"If 4D space is curved in a GR-like fashion what do we really mean by this?"
The wizard smiles slightly. "I told you this once a while ago, but you'll probably understand it a lot better now. Here's an operational definition of the sentence `spacetime is curved': we say spacetime is curved at the point p if, in a very small neighborhood of p, initially comoving geodesics accelerate relative to one another."
"I could make that more precise but actually I already did. Remember those coffee grounds? We set them up very carefully to follow `initially comoving geodesics', and then a little ball of them started rotating and changing shape. If, regardless of the velocity v of the central coffee ground at the point P, the ball does NOT start rotating or change shape, then space is flat at P. Moreover, every bit of information about the Riemann curvature tensor at P can be recovered from knowing how these balls change shape (for all possible velocities v). So we can quite rightly say: `Operationally, curvature of spacetime is just a way of talking about the relative acceleration of initially comoving nearby geodesics.'
"And that's good, right, because curvature of spacetime is all about GRAVITY, and gravity is what you need to understand about WHAT HAPPENS IN FREE FALL."
Oz nods. "An unaccelerated body follows its tangent vector. Since we have been told this is a geodesic, this would seem right."
The wizard raises his eyebrows slightly. "Yeah, sort of. More precisely: it follows a geodesic, which is a curve whose own tangent vector is parallel translated along itself."
"I have a few other ideas I want to bounce off of you," says Oz, "I think is is valid to describe distances measured along an (arbitarily long) path. Of course you won't get the same distance if you choose a different path, but a distance as measured along a path is a valid distance. I think this is valid even if the path has accelerations along it. <Down, Wiz, down.> Indeed I suspect that this is the ONLY valid distance you can define. Hmmm, or is it simply the definition of a 'distance'. Note that this path need not be a geodesic. Is this right???"
The wizard's eyes light up. "WOW! YES! You seem to be seeing exactly why I would always scold you for talking about the distance between P and Q without specifying a path between P and Q! YES! The `arclength' or `proper time' along a path is well-defined both operationally and in our mathematical formalism for GR. Operationally: if the path is spacelike you can use rulers; if it is timelike you can use watches. Mathematically: the metric lets us compute the length of a tangent vector, and we integrate this length along a given path from P to Q to compute the length of the path."
"And YES! Distance between points is ONLY defined given a path between them; this is what distance means in GR."
Oz continued: "Also: In a torsion-free GR then an infinitesimal sphere of massless test points surrounding an (even more) infinitesimal speck of momentum-flow will behave with spherical symmetry under 'control' of the Ricci tensor. We note that this essentially says that the *local* spacetime in this infinitesimal volume can be considered flat. We also note that this is also the case for a torsion-free parallel transportation: it is in locally flat spacetime. This sort of implies that GR models a world where a suitably small piece of spacetime can be considered locally flat."
"If I understand you aright, yes. This is what they call the `equivalence principle'."
Leaning back expansively, Oz says "On a philosophical note one wonders if a black hole horizon is such a place for example, and if any extension of GR will remove this simplification resulting in a truly complex piece of mathematics that will require new tools to manage it."
The wizard does not look pleased. "Well, in GR the black hole horizon is very much `such a place'. We could be on a black horizon right now and not know it if the black hole was big enough!!! I would hate to think this version of the equivalence principle was a `simplification' in a `bad' sense. I would prefer to think it's an important principle which is trying to tell us something."
"We don't seem to have any relativistic effects modelled into our version of GR yet. I expect they are hidden in the notation somewhere or maybe we need to discuss geodesics to find them. I have a feeling that the math may become complex at this point. <Oh dear>."
"Huh??? ALL the relativistic effects you can think of are built into general relativity as I've described it to you; we just need to tease them out. Luckily, the conceptually hard part is to learn all the geometry you need to understand Einstein's equation, and you have almost done all that! Then you are in the position to look at some solutions and know what they mean! For example, I can go into my back room, solve Einstein's equation, work out some geodesics, and show them to you. Then you'll see what the big bang is really like, or black holes, or whatever. This will be fun and easy in comparison with what we've been doing. But just remember: never ever try to go into that back room. It's VERY DANGEROUS in there... lots of nasty, scary MATHEMATICS back there."
Oz adds: "It's kind of interesting that considering energy as momentum flow in the time direction, we can dispense mentally with both mass and energy. We only need to consider momentum flow to describe space curvature, and everything else. Indeed it would be 'nice' to remove that nasty -1 in our metric and make it +1 which I suspect would make us view the momentum flow in the time direction as something slightly different. Has anybody done this, and how would you view momentum flow in the time direction if it had a metric of (+,+,+,+)? (I.e. still modeling the real world)."
The wizard nods. "A chap named Hawking did that once. He called this trick `imaginary time' because (it)^2 = -t^2, so you can get rid of the minus sign in the metric by making the substitution t -> it. In the world of imaginary time, time is no different from space."
"Why did he do this? Well, he was wondering about the question: `what happened right at the moment of the big bang, or before?' Of course, classically this question doesn't make sense at all. But what about when you take quantum gravity into account? That's what Hawking was wondering about."
"Unfortunately, to answer this question, Hawking had to go way back into that back room where we keep the mathematical machinery. [The wizard gestures with his staff to the curtain, which looks blacker than ever, shadows seeping from it and filling the room. Oz suddenly notices it has grown very late and is dark outside.] Way, way back where they keep the REALLY scary mathematics, stuff you wouldn't believe. And unfortunately to understand his answer, you'd have to go in there too. Because, you see, he never came out!"
"And if you went in too --- not that you'd ever even think of it, of course --- but if, *if* you went in, and went THAT far back, it's very likely YOU TOO MIGHT NEVER COME OUT AGAIN."
Deep sinister laughter emanates from behind the curtain. Oz bids a hasty goodbye and runs all the way home.