Tag: speed of light

The Bottom of Time

On By rolcottIn Astronomy: Techniques, Cosmology, James Webb Space Telescope, Uncategorized2 Comments

“Cathleen, one of my Astronomy magazines had an article, claimed that James Webb Space Telescope can see back to the Big Bang. That doesn’t seem right, right?”

“You’re right, Al, it’s not quite right. By our present state of knowledge JWST‘s infrared perspective goes back only 98% of the way to the Bang. Not quite the Bottom of Time, but close.”

“Whaddaya mean, ‘Bottom of Time‘? I’ve heard people talking about how weird it musta been before the Big Bang. And how can JWST see back in time anyway? Telescopes look across space, not time.”

“So many questions, Mr Feder, and some hiding behind others. That’s his usual mode, Cathleen. Care to tag-team?”

“You’re on, Sy. Well, Mr Feder. The ‘look back in time‘ part comes from light not traveling infinitely fast. We’ve known that for three centuries, ever since Rømer—”

“Roamer?”

“Ole Rømer, a Danish scientist who lived in Newson’s time. Everyone knew that Jupiter’s innermost large moon Io had a dependably regular orbit, circling Jupiter every 49½ hours like clockwork. Rømer was an astronomer when he wasn’t tutoring the French King’s son or being Copenhagen’s equivalent of Public Safety Commissioner. He watched Io closely, kept notes on exactly when she ducked behind Jupiter and when she reappeared on the other side. His observed timings weren’t quite regular, generally off by a few minutes. Funny thing was, the irregularities correlated with the Earth‑Jupiter distance — up to 3½ minutes earlier than expected when Earth in its orbit was closest to Jupiter, similarly late when they were far apart. There was a lot of argument about how that could be, but Rømer, Huygens, even Newton, all agreed that the best explanation was that we only see Io’s passage events after light has taken its time to travel from there to here.”

“Seems reasonable. Why should people argue about that?”

“The major sticking point was the speed that Huygens calculated from Rømer’s data. We now know it’s 186000 miles or 300000 kilometers or one lightsecond per second. Different ways of stating the same quantity. Huygens came up with a somewhat smaller number but still. The establishment pundits had been okay with light transmission being instantaneous. Given definite numbers, though, they had trouble accepting the idea that anything physical could go that fast.”

“Tag, my turn. Flip that distance per time ratio upside down — for every additional lightsecond of distance we’re looking at events happening one second farther into the past. That’s the key to JWST‘s view into the long‑ago. Al, you got that JWST‘s infrared capabilities will beat Hubble‘s vis‑UV ones for distance. Unless there’s something seriously wrong with Einstein’s assumption that lightspeed’s an absolute constant throughout spacetime, we expect JWST to give us visibility to the oldest free photons in the Universe, just 379000 years upward from the Big Bang.”

“Wait, I heard weaseling there. Free photons? Like you gotta pay for the others?”

“Ha, ha, Mr Feder. During those first 379000‑or‑so years, we think the Universe was so hot and so dense that no photon’s wave had much of a chance to spread out before it encountered a charged something and got absorbed. At last, things cooled down enough for atoms to form and stay in one piece. Atoms are neutral. Quantum rules restrict their interaction to only photons that have certain wavelengths. The rest of the photons, and there’s a huge number of them, were free to roam the expanding Universe until they happen to find a suitable absorber. Maybe someone’s eye or if we’re lucky, a sensor on JWST or some other telescope.”

Thanks for this to George Derenburger

“What about before the 300‑and‑something thousand years? Like, before Year Zero? Musta been weird, huh?”

“Well, there’s a problem with that question. You’re assuming there was a Year Minus‑One, but that’s just not the case.”

“Why not? Arithmetic works that way.”

“But the Universe doesn’t. Stephen Hawking came up with a good way to think about it. What on Earth is south of the South Pole?”

“Eeayahh … nope. Can’t get any further south than that.”

“Well, there you are, so to speak. Time’s bottom is Year Zero and you can’t get any further down than that. We think.”

~~ Rich Olcott

Speed Limit

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

“Wait, Sy, there’s something funny about that Lorentz factor. I’m riding my satellite and you’re in your spaceship to Mars and we compare notes and get different times and lengths and masses and all so we have to use the Lorentz factor to correct numbers between us. Which velocity do we use, yours or mine?”

“Good question, Vinnie. We use the difference between our two frames. We can subtract either velocity from the other one and replace v with that number. Strictly speaking, we’d subtract velocity components perpendicular to the vector between us. If I were to try to land on your satellite I’d have to expend fuel and energy to change my frame’s velocity to yours. When we matched frames the velocity difference would be zero, the Lorentz factor would be 1.0 and I’d see your solar array as a perfect 10×10‑meter square. Our clocks would tick in sync, too.”

