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 Top Choice

Al grabs me as I step into his coffee shop. “Sy, ya gotta stop Vinnie, he’s using up paper napkins again, and he’s making a mess!”

Sure enough, there’s Vinnie at his usual table by the door. He’s got a kid’s top, a big one, spinning on a little stand. He’s methodically dropping crumpled-up paper wads onto it and watching them fly off onto the floor. “Hey, Vinnie, what’s the project?”

“Hi, Sy. I’m trying to figure how come these paper balls are doing a circle but when they fly off they always go in a straight line, at least at first. They got going-around momentum, right, so how come they don’t make a spiral like stars in a galaxy?”

Astronomy professor Cathleen’s standing in the scone line. She never misses an opportunity to correct a misconception. “Galaxy stars don’t spray out of the center in a spiral, Vinnie. Like planets going around a star, stars generally follow elliptical orbits around the galactic center. A star that’s between spiral arms now could be buried in one ten million years from now. The spiral arms appear because of how the orbits work. One theory is that the innermost star orbits rotate their ellipse axes more quickly than the outer ones and the spirals form where the ellipses pile up. Other theories have to do with increased star formation or increased gravitational attraction within the pile-up regions. Probably all three contribute to the structures. Anyhow, spirals don’t form from the center outward.”

My cue for some physics. “What happens in a galaxy is controlled by gravity, Vinnie, and gravity doesn’t enter into what you’re doing. Except for all that paper falling onto Al’s floor. There’s no in-plane gravitational or electromagnetic attraction in play when your paper wads leave the toy. Newton would say there’s no force acting to make them follow anything other than straight lines once they break free.”

“What about momentum? They’ve got going-around momentum, right, shouldn’t that keep them moving spirally?”

I haul out Old Reliable for a diagram. “Thing is, your ‘going-around momentum,’ also known as ‘angular momentum,’ doesn’t exist. Calm down, Vinnie, I mean it’s a ‘fictitious force‘ that depends on how you look at it.”

“Is this gonna be frames again?”

“Yup. Frames are one of our most important analytical tools in Physics. Here’s your toy and just for grins I’ve got it going around counterclockwise. That little white circle is one of your paper wads. In the room’s frame that wad in its path is constantly converting linear momentum between the x-direction and the y-direction, right?”

“East-West to North-South and back, yeah, I get that.”

“Such a mess to calculate. Let’s make it easier. Switch to the perspective of a frame locked to the toy. In that frame the wad can move in two directions. It can fly away along the radial direction I’ve called r, or it can ride along sideways in the s-direction.”

“So why hasn’t it flown away?”

“Because you put some spit on it to make it stick — don’t deny it, I saw you. While it’s stuck, does it travel in the r direction?”

“Nope, only in the s direction. Which should make it spiral like I said.”

“I’m not done yet. One of Newton’s major innovations was the idea of infinitesimal changes, also known as little-bits. The s-direction is straight, not curved, but it shifts around little-bit by little-bit as the top rotates. Newton’s Laws say force is required to alter momentum. What force influences the wad’s s-momentum?”

“Umm … that line you’ve marked c.”

“Which is the your spit’s adhesive force between the paper and the top. The wad stays stuck until the spit dries out and no more adhesion so no more c-force. Then what happens?”

“It flies off.”

“In which direction?”

“Huh! In the r-direction.”

“And in a straight line, just like Newton said. What you called ‘going-around momentum’ becomes ‘radial momentum’ and there’s no spiraling, right?”

“I guess you’re right, but I miss spirals.”

Al comes over with a broom. “Now that’s settled, Vinnie, clean up!”

~~ Rich Olcott

  • Thanks for the question, Jen Keeler. Stay tuned.

Gravity’s Real Rainbow

Some people are born to scones, some have scones thrust upon them.  As I stepped into his coffee shop this morning, Al was loading a fresh batch onto the rack.  “Hey, Sy, try one of these.”

“Uhh … not really my taste.  You got any cinnamon ones ready?”

“Not much for cheddar-habañero, huh?  I’m doing them for the hipster trade,” waving towards all the fedoras on the room.  “Here ya go.  Oh, Vinnie’s waiting for you.”

I navigated to the table bearing a pile of crumpled yellow paper, pulled up a chair.  “Morning, Vinnie, how’s the yellow writing tablet working out for you?”

“Better’n the paper napkins, but it’s nearly used up.”

“What problem are you working on now?”

“OK, I’m still on LIGO and still on that energy question I posed way back — how do I figure the energy of a photon when a gravitational wave hits it in a LIGO?  You had me flying that space shuttle to explain frames and such, but kept putting off photons.”

