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. The spin connection occurs in two common forms: the Levi-Civita spin connection, when it is derived from the Levi-Civita connection, and the affine spin connection, when it is obtained from the affine connection. The difference between the two of these is that the Levi-Civita connection is by definition the unique torsion-free and metric-preserving connection, whereas the affine connection (and so the affine spin connection) may contain torsion.
Definition Let e μ a {\displaystyle e_{\mu }^{\;\,a}} be the local Lorentz frame fields or vierbein (also known as a tetrad), which is a set of orthonormal space time vector fields that diagonalize the metric tensor
g μ ν = e μ a e ν b η a b , {\displaystyle g_{\mu \nu }=e_{\mu }^{\;\,a}e_{\nu }^{\;\,b}\eta _{ab},}
where g μ ν {\displaystyle g_{\mu \nu }} is the spacetime metric and η a b {\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 g μ ν {\displaystyle g_{\mu \nu }} , when written in terms of the basis e μ a {\displaystyle e_{\mu }^{\;\,a}} , is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e. g μ ν {\displaystyle g^{\mu \nu }} or g μ ν {\displaystyle g_{\mu \nu }} . The Latin or "Lorentzian" vierbein indices can be raised or lowered by η a b {\displaystyle \eta ^{ab}} or η a b {\displaystyle \eta _{ab}} respectively. For example, e μ a = g μ ν e ν a {\displaystyle e^{\mu a}=g^{\mu \nu }e_{\nu }^{\;\,a}} and e ν a = η a b e ν b {\displaystyle e_{\nu a}=\eta _{ab}e_{\nu }^{\;\,b}}
The torsion-free spin connection is given by
ω μ a b = e ν a Γ σ μ ν e σ b + e ν a ∂ μ e ν b = e ν a Γ σ μ ν e σ b − e ν b ∂ μ e ν a , {\displaystyle \omega _{\mu }^{\ ab}=e_{\nu }^{\ a}\Gamma _{\ \sigma \mu }^{\nu }e^{\sigma b}+e_{\nu }^{\ a}\partial _{\mu }e^{\nu b}=e_{\nu }^{\ a}\Gamma _{\ \sigma \mu }^{\nu }e^{\sigma b}-e^{\nu b}\partial _{\mu }e_{\nu }^{\ a},}
… excerpt ends here. Continue reading the full article.
