Tag: Inertial frame

The Battle of The Entropies

On By rolcottIn Physics: Black Holes and Relativity, Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

(the coffee-shop saga continues)  “Wait on, Sy, a black hole is a hollow sphere?”

I hadn’t noticed her arrival but there was Jennie, standing by Vinnie’s table and eyeing Jeremy who was sill eyeing Anne in her white satin.white satin and 2 elephants“That’s not quite what I said, Jennie.  Old Reliable’s software and and I worked up a hollow-shell model and to my surprise it’s consistent with one of Stephen Hawking’s results.  That’s a long way from saying that’s what a black hole is.”

“But you said some physicists say that.  Have they aught to stand on?”

“Sort of.  It’s a perfect case of ‘depends on where you’re standing.'”

Vinnie looked up.  “It’s frames again, ain’t it?”

“With black holes it’s always frames, Vinnie.  Hey, Jeremy, is a black hole something you could stand on?”

“Nosir, we said the hole’s event horizon is like Earth’s orbit, just a mathematical marker.  Except for the gravity and  the  three  Perils  Jennie and you and me talked about, I’d slide right through without feeling anything weird, right?”

“Good memory and just so.  In your frame of reference there’s nothing special about that surface — you wouldn’t experience scale changes in space or time when you encounter it.  In other frames, though, it’s special.  Suppose we’re standing a thousand miles away from a solar-size black hole and Jeremy throws a clock and a yardstick into it.  What would we see?”

“This is where those space compression and time dilation effects happen, innit?”

“You bet, Jennie.  Do you remember the formula?”

“I wrote it in my daybook … Ah, here it is —Schwarzchild factorMy notes say D is the black hole’s diameter and d is another object’s distance from its center.  One second in the falling object’s frame would look like f seconds to us.  But one mile would look like 1/f miles.  The event horizon is where d equals the half-diameter and f goes infinite.  The formula only works where the object stays outside the horizon.”

“And as your clock approaches the horizon, Jeremy…?”

“You’ll see my clock go slower and slower until it sto —.  Oh.  Oh!  That’s why those physicists think all the infalling mass is at the horizon, the stuff falls towards it forever and never makes it through.”

“Exactly.”

“Hey, waitaminute!  If all that mass never gets inside, how’d the black hole get started in the first place?”

“That’s why it’s only some physicists, Vinnie.  The rest don’t think we understand the formation process well enough to make guesses in public.”

“Wait, that formula’s crazy, Sy.  If something ever does get to where d is less than D/2, then what’s inside the square root becomes negative.  A clock would show imaginary time and a yardstick would go imaginary, too.  What’s that about?”

“Good eye, Anne, but no worries, the derivation of that formula explicitly assumes a weak gravitational field.  That’s not what we’ve got inside or even close to the event horizon.”

“Mmm, OK, but I want to get back to the entropy elephant.  Does black hole entropy have any connection to the other kinds?”

“Strutural, mostly.  The numbers certainly don’t play well together.  Here’s an example I ran up recently on Old Reliable.  Say we’ve got a black hole twice the mass of the Sun, and it’s at the Hawking temperature for its mass, 12 billionths of a Kelvin.  Just for grins, let’s say it’s made of solid hydrogen.  Old Reliable calculated two entropies for that thing, one based on classical thermodynamics and the other based on the Bekenstein-Hawking formulation.”Entropy calculations“Wow, Old Reliable looks up stuff and takes care of unit conversions automatically?”

“Slick, eh, Jeremy?  That calculation up top for Schem is classical chemical thermodynamics.  A pure sample of any element at absolute zero temperature is defined to have zero entropy.  Chemical entropy is cumulative heat capacity as the sample warms up.  The Hawking temperature is so close to zero I could treat heat capacity as a constant.

“In the middle section I calculated the object’s surface area in square Planck-lengths lP², and in the bottom section I used Hawking’s formula to convert area to B-H entropy, SBH.  They disagree by a factor of 1033.”

