Spin connection

In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations.

Let ${\displaystyle e_{\mu }^{\ a}}$ be the local Lorentz frame fields or vierbein (also known as a tetrad), which is a set of orthogonal space time vector fields that diagonalize the metric tensor

where ${\displaystyle g_{\mu \nu }}$ is the spacetime metric and ${\displaystyle \eta _{ab}}$ is the Minkowski metric. Here, Latin letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices. This simply expresses that ${\displaystyle g_{\mu \nu }}$, when written in terms of the basis ${\displaystyle e_{\mu }^{\ a}}$, is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e. ${\displaystyle g^{\mu \nu }}$ or ${\displaystyle g_{\mu \nu }}$. The Latin or "Lorentzian" vierbein indices can be raised or lowered by ${\displaystyle \eta ^{ab}}$ or ${\displaystyle \eta _{ab}}$ respectively. For example, ${\displaystyle e^{\mu a}=g^{\mu \nu }e_{\nu }^{\ a}}$ and ${\displaystyle e_{\nu a}=\eta _{ab}e_{\nu }^{\ b}}$

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