Rockfall

<continued>  The coffee shop crowd had gotten rowdy in response to my sloppy physics, but everyone hushed when I reached for my holster and drew out Old Reliable.  All had heard of it, some had seen it in action — a maxed-out tablet with customized math apps on speed-dial.

“Let’s take this nice and slow.  Suppose we’ve got an non-charged, non-spinning solar-mass black hole.  Inside its event horizon the radius gets weird but let’s pretend we can treat the object like a simple sphere.  The horizon’s half-diameter, we’ll call it the radius, is rs=2G·M/c²G is Newton’s gravitational constant, M is the object’s mass and c is the speed of light.  Old Reliable says … about 3 kilometers.  Question is, what happens when we throw a rock in there?  To keep things simple, I’m going to model dropping the rock gentle-like, dead-center and with negligible velocity relative to the hole, OK?”

<crickets>

“Say the rock has the mass of the Earth, almost exactly 3×10-6 the Sun’s mass.  The gravitational potential energy released when the rock hits the event horizon from far, far away would be E=G·M·m/rs, which works out to be … 2.6874×1041 joules.  What happens to that energy?”falling rock and black hole

rs depends on mass, Mr Moire, so the object will expand.  Won’t that push on what’s around it?”

“You’re thinking it’d act like a spherical piston, Jeremy, pushing out in all directions?”

“Yeah, sorta.”

“After we throw in a rock with mass m, the radius expands from rs to rp=2G·(M+m)/c².  I set m to Earth’s mass and Old Reliable says the new radius is … 3.000009 kilometers.  Granted the event horizon is only an abstract math construct, but suppose it’s a solid membrane like a balloon’s skin.  When it expands by that 9 millimeters, what’s there to push against?  The accretion disk?  Those rings might look solid but they’re probably like Saturn’s rings — a collection of independent chunks of stuff with an occasional gas molecule in-between.  Their chaotic orbits don’t have a hard-edged boundary and wouldn’t notice the 9-millimeter difference.  Inward of the disk you’ve got vacuum.  A piston pushing on vacuum expends zero energy.  With no pressure-volume work getting done that can’t be where the infall energy goes.”

“How about lift-a-weight work against the hole’s own gravity?”

“That’s a possibility, Vinnie.  Some physicists maintain that a black hole’s mass is concentrated in a shell right at the event horizon.  Old Reliable here can figure how much energy it would take to expand the shell that extra 9 millimeters.  Imagine that simple Newtonian physics applies — no relativistic weirdness.  Newton proved that a uniform spherical shell’s gravitational attraction is the same as what you’d get from having the same mass sitting at the shell’s geometric center.  The gravitational pull the shell exerts on itself originally was E=G·M²/rs.  Lifting the new mass from rs to rp will cost ΔE=G·(M+m)²/r– G·M²/rs.  When I plug in the numbers…  That’s interesting.”

Vinnie’s known me long enough to realize “That’s interesting” meant “Whoa, I certainly didn’t expect THAT!

“So what didja expect and whatcha got?”

“What I expected was that lift-it-up work would also be just a small fraction of the infall energy and the rest would go to heat.  What I got for ΔE here was 2.6874×1041 joules, exactly 100% of the input.  I wonder what happens if I use a bigger planet.  Gimme a second … OK, let’s plot a range …  How ’bout that, it’s linear!”ep-es

“Alright, show us!”

All the infall energy goes to move the shell’s combined mass outward to match the expanded size of the event horizon.  I’m amazed that such a simple classical model produces a reasonable result.”

“Like Miss Plenum says, Mr Moire, sometimes the best science comes from surprises.”

“I wouldn’t show it around, Jeremy, except that it’s consistent with Hawking’s quantum-physics result.”

“How’s that?”

“Remember, he showed that a black hole’s temperature varies as 1/M.  We know that temperature is ΔE/ΔS, where the entropy change ΔS varies as .  We’ve just found that ΔE varies as M.  The ΔE/ΔS ratio varies as M/M²=1/M, just like Hawking said.”

