Three Perils for a Quest(ion), Part 2

Eddie came over to our table.  “Either you folks order something else or I’ll have to charge you rent.”  Typical Eddie.

“Banana splits sound good to you two?”

[Jeremy and Jennie] “Sure.”

“OK, Eddie, two banana splits, plus a coffee, black, for me.  And an almond biscotti.”

“You want one, that’s a biscotto.”

“OK, a biscotto, Eddie.  The desserts are on my tab.”

“Thanks, Mr Moire.”

“Thanks, Sy.  I know you want to get on to the third Peril on Jeremy’s Quest for black hole evaporation, but how does he get past the Photon Sphere?”

“Yeah, how?”

“Frankly, Jeremy, the only way I can think of is to accept a little risk and go through it really fast.  At 2/3 lightspeed, for instance, you and your two-meter-tall suit would transit that zero-thickness boundary in about 10 nanoseconds.  In such a short time your atoms won’t get much out of position before the electromagnetic fields that hold your molecules together kick back in again.”

“OK, I’ve passed through.  On to the Firewall … but what is it?”

“An object of contention, for one thing.  A lot of physicists don’t believe it exists, but some claim there’s evidence for it in the 2015 LIGO observations.  It was proposed a few years ago as a way out of some paradoxes.”

“Ooo, Paradoxes — loverly.  What’re the paradoxes then?”

“Collisions between some of the fundamental principles of Physics-As-We-Know-It.  One goes back to the Greeks — the idea that the same thing can’t be in two places at once.”

“Tell me about it.  Here’s your desserts.”

“Thanks, Eddie.  The place keeping you busy, eh?”

“Oh, yeah.  Gotta be in the kitchen, gotta be runnin’ tables, all the time.”

“I could do wait-staff, Mr G.  I’m thinking of dropping track anyway, Mr Moire, 5K’s don’t have much in common with base running which is what I care about.  How about I show up for work on Monday, Mr G?”

“Kid calls me ‘Mr’ — already I like him.  You’re on, Jeremy.”

“Woo-hoo!  So what’s the link between the Firewall and the Greeks?”

Link is the right word, though the technical term is entanglement.  If you create two particles in a single event they seem to be linked together in a way that really bothered Einstein.”

“For example?”Astronaut and biscotti
“Polarizing sunglasses.  They depend on a light wave’s crosswise electric field running either up-and-down or side-to-side.  Light bouncing off water or road surface is predominately side-to-side polarized, so sunglasses are designed to block that kind.  Imagine doing an experiment that creates a pair of photons named Lucy and Ethel.  Because of how the experiment is set up, the two must have complementary polarizations.  You confront Lucy with a side-to-side filter.  That photon gets through, therefore Ethel should be blocked by a side-to-side filter but should go through an up-and-down filter.  That’s what happens, no surprise.  But suppose your test let Lucy pass an up-and-down filter.  Ethel would pass a side-to-side filter.”

“But Sy, isn’t that because each photon has a specific polarization?”

“Yeah, Jennie, but here’s the weird part — they don’t.  Suppose you confront Lucy with a filter set at some random angle.  There’s only the one photon, no half-way passing, so either it passes or it doesn’t.  Whenever Lucy chooses to pass, Ethel usually passes a filter perpendicular to that one.  It’s like Ethel hears from Lucy what the deal was — and with zero delay, no matter how far away the second test is executed.  It’s as though Lucy and Ethel are a single particle that occupies two different locations.  In fact, that’s exactly how quantum mechanics models the situation.  Quite contrary to the Greeks’ thinking.”

“You said that Einstein didn’t like entanglement, either.  How come?”

“Einstein published the original entanglement mathematics in the 30s as a counterexample against Bohr’s quantum mechanics.  The root of his relativity theories is that the speed of light is a universal speed limit.  If nothing can go faster than light, instantaneous effects like this can’t happen.  Unfortunately, recent experiments proved him wrong.  Somehow, both Relativity and Quantum Mechanics are right, even though they seem to be incompatible.”

