Category: General

Dodeca to The Rescue

On By rolcottIn Cosmology, General: Fun Stuff, Physics: Black Holes and Relativity, UncategorizedLeave a comment

Days are getting warmer, so my walk around the park’s lake is at dawn. I’m enjoying the view when my phone blares out <bomPAH-dadadadaDEEdah>. Of course, the caller‑ID display shows 710‑555‑1701. “Good morning, Ms Baird. Surprised to hear from you again so soon. How is Project Lonesome coming along?”

“It’s been saved, thanks to your knowledge of antique Physics.”

“Antique??!?”

“Lenz’s Law was already nearly 200 years old in your time, Mr Moire, and I’m another 200 years on. Our engineers have to learn exponentially more than yours do. Things get lost in the welter so we missed the magnetic squirt‑out stumbling block. Anyway, armed with your hint I was able raise the issue early enough in the design phase that we could adjust.”

“Repulsion between opposed magnetic fields is pretty basic Physics. How will your people get around that?”

“By going from ten anti‑matter factory sites to twelve.”

“Huh? Your original design had ten current‑carrying tractor/pressor beams arrayed around the equator of Planetoid Road. That was a problem because currents in the beams would induce their own magnetic fields. The combined field would be perpendicular to the equator. Lenz’s Law says induced fields are in opposition to the black hole’s external field. Field‑field pressure would squeeze Road right out of the system. Twelve beams around the equator would make an even stronger magnetic dipole. How would that solve the problem?”

“It wouldn’t, but that’s not the configuration we’re looking at. Do you know what a dodecahedron is?”

“One of the Platonic solids — twelve identical pentagonal faces, lots of symmetry.”

“Right. So we’re going to center each AM factory in its own face of a dodecahedron plotted onto Road’s surface.”

“Ah, the symmetry’s nearly spherical so the induced fields won’t combine as a dipole. Cute.”

“It’s better than that, Mr Moire. The currents are pulsed, remember, and we didn’t expect to have all the factories running full‑out all the time. By varying individual pulse rates or even shutting down one or two factories we can control the strength and direction of our combined magnetic field. We can tweak that vector relative to the black hole’s field lines as needed to stabilize our position. We could even steer the planetoid around the system if we have to. All with zero fuel expenditure. Wins all around!”

“What about the spaceport?”

“Spaceport?”

“You’re making a lot of AM product here, and you’re going to need to ship it out to wherever Star Fleet needs it. That means Road will need a port and docking facilities and everything that goes with that. I had assumed that’d be at one of the poles. Zero rotational speed up there would would make for easy landings and take‑offs. But the new configuration would have the worldlet tumbling arbitrarily. No fixed poles. Where will you put the docks? For that matter, how else could you build and staff the factories in the first place?”

“That we did think about. Not even the original design included on‑planetoid docking. Landing a ship at either magnetic field convergence point would have been too risky. That’s why we were originally going to site all the AM factories at the equator. Putting them somewhere else is no problem because each factory will be assembled in space. Space tugs can drop it into any position and then flit away. Every AM factory will be fully automated, including the robots that will repair whatever needs repairing, so no staffing required.”

“But how will you get the AM away from the planetoid?”

“The usual way. The AM will be magnetically suspended in massive gamma bixonate cannisters transmitted by industrial‑scale transporters. The bixonate blocks the black hole’s field. Transporter range is limited near Lonesome’s steep gravity well, but we can orbit a transshipment facility close enough to be useful but far enough to be safe.”

“What if a transporter malfunctions? You wouldn’t want AM spraying around the transshipment deck.”

“Reception transporters for AM cargoes work from the outside in so the cannister’s integrity is complete before the AM signal exits the buffer.”

“I sense a promotion in your future.”

~ Rich Olcott

  • Thanks to Alex and Xander for keeping me on‑topic.

