Tag: Thermodynamics

Red Harvest

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

<continued> Al’s coffee shop was filling up as word got around about Anne in her white satin.  I saw a few selfie-takers in the physics crowd surreptitiously edge over to get her into their background.  She was busy thinking so she didn’t notice.  “The entropy-elephant picture is starting to come together, Sy.  We started out with entropy measuring accumulated heat capacity in a steam engine.”

“That’s where Carnot started, yes.”

“But when Jeremy threw that hot rock into the black hole” <several in the astronomy crew threw startled looks at Jeremy>, “its heat energy added to the black hole’s mass, but it should have added to the black hole’s entropy, too.  ‘Cause of Vinnie’s Second Law.”white satin and black hole 3

Vinnie looked up.  “Ain’t my Second Law, it’s thermodynamics’ Second Law.  Besides, my version was ‘energy’s always wasted.’  Sy’s the one who turned that into ‘entropy always increases.'”

“So anyway, black holes can’t have zero entropy like people used to think.  But if entropy also has to do with counting possibilities, than how does that apply to black holes?  They have only one state.”

“That’s where Hawking got subtle.  Jeremy, we’ve talked about how the black hole’s event horizon is a mathematical abstraction, infinitely thin and perfectly smooth and all that.”

“Yessir.”

“Hawking moved one step away from that abstraction.  In essence he said the  event horizon is surrounded by a thin shell of virtual particles.  Remember them, Jeremy?”

“Uh-huh, that was on my quest to the event horizon.  Pairs of equal and opposite virtual particles randomly appear and disappear everywhere in space and because they appear together they’re entangled and if one of them dips into the event horizon then it doesn’t annihilate its twin which — Oh!  Random!  So what’s inside the event horizon may have only one state, so far as we know, but right outside the horizon any point may or may not be hosting, can I call it an orphan particle?  I’ll bet that uncertainty give rise to the entropy, right?”

<finger-snaps of approval from the physics side of the room>

“Well done, Jeremy!  ‘Orphan’ isn’t the conventional term but it gets the idea across.”

“Wait, Sy.  You mentioned that surface area and entropy go together and now I see why.  The larger the area, the more room there is for those poor orphans.  When Jeremy’s rock hit the event horizon and increased the black hole’s mass, did the surface area increase enough to allow for the additional entropy?” <more finger-snapping>

“Sure did, Anne.  According to Hawking’s calculation, it grew by exactly the right amount.  Mass and area both grow as the square of the diameter.”

“How come not the radius?”

“Well , Vinnie, the word ‘radius‘ is tricky when you’re discussing black holes.  The event horizon is spherical and has a definite diameter — you could measure it from the outside.  But the sphere’s radius extends down to the singularity and is kind of infinite and isn’t even strictly speaking a distance.  Space-time is twisted in there, remember, and that radial vector is mostly time near its far end.  On the other hand, you could use ‘radius‘ to mean ‘half the diameter‘ and you’d be good for calculating effects outside the event horizon.”

“OK, that’s the entropy-area connection, but how does temperature tie in with surface gravity?”

“They’re both inversely dependent on the black hole’s mass.  Let’s take surface gravity first, and here when I say ‘r‘ I’m talking ‘half-diameter,‘ OK?”

“Sure.”

“Good.  Newton taught us that an object with mass M has a gravitational attraction proportional to M/r².  That still holds if you’re not inside the event horizon.  Now, the event horizon’s r is also proportional to the object’s mass so you’ve got M/M² which comes to 1/M.  With me?”

“Yeah.”

“Hawking used quantum physics to figure the temperature thing, but here’s a sloppy short-cut.  Anne, remember how we said that entropy is approximately heat capacity divided by temperature?”

“Mm-hmm.”

“The shell’s energy is mostly heat and proportional to M.  We’ve seen the shell’s entropy is proportional to M².  The temperature is heat divided by entropy.  That’s proportional to M/M² which is the same 1/M as surface gravity.” <boos from all sides>. “Hey, I said it was sloppy.”

