Just as Oz was about to reply to the dented apprentice's last remark, it befell that a stray piece of spacetime dilation drifted in from one of the wizard's experiments in his fell castle, high up on the troll infested mountain on a bleak outcropping. As this disturbance dropped into the cave the part where Oz stood kind of slowed down, then seemed to black out altogether! Only a few faint sparks jumping across the seams in erg's rusty armor told him that the odd microwave and radio signal were still coming from that part of the room.
"Oh great", erg thought, "Just what we need; a usenet timelag" (Uniform Spacelike Entrapment; a wizardly technology using the coarse woven fabric of space as a net to catch the unwary. Stray blobs of redshift were either an industrial waste products or failed prototypes, it's not clear which.)
"Well", mused the knight erring, "normally it's bad form to hold a conversation with one's self, but there is no telling how long this could last, or in whose proper time." So, looking around the visible portion of the cave's floor, and careful not to put his hand in the menacing looking red aura surrounding the missing region, he came upon the fell paper:
>"Dear Oz --
>If you wish to learn more general relativity I am afraid you will need
>to pass a test of your valor. So: answer me the following questions:
"Hmm", thinks the sorcerer's lesser apprentice, "I wonder what these wedge shaped runes in the margin mean? Maybe if I held it sideways..."
A image of the wizard suddenly appeared on the sheet, laughing in a rather unpleasant manner. "And stop saying 'fell', nitwit!"
"Yow!" he remarked, dropping the paper. "Maybe I should just try to answer the questions. Let's see...
"1. Explain why, when the energy density within a region of space is sufficiently large, a black hole must form, no matter how much the pressure of whatever substance lying within that region attempts to resist the collapse."
I bet the "no matter how much the pressure" is kind of a red-herring. In General Relativity pressure is just a form of energy, and a very high pressure is just the same as a very high energy density; so "pressure" sort of gets hoisted on its own petard: An extremely high pressure *provides* its own containment. Or something.
Somehow I feel the answer should involve geodesics... It's a most peculiar thing. A blackhole is a region in space where geodesics can get *into* but cannot get *out*. But soft... isn't a geodesic a geodesic in either direction? Of course it is. But one direction is "physical" and the other is not. In particular, one direction of a timelike geodesic describes a particle moving "forward in time" and the other, well, you know... So GR evidently has an arrow of time build into it. Fair statement? And what about "spacelike" geodesics. I suppose they would represent a superluminal trajectory in all reference frames... or should I say, all "proper" reference frames, the way timelike geodesics represent sub-luminal trajectories in all proper reference frames...?
Is there an anti-black hole? A region of space(time) where stuff can only *leave* from, but never get back in? Sure! The complement of a blackhole is such a region... but is it strange that all blackholes are roughly spherica in space, and all "anti-blackholes" are the complement of such regions, or in fact the *one* anti-blackhole is the complement of all such regions...
What happens to geodesics after they get *into* a blackhole? I suppose they all wind up at a singularity. Is a black hole a mini big bang in reverse? What if we embedded another big bang into *this* universe... would that look like a spherical anti-blackhole...?
But all this makes blackholes sound like complicated global results about tracing geodesics... I don't think it was supposed to be this complicated...
"2. Explain how, in the standard big bang model, where the universe is homogeneous and isotropic --- let us assume it is filled with some fluid (e.g. a gas) --- the curvature of spacetime at any point is determined at each point."
Oh damn. This is *just* what got me started on all that analyticity stuff. *Part* of the curvature of spacetime at each point is dertermined by local conditions. I think it was 10 of the 20 independent components of the Riemann tensor, the 10 in the Ricci, the Weyl being free. So... maybe in the standard big bang model, we just set the Weyl equal to zero? Why? It seems too simple. Maybe it turns out that the partial differential equations governing these components are the non-elliptic ones? That they therefore represent something propagating, rather than something that is locally evolving? After all, the "standard big bang model" must just be a name for a simplified pet solution to the general equations... but how do we enforce this simplification. Boundary conditions? Arbitrary assignment?
"3. In the big bang model, what happens to the Ricci tensor as you go back in past all the way to the moment of creation."
Um... becomes singular? Somehow I think a little more detail is called for. No doubt the density of stuff becomes infinite, so the Ricci becomes infinite, and so forth. I know! It's a trick question. "Happens" implies evolution forward in time... Nothing "happens" going back in the past. :-)
Suddenly it becomes clear our apprentice hasn't a clue how to answer these questions quantitatively, or even sharply qualitatively. Perhaps one of the spectators will throw a rock with a clue wrapped around it into the fantasy world of G. Wiz.
But finally the time shift seems to be clearing up, so let's hear what Oz has to say...