A moment of shocked silence, and then…

~~ Rich Olcott

Through The Looking Glass, Darkly

On By rolcottIn General: Fun Stuff, Mathematics: Dimensions, Physics: Entropy and Thermodynamics, Uncategorized1 Comment

The Acme Building is quiet on summer evenings.  I was in my office, using the silence to catch up on paperwork.  Suddenly I heard a fizzing sound.  Naturally I looked around.  She was leaning against the door frame.

White satin looked good on her, and she looked good in it.  A voice like molten silver — “Hello, Mr Moire.”White satin and chessboard 1

“Hello yourself.  What can I do for you?”

“I’m open to suggestions, but first you can help me find myself.”

“Excuse me, but you’re right here.  And besides, who are you?”

“Not where I am but when I am.  Anne.”

“You said it right the first time.”

“No, no, my name is Anne.  At the moment.  I think.  Oh, it’s so confusing when your memory works in circles but not very well.  Do you have the time?”

“Well, I was busy, but you’re here and much more interesting.”

“No, I mean, what time is it?”

I showed her my desk clock — date, time, even the phase of the moon.

“Half past gibbous already?  Oh, bread-and-butter…”

“Wait — circles?  Time’s one-dimensional.  Clock readings increase or decrease, they don’t go sideways.”

“You don’t know Time as well as I do, Mr Moire.  It’s a lot more complicated than that.  Time can be triangular, haven’t you noticed?”

“Can’t say as I have.”

“That paperwork you’re working on, are you near a deadline?”

“Nah.”

“And given that expanse of time, you feel free to permit distractions.  There are so many distractions.”

“You’re very distracting.”

“Thank you, I guess.  But suppose you had an important deadline coming up tomorrow.   That broad flow of possibilities at the beginning of the project has narrowed to just two — finish or don’t finish.  Your Time has closed in on you.”

“So you’re saying we can think of Time as two-dimensional.  The second dimension being…?”

“I don’t know.  I just go there.  That’s the problem.”

“Hmm… When you do, do you feel like you’re turning left or right?”

“No turning or moving forward or backward.  Generally I have to … umm… ‘push’ like I’m going uphill, but that only works if there’s a ‘being pushed’ when I get past that.  Otherwise I’m back where I started, whatever that means.”

“What do you see?  What changes during the episode?”

“Little things. <brief fizzing sound.  She … flickered.>  Like ‘over there’ you’re wearing a bright green T-shirt instead of what you’re wearing here.  And you’re using pen-and-paper instead of that laptop.  Green doesn’t suit you.”

“I know, which is why there’s nothing green in my wardrobe, here.  But that gives me an idea.  Did you always have to ‘push’ to get ‘over there’?”

“Usually.”

“Fine.  OK, I’m going to flip this coin.  While it’s in the air, ‘push’ just lightly and come back to tell me which way the coin fell.”

<fizzing> “Heads.”

“It’s tails here.  OK, we’re going to do that again but this time ‘push’ much harder.”

<louder fizzing> “That was weird.  Your coin rolled off the desk and landed on edge in a crack in the floor so it’s not heads or tails.”

“AaaHAH!”Coins 1

“?”

“Your ‘over theres’ have different levels of probability than ‘over here.’  They’re different realities.  Actually, I’ll bet you travel across ranges of probability.  Or tunnel through them, maybe.  That’d why you have to ‘push’ to get past something that’s less probable in order to get to something that’s more probable.  Like getting past a reality where the coin can just hang in the air or fly apart.”

“I’ve done that.  Once I sneezed while ‘pushing’ and wound up sitting at a tea party where the cream and sugar just refused to stir into the tea.  When I ‘pushed’ from there I practically fell into a coffee shop where the coffee was well-behaved.”

“Case closed.  Now I can answer your question.  Spacewise, you’re in my office on the twelfth floor.  Timewise, I just showed you my clock.  As for which reality, you’re in one with a very high probability because, well, you’re here.”

“So provincial.  Oh, Mr Moire, how little you know.” <fizzing>

On the 12th floor of the Acme Building, high above the city, one man still tries to answer the Universe’s persistent questions — Sy Moire, Physics Eye.