“And this collision is why there’s a problem with black hole evaporation?”

“It’s one of the collisions.”

“There’s more?  Loverly.”

~~ Rich Olcott

Baseball And The Virtual Particle

Al was pouring my mugful of his morning blend (“If it doesn’t wake you up we’ll call the doctor“) when Jeremy stepped into the counter.  “Hi, Mr Moire.  I’m still trying to get my head around that virtual particle thing.  Hi, Al, a large decaf, please, double sugar, three creamers.  It looks like the shorter amount of time you give a particle to happen, the bigger it can get, but that doesn’t make sense because I’d think the longer you wait the more likely it’s gonna happen.  Thanks, Al.”

“Take a breath to blow on that coffee, Jeremy, or you’ll burn your tongue.  Hmm…  Word is your batting average is running about 250 these days.  That right?”

“Yessir.  I didn’t know you’re keeping track.”

“Keeping my ears open is part of my job.  So you’re hitting about once every four at-bats.  That gives Coach an estimate of when you’ll get your next hit.  What’s your slugging average?”

“What’s a slugging average?”

“Your total number of batted-on bases, divided by your at-bats, times a thousand ’cause sports writers don’t do decimal points.  You get one count in the numerator for a single, two for a double and so on.”

“Lemme think.  If I’m doing 250 overall and about half are singles and the other half are doubles that’d give me an SA of … about 375.”

“Pretty good.  So does that number tell Coach anything about when to expect another double?”

“Mmm, no, but what does that have to do with my virtual particle question?”

“In each case you’ve got a pair of statistics that tell you some things and hide other things.  Batting averages and your wait-time notion are about when to expect an event of some sort to occur.  You could hit another single or you could tag a homer — all Coach knows is that you should be able to get on base about once every four at-bats.”

“What about the other statistics?”

“They’re the flip side, sort of.  You could think of the SA as batting potential.  If you hit homers all the time your SA would be 4000.  If you whiff every pitch your SA would be zero.  Anything between those extremes tells Coach something about your productivity but nothing about when you’re going to produce.  Energy uncertainty works the same way for virtual particles.  If you’re doing long-duration energy evaluations you can be pretty sure that any single measurement will be close to the long-term average.  You might possibly see a significant deviation from that average but only if you check just the right brief interval.”Virtual baseball

“And for the particles in that empty space?”

“If you’re looking long-term, no particles.  That’s what ’empty’ means.  When there’s definitely nothing in a volume of space it makes sense to say its energy is zero because particles have mass and therefore embody energy.  But a particle might show up and go away after a very brief interval without significantly affecting that long-term average.  Quantum theory doesn’t say it will show up, just that it might.”

“So does it?”

“Oh yes, in space, in the lab and in commerce.  One explanation for your cell phone’s NFC function hinges on virtual radio-frequency photons being exchanged between devices.”

“Wait.  If a virtual particle shows up in that empty space, then it’s not empty any more and its energy isn’t zero any more, is it?”

“You’ve just discovered one aspect of zero-point energy, the quantum prediction that every system, even empty space, contains a non-zero minimum amount of energy.  People have thought about tapping that energy to power perpetual motion machines.”

“That’d be cool — the ultimate renewable.”

“Wouldn’t it, though?  But no can do, for a couple of reasons.  Virtual particles, by their nature, are random phenomena.  You can’t depend upon what kind of particle might show up, or when, nor how long it might hang around.  It’s not like NFC where antennas generate the particles.  The other issue is that ‘minimum’ means minimum.  If you could pull energy out of that space you’d lower its energy content and drop it below the minimum…. What’s the grin about?”

“Just wondering how they’d score hitting a virtual ball that disappears before the fielder catches it.”

~~ Rich Olcott


No knock, the door just opened suddenly.

“Hello, Jeremy.  Rule of Three?”

