# Thinking in Spacetime

The Open Mic session in Al’s coffee shop is still going string. The crowd’s still muttering after Jeremy stuck a pin in Big Mike’s “coincidence” balloon when Jim steps up. Jim’s an Astrophysics post‑doc now so we quiet down expectantly. “Nice try, Mike. Here’s another mind expander to play with. <stepping over to the whiteboard> Folks, I give you … a hypotenuse. ‘That’s just a line,’ you say. Ah, yes, but it’s part of some right triangles like … these. Say three different observers are surveying the line from different locations. Alice finds her distance to point A is 300 meters and her distance to point B is 400. Applying Pythagoras’ Theorem, she figures the A–B distance as 500 meters. We good so far?”

A couple of Jeremy’s groupies look doubtful. Maybe‑an‑Art‑Major shyly raises a hand. “The formula they taught us is a2+b2=c2. And aren’t the x and y supposed to go horizontal and vertical?”

“Whoa, nice questions and important points. In a minute I’m going to use c for the speed of light. It’s confusing to use the same letter for two different purposes. Also, we have to pay them extra for double duty. Anyhow, I’m using d for distance here instead of c, OK? To your next point — Alice, Bob and Carl each have their own horizontal and vertical orientations, but the A–B line doesn’t care who’s looking at it. One of our fundamental principles is that the laws of Physics don’t depend on the observer’s frame of reference. In this situation that means that all three observers should measure the same length. The Pythagorean formula works for all of them, so long as we’re working on a flat plane and no-one’s doing relativistic stuff, OK?”

Tentative nods from the audience.

“Right, so much for flat pictures. Let’s up our game by a dimension. Here’s that same A–B line but it’s in a 3D box. <Maybe‑an‑Art‑Major snorts at Jim’s amateur attempt at perspective.> Fortunately, the Pythagoras formula extends quite nicely to three dimensions. It was fun figuring out why.”

Jeremy yells out. “What about time? Time’s a dimension.”

“For sure, but time’s not a length. You can’t add measurements unless they all have the same units.”

“You could fix that by multiplying time by c. Kilometers per second, times seconds, is a length.” His groupies go “Oooo.”

“Thanks for the bridge to spacetime where we have four coordinates — x, y, z and ct. That makes a big difference because now A and B each have both a where and a when — traveling between them is traveling in space and time. Computationally there’s two paths to follow from here. One is to stick with Pythagoras. Think of a 4D hypercube with our A–B line running between opposite vertices. We’re used to calculating area as x×y and volume as x×y×z so no surprise, the hypercube’s hypervolume is x×y×z×(ct). The square of the A–B line’s length would be b2=(ct)2+d2. Pythagoras would be happy with all of that but Einstein wasn’t. That’s where Alice and Bob and Carl come in again.”

“What do they have to do with it?”

“Carl’s sitting steady here on good green Earth, red‑shifted Alice is flying away at high speed and blue‑shifted Bob is flashing toward us. Because of Lorentz contractions and dilations, they all measure different A–B lengths and durations. Each observer would report a different value for b2. That violates the invariance principle. We need a ruggedized metric able to stand up to that sort of punishment. Einstein’s math professor Hermann Minkowski came up with a good one. First, a little nomenclature. Minkowski was OK with using the word ‘point‘ for a location in xyz space but he used ‘event‘ when time was one of the coordinates.”

“Makes sense, I put events on my calendar.”

“Good strategy. Minkowski’s next step quantified the separation between two events by defining a new metric he called the ‘interval.’ Its formula is very similar to Pythagoras’ formula, with one small change: s2=(ct)2–d2. Alice, Bob and Carl see different distances but they all see the same interval.”

Minus? Where did that come from?”

~~ Rich Olcott

# Disentangling 3-D Plaid

Our lake-side jog has slowed to a walk and suddenly Mr Feder swerves off the path to thud onto a park bench. “I’m beat.”

Meanwhile, heavy footsteps from behind on the gravel path and a familiar voice. “Hey, Sy, you guys talking physics?”

“Well, we were, Vinnie. Waves, to be exact, but Feder’s faded and anyway his walk wasn’t fast enough to warm me up.”

“I’ll pace you. What’d I miss?”

“Not a whole lot. So many different kinds of waves but physicists have abstracted them down to a common theme — a pattern that moves through space.”

“Haw — flying plaid.”

“That image would work if each fiber color carried specific values of energy and momentum and the cross-fibers somehow add together and there’s lots of waves coming from all different directions so it’s 3-D.”

“Sounds complicated.”

