How Rijkl Measures the Curvature of Space

I went many years wondering what Rijkl meant. In 1956 I read a short mimeographed note by Penrose describing the meaning of the Christoffel symbol Γijk. With that insight I think I can describe Rijkl even easier.

You live at some spot in an n dimensional Riemannian manifold and have a convenient coordinate system. Each of the indexes i, j, k and l take on one of n values; there are n4 such combinations and as many values for Rijkl. For each such set of values Rijkl can be evaluated by the following simple procedure. Travel in direction k until coordinate xk is increased by dxk. (You choose dxk, but see guidance at end.) The other coordinates will not have changed. Next travel indirection l for distance dxl. Now travel backwards in the k direction but a distance −dxk. Then once more in the direction l for distance −dxl. You will arrive back near to where you left. That was just practice.

Now travel the same route again but carry along a vector V that initially points in the direction j with magnitude dxj. (You choose dxj too.) At each step in the journey do not turn V that you carry, even though the coordinates may themselves turn which will cause the coordinate description of V to change. This would be a big effect if you were walking near the North pole. There are several ways that the meaning of this can be defined which we do not explore here. When you get back compare the V you carried with a copy that you left behind. The difference between the vector you started with and the one you brought back will be a new vector with components D1, ...Dn.
Rijkl = Di/(dxjdxkdxl).

You must choose the size of dxj etc. and this is a practical tradeoff—too small and D will be too small to measure, too large and your curvature result won’t really be local. If your manifold is mathematically defined then you should let the dx’s tend to zero and take the limit.

If Rijkl = gRβjkl then
Rijkl = −Rjikl = −Rijlk = Rklij
Rijkl + Rikjl + Riljk = 0. These constraints allow only n2(n2 − 1)/12 degrees of freedom in an n dimensional space. Here is a table:
nn4n2(n2 − 1)/12
2161
3816
425620
562550
Why do you use such inefficient notation? Isn’t there a better way to describe 4D curvature than writing 256 numbers when you need only 20? Because the tensor notation simplifies all the useful equations. We find tricks to avoid writing 256 numbers, even in computers.

Γijk tells how fast the coordinates turn. It is thus about the coördinate system, and only indirectly about the space.


See John Baez’s more extensive stuff along the same lines.