Then Jennie got into the conversation.

~~ Rich Olcott

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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

Gozer, The Stay Puft Black Hole

We’re downstairs at Eddie’s Pizza.  Vinnie orders his usual pepperoni.  In memory of Sam Panapoulos, I order a Hawaiian.  Then we’re back to talking black holes.

“I been thinking, Sy.  These regular-size black holes, the ones close to the Sun’s mass, we got a lot of ’em?”

“I’ve seen an estimate of 50,000 in the Milky Way Galaxy so you could say they’re common.  That’s one way to look at it.  The other way is to compare 50,000 with the 250 billion stars in the galaxy.  One out of 5,000,000 is a black hole, so they’re rare.  Your choice, Vinnie.”

“But all three confirmed LIGO signals were the next bigger size range, maybe 10 to 30 solar masses; two of ’em hittin’ each other and they’ve all been more than a billion lightyears away.  How come LIGO doesn’t see the little guys that are close to us?”

“Darn good question.  Lessee… OK, I’ve got several possibilities.  Maybe the close-in little guys do collide, but the signal’s too weak for us to detect.  But we can put numbers to that.  In each LIGO event we’ve seen, the collision released about 10% of their 40-to-60-Sun total mass-energy in the form of gravitational waves.  LIGO’s just barely able to detect that, right?”

“They were excited they could, yeah.”

“So if a pair of little-guy black holes collided they’d release about 10% of two makes 0.2 solar masses worth of energy.  That’d be way below our detection threshold if the collision is a billion light-years away.  But we’re asking about collisions inside the Milky Way.  Suppose the collision happened near the center, about 26,000 lightyears from us.  Signal strength grows as the square of how close the source is, so multiply that ‘too weak to detect’ wave by (1 billion/26000)² =15×1012, fifteen quadrillion.  LIGO’d be deafened by a signal that strong.”

“But LIGO’s OK, so we can rule that out.  Next guess.”

“Maybe the signal’s coming in at the wrong frequency.  The equations say that just before a big-guy collision the two objects circle each other hundreds of times a second.  That frequency is in the lower portion of the 20-20,000 cycles-per-second human audio range.  LIGO’s instrumentation was tuned to pick up gravitational waves between 30 and 7,000.  Sure enough, LIGO recorded chirps that were heard around the world.”

“So what frequency should LIGO be tuned to to pick up little-guy collisions?”

“We can put numbers to that, too.  Physics says that at a given orbit radius, revolution frequency varies inversely with the square root of the mass.   The big-guy collisions generated chirps between 100 and 400 cps.  Little guy frequencies would be f2/f50=√(50/2)=5 times higher, between 500 and 2000 cps.  Well within LIGO’s range.”

“We don’t hear those tweets so that idea’s out, too.  What’s your third try?”

“Actually I like this one best.  Maybe the little guys just don’t collide.”

“Why would you like that one?”

“Because maybe it’s telling us something.  It could be that they don’t collide simply because they’re surrounded by so many other stars that they never meet up.  But it also could be that binary black holes, the ones that are fated to collide with each other, are the only ones that can grow beyond 10 solar masses.  And I’ve got a guess about how that could happen.”

“Alright, give.”

“Let’s start with how to grow a big guy.  Upstairs we talked about making little guys.  When a star’s core uses up one fuel, like hydrogen, there’s an explosive collapse that sets off a hotter fuel, like helium, until you get to iron which doesn’t play.  At each step, unburnt fuel outside the core gets blown away.  If the final core’s mass is greater than about three times the Sun’s you wind up with a black hole.  But how about if you don’t run out of fuel?”

“How can that happen?  The star’s got what it’s got.”Binary protoBHs

“Not if it’s got close neighbors that also expel unburnt fuel in their own burn-collapse cycles.  Grab their cast-off fuel and your core can get heavier before you do your next collapse.  Not impossible in a binary or cluster where all the stars are roughly the same age.  Visualize kids tossing marshmallows into each other’s mouths.”

“Or paying for each other’s pizzas.  And it’s your turn.”

