Stop worrying about tensor calculus for now!
Alabran said:
Let's see if my understanding is correct.
Ricci Tensor: The rate at which the difference in coordinates of two points (t,1,2,3) change in respect to... themselves...
No, that doesn't even make sense.
Since we have agreed (yes?) that you can't try to understand the notion of tensor in a few days, much less explain it to other AP high school students and still have time left to talk about the EFE, you should avoid the word "tensor" entirely! When we pointed you at Baez and Bunn, we meant (or at least I meant) that you should just try to get the physical picture they are suggesting, which doesn't even require mention of the word "tensor". To wit: after explaining world lines and timelike/null/spacelike vectors/paths as per Geroch, you can say something like this:
noone said:
How did Einstein distinguish between "gravitational force", which by his elevator experiment must be as fictitious as "centrifugal force", and genuine forces which cannot be made to vanish by a change of coordinates? He adopted Minkowski's redefinition of acceleration of a particle as the path curvature of its world line in spacetime. Then he sought to represent the gravitational field entirely in terms of the curvature of spacetime, so that any particle experiencing no genuine forces (in Newtonian terms, no "nongravitational forces") has a world line which is a timelike geodesic. As John Wheeler puts it: "the curvature of spacetime tells freely-falling matter how to move".
To complete the picture, Einstein needed to specify how a given amount of mass-energy creates a specific amount of spacetime curvature. This information is contained in the Einstein field equation (EFE). As Wheeler puts it: "matter tells spacetime how to curve".
Physically speaking, this is what the EFE says: imagine that we have a small sphere of freely falling coffee grounds ("test particles") which are initially comoving with respect to each other. Suppose that there is a bit of fluid inside this sphere, but that the particles are still able to move freely (without experiencing any accelerations). Then all other things being equal, the gravitational attraction of this mass-energy should cause the sphere to contract. The EFE says that that volume changes like
<br />
\frac{\ddot{V}}{V} = \rho + 3 \, p<br />
This formula captures both parts of Wheeler's slogan: "the curvature of spacetime tells freely falling matter (the coffee grounds) how to move, and matter (the blob of fluid inside the sphere of coffee grounds) tells spacetime how to curve".
Compare Newtonian gravitation, which is nonrelativistic, where the pressure term is absent. This means that pressure increases the gravitational attraction; in sufficiently extreme situations, such as a collapsing supernova core, this effect is sufficient to overcome extreme pressure and the result is complete collapse to form a black hole.
Here, we need to use a
small sphere of coffee grounds because in curved manifolds, large spheres can have a very complicated relation between volume and surface area. But small spheres are pretty much just like the euclidean ones you know from high school solid geometry. (Well, you probably don't know, because American schools stopped teaching solid geometry about 1920. Physics/math students need to somehow acquire this essential knowledge out of class...)
Alabran said:
The explanation on the article explained it in terms of a spherical cluster of points centered on point P (coffee grounds, specifically), each initially comoving with each-other. As this ball continues to move, each individual point is affected differently because it's specific path through the curvature of space-time is slightly different. The article explained that the "Ricci Tensor" is then, the rate at which such a sphere would change volume, though I don't understand what that is in respect to since time itself (to my understanding) would change slightly between the points.
Forget the Ricci tensor and focus on the physical intutition. It's not that different from Newtonian intuition, which is partly the point.
Alabran said:
A Ricci Scalar seems like it would be easy to understand once I grasp what a Ricci Tensor is from my basic knowledge of scalars and vectors. Since "Speed" is the magnitude of "Velocity", seems that the Ricci Scalar would than the be magnitude of Ricci Tensor, Using the ensteinian summation technique.
The Ricci tensor is just a trace of the Riemann tensor, and the Ricci scalar is just the trace of the Ricci tensor. Since you haven't studied linear operators, you probably don't know what traces or eigenthings are yet, so I'd put all this aside for now.
For future reference, Baez and Bunn are "secretly" talking about something called the Raychaudhuri equation, which involves something called the kinematical decomposition of the covariant gradient of a vector field and something called the Bel decomposition of the Riemann curvature tensor. But you will need more time and more mathematical/physical background than you currently have to really understand all that!
I already suggested that you just say that because gtr is local field theory, the Newtonian force law is replaced by a focus on the tidal accelerations (these bend world lines since acceleration is path curvature) then carry out a Newtonian computation of the tidal accelerations in a spherically symmetric gravitational field, which is accessible to AP physics students. That satisfies the requirement that you do some math, and then you can discuss conformal diagrams and the Vaidya thought experiment on the formation of a black hole by a collapsing shell of EM radiation.
Alabran said:
I can speculate on the meaning of the meaning of the Weyl Tensor, though I haven't actually checked it yet.
The Riemann tensor decomposes into the sum of a "trace part" built out of the Ricci tensor, plus the Weyl tensor. Since the EFE equates the Ricci tensor to the stress-energy tensor (this is equivalent to the physical picture above, but since you don't know what tensors are you can't expect to really understand this yet), it implies that the Ricci part is the part which is directly proportional to the immediate presence of matter (plus the field energy of any EM field). Since outside a star we have essentially vacuum, the EFE must imply that if the Ricci part of curvature is nonzero, it can generate nonzero Weyl curvature, which is the part which can be nonvanishing in a vacuum.
In particular, the field outside a spherically symmetric object like a star or black hole is a vacuum field, so the curvature there is entirely Weyl curvature. In contrast, in the simplest cosmological models, the FRW models, we imagine that the entire universe is filled with dust (pressureless fluid). Then the curvature is entirely Ricci curvature.
A very important prediction of gtr is that when you wiggle bits of mass-energy in the right way (for example, extending and contracting a pneumatic arm changes the mass distribution in the right way), you create gravitational radiation, ripples of Weyl curvature which propagate outward at the speed of light and which can propagate across a vacuum region.
If you wanted to mention this, you could talk about "Ricci curvature" and "Weyl curvature" without mentioning tensors. The important point is that only Weyl curvature can be nonzero in vacuum regions, and there is a differential equation which relates the two types of curvature.
You would have to preface all this by explaining spacetime, lightcones, worldlines, timelike/null/spacelike vectors, and path curvature, which is alot, so I doubt you'd have time in three periods, given the level. Path curvature is \frac{d\theta}{ds}, where in two-dimensional Riemannian geometry, d\theta is the small angle through which the tangent vector to the curve turns as you move small distance ds along the curve. In 4-d Lorentzian geometry its a bit more complicated but you don't have time to fuss with that.
I have the awful feeling I am starting to write your talk for you...
.)
That's the worst thing you can possibly do at this point if you want to give a good talk which minimizes unintentional disinformation, which I certainly hope you wish to avoid! First, WP is too unreliable for students to use as an information resource Second, that particular article is just about the worst source for an AP high school student which I can imagine, since it focuses on math which everyone is telling you to avoid trying to learn and talk about on Thursday (and in any case, doesn't even try to explain said math, much less the physical meaning of the EFE, which is the heart of gtr). 