“OK, now there’s another thing. That Lorentz formula compares our subtracted speeds to lightspeed c. What do we subtract to get c?”

“Deep question. That’s one of Einstein’s big insights. Suppose from my Mars‑bound spaceship I send out one light pulse toward Mars and another one in the reverse direction, and you’re watching from your satellite. No matter how fast my ship is traveling, Einstein said that you’d see both pulses, forward and backward, traveling at the same speed, c.”

“Wait, shouldn’t that be that your speed gets added to one pulse and subtracted from the other one?”

“Ejected mass works that way, but light has no mass. It measures its speed relative to space itself. What you subtract from c is zero. Everywhere.”

“OK, that’s deep. <pause> But another ‘nother thing—”

“For a guy who doesn’t like equations, you’re really getting into this one.”

“Yeah, as I get up to speed it grows on me. HAW!”

“Nice one, you got me. What’s the ‘nother thing?”

“I remembered how velocity is speed and direction but we’ve been mixing them together. If my satellite’s headed east and your spaceship’s headed west, one of us is minus to the other, right? We’re gonna figure opposite v‑numbers. How’s that work out?”

“You’re right. Makes no difference to the Lorentz factor because the square of a negative difference is the same as the square of its positive twin. You bring up an important point, though — the factor applies to both of us. From my frame, your clock is running slow. From your frame, mine’s the slow one. Einstein’s logic says we’re both right.”

“So we both show the same wrong time, no problem.”

“Nope, you see my clock running slow relative to your clock. I see exactly the reverse. But it gets worse. How about getting your pizza before you order it?”

“Eddie’s good, he ain’t that good. How do you propose to make that happen?”

“Well, I don’t, but follow me here. <working numbers on Old Reliable> Suppose we’re both in spaceships. I’m loafing along at 0.75c relative to Eddie’s pizza place on Earth and your ship is doing 3c. Also, suppose that we can transmit messages and mass much faster than lightspeed.”

“Like those Star Trek transporters and subspace radios.”

“Right. OK, at noon on my personal clock you tell me you’ve ordered pizza so I get one, too. Eddie slaps both our pizzas into his transporter 10 minutes later. The math works out that according to my clock you get your pizza 8.9 minutes before you put in your order. You like that?”

“Gimme a sec … nah, I don’t think so. If I read that formula right with v1 being you and v2 being me, if you run that formula for what I’d see with my velocity on the bottom, that’s a square root of a minus which can’t be right.”

“Yup, the calculation gives an imaginary number, 4.4i minutes, whatever that means. So between us we have two results that are just nonsense — I see effect before cause and you see a ridiculous time. To avoid that sort of thing, Einstein set his speed limit for light, gravity and information.”

“I’m willing to keep under it if you are.”

“Deal.”

~~ Rich Olcott

Three-speed Transmission

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

“Have I got this straight, Sy? You’re saying that prisms throw rainbows because light goes slower through glass than in air and that bends the beam, but every frequency lightwave bends a different amount. Also you’re saying all the bending happens when speeds switch at the glass face, not inside the glass. Am I right so far?”

“Perfect, Vinnie, but you skipped an important detail.”

“Which one?”

“Snell’s ‘index of refraction‘, the ratio of wave speed in vacuum to wave speed in the medium. The higher the frequency, the higher the speed in the medium so the index decreases towards 1.0. The definition lets us calculate wave speed in the medium from that frequency’s refraction index. For most materials the index is usually greater than 1.0, meaning that the speed inside the material is usually slower than in space.”

“Still using those ‘most‘ and ‘usually‘ weasel‑words.”

“Guilty as charged, because we’ve finally gotten to the ‘multiple speeds of light‘ thing. Which means I need more precise wording. The wave speed we’ve been talking about so far applies to a specific part of the wave, say the peak or trough. Those are wave phases, so I’m going to call that speed the ‘phase speed‘, OK?”

“Fine with me.”

“Good, because the second speed is different. Among his many important contributions, Lord Rayleigh pointed out that you can’t have a pulse that’s one pure frequency. A single‑frequency wave never starts and never ends. Do you remember the time I combined waves to draw a camel?”

“You did, mostly, but there was funny stuff at his nose and butt.”

“Because I only included about a hundred component waves. It’d take many more to kill those boundary zig‑zags. Any finite wave has the same issue. Rayleigh said that an individual wave has a phase speed, but any ‘peculiarity,’ like a pulse rise or fall, could only be created by a group of waves. The peculiarity could travel at a different speed from the component waves, like a pair of scissors where the cutting point moves faster than either blade.”

