# Layer Upon Layer

“Excuse me, profesora, you wanted me to come to your office?”

“Yes, Maria. Come in, please. I wanted to have a chat with you before you give your class presentation tomorrow.”

“I am a little nervous about it.”

“I thought you might be. I wanted to help with that. I’ll start by saying that your English language skills have gotten much better than you give yourself credit for. Better yet, you’ll be speaking before friends who want you to succeed. I’m sure you’ll do fine. I think if we go over your material together you’ll be more confident. Come open your laptop on my desk where we can both see it. Now bring up your first slide.”

“Yes, profesora. Already you know that the title of my presentation is ‘The Structure of The Sun.’ I only have one slide, this one, that shows a slice of a star like our Sun.”

“How did the star get that way?”

“It condensed from a galactic gas cloud that was mostly hydrogen. I plan to talk about that with waving of the hands because a good picture of it needs to be in motion and I don’t know how to do that yet.”

“Fair enough, just don’t skip over it. Beginnings are important. Now talk me through your diagram.”

“It starts in the middle ¿see the fusion zone? where protons, that’s hydrogen atoms without their electrons, are squeezed together to release energy and make alpha particles, that’s helium atoms without their electrons. The protons have the same charge so they push each other away, but they are beneath many kilometers of mass that push them together. Also, the temperature is very hot, tens of millions of degrees. Hot atoms move fast, so when the protons are pushed together it happens with enough force and speed .. sorry, I need a word, superar?”

“Overcome.”

“Thank you. The protons are pushed together with enough force and speed to overcome the charge barrier. The actual reactions are complicated. At the end there is an alpha particle, four times heavier than a proton, and there is much more energy than the overcoming used up. The fusion zone makes heat and the heavy alpha particles fall down into the ash zone. The heat must go somewhere. Already the center is hotter so the new heat goes upward into the radiation zone.”

“And it’s called that because…?”

“Because atom motion is so, mm, frantic?”

“Good word.”

“… So frantic that there’s no moving in the same direction together, no convection like when steam rises over boiling water. Heat can only travel by convection, conduction or radiation. If there is no convection, moving heat must go neighbor‑to‑neighbor by conduction which is collision or by radiation which is photons jumping between atoms again and again until they escape. I have read that one photon’s energy can take 10000 years to cross the radiation zone.”

“So how is the next zone different?”

“It is much higher up from the center, nearly ¾ of the way to the surface. The pressure is 100 times less than in the fusion zone. The atoms have more room to move around together and form winds to carry the heat up by convection. But they can’t only go up, they have to come down, too, and that’s why my drawing has loops.”

“Is there a name for the loops?”

“Oh, yes, they are called Bénard cells and they’re very much like what I see looking into a pot of water just before it boils.”

“What’s the orange above the convection zone?”

“That’s the part of the Sun that we see, the photosphere that emits light in a continuous spectrum. The Fraunhofer lines, the dark lines in the astronomer’s spectrum, are the shadows of atoms high in in the photosphere that absorb only certain colors. I was surprised to learn how narrow the photosphere is, not even 0.02% of the Sun’s radius. Anyway, that’s my presentation, but now I have a question. The Sun’s fuel is hydrogen. The books say when the Sun runs out of fuel it will eject much of its hydrogen mass and collapse to a white dwarf. So it didn’t run out of fuel, yes?”

~~ Rich Olcott

# Generation(s) of Stars

“How’re we gonna tell, Mr Moire?”

“Tell what, Jeremy?”

“Those two expanding Universe scenarios. How do we find out whether it’s gonna be the Big Rip or the Big Chill?”

“The Solar System will be recycled long before we’d have firm evidence either way. The weak dark energy we have now is most effective at separating things that are already at a distance. In the Big Rip’s script a brawnier dark energy would show itself first by loosening the gravitational bonds at the largest scale. Galaxies would begin scattering into the voids between the multi‑galactic sheets and filaments we’ve been mapping. Only later would the galaxies themselves release their stars to wander off and dissolve when dark energy gets strong enough to overcome electromagnetism.”

“How soon will we see those things happen?”