~~ Rich Olcott

Three Perils for a Quest(ion), Part 1

On By rolcottIn Physics: Black Holes and Relativity, Physics: Quantum Mechanics, Uncategorized2 Comments

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

On By rolcottIn Physics: Black Holes and Relativity, UncategorizedLeave a comment

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 factor.  A 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

Questions, Meta-questions and Answers

On By rolcottIn Physics: Black Holes and Relativity, Uncategorized2 Comments

<We rejoin Sy and Vinnie in the library stacks…> “Are you boys discussing me?”

<unison> “Oh, hi, Ramona.”

“Actually no, Ramona, we were discussing relativistic time dilation.”

“I know that, Sy, I’ve been reading your posts. Now I’ve got a question.”

“But how…?  Never mind.  Guess I’d better watch my writing.  What can I do for you?”

“You and Vinnie have been going on about kinetic time dilation and gravitational time dilation like they’re two separate things, right?”

“That’s how we’ve treated them, right, but the textbooks do the same.  The velocity-dependent time-stretch equation, tslow/tfast = √[1-(v²/c²)], comes out of Einstein’s Special Theory of Relativity. The gravity-dependent equation, tslow/tfast = √[1-(2G·M/r·c²)], came from his General Theory of Relativity.”

“But there’s no rule that says an object can’t be moving rapidly while it’s in a gravitational field, is there?  That Endurance spacecraft orbiting the black hole in the Interstellar movie certainly seemed to be in that situation.”

“No question, Ramona.  General Relativity’s just more, er, general.”

“Fine, but shouldn’t they work together?”

That got Vinnie started.  “Yeah, Sy, I started this with LIGO and gravity but you and those space shuttles got me into this speed thing.  How do you bridge ’em?”

“Not easily.  Einstein set the rules of the game when he wrote down his fundamental equations.  Physicists and mathematicians have been trying to solve them ever since.  Schwarzchild found the first solution within a year after the equations hit the streets, but he did the simplest possible system — a non-rotating spherical object with no electrical charge and alone in the Universe.  It took another half-century before Kerr and friends figured out how to handle rotating spheres with an electric charge, but even those objects are assumed to be isolated from all other masses.  Mm … how do you figure velocity, Vinnie?”

“Distance divided by time, easy.”

“Not quite that easy.  The equations say that if you’re close to a massive object, space gets compressed, time gets stretched, and the time and space dimensions get scrambled.  Literally.  Time near a Schwarzchild object points inward as you approach the sphere’s center, and don’t ask me how to visualize that.  A Kerr object has a belt around its equator where time runs backwards.  Craziness.”

“Well, how about if I’m not that close?”

“That’s easier to answer, Ramona.  Suppose the three of us are each flying at safe distances from some heavy object with mass M.  I’m farthest away so I’m holding the fastest clock.  We’ll compare Vinnie’s and your clocks to mine.  OK?”3-clocks

“Sure, why not?”

“Fine.  Now, Vinnie, you’re closer in, resting on the direct line between me and the object.  You’re at distance r from it.  How fast does your clock run?”

“Uhhh…  We’re both on that same radial line so we’re in the same inertial frame, no kinetic effect.  I suppose you see it ticking slower because of the gravitational effect.”

“M-hm, and my clock ticks how often between ticks of yours?”

“You want the equation, huh?  All right, it’s tvinnie/tsy = √[1-(2G·M/r·c²)].”

“You’re reading my mind with those subscripts.  Now, Ramona, you’re at that same distance from the object but you’re in orbit around it.  Measured against Vinnie’s position you’ve got velocity v.  How fast is his clock ticking compared to yours?”

“Mmm…  We’re at the same level in the gravity field, so the gravitational thing makes no difference.  So … tramona/tvinnie = √[1-(v²/c²)].  Aaand, he’d see my clock running slow by the same amount. That’s weird.”

“Weird but true.  Last step — Ramona, you’re deeper in the gravitational field and you’re speeding away from me, so tramona/tsy=(tramona/tvinnie)*(tvinnie/tsy)=√[1-(2G·M/r·c²)]*√[1-(v²/c²)] covers both.”

“OK, that’s settled.  Back to Vinnie’s original question.  LIGOs are set in concrete, their velocities are zero so LIGO signals are all about gravity, right?”

“Right.”

