof suddenly tied together a bunch of Physics specialties that had grown up separately — cosmology, relativity, thermodynamics, electromagnetism, optics, classical Newtonian mechanics, fluid mechanics, nuclear physics, even string theory—”

“Very large to very small, I get that, but how can one theory have that range? And what’s a conserved quantity?”

“It’s theorem, not theory, and it capped two centuries of theoretical development. Conserved quantities are properties that don’t change while a system evolves from one state to another. Newton’s First Law of Motion was about linear momentum as a conserved quantity. His Second Law, F=ma, connected force with momentum change, letting us understand how a straight‑line system evolves with time. F=ma was our first Equation of Motion. It was a short step from there to rotational motion where we found a second conserved quantity, angular momentum, and an Equation of Motion that had exactly the same form as Newton’s first one, once you converted from linear to angular coordinates.”

“Converting from x-y to radius-angle, I take it.”

“Exactly, Anne, with torque serving as F. That generalization was the first of many as physicists learned how to choose the right generalized coordinates for a given system and an appropriate property to serve as the momentum. The amazing thing was that so many phenomena follow very similar Equations of Motion — at a fundamental level, photons and galaxies obey the same mathematics. Different details but the same form, like a snowflake rotated by 60 degrees.”

“Ooo, lovely, a really deep symmetry!”

“Mm-hm, and that’s where Noether came in. She showed that for a large class of important systems, smooth continuous symmetry along some coordinate necessarily entails a conserved quantity. Space‑shift symmetry implies conservation of momentum, time‑shift symmetry implies conservation of energy, other symmetries lock in a collection of subatomic quantities.”

“Symmetry explains a lot, mm-hm.”

~~ Rich Olcott

Edged Things and Smooth Things

On By rolcottIn Cosmology, Mathematics, Physics: Fundamentals, Uncategorized2 Comments

“Yeughh, Sy, that whirling, the entire Universe spinning around me in every direction at once.”

“Well, you were at a point of spherical symmetry, Anne.”

“There’s that word ‘symmetry‘ again. Right side matches left side, what else is there to say?”

“A whole lot, especially after the mathematicians and physicists started playing with the basic notion.”

“Which is?”

“Being able to execute a transformation without making a relevant difference.”

“Relevant?”

“To the context. Swapping the king of spades for the king of hearts would be relevant in some card games but not others, right? If it doesn’t affect the play or the scoring, swapping those two when no‑one’s looking would be a legitimate symmetry operation. Spin a snowflake 60° and it looks the same unless you care exactly where each molecule is. That’s rotational symmetry, but there’s lots of geometric symmetry operations — reflections, inversions, glides, translations—”

“Translation is a symmetry operation?”

“In this connection, ‘translation‘ means movement or swapping between two different places in space. The idea came from crystals. Think of a 3D checkerboard, except the borderlines aren’t necessarily perpendicular. Perfect crystals are like that. Every cube‑ish cell contains essentially the same arrangement of atoms. In principle you could swap the contents of any two cells without making a difference in any of the crystal’s measurable properties. That’d be a translation symmetry operation.”

“Glides make me think of ice skating.”

“The glide operation makes me think of a chess knight’s move — a translation plus a reflection across the translation path. Think of wet footprints crossing a dry floor. That’s one example of combining operations to create additional symmetries. You can execute 48 unique symmetry operations on a cube even without the translation‑related ones. In my grad school’s crystallography class they taught us about point group and wallpaper and space group symmetries. It blew me away — beautiful in both mathematical and artistic senses. You’ve seen M C Escher’s art?”

“Of course, I love it. I pushed into his studio once to watch him work but he spotted me and shouted something Dutch at me. I’ve wondered what he thought when I pushed out of there.”

“His pieces drew heavily on geometric symmetries. So did Baroque art, music and architecture.”

“Music? Oh, yes — they had motifs and whole sections you could swap, and rhythm patterns and tunes you could read forwards and backwards like in a mirror… We’ve come a long way from snowflake symmetry, haven’t we?”

“We’re just getting started. Here’s where the Physics folks generalized the idea. Your unfortunate experience in space is right on the edge of what most people consider as symmetry. Were you impressed with the cube’s 48 operations?”