The Polar Peril

On By rolcottIn Cosmology, General: Fun Stuff, Physics: Black Holes and Relativity, UncategorizedLeave a comment

This park bench is perfectly situated for watching the summer solstice sun setting across the lake. I’m enjoying the view when <bomPAH-dadadadaDEEdah> blares from my phone. Number 710‑555‑1701 on the caller‑ID display evokes memories of a sultry stare and pointed ears…. “Good evening, Ms Baird. What’s new in the 23rd century?”

“Not much that I can tell you about, considering the Prime Directive. We’re taking a second shot at Project Lonesome. You may recall we’d identified an isolated black hole we called Lonesome. We were going to use two orbiting planetoids, Pine and Road, to generate electricity from its spinning electromagnetic field. Star Fleet’s anti‑matter synthesizers need that power.”

“Yes, I remember. Three‑body gravitational instability cost you Pine and your rank. Have you recovered the rank?”

“I’m only up to Lieutenant Commander so far, but part of the deal was that I’m back on Invigilator for a second shot at Lonesome. This time we’ve simplified the design. We had to, we only have Road to work with.”

“If I recall correctly, the idea was that the rotating field would induce electrical currents within conducting tractor beams strung between Pine and Road. How are you going to do that with only one orbiter? Relativity would be a challenge if you string something down to Lonesome.”

“We’re not stupid, Mr Moire, and we’re quite familiar with Relativity’s effects. I’ll give you an example your science is ready for. General Relativity says that time runs more slowly the deeper you go into a gravity well, yes?”

“Indeed. Our GPS time‑signal calibrations include an Einstein correction for reduced Earth gravity at satellite height. We have a movie, Interstellar, where gravitational time dilation was a major plot element.”

“A classic film. Even now we watch it occasionally. Consider this: on its own timeline, a star within its gravity well is younger than remote measurements say it is. The more massive the star, the greater the difference between experienced and apparent age.”

“Ooo, nice.”

“Counteracting gravitational discrepancies was a major challenge when we were developing transporter technology. Physics just doesn’t like teleporting a material object between different timescales, even through subspace. We finally gave up on that approach. What worked was transcribing the object into a subspace signal between pattern buffers. In the receiving environment we rebuild from the buffered signal and amorphous matter. Sometimes roundabout is the best path.”

“I’m with Bones on that—”

“Who?”

“Doctor Leonard McCoy.”

“Never heard of him. Must have been before my time.”

“Anyway, he’ll refuse to use transporters, convinced they’ll spray his atoms all over the Universe.”

“Way, way before my time, then. These days they’re as safe as your airplanes. Safer.”

“So what’s the new design?”

“The plan is to site ten plants around Road‘s equator. Each plant will have a free‑standing AM production facility teamed with two conductive tractor/pressor beams locked onto a floating plate of gamma‑phase bixonate. I’m allowed to give you those details because it’ll be a century before you’ll have bixonate production technology, much less gamma restructuring capability.”

“Why the special material?”

“At the nanoscale level, gamma bixonate’s surface is an array of corner‑cube mirrors that perfectly reflect electron waves back to their source. Meanwhile, the jagged edges between nano‑cubelets topologically obstruct local coherent magnetic fields. The combination is key to my Project Lonesome architecture. The power production cycle starts with injecting pulses of spin‑up electrons into one beam’s center where they’re trapped and streamed upwards. The black hole’s rotating magnetic field accelerates the electrons until they hit the bixonate nano‑cubelets that reflect them down the other beam. The reflection operation also flips the electron spin state so now the field accelerates them downward towards the factory’s energy receptors.”

“Double the acceleration effect.”

“Mm‑hm. We expect to achieve energies in the high exa‑electronvolt range, about a billion times what your CERN instrument can do. That’s plenty for AM synthesis.”

“One question.”

“Yes?”

“Have your engineers considered Lenz’ Law? Your immense electric currents will generate their own magnetic fields in opposition to the black hole’s. The induced fields will squirt Lone right out of the system.”

A sudden shocked pause, then the connection drops.

~ Rich Olcott

  • Thanks to Xander for a deeper Time question.

