Hyperbolas But Not Hyperbole

Minus? Where did that come from?”

<Gentle reader — If that question looks unfamiliar, please read the preceding post before this one.>

Jim’s still at the Open Mic. “A clever application of hyperbolic geometry.” Now several of Jeremy’s groupies are looking upset. “OK, I’ll step back a bit. Jeremy, suppose your telescope captures a side view of a 1000‑meter spaceship but it’s moving at 99% of lightspeed relative to you. The Lorentz factor for that velocity is 7.09. What will its length look like to you?”

“Lorentz contracts lengths so the ship’s kilometer appears to be shorter by that 7.09 factor so from here it’d look about … 140 meters long.”

“Nice, How about the clocks on that spaceship?”

“I’d see their seconds appear to lengthen by that same 7.09 factor.”

“So if I multiplied the space contraction by the time dilation to get a spacetime hypervolume—”

“You’d get what you would have gotten with the spaceship standing still. The contraction and dilation factors cancel out.”

“How about if the spaceship went even faster, say 99.999% of lightspeed?”

“The Lorentz factor gets bigger but the arithmetic for contraction and dilation still cancels. The hypervolume you defined is always gonna be just the product of the ship’s rest length and rest clock rate.”

His groupies go “Oooo.”

One of the groupies pipes up. “Wait, the product of x and y is a constant — that’s a hyperbola!”

“Bingo. Do you remember any other equations associated with hyperbolas?”

“Umm… Yes, x2–y2 equals a constant. That’s the same shape as the other one, of course, just rotated down so it cuts the x-axis vertically.”

Jeremy goes “Oooo.”

Jim draws hyperbolas and a circle on the whiteboard. That sets thoughts popping out all through the crowd. Maybe‑an‑Art‑major blurts into the general rumble. “Oh, ‘plus‘ locks x and y inside the constant so you get a circle boundary, but ‘minus‘ lets x get as big as it wants so long as y lags behind!”

Another conversation – “Wait, can xy=constant and x2–y2=constant both be right?”
  ”Sure, they’re different constants. Both equations are true where the red and blue lines cross.”

A physics student gets quizzical. “Jim, was this Minkowski’s idea, or Einstein’s?”

“That’s a darned good question, Paul. Minkowski was sole author of the paper that introduced spacetime and defined the interval, but he published it a year after Einstein’s 1905 Special Relativity paper highlighted the Lorentz transformations. I haven’t researched the history, but my money would be on Einstein intuitively connecting constant hypervolumes to hyperbolic geometry. He’d probably check his ideas with his mentor Minkowski, who was on the same trail but graciously framed his detailed write‑up to be in support of Einstein’s work.”

One of the astronomy students sniffs. “Wait, different observers see the same s2=(ct)2d2 interval between two events? I suppose there’s algebra to prove that.”

“There is.”

“That’s all very nice in a geometric sort of way, but what does s2 mean and why should we care whether or not it’s constant?”

“Fair questions, Vera. Mmm … you probably care that intervals set limits on what astronomers see. Here’s a Minkowski map of the Universe. We’re in the center because naturally. Time runs upwards, space runs outwards and if you can imagine that as a hypersphere, go for it. Light can’t get to us from the gray areas. The red lines, they’re really a hypercone, mark where s2=0.”

From the back of the room — “A zero interval?”

“Sure. A zero interval means that the distance between two events exactly equals lightspeed times light’s travel time between those events. Which means if you’re surfing a lightwave between two events, you’re on an interval with zero measure. Let’s label Vera’s telescope session tonight as event A and her target event is B. If the A–B interval’s ct difference is greater then its d difference then she can see Bif the event is in our past but not beyond the Cosmic Microwave Background. But if a Dominion fleet battle is approaching us through subspace from that black dot, we’ll have no possible warning before they’re on us.”

Everyone goes “Oooo.”

~~ Rich Olcott

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

Superman flying at lightspeed? Umm… no

Back when I was in high school I did a term paper for some class (can’t remember which) ripping the heck out of Superman physics.  Yeah, I was that kind of kid.  If I recall correctly, I spent much of it slamming his supposed vision capabilities — they were fairly ludicrous even to a HS student and that was many refreshes of the DC universe ago.FTL SupermanBut for this post let’s consider a trope that’s been taken off the shelf again and again since those days, even in the movies.  This rendition should get the idea across — Our Hero, in a desperate effort to fix a narrative hole the writers had dug themselves into, is forced to fly around the Earth at faster-than-light speeds, thereby reversing time so he can patch things up.

So many problems…  Just for starters, the Earth is 8000 miles wide, Supey’s what, 6’6″?, so on this scale he shouldn’t fill even a thousandth of a pixel.  OK, artistic license.  Fine.

Second problem, only one image of the guy.  If he’s really passing us headed into the past we should see two images, one coming in feet-first from the future and the other headed forward in both space and time.  Oh, and because of the Doppler effect the feet-first image should be blue-shifted and the other one red-shifted.

Of course both of those images would be the wrong shape.  The FitzGerald-Lorentz Contraction makes moving objects appear shorter in the direction of motion.  In other words, if the Man of Steel were flying just shy of the speed of light then 6’6″ tall would look to us more like a disk with a short cape.

Tall-Dark-And-Muscular has other problems to solve on his way to the past.  How does he get up there in the first place?  Back in the day, DC explained that he “leaped tall buildings in a single bound.”   That pretty much says ballistic high-jump, where all the energy comes from the initial impulse.  OK, but consider the rebound effects on the neighborhood if he were to jump with as much energy as it would take to orbit a 250-lb man.  People would complain.

Remember Einstein’s famous E=mc²?  That mass m isn’t quite what most people think it is.  Rather, it’s an object’s rest mass m0 modified by a Lorentz correction to account for the object’s kinetic energy.   In our hum-drum daily life that correction factor is basically 1.00000…   When you get into the lightspeed ballpark it gets bigger.

Here’s the formula with the Lorentz correction in red: m=m0/√[1-(v/c)2].  The square root nears zero as Superman’s velocity v approaches lightspeed c.  When the divisor gets very small the corrected mass gets very large.  If he got to the Lightspeed Barrier (where v=c) he’d be infinitely massive.

So you’ve got an infinite mass circling the Earth about 7 times a second — ocean tides probably couldn’t keep up, but the planet would be shaking enough to fracture the rock layers that keep volcanoes quiet.  People would complain.

Of course, if he had that much mass, Earth and the entire Solar System would be orbiting him.

On the E side of Einstein’s equation, Superman must attain that massive mass by getting energy from somewhere.  Gaining that last mile/second on the way to infinity is gonna take a lot of energy.

But it’s worse.  Even at less than lightspeed, the Kryptonian isn’t flying in a straight line.  He’s circling the Earth in an orbit.  The usual visuals show him about as far out as an Earth-orbiting spacecraft.  A GPS satellite’s stable 24-hour orbit has a 26,000 mile radius so it’s going about 1.9 miles/sec.  Superman ‘s traveling about 98,000 times faster than that.  Physics demands that he use a powered orbit, continuously expending serious energy on centripetal acceleration just to avoid flying off to Vega again.

The comic books have never been real clear on the energy source for Superman’s feats.  Does he suck it from the Sun?  I sure hope not — that’d destabilize the Sun and generate massive solar flares and all sorts of trouble.

Not even the DC writers would want Superman to wipe out his adopted planet just to fix up a plot point.

~~ 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