“I suppose. I haven’t had time to think about it.”

“A sphere has an infinite number. You could pick any of an infinite number of lines through its center. Each is an axis for an infinite number of rotational symmetries. Times two because there’s an inversion point at the center so the rotation could go in either direction. Then each line is embedded in an infinite number of reflection planes.”

“Goodness, no wonder I was dizzy. But it’s still geometry. What was the edge that the physicists went past?”

“The border between step‑at‑a‑time discrete symmetries and continuous ones. Rotate that snowflake 60° and you’ve got a match; anything not a multiple of 60° won’t pair things up. Across the border, some of the most important results in modern Physics depend on continuous symmetries.”

“How can you even have a continuous symmetry?”

“Here, I’ll draw a circle on this square of paper. I can rotate the square by 90, 180 or 270 degrees and everything’s just the way it was. But if the square’s not relevant because we’re only interested in the circle, then I can rotate the paper by any amount I like and it’s a no‑difference transformation, right?”

“Continuous like on an infinite line but it’s wrapped around.”

“Exactly, and your infinite line is another example — any translation along that line, by a mile or a millimeter, is a perfectly good symmetry operation.”

“Ooo, and time, too. I experience time as an infinite line.”

“So does everyone. but most only travel in one direction.”

~~ Rich Olcott

Three Ways To Get Dizzy

On By rolcottIn Cosmology, Mathematics, Physics: Fundamentals, UncategorizedLeave a comment

<FZzzzzzzzzzzzzzzzzzzzzzzttt!> “Urk … ulp … I need to sit down, quick.”

“Anne? Welcome back, the couch is over there. Goodness, you do look a little green. Can I get you something to drink?”

“A little cool water might help, thanks.”

“Here. Just sit and breathe. That wasn’t your usual fizzing sound when you visit my office. When you’re ready tell me what happened. Must have been an experience, considering some of your other superpower adventures. Where did you ‘push‘ to this time?”

“Well, you know when I push forward I go into the future and when I push backward I go into the past. When I push up or down I get bigger or smaller. You figured out how pushing sideways kicks me to alternate probabilities. And then <shudder> there was that time I found a new direction to push and almost blew up the Earth.”

“Yes, that was a bad one. I’d think you’ve pretty well used up all the directions, though.”

“Not quite. This time I pushed outwards, the same in every direction.”

“Creative. And what happened?”

“Suddenly I was out in deep space, just tumbling in the blackness. There wasn’t an up or down or anything. I couldn’t even tell how big I was. I could see stars way off in the distance or maybe they were galaxies, but they were spinning all crazy. It took me a minute to realize it was me that was spinning, gyrating in several ways at once. It was scary and nauseating but I finally stopped part of it.”

“Floating in space with nothing to kill your angular momentum … how’d you manage to stabilize yourself at all?”

“Using my push superpower, of course. The biggest push resistance is against the past. I pulled pastward from just my shoulders and that stopped my nose‑diving but I was still whirling and cart‑wheeling. I tried to stop that with my feet but that only slowed me down and I was getting dizzy. My white satin had transformed into a spacesuit and I definitely didn’t want to get sick in there so I came home.”

“How’d you do that?”

“Oh, that was simple, I pulled inward. I had to um, zig‑zag? until I got just the right amount.”

“That explains the odd fizzing. I’m glad you got back. Looks like you’re feeling better now.”

“Mostly. Whew! So, Mr Physicist Sy, help me understand it all. <her voice that sounds like molten silver> Please?”

“Well. Um. There’s a couple of ways to go here. I’ll start with degrees of freedom, okay?”

“Whatever you say.”

“Right. You’re used to thinking in straight‑line terms of front/back, left/right and up/down, which makes sense if you’re on a large mostly‑flat surface like on Earth. In mathspeak each of those lines marks an independent degree of freedom because you can move along it without moving along either of the other two.”

“Like in space where I had those three ways to get dizzy.”

“Yup, three rotations at right angles to each other. Boatmen and pilots call them pitch, roll and yaw. Three angular degrees of freedom. Normal space adds three x-y-z straight‑line degrees, but you wouldn’t have been able to move along those unless you brought along a rocket or something. I guess you didn’t, otherwise you could have controlled that spinning.”