By Her Bootstraps

On By rolcottIn General: Fun Stuff, Physics: Einstein, UncategorizedLeave a comment

A rainy day and business is slow. Vinnie and I are playing cards in my office. He lays down a queen of hearts and gets a distant look in his eye. “I wonder where she is now. Or when.”

That ‘or when’ phrase told me who he’s thinking about — Anne, with her white satin attire and molten‑silver voice. “I’m always wondering about that, Vinnie, but it’s hard to apply the word ‘now’ to a time‑traveler. She’s visited so many time periods. When you say ‘now’ you’re covering a lot of separated moments from our perspective, even if there’s only one continuous series from hers.”

“Good point, Sy. I still have trouble believing she’s real and not some figment of your imagination. Yeah, Eddie and Cathleen and Susan and me, we’ve all met her. But like Cal said the other day, it looks like she breaks some pretty fundamental rules. Conservation of Energy, for one.”

“How do you figure that?”

“Stands to reason. Einstein says mass is energy, right?”

“Mm-hm.”

“Anne has mass, right?”

“Mm-hm.”

“So the Universe is moving along, one second per second as usual, with however much total energy it’s got. Suddenly Anne appears when there wasn’t an Anne before. That’s a burst of energy that wasn’t in the Universe up to then. Later on, when she feels like it, she pops out to some other time or Universe and her mass‑energy disappears. Where’s the Conservation?”

“I can think of two possibilities. First, maybe Anne is a virtual particle. According to Heisenberg’s Uncertainty Principle, virtual particles are always popping into and out of existence even in empty space. In this application, ‘uncertainty’ is the difference between ‘there’ and ‘not there.‘ How long a virtual particle lasts depends on how much energy it packs — sneaky little ones can hang around for a while, big ones generally check out quickly. Stephen Hawking said that black holes evaporate by eating the anti‑particle partners of virtual particle pairs created at the Event Horizon.”

“Put numbers to that.”

<poking at Old Reliable’s keypad> “Let’s see… I have no idea how much she weighs, say it’s about 50 kilograms. For an object with that mass, the minimum expected lifetime would be about 10‑53 second. Anne’s lifetime is enormously longer than that, thank goodness, so we can probably rule out the virtual particle option. On the other hand, maybe Anne somehow keeps a separate set of books. Maybe she conserves her own personal mass‑energy which doesn’t count in whatever Universe she’s visiting. As a physicist I’m not willing to guess how she’d do that.”

“I like the second one. Makes sense, sorta. But Cal came up with another difficulty, though.”

“Which is?”

“Chains of cause and effect. You’ve said keeping the links in the right order was a big part of Einstein’s Relativity theory.”

“That’s true. Any observer should see a distant cause precede its effect no matter the observer’s speed relative to the event locations. That imperative means that the Universe has Lorentz symmetry which is the basis for Special Relativity. That’s a fundamental.”

“Right. So what blocks time‑traveling Anne from looping back to cause or uncause something? Killing off Hitler while he was still painting houses, for instance, or getting Columbus to schedule his trip a little later into hurricane season. Anyone else in the Universe would see Anne’s effect happening before she went back to do that.”

<fffshwwwPOP!!> A molten‑silver voice behind me. “Easy answer, Vinnie. And no, I’m not in Sy’s imagination, see?”

“Whoa! Talk about timing. Hi, Anne. You look way too good for Sy’s imagination. What’s your easy answer?”

“Pure self-interest. Ever read Heinlein’s story, By His Bootstraps? My timeline is so twisted back and forth I don’t have any idea where I started from. Sometimes I meet myself and I don’t know whether I’m coming or going. We never know what to say to each other. If I were to change anything, anywhere or anywhen, it’d probably get in the way of my being me. My strict rule is, ‘Always an observer, never a participant.‘ It’s worked so far.”

~ Rich Olcott

  • Thanks to Alex for getting me started on this topic.

