Maybe It’s Just A Coincidence

Raucous laughter from the back room at Al’s coffee shop, which, remember, is situated on campus between the Physics and Astronomy buildings. It’s Open Mic night and the usual crowd is there. I take a vacant chair which just happens to be next to the one Susan Kim is in. “Oh, hi, Sy. You just missed a good pitch. Amanda told a long, hilarious story about— Oh, here comes Cap’n Mike.”

Mike’s always good for an offbeat theory. “Hey, folks, I got a zinger for you. It’s the weirdest coincidence in Physics. Are you ready?” <cheers from the physicists in the crowd> “Suppose all alone in the Universe there’s a rock and a planet and the rock is falling straight in towards the planet.” <turns to Al’s conveniently‑placed whiteboard> “We got two kinds of energy, right?”

Potential Energy    Kinetic Energy

Nods across the room except for Maybe-an-Art-major and a couple of Jeremy’s groupies. “Right. Potential energy is what you get from just being where you are with things pulling on you like the planet’s gravity pulls on the rock. Kinetic energy is what potential turns into when the pulls start you moving. For you Physics smarties, I’m gonna ignore temperature and magnetism and maybe the rock’s radioactive and like that, awright? So anyway, we know how to calculate each one of these here.”

PE = GMm/R    KE = ½mv²

“Big‑G is Newton’s gravitational constant, big‑M is the planet’s mass, little‑m is the rock’s mass, big‑R is how far apart the things are, and little‑v is how fast the rock’s going. They’re all just numbers and we’re not doing any complicated calculus or relativity stuff, OK? OK, to start with the rock is way far away so big‑R is huge. Big number on the bottom makes PE’s fraction tiny and we can call it zero. At the same time, the rock’s barely moving so little‑v and KE are both zero, close enough. Everybody with me?”

More nods, though a few of the physics students are looking impatient.

“Right, so time passes and the rock dives faster toward the planet Little‑v and kinetic energy get bigger. Where’s the energy coming from? Gotta be potential energy. But big‑R on the bottom gets smaller so the potential energy number gets, wait, bigger. That’s OK because that’s how much potential energy has been converted. What I’m gonna do is write the conversion as an equation.

GMm/R=½mv²

“So if I tell you how far the rock is from the planet, you can work the equation to tell me how fast it’s going and vice-versa. Lemme show those straight out…”

v=(2GM/R)    R=2GM/v²

Some physicist hollers out. “The first one’s escape velocity.”

“Good eye. The energetics are the same going up or coming down, just in the opposite direction. One thing, there’s no little‑m in there, right? The rock could be Jupiter or a photon, same equations apply. Suppose you’re standing on the planet and fire the rock upward. If you give it enough little‑v speed energy to get past potential energy equals zero, then the rock escapes the planet and big‑R can be whatever it feels like. Big‑R and little‑v trade off. Is there a limit?”

A couple of physicists and an astronomy student see where this is going and start to grin.

“Newton physics doesn’t have a speed limit, right? They knew about the speed of light back then but it was just a number, you could go as fast as you wanted to. How about we ask how far the rock is from the planet when it’s going at the speed of light?”

R=2GM/

Suddenly Jeremy pipes up. “Hey that’s the Event Horizon radius. I had that in my black hole term paper.” His groupies go “Oooo.”

“There you go, Jeremy. The same equation for two different objects, from two different theories of gravity, by two different derivations.”

“But it’s not valid for lightspeed.”

“How so?”

“You divided both sides of your conversion equation by little‑m. Photons have zero mass. You can’t divide by zero.”

Everyone in the room goes “Oooo.”

~~ Rich Olcott

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

A Perspective on Gravity

“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.

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