Category: Physics: Black Holes and Relativity

“Hot Jets, Captain Neutrino!”

On By rolcottIn Astronomy: Multi-messenger, Astronomy: Techniques, Cosmology, Physics: Black Holes and Relativity, Physics: Light and Relativity, UncategorizedLeave a comment

“Hey, Cathleen, while we’re talking IceCube, could you ‘splain one other thing from that TV program?”

“Depends on the program, Al.”

“Oh, yeah, you weren’t here when we started on this.  So I was watching this program and they were talking about neutrinos and how there’s trillions of them going through like my thumbnail every second and then IceCube saw this one neutrino that they’re real excited about so what I’m wondering is, what’s so special about just that neutrino? How do they even tell it apart from all the others?”

“How about the direction it came from, Cathleen?  We get lotsa neutrinos from the Sun and this one shot in from somewhere else?”

SMBH jet and IceCube
Images from NASA and JPL-Caltech

“An interesting question, Vinnie.  The publicity did concern its direction, but the neutrino was already special.  It registered 290 tera-electron-volts.”

“Ter-what?”

“Sorry, scientific shorthand — tera is ten-to-the-twelfth.  A million electrons poised on a million-volt gap would constitute a Tera-eV of potential energy.  Our Big Guy had 290 times that much kinetic energy all by himself.”

“How’s that stack up against other neutrinos?”

“Depends on where they came from.  Neutrinos from a nuclear reactor’s uranium or plutonium fission carry only about 10 Mega-eV, wimpier by a factor of 30 million.  The Sun’s primary fusion process generates neutrinos peaking out at 0.4 MeV, 25 times weaker still.”

“How about from super-accelerators like the LHC?”

“Mmm, the LHC makes TeV-range protons but it’s not designed for neutrino production.  We’ve got others that have been pressed into service as neutrino-beamers. It’s a complicated process — you send protons crashing into a target.  It spews a splatter of pions and K-ons.  Those guys decay to produce neutrinos that mostly go in the direction you want.  You lose a lot of energy.  Last I looked the zippiest neutrinos we’ve gotten from accelerators are still a thousand times weaker than the Big Guy.”

I can see the question in Vinnie’s eyes so I fire up Old Reliable again.  Here it comes… “What’s the most eV’s it can possibly be?”  Good ol’ Vinnie, always goes for the extremes.

“You remember the equation for kinetic energy?”

“Sure, it’s E=½ m·v², learned that in high school.”

“And it stayed with you.  OK, and what’s the highest possible speed?”

“Speed o’ light, 186,000 miles per second.”

“Or 300 million meters per second, ’cause that’s Old Reliable’s default setting.  Suppose we’ve got a neutrino that’s going a gnat’s whisker slower than light.  Let’s apply that formula to the neutrino’s rest mass which is something less than 1.67×10-36 kilograms…”Speedy neutrino simple calculation“Half an eV?  That’s all?  So how come the Big Guy’s got gazillions of eV’s?”

“But the Big Guy’s not resting.  It’s going near lightspeed so we need to apply that relativistic correction to its mass…Speedy neutrino relativistic calculation“That infinity sign at the bottom means ‘as big as you want.’  So to answer your first question, there isn’t a maximum neutrino energy.  To make a more energetic neutrino, just goose it to go even closer to the speed of light.”

“Musta been one huge accelerator that spewed the Big Guy.”

“One of the biggest, Al.”  Cathleen again.  “That’s the exciting thing about what direction the particle came from.”

“Like the North Pole or something?”

“Much further away, much bigger and way more interesting.  As soon as IceCube caught that neutrino signal, it automatically sent out a “Look in THIS direction!” alert to conventional observatories all over the world.  And there it was — a blazar, 5.7 billion lightyears away!”

“Wait, Cathleen, what’s a blazar?”

“An incredibly brilliant but highly variable photon source, from radio frequencies all the way up to gamma rays and maybe cosmic rays.  We think the thousands we’ve catalogued are just a fraction of the ones within range.  We’re pretty sure that each of them depends on a super-massive black hole in the center of a galaxy.  The current theory is that those photons come from an astronomy-sized accelerator, a massive swirling jet that shoots out from the central source.  When the jet happens to point straight at us, flash-o!”

“Duck!”

“I wouldn’t worry about a neutrino flood.  The good news is IceCube’s signal alerted astronomers to check TXS 0506+056, a known blazar, early in a new flare cycle.”

“An astrophysical fire alarm!”

~~ Rich Olcott

Bigger than you’d think

On By rolcottIn Astronomy: Techniques, Cosmology, Physics: Black Holes and Relativity, Physics: Light and Relativity, Physics: Quantum Mechanics, UncategorizedLeave a comment

Al’s coffee shop, the usual mid-afternoon crowd of chatterers and laptop-tappers.  Al’s walking his refill rounds, but I notice he’s carrying a pitcher rather than his usual coffee pot.  “Hey, Al, what’s with the hardware?”

