# Christoffel Symbol and Spin Connection

I'm currently studying general relativity from Sean Carroll's book Spacetime and Geometry: An Introduction to General Relativity. In this book I found that, in a simple way, Christoffel symbol is a connection coefficient which appears because of the Levi-Civita connection (the covariant derivative). So, Christoffel symbol is not a connection. However, I also read in the appendix about the non-coordinate bases (tetrad), that there is a new object called spin connection, which also appears because of the covariant derivative with respect to the Lorentzian indices. This object is a connection.

My question is, why Christoffel symbol is not a connection, but spin connection is a connection, even though those objects come from the similar definition of covariant derivative?

• This question (v3) seems to (at least partially) boil down to different semantic conventions: Some authors would call, say, the $\Gamma$ symbol or the $\omega$ symbol for a connection and others would call $\nabla$ a connection. Commented May 23, 2018 at 5:59

For example, if $T$ is a type (2,0) tensor field, then it is common to refer to its components $T^{\mu\nu}$ as a tensor field.
In essence, the connection coefficients are the components of a connection. So it is also customary to refer to $\Gamma^\rho_{\mu\nu}$ or $\omega^{\ a}_{\mu\ b}$ as a connection.
Do note that the exact definition of the object themselves are subject to convention. There are authors who refer to different things as a connection, than the $\nabla$ covariant derivative. In fact, there are general connections that do not admit a representation as a covariant derivative. In general, a connection is always a rule to define parallel transport of locally defined geometric objects along curves. But the exact way of defining them differ from author to author.
Also, the Levi-Civita connection and the "spin connection" are essentially the same. The Christoffel symbols $\Gamma^\rho_{\mu\nu}$ are the components of $\nabla$ when taken with respect to a holonomic (coordinate-) frame, while the "spin connection" coefficients $\omega^{\ a}_{\mu\ b}$ are the components of $\nabla$ when taken with respect to an orthonormal frame.