“Huh?  No, I was down the hall just now when I saw you go into your office so I knew you hadn’t gotten busy with something yet.  Sir.  What’s the Rule of Three?”

“Never mind.  You’re up here about virtual particles, I guess.”

“Yessir.  You said they’re ‘now you might see them, now you probably don’t.’  What’s that about and what do they have to do with abstraction and Einstein’s ‘underlying reality’?”

“What have you heard about Heisenberg’s Uncertainty Principle?”

“Ms Plenum says you can’t know where you are and how fast you’re going.”

“Ms Plenum’s got part of the usual notion but she’s missing the idea of simultaneous precision and a few other things.  Turns out you CAN know approximately where you are AND approximately how fast you’re going at a particular moment, but you can’t know both things precisely.  There’s going to be some imprecision in both measurements.  Think about Coach using a radar gun to track a thrown baseball.  How does radar work?”

“It bounces a light beam off of something and measures the light’s round-trip travel time.  I suppose it multiplies by the speed of light to convert time to distance.”

“Good.  Now how does it get the ball’s speed?”

“Uhh… probably uses two light pulses a certain time apart and calculates the speed as distance difference divided by time difference.”

“Got it in one.  Now, suppose that a second after the ball’s thrown the radar says the ball is 61 feet away from the plate and traveling at 92 mph.  Air resistance acts to slow the ball’s flight so that 92 is really an average.   Maybe it was going 92.1 mph at the first radar pulse and 91.9 mph at the second pulse.  So that reported speed has an 0.2 mph range of uncertainty.”

“Oh, and neither of the two pulses caught the ball at exactly 61 feet so that’s uncertain, too, right?”

“There you go.  We know the two averages, but each of them has a range.  The Uncertainty Principle says that the product of those two ranges has to be greater than Planck’s constant, 10-34 Joule·second.  Plugging that Joule-fraction and the mass of an electron into Einstein’s E=mc², we restate the constant as about 10-21 of an electron-second.  Those are both teeny numbers — but they’re not zero.”

“So speed and location make an uncertainty pair.  Are there others?”Zebras“A few.  The most important for this discussion is energy and time.”

“Wait a minute, those two can’t be linked that way.”

“Why not?”

“Well, because … umm … speed is change of location so those two go together, but energy isn’t change of time.  Time just … goes, and adding energy won’t make it go faster.”

“As a matter of fact, there are situations where adding energy makes time go slower, but that’s a couple of stories for another day.  What we’re talking about here is uncertainty ranges and how they combine.  Quantum theory says that if a given particle has a certain energy, give or take an energy range, and it retains that energy for a certain duration, give or take a time range, then the product of the two ranges has to be larger than that same Planck constant.   Think about a 1-meter cube of empty space out there somewhere.  Got it?


“Suppose a particle appeared and then vanished somewhere in that cube sometime during a 1-second interval.  What’s the longest time that particle could have existed?”

“Easy — one second.”

“How about the shortest time?”

“Zero.  Wait, it’d be the smallest possible non-zero time, wouldn’t it?”

“Good catch.  So what’s the time uncertainty?”

“One second minus that tiniest bit of time.”

“And what’s the corresponding energy range?”

“That constant number that I forget.”

“10-21 electron-second’s worth.  Now let’s pick a shorter interval.  What’s the mass range for a particle that appears and disappears sometime during the 10-19 second it takes a photon to cross a hydrogen atom?”

“That’s 10-21 electron-second divided by 10-19 second, so it’d be, like, 0.01 electron.”

“How about 1% of that 10-19 second?”

“Wow — that’d be a whole electron.”

“A whole electron’s worth of uncertainty.  But is the electron really there?”

“Probably not, huh?”

“Like I said, ‘Now you probably don’t’.”

~~ Rich Olcott

Abstract Horses

It was a young man’s knock, eager and a bit less hesitant than his first visit.

“C’mon in, Jeremy, the door’s open.”

“Hi, Mr Moire, it’s me, Jerem…  How did ..?  Never mind.  Ready for my black hole questions?”

