The Ricci Tensor

We can treat our wonderful metric, g_ab as a matrix, and hence invert it. This needs a bit of explaining - the inverse of a matrix is a another matrix which when the two are multiplied together gives 1 (well, a matrix whose diagonal components are 1). Matrix multiplication is a tricky process which most of you will learn in Linear Algebra (or may already have) in University, for now just know that this is like taking a tensor and flipping it around. We use this to produce some interesting results, leading up to Einstein's equation.

The first thing we encounter is the Ricci tensor, which we obtain by yet another mathematical trick of making one of the subscripts of the Riemann the same as its superscript (one of the steps in contraction), and we obtain the following:

Now we know what it is mathematically, but what does it mean? Imagine a ball of particles going through space - they could be anything, atoms, coffee grounds, baseballs. If we take the velocity vector of one of the particles (which has four components because we're working in space-time) and parallel transport it over to another we obtain its velocity vector - thus the particles are "co-moving" (a fancy, more precise way of saying they're moving with the same speed in the same direction). If the space were flat, the ball would continue on in this state indefinately.

However, the particles generate their own gravitational field and hence they will be deflected from each other according to the Riemann tensor and the geodesic deviation equation, which defines, in terms of the Riemann tensor, how initially co-moving particles will accelerate away from each other. The ball will begin to shrink and change shape - the rates of which are captured in the Ricci and Weyl tensors. The Weyl tensor specifies things external to the sources we're talking about - say another planet a few hundred thousand kilometres away, things with nothing to do with the current source of gravitation but which still affect the gravitational field (this describes how the shape of our ball changes). The Ricci tensor consists of ten components (half those of the complete Riemann tensor) and is dictated by the rate of change of the volume of our little ball of particles.

Formally, the second time derivative of the volume of the ball is approximately -R_abv^av^b times the volume of the ball when it was co-moving. The Ricci tensor tracks how the general paths of particles in space-time are deflected due to curvature.

Now we make use of the inverted metric and define the Ricci Scalar. We use a process known as raising an index, and first obtain the following from the Ricci tensor: