Spin Connection - Definition

Definition

Let us first introduce the local Lorentz frame fields or vierbein (also known as a tetrad), this is basically four orthogonal space time vector fields labeled by . Orthogonal meaning

where is the inverse matrix of is the spacetime metric and is the Minkowski metric. Here, capital letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices. The spacetime metric can be expressed by

which simply expresses that, when written in terms of the basis, is locally flat.

The spin connection defines a covariant derivative on generalized tensors. For example its actin on is

where is the affine connection. The connection is said to be compatible to the vierbein if it satisfies

The spin connection is then given by:

where we have introduced the dual-vierbein satisfying and . We expect that will also annihilate the Minkowski metric ,

This implies that the connection is anti-symmetric in its internal indices, .

By substituting the formula for the affine connection written in terms of the, the spin connection can be written entirely in terms of the ,

To directly solve the compatibility condition for the spin connection, one can use the same trick that was used to solve for the affine connection . First contract the compatibility condition to give

.

Then, do a cyclic permutation of the free indices and, and add and subtract the three resulting equations:

where we have used the definition . The solution for the spin connection is

.

From this we obtain the same formula as before.