What the hell is a Christoffel symbol?

I have to admit that I’ve gone a long time without really understanding the physical interpretation of the Christoffel symbols of a connection. In fact, there is an interpretation that, in the special case of the Christoffel symbols for the Levi-Civita connection in polar coordinates on Euclidean space, could be understood by me at age 16, after I took an intro physics class (though I definitely wouldn’t understand the relativistic or Yang-Millsy stuff). Here I want to record it. As usual, I’m pretty sure that everything here is very well-known, but I want to write it all down for my own intuition.

Let D be the covariant derivative of a connection on a vector bundle E. Given a coordinate frame e, one defines the Christoffel symbols by $D_j e_k = {\Gamma^i}_{jk} e_i$ . Here and always we use Einstein’s convention.

The Levi-Civita connection. Suppose E is the tangent bundle of spacetime and D is the Levi-Civita connection of the metric. Then for any free-falling particle with velocity v and acceleration a, one has the relativistic form of Newton’s first law of motion $a^k + {\Gamma^k}_{ij} v^iv^j = 0$ , which to mathematicians is more popularly known as the geodesic equation. It says that the “acceleration” in the coordinate frame e is entirely due to the fact that e itself is an accelerated frame.

Viewing $\Gamma^k$ as a bilinear form, we can rewrite Newton’s first law as $a^k = -\Gamma^k(v, v)$ , which now resembles Newton’s second law with unit mass. Indeed, the acceleration of the particle is given exactly by a quantity $-\Gamma^k(v, v) e_k$ which can be reasonably interpreted as a “force”. For example, one could consider the case that the spatial origin is a particle P which is orbiting around a point. If one believes that P really is “inertial”, then they will measure a fictitious force — the centrifugal force — acting on all objects. In general relativity, moreover, I think that the notion of “inertial” is ill-defined. In this case, if v is timelike then $\Gamma^k(v, v)$ is the acceleration due to gravity. In particular these fictitious forces all scale linearly with mass, because the geodesic equation does not have a mass factor and so we need to cancel out the factor of mass in the law $F = ma$ .

It will be convenient to go to another level of abstraction and view $\Gamma: T_pM \otimes T_pM \to T_pM$ as a quadratic form valued in the tangent space. In other words it is tempting to think of $\Gamma$ as a section of $T^*M \otimes T^*M \otimes TM$ . This of course presupposes that M has a trivial tangent bundle, since the Christoffel symbols are only defined locally. Putting our doubts aside, this is equivalent to thinking of $\Gamma$ as a section of $T^*M \otimes \text{End } TM$ .

Connections on G-bundles. Let me remind you that if G is a Lie group, then a G-bundle is a bundle of representations of G. Thus we can view quotients of G and its Lie algebra $\mathfrak g$ as both subsets of End E, whenever E is a G-bundle. By a gauge transformation of a G-bundle E one means a section of End E which is in fact a section of G. Thus gauge transformations act on E (and so also on End E, etc.)

If E is a G-bundle, by a covariant derivative on E I mean a covariant derivative whose Christoffel symbols $\Gamma$ are not just sections of $T^*M \otimes \text{End } E$ but in fact are sections of $T^*M \otimes \mathfrak g$ . (Briefly, the Christoffel symbols are $\mathfrak g$ -valued 1-forms.) In this case, if we have two covariant derivatives D, D’ which lie in the same orbit of the gauge transformations, we call D, D’ gauge-equivalent. We tend to think of covariant derivatives of G-bundles (modulo gauge-equivalence) as describing physical theories.

For example, consider the trivial U(1)-bundle E. This is the trivial line bundle equipped with the canonical action of U(1) on the complex numbers. A covariant derivative D on E is defined by locally giving Christoffel symbols which are $\mathfrak u(1)$ -valued 1-forms — in other words, imaginary 1-forms. A gauge transformation, then, is defined by adding an imaginary exact 1-form to the Christoffel symbols. We interpret the Christoffel symbols A as (i times) potentials for the electromagnetic field. In fact, one can take the exterior derivative of A and obtain a closed 2-form $F = dA$ , which one can view as the Faraday tensor. The fact that one can add an exact 1-form to A is exactly the gauge invariance of the Maxwell equation $*d*dA = j$ where j is the current 1-form.

So what is D in the case of electromagnetism? It acts on sections as $D_i = \partial_i + A_i$ . So for a function u (i.e. a section of the trivial bundle E) on M, $(D_i - \partial_i)u$ weights u according to the strength of the electromagnetic potential. This is mainly interesting when u is a constant function, in which case $Du = uA$ is the potential rescaled by u.

I think that the takeaway here is: the Christoffel symbols are a fictitious and local V-valued 1-form, where V is some vector bundle ( $V = \mathfrak g$ or $V = TM \otimes T^*M$ above). In any particular case they should have a nice physical interpretation but I don’t think one can interpret the Maxwell-Yang-Mills case and the Levi-Civita case as one and the same.