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Concept
Spin connection
Summary
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 connection, whereas the affine connection (and so the affine spin connection) may contain torsion.
Let 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
where is the spacetime metric and is the Minkowski metric. Here, Latin letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices. This simply expresses that , when written in terms of the basis , is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e. or . The Latin or "Lorentzian" vierbein indices can be raised or lowered by or respectively. For example, and
The torsion-free spin connection is given by
where are the Christoffel symbols. This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.
Note that using the gravitational covariant derivative of the contravariant vector . The spin connection may be written purely in terms of the vierbein field as
which by definition is anti-symmetric in its internal indices .
Official source
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