~~ Rich Olcott

Rockin’ Round The Elephant

On By rolcottIn Physics: Black Holes and Relativity, Physics: Entropy and Thermodynamics, Uncategorized8 Comments

<continued…>  “That’s what who said?  And why’d he say that?”

“That’s what Hawking said, Al.  He’s the guy who first applied thermodynamic analysis to black holes.  Anyone happen to know the Three Laws of Thermodynamics?”

Vinnie pipes up from his table by the coffee shop door.  “You can’t win.  You can’t even break even.  But you’ll never go broke.”

“Well, that’s one version, Vinnie, but keep in mind all three of those focus on energy.  The First Law is Conservation of Energy—no process can create or destroy energy, only  transform it, so you can’t come out ahead.  The Second Law is really about entropy—”

“Ooo, the elephant!”white satin and black hole 2

“Right, Anne.  You usually see the Second Law stated in terms of energy efficiency—no process can convert energy to another form without wasting some of it. No breaking even.  But an equivalent statement of that same law is that any process must increase the entropy of the Universe.”

“The elephant always gets bigger.”

“Absolutely.  When Bekenstein and Hawking thought about what would happen if a black hole absorbed more matter, worst case another black hole, they realized that the black hole’s surface area had to follow the same ‘Never decrease‘ rule.”

“Oh, that Hawking!  Hawking radiation Hawking!  The part I didn’t understand, well one of the parts, in that “Black Holes” Wikipedia article!  It had to do with entangled particles, didn’t it?”

“Just caught up with us, eh, Jeremy?  Yes, Stephen Hawking.  He and Jacob Bekenstein found parallels between what we can know about black holes on the one hand and thermodynamic quantities on the other.  Surface area and entropy, like we said, and a black hole’s mass acts mathematically like energy in thermodynamics.  The correlations were provocative ”

“Mmm, provocative.”

“You like that word, eh, Anne?  Physicists knew that Bekenstein and Hawking had a good analogy going, but was there a tight linkage in there somewhere?  It seemed doubtful.”

“Nothin’ to count.”

“Wow, Vinnie.  You’ve been reading my posts?”

“Sure, and I remember the no-hair thing.  If the only things the Universe can know about a black hole are its mass, spin and charge, then there’s nothing to figure probabilities on.”

“Exactly.  The logic sequence went, ‘Entropy is proportional to the logarithm of state count, there’s only one state, log(1) equals zero,  so the entropy is zero.’  But that breaks the Third Law.  Vinnie’s energy-oriented Third Law says that no object can cool to absolute zero temperature.  But an equivalent statement is that no object can have zero entropy.”

“So there’s something wrong with black hole theory, huh?”

“Which is where our guys started, Vinnie.  Being physicists, they said, ‘Suppose you were to throw an object into a black hole.  What would change?’”

“Its mass, for one.”

“For sure, Jeremy.  Anything else?”

“It might not change the spin, if you throw right.”

“Spoken like a trained baseball pitcher.  Turns out its mass governs pretty much everything about a black hole, including its temperature but not spin or charge.  Once you know the mass you can calculate its entropy, diameter, surface area, surface gravity, maximum spin, all of that.  Weird, though, you can’t easily calculate its volume or density — spatial distortion gets in the way.”

“So what happens to all those things when the mass increases?”

“As you might expect, they change.  What’s interesting is how each of them change and how they’re linked together.  Temperature, for instance, is inversely proportional to the mass and vice-versa.  Suppose, Jeremy, that you threw two big rocks, both the same size, into a black hole.  The first rock is at room temperature and the other’s a really hot one, say at a million degrees.   What would each do?”

“The first one adds mass so from what you said it’d drop the temperature.  The second one has the same mass, so I don’t see, wait, temperature’s average kinetic energy so the hot rock has more energy than the other one and Einstein says that energy and mass are the same thing so the black hole gets more mass from the hot rock than from the cold one so its temperature goes down … more?  Really?”

