Breaking Up? Not So Hard

<transcript of smartphone dictation by Sy Moire, hard‑boiled physicist>
Day 173 of self‑isolation….
Perfect weather for a brisk solitary walk, taking the park route….
There’s the geese. No sign of Mr Feder, just as well….

Still thinking about Ms Baird and her plan for generating electric power from a black hole named Lonesome….
Can just hear Vinnie if I ever told him about this which I can’t….
“Hey, Sy, nothin’ gets out of a black hole except gravity, but she’s using Lonesome‘s magnetic field to generate electricity which is electromagnetic. How’s that happen?”
Good question….

Hhmph, that’s one angry squirrel….
Ah, a couple of crows pecking the ground under its tree. Maybe they’re too close to its acorn stash….

We know a black hole’s only measurable properties are its mass, charge and spin….
And maybe its temperature, thanks to Stephen Hawking….
Its charge is static — hah! cute pun — wouldn’t support continuous electrical generation….
The Event Horizon hides everything inside — we can’t tell if charge moves around in there or even if it’s matter or anti‑matter or something else….
The no‑hair theorem says there’s no landmarks or anything sticking out of the Event Horizon so how do we know the thing’s even spinning?

Ah, we know a black hole’s external structures — the jets, the Ergosphere belt and the accretion disk — rotate because we see red- and blue-shifted radiation from them….
The Ergosphere rotates in lockstep with Lonesome‘s contents because of gravitational frame-dragging….
Probably the disk and the jets do, too, but that’s only a strong maybe….
But why should the Ergosphere’s rotation generate a magnetic field?

How about Newt Barnes’ double‑wheel idea — a belt of charged light‑weight particles inside a belt of opposite‑charged heavy particles all embedded in the Ergosphere and orbiting at the black hole’s spin rate….
Could such a thing exist? Can simple particle collisions really split the charges apart like that?….

OK, fun problem for strolling mental arithmetic. Astronomical “dust” particles are about the size of smoke particles and those are about a micrometer across which is 10‑6 meter so the volume’s about (10‑6)3=10‑18 cubic meter and the density’s sorta close to water at 1 gram per cubic centimeter or a thousand kilograms per cubic meter so the particle mass is about 10‑18×103=10‑15 kilogram. If a that‑size particle collided with something and released just enough kinetic energy to knock off an electron, how fast was it going?

Ionization energy for a hydrogen atom is 13 electronvolts, so let’s go for a collision energy of at least 10 eV. Good old kinetic energy formula is E=½mv² but that’s got to be in joules if we want a speed in meters per second so 10 eV is, lemme think, about 2×10‑18 joules/particle. So is 2×2×10‑18/10‑15 which is 4×10‑3 or 40×10‑4, square root of 40 is about 6, so v is about 6×10‑2 or 0.06 meters per second. How’s that compare with typical speeds near Lonesome?

Ms Baird said that Lonesome‘s mass is 1.5 Solar masses and it’s isolated from external gravity and electromagnetic fields. So anything near it is in orbit and we can use the circular orbit formula v²=GM/r….
Dang, don’t remember values for G or M. Have to cheat and look up the Sun’s GM product on Old Reliable….
Ah-hah, 1.3×1020 meters³/second so Lonesome‘s is also near 1020….
A solar‑mass black hole’s half‑diameter is about 3 kilometers so Lonesome‘s would be about 5×103 meters. Say we’re orbiting at twice that so r‘s around 104 meters. Put it together we get v2=1020/104=1016 so v=108 meters/sec….
Everything’s going a billion times faster than 10 eV….
So yeah, no problem getting charged dust particles out there next to Lonesome….

Just look at the color in that tree…
Weird when you think about it. The really good color is summertime chlorophyll green when the trees are soaking up sunlight and turning CO2 into oxygen for us but people get excited about dying leaves that are red or yellow…

Well, now. Lonesome‘s Event Horizon is the no-going-back point on the way to its central singularity which we call infinity because its physics are beyond anything we know. I’ve just closed out another decade of my life, another Event Horizon on my own one‑way path to a singularity…

Hey! Mr Feder! Come ask me a question to get me out of this mood.