“Can’t argue with that, Vinnie, but there’s a reason.  Photons are different from atoms and such because they’ve got zero mass.  Not just nearly massless like neutrinos, but exactly zero.  So — do you remember Newton’s formula for momentum?”

“Yeah, momentum is mass times the velocity.”

“Right, so what’s the momentum of a photon?”

“Uhh, zero times speed-of-light.  But that’s still zero.”

“Yup.  But there’s lots of experimental data to show that photons do carry non-zero momentum.  Among other things, light shining on an an electrode in a vacuum tube knocks electrons out of it and lets an electric current flow through the tube.  Compton got his Nobel prize for that 1923 demonstration of the photoelectric effect, and Einstein got his for explaining it.”

“So then where’s the momentum come from and how do you figure it?”

“Where it comes from is a long heavy-math story, but calculating it is simple.  Remember those Greek letters for calculating waves?”

(starts a fresh sheet of note paper) “Uhh… this (writes λ) is lambda is wavelength and this (writes ν) is nu is cycles per second.”

“Vinnie, you never cease to impress.  OK, a photon’s momentum is proportional to its frequency.  Here’s the formula: p=h·ν/c.  If we plug in the E=h·ν equation we played with last week we get another equation for momentum, this one with no Greek in it:  p=E/c.  Would you suppose that E represents total energy, kinetic energy or potential energy?”

“Momentum’s all about movement, right, so I vote for kinetic energy.”

“Bingo.  How about gravity?”

“That’s potential energy ’cause it depends on where you’re comparing it to.”

light-in-a-gravity-well“OK, back when we started this whole conversation you began by telling me how you trade off gravitational potential energy for increased kinetic energy when you dive your airplane.  Walk us through how that’d work for a photon, OK?  Start with the photon’s inertial frame.”

“That’s easy.  The photon’s feeling no forces, not even gravitational, ’cause it’s just following the curves in space, right, so there’s no change in momentum so its kinetic energy is constant.  Your equation there says that it won’t see a change in frequency.  Wavelength, either, from the λ=c/ν equation ’cause in its frame there’s no space compression so the speed of light’s always the same.”

“Bravo!  Now, for our Earth-bound inertial frame…?”

“Lessee… OK, we see the photon dropping into a gravity well so it’s got to be losing gravitational potential energy.  That means its kinetic energy has to increase ’cause it’s not giving up energy to anything else.  Only way it can do that is to increase its momentum.  Your equation there says that means its frequency will increase.  Umm, or the local speed of light gets squinched which means the wavelength gets shorter.  Or both.  Anyway, that means we see the light get bluer?”

“Vinnie, we’ll make a physicist of you yet.  You’re absolutely right — looking from the outside at that beam of photons encountering a more intense gravity field we’d see a gravitational blue-shift.  When they leave the field, it’s a red-shift.”

“Keeping track of frames does make a difference.”

Al yelled over, “Like using tablet paper instead of paper napkins.”

~~ Rich Olcott

Scone but not forgotten

Al grabbed me as I stepped into his coffee shop.  “Sy, you gotta help me!”

“What’s the trouble, Al?”

“It’s Vinnie.  He’s over there, been scribbling on paper napkins all morning.  I’m running out of napkins, Sy!”

I grabbed a cinnamon scone from the rack and a chair at Vinnie’s table.  “What’s keeping you so busy, Vinnie?”  As if I didn’t know.

LIGO, of course.  Every time I think I understand how the machine works something else occurs to me and it slips outa my hands.”

“How about you explain it to me.  Sometimes the best way to find an answer is to describe the problem to someone else.”

Interferometer 1
Vinnie’s paper napkin #1

(grabbing a napkin near the bottom of one stack) “All right, Sy, I sketched the layout here.  You got these two big L-shaped machines out in the middle of two nowheres 2500 miles apart.  Each L is a pair of steel pipes 2½ miles long.  At the far end of each arm there’s a high-tech stabilized mirror.  Where the two arms meet there’s a laser rigged up to shoot beams down both arms.  There’s also a detector located where the reflected beams join up and cancel each other out unless there’s a gravity wave going past.  Am I good so far?”

“Yeah, that’s pretty much the diagram you see in the books, except it’s gravitational waveGravity waves are something else.”

interferometer-4
Paper napkin #2

“Whatever.  So, here’s a sketch of where I was at when I asked you that first question.  See, I copied my original sketch onto another napkin and stretched it a little where the black circle is to show what a gravitational wave would do in stretch phase.  Ignore the little rips.”

“What rips?”

“Uh, thanks.  Anyway, I was thinking the gravitational wave that stretches the x-beam would also stretch the x-pipe so they couldn’t use the light wave to measure the pipe it’s in.  But LIGO works so that’s wrong thinkin’.