“If they happen. Plan on 188 billion years or so, depending on how fast dark energy strengthens. The Rip itself would take about 2 billion years, start to finish. Remember, our Sun will go nova in only five billion years so even the Rip scenario is far, far future. I prefer the slower Chill story where the Cosmological Constant stays constant or at least the w parameter stays on the positive side of minus‑one. Weak dark energy doesn’t mess with large gravitationally‑bound structures. It simply pushes them apart. One by one galaxies and galaxy clusters will disappear beyond the Hubble horizon until our galaxy is the only one in sight. I take comfort in the fact that our observations so far put w so close to minus‑one that we can’t tell if it’s above or below.”

“Why’s that?”

“The closer (w+1) approaches zero, the longer the timeline before we’re alone. We’ll have more time for our stars to complete their life cycles and give rise to new generations of stars.”

“New generations of stars? Wow. Oh, that’s what you meant when you said our Solar System would be recycled.”

“Mm-hm. Think about it. Back when atoms first coalesced after the Big Bang, they were all either hydrogen or helium with just a smidgeon of lithium for flavor. Where did all the other elements come from? Friedmann’s student George Gamow figured that out, along with lots of other stuff. Fascinating guy, interested in just about everything and good at much of it. Born in Odessa USSR, he and his wife tried twice to defect to the West by kayak. They finally made it in 1933 by leveraging his invitation to Brussels and the Solvay Conference on Physics where Einstein and Bohr had their second big debate. By that time Gamow had produced his ‘liquid drop‘ theory of how heavy atomic nuclei decay by spitting out alpha particles and electrons. He built on that theory to explain how stars serve as breeder reactors.”

“I thought breeder reactors are for turning uranium into plutonium for bombs. Did he have anything to do with that?”

“By the start of the war he was a US citizen as well as a top-flight nuclear theorist but they kept him away from the Manhattan Project. That undoubtedly was because of his Soviet background. During the war years he taught university physics, consulted for the Navy, and thought about how stars work. His atom decay work showed that alpha particles could escape from a nucleus by a process a little like water molecules in a droplet bypassing the droplet’s surface tension. For atoms deep inside the Sun, he suggested that his droplet process could work in reverse. He calculated the temperatures and pressures it would take for gravity to force alpha particles or electrons into different kinds of nuclei. The amazing thing was, his calculations worked.”

“Wait — alpha particles? Where’d they come from if the early stars were just hydrogen and helium?”

“An alpha particle is just a helium atom with the electrons stripped off. Anyway, with Gamow leading the way astrophysicists figured out how much of which elements a given star would create by the time it went nova. Those elements became part of the gas‑dust mix that coalesces to become the next generation of stars. We may have gone through 100 such cycles so far.”

“A hundred generations of stars. Wow.”

~~ Rich Olcott

“A strawberry scone and my usual black, Al.”

“Sure thing, Sy, comin– Hiya, Cathleen, see my new poster? Event Horizon Telescope pictures of the two big‑guy black holes we’ve actually seen so far. Those white-hot blobs buried in those red rings. Ain’t it a beaut? What’ll you have?”

“They’re certainly wonderful graphics, Al. I’ll have a caramel latte, please, with a plain scone.” I’m waiting for it, because Cathleen never passes up a teachable moment. Sure enough — “Of course, neither one actually looks like that or represents what you think. Those images were created from radio waves, not visible light or even infrared. The yellows and whites don’t represent heat, and that darkness in the middle isn’t the black hole.”

“Whoa, don’t harsh Al’s happy, Cathleen. Maybe just go at it a step at a time?”

<sigh> “You’re right, Sy. Sorry, Al, I just get frustrated when press‑agent science gets in the way of the real stuff which is already interesting on its own. For instance, I haven’t seen anything in the pop‑sci press about the EHT people using the same 2017 data to produce both images, even though the two objects are almost 90° apart in the sky. I think about our optical telescopes and the huge high-tech motors it takes to point them in the right direction. These guys just re-work their data and they’re good for another round.”

“It’s a cute trick, alright, Cathleen, steering a distributed telescope with arithmetic.”

“OK, you guys are over my head — distributed telescope?”