“Yup.  Weird, huh?”

“How’s that work?”

“That’s what they asked.”

~~ Rich Olcott

Enter the Elephant, stage right

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

“Anne?”

“Mm?”

“Remember when you said that other reality, the one without the letter ‘C,’  felt more probable than this one?”

“Mm-mm.”

“What tipped you off?”

“Now you’re asking?”

“I’m a physicist, physicists think about stuff.  Besides, we’ve finished the pizza.”

<sigh> “This conversation has gotten pretty improbable, if you ask me.  Oh, well.  Umm, I guess it’s two things.  The more-probable realities feel denser somehow, and more jangly. What got you on this track?”

“Conservation of energy.  Einstein’s E=mc² says your mass embodies a considerable amount of energy, but when you jump out of this reality there’s no flash of light or heat, just that fizzing sound.  When you come back, no sudden chill or things falling down on us, just the same fizzing.  Your mass-energy that has to go to or come from somewhere.  I can’t think where or how.”

“I certainly don’t know, I just do it.  Do you have any physicist guesses?”

“Questions first.”

“If you must.”

“It’s what I do.  What do you perceive during a jump?  Maybe something like falling, or heat or cold?”

“There’s not much ‘during.’  It’s not like I go through a tunnel, it’s more like just turning around.  What I see goes out of focus briefly.  Mostly it’s the fizzy sound and I itch.”

“Itch.  Hmm…  The same itch every jump?”

“That’s interesting.  No, it’s not.  I itch more if I jump to a more-probable reality.”

“Very interesting.  I’ll bet you don’t get that itch if you’re doing a pure time-hop.”

“You’re right!  OK, you’re onto something, give.”

“You’ve met one of my pet elephants.”

“Wha….??”White satin and elephant

“A deep question that physics has been nibbling around for almost two centuries.  Like the seven blind men and the elephant.  Except the physicists aren’t blind and the elephant’s pretty abstract.  Ready for a story?”

“Pour me another and I will be.”

“Here you go.  OK, it goes back to steam engines.  People were interested in getting as much work as possible out of each lump of coal they burned.  It took a couple of decades to develop good quantitative concepts of energy and work so they could grade coal in terms of energy per unit weight, but they got there.  Once they could quantify energy, they discovered that each material they measured — wood, metals, water, gases — had a consistent heat capacity.  It always took the same amount of energy to raise its temperature across a given range.  For a kilogram of water at 25°C, for instance, it takes one kilocalorie to raise its temperature to 26°C.  Lead and air take less.”

“So where’s the elephant come in?”

“I’m getting there.  We started out talking about steam engines, remember?  They work by letting steam under pressure push a piston through a cylinder.  While that’s happening, the steam cools down before it’s puffed out as that classic old-time Puffing Billy ‘CHUFF.’  Early engine designers thought the energy pushing the piston just came from trading off pressure for volume.  But a guy named Carnot essentially invented thermodynamics when he pointed out that the cooling-down was also important.  The temperature drop meant that heat energy stored in the steam must be contributing to the piston’s motion because there was no place else for it to go.”

“I want to hear about the elephant.”

“Almost there.  The question was, how to calculate the heat energy.”

“Why not just multiply the temperature change by the heat capacity?”

“That’d work if the heat capacity were temperature-independent, which it isn’t.  What we do is sum up the capacity at each intervening temperature.  Call the sum ‘elephant’ though it’s better known as Entropy.  Pressure, Volume, Temperature and Entropy define the state of a gas.  Using those state functions all you need to know is the working fluid’s initial and final state and you can calculate your engine.  Engineers and chemists do process design and experimental analysis using tables of reported state function values for different substances at different temperatures.”

“Do they know why heat capacity changes?”

“That took a long time to work out, which is part of why entropy’s an elephant.  And you’ve just encountered the elephant’s trunk.”

“There’s more elephant?”

“And more of this.  Want a refill?”

~~ Rich Olcott