~~ Rich Olcott

Prelude to A Waltz

An excited knock, but one I recognize.  In comes Vinnie, waving his fresh copy of The New York Times.

LIGO‘s done it again!  They’ve seen another black hole collision!”

“Yeah, Vinnie, I’ve read the Abbott-and-a-thousand paper.  Three catastrophic collisions detected in less than two years.  The Universe is starting to look like a pretty busy place, isn’t it?”

“And they all involve really big black holes — 15, 20, even 30 times heavier than the Sun.  Didn’t you once say black holes that size couldn’t exist?”

“Well, apparently they do.  Of course the physicists are busily theorizing how that can happen.  What do you know about how stars work, Vinnie?”

“They get energy from fusing hydrogen atoms to make helium atoms.”

“So far, so good, but then what happens when the hydrogen’s used up?”

“They go out, I guess.”

“Oh, it’s a lot more exciting than that. Does the fusion reaction happen everywhere in the star?”

“I woulda said, ‘Yes,’ but since you’re asking I’ll bet the answer is,  ‘No.'”

“Properly suspicious, and you’re right.  It takes a lot of heat and pressure to force a couple of positive nuclei close enough to fuse together despite charge repulsion.  Pressures near the outer layers are way too low for that.  For our Sun, for instance, you need to be 70% of the way to the center before fusion really kicks in.  So you’ve got radiation pressure from the fusion pushing everything outward and gravity pulling everything toward the center.  But what’s down there?  Here’s a hint — hydrogen’s atomic weight is 1, helium’s is 4.”

“You’re telling me that the heavier atoms sink to the center?”

“I am.”

“So the center builds up a lot of helium.  Oh, wait, helium atoms got two protons in there so it’s got to be harder to mash them together than mashing hydrogens, right?”Star zones
“And that’s why that region’s marked ash zone in this sketch.  Wherever conditions are right for hydrogen fusion, helium’s basically inert.  Like ash in a campfire it just sinks out of the way.  Now the fire goes out.  What would you expect next?”

“Radiation pressure’s gone but gravity’s still there … everything must slam inwards.”

Slam is an excellent word choice, even though the star’s radius is measured in thousands of miles.  What’s the slam going to do to the helium atoms?”

“Is it strong enough to start helium fusion?”

“That’s where I’m going.  The star starts fusing helium at its core.  That’s a much hotter reaction than hydrogen’s.  When convective zone hydrogen that’s still falling inward meets fresh outbound radiation pressure, most of the hydrogen gets blasted away.”

“Fusing helium – that’s a new one on me.  What’s that make?”

“Carbon and oxygen, mostly.  They’re as inert during the helium-fusion phase as helium was when hydrogen was doing its thing.”

“So will the star do another nova cycle?”

“Maybe.  Depends on the core’s mass.  Its gravity may not be intense enough to fuse helium’s ashes.  In that case you wind up with a white dwarf, which just sits there cooling off for billions of years.  That’s what the Sun will do.”

“But suppose the star’s heavy enough to burn those ashes…”

“The core’s fresh light-up blows away infalling convective zone material.  The core makes even heavier atoms.  Given enough fuel, the sequence repeats, cycling faster and faster until it gets to iron.  At each stage the star has less mass than before its explosion but the residual core is more dense and its gravity field is more intense.  The process may stop at a neutron star, but if there was enough fuel to start with, you get a black hole.”

“That’s the theory that accounts for the Sun-size black holes?”

“Pretty much.  I’ve left out lots of details, of course.  But it doesn’t account for black holes the size of 30 Suns — really big stars go supernova and throw away so much of their mass they miss the black-hole sweet spot and terminate as a neutron star or white dwarf.  That’s where the new LIGO observation comes in.  It may have clued us in on how those big guys happen.”

“That sketch looks like a pizza slice.”

“You’re thinking dinner, right?”

“Yeah, and it’s your turn to buy.”

~~ Rich Olcott

Three Perils for a Quest(ion), Part 3

“Things are finally slowing down.  You folks got an interesting talk going, mind if I join you?  I got biscotti.”

