Can Gravity Be Understood as a Gauge Theory Through Rovelli's LQG Approach?

[Klaus_Hoffmann]

GR as a Gauge theory ??

don't know if this is true or not, but i have been reading books by ROvelli (LQG) or 'Gauge theories' the question is could we study Gravity as the set of functions $$A_{\mu}^{I}(x)$$

Then we write the Einstein Lagrangian (or similar) as:

$$\mathcal L = F_{ab}^{I}F^{I}_{ab}$$ (sum over I=0,1,2,3)

$$F_{ab}= \partial _{a}A^{I}_{b}-\partial _{b}A^{I}_{a}-\Gamma_{jk}^{i}A_{j}^{I}(x) A_{k}^{I}(x)$$

I think Rovellli in his LQG theory used this representation... then $$(\partial_{0}A_{\mu}^{I}$$ is the Kinetic part of Lagrangian and

$$dA_{\mu}^{I}$$ (d- exterior derivative) represents the potential.

then how would it read the Einstein Field equation and the Riemann or similar tensors ??

 

[Demystifier]

Yes, you can formulate gravity in that way. However, then the Lagrangian is not quadratic in F, but linear in F. In addition, you have one additional independent field - the tetrad (corresponding to the metric tensor itself), which does not have an analog in Yang-Mills theories. Having two independent fields, you obtain two set of equations of motion. One is the Einstein equation, while the other is a relation between the connection A and the metric derivatives. In the absence of matter, this relation is the same as in GR. In the case of matter with spin, the connection gets additional terms, describing geometry with torsion. This is the so-called Einstein-Cartan theory of gravity.