“Sounds like carrier wave and sidebands on my ham radio. But if different frequencies have different speeds they’d get all out of sync with each other. How does a photon stay in one piece?”

“The vacuum is non-dispersive — the photon’s component waves all travel at the same speed and stay together. If a medium absorbs some frequency, that makes it dispersive and that changes things.”

“Ah, that’s why you hedged about transparency.”

“Exactly. Throw in a few absorbing atoms, like cobalt that absorbs red or gold that absorbs blue, and you get interesting effects from your sideband components interacting. Skipping some math, the bottom line is simple and cute. The group speed’s equation is just like the phase speed’s except there’s a positive or negative correction term in the denominator.”

“Sy, I don’t like equations, remember? I suppose f is frequency in your correction term but what’s slope?”

“That’s a measure of how rapidly the index changes as the frequency changes. For most frequencies and most media, the slope is very slightly negative because the index slowly descends towards 1.0 at high energies. The vg fraction’s denominator stays just less than nf so the group goes slightly faster than the phase. Near an absorption line, though, things get sloppy. Waves that are just a little off the absorber’s favorite frequency can still interact with it. That changes their speed and the ‘corrected’ refraction index.”

“Gimme a sec … guess I’m OK with the positive slopes but there’s that yellow part where the slope is negative. Wouldn’t that make the fraction’s bottom smaller and the group speed higher?”

“Certainly. In fact, under the right conditions the denominator can be less than 1.0. That pushes the group speed above c — faster than light in vacuum, even though the component waves all run slower than vacuum lightspeed. It’s only the between‑component out‑of‑syncness relationship that scissors along beyond c.”

“You said there’s a third speed?”

“Signals. In a dispersive medium those sideband waves get chaotic and can’t carry information. Wave theory and Einstein agree — chaos may be able to travel faster than light, but information can’t.”

~~ Rich Olcott

Trombones And Echoes

On By rolcottIn Physics: Foundations of Measurement, Physics: Waves, UncategorizedLeave a comment

Vinnie’s fiddling with his Pizza Eddie’s pizza crumbs. “Hey, Sy, so we got the time standard switched over from that faked 1900 Sun to counting lightwave peaks in a laser beam. I understand why that’s more precise ’cause it’s a counting measure, and it’s repeatable and portable ’cause they can set up a time laser on Mars or wherever that uses the same identical kinds of atoms to do the frequency stuff. All this talk I hear about spacetime, I’m thinkin’ space is linked to time, right? So are they doing smart stuff like that for measuring space?”

“They did in 1960, Vinnie. Before that the meter was defined to be the distance between two carefully positioned scratches on a platinum-iridium bar that was lovingly preserved in a Paris basement vault. In 1960 they went to a new standard. Here, I’ll bring it up on Old Reliable. By the way, it’s spelled m-e-t-e-r stateside, but it’s the same thing.”

“Mmm… Something goofy there. Look at the number. You’ve been going on about how a counted standard is more precise than one that depends on ratios. How can you count 0.73 of a cycle?”

“You can’t, of course, but suppose you look at 100 meters. Then you’d be looking at an even 165,076,373 of them, OK?”

“Sorta, but now you’re counting 165 million peaks. That’s a lot to ask even a grad student to do, if you can trust him.”

“He won’t have to. Twenty-three years later they went to this better definition.”

“Wait, that depends on how accurate we can measure the speed of light. We get more accurate, the number changes. Doesn’t that get us into the ‘different king, different foot-size’ hassle?”

“Quite the contrary. It locks down the size of the unit. Suppose we develop technology that’s good to another half-dozen digits of precision. Then we just tack half-a-dozen zeroes onto that fraction’s denominator after its decimal point. Einstein said that the speed of light is the same everywhere in the Universe. Defining the meter in terms of lightspeed gives us the same kind of good-everywhere metric for space that the atomic clocks give us for time.”

“I suppose, but that doesn’t really get us past that crazy-high count problem.”

“Actually, we’ve got three different strategies for different length scales. For long distances we just use time-of-flight. Pick someplace far away and bounce a laser pulse off of it. Use an atomic clock to measure the round-trip time. Take half that, divide by the defined speed of light and you’ve got the distance in meters. Accuracy is limited only by the clock’s resolution and the pulse’s duration. The Moon’s about a quarter-million miles away which would be about 2½ seconds round-trip. We’ve put reflectors up there that astronomers can track to within a few millimeters.”

“Fine, but when distances get smaller you don’t have as many clock-ticks to work with. Then what do you do?”