Ramona links arms with him.  “Let’s go dancing.”  Then she gives me the eye.  “Sugarlumps, Sy?  Really?”

On the 12th floor of the Acme Building, high above the city, one man still tries to answer the Universe’s persistent questions — Sy Moire, Physics Eye.

~~ Rich Olcott

Weight And Wait, Two Aspects of Time

On By rolcottIn Physics: Black Holes and Relativity, Physics: Light and Relativity, UncategorizedLeave a comment

I was deep in the library stacks, hunting down a journal article so old it hadn’t been digitized yet.  As I rounded the corner of Aisle 5 Section 2, there he was, leaning against a post and holding a clipboard.

“Vinnie?  What are you doing here?”

“Waiting for you.  You weren’t in your office.”

“But how…?  Never mind.  What can I do for you?”

“It’s the time-dilation thing.  You said that there’s two kinds, a potential energy kind and a kinetic energy kind, but you only told me about the first one.”

“Hey, Ramona broke up that conversation, don’t blame me.  You got blank paper on that clipboard?”

“Sure.  Here.”

“Quick review — we said that potential energy only depends on where you are.  Suppose you and a clock are at some distance r away from a massive object like that Gargantua black hole, and my clock is way far away.  I see your clock ticking slower than mine.  The ratio of their ticking rates, tslow/tfast = √[1-(2G·M/r·c²)], only depends on the slow clock’s position.  Suppose you move even closer to the massive object.  That r-value gets smaller, the fraction inside the parentheses gets closer to 1, the square root gets smaller and I see your clock slow down even more.  Sound familiar?”

“Yeah, but what about the kinetic thing?”time-and-the-rovers

“I’m getting there.  You know Einstein’s famous EEinstein=m·c² equation.  See?  The formula contains neither a velocity nor a position.  That means EEinstein is the energy content of a particle that’s not moving and not under the influence of any gravitational or other force fields.  Under those conditions the object is isolated from the Universe and we call m its rest mass.  We good?”

“Yeah, yeah.”

“OK, remember the equation for gravitational potential energy?”

“E=G·M·m/r.”

“Let’s call that Egravity.  Now what’s the ratio between gravitational potential energy and the rest-mass energy?”

“Uh … Egravity/EEinstein = G·M·m/r·m·c² = G·M/r·c². Hey, that’s exactly half the fraction inside the square root up there. tslow/tfast = √[1-(2 Egravity/EEinstein)].  Cool.”

“Glad you like it.  Now, with that under our belts we’re ready for the kinetic thing.  What’s Newton’s equation for the kinetic energy of an object that has velocity v?”

“E=½·m·v².”

“I thought you’d know that.  Let’s call it Ekinetic.  Care to take a stab at the equation for kinetic time dilation?”

“As a guess, tslow/tfast = √[1-(2 Ekinetic/EEinstein)]. Hey, if I plug in the formulas for each of the energies, the halves and the mass cancel out and I get tslow/tfast = √[1-2(½m·v²/m·c²)] = √[1-(v²/c²)].  Is that it?”

“Close.  In Einstein’s math the kinetic energy expression is more complicated, but it leads to the same formula as yours.  If the velocity’s zero, the square root is 1.0 and there’s no time-slowing.  If the object’s moving at light-speed (v=c), the square root is zero and the slow clock is infinitely slow.  What’s interesting is that an object’s rest energy acts like a universal energy yardstick — both flavors of time-slowing are governed by how the current energy quantity compares to EEinstein.”

“Wait — kinetic energy depends on velocity, right, which means that it’ll look different from different inertial frames.  Does that mean that the kinetic time-slowing depends on the frames, too?”

“Sure it does.  Best case is if we’re both in the same frame, which means I see you in straight-line motion.  Each of us would get the same number if we measure the other’s velocity.  Plug that into the equation and each of us would see the same tslow for the other’s clock.  If we’re not doing uniform straight lines then we’re in different frames and our two dilation measurements won’t agree.”

“… Ramona doesn’t dance in straight lines, does she, Sy?”

“That reminds me of Einstein’s quote — ‘Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That’s relativity.‘  You’re thinking curves now, eh?”

“Are you boys discussing me?”

