On the left arm are fresh troll-fist sized dents in the armor, and on the helm too, underneath which a bit of cloth is wrapped around his forehead, matted with what looks like fresh blood. From his belt however hangs a fresh troll pelt (rock wool), still dripping petroleum; this time the fight was not bloodless. (Trolls are about the consistency of slate, it turns out, and cleave quite nicely).
Oz and the wizard simply stare in astoundment as erg clumps across the floor, partially dragging one steel boot on the quartzite, pulls out a stool and heavily drops himself on it, bowing the legs and threatening to splinter the wood. Not until he has pulled off his battered helmet does he begin to speak...
"Look WIZARD, we have got to get some things straight. You pulled me out of my happy life in Mundania, and stopped many interesting conversations I was having in mid-stream, to come here and "help" you. Well, I sat down in that cave, trying to digest all those tablets, frame some nice questions, and maybe even get inside Oz's brilliant if erratic mind (here he glanced at his companion, who smiled back at him with thinly concealed daggers). That would be fine, except you forgot one thing; when you conjured me here, you forgot to conjure me out of my mundane responsibilities, so every day have to go back to the underworld, to pursue that line of penance called a JOB. So there just isn't time! So damnitall, I'm just going to jump into your conversation with Oz here in mid-stream too, and if I happen to ask something you've just answered (here some inner warning made erg back off his tack a bit), why then maybe my friend here will be kind enough to clue me in <further exchange of frozen smiles>. So if you want to turn me into a newt, then b*g*rall, I don't really care at the moment! Otherwise, you are going to give me some answers!"
A shadow of anger flickered over the wizard's face, and Oz involuntarily tensed, waiting for the thunderclap and smell of burning flesh. But then the wizard relaxed a bit, and a thin smile replaced the scowl. This one showed some spirit, and might be useful for something after all!
"Well, my young friend", he began, eyeing erg's graying strands as Oz unconsiously let out his breath, "I was wondering when you would decide to leave that damp cave and come up and talk to us. I didn't pull you out of Mundania to sit down there are play with the trolls, you know..."
Here erg angered momentarily, as the wizard ironically admired the state of his armor, but he too inhaled and relaxed, and began with a somewhat frozen looking smile...
"So just what questions can we answer for you..."
Then, as luck would have it, having fully riveted their attention for a nanosecond, his mundane job began calling him. Curses! He had better think fast, and get in some questions as he faded from the cell...
"Ok, just what *is* this "geodesic deviation equation" you keep talking about -- in words. Is that the same thing as the parable of the coffee grounds...? And that question about "calculating the curvature at each point of spacetime", what do you mean "calculate"? I thought you mentioned something about the big bang, a *model*, and I assumed you meant "calculate from the model", but OZ here is trundling out all sorts of experimental apparatus, and that seems to meet with your approval. Did you say 'calculate' or 'measure'? Does anyone "measure" the curvature of space in the laboratory?
Erg is receding, and Ed is popping back into his chair by the computer...
"You haven't seen the last of me.... but next time I'm using the front door... n o m o r e t r o l l s ..."
The last almost inaudible, as he vanishes from the cell with a sough.
"Damn!" cried the wizard. "That whimsical Courier copied Ed Green into the fantasy realm using her Applied Strategem to Corral Innocent Inebriates... which would have been fine... but she forgot to remove the copy in Mundania! I suspect he'll keep fading in and out like that. In Mundania, you see, they have this appalling system of `jobs' which keeps people from concentrating on physics, mathematics, and all the other really important work..." G. Wiz shook his head in dismay.
Oz was about to point out that G. Wiz had an enormous staff of servants, including Oz, who did nothing but follow orders... didn't those count as `jobs'? But he had learnt by now to hold his tongue.
"Ah, so what am I supposed to do now?" asked the Wizard in annoyance. "He asks some reasonable questions and then he fizzles out into nothingness before I can answer! I know. Write this down, Oz, and give it to erg when he comes back. Or maybe erg can still see us from Mundania, using all that fancy software they have out there. Anyway, write it down, just in case." He tosses Oz a quill and hands him a piece of parchment. Oz sits down on a chair and writes as the Wizard answers erg's queries.
"Okay, what is the geodesic deviation equation? The geodesic deviation equation describes the relative acceleration of nearby geodesics. Note that since this is, ahem, general RELATIVITY, it is pointless to speak of the acceleration of a particle in free fall, except by comparing it to some other nearby particle in free fall. That's why our equation describes RELATIVE acceleration. It shows that this relative acceleration is entirely due to the curvature of spacetime, that is, the Riemann tensor. And let me give the equation. (I know, erg wanted words, but the equation is so blitheringly simple that I'm sure he'll understand it if he puts his mind to it, and forgets that accursed `job' for just a minute.) Suppose we have two particles in free fall right next to each other, and initially at rest relative to each other. Suppose the velocity of the first one is initially equal to v. Here v is a tangent vector to 4-dimensional spacetime, as always; it's a "4-vector" pointing tangent to the worldline of the particle. And suppose the vector pointing from the first particle to the second is w. Then the acceleration of the second particle relative to the first is
A = R(v,w,v).
Here R is the Riemann tensor, which eats three vectors and spits out one."
"I derived this for Oz the other the other day." Oz silently cursed at having to keep scribbling this incessant stream of verbiage while being referred to in the third person. "By the way, there could be a sign error here, I have been tracking down a certain minus sign for weeks now and I still don't know where it comes from."
"And yes, this is indeed deeply related to the parable of the coffee grounds; the picture, as always, is this..." Here he reached over and drew it directly on the parchment:
| | | | v^ ^ | w | P->-----QThe two coffee grounds start at P and Q, with their worldlines initially parallel, but their worldlines `deviate' and accelerate relative to one another as given by the equation."
"And now for your other question: how come, when I asked Oz to CALCULATE the Riemann curvature at each point of spacetime in the big bang, he meets with approval when he trundles out a big batch of experimental apparatus and attempts to MEASURE it?" The wizard grinned. "Well, I am glad that though my thunderbolts of correction are all too obvious, my kinder, gentler side does not go completely unnoticed... even if it's not always recognized for what it is. You see, I actually said to DETERMINE the curvature, that was my exact word. Of course I meant for Oz to CALCULATE it, but when Oz got it into his head to MEASURE it I went along for the ride, since after all a little bit of thought on how one might do that is not altogether a bad thing. As it turned out, his experimental setup turned out to be an only slightly modified version of the above picture. This gave me the opportunity to slyly introduce the geodesic deviation equation. Indeed, by this point Oz was so possessive of the idea that he practically exploded in rage and static electricity when I suggested he bone up on the definition of Riemann curvature, to get it all worked out." Oz decided not to write down that last sentence. The wizard looked knowingly at Oz but let it pass.
"And, you see, it all worked out in the end. Apprentices may seem to fumble and stumble, yet surprisingly often they still manage to pass the tests set for them. It's all part of the `learning process', just like getting fried to a crisp for getting things wrong. Oz wound up figuring out the Ricci tensor after all, and then the Weyl tensor (though I must admit I have some worries on that... I may have screwed up a bit), and then he learned how to glue them together to get the Riemann tensor, so he triumphed in the end."
"Now for your third main question: does anyone really measure the curvature of spacetime in the laboratory? YES! Whenever we measure a "tidal force", we are measuring the relative acceleration of geodesics, and hence, by the geodesic deviation equation, we are measuring spacetime curvature. More generally, ANY measurement of the effects of gravity can be construed as a measurement of spacetime curvature, possibly of a rather indirect sort."
Oz sighed. G. Wiz had stopped moments before the parchment had run out.