Fibonacci’s Legacy

On By rolcottIn General: Fun Stuff, Mathematics, Mathematics: Logs and Exponentials, Uncategorized3 Comments

Of course you’ve seen images featuring the Fibonacci spiral, maybe one that overlays the spiral onto a snail shell or the whorls of a sunflower head. As you see, it’s derived from a series of squares, the side of each equal to the sum of the sides of the next two smaller ones. The spiral smoothly connects opposite corners of successive squares. Fibonacci didn’t invent the spiral, but he did discover the {1, 1, 2, 3, 5, 8, 13, 21, 34, …} series it exploits. He’s famous for the series, but that was as dust before his other major contribution to our world.

Leonardo was the foremost mathematician of 13th‑century Italy. His father was Guglielmo Bonacci so Leonardo was figli di (son of) Bonacci, hence fi’Bonacci or Fibonacci. As a citizen of the Pisan Republic he was often cited as Leonardo di Pisa, preceding the more famous Leonardo of Vinci (a district in Florence, 90 kilometers east of Pisa) by nearly three centuries.

In 1202 Fibonacci wrote a ground‑breaking mathematics textbook, Liber Abaci, that included many solved examples of problems from trade and finance. His famous rabbit problem was part of the mix.

Fibonaci’s rabbits, diagram by “Romain” under CCS-A 4.0 license

Suppose you’ve got a pair of immortal rabbits that multiply like clockwork — each pair more than a month old produces one additional pair per month. How many pairs would you have after n months? The figure shows the first few stages of the process. At the end of any month you’ve got all the start‑of‑month rabbits plus one pair for each new pair that reached one‑month maturity. 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21, … That is Fibonacci’s Series.

Australia discovered to its sorrow how rapidly the Series’ numbers can rise. After the first few generations the rabbit population grows literally exponentially (Those two words are misused way too often but they each apply here.).

Exponential growth appears when a process operates periodically on something to produce more of the same thing. The periodicity can be timewise, like seasonal years or generations of living things, or spacewise, like successive outward divisions of the cells that give rise to sunflower seeds or snail shells. If the something comes in units, like rabbit pairs, the growth comes in integers. If the units are small and the numbers are large, like yeast cells in a beer vat, it’s easier to express the growth as a percentage. The small‑unit numbers eventually increase as en/R, where R=½(1+√5). Compound interest? Same mechanism, same mathematics, just more arithmetic.

Back to the 12th century. Papa Bonacci was a merchant engaged in commerce between Pisa and Algiers. Young Leonardo grew up in his Dad’s North African trading post, first observing and later negotiating with Arabian traders who used a number system they’d gotten from Hindu merchants. He was smart enough to appreciate the power inherent in the Hindu‑Arabic number system (powers‑of‑10 positional notation, with a zero indicator to mark each slot where a number lacks a contribution from that 10‑power). In later years he wrote it up, in Liber Abaci.

At the dawn of the 13th century, Europe had been using Roman numerals for a millennium. The Italian trading empires of the 15th century could not have been built without the rigor, discipline and reliability of double‑entry bookkeeping. Can you imagine doing double‑entry accounting, much less compound interest calculations, with Roman numerals? I can’t, either. Europe’s commercial interests simply couldn’t have developed without the trustworthy record‑keeping techniques based on Liber Abaci‘s arithmetic principles. It’s easy to find realistic through‑lines from Fibonacci’s book to most of modern technology.

Today we say that computing is all ones and zeros. Fibonacci brought us the zeroes.

Rabbit image by JJ Harrison,
under CC ASA 3.0 license
  • 547 is a prime number. This the 547th post in this blog, a suitable spot for me to hit “Pause” so I can free up time for another project. That one promises plenty of rabbit holes like the ones I’ve explored while having fun researching topics here. There are limits. In the future I’ll be posting here irregularly rather than on the weekly schedule I’ve kept to for a decade. Stay in touch.