“As complicated as the sound from a symphony.”

“I prefer dixieland.”

“Same principle. Trumpet, trombone, clarinet, banjo — many layers of harmony but you can choose to tune in on just one line. That’s a clue to how physicists un-complicate waves.”

“How so?”

“Back in the early 19th century, Fourier showed that you can think about any continuous variation stream, no matter how complicated, in terms of a sum of very simple variations called sine waves. You’ve seen pictures of a sine wave — just a series of Ss laid on their sides and linked together head-to-tail.”

“Your basic wiggly line.”

“Mm-hm, except these wiggles are perfectly regular — evenly spaced peaks, all with the same height. The regularity is why sine waves are so popular. Show a physicist something that looks even vaguely periodic and they’ll immediately start thinking sine wave frequencies. Pythagoras did that for sound waves 2500 years ago.”

“Nah, he couldn’t have — he died long before Fourier.”

“Good point. Pythagoras didn’t know about sine waves, but he did figure out how sounds relate to spatial frequencies. Pluck a longer bowstring, get a lower note. Pinch the middle of a vibrating string. The strongest remaining vibration in the string sounds like the note from a string that’s half as long. Pythagoras worked out length relationships for the whole musical scale.”

“You said ‘spacial frequency’ like there’s some other kind.”

“There is, though they’re closely related. Your ear doesn’t sense the space frequency, the distance between peaks. You sense the time between peaks, the time frequency, which is the space frequency, peaks per meter, times how fast the wave travels, meters per second. See how the units work out?”

“Cute. Does that space frequency/time frequency pair-up work for all kinds of waves?”

“Mostly. It doesn’t work for standing waves. Their energy’s trapped between reflectors or some other way and they just march in place. Their time frequency is zero peaks per second whatever their peaks per meter space frequency may be. Interesting effects can happen if the wave velocity changes, say if the wave path crosses from air to water or if there’s drastic temperature changes along the path.”

“Hah! Mirages! Wait, that’s light getting deflected after bouncing off a hot surface into cool air. Does sound do mirages, too?”

“Sure. Our hearing’s not sharp enough to notice sonic deflection by thermal layering in air, but it’s a well-known issue for sonar specialists. Echoes from oceanic cold/warm interfaces play hob with sonar echolocation. I’ll bet dolphins play games with it when the cold layer’s close enough to the surface.”

“Those guys will find fun in anything. <pause> So Pythagoras figured sound frequencies playing with a bow. Who did it for light?”

“Who else? Newton, though he didn’t realize it. In his day people thought that light was colorless, that color was a property of objects. Newton used the rainbow images from prisms to show that color belonged to light. But he was a particle guy. He maintained that every color was a different kind of particle. His ideas held sway for over 150 years until Fresnel convinced the science community that lightwaves are a thing and their frequencies determine their color. Among other things Fresnel came up with the math that explained some phenomena that Newton had just handwaved past.”

“Fresnel was more colorful than Newton?”

“Uh-uh. Compared to Newton, Fresnel was pastel.”

~~ Rich Olcott

# The Solar System is in gear

Pythagoras was onto far more than he knew.  He discovered that a stretched string made a musical tone, but only when it was plucked at certain points.  The special points are those where the string lengths above and below the point are in the ratio of small whole numbers — 1:1, 1:2, 2:3, ….  Away from those points you just get a brief buzz.  All of Western musical theory grew out of that discovery.

The underlying physics is straightforward.  The string produces a stable tone only if its motion has nodes at both ends, which means the vibration has to have a whole number of nodes, which means you have to pluck halfway between two of the nodes you want.  If you pluck it someplace like 39¼:264.77 then you excite a whole lot of frequencies that fight each other and die out quickly.

That notion underlies auditorium acoustics and aircraft design and quantum mechanics.  In a way, it also determines where objects reside in the Solar System.

If you’ve got a Sun with only one planet, that planet can pick any orbit it wants — circular or grossly elliptical, close approach or far, constrained only by the planet’s kinetic energy.

If you toss in a second planet it probably won’t last long — the two will smash together or one will fall into the Sun or leave the system.  There are half-a-dozen Lagrange points, special configurations like “all in a straight line” where things are stable.  Other than those, a three-body system lives in chaos — not even a really good computer program can predict where things will be after a few orbits.

Add a few more planets in a random configuration and stability goes out the window — but then something interesting happens.  It’s the Chladni effect all over again.  Planets and dust and everything go rampaging around the system.  After a while (OK, a billion years or so) sweet-spot orbits start to appear, special niches where a planet can collect small stuff but where nothing big comes close enough to break it apart.  It’s not like each planet seeks shelter, but if it finds one it survives.

