Christoffel symbols

Gauge theory has its roots in general relativity, and we have to start there if we want to properly understand it. (Also general relativity will become our friend and we will come back to it to compare ideas from gauge theory.) An excellent book to learn general relativity is Gravitation and I was very fortunate to learn it as a graduate student directly from Professor Misner. The book's famous saying is: "spacetime tells matter how to move; matter tells spacetime how to curve".

What does it mean spacetime curves? Here I recall an explanation from Gravitation which made a strong impression on me at the time. If you throw a ball you see its curved trajectory as it is attracted by Earth. Similarly, a bullet's trajectory is also curved down, but by a very tiny amount. Why is this huge discrepancy in the amount of curvature in each case? Naively one may say that because of the speed, but how about the equivalence principle? The mass should make no difference in what it basically amounts to a free fall motion where the gravitational effects vanish along the trajectory in an "Einstein's elevator". Therefore one would expect that the curvature is the same!!! And it is the same! Only if one thinks in 3+1 spacetime dimensions, not in 3 space dimensions. Here is the picture from page 33 of the book which explains it all:

Space-time, or in general any Riemannian space is characterized by a metric tensor. When the space is curved, the ordinary derivative has to be modified to take into account changes to the local metric. The corrected derivation operation is called covariant derivative. The difference between the ordinary derivative \( \partial \) and the covariant derivative \( D \) is the Christoffel symbol \( \Gamma \):

\( D_\rho f^\alpha = \partial_\rho f^\alpha + \Gamma^{\alpha}_{\rho \sigma} f^\sigma \)

\( \Gamma \) is also called the affine connection because it helps define the parallel transport along geodesic lines which locally are the straight lines in the curved space ("spacetime tells matter how to move"). Along geodesics, the covariant derivative \( D \) is zero. Gravity does not bend light, There is no dynamical explanation of light bending, but a kinematic one. Light always continues to move locally in a straight line, but specetime itself is warped by the presence of mass. The geodesics are affected by mass. If you ever traveled in an airplane, the trajectory from the starting point to the destination appears curved on the TV screens in the plane. The plane always goes straight from point A to point B on the shortest path to save fuel, but the trajectory appears curved because Earth is not flat, but a curved space (a sphere). The apparent trajectory curvature is encoded by the Christoffel symbols. 

So what is then the covariant derivative of a more complex object like say a tensor \( T^{\alpha \beta}\)?

\( D_{\rho} T^{\alpha \beta} = \partial_\rho T^{\alpha \beta} + \Gamma^{\alpha}_{\rho \sigma} T^{\sigma \beta} + \Gamma^{\beta}_{\rho \sigma} T^{\alpha \sigma} \)

This formula is easily generalized in case of an arbitrary tensor of n indexes: \( \Gamma\) appears n times there. We can apply this to the metric tensor \( g_{\mu \nu}\) and after some algebraic manipulations solve for \( \Gamma \):

\( \Gamma^{\sigma}_{\mu \nu} = \frac{1}{2} g^{\sigma \rho} ( \frac{\partial g_{\rho \mu}}{\partial x^{\nu}} + \frac{\partial g_{\rho \nu}}{\partial x^{\mu}} - \frac{\partial g_{\mu \nu}}{\partial x^{\rho}})\)

The moral of the story is that on Riemann spaces, because the Christoffel symbols depend on the metric, the covariant derivative depends on the metric as well and everything is defined using intrinsic notions. There is only one "fly in the ointment": \( \Gamma \) itself is not a tensor. Why? Because it depends on the partial derivative of the metric tensor with respect of the coordinate system. Change the coordinate system and you get a different \( \Gamma \). What we need is a better mathematical object which changes nicely when the coordinate system is changed. In other words, we need the curvature tensor.

Next time we'll introduce the curvature tensor and we'll talk about a key physical principle: curvature is force.