“The EHT Collaboration works with eight radio telescopes scattered across the world. The signal from any point in the sky has a different time offset at each telescope depending on the angle to the point. If you know the baseline between each pair of scopes and you’ve got really good clocks keeping track of time at each location, when you combine the data from all eight locations it’s just arithmetic to pick out matching signals at the right set of offsets for any point of origin.”

“A lot of arithmetic, Cathleen.”

“I’ll give you that, Sy. Al, it took the researchers and some hefty compute facilities two years to boil down the data for the M87 monster. In principle, when they wanted to inspect the Milky Way’s beast all they had to do was run through the same data selecting for signal matches at the offsets pointing to Sgr A*. Awesome tech, huh?”

“Awesome, yeah, but if the colors aren’t heat, what are they?”

“Electron density, mostly. Your red‑and‑yellow Jupiter poster over there is like most heat maps. Researchers figure a pixel’s temperature by comparing data from multiple wavelengths with the Planck curve or some other calibrated standard. These images, though, came from a single wavelength, 1.3 millimeters. Light at shorter wavelengths can’t get past the dust, longer wavelengths can’t give us the image resolution. Millimeters waves are in the radio part of the spectrum — too low‑energy to detect moving charge inside atoms or between molecule components. The only thing that can give off those photons is free‑floating electrons. The brightest pixels have the most electrons.”

“So the hole isn’t the black hole?”

“Depends on your definition, I suppose. Everyone visualizes that black sphere, the event horizon, when they think ‘black hole.’ That’s not what the dark patches are. By my definition, though, a ‘black hole‘ is the whole package — central mass, event horizon, ergosphere if it’s spinning, a jet maybe and everything else that’s associated with the mass. It’s as much a collection of processes as a thing. Anyhow, the bright stuff in these images does come from accretion disks.”

“The dark patch is the disk’s inside edge?”

“Nope, it’s the shadow of the photon sphere. Before you ask, that’s a light‑trapping shell 1½ times the horizon’s diameter. Depending on its angle of approach, a photon that touches the sphere either spirals inward, orbits forever, or swerves outward. Going straight doesn’t happen. The shadow memorializes Earth‑bound photons that bounced away from us.”

“I guess my happy’s back, Cathleen, but it’s different.”

“You’re welcome, Al. Now how about the coffee and scones we asked for?”

~~ Rich Olcott

# Spitzer versus Webb

Why have I written so many posts about JWST? This is why.

~~ Rich Olcott

# Pushing It Too Far

It’s like he’s been taking notes. Mr Feder’s got a gleam in his eye and the corner of his mouth is atwitch. “You’re not getting off that easy, Cathleen. You said that Earendell star’s 66 trillion lightyears away. Can’t be, if the Universe’s only 14 billion years old. What’s going on?”

“Oops, did I say trillion? I meant billion, of course, 109 not 1012. A trillion lightyears would be twenty times further than the edge of our observable universe.”

“Hmph. Even with that fix it’s goofy. Sixty-six billion is still what, five times that 14 billion year age you guys keep touting. I thought light couldn’t travel that far in that time.”

“I thought the Universe is 93 billion light years across.”
”That’s diameter, and it’s just the observable universe.”
”Forty-seven billion radially outward from us.”
”None of that jibes with 14 billion years unless ya got stuff goin’ faster than light.”

“Guys, guys, one thing at a time. About that calculation, I literally did it on the back of an envelope, let’s see if it’s still in my purse … Nope, must be on my office desk. Anyhow, distance is the trickiest part of astronomy. The only distance‑related thing we can measure directly is z, that redshift stretch factor. Locate a familiar pattern in an object’s spectrum and see where its wavelength lies relative to the laboratory values. The go‑to pattern is hydrogen’s Lyman series whose longest wavelength is 121 nanometers. If you see the Lyman pattern start at 242 nanometers, you’ve got z=2. The report says that the lens is at z=2.8 and Earendel’s galaxy is at z=6.2. We’d love to tie those back to distance, but it’s not as easy as we’d like.”

“It’s like radar guns, right? The bigger the stretch, the faster away from us — you should make an equation outta that.”