<unison> “Oh, hi, Ramona.”

~~ Rich Olcott

Three Body Problems

On By rolcottIn Astronomy: Stellar Science, Gravitational astronomy, Physics: Black Holes and Relativity, UncategorizedLeave a comment

The local science museum had a showing of the Christopher Nolan film Interstellar so of course I went to see it again.  Awesome visuals and (mostly) good science because Nolan had tapped the expertise of Dr Kip Thorne, one of the primary creators of LIGO.  On the way out, Vinnie collared me.

“Hey, Sy, ‘splain something to me.”

“I can try, but first let’s get out of the weather.  Al’s coffee OK with you?”

“Yeah, sure, if his scones are fresh-baked.”

Al saw me walking in.  “Hey, Sy, you’re in luck, I just pulled a tray of cinnamon scones out of the oven.”  Then he saw Vinnie.  “Aw, geez, there go my paper napkins again.”

Vinnie was ready.  “Nah, we’ll use the backs of some ad flyers I grabbed at the museum.  And gimme, uh, two of the cinnamons and a large coffee, black.”

“Here you go.”

At our table I said, “So what’s the problem with the movie?”

“Nobody shrank.  All this time we been talking about how things get smaller in a strong gravity field.  That black hole, Gargantua, was huge.  The museum lecture guy said it was like 100 million times as heavy as the Sun.  When the people landed on its planet they should have been teeny but everything was just regular-size.  And what’s up with that ‘one hour on the planet is seven years back home’ stuff?”

“OK, one thing at a time.  When the people were on the planet, where was the movie camera?”

“On the planet, I suppose.”

“Was the camera influenced by the same gravitational effects that the people were?”

“Ah, it’s the frames thing again, ain’t it?  I guess in the on-planet inertial frame everything stays the relative size they’re used to, even though when we look at the planet from our far-away frame we see things squeezed together.”

(I’ve told you that Vinnie’s smart.)  “You got it.  OK, now for the time thing.  By the way, it’s formally known as ‘time dilation.’  Remember the potential energy/kinetic energy distinction?”

“Yeah.  Potential energy depends on where you are, kinetic energy depends on how you’re moving.”

“Got it in one.  It turns out that energy and time are deeply intertwined all through physics.  Would you be surprised if I told you that there are two kinds of time dilation, one related to gravitational potential and the other to velocity?”

“Nothing would surprise me these days.  Go on.”

“The gravity one dropped out of Einstein’s Theory of Special Relativity.  The velocity one arose from his General Relativity work.”  I grabbed one of those flyers.  “Ready for a little algebra?”

“Geez.  OK, I asked for it.”gargantua-3
“You certainly did.  I’ll just give you the results, and mind you these apply only near a non-rotating sphere with no electric charge.  Things get complicated otherwise.  Suppose the sphere has mass M and you’re circling around it at a distance r from its geometric center.  You’ve got a metronome ticking away at n beats per your second and you’re perfectly happy with that.  We good?”

“So far.”

“I’m watching you from way far away.  I see your metronome running slow, at only n√[1-(2 G·M/r·c²)] beats per my second.  G is Newton’s gravity constant, c is the speed of light.  See how the square root has to be less than 1?”

“Your speed of light or my speed of light?”

“Good question, considering we’re talking about time and space getting all contorted, but Einstein guarantees that both of us measure exactly the same speed.  So anyway, in the movie both the Miller’s Planet landing team and that poor guy left on good ship  Endurance are circling Gargantua.  Earth observers would see both their clocks running slow.  But Endurance is much further out (larger r, smaller fraction) from Gargantua than Miller’s Planet is.  Endurance’s distance gave its clock more beats per Earth second than the planet gets, which is why the poor guy aged so much waiting for the team to return.”

“I wondered about that.”

Then we heard Ramona’s husky contralto.  “Hi, guys.  Al said you were back here talking physics.  Who wants to take me dancing?”

We both stood up, quickly.

“Whee, this’ll be fun.”

~~ Rich Olcott

Gravity’s Real Rainbow

On By rolcottIn Astronomy: Multi-messenger, Astronomy: Stellar Science, Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Waves, UncategorizedLeave a comment

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

LIGO and lambda and photons, oh my!

