Straight

In class tomorrow I’ll introduce one of my favourite equations:

$$\frac{d^2x^\sigma}{d\tau^2}+\Gamma^\sigma_{\rho\nu}\frac{dx^\rho}{d\tau}\frac{dx^\nu}{d\tau}= 0\ .$$

… Wait – Where did everyone go?!

Come back! I’m not expecting you to know what it means, I just wanted to talk a bit with it sort of … nearby. If you consider yourself a bit intimidated by mathematics, be assured that it won’t bite. (No more than a piece of sheet music lying nearby will harm someone who has not learned to read music.)

It turns out that it is pretty geometry! In the equation, we’ve the object
$$\Gamma^{\sigma}_{\rho\nu}\equiv\frac12 g^{\sigma\mu}\Biggl(\frac{\partial g_{\mu\nu}}{\partial x^\rho}+\frac{\partial g_{\mu\rho}}{\partial x^\nu}-\frac{\partial g_{\rho\nu}}{\partial x^\mu}\Biggr)\ ,$$
called the “Christoffel symbols”. The set of objects $$g_{\mu\nu}$$ (the “metric”) actually encode the properties of the space you wish to study (like the plane, or the sphere), and the equation at the top tells you what are the “straight lines” in that space. Well, in the plane (like your desktop) they are straight lines, while in other spaces they are the analogue of straight lines – if you want to go from one point to another point somewhere else in the space and desire to travel along the shortest path to do so, you want to follow such a line. It is called a “geodesic”. The equation is commonly called the geodesic equation.

You know such lines, intuitively, in a non-trivial example. Next time you look at a globe (wait, does anyone but me look at maps and globes any more? I love them!), you’ll probably see examples of those lines drawn in. They are the “great circles”, the lines of longitude, and the equator. (Image used with permission.)

I just made a class worksheet that guides one through a bit of playing with this equation to get the class excited about how to use it by exploring an example together. Their first task will be to work on the two dimensional plane (in various coordinate systems), and then for homework they will actually prove that the lines of longitude and the equator are indeed geodesics for the two dimensional sphere.

What has all this got to do with the price of eggs? Well, this is a General Relativity class. We’ve just done (Monday’s class) Einstein’s realisation (his “happiest thought”) – The Equivalence Principle: Gravity is, locally, the same as acceleration.

We’re about to realize that moving under the influence of gravity is something deliciously simple: Following a geodesic in the geometry of spacetime. The equation at the top describes that.

I think that there are few things more beautiful than this in all of science.

-cvj

7 Responses to Straight

1. Rick says:

Clifford,
Are you using Hartle’s book for this class? Just curious.

2. Clifford says:

That is the book they are recommended to use. I love the approach Jim uses. I was one of the early testers of the book back when it was a set of notes that he circulated. So I built up my own notes and so forth from way back then, (I started an undergrad GR course when I was at Kentucky, and again on coming here to USC) rebuilding and refining each time I teach.

Cheers,

-cvj

3. PhilipAShane says:

Awesome. So much more interesting than the price of eggs!

4. Kevin Tah N says:

How are these structures modified when thinking of general relativity as a generalization of Yang-Mills. Is there really more power in taking such an approach?

5. Clifford says:

Well, that object, the Christoffel symbols, becomes (after some dressing) the analogue of the gauge connection in Yang-Mills. You write the whole thing as a having a gauge group where the Lorentz group is the gauge group. There are lots of good books and papers on this that go through the analogy. It is a fun game to play. I’ve not seen anything compelling that says it is more powerful an approach. Certainly at the classical level I think I makes the theory less beautiful. People pursue it as an attempt to find a good framework to begin to quantise the theory. The jury is still out on what exactly you get for that.

Cheers,

-cvj

6. Anonymous Snowboarder says:

C – A couple years ago I snagged an older globe from my parents home. After a tiny bit of fixing, it was good as old… back in the USSR! Brings back fond memories of cold war Olympics and rooting against those East German lugers!

7. Clifford says:

ðŸ™‚

[…] Yeah I’m back in the U.S.S.R.
You don’t know how lucky you are boys
Back in the U.S.S.R.

Well the Ukraine girls really knock me out
They leave the West behind
And Moscow girls make me sing and shout
That Georgia’s always on my mind […]

(From “Back in the USSR”, by the Beatles…)

-cvj