That’s it. The class is over… I have to admit that I’m pretty sad to see the end of it, although I’m very very tired. It was such a great group. (I’ll be toasting the end of it all with some of the splendid stuff to the right.)
Recall that we stepped away from black holes. After a look at cosmology for some lectures, where we understood the role of four crucial components in determining a universe’s properties (curvature, matter, radiation, and vacuum energy) we dove back into formalism for a short while (one lecture) to develop a little more the tools we needed to properly under stand how to formulate Einstein’s field equations.
It did not take long… You need only the idea that it makes sense to formulate everything in terms of objects that allow you to express the full sense of an equation in any coordinate system you care to write. Once that is done (the objects are called tensors, and the idea and how they work is pretty simple to get to grips with) the key to formulating the field equations of gravity is to have a look at the structure of other familiar systems. The field equations of electromagnetism (Maxwell’s equations) and the field equations for Newton’s formulation of gravity give the required clues. A rummage around the geometry to find the appropriate object to express the physics in terms of uncovers the Riemann tensor and its cousins (“contractions” to get Ricci and so forth), and you’re almost there. A step back to learn how to package energy and stresses and pressures into one utterly cool object called the stress-energy tensor gives the final pieces of the puzzle, and the full Einstein equations (right) are done! (It’s ok. It is just for decoration, I’m not expecting you to follow the details and everything I said, nor understand the equation. If you have not studied this before, a single blog post is not the place to do it. I’m just giving impressions…)
They also had a session of two hours where the class actually played (using a guided worksheet) with the equations and unpacked everything for various two dimensional examples allowed them to get a feel for how the equations actually work, and how the various solutions (Schwarzschild and Kerr for stars and black holes, Friedman-Robertson-Walker for cosmology…) end up being solutions to the field equations. They seemed to enjoy that.
Of course, I would be neglectful if I did not spend a moment or two telling them that in the “old days” we actually worked out everything by hand for full four dimensional solutions, and not just th two dimensional cases I gave them. I recalled with some fondness the large green and white Z-fold computer paper tat was just so great to calculating on the back of because it was so large and you could spread out and write small and see your whole computation laid out in front of you… Sigh. They responded with good humour by mumbling things like “yeah, and you didn’t have shoes and you had to walk several miles uphill and…” at which I laughed and joined in with “yeah and carry water and so forth up the hill, all the while calculating the components of the Riemann tensor for the Schwarzschild geometry…”
Yeah, we laughed. It was a lot of fun. (I thought it best not to mention to them that I actually did walk uphill carrying water for household use at some points in my childhood. Wasn’t calculating things for General Relativity at the time though, I’ll admit. That would come well over a decade later.)
Well, it is time to turn my attention to pressing matters, such as writing a big report on my committee’s activities and findings over the last nine months. It is due soon, so the pressure’s not off yet. Oh, and I’ve got to write the final for this class. Oh no… a bunch of other things I’ve been neglecting are coming into focus in my mind now that the teaching component of the semester is over… Sigh.
Well, anyway, it is the end of a long day, and moving into the night. I’ve just come back to my area of town from a long meeting in “beautiful downtown Burbank” and am sitting in a Pinkberry for a while before buying a few groceries and walking home. I’m going to savour these simple tasks for a while before re-engaging…
Some Related Asymptotia Posts (not exhaustive):