“Got iced coffee here, Sy.  It’s hot out, people want to cool down.  Besides, this is in honor of IceCube.”

“Didn’t realize you’re gangsta fan.”

“Nah, not the rapper, the cool experiment down in the Antarctic.  It was just in the news.”

“Oh?  What did they say about it?”

“It’s the biggest observatory in the world, set up to look for the tiniest particles we know of, and it uses a cubic mile of ice which I can’t think how you’d steer it.”

A new voice, or rather, a familiar one. “One doesn’t, Al.”
Neutrino swirl 1“Hello, Jennie.  Haven’t seen you for a while.”

“I flew home to England to see my folks.  Now I’m back here for the start of the Fall term.  I’ve already picked a research topic — neutrinos.  They’re weird.”

“Hey, Jennie, why are they so tiny?”

“It’s the other way to, Al.  They’re neutrinos because they’re so tiny.  Sy would say that for a long time they were simply an accounting gimmick to preserve the conservation laws.”

“I would?”

“Indeed.  People had noticed that when uranium atoms give off alpha particles to become thorium, the alpha particles always have about the same amount of energy.  The researchers accounted for that by supposing that each kind of nucleus has some certain quantized amount of internal energy.  When one kind downsizes to another, the alpha particle carries off the difference.”

“That worked well, did it?”

“Oh, yes, there are whole tables of nuclear binding energy for alpha radiation.  But when a carbon-14 atom emits a beta particle to become nitrogen-14, the particle can have pretty much any amount of energy up to a maximum.  It’s as though the nuclear quantum levels don’t exist for beta decay.  Physicists called it the continuous beta-spectrum problem and people brought out all sorts of bizarre theories to try to explain it.  Finally Pauli suggested maybe something we can’t see carries off energy and leaves less for the beta.  Something with no charge and undetectable mass and the opposite spin from what the beta has.”

“Yeah, that’d be an accounting gimmick, alright.  The mass disappears into the rounding error.”

“It might have done, but twenty years later they found a real particle.  Oh, I should mention that after Pauli made the suggestion Fermi came up with a serious theory to support it.  Being Italian, he gave the particle its neutrino name because it was neutral and small.”

“But how small?”

“We don’t really know, Al.  We know the neutrino’s mass has to be greater than zero because it doesn’t travel quite as fast as light does.  On the topside, though, it has to be lighter than than a hydrogen atom by at least a factor of a milliard.”

“Milliard?”

“Oh, sorry, I’m stateside, aren’t I?  I should have said a billion.  Ten-to-the-ninth, anyway.”

“That’s small.  I guess that’s why they can sneak past all the matter in Earth like the TV program said and never even notice.”

This gives me an idea.  I unholster Old Reliable and start to work.

“Be right with you… <pause> … Jennie, I noticed that you were being careful to say that neutrinos are light, rather than small.  Good careful, ’cause ‘size’ can get tricky at this scale.  In the early 1920s de Broglie wrote that every particle is associated with a wave whose wavelength depends on the particle’s momentum.  I used his formula, together with Jennie’s upper bound for the neutrino’s mass, to calculate a few wavelength lower bounds.Neutrino wavelength calcMomentum is velocity times mass.  These guys fly so close to lightspeed that for a long time scientists thought that neutrinos are massless like photons.  They’re not, so I used several different v/c ratios to see what the relativistic correction does.  Slow neutrinos are huge, by atom standards.  Even the fastest ones are hundreds of times wider than a nucleus.”

“With its neutrino-ness spread so thin, no wonder it’s so sneaky.”

“That may be part of it, Al.”

“But how do you steer IceCube?”

~~ Rich Olcott

Rhythm Method

On By rolcottIn Astronomy: Multi-messenger, Astronomy: Stellar Science, Cosmology, Physics: Black Holes and Relativity, Physics: Light and Relativity, Physics: Newton and Gravity, UncategorizedLeave a comment

A warm Summer day.  I’m under a shady tree by the lake, watching the geese and doing some math on Old Reliable.  Suddenly a text-message window opens up on its screen.  The header bar says 710-555-1701.  Old Reliable has never held a messaging app, that’s not what I use it for.  The whole thing doesn’t add up.  I type in, Hello?

Hello, Mr Moire.  Remember me?

Suddenly I do.  That sultry knowing stare, those pointed ears.  It’s been a year.  Hello, Ms Baird.  What can I do for you?

Another tip for you, Mr Moire.  One of my favorite star systems — the view as you approach it at near-lightspeed is so ... meaningful.  Your astronomers call it PSR J0337+1715.

