Claire Gray: Nerdy Epiphany

So just for fun I've been going to the 4th year's General Relativity lectures, held on Monday and Friday mornings. Why? Cause it's interesting, and it's amazing how much you forget between one year and the next. Also, the syllabus is slightly different this year; last year we looked a lot at geodesics and descriptions of different types of curved spacetimes, but we never actually got to deriving the field equations. This year they are doing that, another reason for me to want to attend the lectures.

Anyway, up till now we've been covering pretty much the same stuff as we did last year, only in different orders and different ways. Today, though, we got to something new that made me really excited in a completely nerdy way.

But first, a little bit of GR and personal history... In differential geometry there's this thing called the Riemann Curvature Tensor
Curvature_tensor_1 (if it's hard to see, those indices are the greek letters mu, nu, rho and sigma)

which basically describes the curvature of a manifold, and is used in GR to describe the curvature of space.

Now, in third year Computational and Applied Maths, we had a course on tensor analysis. We didn't really understand what was going on, to tell the truth, but we just learnt all the rules for index notation and worked from there. One hot stuffy afternoon we were semiconsciously making our way through a lecture, when our lecturer threw in what was probably supposed to inspire us that what we were learning was so cool... He went (something along the lines of):

"And what Einstein did was take the Riemann tensor R mu nu rho sigma and changed the third index to mu, so it was R mu nu mu sigma, and hence got his field equations!"

Needless to say, there was no resounding "wow!". In fact, it was more like a resounding "Huh???"

Okay, back to this morning's lecture. R.... as it's shown above, for a 4-dimensional spacetime, has 256 components. This is rather nasty, so we'd want to reduce the number of components to make things easier for us. The way we do this is by "contracting" the tensor - summing over identical indices a la Einstein.

So after quite a bit of maths, obtaining and using R....'s (anti)symmetry relations and it's other properties, it turns out that the only non-trivial way to contract the tensor is exactly the way we "huh?"-ed about two years ago. Only now, after going through the math, I (think I) understand why. I was so excited, it made my day. Can you say NERD?!

Oh yes, the new tensor you get from contracting R.... in just the way we did above is called the Ricci tensor. And after a bit more mathematical fun on this one, you can define the aptly-named Einstein tensor, which appears in ... ta-da! ... the Einstein Field Equation.

But we're not there yet.