“They have, Mr Feder, but Doppler’s simple linear relationship is only good for small z, near zero. If z‘s greater than 0.1 or so, relativity’s in play and things get complicated.”

“Wait, the Hubble constant ties distance to speed. That was Hubble’s other big discovery. Old Reliable here says it’s something like 70 kilometers per second for every megaparsec distance. What’s that in normal language? <tapping keys> Whoa, so for every lightyear additional distance, things fly away from us about an inch per second faster. That’s not much.”

“True, Sy, but remember we’re talking distant, barely observable galaxies that are billions of lightyears away. Billions of inches add up. Like with the Doppler calculation, you get startling numbers if you push a simple linear relation like this too far. As an extreme example, your Hubble rule says that light from a galaxy 15 billion lightyears away will never reach us because Hubble Flow moves them away faster than photons fly toward us. We don’t know if that’s true. We think Hubble’s number changes with time. Researchers have built a bucketful of different expansion models for how that can happen; each of them makes different predictions. I’m sure my 66 came from one of those. Anyhow, most people nowadays don’t call it the Hubble constant, it’s the Hubble parameter.”

“Sixty-six or forty-seven or whatever, those diameters still don’t jibe with how long the light’s had a chance to travel.”

“Sy, care to take this? It’s more in your field than mine.”

“Sure, Cathleen. The ‘edge of the observable universe‘ isn’t a shell with a fixed diameter, it simply marks the take-off points for the oldest photons to reach us so far. Suppose Earendel sent us a photon about 13 billion years ago. The JWST caught it last night, but in those 13 billion years the universe expanded enough to insert twenty or thirty billion lightyears of new space between between here and Earendel. The edge is now that much farther away than when the photon’s journey started. A year from now we’ll be seeing photons that are another year older, but the stars they came from will have flown even farther away. Make sense?”

“A two-way stretch.”

“You could say that.”

~~ Rich Olcott

• Thanks to my brother Neil, who pointed out the error and asked the question.

# A Thumbtack in A Needlestack

“What’re the odds?”

“Odds on what, Vinnie?”

“A gazillion galaxies out there, only 41 lensing galaxy clusters, but one of them shows us a singleton star. I mean, what’s special about that star? What are the odds?”

I can’t help it. “Astronomical, Vinnie.”

Cathleen punches my shoulder, hard. “Sy Moire, you be ashamed of yourself. That pun was ancient a century ago. Vinnie, the odds are better than they seem. We didn’t just stumble on Earendel and the Sunrise Arc, we found them in a highly targeted Big Data search for things just like that — objects whose light was extremely stretched and also gravitationally bent in our direction. The Arc’s lensing galaxy cluster has a spherical effect, more or less, so it also acts on light from other far-away objects and sends it in other directions. It even bends an image of our Milky Way towards Earendel’s galaxy.”

“I call weaseling — you used ‘more or less‘.”

“Guilty as charged, Vinnie. A nice, spherical black hole is the simplest case of gravitational lensing — just one mass at the center of its simple light‑bending gravity field. Same thing for a single star like our Sun. Clusters are messy. Tens or hundreds of billion‑star galaxies, scattered at random angles and random positions about their common center of mass. The combined gravity field is lumpy, to say the least. Half of that research paper is devoted to techniques for estimating the field and its effects on light in the region around the Arc.”

“I guess they had to get 3D positions for all the galaxies in the cluster. That’d be a lot of work.”

“It would, Al, but that’s beyond what current technology can do. Instead, they used computer models to do — get this, Sy — curve fitting.”

<chuckle> “Good one, Cathleen.”

“What’s so funny?”

“There’s a well-established scientific technique called ‘curve fitting.’ You graph some data and try to find an equation that does a respectable job of running through or at least near your data points. Newton started it, of course. Putting it in modern terms, he’d plot out some artillery data and say, ‘Hmm, that looks like a parabola H=h+v·t+a·t2. I wonder what values of h, v and a make the H-t curve fit those measurements. Hey, a is always 32 feet per second per second. Cool.’ Or something like that. Anyhow, Cathleen’s joke was that the researchers used curve fitting to fit the Sunrise Arc’s curve, right?”