“You go to something that doesn’t depend on clock-ticks but is still connected to that constant speed of light. Here, this video on Old Reliable ought to give you a clue.”

“OK, the speed which is a constant is the number of peaks that’s the frequency times the distance between them that’s the wavelength. If I know a wavelength then arithmetic gets me the frequency and vice-versa. Fine, but how do I get either one of them?”

“How do you tune a trombone?”

“Huh? I suppose you just move the slide until you get the note you want.”

“Yup, if a musician has good ear training and good muscle memory they can set the trombone’s resonant tube length to play the right frequency. Table-top laser distance measurements use the same principle. A laser has a resonant cavity between two mirrors. Setting the mirror-to-mirror distance determines the laser’s output. When you match the cavity length to something you want to measure, the laser beam frequency tells you the distance. At smaller scales you use interference techniques to compare wavelengths.”

Vinnie gets a gleam in his eye. “Time-of-flight measurement, eh?” He flicks a pizza crumb across the room.

In a flash Eddie’s standing over our table. “Hey, hotshot, do that again and you’re outta here!”

“Speed of light, Sy?”

“Pretty close, Vinnie.”

~~ Rich Olcott

The Speeds of Light

On By rolcottIn Astronomy: Techniques, Physics: Electromagnetism, Physics: Fundamentals, Physics: Light and Relativity, UncategorizedLeave a comment

“I don’t give up easy, Sy.”

“I know that, Vinnie.  Still musing about lightwaves and how they’re all an electron’s fault?”

“Yeah.  Hey, can your OVR app on Old Reliable grab a shot from this movie running on my smartphone?”

“We can try … got it.  Now what?”

“I wanna try mixing that with your magnetic field picture.”

“I’ll bring that up … Here, have at it.”

“Umm … Nice app, works very intuitive-like …  OK, see this?”Electrons and lightwave

“Ah.  It’s a bit busy, walk me through what’s in there.”

“OK. First we got the movie’s lightwave.  The ray’s running along that black arrow, see?  Some electron back behind the picture is going up and down to energize the ray and that makes the electric field that’s in red that makes other electrons go up and down, right?”

“That’s the red arrow, hmm?”

“Yeah, that electron got goosed ’cause it was standing in the way.  It follows the electric field’s direction.  Now help me out with the magnetic stuff.”

“Alright.  The blue lines represent the lightwave’s magnetic component.  A lightwave’s magnetic field lines are always perpendicular to its electric field.  Magnetism has no effect on uncharged particles or motionless charged particles.  If you’re a moving charged particle, say an electron, then the field deflects your trajectory.”

“This is what I’m still trying to wrap my head around.  You say that the field’s gonna push the particle perpendicular to the field and to the particle’s own vector.”

“That’s exactly what happens.  The green line, for instance, could represent an electron that crossed the magnetic field.  The field deflected the electron’s path upwards, crossways to the field and the electron’s path.  Then I suppose the electron encountered the reversed field from the lightwave’s following cycle and corrected course again.”

“And the grey line?”

“That’d be an electron crossing more-or-less along the field.  According to the Right Hand Rule it was deflected downward.”

“Wait.  We’ve got two electrons on the same side of the field and they’re deflected in opposite directions then correct back.  Doesn’t that average out to no change?”

“Not quite.  The key word is mostly.  Like gravity fields, electromagnetic fields get weaker with distance.  Each up or down deflection to an electron on an outbound path will be smaller than the previous one so the ‘course corrections’ get less correct.  Inbound electrons get deflected ever more strongly on the way in, of course, but eventually they become outbound electrons and get messed up even more.  All those deflections produce an expanding cone of disturbed electrons along the path of the ray.”

“Hey, but when any electron moves that changes the fields, right?  Wouldn’t there be a cone of disturbed field, too?”

“Absolutely.  The whole process leads to several kinds of dispersion.”

“Like what?”

“The obvious one is simple geometry.  What had been a simple straight-line ray is now an expanding cone of secondary emission.  Suppose you’re an astronomer looking at a planet that’s along that ray, for instance.  Light’s getting to you from throughout the cone, not just from the straight line.  You’re going to get a blurred picture.”

“What’s another kind?”

“Moving those electrons around extracts energy from the wave.  Some fraction of the ray’s original photons get converted to lower-energy ones with lower frequencies.  The net result is that the ray’s spectrum is spread and dispersed towards the red.”

“You said several kinds.”

“The last one’s a doozy — it affects the speeds of light.”

