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.
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
The Greek vierbein indices can be raised or lowered by the metric, i.e.
The Latin or "Lorentzian" vierbein indices can be raised or lowered by
The spin connection may be written purely in terms of the vierbein field as[1]
In the Cartan formalism, the spin connection is used to define both torsion and curvature.
These are easiest to read by working with differential forms, as this hides some of the profusion of indexes.
The primary difference is that these retain the indexes on the vierbein, instead of completely hiding them.
More narrowly, the Cartan formalism is to be interpreted in its historical setting, as a generalization of the idea of an affine connection to a homogeneous space; it is not yet as general as the idea of a principal connection on a fiber bundle.
It serves as a suitable half-way point between the narrower setting in Riemannian geometry and the fully abstract fiber bundle setting, thus emphasizing the similarity to gauge theory.
for the orthonormal coordinates on the cotangent bundle, the affine spin connection one-form is
The first Bianchi identity is obtained by taking the exterior derivative of the torsion:
The covariant derivative for a generic differential form
It is easy to deduce by raising and lowering indices as needed that the frame fields defined by
This is also deduced by taking the gravitational covariant derivative
To directly solve the compatibility condition for the spin connection
The spin connection arises in the Dirac equation when expressed in the language of curved spacetime, see Dirac equation in curved spacetime.
This fact is utilized by employing tetrad fields describing a flat tangent space at every point of spacetime.
We wish to construct a generally covariant Dirac equation.
We have introduced local Lorentz transformations on flat tangent space generated by the
This means that the partial derivative of a spinor is no longer a genuine tensor.
and is a genuine tensor and Dirac's equation is rewritten as
The generally covariant fermion action couples fermions to gravity when added to the first order tetradic Palatini action,
The tetradic Palatini formulation of general relativity which is a first order formulation of the Einstein–Hilbert action where the tetrad and the spin connection are the basic independent variables.
In the 3+1 version of Palatini formulation, the information about the spatial metric,
and we obtain a formula similar to the one given above but for the spatial spin connection
The spatial spin connection appears in the definition of Ashtekar–Barbero variables which allows 3+1 general relativity to be rewritten as a special type of
as the configuration variable, the conjugate momentum is the densitized triad