“They did that, Sy. The underlying physical model, something called ‘caustic optics,’ says that—”

“Caustic like caustic soda? I got burnt by that stuff once.”

“That’s the old name for sodium hydroxide, Vinnie. It’s a powerful chemical and yeah, it can give you trouble if you’re not careful. That name and caustic optics both come from the Greek word for burning. The optics term goes back to using a lens as a burning glass. See those focused patterns of light next to your water glass? Each pattern is a caustic. The Arc’s lensing cluster’s like any light‑bender, it’s enclosed in a caustic perimeter. Light passing near the perimeter gets split, the two parts going to either side of the perimeter. The Earendel team’s curve‑fitting project asked, ‘Where must the caustic perimeter be to produce these duplicate galaxy images neighboring the Arc?‘ The model even has that bulge from the gravity of a nearby foreground galaxy.”

“And the star?”

“Earendel seems to be smack on top of the perimeter. Any image touching that special line is intensified way beyond what it ought to be given the source’s distance from us. It’s a pretty bright star to begin with, though. Or maybe two stars.”

“Wait, you don’t know?”

“Not yet. This study pushed the boundaries of what Hubble can do for us. We’re going to need JWST‘s infrared instruments to nail things down.”

Al’s in awe. “Wow — that caustic’s sharp enough to pick one star out of a galaxy.”

“Beat the astronomical odds, huh?”

~~ Rich Olcott

# A Needle in A Needlestack

“How’d they find that far-away star, Cathleen? Seems like you’d have to know just where to point your telescope.”

“It’s worse than that, Al, first you’ve got to find that telescope, or more precisely, its lens. We can’t simply swing a black hole or galaxy cluster into position for a good look at something interesting. No, we have to discover lensing objects that magnify good stuff beyond them. The good news is that some of those are out there, but the bad news is that the sky is cluttered with far more objects that don’t play the game we want. This research team appears to have hit paydirt but they did it with humungous power shovels and heavy‑duty panning techniques.”

“Impressive metaphor, Cathleen. Could you un‑metaphor it for us?”

“Sure, Sy. The power shovels are Hubble and Spitzer, both of which piled up beaucoodles of data from decades of infrared observing time.”

“I thought Hubble was designed for visible and UV surveillance.”

“It is, mostly, but since 2009 its instrument suite included WFC3, a camera that’s sensitive out to 1700 nanometers and covers a square 2 arcminutes on a side. That’s a lot, by big‑telescope astronomy standards.”

“Wait, arcminutes?”

“That’s right, Mr Feder. We astronomers have trouble with distances but we’re good at measuring angles. The Moon’s about a degree across. One degree is sixty arcminutes, next step down is sixty arcseconds per arcminute. After that we go semi‑metric, milliarcseconds and so forth. One WFC3 pixel records a patch of sky 130 milliarcseconds across. JWST‘s NIRCam instrument has a resolution twice as sharp. Anyway, Hubble‘s 1700‑nanometer limit is plenty good enough to pick up 120‑nanometer hydrogen light that’s been stretched out by a factor of z=2.8. Distance and stretch correlate; the lens that highlighted Earendel and its Sunrise Arc for NASA and Vinnie is that far away.”

“How far away?”

“It’s tricky to answer that. The spectra we see let us measure an object’s z‑factor, which by way of the Doppler effect tells us how fast the object is moving away. Hubble’s constant ties that to distance, sort of. My convenient rule of thumb is that an object whose z is near 2 is running away at 80% of lightspeed and on the average is about 55 trillion lightyears from us but don’t quote me because relativity complicates matters. Using the same dicey calculation I estimated the lens and Earendel velocities at 87% and 96% of lightspeed, which would put their ‘proper distances‘ around 60 and 66 trillion lightyears away. And no, I’m not going to go into ‘proper distance‘ versus ‘comoving distance‘.”

“Let’s get back to your metaphor, Cathleen. I get that Hubble and Spitzer and such generated a ton of data. What’s the panning part about?”