It’s a matter of simple arithmetic and synchrony.  Suppose you’re in a 600-day orbit.  Neighbor Fred looking for a good spot to occupy could choose your same 600-day orbit but on the other side of the Sun from you.  But that’s a hard synchrony to maintain — be off by a few percent and in just a few years, SMASH!

The next safest place would be in a different orbit but still somehow in synchrony with yours.  Inside your orbit Fred has to go faster and therefore has a shorter orbital period than yours.  Suppose Fred’s year is exactly 300 days (a 2:1 period ratio, like a 2:1 gear ratio).  Every six months he’s sort-of close to you but the rest of the time he’s far away.

Our Solar System does seem to have developed using gear-year logic.  Adjacent orbital years are very close to being in whole-number ratios.  Mercury, for instance, circles the Sun in about 88 days.  That’s just 2% away from 2/5 of Venus’s 225¾ days.

This table shows year-lengths for the Sun’s most prominent hangers-on, along with ratios for adjacent objects.  For the “ideal” ratios I arbitrarily picked nearby whole-number multiples of 2.  I calculated how long each object’s year “should” be compared to its lower neighbor — the average inaccuracy across all ten objects is only 0.18%.

##### 100%

The usual rings-around-the-Sun diagram doesn’t show the specialness of the orbits we’ve got.  This chart shows the four innermost planets in their “ideal” orbits, properly scaled and with approximately the right phases.  I used artistic license to emphasize the gear-like action by reversing Earth’s and Mercury’s direction.   Earth and Mars are never near each other, nor are Earth and Venus.

It doesn’t show up in this video’s time resolution, but Venus and Mercury demonstrate another way the gears can work.  Mercury nears Venus twice in each full 5-year cycle, once leading and once trailing.  The leading pass slows Mercury down (raising it towards Venus), but the trailing pass speeds it up again.  Net result — safe!

~~ Rich Olcott

# You can’t get there from here

In this series of posts I’ve tried to get across several ideas:

• By Einstein’s theories, in our Universe every possible combination of place and time is an event that can be identified with an “address” like (ct,x,y,z) where t is time, c is the speed of light, and x, y and z are spatial coordinates
• The Pythagorean distance between two events is d=√[(x1-x2)2+(y1-y2)2+(z1-z2)2]
• The Minkowski interval between two events is √[(ct1-ct2)2 d2]
• When i=√(-1) shows up somewhere, whatever it’s with is in some way perpendicular to the stuff that doesn’t involve i

That minus sign in the third bullet has some interesting implications.  If the time term is bigger then the spatial term, then the interval (that square root) is a real number.  On the other hand, if the time term is smaller, the interval is an imaginary number and therefore is in some sense perpendicular to the real intervals.

We’ll see what that means in a bit.  But first, suppose the two terms are exactly equal, which would make the interval zero.  Can that happen?

Sure, if you’re a light wave.  The interval can only be zero if d=(ct1-ct2).  In other words, if the distance between two events is exactly the distance light would travel in the elapsed time between the same two events.

In this Minkowski diagram, we’ve got two number lines.  The real numbers (time) run vertically (sorry, I know we had the imaginary line running upward in a previous post, but this is the way that Minkowski drew it).  Perpendicular to that, the line of imaginary numbers (distance) runs horizontally, which is why that starscape runs off to the right.
The smiley face is us at the origin (0,0,0,0).  If we look upwards toward positive time, that’s the future at our present locationLooking downward to negative time is looking into the past.  Unless we move (change x and/or y and/or z), all the events in our past and future have addresses like (ct,0,0,0).

What about all those points (events) that aren’t on the time axis?  Pick one and use its address to figure the interval between it and us.  In general, the interval will be a complex number, part real and part imaginary, because the event you picked isn’t on either axis.

The two orange lines are special.  Each of them is drawn through all the events for which the distance is equal to ct.  All the intervals between those events and us are zero.  Those are the events that could be connected to us by a light beam.

The orange connections go one way only — someone in our past (t less than zero) can shine a light at us that we’ll see when it reaches us.  However, if someone in our future tried to shine a light our way, well, the passage of time wouldn’t allow it (except maybe in the movies).  Conversely, we can shine a light at some spatial point (x,y,z) at distance d=√(x2+y2+z2) from us, but those photons won’t arrive until the event in our future at (d,x,y,z).