So of course I head over to Al’s coffee shop after erasing everything but that astronomical designation.  As I hoped, Cathleen and a few of her astronomy students are on their mid-morning break.  Cathleen winces a little when she sees me coming.  “Now what, Sy?  You’re going to ask about blazars and neutrinos?”

I show her Old Reliable’s screen.  “Afraid not, Cathleen, I’ll have to save that for later.  I just got a message about this star system.  Recognize it?”

“Why, Sy, is that a clue or something?  And why is the lettering in orange?”

“Long story.  But what can you tell me about this star system?”

“Well, it’s probably one of the most compact multi-component systems we’re ever going to run across.  You know what compact objects are?”

“Sure.  When a star the size of our Sun exhausts most of its hydrogen fuel, gravity wins its battle against heat.  The star collapses down to a white dwarf, a Sun-full of mass packed into a planet-size body.  If the star’s a bit bigger it collapses even further, down to a neutron star just a few miles across.  The next step would be a black hole, but that’s not really a star, is it?”

“No, it’s not.  Jim, why not?”

“Because by definition a black hole doesn’t emit light.  A black hole’s accretion disk or polar jets might, but not the object itself.”

“Mm-hm.  Sy, your ‘object’ is actually three compact objects orbiting  around each other.  There’s a neutron star with a white dwarf going around it, and another white dwarf swinging around the pair of them.  Vivian, does that sound familiar?”

“That’s a three-body system, like the Moon going around the Earth and both going around the Sun.  Mmm, except really both white dwarfs would go around the neutron star because it’s heaviest and we can calculate the motion like we do the Solar System.”

“Not quite.  We can treat the Sun as motionless because it has 99% of the mass.  J0337+1715’s neutron star doesn’t dominate its system as much as the Sun does ours.  That outermost dwarf has 20% of its system’s mass.  Phil, what does that suggest to you?”

“It’d be like Pluto and Charon.  Charon’s got 10% of their combined mass and so Pluto and Charon both orbit a point 10% of the way out from Pluto.  From Earth we see Pluto wobbling side to side around that point.  So the neutron star must wobble around the point 20% outward towards the heavy dwarf.  Hey, star-wobble is how we find exoplanets.  Is that what this is about, Mr Moire?  Did someone measure its red-shift behavior?”PSR J0337+1715Cathleen saves me from answering.  “Not quite.  The study Sy’s chasing is actually a cute variation on red-shift measurements.  That ‘PSR‘ designation means the neutron star is a pulsar.  Those things emit electromagnetic radiation pulses with astounding precision, generally regular within a few dozen nanoseconds.  If we receive slowed-down pulses then the object’s going away; sped-up and it’s approaching, just like with red-shifting.  The researchers  derived orbital parameters for all three bodies from the between-pulse durations.  The heavy dwarf is 200 times further out than the light one, for instance.  Not an easy experiment, but it yielded an important result.”

My ears perk up.  “Which was…?”

“The gravitational force between the pulsar and each dwarf was within six parts per million of what Newton’s Laws prescribe.  That observation rules out whole classes of theories that tried to explain galaxies and galaxy clusters without invoking dark matter.”

Cool, huh?

Uh-huh.

~~ Rich Olcott

RIP, Dr Hawking

On By rolcottIn General: Serious Stuff, Physics: Black Holes and Relativity, Uncategorized1 Comment

Today I depart from my normal schedule and the current story line and science line.  A giant has left us and I want to pay proper tribute.

Dr Stephen Hawking enjoyed telling people of his fortunate birth date, exactly 300 years after Galileo Galilei passed away.  He liked a good joke, and I think he’d be tickled with this additional connection to the man whose work made Hawking’s work possible:
RIP Hawking

The equation in the center of this cut is Hawking’s favorite result, which he wanted to be carved on his gravestone.  It links a black hole’s entropy (S) to its surface area (A).  The other letters denote a collection of constants that have been central to the development of theoretical Physics over the past century and a half:

  • k is Boltzmann’s constant, which links temperature with kinetic energy
  • c is the speed of light, the invariance of which led Einstein to Relativity
  • G is Newton’s universal gravitational constant
  • h is Planck’s constant, the “quantum of action”

Hawking spent much of his career thinking deeply about the implications of Einstein’s concepts.  Newton’s equations support excellent descriptions of everyday physical motions, from the fall of raindrops to the orbits of solar systems.  Einstein’s equations led to insights about conditions at the most extreme — velocities near lightspeed, masses millions of times the Sun’s but packed into a volume only a few dozen miles wide.

But Hawking also pondered extremes of the ultimately large and the ultimately small — the edge of the Universe and distances far smaller than atomic nuclei.  Because his physical condition prohibited speech or quick jottings, he was forced to develop extraordinary powers of concentration and visualization that enabled him to encapsulate in a few phrases insights that would take others books to develop and communicate.