“I’ll do what I can, Jeremy, but mind you, even the cosmologists are still having a hard time understanding them.  What’s your first question?”

“I read where nothing can escape a black hole, not even light, but Hawking radiation does come out because of virtual particles and what’s that about?”

“That’s a very lumpy question.  Let’s unwrap it one layer at a time.  What’s a particle?”

“A little teeny bit of something that floats in the air and you don’t want to breathe it because it can give you cancer or something.”

“That, too, but we’re talking physics here.  The physics notion of a particle came from Newton.  He invented it on the way to his Law of Gravity and calculating the Moon’s orbit around the Earth.  He realized that he didn’t need to know what the Moon is made of or what color it is.  Same thing for the Earth — he didn’t need to account for the Earth’s temperature or the length of its day.  He didn’t even need to worry about whether either body was spherical.  His results showed he could make valid predictions by pretending that the Earth and the Moon were simply massive points floating in space.”

Accio abstractify!  So that’s what a physics particle is?”

“Yup, just something that has mass and location and maybe a velocity.  That’s all you need to know to do motion calculations, unless the distance between the objects is comparable to their sizes, or they’ve got an electrical charge, or they move near lightspeed, or they’re so small that quantum effects come into play.  All other properties are irrelevant.”

“So that’s why he said that the Moon was attracted to Earth like the apple that fell on his head was — in his mind they were both just particles.”

“You got it, except that apple probably didn’t exist.”

“Whatever.  But what about virtual particles?  Do they have anything to do with VR goggles and like that?”

“Very little.  The Laws of Physics are optional inside a computer-controlled ‘reality.’  Virtual people can fly, flow of virtual time is arbitrary, virtual electrical forces can be made weaker or stronger than virtual gravity, whatever the programmers decide will further the narrative.  But virtual particles are much stranger than that.”

“Aw, they can’t be stranger than Minecraft.  Have you seen those zombie and skeleton horses?”Horses

“Yeah, actually, I have.  My niece plays Minecraft.  But at least those horses hang around.  Virtual particles are now you might see them, now you probably don’t.  They’re part of why quantum mechanics gave Einstein the willies.”

“Quantum mechanics comes into it?  Cool!  But what was Einstein’s problem?  Didn’t he invent quantum theory in the first place?”

“Oh, he was definitely one of the early leaders, along with Bohr, Heisenberg, Schrödinger and that lot.  But he was uncomfortable with how the community interpreted Schrödinger’s wave equation.  His row with Bohr was particularly intense, and there’s reason to believe that Bohr never properly understood the point that Einstein was trying to make.”

“Sounds like me and my Dad.  So what was Einstein’s point?”

“Basically, it’s that the quantum equations are about particles in Newton’s sense.  They lead to extremely accurate predictions of experimental results, but there’s a lot of abstraction on the way to those concrete results.  In the same way that Newton reduced Earth and Moon to mathematical objects, physicists reduced electrons and atomic nuclei to mathematical objects.”

“So they leave out stuff like what the Earth and Moon are made of.  Kinda.”

“Exactly.  Bohr’s interpretation was that quantum equations are statistical, that they give averages and relative probabilities –”

“– Like Schrödinger’s cat being alive AND dead –”

“– right, and Einstein’s question was, ‘Averages of what?‘  He felt that quantum theory’s statistical waves summarize underlying goings-on like ocean waves summarize what water molecules do.  Maybe quantum theory’s underlying layer is more particles.”

“Are those the virtual particles?”

“We’re almost there, but I’ve got an appointment.  Bye.”

“Sure.  Uhh… bye.”

~~ Rich Olcott

Three Body Problems

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

LIGO and lambda and photons, oh my!

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.

“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 (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.”


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

Does a photon experience time?

My brother Ken asked me, “Is it true that a photon doesn’t experience time?”  Good question.  As I was thinking about it I wondered if the answer could have implications for Einstein’s bubble.

