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

Dimensional venturing, Part 2 – Twirling in 4-space

Last week we introduced the tesseract, which is to a cube what a cube is to a square — an extension into one more dimension.  That’s why it’s also called a hypercube.  The first tesseract diagrams I ever saw were so confusing — they looked like lots of overlapped squares tied together with lines that didn’t make much sense.  I wondered, “Wouldn’t it be easier to understand a tesseract if I could see it rotating?”

Years later computers and I had both moved ahead to where I could generate the pictures you see in this post.  What I learned while doing that was that 4-D figures have two equators.  In four dimensions, it’s possible for something to rotate in two perpendicular directions at the same time.  Read on and please don’t mind my doggerel — it doesn’t bite.

line2c The LINE is just a single stroke,
a path from here to there.
Stretch it out beside itself
and you will have a SQUARE.
Where’s its face when it turns around?
Gone, ’cause its back’s not there.
square2c
cube2c The CUBE’s a square
made thick, you see.
Length, breadth and depth
comprise a full 3-D.
Add yet a thickness more,
crosswise all to X, Y, Z.
A TESSERACT on a corner spins
but an XY-slice is all we see.
tess2cxy
tess2czw But the axis, too, can rotate through
a path that’s drawn invisibly.
Four faces grow and shrink in place —
it’s hard to do that physically.
This tesseract is tumbling ’bout
two equators perpendicular.
Were I in such a state, I vow,
I’d be giddy, even sickular.
tess4cxyzw

In the 4-D views, when one of the tesseract’s cubical faces appears to disappear into an adjacent face, what’s actually happening is that the face is sliding past the other face along that fourth dimension (which I called W because why not?)

You’re looking at a two-dimensional picture of the three-dimensional projection of a four-dimensional object as it moves in 5-space (X, Y, Z, W, and time — if it didn’t move in time then it couldn’t be spinning).

Next week — Herr Klein’s bottle, or rather flask, or rather surface.

~~ Rich Olcott

Dimensional venturing, Part 1 – What’s 4-D?

Whenever a science reporter uses the phrase “string theory,” it’s invariably accompanied by a sentence about tiny strings vibrating in 10 or 11 dimensions. Huh? How can you have more than three? And what does it really mean to say that that comix villain comes from the 4th dimension?  Actually, we live in many dimensions, though it’s not easy to visualize them all at once. Let’s get some practice.

Right now, you’re reading along a line, a one-dimensional path from left to right. Imagine a point drawing a straight line about a foot in front of you. Let that line just hang out there in the air, glowing a gentle green color, with one “edge” (the line itself) and two “corners” (its ends).

As you read down the page, you traverse a series of lines laid out next to each other in the two-dimensional plane of the page. Imagine your green line moving upward, leaving a plane of yellow sparkles behind it. Stop when you’ve got a sparkly yellow square in front of you showing its one face, four edges (one green, three yellow) and four corners (two green, two yellow). Let’s put some red paint on one of those yellow edges.Cube

Stack up enough printed pages and you’re got a 3-dimensional book. Imagine that nice yellow square moving away from you until you’ve got a friendly cube hanging out in the air. Our original line, the green edge, has produced a green face going into the distance. The red edge has built a pink face. All together, the cube has 8 corners, 12 edges and 6 faces. OK, now make your cube disappear.

But we’re not done yet. Time is a dimension. Consider that cube. Before you dreamed it up – nothing. Then suddenly a cube. Then nothing again. During the interval the cube was floating in front of you, the green line was tracing out a green face in time. The pink face was drawing a pink cube. The whole cube, from when it started to exist until it went away, traced out a four-dimensional figure called a tesseract, also called a 4-cube or hypercube. The tesseract was bounded by a cube at the beginning, six cubes while it existed (one from each face of the initial cube), and a cube at the end of its time, for a total of eight.

Just for grins, count up the faces, edges and corners for yourself.

But wait, there’s more. The tesseract doesn’t just sit there, it can spin. Being four-dimensional, it can spin in a surprising way. We’ll get to that next week.

~~ Rich Olcott