“‘Speeds,’ plural?”ripples in a wave

“There’s the speed of field’s ripples, and there’s the speed of the whole signal, say when a star goes nova.  Here’s a picture I built on Old Reliable.  The gold line is the electric field — see how the ripples make the red electron wobble?  The green dots on the axis give you comparison points that don’t move.  Watch how the ripples move left to right just like the signal does, but at their own speed.”

“Which one’s Einstein’s?”

“The signal.  Its speed is called the group velocity and in space always runs 186,000 mph.  The ripple speed, technically it’s the phase velocity, is slower because of that extracted-and-redistributed-energy process.  Different frequencies get different slowdowns, which gives astronomers clues about the interstellar medium.”

“Clues are good.”

~~ Rich Olcott

Gravity’s Real Rainbow

On By rolcottIn Astronomy: LIGO, Astronomy: Multi-messenger, Astronomy: Stellar Science, 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: LIGO, Astronomy: Multi-messenger, 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

Superluminal Superman

On By rolcottIn Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

Comic book and movie plotlines often make Superman accelerate up to lightspeed and travel backward in time.  Unfortunately, well-known fundamental Physics principles forbid that.  But suppose Green Lantern or Dr Strange could somehow magic him past the Lightspeed Barrier.  Would that let him do his downtimey thing?

Light_s hourglass
Light’s Hourglass

A quick review of Light’s Hourglass.  According to Einstein we live in 4D spacetime.  At any moment you’re at a specific time t relative to some origin time t=0 and a specific 3D location (x,y,z) relative to a spatial origin (0,0,0).  Your spacetime address is (ct,x,y,z) where c is the speed of light.  This diagram shows time running vertically into the future, plus two spatial coordinates x and y.  Sorry, I can’t get z into the diagram so pretend it’s zero.

The two cones depict all the addresses which can communicate with the origin using a flash of light.  Any point on either cone is at just the right distance d=√(x²+y²+z²) to match the distance that light can travel in time t.  The bottom cone is in the past, which is why we can see the light from old stars.  The top cone is in the future, which is why we can’t see light from stars that aren’t born yet.

If he obeys the Laws of Physics as we know them, Superman can travel anywhere he wants to inside the top cone.  He goes upward into the future at the rate of one second per second, just like anybody.  On the way, he can travel in space as far from (ct,0,0,0) as he likes so long as it’s not farther than the distance that light can travel the same route at his current t.

From our perspective, the Hourglass is a stack of circles (spheres in 3D space) centered on (ct,0,0,0).  From Supey’s perspective at time t he’s surrounded by a figure with radius ct that Physics won’t let him break through.  That’s his Lightspeed Barrier, like the Sonic Barrier but 900,000 times faster.

Suppose Green Lantern has magicked Supey up to twice lightspeed along the x-axis.  At moment t, he’s at (ct,2ct,0,0), twice as far as light can get.  In the diagram he’s outside the top cone but above the central disk.

Now GL pours on the power to accelerate Superman.  Each increment gets the Man of Steel closer to that disk.  He’s always “above” it, though, because he’s still moving into the future.  Only if he were to get to infinite speed could he reach the disk.

However, at infinite speed he’d go anywhere/everywhere instantaneously which would be confusing to even his Kryptonian intellect.  On the way he might run into things (stars, black holes,…) with literally zero time to react.

But the plotlines have Tall-Dark-and-Muscular flying into the past, breaching that disk and traveling downwards into the bottom cone.  Can GL make that happen?

Enter the Lorentz correction.  If you have rest mass m0 and you’re traveling at speed v, your effective mass is m=m0/√[1-(v/c)²]. That raises a couple of issues when you exceed lightspeed.

Suppose GL decelerates Superluminal Supey down towards lightspeed.  The closer he approaches c from higher speeds, the smaller that square root gets and the greater the effective mass.  It’s the same problem Superman faced when accelerating up to lightspeed.  That last mile per second down to c requires an infinite amount of braking energy — the Lightspeed Barrier is impermeable in both directions.

The other problem is that if v>c there’s a negative number inside that square root.    Above lightspeed, your effective mass becomes Bombelli-imaginary.  Remember Newton’s famous F=m·a?  Re-arrange it to a=F/m.  A real force applied to an object with imaginary mass produces an imaginary acceleration.  “Imaginary” in Physics generally means “perpendicular in some sense” and remember we’re in 4D here with time perpendicular to space.

GL might be able to shove Superman downtime, but he’d have to

  1. squeeze inward at hiper-lightspeed with exactly the same force along all three spatial dimensions, to make sure that “perpendicular” is only along the time axis
  2. start Operation Squish at some time in his own future to push towards the past.

Nice trick.  Would Superman buy in?

~~ Rich Olcott