On By rolcottIn Astronomy: Multi-messenger, Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Light and Relativity, Physics: Quantum Mechanics, UncategorizedLeave a comment

I was walking my daily constitutional when Al waved me into his coffee shop.  “Sy, he’s at it again with the paper napkins.  Do something!”

I looked over.  There was Vinnie at his table, barricaded behind a pile of crumpled-up paper.  I grabbed a chair.

“Morning, Vinnie.  Having fun?”

“Greek letters.  Why’d they have to use Greek letters?”

The question was both rhetorical and derivative so I ignored it.  There were opened books under the barricade — upper-level physics texts.  “How come you’re chasing through those books?”

“I wanted to follow up on how LIGO operates with photons after we talked about all that space shuttle stuff.  But geez, Sy!”

“You’re a brave man, Vinnie.  So,  which letters are giving you trouble?”

“These two, that look kinda like each other upside down.” He pointed to one equation, λ=c/ν.

“Ah, wavelength equals the speed of light divided by the frequency.”

“How do you do that?”

“Some of those symbols go way back.  You just get used to them.  Most of them make sense when you learn the names for the letters — lambda (λ) is the peak-to-peak length of a lightwave, and nu (ν) is the number of peaks per second.  If it makes you feel any better, I’ve yet to meet a physicist who can write a zeta (ζ) — they generally just draw a squiggle and move on.”

“And there’s this other equation,” pointing to E=h·ν.  “What’s that about?”

“Good eye.  You just picked two equations that are fundamental to LIGO’s operation.  If a lightwave has frequency ν, the equations tell us two things about it — its energy is h·ν (h is Planck’s constant, 6.6×10-34 Joule-seconds), and its wavelength is c/ν (c is the speed of light).  For instance, yellow light has a frequency near 520×1012/sec.  One photon carries 3.8×10-40 Joules of energy.  Not much, but it adds up when a light beam contains lots of photons.  The same photon has a wavelength near 580×10-9 meters traveling through free space.”

“So what happens when one of those photons is in a LIGO beam?  Won’t a gravitational wave’s stretch-and-squeeze action mess up its wave?”

paper-napkin-waveI smoothed out one of Vinnie’s crumpled napkins. As I folded it into pleats and scooted it along the table I said, “Doesn’t mess up the wave so much as change the way we think about it.  We’re used to graphing out a spatial wave as an up-and-down pattern like this that moves through time, right?”

“That’s a lousy-looking wave.”

time-and-space-and-napkin
As the napkin moves through space,
the upper graph shows the height of its edge
above the observation point.

“It’s a paper napkin, f’pitysake, and I’m making a point here. Watch close.  If you monitor a particular point along the wave’s path in space and track how that point moves in time, you get the same profile except we draw it along the t-axis instead of along a space-axis.  See?”

“Hey, the time profile is the space profile going backwards.  Oh, right, it’s goin’ into the past ’cause it’s a memory.”

“That’s one of those things that people miss.  If you only draw sine waves, they’re the same in either direction.  The important point is that although timewaves and spacewaves have the same shape, they’ve got different meanings.  The timewave is directly connected to the wave’s energy by that E equation.  The spacewave is indirectly connected, because your other equation there scales it by the local speed of light.”

“Come again?  Local speed of light?  I thought it was 186,000 miles per second everywhere.”

“It is, but some of those miles are shorter than others.  Near a heavy mass, for instance, or in the compression phase of a gravitational wave, or inside a transparent material.  If you’re traveling in the lightwave’s inertial frame, you see no variation.  But if you’re watching from an independent inertial frame, you see the lightwave hit a slow patch.  Distance per cycle gets shorter.  Like that lambda-nu equation says, when c gets smaller the wavelength decreases.”

Al walked over.  “Gotcha a present, Vinnie.  Here’s a pad of yellow writing paper.  No more napkins, OK?”

“Uhh, thanks.”

“Don’t mention it.”

~~ Rich Olcott

Scone but not forgotten

On By rolcottIn Gravitational astronomy, Physics: Black Holes and Relativity, Uncategorized3 Comments

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 wave.  Gravity 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