Hawking wrote books, too, of course, of a quality and clarity that turned his name and Science into watchwords for the general public as well as the physics community.  By his life and how he lived it he was an inspiration to many, abled and otherwise.  Science needs its popularizers, though some in the field deprecate them as hangers-on.  Hawking managed to bridge that gap with ease and grace, a giant with standing on either side.

Requiescat in pace, Dr Hawking.  Thank you.

~~ Rich Olcott

Gravity from Another Perspective

On By rolcottIn Physics: Black Holes and Relativity, UncategorizedLeave a comment

“OK, we’re looking at that robot next to the black hole and he looks smaller to us because of space compression down there.  I get that.  But when the robot looks back at us do we look bigger?”

We’re walking off a couple of Eddie’s large pizzas.  “Sorry, Mr Feder, it’s not that simple.  Multiple effects are in play but only two are magnifiers.”

“What isn’t?”

“Perspective for one.  That works the same in both directions — the image of an object shrinks in direct proportion to how far away it is.  Relativity has nothing to do with that principle.”

“That makes sense, but we’re talking black holes.  What does relativity do?”

“Several things, but it’s complicated.”

“Of course it is.”

“OK, you know the difference between General and Special Relativity?”

“Yeah, right, we learned that in kindergarten.  C’mon.”

“Well, the short story is that General Relativity effects depend on where you are and Special Relativity effects depend on how fast you’re going.  GR says that the scale of space is compressed near a massive object.  That’s the effect that makes our survey robot appear to shrink as it approaches a black hole.  GR leaves the scale of our space larger than the robot’s.  Robot looks back at us, factors out the effect of perspective, and reports that we appear to have grown.  But there’s the color thing, too.”

“Color thing?”

“Think about two photons, say 700-nanometer red light, emitted by some star on the other side of our black hole.  One photon slides past it.  We detect that one as red light.  The other photon hits our robot’s photosensor down in the gravity well.  What color does the robot see?”

“It’s not red, ’cause otherwise you wouldn’t’ve asked me the question.”

“Check.”

“Robot’s down there where space is compressed…  Does the lightwave get compressed, too?”

“Yup.  It’s called gravitational blue shift.  Like anything else, a photon heading towards a massive object loses gravitational potential energy.  Rocks and such make up for that loss by speeding up and gaining kinetic energy.  Light’s already at the speed limit so to keep the accounts balanced the photon’s own energy increases — its wavelength gets shorter and the color shifts blue-ward.  Depending on where the robot is, that once-red photon could look green or blue or even X-ray-colored.”

“So the robot sees us bigger and blue-ish like.”Robots and perspective and relativity 2“But GR’s not the only player.  Special Relativity’s in there, too.”

“Maybe our robot’s standing still.”

“Can’t, once it gets close enough.  Inside about 1½ diameters there’s no stable orbit around the black hole, and of course inside the event horizon anything not disintegrated will be irresistibly drawn inward at ever-increasing velocity.  Sooner or later, our poor robot is going to be moving at near lightspeed.”

“Which is when Special Relativity gets into the game?”

“Mm-hm.  Suppose we’ve sent in a whole parade of robots and somehow they maintain position in an arc so that they’re all in view of the lead robot.  The leader, we’ll call it RP-73, is deepest in the gravity well and falling just shy of lightspeed.  Gravity’s weaker further out — trailing followers fall slower.  When RP-73 looks back, what will it see?”

“Leaving aside the perspective and GR effects?  I dunno, you tell me.”

“Well, we’ve got another flavor of red-shift/blue-shift.  Speedy RP-73 records a stretched-out version of lightwaves coming from its slower-falling followers, so so it sees their colors shifted towards the red, just the opposite of the GR effect.  Then there’s dimming — the robots in the back are sending out n photons per second but because of the speed difference, their arrival rate at RP-73 is lower.  But the most interesting effect is relativistic aberration.”

“OK, I’ll bite.”

“Start off by having RP-73 look forward.  Going super-fast, it intercepts more oncoming photons than it would standing still.”

“Bet they look blue to it, and really bright.”

“Right on.  In fact, its whole field of view contracts towards its line of flight.  The angular distortion continues all the way around.  Rearward objects appear to swell.”

“So yeah, we’d look bigger.”

“And redder.  If RP-73 is falling fast enough.”

~~ Rich Olcott

  • Thanks to Timothy Heyer for the question that inspired this post.

A Perspective on Gravity

On By rolcottIn Physics: Black Holes and Relativity, Uncategorized2 Comments

“I got another question, Moire.”

“Of course you do, Mr Feder.”

“When someone’s far away they look smaller, right, and when someone’s standing near a black hole they look smaller, too.  How’s the black hole any different?”

“The short answer is, perspective depends on the distance between the object and you, but space compression depends on the distance between the object and the space-distorting mass.  The long answer’s more interesting.”

