Having now defined the Riemann tensor, which provides one measure of the curvature of spacetime, the next step towards general relativity is to define two variations on it.

The first object that we can define is called the Ricci Tensor, and it is closely related to the Riemann tensor that we defined last time. Recall that the Riemann tensor was defined as function

B = R(A,U,V)
in which A is the vector being moved, and U and V represent two other vectors that define the two paths along which we are moving A. As before, B is the output of the function and represents the different between moving A along U and then V, versus moving A along V and then U.

Unfortunately the Ricci tensor doesn't have quite such a simple intuitive definition, but it can be thought of as an average of the Riemann tensor. Suppose we have four vectors that define a coordinate system, denoted by xi, where i = 0,1,2,3,4. We can then ask how much the xi-component of A changes if one of the path vectors is xi. Or in other words, if we write B as a sum of components along the basis vectors xi,

R(A,U,V) = R0(A,U,V) x0 + R1(A,U,V) x1 + R2 (A,U,V) x2 + R3(A,U,V) x3

then what is the value of Ri (A, xi, V)? 

The Ricci tensor can then be defined as the sum over the four components of R that have been defined in this way:

Ri(A,V) = R0 (A, x0, V) + R1 (A, x1, V) + R2 (A, x2, V) + R3 (A, x3, V)

And as we will see in the next article, it is this object that will be the core of Einstein's equations of general relativity.