“OK, next is for after we talked about inertial frames.  Took me a few tries to get it like I want it and I wound up having to do two sketches, one for each frame.”  He grabbed a couple more napkins from different stacks.

interferometer-5lp
Paper napkins #37 and #59

“I didn’t do the yellow wiggles ’cause that got confusing and besides I don’t do wiggly lines so good.  Point is, the space-stretch only shows up in the laboratory inertial frame.  The light waves move with space so they don’t notice the difference, right?”

“Well, I wouldn’t want to put it that way in court, Vinnie, but it’s a pretty good description.”

“So the light waves bop along at 186,000 miles per second in their frame, but from the machine’s perspective those are stretched miles so the guy running the machine thinks those photons are faster than the ones in the other pipe.  And that difference in speed gets the yellow lines out of phase with the blue ones and the detector rings a bell or something, right?”

“It’s even better than that.” I reached for another napkin, caught Al’s eye on me and grabbed an envelope from my coat pocket instead. “Remember how a gravitational wave works in two directions perpendicular to the wave’s line of travel?”

interferometer-5d
On the back of an envelope

“Yeah, so?”

“So at the same moment that the wave is stretching space in the x-direction, it’s squeezing space in the y-direction.  LIGO’s detection scheme monitors the difference between the two returning beams.  As I’ve drawn it here using the detector’s inertial frame, the x-beam is going fast AND the y-beam is going slow so the detector sees twice the phase difference. A few milliseconds later they’ll switch because the x-direction will get squeezed while the y-direction gets stretched.  And yeah, a bell does ring but only after some computers munch on the data and subtract out environmental stuff like temperature swings and earthquakes and the janitor’s footsteps.”

“Uh-huh, I think I got it.” Turning in his chair, “Hey, Al, bring Sy here another scone, on me.  And put the one he’s got on my tab, too.”

“Thanks, Vinnie.”

“Don’t mention it.”

~~ Rich Olcott

Three ways to look at things

A familiar shadow loomed in from the hallway.

“C’mon in, Vinnie, the door’s open.”

“I brought some sandwiches, Sy.”

“Oh, thanks, Vinnie.”

“Don’t mention it.    An’ I got another LIGO issue.”

“Yeah?”

“Ohh, yeah.  Now we got that frame thing settled, how does it apply to what you wrote back when?  I got a copy here…”

The local speed of light (miles per second) in a vacuum is constant.  Where space is compressed, the miles per second don’t change but the miles get smaller.  The light wave slows down relative to the uncompressed laboratory reference frame.

“Ah, I admit I was a bit sloppy there.  Tell you what, let’s pretend we’re piloting a pair of space shuttles following separate navigation beams that are straight because that’s what light rays do.  So long as we each fly a straight line at constant speed we’re both using the same inertial frame, right?”

“Sure.”

“And if a gravity field suddenly bent your beam to one side, you’d think you’re still flying straight but I’d think you’re headed on a new course, right?”

“Yeah, because now we’d have different inertial frames.  I’d think your heading has changed, too.”two-shuttles

“So what does the guy running the beams see?”

“Oh, ground-pounders got their own inertial frame, don’t they?  Uhh… He sees me veer off and you stay steady ’cause the gravity field bent only my beam.”

“Right — my shuttle and the earth-bound observer share the same inertial frame, for a while.”

“A while?”

“Forever if the Earth were flat because I’d be flying straight and level, no threat to the shared frame.  But the Earth’s not flat.  If I want to stay at constant altitude then I’ve got to follow the curve of the surface rather than follow the light beam straight out into space.  As soon as I vector downwards I have a different frame than the guy on the ground because he sees I’m not in straight-line motion.”

“It’s starting to get complicated.”

“No worries, this is as bad as it gets.  Now, let’s get back to square one and we’re flying along and this time the gravity field compresses your space instead of bending it.  What happens?  What do you experience?”

“Uhh… I don’t think I’d feel any difference.  I’m compressed, the air molecules I breath are compressed, everything gets smaller to scale.”

“Yup.  Now what do I see?  Do we still have the same inertial frame?”

“Wow.  Lessee… I’m still on the beam so no change in direction.  Ah!  But if my space is compressed, from your frame my miles look shorter.  If I keep going the same miles per second by my measure, then you’ll see my speed drop off.”

“Good thinking but there’s even more to it.  Einstein showed that space compression and time dilation are two sides of the same phenomenon.  When I look at you from my inertial frame, your miles appear to get shorter AND your seconds appear to get longer.”

“My miles per second slow way down from the double whammy, then?”

“Yup, but only in my frame and that other guy’s down on the ground, not in yours.”