Author’s note — Yes, ambient radiation in Lonesome‘s immediate vicinity probably would account for far more ionization than physical impact, but this was a nice exercise in estimation and playing with exponents and applied physical principles.

~~ Rich Olcott

Rockin’ Round The Elephant

<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

No-hair today, grown tomorrow

It was a classic May day, perfect for some time by the lake in the park.  I was watching the geese when a squadron of runners stampeded by.   One of them broke stride, dashed my way and plopped down on the bench beside me.  “Hi, Mr Moire. <pant, pant>”

“Afternoon, Jeremy.  How are things?”

“Moving along, sir.  I’ve signed up for track, I think it’ll help my base-running,  I’ve met a new girl, she’s British, and that virtual particle stuff is cool but I’m having trouble fitting it into my black hole paper.”

“Here’s one angle.  Nobelist Gerard ‘t Hooft said, ‘A particle is fundamental when it’s useful to think of it as fundamental.‘  In that sense, a black hole is a fundamental particle.  Even more elementary than atoms, come to think of it.”

“Huh?”

“It has to do with the how few numbers you need to completely specify the particle.  You’d need a gazillion terabytes for just the temperatures in the interior and oceans and atmosphere of Earth.  But if you’re making a complete description of an isolated atom you just need about two dozen numbers — three for position, three for linear momentum, one for atomic number (to identify which element it represents), one for its atomic weight (which isotope), one for its net charge if it’s been ionized, four more for nuclear and electronic spin states, maybe three or four each for the energy levels of its nuclear and electronic configuration.  So an atom is simpler than the Earth”

“And for a black hole?”

“Even simpler.  A black hole’s event horizon is smooth, so smooth that you can’t distinguish one point from another.  Therefore, no geography numbers.  Furthermore, the physics we know about says whatever’s inside that horizon is completely sealed off from the rest of the universe.  We can’t have knowledge of the contents, so we can’t use any numbers to describe it.  It’s been proven (well, almost proven) that a black hole can be completely specified with only eleven numbers — one for its total mass-energy, one for its electric charge, and three each for position, linear momentum and angular momentum.  Leave out the location and orientation information and you’ve got three numbers — mass, charge, and spin.  That’s it.”

“How about its size or it temperature?”

“Depends how you measure size.  Event horizons are spherical or nearly so, but the equations say the distance from an event horizon to where you’d think its center should be is literally infinite.  You can’t quantify a horizon’s radius, but its diameter and surface area are both well-defined.  You can calculate both of them from the mass.  That goes for the temperature, too.”

“How about if it came from antimatter instead of matter?”

“Makes no difference because the gravitational stresses just tear atoms apart.”

“Wait, you said, ‘almost proven.’  What’s that about?”no hair 1

“Believe it or not, the proof is called The No-hair Theorem.  The ‘almost’ has to do with the proof’s starting assumptions.  In the simplest case, zero change and zero spin and nothing else in the Universe, you’ve got a Schwarzchild object.  The theorem’s been rigorously proven for that case — the event horizon must be perfectly spherical with no irregularities — ‘no hair’ as one balding physicist put it.”

“How about if the object spins and gets charged up, or how about if a planet or star or something falls into it?”

“Adding non-zero spin and charge makes it a Kerr-Newman object.  The theorem’s been rigorously proven for those, too.  Even an individual infalling mass has only a temporary effect.  The black hole might experience transient wrinkling but we’re guaranteed that the energy will either be radiated away as a gravitational pulse or else simply absorbed to make the object a little bigger.  Either way the event horizon goes smooth and hairless.”

“So where’s the ‘almost’ come in?”

“Reality.  The region near a real black hole is cluttered with other stuff.  You’ve seen artwork showing an accretion disk looking like Saturn’s rings around a black hole.  The material in the disk distorts what would otherwise be a spherical gravitational field.  That gnarly field’s too hairy for rigorous proofs, so far.  And then Hawking pointed out the particle fuzz…”

~~ Rich Olcott