About an hour later, Oz returned.
"So," asked the Wiz. "Have you figured it out yet? I haven't got all day, you know, just to hear your efforts to crack a nut you should have cracked long ago by now."
Oz stared at the Wizard. He was speechless. Slowly he placed both hands on his hips and, careless of bodily danger, faced the Wizard up. He was cool and controlled, but you could feel the exasperation boiling up inside him. He had just worked through a whole lotta junk, and he suddenly realised the wizard just wanted the old fashioned stuff.
Oz opened his mouth and spoke:
"Hey Wiz, now you just listen here and cop a dose of this!
We have:
R00 = (1/2)(E + 3P)
Well, the momentum flow in our little ball of particles would seem to be increasing but I would guess that the ball is isotropic and uniform in the spacial dimensions so the above looks a reasonable guide to the curvature. We note that the density (E & P) are getting bigger and bigger as the ball contracts, so the curvature is getting bigger and bigger too.
After a while our little ball of coffee grounds (iron filings would have been better) is so dense that the grounds start bumping into each other. Some of them at the surface get ejected and start to head away from the ball. Unfortunately for them they are now in very curved space. Since the ejected grounds now follow geodesics away from the ball we can use our little formula:
d2V/dt2 = -(1/2)(E + 3P)
except this time as the particle speeds outwards the volume of space enclosing the mass is increasing so the effective density is decreasing and so d2V/dt2 is decreasing as it heads outwards. (Well, nobody has actually said this, indeed they implied that it was a naughtly thing to say, however this way we should get a definitive fireball answer one way or the other).
Now some lucky speedy little grounds will escape all the way to infinity, but other slower doomed grounds will find that their velocity wrt the main ball eventually becomes negative and the curvature bends them round to head inexorably back to the still-contracting main ball. In this way the velocity between grounds will be reduced. Essentially by evaporation of high velocity coffee grounds, to below the ever increasing escape velocity of the ball. We note in passing that these coffee grounds feel no acceleration as they travel along their curved path.
So the relative velocities of the coffee grounds (note that they are infinitely small coffee grounds) within the ball can increase as the curvature increases. As the ball gets smaller it takes faster and faster coffee grounds to escape from the increasingly curved spacetime that surrounds the ball. Now let's give these relative velocities a more useful name, say 'temperature'.
In passing it's clear that any given curvature will have some characteristic temperature where it will be in equilibrium. In effect all the coffee grounds are orbiting the centre. It will inevitably lose some energy continually by evaporation. If you had an energy source that put heat into the ball at the rate it is lost, it could sit at that curvature until the power source ran out. It would then continue to contract. These are called 'suns'. Lo and behold we note that low density suns have a low temperature, and high density suns have a high temperature. This is why a red giant is red, it has very low density. Massive stars have to have a high temperature because of their high density. We ignore neutron stars and dwarves because they have a different mechanism to prevent collapse.
However our coffee grounds have no such source. They continue to contract and the temperature continues to rise as the coffee grounds have to get faster and faster to escape to infinity due to the ever increasing curvature of the ever increasing density of the ever decreasing ball. I feel that there should be a mechanism that allows the ball to completely evaporate, but starting where we started I am not too sure there is one.
One might wonder if there is a maximum velocity. If there were then at some point the curvature gets so curved that nothing can escape to infinity. This would be the event horizon, and I *still* think that is where the curvature is one (or something like that). The worldline of a photon then goes from 45 degrees to being parallel to the time axis.
Anyway, our hot little ball of incinerated coffee grounds would still continue to contract, and contract, and contract, until it was infinitely small and the space around it infinitely curved. I guess this counts as a 'blow up' even for ornery old Wizards. HUMPF!"
Oz paused for breath. He had let it all come out. Blow the infernal Ricci's and Riemann's and Einsteinian giff-gaff. He just said it how he saw it. You could see the defiance all over his face.
"AND while we're at it, " bellowed Oz in intense irritation, "this is nothing I couldn't have said before this course started (except for swapping curvature for 'gravitational field'). It's a lot easier, of course, if I don't have to pull this stuff out of equations I only partly understand (well, you gotta be optimistic). So where have I used my newfound knowledge?"
"Nowhere!" he cried.
The strain all became too much for Oz and he fell on the quartzite floor and sobbed as he waited for the fireball to end all fireballs.
But all he heard was the following comment. "Very good as far as it goes. But there *is* something you have learned, something very particular to general relativity, which is crucial to the formation of black holes. Think! What is it?"
Disgusted, Oz got up and walked out. Continued...