05 January 2007

A complex story, part 4

(See parts 1, 2, and 3.)

Everybody literally sees the world from a different point of view. Each person is standing in a different location and looking out in a different direction from everyone else. But all viewpoints share certain similarities. If you and I are near one another, we'll see the same events happen in the same order, and although we may differ in our use of the words “right” and “left”, if we're watching something from opposite sides, we'll at least agree on the distances between things. If I see two people holding hands, you'll never see them on separate sides of the street at the same time, no matter where you're standing. All the different viewpoints preserve certain observed properties: distances, angles, durations, causality, and so on.

Mathematically, we can write this in two equations. For each of us, every event has a measurable position in space (x, y, z) and time (t). If we put my observations on the left-hand side and yours on the right, they will match.

We agree on distances: x2 + y2 + z2 = x'2 + y'2 + z'2

We agree on durations: t = t'

Even if I'm in a car doing eighty and you're sitting on the sidewalk enjoying an ice cream cone, we'll agree on the distances between and durations of any events we both happen to witness as I zoom by.

...Or so everyone thought. Don't get me wrong, this is a lovely picture. Mathematically, it's your basic three-dimensional Euclidean geometry, plus a separate dimension for time. All our viewpoints are identical except for a bit of spacial displacement and rotation. There's only one problem. This isn't how the universe really behaves.

1887 was the year of the famous Michelson-Morley experiment, which blew this nice, simple Newtonian view all to hell. For twenty years, confusion reigned. By 1905, a mere eyeblink in academic terms, physics had righted itself, now with a totally new model of space and time.

The new theory was called special relativity. It was built on brilliant new insights from Hendrik Lorentz, Henri PoincarĂ©, and Albert Einstein. And it went something like this: Two observers traveling at incredible velocities (relative to one another) actually do not agree on distances, angles, durations, or even the relative time-order of events. But they will agree on something even more fundamental: the basic laws of nature, including laws of motion, causality, and—in particular—the speed of light.

This had the advantage of being, you know, consistent with experiment. But geometrically, it was awfully weird. It wrecked the two equations above. Individual viewpoints were not simple spacial rotations and translations of one another. They were, uh, Lorentz transformations. Yeah. It was two more years before geometry caught up with physics.

In 1907, Hermann Minkowski discovered a kind of geometry (a four-dimensional manifold) that exactly describes the spacetime of special relativity. That is, Minkowski space is the actual geometry of the universe around us, according to relativity. Minkowski's geometry succeeded by treating space and time as interrelated. For example, in Minkowski space:

We may not agree on the spatial distance between two events: x2 + y2 + z2x'2 + y'2 + z'2

We may not agree on the passage of time: tt'

But we will agree on a particular mathematical combination of the two: x2 + y2 + z2 - ct2 = x'2 + y'2 + z'2 - ct'2

(Here c is the speed of light.)

Now comes the controversial, beautiful part. Define a variable w as ict. We're going to use w as our time coordinate, instead of t. Then the last equation above becomes:

x2 + y2 + z2 + w2 = x'2 + y'2 + z'2 + w'2

This looks a lot like our original equation for distance. And in fact this equation describes basic Euclidean geometry in four dimensions. Time becomes just another spacial dimension. All viewpoints are again simple rotations and translations of one another—not in three-dimensional space, but in four-dimensional spacetime.

Here the role of the complex numbers is to provide a new way of looking at the geometry of the universe.

But... what does it all mean? Is time really an imaginary dimension? What does it mean for three dimensions to be real numbers and one to be an imaginary number? These questions are, in a way, the same questions RT asked me months ago, the questions that got me interested in telling this story. What are the imaginary numbers? Do they exist? Do they appear in nature? I don't think anyone really knows. Einstein found the ict trick interesting at least (he mentions it twice in his short book Relativity, which by the way I enthusiastically recommend), but some physicists think it's a red herring. Maybe we're just dressing the universe up to look more comfortable and familiar.


JJD said...

A really deep connection between complex numbers and the distribution of primes is the subject of the Riemann Hypothesis. Maybe a posting on this could be part i of the series....

Davis said...

This is a nice series of pieces. I just have one comment:

Do they exist? Do they appear in nature? I don't think anyone really knows.

After studying math for a very long time, I've come to see these as essentially meaningless questions. They implicitly assume the objects of mathematics refer to things out there in the "real world" (a sort of naive Platonistic view of math).

This seems like an odd view to me -- where in the real world can I point to something and say "there's pi! There's 2^e!"? (2^e should seem pretty weird, if you sit and try to figure out what it's supposed to mean.) Rather, it seems that the objects of math are simply things we define in a rigorous manner. Occasionally find things in the real world that are remarkably well-modeled by these objects, and sometimes things in the real world inspire new definitions, but it feels wrong to say the real-world things are the same as the mathematical objects.

jto said...

OK, but you realize how bizarre it is that such a natural thing should feel wrong to you, right? Dude, you put scare-quotes around the phrase real world. Does that not give you pause?

Of course I don't mean to say that pi exists as a physical object in nature, any more than buildings or bees or blogs. None of these things are terms in the fundamental laws of physics. Well, except pi...

Sorry. What I mean is, we have all these words, and they describe things or phenomena in nature. The question of whether, say, altruism describes anything in the real world is a fair question. The question I was asked is whether the complex numbers describe anything in nature. It seems a perfectly natural question to me.

The major theme of this blog from now on will be the sheer strangeness of mathematics and the mathematician's way of seeing things. You're doing nothing to dispel that. Not that I would have it any other way. :)