When Einstein was a grad student in Göttingen, he skipped out on most of the classes given by his math professor Hermann Minkowski.  Then in 1905 Einstein’s Special Relativity paper scooped some work that Minkowski was doing.  In response, Minkowski wrote his own paper that supported and expanded on Einstein’s.  In fact, Minkowski’s contribution changed Einstein’s whole approach to the subject, from algebraic to geometrical.

But not just any geometry, four-dimensional geometry — 3D space AND time.  But not just any space-AND-time geometry — space-MINUS-time geometry.  Wait, what?Pythagoras1

Early geometer Pythagoras showed us how to calculate the hypotenuse of a right triangle from the lengths of the other two sides. His a2+b2 = c2 formula works for the diagonal of the enclosing rectangle, too.

Extending the idea, the body diagonal of an x×y×z cube is √(x2+y2+z2) and the hyperdiagonal  of a an ct×x×y×z tesseract is √(c2t2+x2+y2+z2) where t is time.  Why the “c“?  All terms in a sum have to be in the same units.  x, y, and z are lengths so we need to turn t into a length.  With c as the speed of light, ct is the distance (length) that light travels in time t.

But Minkowski and the other physicists weren’t happy with Pythagorean hyperdiagonals.  Here’s the problem they wanted to solve.  Suppose you’re watching your spacecraft’s first flight.  You built it, you know its tip-to-tail length, but your telescope says it’s shorter than that.  George FitzGerald and Hendrik Lorentz explained that in 1892 with their length contraction analysis.

What if there are two observers, Fred and Ethel, each of whom is also moving?  They’d better be able to come up with the same at-rest (intrinsic) size for the object.

Minkowski’s solution was to treat the ct term differently from the others.  Think of each 4D address (ct,x,y,z) as a distinct event.  Whether or not something happens then/there, this event’s distinct from all other spatial locations at moment t, and all other moments at location (x,y,z).

To simplify things, let’s compare events to the origin (0,0,0,0).  Pythagoras would say that the “distance” between the origin event and an event I’ll call Lucy at (ct,x,y,z) is √(c2t2+x2+y2+z2).

Minkowski proposed a different kind of “distance,” which he called the interval.  It’s the difference between the time term and the space terms: √[c2t2 + (-1)*(x2+y2+z2)].

If Lucy’s time is t=0 [her event address (0,x,y,z)], then the origin-to-Lucy interval is  √[02+(-1)*(x2+y2+z2)]=i(x2+y2+z2).  Except for the i=√(-1) factor, that matches the familiar origin-to-Lucy spatial distance.

Now for the moment let’s convert the sum from lengths to times by dividing by c2.  The expression becomes √[t2-(x/c)2-(y/c)2-(z/c)2].  If Lucy is at (ct,0,0,0) then the origin-to-Lucy interval is simply √(t2)=t, exactly the time difference we’d expect.

Finally, suppose that Lucy departed the origin at time zero and traveled along x at the speed of light.   At any time t, her address is (ct,ct,0,0) and the interval for her trip is √[(ct)2-(ct)2-02-02] = √0 = 0.  Both Fred’s and Ethel’s clocks show time passing as Lucy speeds along, but the interval is always zero no matter where they stand and when they make their measurements.

Feynman diagramOne more step and we can answer Ken’s question.  A moving object’s proper time is defined to be the time measured by a clock affixed to that object.  The proper time interval between two events encountered by an object is exactly Minkowski’s spacetime interval.  Lucy’s clock never moves from zero.

So yeah, Ken, a photon moving at the speed of light experiences no change in proper time although externally we see it traveling.

Now on to Einstein’s bubble, a lightwave’s spherical shell that vanishes instantly when its photon is absorbed by an electron somewhere.  We see that the photon experiences zero proper time while traversing the yellow line in this Feynman diagram.  But viewed from any other frame of reference the journey takes longer.  Einstein’s objection to instantaneous wave collapse still stands.

~~ Rich Olcott