“Why would I have carried a rocket when I didn’t know where I was going? Anyhow, my push‑power can drive my straight‑line motion except I didn’t know where I was and that awful spinning had me discombobulated”

“Frankly, I’m glad I don’t know how you feel. Anyhow, if measurable motion is defined along a degree of freedom the measurement is called a coordinate. Simple graphs have an x-coordinate and a y-coordinate. An origin plus almost any three coordinates makes a coordinate system able to locate any point in space. The Cartesian x-y-z system uses three distances or you can have two distances and an angle, that’s cylindrical coordinates, or two angles and one distance and that’s polar coordinates.”

“Three angles?”

“You don’t know where you are.”

<shudder>
 <shudder>

~~ Rich Olcott

A Neutral Party

On By rolcottIn Cosmology, UncategorizedLeave a comment

“Hi, folks, sorry I’m late to the party. What are we arguing about and which side am I on?”

“Hi, Vinnie. We started out talking about neutrality and Jim proved that we’re electrically neutral otherwise we’d spray ourselves apart because of like‑charge repulsons.”

“Yeah, an’ then we got into the Standard Module picture here and how it’s weird that the electron charge exactly cancels out the quark mixture in a proton even though electrons don’t have quarks and quarks don’t have exact charges.”

Jim’s on it. “Almost, Eddie. Quarks have exact charges, but they’re exact fractions. They just add up when you mix three of them to make a particle. Two of them, sometimes. Up‑quark, up‑quark and down‑quark is two‑thirds plus two‑thirds minus one‑third equals one. That’s one proton, exactly opposing one electron’s charge.”

Vinnie’s good at mental math. “What happens when you mix one‑third plus one‑third minus two‑thirds which is zero?”

“Two downs and an up. That’s a neutron.”

“Ups, downs, electrons, protons, neutrons — except for the neutrino the first column’s pretty much atoms, right? What’s with those other boxes?”

“We only see evidence for the other purple‑box quarks in collider records or nuclear reactions. Same for the muon and tau. They’re all way too unstable to contribute much to anything that hangs around. The guys in the red and gold boxes aren’t building blocks, they’re more like glue that holds everything else together. The green‑box neutrinos at the bottom are just weird and we’ll probably be a long time figuring them out.”

“Says here that neutrinos have zero charge, and so do most of the force thingies. Is that really zero or is it just too small to measure?”

“A true Chemistry‑style question, Susan. Charges we can count but you’re right, energy exchanges in a process have to be measured. The zero charges are really zero. For example, Pauli dreamed up the neutrino as an energy‑accounting trick for a nuclear process where all the charges went to known products but there was energy left over. If they existed at all, neutrinos could carry away that energy but they had to have zero charge. A quarter‑century later we detected some and they fit all the requirements.”

Vinnie perks up. “Zero charge so they doesn’t interact with light, teeny mass per each but there’s a hyper‑gazillion of them out there which oughtta add up to a lot of mass. Could neutrinos be what dark matter is?”

“Some researchers thought that for a while but the idea hasn’t held up to inspection. The neutrinos we know about come to about 1% of dark matter’s mass. Some people think there may be a really heavy fourth kind of neutrino that would make up the difference, but it’s a long shot and there’s no firm evidence for it so far. Dark matter doesn’t interact with photons, photons interact with electric charge, quarks have electric charge. If you’ve got quarks you’re not dark matter.”

“How about neutrons floating around?”
 ”Those molecular clouds I’ve read about Aren’t they neutral? Are there neutral stars?”
  ”How about neutron stars and black holes?”
   ”What’s a neutron star?”

“All good questions. Free neutrons are a bad bet, Vinnie — unless they’re bound with protons they usually emit an electron and become a proton within an hour. Susan, electrostatic forces would overwhelm gravity so we believe stars and molecular clouds must be electrically neutral or close to it. Anyway, stars and clouds can’t be dark matter because they’ve got quarks. Eddie, what do you suppose happens when a star uses up the fuel that keeps it big?”

“Since you ask it that way, I suppose it caves in.”

