Wednesday, April 15, 2009

Wednesday Math, Vol. 67: The Elegant Universe

Last July, I wrote a Wednesday math post that gave a summary of the position of physicists who are opposed to the ideas and practice of string theory. Today, I will give the string theorists time to present their case, which is presented brilliantly by physicist Brian Greene in his 1999 book The Elegant Universe.

Yesterday, I wrote a post about what crazy people believe. I even got an actual crazy person to post a comment! Well, people like me, whose craziness can be debated, we believe something, too.

Deep down, I believe math works.

You may not think the word "believe" is needed. It takes no faith to see that 12-5 = 7, which is to say if I bought a dozen eggs and I've used five, there should be seven left.

But when we deal with the real world and physics, the math gets trickier and the things being described more complex. Forces like gravity and electromagnetism aren't just added and subtracted. Geometry starts getting involved, and Einstein proved that the geometry is taking place in a universe that isn't nice and flat, but that bends and warps. The math gets a lot fancier, but it still works.

Here's where the belief comes in. Math is pristine and clean and the world is messy, but math still works. That's surprising. Math can deal easily with the concepts of infinitely big and infinitesimally small, but the best evidence in physics is those things never happen. Here is where belief in math becomes an article of faith.

Ideas in physics change, and when dealing on a human scale, they change rather quickly. Einstein's equations could work in a universe that was expanding or shrinking or standing still. He believed it was standing still, so he chose the "cosmological constant" that would make his equations show a static universe. It was really a fudge factor and that isn't elegant. Even in his lifetime, the evidence for an expanding universe was becoming clearer. By the 1960's, when radio telescopes detected low level background radiation in every direction you could point a telescope, physicists agreed that the "sound" they heard was the echo of the big bang, and that the expanding universe wasn't up for debate anymore. It may have gone against an intuition of Einstein's, but the reality of an expanding universe let us get rid of an arbitrary cosmological constant and general relativity was its elegant self once more.

There is no elegance in quantum mechanics. It just works, and insanely well. John Von Neumann, the great Hungarian genius who is credited with first completing the equations that are the underpinnings of the quantum theory, has a quote. "Young man, in mathematics you don't understand things. You just get used to them." Once the rules of sub-atomic particles were turned into math, one obvious rule of the macroscopic world, the world big enough for us to see, had to be thrown away. While two things couldn't be in the same place at the same time, some things like photons could be in two places at once, and simple experiments proved it.

And then there was physics' dirty secret. The elegant relativity and inelegant quantum mechanics couldn't both be true at the same time. One of the theories, both of them brilliant at explaining the world at vastly different scales, was incorrect.

String theory hopes to reconcile the irreconcilable by saying the universe isn't what we think it is. Instead of three spatial dimensions and a fourth dimension of time to explain the position of a thing in spacetime, there are six more spatial dimensions we can't see. Or maybe seven. No string theory tells us there is another time dimension, which I think could be really cool.

And herein lies one of the big problems of string theory, and Brian Greene to his credit doesn't hide it. There are currently five different ten dimensional string theories and one eleven dimension string theory. Some of them are "duals" of one another, which means we could be looking at the same thing in a different state, the way water is the same as ice is the same as steam, just at different temperatures. There is also a hope to unite all the string theories into a unified theory called M-theory. Still, there is no denying that string theory has opened a Pandora's box. Relativity is loved for its simplicity, the fact that if we accept the curvature of space, some confusing things like gravity have to work the way they do because it's just geometry. String theory has an overabundance of possible ways the universe can be, and physicists have been spoiled into thinking that it really should work exactly one way with no "special rules".

Since I am only writing a blog and The Elegant Universe is a well-written book of about 400 pages, clearly I am simplifying the case quite a bit. Lemme 'splain, as well as I can, the idea of hidden dimensions.

As I said before, even in quantum mechanics two things can't be in the same place at the same time. In math, a line is considered to be one dimensional, with only length, but no width or depth. It is supposed to be infinitesimally thin in two different directions. If we think of this ant on a garden hose as a point on a line, no other ant can be exactly where this one is. But the surface of a garden hose has two dimensions, and there can be another ant at the exact same place in the left-right dimension on the side of the garden hose we can't see. While nothing in the universe is truly infinite, we can make the garden hose as long as we like, at least as a thought experiment, so the length is macroscopic. In string theory, the hidden dimension is the circumference of the garden hose, shrunk down very tiny, down to less than the Planck length, a smallness far beyond human scale at 10 to the power of -33 centimeters. Still, it counts as a dimension, and in some equations even this tiny thing cannot be ignored.

One of the things that makes string theory "elegant" is that these tiny dimensions mean that the big bang didn't start from something infinitesimal, but instead from a thing that has an incredibly small but measurable size. Here again is where mathematics becomes a faith. Ever since the Greeks, we have accepted the ideas of points and lines and planes, things with zero or one or two dimensions embedded in the three macroscopic dimensions we see around us. We need these concepts to do the math, they do not conform to the physical world in any way, but still math works.

Ben Franklin said beer is proof that God wants us to be happy. I believe math is proof that God wants us to figure out this world we live in, and for some of us, that makes us happy, too.

(UPDATE: Changes made to reflect the corrections of my good friend Ken Rose. Also, this is my 1,000th post. Yay!)


dguzman said...

Sometimes I want to be a photon so I can be in two places at once.

The rest just blew my mind. But I sure do love your last paragraph.

ken said...

The cosmological constant isn't really a fudge. Mathematically it's a constant of integration, which is one of those things folks like to ignore, but it's not really right to do so.
And btw, the expansion of the universe was pretty well demonstrated well before the discovery of the cosmic background radiation. That killed steady state and supported the big bang, but expansion was known previously.
Verification word is "proles".

Matty Boy said...

Thanks, Ken. I made a few re-writes, and also added that this was my 1,000th post.