Geometric meaning of spin connection

Question

A very short question: Does the spin connection that we encounter in General Relativity $$\omega_{\mu,ab}$$ have a geometric meaning? I am supposing it does because it comes from mathematical terms that take geometric parts on a manifold but I am not sure how to visualize that?

Answer 1

The spin connection has a geometric meaning - it is a connection associated to a particular non-coordinate frame, which diagonalizes the metric. Here's how:

Let $M$ be our spacetime. The tangent bundle $TM$ may be thought of as the associated bundle to an $\mathrm{SO}(n)$-principal bundle, where the $\mathrm{SO}(n)$ matrices represent ordered orthonormal bases at every point (every column vector is orthonormal to every other in such a matrix, which is the way in which it represents a basis).

The spin connection is now just a $\mathfrak{so}(n)$-valued connection 1-form $\omega$ on $TM$, which may locally be expanded as $$ \omega = \omega_\mu \mathrm{d}x^\mu = {{\omega_\mu}^a}_b {T^b}_a\mathrm{d}x^\mu$$ and the ${{\omega_\mu}^a}_b$ the the connection coefficients physicists usually deal with, and the ${T^b}_a$ are a basis for the $\mathfrak{so}(n)$ matrices, usually the simple antisymmetric matrices with two non-zero entries own would always write down.

Usually, we think of tangent vectors as being expanded as $v = v^\mu \partial_\mu$, so the natural basis at every point is given by the coordinates, which may be arbitrarily ugly. in particular, the metric is $g_{\mu\nu}$. We now want to (locally) change frames such that the metric becomes the standard diagonal metric $\eta_{\mu\nu}$ ¹ because that one is evidently easier to work with. Such a (local) change of frames is given by a linear invertible map $$ e : TM \to TM$$ which is given in components by ${e^a}_\mu$ with $b^a = e^a{}_\mu v^\mu$ for $v$ the components in the coordinate basis and $b$ the components in the diagonal basis. $e$ is called the vielbein. Since $TM$ carries the natural Levi-Civita connection given by the Christoffel symbols $\Gamma$, we get a connection on the bundle by $$ \omega = e \Gamma e^{-1} + e \mathrm{d}e^{-1}$$ or, in components, $$ {{\omega_\mu}^a}_b = {e^\nu}_b {\Gamma^\lambda}_{\mu\nu}{e^a}_\lambda - {e^\nu}_b \partial_\mu {e^a}_\nu$$ which is how one obtains the spin connection. We may think of the spin connection as describing the Levi-Civita connection in a "moving frame" whose motion is given by the vielbein such that the metric takes the simple form we are used to from Euclidean/Minkowski space.

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^{1$\mathrm{SO}(n)$ is the Riemannian, $\mathrm{SO}(1,n-1)$ the Lorentzian case, but there's not much of a difference in the description we have here.}