~ Rich Olcott

Cozy Numbers

On By rolcottIn General: Fun Stuff, Mathematics: Dimensions, UncategorizedLeave a comment

If you’ve followed this blog for more than a year (in which case, thanks), you’ve probably noticed that I like numbers. As a kid I got a thrill when I figured out why (111 111 111)² is the amazingly symmetrical 12 345 678 987 654 321. Like many other numberphiles I’m fascinated by special categories like primes (no divisor other than unity and the number itself) and figurate numbers (counting the dots that make a geometrical pattern like concentric hexagons).

Every number has charms of its own (2025 is a delicious example), but 2026 doesn’t seem like it’s particularly special. 2026 is 2×1013, so it’s not prime nor is it the square or power of some smaller number. It’s not palindromic like 1991 and 2002 are. In binary it’s 111 1110 10102 — not very interesting. It can be written as 201013 , which is just silly. But 2026 does turn out to be special, in a special way — it’s a Beprisque number and they’re rare.

The On-line Encyclopedia of Integer Sequences®, a.k.a. OEIS, does for whole numbers what Bartlett’s Familiar Quotations does for words: it’s a venerable compendium of usage in context, with citations. An OEIS search for “2026” found 270 sequences that include the number. Most are technical or abstruse to a fault. Entry A118454, for example, identifies 2026 as the algebraic degree of the onset of the logistic map’s 11th bifurcation. Don’t ask.

Among my search hits was A216840, “palindromic numbers of 3 digits in two bases.” It’s cool that 202610 is bcb13 and also a4a14. Unfortunately, my math software’s number‑base converter has only 36 available digits (0‑9, a‑z) so it can’t display numbers in base‑37 or worse.

My favorite entry in the set is A163492: Beprisque numbers are integers such that their two immediate neighbors on the number line are a perfect square and a prime. 10 is Beprisque because it’s nestled between 9=3² and 11 (a prime). So is 2026, sitting between 2025=45² and 2027 (prime).

“Beprisque” — looks kind of French, doesn’t it? Maybe named after one of those 18th century scientist‑aristocrats that Robespierre killed off? Nope, it’s simply a mash‑up of “between prime and square.”

This being a New Year post, I couldn’t resist mentioning the New Year monkey. Every year the financial press services trot the poor thing out, give it a handful of darts with instructions to fling them it at the Wall Street Journal‘s stock listing or some such. The monkey’s acumen is assessed by how much its dart‑selected equities have grown (or not) over the past year. Usually the monkey’s score rates near or even better than that of the “professional” stock pickers, which is supposed to tell us something.

Suppose we were to direct the monkey to toss its darts at the number line. How well would it do at hitting Beprisque numbers? How dense are the primes and squares and what are the odds that one from one category is exactly two seats away from one from the other?

It depends on how much of the line the monkey can aim at. The range between 1 and 100 has 10 squares (10% coverage), 25 primes (25% coverage) and 9 Beprisques (1, 2, 3, 8, 10, 24, 48, 80, 82). Nine out of a hundred is 9% coverage; looks good. But widen the range to 1000 and the coverages drop — 31 squares (3%), 168 primes (17%) and 17 Beprisques — 10 times the range but fewer than twice the hits. Extend the number line to 1 000 000 and the percentages get even worse: 0.1% squares, 7.9% primes and only 0.022 6% Beprisques. The Beprisque coverage drops to 0.007 2% if the monkey’s target range shifts to the span between one and two million.

Why so few? On the average primes are further apart for bigger numbers because the prime count grows about 15% more slowly than the range does. Meanwhile, the separation between successive squares grows linearly as the squared number increases:
  (n+1)² – n² = n²+2n+1 – n² = 2n+1
The two trends conspire to thin out those cozy sequences. Give the poor monkey an infinite range and its odds on a square‑Beprisque‑prime triplet evaporate like a certain team’s championship hopes.

~ Rich Olcott

Does Tomorrow Exist?

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

Power’s back on. The elevator lets us out on the second floor where we proceed into Eddie’s Pizza Place. We order, find a table, and Cathleen cocks an eyebrow. “So, Anne, you’re a time‑traveler?”