The rest of the Minkowski diagram could do for a Venn diagram.  We at (0,0,0,0) can do something that will cause something to happen at (ct,x,y,z) to the left of the top orange line.  However, we won’t be able to see that effect until we time-travel forward to its t.  That region is “reachable but not seeable.”

Similarly, events to the left of the bottom orange line can affect us (we can see stars, for instance) but they’re in our past and we can’t cause anything to happen then/there.  The region is “seeable but not reachable.”

Then there’s the overlap, the segment between the two orange lines.  Events there are so far away in spacetime that the intervals between them and us are imaginary (in the mathematical sense).  To put it another way, light can’t get here from there.  Neither can cause and effect.

Physicists call that third region space-like, as opposed to the two time-like regions.  Without a warp drive or some other way around Einstein’s universal speed limit, the edge of “space-like” will always be The Final Frontier.

~~ 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?

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.

One 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

# Is cyber warfare imaginary?

Rule One in hooking the reader with a query headline is: Don’t answer the question immediately.  Let’s break that one.  Yes, cyber warfare is imaginary, but only for a certain kind of “imaginary.”  What kind is that, you ask.  AaaHAH!

It all has to do with number lines.  If the early Greek theoreticians had been in charge, the only numbers in the Universe would have been the integers: 1, 2, 3,….  Life is simple when your only calculating tool is an abacus without a decimal point.  Zero hadn’t been invented in their day, nor had negative numbers.

Then Pythagoras did his experiments with harmony and harp strings, and the Greeks had to admit that ratios of integers are rational.

More trouble from Pythagoras: his a2+b2=c2 equation naturally led to c=√(a2+b2).  Unfortunately, for most integer values of a and b, c can’t be expressed as either an integer or a ratio of integers.  The Greeks labeled such numbers (including π) as irrational and tried to ignore them.

Move ahead to the Middle Ages, after Europe had imported zero and the decimal point from Brahmagupta’s work in India, and after the post-Medieval rise of trade spawned bookkeepers who had to cope with debt.  At that point we had a continuous number line running from “minus a whole lot” to “plus you couldn’t believe” (infinity wasn’t seriously considered in Western math until the 17th century).

By then European mathematicians had started playing around with algebraic equations and had stumbled into a problem.  They had Brahmagupta’s quadratic formula (you know, that [-b±√(b2-4a·c)]/2a thing we all sang-memorized in high school).  What do you do when b2 is less than 4a·c and you’re looking at the square root of a negative number?

Back in high school they told us, “Well, that means there’s no solution,” but that wasn’t good enough for Renaissance Italy.  Rafael Bombelli realized there’s simply no room for weird quadratic solutions on the conventional number line.  He made room by building a new number line perpendicular to it.  The new line is just like the old one, except everything on it is multiplied by i=√(-1).

(Bombelli used words rather than symbols, calling his creation “plus of minus.”  Eighty years later, René Descartes derisively called Bombelli’s numbers “imaginary,” as opposed to “real” numbers, and pasted them with that letter i.  Those labels have stuck for 380 years.  Except for electricity theoreticians who use j instead because i is for current.)

Suppose you had a graph with one axis for counting animal things and another for counting vegetable things.  Animals added to animals makes more animals; vegetables added to vegetables makes more vegetables.  If you’ve got a chicken, two potatoes and an onion, and you share with your buddy who has a couple of carrots, some green beans and another onion, you’re on your way to a nice chicken stew.

Needs salt, but that’s on yet another axis.

Bombelli’s rules for doing arithmetic on two perpendicular number lines work pretty much the same.  Real numbers added to reals make reals, imaginaries added to imaginaries make more imaginaries.  If you’ve got numbers like x+i·y that are part real and part imaginary, the separate parts each follow their own rule.  Multiplication and division work, too, but I’ll let you figure those out.

The important point is that what happens on each number line can be specified independently of what happens on the other, just like the x and y axes in Descartes’ charts.  Together, Bombelli’s and Descartes’ concepts constitute a nutritious dish for physicists and mathematicians.

Scientists love to plot different experimental results against each other to see if there’s an interesting relationship in play.  For certain problems, for example, it’s useful to plot real-number energy of motion (kinetic energy) against some other variable on the i-axis.

Two-time Defense Secretary Donald Rumsfeld used to speak of “kinetic warfare,” where people get killed, as opposed to the “non-kinetic” kind.  Apparently, he would have visualized cyber somewhere up near the i-axis.  In that scheme, cyber warriors with their ones and zeros are Bombelli-imaginary even if they’re real.

~~ Rich Olcott