“Wait!  If my space is compressed, what happens to the space around what got compressed?  Doesn’t the compression immediately suck in the rest of the Universe?”

“Einstein’s got that covered, too.  He showed that gravity doesn’t act instantaneously.  Whenever your space gets compressed, the nearby space stretches to compensate (as seen from an independent frame, of course).  The edge of the stretching spreads out at the speed of light.  But the stretch deformation gets less intense as it spreads out because it’s only offsetting a limited local compression.”

“OK, let’s get back to LIGO.  We got a laser beam going back and forth along each of two perpendicular arms, and that famous gravitational wave hits one arm broadside and the other arm cross-wise.  You gonna tell me that’s the same set-up as me and you in the two shuttles?”

“That’s what I’m going to tell you.”

“And the guy on the ground is…”

“The laboratory inertial reference.”

“Eat your sandwich, I gotta think about this.”

(sounds of departing footsteps and closing door)

“Don’t mention it.”

~~ Rich Olcott

A Shift in The Flight

I heard a familiar squeak from the floorboard outside my office.

“C’mon in, Vinnie, the door’s open.  What can I do for you?”

“I still got problems with LIGO.  I get that dark energy and cosmic expansion got nothin’ to do with it.  But you mentioned inertial frame and what’s that about?”

earth-moon“Does the Moon go around the Earth or does the Earth go around the Moon?”

“Huh?  Depends on where you are, I guess.”

“Well, there you are.”

“Waitaminnit!  That can’t be all there is to it!”

“You’re right, there’s more.  It all goes back to Newton’s First Law.”  (showing him my laptop screen)  “Here’s how Wikipedia puts it in modern terms…”

In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.

“That’s really a definition rather than a Law.  If you’re looking at an object and it doesn’t move relative to you or else it’s moving at constant speed in a straight line, then you and the object share the same inertial frame.  If it changes speed or direction relative to you, then it’s in a different inertial frame from yours and Newton’s Laws say that there must be some force that accounts for the difference.”

“So another guy’s plane flying straight and level with me has a piece of my inertial frame?”

“Yep, even if you’re on different vectors.  You only lose that linkage if either airplane accelerates or curves off.”

“So how’s that apply to LIGO’s laser beams?  I thought light always traveled in straight lines.”

“It does, but what’s a straight line?”

“Shortest distance between two points — I been to flight school, Sy.”

“Fine.  So if you fly from London to Mexico City on this globe here you’d drill through the Earth?”mex-atl-jfk-lgw

“Of course not, I’d take the Great Circle route that goes through those two cities.  It’s the shortest flight path.  Hey, how ’bout that, the circle goes through NYC and Atlanta, too.”

“Cool observation, but that line looks like a curve from where I sit.”

“Yeah, but you’re not sittin’ close to the globe’s surface.  I gotta fly in the flight space I got.”

“So does light.  Photons always take the shortest available path, though sometimes that path looks like a curve unless you’re on it, too.  Einstein predicted that starlight passing through the Sun’s gravitational field would be bent into a curve.  Three years later, Eddington confirmed that prediction.”

“Light doesn’t travel in a straight line?”

“It certainly does — light’s path defines what is a straight line in the space the light is traveling through.  Same as your plane’s flight path defines that Great Circle route.  A gravitational field distorts the space surrounding it and light obeys the distortion.”

“You’re getting to that ‘inertial frames’ stuff, aren’t you?”

“Yeah, I think we’re ready for it.  You and that other pilot are flying steady-speed paths along two navigation beams, OK?”

“Navigation beams are radio-frequency.”

“Sure they are, but radio’s just low-frequency light.  Stay with me.  So the two of you are zinging along in the same inertial frame but suddenly a strong gravitational field cuts across just your beam and bends it.  You keep on your beam, right?”

“I suppose so.”

“And now you’re on a different course than the other plane.  What happened to your inertial frame?”

“It also broke away from the other guy’s.”

“Because you suddenly got selfish?”

“No, ’cause my beam curved ’cause the gravity field bent it.”

“Do the radio photons think they’re traveling a bent path?”

“Uh, no, they’re traveling in a straight line in a bent space.”

“Does that space look bent to you?”

“Well, I certainly changed course away from the other pilot’s.”

“Ah, but that’s referring to his inertial frame or the Earth’s, not yours.  Your inertial frame is determined by how those photons fly, right?  In terms of your frame, did you peel away or stay on-beam?”

“OK, so I’m on-beam, following a straight path in a space that looks bent to someone using a different inertial frame.  Is that it?”

“You got it.”

(sounds of departing footsteps and closing door)

“Don’t mention it.”

~~ Rich Olcott