If you click on some of the capitalized concepts, you will jump to the corresponding place in the longer outline. In some cases, if you click on them again from there, you will jump to a still more thorough explanation.
A TENSOR of "rank (1,k)" at a point p of spacetime is a function that takes as input a list of k tangent vectors at the point p and returns as output a tangent vector at the point p. The output must depend linearly on each input.
Take the vector w, and parallel transport it around a wee parallelogram whose two edges point in the directions εu and εv , where ε is a small number. The vector w comes back a bit changed by its journey; it is now a new vector w'. We then have
w' - w = -ε2R(u,v,w) + terms of order ε3
Thus the Riemann tensor keeps track of how much parallel translation around a wee parallelogram changes the vector w.
g(v,w) = gab va wb
for some matrix of numbers gab, where as usual we sum over the repeated indices a,b, following the EINSTEIN SUMMATION CONVENTION. Another way to think of it is that our coordinates give us a basis of tangent vectors at p, and gab is the inner product of the basis vector pointing in the xa direction and the basis vector pointing in the xb direction.
Similarly, the vector R(u,v,w) has components
R(u,v,w)a = Rabcd ub vc wd
where we sum over the indices b,c,d.
Okay: starting from the Riemann tensor, which has components Rabcd, we now define the RICCI TENSOR to have components
Rbd = Rcbcd
where as usual we sum over the repeated index c. Then we "RAISE AN INDEX" and define
Rad = gab Rbd,
and then we define the RICCI SCALAR by
R = Raa
The Riemann tensor knows everything about spacetime curvature, but these gadgets distill certain aspects of that information which turn out to be important in physics. Finally, we define the Einstein tensor by
Gab = Rab - (1/2)R gab.
You still should not feel you understand why I am defining it this way!! Don't worry! That will take a bit longer to explain. But we are almost at Einstein's equation; all we need now is:
T(u,v) = Tab ua vb
In coordinates where x0 is the time direction t while x1, x2, x3 are the space directions (x,y,z), and the metric looks like the usual Minkowski metric (at the point in question) we have the following physical interpretation of the components Tab:
The top row of this 4x4 matrix, keeps track of the density of energy - that's T00 - and the density of momentum in the x,y, and z directions - those are T01, T02, and T03 respectively. This should make sense if you remember that "density" is the same as "flow in the time direction" and "energy" is the same as "momentum in the time direction". The other components of the stress-energy tensor keep track of the flow of energy and momentum in various spatial directions.
G = T
or if you like coordinates and more standard units,
Gab = 8 π k/c2 Tab
where k is Newton's gravitational constant and c is the speed of light. This equation says how the flow of energy and momentum through any given point of spacetime affect the curvature of spacetime there.
But what does Einstein's equation mean? Well, it turns out that that a = b = 0 component of this equation can be translated into plain English as follows:
Take any small ball of initially comoving test particles in free fall. As time passes, the rate at which the ball begins to shrink in volume is proportional: to the energy density at the center of the ball, plus the flow of x-momentum in the x direction there, plus the flow of y-momentum in the y direction, plus the flow of z-momentum in the z direction.Even better, if this holds in every coordinate system, Einstein's equation holds. So, all of general relativity can be recovered from this one sentence if one really thinks hard. In practice, though, it's best to master tensors and learn how to calculate with them to extract the predictions of general relativity.
For more details, go to the longer course outline.