“And you’re gonna tell it to me, right?”

“Of course.  I never let a teachable moment pass by.  Remember the August eclipse?”

“Do I?  I was stuck in that traffic for hours.”

“How’s it work then?”

“The eclipse?  The Moon gets in front of the Sun and puts us in its shadow. ‘S weird how they’re both the same size so we can see the Sun’s corundum and protuberances.”

“Corona and prominences.  Is the Moon really the same size as the Sun?”

“Naw, I know better than that.  Like they said on TV, the Moon’s about ¼ the Earth’s width and the Sun’s about 100 times bigger than us.  It’s just they look the same size when they meet up.”

“So the diameter ratio is about 400-to-1.  Off the top of your head, do you know their distances from us?”

“Millions of miles, right?”

“Not so much, at least for the Moon.  It’s a bit less than ¼ of a million miles away.  The Sun’s a bit less than 100 million miles away.”

“I see where you’re going here — the distances are the same 400-to-1 ratio.”

“Bingo.  The Moon’s actual size is 400 times smaller than the Sun’s, but perspective reduces the Sun’s visual size by the same ratio and we can enjoy eclipses.  Let’s try another one.  To keep the arithmetic simple I’m going to call that almost-100-million-mile distance an Astronomical Unit.  OK?”

“No problemo.”

“Jupiter’s diameter is about 10% of the Sun’s, and Jupiter is about 5 AUs away from the Sun.  How far behind Jupiter would we have to stand to get a nice eclipse?”

“Oh, you’re making me work, too, huh?  OK, I gotta shrink the Sun by a factor of 10 to match the size of Jupiter so we gotta pull back from Jupiter by the same factor of 10 times its distance from the Sun … fifty of those AUs.”

“You got it.  And by the way, that 55 AU total is just outside the farthest point of Pluto’s orbit.  It took the New Horizons spacecraft nine years to get there.  Anyhow, perspective’s all about simple ratios and proportions, straight lines all the way.  So … on to space compression, which isn’t.”

“We’re not going to do calculus, are we?”

“Nope, just some algebra.  And I’m going to simplify things just a little by saying that our black hole doesn’t spin and has no charge, and the object we’re watching, say a survey robot, is small relative to the black hole’s diameter.  Of course, it’s also completely outside the event horizon or else we couldn’t see it.  With me?”

“I suppose.”

“OK, given all that, suppose the robot’s as-built height is h and it’s a distance r away from the geometric center of an event horizon’s sphere.  The radius of the sphere is rs.  Looking down from our spaceship we’d see the robot’s height h’ as something smaller than h by a factor that depends on r.  There’s a couple of different ways to write the factor.  The formula I like best is h’=h√[(r-rs)/r].”

“Hey, (r-rs) inside the brackets is the robot’s distance to the event horizon.”

“Well-spotted, Mr Feder.  We’re dividing that length by the distance from the event horizon’s geometric center.  If the robot’s far away so that r>>rs, then (r-rs)/r is essentially 1.0 and h’=h.  We and the robot would agree on its height.  But as the robot closes in, that ratio really gets small.  In our frame the robot’s shrinking even though in its frame its height doesn’t change.”

“We’d see it getting smaller because of perspective, too, right?”

“Sure, but toward the end relativity shrinks the robot even faster than perspective does.”

“Poor robot.”

~~ Rich Olcott

  • Thanks to Carol, who inspired this post by asking Mr Feder’s question but in more precise form.

Meanwhile, back at the office

On By rolcottIn Physics: Black Holes and Relativity, Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

Closing time.  Anne and I stroll from Al’s coffee shop back to the Acme Building.  It’s a clear night with at least 4,500 stars, but Anne’s looking at the velvet black between them.

“What you said, Sy, about the Universe not obeying Conservation of Energy — tell me more about that.”

“Aaa-hmmm … OK.  You’ve heard about the Universe expanding, right?”

“Ye-es, but I don’t know why that happens.”

“Neither do the scientists, but there’s pretty firm evidence that it’s happening, if only at the longest scales.  Stars within galaxies get closer together as they radiate away their gravitational energy.  But the galaxies themselves are getting further apart, as far out as we can measure.”

“What’s that got to do with Conservation of Energy?”

“Well, galaxies have mass so they should be drawn together by gravity the way that gravity pulls stars together inside galaxies.  But that’s not what’s happening.  Something’s actively pushing galaxies or galaxy clusters away from each other.  Giving the something a name like ‘dark energy‘ is just an accounting gimmick to pretend the First Law is still in effect at very large distances — we don’t know the energy source for the pushing, or even if there is one.  There’s a separate set of observations we attribute to a ‘dark energy‘ that may or may not have the same underlying cause.  That’s what I was talking about.”Fading white satin

We’re at the Acme Building.  I flash my badge to get us past Security and into the elevator.  As I reach out to press the ’12’ button she puts her hand on my arm.  “Sy, I want to see if I understand this entropy-elephant thing.  You said entropy started as an accounting gimmick, to help engineers keep track of fuel energy escaping into the surroundings.  Energy absorbed at one temperature they called the environment’s heat capacity.  Total energy absorbed over a range of temperatures, divided by the difference in temperature, they called change in entropy.”