“Well, in the old days it would have been hired hands and graduate students spending years peering at dots on photographic plates. These days it’s computers, thank Heaven. The research team used a series of programs to filter their digital data. The software had to decide which dots are stars or noise specks and which are galaxies or arcs. Then it picked out the reddest red galaxy images, then clusters of galaxy images at the same redness level that are near each other in space, then clusters with arcs around them. I said that WFC3 covers a square 2 arcminutes on a side, remember? The sky, both hemispheres, contains almost 2½ million squares like that, although the surveys didn’t get all of them. Anyhow, after burning through cubic acres of computer time the team found 41 deep red lensing clusters.”

“Only 41.”

“Yup.”

We ponder that for a minute, then Vinnie pipes up. “Wait, the dots are in color?”

“No, but these images are generally taken through a filter that transmits only a known narrow wavelength range, infrared or whatever. Using relative dot intensity at several different wavelengths you can create ‘false color‘ images. When you find something, you know where to point spectroscopic tools to be sure you’ve found the good stuff.”

“Like a star shining less than a billion years after the Big Bang.”

“Paydirt.”

~~ Rich Olcott

# When The Stars Are Aligned Right

Cathleen and I are chatting when Vinnie bursts into the coffee shop waving a newspaper. “New news, guys, they’ve just announced Hubble spotted the farthest‑away star. How about that? Think what JWST will be able to do!”

Cathleen raises an eyebrow. “Sounds like press release science. What else do they say?”

“Not a whole lot. Lessee… These guys went through old Hubble data and found a piece of an Einstein ring which I don’t know what that is and partway along the ring is a star and somehow they figured out it’s 50 times heavier than the Sun and 12 billion years old and it’s the farthest star they’ve ever seen and that’s why NASA’s all excited.”

“Do you believe all that?”

“Maybe the NASA PR people do?”

“Maybe. I just read the technical paper behind that announcement. The authors themselves aren’t absolutely sure. The paper’s loaded with supporting evidence and ‘how we did it‘ details but it’s also loaded with caveats. The text includes a string of alternative explanations for their observations, winding up with a typical ‘we await further evidence from JWST‘ statement. Reads a lot more like real science. Besides, we’ve already seen more distant stars but they’re all jumbled together inside their very distant galaxies.”

“It’s a gravitational lensing effect. Sy, does Old Reliable still have a copy of that graphic you did about gravitational lensing?”

“That was years ago. Let me check… Uh‑huh, here it is.”

“Thanks. Vinnie, you know how a prism changes light’s direction.”

“Sy and me, we talked about how a prism bends light when light crosses from air to glass or the other way ’cause of the different speed it goes in each material. Uhh, if I remember right the light bends toward the slower speed, and you get more bend with shorter wavelengths.”

“Bingo, Vinnie. Gravitational lensing also bends light, but the resemblance ends there. The light’s just going through empty space, not different media. What varies is the shape of spacetime itself. Say an object approaches a heavy mass. Because of relativity the space it moves through appears compressed and its time is dilated. Compressed distance divided by dilated time means reduced velocity. Parts of a spread‑out lightwave closest to the mass slow down more than parts further way so the whole wave bends toward the heavy mass. Okay?”

“Hold on. Umm, so in your picture light coming towards us from that galaxy doesn’t get blocked by that black thingy, the light bends around it on both sides and focuses in on us?”

“Exactly. Now carry it further. The diagram cuts a flat 2D slice along round 3D spatial reality. Those yellow lines really are cones. Three‑sixty degrees around the black blob, the galaxy’s light bends by the same amount towards the line between us and the blob. Your Einstein ring is a cut across the cone, assuming that the galaxy, the blob and Earth are all exactly on the same straight line. If the galaxy’s off‑center the picture isn’t as pretty — you only get part of a ring, like those red arcs in Sy’s diagram.”

“A galactic rainbow. That ought to be awesome!”

“Well it would be, but there’s another difference between prisms and blobs. Rainbows happen because prisms and raindrops bend short‑wavelength colors more than longer ones, like you said. Gravitational lensing doesn’t care about wavelength. Wavelengths do shift as light traverses a gravitational well but the outbound red shift cancels the inbound blue shift.. Where gravity generates an Einstein ring, all wavelengths bend through the same angle. Which is a good thing for bleeding‑edge astronomy researchers.”

