Generally Covariant Dirac equation: The spin connection

Wikipedia, an answer on stackexchange and a few papers in the Arxiv I've found all have different definitions of the spin connection found in the Dirac equation. Can anyone please tell me what the correct definition is in terms of derivatives in local coordinates, tetrads and Christoffel symbols?

The one I've been using is from wikipedia:

$$\nabla_\mu - \partial_\mu = \frac{1}{8}\omega_\mu^{ab}[\gamma_a,\gamma_b]$$ But this answer appears very different.

As well does anyone know what the spin connection should be in the Schwarzchild metric?

• The link you provided is the one - dressed up to look like a Christoffel connection. I believe the Schwarzchild metric is a stationary metric, i.e., independent of time. – Cinaed Simson Apr 11 at 23:14