The elevator lets us out on my floor and we walk to door 1217.  “You’ve got it right so far, Anne.  Then what?”

“Then the chemists realized that you can predict how lots of systems will work from only knowing a certain set of properties for the beginning and end states.  Pressure, volume, chemical composition, whatever, but also entropy.  But except for simple gases they couldn’t predict heat capacity or entropy, only measure it.”

My key lets us in.  She leans back against the door frame.  “That’s where your physicists come in, Sy.  They learned that heat in a substance is actually the kinetic energy of its molecules.  Gas molecules can move around, but that motion’s constrained in liquids and even more constrained in solids.  Going from solid to liquid and from liquid to gas absorbs heat energy in breaking those constraints.  That absorbed heat appears as increased entropy.”

She’s lounging against my filing cabinet.  “The other way that substances absorb heat is for parts of molecules to rotate and vibrate relative to other parts.  But there are levels.  Some vibrations excite easier than others, and many rotations are even easier.  In a cold material only some motions are active.  Rising temperature puts more kinds of motion into play.  Heat energy spreads across more and more sub-molecular absorbers.”

She’s perched on the edge of my desk.  “Here’s where entropy as possibility-counting shows up.  More heat, more possibilities, more entropy.  Now we can do arithmetic and prediction instead of measuring.  Anything you can count possibilities for you can think about defining an entropy for, like information bits or black holes or socks.  But it’ll be a different entropy, with its own rules and its own range of validity.  … And…”Riding the Elephant

She’s looming directly over me.  Her dark eyes are huge.

“And…?”

“When we first met, Sy, you asked what you could do for me.  You’ve helped me see that when I travel across time and probability I’m riding the Entropy Elephant.  I’d like to show my appreciation.  Can you think of a possibility?”

A dark night, in a city that knows how to keep its secrets.  On the 12th floor of the Acme Building, one man still tries to answer the Universe’s persistent questions — Sy Moire, Physics Eye.

~~ Rich Olcott

Thoughts of Chair-man Moire

On By rolcottIn Physics: Black Holes and Relativity, Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

“My apples and orange peels question, Sy,  isn’t that the same as Jeremy’s?  What’s the connection between heat capacity and counting?”

“You’re right, Anne.  Hmm.  Say, Al, all your coffee shop tables came with four chairs apiece, right?”

“Yup, four-tops every one, even in the back room.”

“You neaten them all up, four to a table, in the morning?”

“The night before.  There’s never time in the morning, customers demand coffee first thing.”

“But look, we’ve got six people seated at this table.  Where’d the extra chairs come from?”

“Other tables, of course.  Is this going somewhere?”

“Almost there.  So in fact the state of the room at any time will have some random distribution of chairs to tables.  You know on the average there’ll be four at a table, but you don’t know the actual distribution until you look, right?”

“Hey, we’re counting again.  You’re gonna say that’s about entropy ’cause the difference between four at a table and some other number is all random and there’s some formula to calculate entropy from that.”elephants and chairs

“True, Vinnie, but we’re about to take the next step.  How did these chairs wind up around this table?”

“We pulled them over, Mr. Moire.”

“My point is, Jeremy, we spent energy to get them here.  The more chairs that are out of position — ”

“The higher the entropy, but also the more energy went into the chairs.  It’s like that heat capacity thing we started with, the energy that got absorbed rather than driving the steam engine.”

“Awright, Anne!” from Jeremy <Jennie bristles a bit>, “and if all the chairs are in Al’s overnight position it’s like absolute zero.  Hey, temperature is average kinetic energy per particle so can we say that the more often a chair gets moved it’s like hotter?”

Jennie breaks in.  “Not a bit of it, Jeremy!  The whole metaphor’s daft.  We know temperature change times heat capacity equals the energy absorbed, right, and we’ve got a link between energy absorption and entropy, right, but what about if at the end of the day all the chairs accidentally wind up four at a table?  Entropy change is zero, right, but customers expended energy moving chairs about all day and Al’s got naught to set straight.”

“Science in action, I love it!  Anne and Jeremy, you two just bridged a gap it took Science a century to get across.  Carnot started us on entropy’s trail in 1824 but scientists in those days weren’t aware of matter’s atomic structure.  They knew that stuff can absorb heat but they had no inkling what did the absorbing or how that worked.  Thirty years later they understood simple gases better and figured out that average kinetic energy per particle bit.  But not until the 1920s did we have the quantum mechanics to show how parts of vibrating molecules can absorb heat energy stepwise like a table ‘absorbing’ chairs.  Only then could we do Vinnie’s state-counting to calculate entropies.”

