When they reached the Wizard's keep, the Wiz invited Oz to his room for a cup of coffee. The sun was beginning to rise, and Oz felt rather exhausted and chilled. Luckily the fire was already lit, and the water soon began to boil. "I already had a cup," said the Wiz. "But I could use another. That ridiculous episode you just wrote, where you had us flying through the air at an ungodly speed, was rather wearing."
Oz smiled and said nothing. They both knew, of course, that they were merely fictional characters of their own devising, but it wasn't wise to dwell on it too much.
Grinding some coffee, the Wiz smiled at Oz. "You seemed to be hitting it off pretty well with Rosie."
Oz blushed. "Umm, yes, and the strange thing is, I'm not quite sure why. Some sort of... magic?"
The Wizard raised his eyebrows. "Don't you know? It's the general relativity."
Oz had heard this, but it was hard to believe. "Really?? How does it work?"
"Well, it's simple really. Rosie was one of my students, once upon a time. But you know how hard it is to get jobs in physics. So she took up bartending. She knows a lot more than she lets on - that manner of hers, she just does that to get good tips. I'm sure she'd love nothing more than a good intimate chat about the symmetries of the Riemann tensor. But most of the men down there are uneducated oafs. Well, what do you expect? It's just a typical village. Anyway, she *knows* you've been learning some general relativity... rumors travel fast here... so I'm sure she's been dying to meet you."
Oz didn't know what to say.
"Here, have some coffee. Anyway," said the Wiz, "To please her you're going to need to learn a lot more about tensors than you have so far!"
Oz frowned. So Rosie was a physicist? It seemed hard to believe for some reason. Could the Wiz be saying this just to motivate him to study? Still, Rosie *had* seemed to fall for him awfully easily... he couldn't think of any other explanation.
Unconcerned with Oz's musings, G. Wiz continued, "So: you wanted to look behind the curtain and see how to compute the Riemann curvature of a metric. Let's do it! It's not going to be easy. Gird yourself. It will take a while of trying, but let's get started." He strolled over towards the curtain, and then stopped dead in his tracks.
"Hmm," said the Wiz. "Wait a minute. This calls for some ominous music." He clapped his hands and a waft of low, dissonant cello music drifted over from the back room. A cloud drifted over the sun, and it began to rain.
"Now, before I let back there," said the Wiz, "You must promise not to reveal what you learn to the uninitiated. This knowledge is DANGEROUS." A flash of lightening and a thunderclap served to accentuate his warning.
"Okay, sure, I promise," said Oz, eager to finally see what was back there.
"Hmm," said the Wizard. "You don't seem to be taking this sufficiently seriously. But it's been a long night. I'll just show you some equations now, and then we'll dig into them more deeply later." He pulled back the curtain, and the cello music worked its way to a crescendo. "Come on in!"
Oz stepped in and saw to his surprise that the back room was not too different from the room they had just been in! True, there was a large tome resting on a podium, with six-foot-tall black candles on either side. But besides that, nothing much... a couple of chairs, a desk... but nothing like what he had expected, or what he had seen when he had snuck in there that fateful night. Then he noticed that at the rear of *this* room there was *another* black curtain, just like the one he had stepped through, although even blacker, if possible. Apparently it lead to still another room!
"Oh yes," the Wiz said, following Oz's glance with his eyes. "I moved most of the REALLY dangerous stuff back there so you wouldn't get hurt. Whatever you do, DON'T GO BACK THERE, okay?"
The Wiz motioned to the large tome. "Now, what you'll do in the weeks to come is to work with the following formulas. First, you will familiarize yourself with the metric for a big bang universe... a special case, actually, the "spatially flat" or "critical" case. It looks like this..." He turned a page and Oz saw, in large print, the formula
g = -dt^2 + R(t)^2 (dx^2 + dy^2 + dz^2)
Oz felt slightly scared, but not too scared... it seemed vaguely familiar. He was about to ask a question when the Wiz said "No, not now, we'll talk all about it soon enough. I just want to show you a few things first."
"Then," said the Wizard, "I am going to teach you how to compute the Christoffel symbols, or connection, given any metric."
"Christoffel symbols? Connection? Huh?" asked Oz.
"That is the way we talk about parallel translation, here in the back room," said the Wizard. "Don't worry, I'll explain it all in due time. For now I just want to show you the explicit formula for the Christoffel symbols in terms of the metric. It is..." and he turned the page:
C^a_{cd} = (1/2) g^{ab} (g_{bc,d} + g_{bd,c} - g_{cd,b})
"... that!"
Oz blanched with fear. What were those commas doing down there with the subscripts? What did this all MEAN?
Unperturbed, the Wizard continued, "Usually people write the Christoffel symbols as Gamma, but we're in an ASCII environment here so we'll call them C... for Christoffel, or connection. For now, all you need to know is there exists this explicit formula, but your goal will be to understand it, so that you can apply it to the metric on the previous page."
"And then," the Wiz continued, "we will compute the Riemann tensor from the Christoffel symbols. This was what you wanted, remember? You wanted to compute the Riemann curvature of a metric... well, we do it using the Christoffel symbols as an intermediate step."
Oz wondered why he had ever wanted to know these things. The formula for the connection seemed to expand and grow ever more fearsome as he stared at it.
"So," continued the Wizard, "we will need to learn, and understand, the following formula for the Riemann tensor in terms of the Christoffel symbols." And he turned the page, revealing the following formula, written, it seemed, in blood:
R^a_{bcd} = C^a_{bd,c} - C^a_{cd,b} + C^e_{bd}C^a_{ec} - C^e_{cd}C^a_{eb}
Oz's hair stood on end. He backed away, slowly. "No," he said. "No. There is no way I am EVER going to understand THAT."
The Wizard smiled. "That's what most people say when they first see it. I felt that way myself." But somehow this did not reassure Oz. Another stroke of lightening cracked through the air, and he jumped.
"NO!" cried Oz. "I will NOT understand it! NEVER!"
The Wizard laughed. "I won't force you to, if you decide you don't want to. But remember, I *warned* you not to come back here, but you insisted!"
"It was a mistake! I take it back," said Oz. "Can I go now?"
The Wizard let out a guffaw. "Hey! I'll EXPLAIN all this stuff. When you approach it correctly, it's not so hard. Right now I'm just SHOWING it to you..."
"... just to scare the wits out of me," said Oz.
"Right," agreed the Wizard. "Hey, look, you were expecting thrills and chills! Who am I to let you down? And it's actually a tradition, in general relativity, to scare people with these formulas. Actually, you're having an easy time compared to me, back when I was a kid." He shut the book. "Come on, you've had a long night, go to bed. We'll start when you get rested up. When we're finally done, you'll be able to compute the Riemann curvature of that big bang metric and actually solve Einstein's equation! Think of that! But first, we'll get used to that big bang metric, and also get used to this Christoffel symbol business, and, yes, even get used to that formula for the Riemann tensor. So don't get too scared just yet."
But just at this moment, some sinister laughter poured forth from behind the curtain leading to the back room of the back room. The Wizard frowned and hurled a fireball in the general direction of the curtain. "Not yet," he repeated... and Oz slowly walked out, and then ran down the hallway and out the door to his cave, and fell exhausted on his straw mat, going immediately to sleep.