“Got it in one. If the star’s too big to collapse to be a white dwarf but too small to collapse to be a black hole, it collapses to be a neutron star. Really weird objects — a star‑and‑a‑half of of mass packed into a 10‑kilometer sphere, probably spinning super‑fast and possessing a huge magnetic field. From a ‘what is dark matter?‘ perspective, though, collapsed stars of any sort are still made of quarks and can’t qualify.”

“So what is dark matter then?”

“Good question.”

~~ Rich Olcott

  • Thanks to Alex, who asked a question.

Quarkery

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

Susan, aghast. “But I thought the Standard Model was supposed to be the Theory of Everything.”

Jim, abashed. “A lot of us wish that phrase had never been invented. Against the mass of the Universe it’s barely the theory of anything.”

Me, typecast. “That’s a heavy claim, Jim. Big Physics has put many dollars and fifty years of head time into filling out that elegant table of elementary particles. I remember the celebration when the LHC finally found the Higgs boson in 2012. I’ve read that the Higgs field is responsible for the mass of the Universe.”

“A little bit true, Sy, sort of. We think it’s responsible for about 1% of the mass of all the matter we understand. There’s another mechanism that accounts for the other 99%.”

Eddie, downcast. “I’m lost, guys. What Standard Module are you talking about?”

“Do you remember the Periodic Table of the chemical elements?”

“A little. Science class had big poster up on the wall. Had all kinds of atoms in it, right?”

“Yup. Scientists spent centuries breaking down minerals and compounds to find substances that chemical methods couldn’t break down any further. Those were the chemical elements, things like iron and carbon and oxygen. The Periodic Table arranges elements so as to highlight similarities in how they’ll interact. The Standard Model carries that idea down to the sub‑subatomic level.”

“Wait, sub‑subatomic level?”

“Mm-hm. Chemists would say that ‘subatomic‘ is about electrons, protons and neutrons. Count an atom’s electrons. That and some fairly simple rules can tell you what structure types it prefers to participate in and what it reacts with. Count the protons and neutrons in its nucleus. That gives you its atomic weight and starts you on the road to figuring reaction quantities. That’s all that the chemists need to know about atoms. All due respect, Susan, but physicists want to dig deeper. That’s what the Standard Model is all about.”

“So you’re saying that the protons and neutrons are made of these … quarks and things? Is that what comes out of those collider experiments?”

“No on both, Eddie. You ever whack a light pole with a baseball bat?”

“Sure, who hasn’t?”

“The sounds that came out, do you think the pole was made of them?”

“Course not, and I never bought the Brooklyn Bridge, neither.”

“Calm down, Eddie, just making a point. Suppose before you whacked that pole you’d attached a whole string of sensitive microphones all up and down it, and then when you whacked it you recorded all the vibrations your whack set off. Do you think with the recorded frequencies and a lot of math a good audio engineer could tell you what the pole is made of and how thick the casing is?”

“Maybe.”

“That’s what’s going on with the colliders. They whack particles with other particles, record everything that comes out and use math to work out what must have happened to make that event happen. Theory together with data from a huge number of whacks let people like Heisenberg, Gell‑Mann, Ne’eman and Nishijima to the seventeen boxes in that table.”

“‘Splain those particles to me.”

“Don’t think particles, think collections of properties. The Periodic Table’s ‘iron‘ box is about having 26 electrons and combining with 24 grams of oxygen to form 80 grams of Fe2O3. In the Standard Model table, the boxes are about energy, charge, lifetime, some technical properties, and rules for which can interact with what. We’ve never seen a free‑standing quark particle and there’s good reason to think we never will. We mostly see only two‑ or three‑quark mixtures. Some of the properties, like charge, simply add together. It takes a mixture to make a particle.”

“Then how did they figure what goes into a box?”

“Theoreticians worked to find the minimum set of independent properties that could still describe observations. Different mixtures of up and down quarks, for instance, account for protons, neutrons and many mesons.”

Vinnie, at last. “Hi, folks, sorry I’m late to the party. What are we arguing about and which side am I on?”

Higgs candidate LHC event trace
Electrons (green) and muons (red) exiting the event

~~ Rich Olcott