Christoffel Symbols - Gauge Fields

[tunafish]

Hi everyone! Two question for you ():

1) I know that General relativity may also be seen as a gauge theory, but which kind of gauge group is used there??

2) In the gauge theory wiew the Christoffel symbols $\Gamma^{\alpha}_{\mu\kappa}$ in the covariant derivative $$\nabla_{\mu}\vec{U}=\left(\frac{\partial U^{\alpha}}{\partial x^{\mu}}+U^{\kappa}\Gamma^{\alpha}_{\mu\kappa} \right)\vec{e}_{\alpha}$$ takes the role of the gauge fields, and so I (should) be able to express them in function of the generators of the Lie algebra, but what kind of Lie algebra am I supposed to use? And what are its generators??Thanks for your help! (first post!)

 

[henry_m]

Hi, and welcome to PF!
The clearest way to see GR as a gauge theory is to formulate it in terms of an orthonormal basis of vectors $e_\alpha$, satisfying $g(e_\alpha,e_\beta)=\eta_{\alpha\beta}$. [The components $e_\alpha^\mu$ are sometimes called vierbeins. I use Greek letters from the start of the alphabet to denote the basis vectors and components in terms of them, and from the middle of the alphabet to denote a coordinate basis.]

There is then a gauge symmetry in that we can take a Lorentz transformation independently at each point in spacetime. The gauge group is the Lorentz group.

The covariant derivative can then be expressed as
$$\nabla_\mu U=\nabla_\mu (U^\alpha e_\alpha)=(\partial_\mu U^\alpha)e_\alpha+U^\alpha \nabla_\mu e_\alpha=(\partial_\mu U^\alpha+\Gamma^\alpha_{\mu\beta}U^\beta)e_\alpha$$

Notice that there are a mixture of indices from the coordinates and the orthonormal basis in the Christoffel symbols: the $\alpha,\beta$ indices are very much like the internal, Lie algebra indices in an ordinary gauge theory, and Gamma the gauge fields.

 

[tunafish]

Thanks for your time henry_m, I didn't expect a rensponse so fast!

You obviously say that since the coordinate transformation (an consequentely the change of basis they induce on the manifold) does not affect any phenomena (physic is the same for all observer) the lorentzian group, the group of all the coordinate transormation is the gauge group for GR.

It makes sense: a coordinate transformation doesn't affect the lagrangian, the symmetry is rigid in a minkowsky spacetime and local on a curved manifold, so I need to define a covariant derivative (which "connects" all the tangent spaces by defining when a vector is parallel transported), which has the form I've written above.

In this wiew the Christoffel symbols are effectively gauge field.


All I've written above is what i thought to be true since a few days ago, when I red that the gauge group for GR is SL (of which I've never heard before), which has something to do whit spinors.

How is that?