The wizard frowns and says "Note, it's not the metric itself which is or is not "orthogonal" - which I guess is your new terminology for "diagonal" - instead, its the COMPONENTS of the metric IN A PARTICULAR COORDINATE SYSTEM which are, or aren't, diagonal. We could take the usual round metric on the sphere and work in some screwy wiggly coordinate system, and then its components in that coordinate system would not be diagonal."
Oz nods. "By the way... how do you actually compute the curvature? You went back in the other room and did it... it must have been a lot of work, but surely you can at least explain the basic idea!"
The wizard frowns again and says "I'm sorry, that's complicated. I don't think you're ready for that yet."
Oz becomes upset. "No, no, no, no, no. This really will not do. How do you expect us to get a proper grasp, nay even a basic vague concept, if we can't even see how to work out one element of what is likely the simplest non-trivial Riemann tensor. I know, I don't like it either, but it's no good fudging it. It's just gotta be done. You just have to put guards on the machine, hand out hard hats, dark goggles, and make sure we all stay clear, keep our fingers out of the way, and just watch."
"NO!!!! You must NEVER, NEVER go back into that room where I actually compute things. The machines are VERY dangerous. If they could only cut off your fingers, frankly I wouldn't mind; I'd let you go in. But they can destroy your very soul! I was AFRAID you'd start wanting to go back there."
The wizard pauses, frowns, and fingers his beard fretfully as he ponders what to do.
"Let me show you what I mean. Hold on a second."
He walks over to the black curtain and slips behind it, motioning for Oz to keep his distance. A click is heard and then a long, high creaking noise as of a rusty door opening. Then Oz some thumping around and a loud clang as of a door slamming shut. After some more noise, the wizard lifts the curtain and rolls out a stretcher. Oz is shocked to find on the stretcher a human figure completely covered with white crystals of ice.
"This, my friend, is what I meant. See this poor fellow? He is frozen stiff. The worst thing is, he's still alive under all that ice! Do you want to know how this fellow got that way?"
Oz nods, getting over his shock and gradually moving closer towards the stretcher.
"This was once a student of mine, like you, eager to learn general relativity. He was an excellent student, much better than SOME, and he had progressed to the point where he was writing code to do numerical simulations of Einstein's equation. He was trying to figure out what happened when two black holes collide - a problem, by the way, that is still not fully understood."
"Anyway, he noticed a lot of problems. When he reduced the mesh size - never mind, this is just some jargon - sometimes his answers seemed to converge, other times not. He asked me about it so I suggested that he read a bit on numerical analysis. Oh, had I only known!" The wizard pauses sadly a moment.
"To understand the numerical analysis he realized he needed to learn a bit of analysis. After all, how could you compute the answer to something if you weren't sure the solution existed in the first place?? Pretty soon he was quite an expert on existence and uniqueness for nonlinear hyperbolic PDE. He studied Sobolev spaces, and energy bounds, and the work of Choquet-Bruhat...."
"But as he did he noticed something funny happening. Occaisionally he would feel a slight chill. He disregarded it and kept on working, delving ever more deep into nonlinear analysis. He lost interest in his original goal of simulating black hole collisions. Proving existence of solutions to equations seemed much more interesting than actually solving them. After a while he noticed frost forming on his glasses. He just wiped it off and kept on proving theorems. Unfortunately he failed to notice the icicles growing on his desk.... By the time we found him, it was too late. He was frozen solid, but still thinking about existence and uniqueness of solutions of nonlinear PDE...."
"And here he is, still that way, in a condition of... rigor mortis."
Oz leans forwards and touches the frozen figure with his finger. He feels a strange longing, but also a chill, which seems to seep up his finger and into his heart.
" DON'T DO THAT!!!!!" The wizard raises his staff and aims it at Oz, sending a fireball at him, and wheels the stretcher away from Oz. "BACK!!!! Little do you know the dangers!!!! I am protected from the infectious chill by many magic spells, but you are not. Oh, you fool! Let me put this back into the vault. Stay there." He wheels the stretcher into the back room again, and Oz again hears the clanging of a great metal door being slammed shut. The wizard then reappears....
