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Im 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?

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  • $\begingroup$ 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. $\endgroup$ – Qmechanic May 23 '18 at 5:59
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It is common in physics literature to identify invariant geometric objects with their components in some reference frame.

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.

The only reason why there is any significant difference between the two is that spinor fields can only be represented in orthonormal frames, so to define covariant derivatives of spinors, you need the "spin connection". But this is all just juggling with local component representations, the underlying invariant object is the same.

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