“Yeah, more energy, spread across more steps, hiding more details we don’t know behind an average, more entropy.  But what about Jennie’s point?”

“Science is a stack of interconnected metaphors, Vinnie.  Some are better than others.  The trick is attending to the boundaries where they stop being valid.  Jennie’s absolutely correct that my four-chair argument is only a cartoon for illustrating stepwise energy accumulation.  If Al had a billion tables instead of a dozen or so, the odds on getting everything back to the zero state would disappear into rounding error.”

“How does black hole entropy play into this, Sy?”TSE classical vs BH

“Not very well, actually.  Oh, sure, the two systems have similar structures.  They’ve each got three inter-related central quantities constrained by three laws.  Here, I’ve charted them out on Old Reliable.”

“OK, their Second and Third Laws look pretty much the same, but their First Laws don’t match up.”

“Right, Al.  And even Bekenstein pointed out inconsistencies between classic thermodynamic temperature and what’s come to be called Hawking temperature.  Hawking didn’t agree.  The theoreticians are still arguing.  Here’s a funny one — if you dig deep enough, both versions of the First Law are the same, but the Universe doesn’t obey it.”

“That’s it, closing time.  Everybody out.”

~~ Rich Olcott

Taming The Elephant

On By rolcottIn Physics: Black Holes and Relativity, Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

Suddenly they were all on the attack.  Anne got in the first lick.  “C’mon, Sy, you’re comparing apples and orange peel.  Your hydrogen sphere would be on the inside of the black hole’s event horizon, and Jeremy’s virtual particles are on the outside.”

[If you’ve not read my prior post, do that now and this’ll make more sense.  Go ahead, I’ll wait here.]white satin and 5 elephantsJennie’s turn — “Didn’t the chemists define away a whole lot of entropy when they said that pure elements have zero entropy at absolute zero temperature?”

Then Vinnie took a shot.  “If you’re counting maybe-particles per square whatever for the surface, shouldn’t you oughta count maybe-atoms or something per cubic whatever for the sphere?”

Jeremy posed the deepest questions. “But Mr Moire, aren’t those two different definitions for entropy?  What does heat capacity have to do with counting, anyhow?”

Al brought over mugs of coffee and a plate of scones.  “This I gotta hear.”

“Whew, but this is good ’cause we’re getting down to the nub.  First to Jennie’s point — Under the covers, Hawking’s evaluation is just as arbitrary as the chemists’.  Vinnie’s ‘whatever’ is the Planck length, lP=1.616×10-35 meter.  It’s the square root of such a simple combination of fundamental constants that many physicists think that lP2=2.611×10-70 m², is the ‘quantum of area.’  But that’s just a convenient assumption with no supporting evidence behind it.”

“Ah, so Hawking’s ABH=4πrs2 and SBH=ABH/4 formulation with rs measured in Planck-lengths, just counts the number of area-quanta on the event horizon’s surface.”

“Exactly, Jennie.  If there really is a least possible area, which a lot of physicists doubt, and if its size doesn’t happen to equal lP2, then the black hole entropy gets recalculated to match.”

“So what’s wrong with cubic those-things?”

“Nothing, Vinnie, except that volumes measured in lP3 don’t apply to a black hole because the interior’s really four-dimensional with time scrambled into the distance formulas.  Besides, Hawking proved that the entropy varies with half-diameter squared, not half-diameter cubed.”

“But you could still measure your hydrogen sphere with them and that’d get rid of that 1033 discrepancy between the two entropies.”

“Not really, Vinnie.  Old Reliable calculated solid hydrogen’s entropy for a certain mass, not a volume.”

“Hawking can make his arbitrary choice, Sy, he’s Hawking, but that doesn’t let the chemists off the scaffold.  How did they get away with arbitrarily defining a zero for entropy?”

“Because it worked, Jennie.  They were only concerned with changes — the difference between a system’s state at the end of a process, versus its state at the beginning.  It was only the entropy difference that counted, not its absolute value.”

“Hey, like altitude differences in potential energy.”

“Absolutely, Vinnie, and that’ll be important when we get to Jeremy’s question.  So, Jennie, if you’re only interested in chemical reactions and if it’s still in the 19th Century and the world doesn’t know about isotopes yet, is there a problem with defining zero entropy to be at a convenient set of conditions?”

“Well, but Vinnie’s Second Law says you can never get down to absolute zero so that’s not convenient.”

