When Oz got up the nerve to come back, he found the wizard snoozing on the couch in his study. He hardly seemed to leave that room!
"Ahem...." said Oz.
"Huh? Uh?" G. Wiz thrashed about, rubbed his eyes, and turned to look at Oz. "Oh, it's just you."
"Yes sir. I was just thinking about this big bang problem...."
"You haven't started with the black hole, eh?"
"No... Okay. So I almost worked out the Ricci tensor, and the Riemann is really basically just the Ricci plus the Weyl... you explained how that works the other day... and we are also (at least I am at the moment) setting the Weyl tensor to zero...."
The wizard suddenly seemed much more awake. "Oh yeah? Why are you doing that? I'm not saying it's bad, but explain why you're doing that?" Covering his mouth with his hand, he turned away somewhat and whispered: "Hint: isotropy, isotropy, isotropy."
"Eh? I thought I gave a quasi-plausible reason way way back a long time ago, probably in one of our conversations that never made it into that world-wide web site. My argument went as follows:
a) There ain't no boundary conditions since we include the entire universe. From what has been said about the Weyl tensor, some of it's terms relate to boundary conditions. Maybe tidal effects 'n stuff. These gotta be zero.
b) The universe is uniformly isotrophic so there ain't no waves or other transient stuff wandering about. This should take out another whole load of terms from the Weyl.
I am a mite concerned that the Weyl does include static curvature from distant bits. Hopefully in this situation these end up zero too." Oz grinned nervously.
The wizard stood up. "There's just one thing you should know...."
"Yes?" Oz asked.
The wizard let out a roar, slammed his cane down on the floor, and split one of the flagstones clear in half. "It's ISOTROPIC! Not ISOTROPHIC."
"Right," said Oz, cringing slightly. "I always get that one wrong."
"Well, just think what it means! `Iso' means `equal' and `tropos' means something like "to change, or turn" --- remember how plants that turn to meet the light are `heliotropic', and the layer of atmosphere where the weather keeps turning and churning is called the `troposphere'. So "isostropic" means that no matter how you turn, things are equal.
On the other hand, `trophos' is all about food, growth, and the like: muscles can `atrophy' or `hypertrophy', for example. So `isotrophic' might refer to something like `equally well developed' muscles.
You may argue that the word `catastrophic' means, etymologically, a `downturn' --- but here the suffix is not `trophic' but really `strophic', from `strophein', which also means to turn.
So if you'd said `isostrophic', I would understand and forgive. But `isotrophic'?? NEVER!!" He slammed the staff down again, splitting the flagstone in the other direction.
"Anyway," he continued in a calmer tone, "Your arguments are indeed quasi-plausible. Indeed, Penrose makes a big deal out of his quasi-plausible `Weyl curvature hypothesis', which says that as we approach the big bang, the Weyl curvature must go to zero, for reasons similar to what you say, and so the Weyl curvature serves as a kind of `arrow of time', increasing as time passes.
But this is merely speculation, and your argument is merely quasi-plausible. Why don't you concoct a 100% rigorous argument based on what we know about the Weyl curvature? Remember: the Ricci curvature tells how your little ball of freely falling coffee grounds changes volume, while the Weyl curvature tells how it changes shape into an ellipsoid."
Oz frowned. "Aw, you gave it away. Ok, ok it's a better way of saying it than `boundary conditions' and stuff. The universe couldn't be or stay uniformly isotroph^Hic if Weyl was anything but zero. Almost by definition, well probably actually by definition."
G. Wiz nodded. "Sorry, I should have given you a sneakier hint and let you have more of the fun of figuring it out. Yes, if a little sphere of initially comoving particles in free fall turned into an ellipsoid, there would necessarily be some preferred directions (the axes of the ellipsoid), hence anisotropy."
The wizard smiled and sat back down on his couch. "So, you have worked out the Weyl curvature of the big bang universe and also the Ricci curvature --- given the energy density E and the pressure P. No heavy-duty calculations, just symmetry! Cool, huh? I would say you're almost done with questions 2 and 3:
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 may is determined at each point.
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?
I still would like to hear a bit more on what happens to the components Rii of the Ricci as we approach the moment of creation. (Note: it's standard to use letters like i,j,k to range from 1 to 3. You told me R00 went to infinity, so I'm asking about the other diagonal terms.)
But that's mainly it. You have come so close to completely working out the big bang model, that when you answer question number 1, I will reward you by finishing it up. By the way, I'm tremendously enjoying this. I never really understood the big bang model as well as I do now... I'll have to tell you what I've realized, after you answer question number 1. Black holes!"