“Why’s that, Cathleen?”

“If the effect were wavelength‑dependent we’d have aberration, the astronomer’s nemesis. Images would be smeared out. As it is, all the photons from a point hit the same spot on the sensor and we’ve got something to see.”

“Good point, Sy. Each galactic star emits light in every direction. In effect, the blob collects light over its entire surface area and concentrates that light along the focal line. We get the brightest image when the stars are aligned right.”

~~ Rich Olcott

# Now And Then And There

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.”

“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

# Now And Then

“Alright, I suppose there’s no going down below the Universe’s Year Zero, but what about the other direction? Do you physics guys have a handle on Time’s Top?”

“That’d be Cosmology, Mr Feder. We physicists avoid theorizing about stuff we can’t check against data. Well, except for string theory. The far past leaves clues that astronomers like Cathleen can gather. Sad to say, though, we barely have a handle on Now.”

Cathleen grins. Al and Mr Feder go, “Whaaat?”

“No, really. One of Einstein’s insights was that two observers randomly and independently flying through space won’t be able to agree on whether two external events occurred simultaneously. They can’t even agree on what time it is now.”

“Oh, yeah, I know about that. I’ve read about how the GPS system needs to make corrections to account for what relativity does to the satellite timings.”

“You’re right, Al, but that’s a different issue. Some of that relativistic correction has to do with space compression because of Earth’s mass. The simultaneity problem is strictly about rapid motion and geometry.”

“Wait — geometry?”

“Relativistic geometry, which is a bit different from the kind that Descartes built.”

“Whoa, Sy, slow down there. Descartes was the ‘I think therefore I am‘ guy, right? What’s that got to do with geometry?”

“I guess I got a little ahead of myself there, didn’t I? OK. Yeah, Al, same Descartes. Grew up Catholic in France, was a professional mercenary soldier in the Thirty Years War, wound up fighting first on the Catholic French side and later on fought on the Protestant Dutch side but cross‑over was common, both directions. He realized he was in an ostensibly religious war that was really about who ruled over whom. That may have had something to do with him becoming a professional philosopher who rejected all religious dogmas in favor of what he could learn solely from logic and his own senses. That’s where his famous mantra came from — he started by proving to himself that he existed.”

“Logic led to geometry, I suppose.”

“Indeed, but a new kind, one that required a few innovations that Descartes developed. On the one hand, mathematicians traditionally expressed algebraic problems in words and some of them were doozies, like saying ‘the zenzizenzizenzic‘ where we’d just say x8. We got that simple but <ahem> powerful notation from Descartes. On the geometry side, he’d ditch all the confusing line-ending markers in a diagram like this one. Instead, he’d label the whole line representing a known quantity with a front-of-the-alphabet letter like a or b or c. A line representing an unknown quantity would get its label from the alphabet-trailers like x, y and z. Then he used the same character conventions and his new power notation to write and manipulate algebraic expressions. Those notational inventions were foundational for his bridge between algebraic and geometrical problems. Draw your problem with lines and curves, transform it to algebraic equations, solve that problem exactly, transform it back to geometry and you’re done. Or vice-versa.”

“That goes back to Descartes, huh?”

“Mm-hm. His big innovation, though, arose from a borrow from an early Greek gadget called a mesolabe. He proposed an idealized version that would let someone break a line into exact fractions or compare a length against a unit length. That broke the rules of classical Geometry but setting his mesolabe’s Y‑angle to 90° prompted him to name points by their distance along the x– and y‑axes. That’s the nub of the Cartesian coordinate system — a rectangular grid of numbered straight lines that go on forever. Graph paper, right? Wrap the grid around the Earth and you’ve got latitudes and longitudes. Add more numbered grid lines perpendicular to either grid and you’ve got z‑axis coordinates. Three coordinates let you name any point in space. Newton and all the physicists who came after him until the dawn of the 20th Century assumed Descartes’ nice, stable coordinate system.”

“20th Century — that’s when Einstein came on the scene. He broke that system?”

“Sure did. You’ve heard about bent space?”

“Who hasn’t?”

“Well, fasten your seat belts, it’s going to be a fun ride.”

~~ Rich Olcott