“Good point, but the Ideal Gas Law and other tools let scientists extrapolate experimentally measured properties down to extremely low temperatures.  In fact, the very notion of absolute zero temperature came from experiments where the volume of a  hydrogen or helium gas sample appears to decrease linearly towards zero at that temperature, at least until the sample condenses to a liquid.  With properly calibrated thermometers, physical chemists knocked themselves out measuring heat capacities and entropies at different temperatures for every substance they could lay hands on.”

“What about isotopes, Mr Moire?  Isn’t chlorine’s atomic weight something-and-a-half so there’s gotta be several of kinds of chlorine atoms so any sample you’ve got is a mixture and that’s random and that has to have a non-zero entropy even at absolute zero.”

“It’s 35.4, two stable isotopes, Jeremy, but we know how to account for entropy of mixing and anyway, the isotope mix rarely changes in chemical processes.”

“But my apples and orange peels, Sy — what does the entropy elephant do about them?”

~~ Rich Olcott

The Battle of The Entropies

On By rolcottIn Physics: Black Holes and Relativity, Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

(the coffee-shop saga continues)  “Wait on, Sy, a black hole is a hollow sphere?”

I hadn’t noticed her arrival but there was Jennie, standing by Vinnie’s table and eyeing Jeremy who was sill eyeing Anne in her white satin.white satin and 2 elephants“That’s not quite what I said, Jennie.  Old Reliable’s software and and I worked up a hollow-shell model and to my surprise it’s consistent with one of Stephen Hawking’s results.  That’s a long way from saying that’s what a black hole is.”

“But you said some physicists say that.  Have they aught to stand on?”

“Sort of.  It’s a perfect case of ‘depends on where you’re standing.'”

Vinnie looked up.  “It’s frames again, ain’t it?”

“With black holes it’s always frames, Vinnie.  Hey, Jeremy, is a black hole something you could stand on?”

“Nosir, we said the hole’s event horizon is like Earth’s orbit, just a mathematical marker.  Except for the gravity and  the  three  Perils  Jennie and you and me talked about, I’d slide right through without feeling anything weird, right?”

“Good memory and just so.  In your frame of reference there’s nothing special about that surface — you wouldn’t experience scale changes in space or time when you encounter it.  In other frames, though, it’s special.  Suppose we’re standing a thousand miles away from a solar-size black hole and Jeremy throws a clock and a yardstick into it.  What would we see?”

“This is where those space compression and time dilation effects happen, innit?”

“You bet, Jennie.  Do you remember the formula?”

“I wrote it in my daybook … Ah, here it is —Schwarzchild factorMy notes say D is the black hole’s diameter and d is another object’s distance from its center.  One second in the falling object’s frame would look like f seconds to us.  But one mile would look like 1/f miles.  The event horizon is where d equals the half-diameter and f goes infinite.  The formula only works where the object stays outside the horizon.”

“And as your clock approaches the horizon, Jeremy…?”

“You’ll see my clock go slower and slower until it sto —.  Oh.  Oh!  That’s why those physicists think all the infalling mass is at the horizon, the stuff falls towards it forever and never makes it through.”

“Exactly.”

“Hey, waitaminute!  If all that mass never gets inside, how’d the black hole get started in the first place?”

“That’s why it’s only some physicists, Vinnie.  The rest don’t think we understand the formation process well enough to make guesses in public.”

“Wait, that formula’s crazy, Sy.  If something ever does get to where d is less than D/2, then what’s inside the square root becomes negative.  A clock would show imaginary time and a yardstick would go imaginary, too.  What’s that about?”

“Good eye, Anne, but no worries, the derivation of that formula explicitly assumes a weak gravitational field.  That’s not what we’ve got inside or even close to the event horizon.”

“Mmm, OK, but I want to get back to the entropy elephant.  Does black hole entropy have any connection to the other kinds?”

“Strutural, mostly.  The numbers certainly don’t play well together.  Here’s an example I ran up recently on Old Reliable.  Say we’ve got a black hole twice the mass of the Sun, and it’s at the Hawking temperature for its mass, 12 billionths of a Kelvin.  Just for grins, let’s say it’s made of solid hydrogen.  Old Reliable calculated two entropies for that thing, one based on classical thermodynamics and the other based on the Bekenstein-Hawking formulation.”Entropy calculations“Wow, Old Reliable looks up stuff and takes care of unit conversions automatically?”

“Slick, eh, Jeremy?  That calculation up top for Schem is classical chemical thermodynamics.  A pure sample of any element at absolute zero temperature is defined to have zero entropy.  Chemical entropy is cumulative heat capacity as the sample warms up.  The Hawking temperature is so close to zero I could treat heat capacity as a constant.

“In the middle section I calculated the object’s surface area in square Planck-lengths lP², and in the bottom section I used Hawking’s formula to convert area to B-H entropy, SBH.  They disagree by a factor of 1033.”

A moment of shocked silence, and then…

~~ Rich Olcott