A Wheel in A Wheel

The conversation’s gotten a little dry so I carry our mugs over to Al’s coffee tap for refills. Vinnie’s closest so he gets the first one. “Thanks, Sy. So you say that a black hole has all these other things on the outside — the photon sphere and that weird belt if it’s rotating and the accretion disk and the jets which is what I asked about in the first place.”

Astrophysicist-in-training Newt Barnes gets the second mug. “My point, Vinnie, is they all act together. You can’t look at just one thing. Thanks, Sy. You know, you should’ve paid more attention to the ergosphere.”

“Ergosphere?”

“Yeah, Vinnie, that pumpkin-shaped layer Sy described — actually, more a pumpkin shell. The event horizon and photon sphere take up space inside of it and the accretion disk’s inner edge grazes its equator. The pumpkin is fatter for a more rapidly rotating black hole, but its boundary still dips down to meet the event horizon at the rotational poles. Diagrams usually show it just sitting there but that’s not quite true.”

“It wobbles?”

“No, the shape stays in place, locked to the event horizon just like the diagrams show. What’s inside it, though, is moving like mad. That’s what we’d see from a far-away frame, anyhow.”

Frames again, I knew it. The pumpkin’s got frames?”

“With extreme-gravity situations it’s always frames, Vinnie. The core’s gravity pulls in particles from the accumulation disk. They think they’re going straight. From an outsider’s perspective everything swerves spinwise at the ergosphere’s boundary. Even if a high-speed particle had been aimed in the other direction, it’s going spinwise once it’s inside the ergosphere.”

“Who’s making it do that?”

“Frame-dragging on steroids. We’ve known for a century that gravity from any massive body compresses the local space. ‘Kilometers are shorter near a black hole,’ as the saying goes. If the body is rotating, that counts too, at least locally — space itself joins the spin. NASA’s Gravity B probe detected micromicrodegree-level frame rotation around Earth. The ergosphere, though, has space is twisted so far that the direction of time points spinwise in the same way that it points inwards within the event horizon. Everything has to travel along time’s arrow, no argument.”

“You said ‘local‘ twice there. How far does this spread?”

“Ah, that’s an important question. The answer’s ‘Not as far as you think.’ Everything scales with the event horizon’s diameter and that scales with the mass. If the Sun were a non-rotating black hole, for instance, its event horizon would be only about 6 kilometers across, less than 4 miles. Its photon sphere would be 4.5 kilometers out from the center and the inner edge of its accretion disk would be a bit beyond that. Space compression dies out pretty quick on the astronomical scale — only a millionth of the way out to the orbit of Mercury the effect’s down to just 3% of its strength at the photosphere.”

“How about if it’s rotating?”

“The frame-dragging effect dies out even faster, with the cube of the distance. At the same one-millionth of Mercury’s orbit, the twist-in-space factor is 0.03% of what it is at the photosphere. At planet-orbit distances spin’s a non-player. However, in the theory I’m researching, spin’s influence may go much further.”

“Why’s that?”

“Seen from an outside frame, what’s inside the ergosphere rotates really fast. Remember that stuff coming in from the accretion disk’s particle grinder? It ought to be pretty thoroughly ionized, just a plasma of negative electrons and positive particles like protons and atomic nuclei. The electrons are thousands of times lighter than the positive stuff. Maybe the electrons settle into a different orbit from the positive particles.”

“Further in or further out?”

“Dunno, I’m still calculating. Either way, from the outside it’d look like two oppositely-charged disks, spinning in the same direction. We’ve known since Ørsted that magnetism comes from a rotating charge. Seems to me the ergosphere’s contents would generate two layers of magnetism with opposite polarities. I think what keeps the jets confined so tightly is a pair of concentric cylindrical magnetic fields extruded from the ergosphere. But it’s going to take a lot of math to see if the idea holds water.”

“Or jets.”

~~ 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