“Pull up a chair, Eddie.  You know everybody?”

“You and Jeremy, yeah, but the young lady’s new here.”

“I’m Jennie, visiting from England.”

“Pleased to meetcha.  So from what I overheard, we got Jeremy on some kinda Quest to a black hole’s crust.  He’s passed two Perils.  There’s a final one got something to do with a Firewall.”

“One minor correction, Eddie.  He’s not going to a crust, because a black hole doesn’t have one.  Nothing to stand on or crash into, anyway.  He’s headed to its Event Horizon, which is the next best thing.  If you’re headed inward, the Horizon marks the beginning of where it’s physically impossible to get out.”

“Hotel California, eh?”

“You could say that.  The first two Perils had to do with the black hole’s intense gravitational field.  The one ahead has to do with entangled virtual particles.”

“Entangled is the Lucy-and-Ethel thing you said where two particles coordinate instant-like no matter how far apart they are?”

“Good job of overhearing, there, Eddie.  Jeremy, tell him abut virtual particles.”

“Umm, Mr Moire and I talked about a virtual particle snapping into and out of existence in empty space so quickly that the long-time zero average energy isn’t affected.”

“What we didn’t mention then is that when a virtual pair is created, they’re entangled.  Furthermore, they’re anti-particles, which means that each is the opposite of the other — opposite charge, opposite spin, opposite several other things.  Usually they don’t last long — they just meet each other again and annihilate, which is how the average energy stays at zero.  Now think about creating a pair of virtual particles in the black hole’s intense gravitational field where the creation event sends them in opposite directions.”Astronaut and semi-biscotto
“Umm… if they’re on opposite paths then one’s probably headed into the Horizon and the other is outbound. Is the outbound one Hawking radiation?  Hey, if they’re entangled that means the inbound one still has a quantum connection with the one that escaped!”

“Wait on.  If they’re entangled and something happening to one instantaneously affects its twin, but the gravity difference gives each a different rate of time dilation, how does that work then?”

“Paradox, Jennie!  That’s part of what the Firewall is about.  But it gets worse.  You’d think that inbound particle would add mass to the black hole, right?”

“Surely.”

“But it doesn’t.  In fact, it reduces the object’s mass by exactly each particle’s mass.  That ‘long-time zero average energy‘ rule comes into play here.  If the two are separated and can’t annihilate, then one must have positive energy and the other must have negative energy.  Negative energy means negative mass, because of Einstein’s mass-energy equivalence.  The positive-mass twin escapes as Hawking radiation while the negative-mass twin joins the black hole, shrinks it, and by the way, increases its temperature.”

“Surely not, Sy.  Temperature is average kinetic energy.  Adding negative energy to something has to decrease its temperature.”

“Unless the something is a black hole, Jennie.  Hawking showed that a black hole’s temperature is inversely dependent on its mass.  Reduce the mass, raise the temperature, which is why a very small black hole radiates more intensely than a big one.  Chalk up another paradox.”

“Two paradoxes.  Negative mass makes no sense.  I can’t make a pizza with negative cheese.  People would laugh.”

“Right.  Here’s another.  Suppose you drop some highly-structured object, say a diamond, into a black hole.  Sooner or later, much later really, that diamond’s mass-energy will be radiated back out.  But there’s no relationship between the structure that went in and the randomized particles that come out.  Information loss, which is totally forbidden by thermodynamics.  Another paradox.”

“The Firewall resolves all these paradoxes then?”

“Not really, Jennie.  The notion is that there’s this thin layer of insanely intense energetic interactions, the Firewall, just outside of the Event Horizon.  That energy is supposed to break everything apart — entanglements, pre-existing structures, quantum propagators (don’t ask), everything, so what gets through the horizon is mush.  Many physicists think that’s bogus and a cop-out.”

“So no Firewall Peril?”

“Wanna take the chance?”

