- #1

JosephButler

- 18

- 0

The spin connection comes from the local (gauge) Lorentz symmetry of how we orient vielbeins at each point in space, it contains a manifold index and two tangent space indices. The Christoffel connection comes from the (also local/gauge) diffeomorphism invariance of general relativity, carrying three manifold indices.

My question is how are these two connections related, or are they both independent degrees of freedom?

My background is in particle physics rather than relativity, so I prefer to think about these connections as gauge fields. In this case, I'm confused about the degrees of freedom that people work with when they do perturbative quantum gravity. Why is it that people work with perturbations on the metric rather than perturbations of the Christoffel symbol, which seems to be the "actual" gauge field? (I'm told this has something to do with the Palatini formalism which connects the two?) Further, why don't people treat the spin connection as a physical gauge field?

Thanks!

Joe