Bernhard Riemann
1. The Riemann Tensor
The Einstein field equations,
are usually described as saying that matter tells spacetime how to curve. The right-hand side, \(T_{\mu\nu}\), is the stress-energy tensor; it tells us the matter and energy present. The left-hand side, the Einstein tensor \(G_{\mu\nu}\), is a purely geometric object. But \(G_{\mu\nu}\) is not fundamental, it is actually built from an object that measures curvature. That object is the Riemann tensor, and this post is about where it comes from.
1.1 The question behind the tensor
Curvature sounds like it should be pretty hard to define intrinsically. If you live inside a space and cannot step outside it, how can you tell whether it is curved? There are a few interesting ways which I will explore in other posts, but one way is to transport a vector around a loop and see if it comes back changed. This seems a little odd at first glance. But let’s look at a simple case of curved space: take a vector on the surface of a sphere, pointing north. Slide it up to the pole, keeping it always pointing ``the same way,'' then back down a different meridian, then along the equator to where you started. It comes back upside down. Do the same on a flat plane and it returns exactly as it left. This rotation measures curvature, which can be given a number thanks to the Riemann curvature tensor.
1.2 Curvature as anticommutative derivatives
Okay great. So we have a nice intuitive definition, but it doesn’t really give us an idea on how to actually compute this curvature. Luckily, there is an equivalent statement that is easier to work with. Say we are moving a vector in the \(\beta\) direction, then the \(\gamma\) direction versus \(\gamma\) first, then \(\beta\)' is the same as going two ways around the loop. In the language of covariant derivatives \(\nabla_\mu\), this is asking whether the order matters:
For ordinary partial derivatives this is always zero. For covariant derivatives it might not necessarily be. This let’s us define the Riemann tensor:
Working the two derivatives out and subtracting, most of the terms cancel, and what’s left is a formula made entirely from the Christoffel symbols and their derivatives:
1.3 Why derivatives of the Christoffel symbols?
I will make another post on Christoffel symbols. These are very interesting tensor-like objects (but they are actually fake tensors!) that measure changes in directions with respect to basis vectors or covectors, depending on the kind of symbol. In short, I like to think of them as factors which measure changes in basis vectors with respect to different coordinates and different directions. The point is, at any single point you can always choose coordinates in which every Christoffel symbol vanishes. In GR this is what a freely falling observer experiences locally, and it is why you feel weightless in orbit. But this means that the Christoffel symbols themselves can’t actually measure curvature, because we can change them in a given space, so we will get different values for the same curvature. But what cannot be transformed away is how they change from point to point. If the space is curved, the symbols won’t stay zero as you step to a neighbouring point, and their derivatives survive. This is why the Riemann tensor is built from derivatives of \(\Gamma\), and why
1.4 From Riemann to Einstein
The rest is a process called contraction. Summing the Riemann tensor over one upper and one lower index gives the Ricci tensor \(R_{\alpha\gamma}=R^{\sigma}{}_{\alpha\sigma\gamma}\); contracting again gives the Ricci scalar \(R\); and the combination
is the Einstein tensor we started with. So the single line \(G_{\mu\nu}=8\pi G\,T_{\mu\nu}/c^4\) is, in truth, a statement about vectors failing to return to themselves after a loop. Matter bends spacetime; the Riemann tensor is how spacetime keeps keeps track of it!