~~ Rich Olcott

Three Perils for a Quest(ion), Part 1

Eddie makes great pizzas but Jeremy thinks they stay in the oven just a little too long.  As he crunched an extra-crispy wedge-edge he mused, “Gravity aside, I wonder what it’d be like to land on a black hole.  I bet it’d be real slippery if it’s as smooth as Mr Moire says.”

Jennie cut in.  “Don’t be daft, lad.  Everyone’s read about the spaceman sliding through the event horizon unaware until it’s too late.  Someone far away sees the bloke’s spacetime getting all distorted but in his local frame of reference everything’s right as rain.  Right, Sy?”

“As rain, Jennie, if all you’re concerned about is relativity.  But Spaceman Jeremy has lots of other things to be concerned about on his way to the event horizon.  Which he couldn’t stand on anyway.”

“Why not, Mr Moire?  I mean, I said ‘gravity aside’ so I ought to be able to stand up.”

“Nothing to stand on, Jeremy.  It’d be like trying to stand on Earth’s orbit.”

“Pull the other one, Sy.  How can they be alike?”

“Both of them are mathematical constructs rather than physical objects.  An orbit is an imaginary line that depicts planet or satellite locations.  An event horizon is an imaginary figure enclosing a region with such intense spacetime curvature that time points inward.  They’re abstract objects, not  concrete ones.  But let’s get back to Jeremy’s black hole evaporation quest.  He’ll have to pass three perils.”

“Ooo, a Quest with Perils —  loverly.  What are the Perils then?”

“The Roche Radius, the Photon Sphere and the Firewall.  Got your armor on, Jeremy?”Astronaut and 3xBlack hole

“Ready, Mr Moire.”

“Stand up.  The Roche effect is all about gravitational discrepancy between two points.  The two meter distance between your head and feet isn’t enough for a perceptible difference in downward pull.  However, when we deal with astronomical distances the differences can get significant.  For instance, ocean water on the day side of Earth is closer to the Sun and experiences a stronger sunward pull than water on the night side.”

“Ah, so that’s why we get tides.”

“Right.  Sit, sit, sit.  So in 1849 Édouard Roche wondered how close two objects could get until tidal forces pulled one of them apart.  He supposed the two objects were both just balls of rocks or fluid held together by gravity.  Applying Newton’s Laws and some approximations he got a formula for threshold distance in terms of the big guy’s mass and the little guy’s density.  Suppose you’re held together only by gravity and you’re nearing the Sun feet-first.  Its mass is 2×1030 kg/m³.  Even including your space armor, your average density is about 1.5 kg/m³.  According to Roche’s formula, if you got closer than 8.6×106 kilometers your feet would break away and fall into the Sun before the rest of you would.  Oh, that distance is about 1/7 the radius of Mercury’s orbit so it’s pretty close in.”

“But we’re talking black holes here.  What if the Sun collapses to a black hole?”

“Surprisingly, it’s exactly the same distance.  The primary’s operative property is its mass, not its diameter.  Good thing Jeremy’s really held together by atomic and molecular electromagnetism, which is much stronger than gravity.  Which brings us to his second Peril, the dreaded Photon Sphere.”

“Should I shudder, Sy?”

“Go ahead, Jennie.  The Sphere is another mathematical object, not something physical you’d collide with, Jeremy.  It’s a zero-thickness shell representing where electromagnetic waves can orbit a massive object like a black hole or a neutron star.  Waves can penetrate the shell easily in either direction, but if one happens to fly in exactly along a tangent, it’s trapped on the Sphere.”

“That’s photons.  Why is it a peril to me?”

“Remember that electromagnetism that holds you together?  Photons carry that force.  Granted, in a molecule they’re standing waves rather than the free waves we see with.  The math is impossible, but here’s the Peril.  Suppose one of your particularly important molecules happens to lie tangent to the Sphere while you’re traversing it.  Suddenly, the forces holding that molecule together fly away from you at the speed of light.  And that disruption inexorably travels along your body as you proceed on your Quest.”

[both shudder]

~~ Rich Olcott

The Thin Edge of Infinity

Late in the day, project’s half done but it’s hungry time.  I could head home for a meal and drive back, but instead I board the elevator down to Eddie’s Pizza on the second floor.  The door opens on 8 and Jeremy gets on, with a girl.