“Lots of dimensions, actually. Time, space, probability… Once I accidentally jumped into a Universe where the speed of light was a lot slower. I was floating near a planet in a small system whose sun flared up but it took a long, long time for the flash to reflect off the planet behind me. Funny, I felt stiffer than usual. It was a lot harder to move my arms. I avoid cruising dimensions like that one.”

<The other eyebrow goes up> “Wait, what’s the speed of light got to do with dimensions? And why would it affect moving your arms?”

My cue. “Physics has a long‑standing problem with the speed of light and a dozen or so other fundamental numbers like Newton’s gravitational constant and Einstein’s cosmological constant. We can measure them but we can’t explain why they have the values they do. Okay, the speed of light depends on electric and magnetic force constants, but we can’t explain those, either — the rabbit hole just gets deeper. In practice, our Laws of Physics are a set of equations with blanks for plugging in the measured values. People have suggested that there’s a plethora of alternate universes with the same laws of physics we have but whose fundamental constants can vary from ours. Apparently Anne traveled along a dimension that connects universes with differing values of lightspeed.”

“I suppose. … But the arm‑moving part?”

“An effect of Special Relativity. Newton’s Second Law a=F/m says that an object’s acceleration equals the applied force per unit mass. That works fine in every‑day life but not when the object’s velocity gets close to lightspeed.” <jotting on a paper napkin> “I don’t see Vinnie nearby so here’s the relativistic equation: a=(F/m)×√[1–(v/c)²]. The v/c ratio compares object velocity to lightspeed. The Lorentz factor, that square root, is less than 1.0 for velocities less than lightspeed. This formula says a given amount of force per unit mass produces less acceleration than Newton would expect. How much less depends on how fast you’re already going. In fact, the acceleration boost approaches zero when v approaches c. With me?”

“If your factor’s exactly zero, then even an infinite force couldn’t accelerate you, right? But what’s all that got to do with my arm?”

“Zero acceleration, mm‑hm. Suppose your arm’s rest mass and muscle force per unit mass are the same in the slow‑light universe as they are in ours. The Lorentz factor’s different. Lightspeed in our Universe is 3×108 m/s. Suppose you wave your arm at 10 m/s. Your Lorentz factor here is √[1–(10/3×108)²] which is so close to unity we couldn’t measure the difference. Now suppose ‘over there’ the lightspeed is 20 m/s and you try the same wave. The Lorentz formula works out to √[1–(10/20)²] or about 85%. That wave would cost you about 15% more effort.”

<Both eyebrows down> “Have you tried going forward in time?”

“Sure, but I can’t get very far. It’s like I’ve got an anchor ‘here.’ I can move back ‘here’ from the past, no problem, but when I try to move forward from ‘here’ even a day or so … It’s hard to describe but as I go everything feels fuzzier and then I get queasy and have to stop. Do you have an explanation for that, Sy?”

“Well, an explanation but I can’t tell you it’s correct. Einstein thought it conflicts with Relativity, other people disagree. According to the growing block theory of time, the past and present are set and unchanging but the future doesn’t exist until we get there. Your description sounds like a build on that theory, like maybe the big structures extend a bit beyond us but their quantum details are still chaotic until time catches up with them. There are a few reports of lab experiments that would be consistent with something like that but it’s early days in the research.”

“As the saying goes, ‘Time will tell,’ right, Sy?”

“Mm-hm, lo que será, será.“

~ Rich Olcott

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

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

The Time Is Out of Joint

On By rolcottIn Cosmology, Crazy Theories, UncategorizedLeave a comment

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

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

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

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

“Why was Newton’s Universe special?

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

“It’s frames again, ain’t it?”

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

Jeremy gives me a confused look.

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

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

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

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

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

Vinnie erupts. “HAW! Successive approximation again!”

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

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

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

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

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

Vinnie hoots, “Bo-o-o-ohh-GUS!”

~ Rich Olcott