"So, you may think it a little thing, a trifling thing, to learn how I calculated the Riemann tensor of a sphere, but I assure you it is not. Very few know that secret. And if you learn that, you may be irresistably drawn to mathematics, and thence towards RIGOR, and you too may, like my poor student, perish in the icy splendor thereof."
Oz, abashed, decides not to press the point. But after a long pause, he starts thinking about the Riemann tensor of the sphere again... and says:
"I have to say, in my heart of hearts, that I don't really like this definition very much. Not really. For one thing each leg is not epsilon long. I would rather prefer to see it as going in a little square and finding I was *not* back where I started. Then I would have a little path back to where I *had* started from which would be a vector I could 'easily calculate' and would (I think) give me a measure of curvature. I also rather suspect that it would give me the same measure of curvature. Not the actual vector itself, of course, you would have to fiddle with it a bit to get the curvature."
"Hmm. I'm not sure what you are saying. First of all, as I said, the LENGTH of the edges of the "little square" plays no role in the definition of the curvature. To compute
(where 1 = theta, 2 = phi) we simply take the vector in the theta direction at a point P, parallel translate it from P over to the point whose theta coordinate is epsilon more, then over to the point whose phi coordinate is epsilon more, then over to the point whose theta coordinate is epsilon less, and then back where we started, and we see how much it now points in the phi direction. Then we divide by epsilon2, throw in a minus sign for good luck, and take the limit as epsilon goes to zero."
"There are various things about the above paragraph which are a wee bit subtle and may confuse you... but anyway, I'm just parrotting the recipe for curvature, and if this is confusing, that means perhaps you didn't quite understand the original definition of curvature. Which is nothing to be ashamed of, because often things look simple in the abstract and then mysteriously become confusing in any concrete special case."
Oz picks up some old notes and reads them:
"Let me describe the Ricci scalar, R, in 2d. This is positive at a given point if the surface looks locally like a sphere or ellipsoid there, and negative if it looks like a hyperboloid - or "saddle". If the R is positive at a point, the angles of a small triangle there made out of geodesics add up to a bit more than 180 degrees. If R is negative, they add up to a bit less."
"For example, a round sphere of radius r has Ricci scalar curvature R = 2/r2 at every point."
Oz coughs. "Well, I don't like to mention this, but nobody has mentioned a Ricci SCALAR before."
"WHAT??????????" The wizard, obviously still stressed from the previous incident, goes ballistic. He waves his staff about and shoots fireballs in all four directions of the compass, cursing with anger. "Listen here, Oz! I keep careful notes on EVERYTHING I EVER TELL YOU, so don't say I never mentioned the Ricci scalar. Here's what I said, and I quote...." He ruffles around on his desk through enormous stacks of yellowing papers, pauses, scratches his head, and then yanks out one from the middle of the very tallest pile. "Ahem:
"The EINSTEIN TENSOR. The matrix gab is invertible and we write its inverse as gab. We use this to cook up some tensors starting from the Riemann curvature tensor and leading to the Einstein tensor, which appears on the left side of Einstein's marvelous equation for general relativity. We will do this using coordinates to save time... though later we should do this over again without coordinates. This part is the only profound and mysterious part, at least to me.
Okay, starting from the Riemann tensor, which has components Rabcd, we now define the RICCI TENSOR to have components
Rbd = Rcbcd
where as usual we sum over the repeated index c. Then we "raise an index" and define
Rad = gab Rbd,
and then we define the RICCI SCALAR by
R = Raa
The Riemann tensor knows everything about spacetime curvature, but these gadgets distill certain aspects of that information which turn out to be important in physics. Finally, we define the Einstein tensor by
Gab = Rab - (1/2)R gab."
"Now GET OUT! How do you expect to become a sorcerer at this rate?"
Oz skulks out, thinking the old fellow must have drank too much coffee today... "What an old fart," he mutters.
(For an amusing digression click here.)