“Oh, hi, Mr. Moire.  Didja see I hit a triple in the last game?  What if the Sun became a black hole?  This is that English girl I told you about.”

“Hello, Jennie.”

“Wotcha, Sy.”

“You know each other?”

“Ra-ther.  He wrote me into his blog a year ago.  You were going on about particles then, right, Sy?”

“Right, Jennie, but that was particles confined in atoms.  Jeremy’s interested in larger prey.”

“So I hear.”

The elevator lets us out at Eddie’s place.  We luck into a table, order and resume talking.  I open with, “What’s a particle?”

“Well, Sy, your post with Jeremy says it’s an abstract point with a minimal set of properties, like mass and charge, in a mathematical model of a real object with just that set of properties.”

“Ah, you’ve been reading my stuff.  That simplifies things.  So when can we treat a black hole like a particle?  Did you see anything about that in my archives, Jennie?”

“The nearest I can recall was Professor ‘t Hooft’s statement.  Ermm… if the Sun’s so far away that we can calculate planetary orbits accurately by treating it as a point, then we’re justified in doing so.”

“And if the Sun were to suddenly collapse to a black hole?”

“It’d be a lot smaller, even more like a point.  No change in gravity then.  But wouldn’t Earth be caught up in relativity effects like space compression?’

“Not unless you’re really close.  Space compression around a non-rotating (Schwarzchild) black hole scales by a factor that looks like Schwarzchild factor, where D is the object’s diameter and d is your distance from it.  Suppose the Sun suddenly collapsed without losing any mass to become a Schwarzchild object.  The object’s diameter would be a bit less than 4 miles.  Earth is 93 million miles from the Sun so the compression factor here would be [poking numbers into my smartphone] 1.000_000_04.  Nothing you’d notice.  It’d be 1.000_000_10 at Mercury.  You wouldn’t see even 1% compression until you got as close as 378 miles, 10% only inside of 43 miles.  Fifty percent of the effect shows up in the last 13 miles.  The edge of a black hole is sharper than this pizza knife.”Knife-edges

“How about if it’s spinning?  Ms Plenum referred me to a reading about frame-dragging.”

“Ah, Jeremy, you’re thinking of Gargantua, the Interstellar movie’s strangely lopsided black hole.  I just ran across this report by Robbie Gonzalez.  He goes into detail on why the image is that way, and why it should have looked more like this picture.  Check out the blueshift on the left and the shift into the infra-red on the right.”

better Gargantua

A more accurate depiction of Gargantua.  Image from
James, et al., Class. Quantum Grav. 32 (2015) 065001 (41pp),
licensed under CC BY-NC-ND 3.0

[both] “Awesome!”

“So it’s the spin making the weirdness then, Sy?”

“Yes, ma’am.  If Gargantua weren’t rotating, then the space around it would be perfectly spherical.  As Gonzalez explains, the movie’s plotline needed an even more extreme spacetime distortion than they could get from that.  Dr Kip Thorne, their physics guru, added more by spinning his mathematical model nearly up to the physical limit.”

“I’ll bite, Mr Moire.  What’s the limit?”

“Rotating so fast that points on the equator would be going at lightspeed.  Can’t do that.  Anyhow, extreme spin alters spacetime distortion, which goes from spherical to pumpkin-shaped with a twist.  The radial scaling changes form, too, from Schwarzchild factor to Kerr factorA is proportional to spin.  When A is small (not much spin) or the distance is large those A/d² terms essentially vanish relative to the others and the scaling looks just like the simple almost-a-point Schwarzchild case.  When A is large or the distance is small the A/d² terms dominate top and bottom, the factor equals 1 and there’s dragging but no compression.  In the middle, things get interesting and that’s where Dr Thorne played.”

“So no relativity jolt to Earth.”

“Yep.”

“Here’s your pizzas.”

“Thanks, Eddie.”

[sounds of disappearing pizza]

~~ Rich Olcott