Strange Tetrad Form of Einstein-Hilbert Action

[bolbteppa]

I have seen it the claimed that the Einstein-Hilbert action can be written in terms of a tetrad $e_{\mu} \, ^a$ as
\begin{align}
S &= \int d^n x \, e R(e_{\mu} \, ^a, \omega_{\mu a} \, ^b (e)) \\
&= \int d^n x \, e (T_{ca} \, ^a T^{cb} \, _{b} - \frac{1}{2} T_{ab \ c} T^{ac \ b} - \frac{1}{4} T_{ab \ c} T^{ab \ c}),
\end{align}
where
$$T_{\mu \nu} \, ^a = \partial_{\mu} e_{\nu} \, ^a - \partial_{\nu} e_{\mu} \, ^a$$

and the spin connection is

$$ \omega_{\mu b c} = \frac{1}{2}(e^{\rho} \, _b \partial_{\mu} e_{\rho c} - e^{\rho} \, _c \partial_{\mu} e_{\rho b}) - \frac{1}{2}(e^{\rho} \, _b \partial_{\rho} e_{\mu c} - e^{\rho} \, _c \partial_{\rho} e_{\mu b} ) \\ - \frac{1}{2}(e^{\lambda} \, _b e^{\rho} \, _c \partial_{\lambda} e_{\rho a} - e^{\lambda} \, _c e^{\rho} \, _b \partial_{\lambda} e_{\rho a})e_{\mu} \, ^a ?$$
I have never seen this form of the action before, and have not studied much on tetrad's yet, is there a reference to a derivation/explanation of this form of the action which seems nice and quadratic?

(Edit, fixed initial mistake in action, thank you).

Edit 2, apparently it's a linearized version of teleparallel gravity https://arxiv.org/pdf/hep-th/0304067.pdf

 

[dextercioby]

Who is the C tensor/tensor-like object?

 

[haushofer]

Maybe my insight helps a little bit:

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

Also, check out Samtleben's Introduction to Supergravity, which has a very clear exposure of the vielbein formalism and 1st vs 2nd order formalism.

I haven't checked your second equation, but usually one rewrites the Riemann tensor in terms of the spin connection curvature, and then constructs the Ricci scalar out of it. Your T seems to be the linearized curvature of the Vielbein (the "torsion"), such that the spin connection contributions are higher order and drop out. I'm a bit puzzled by your action (2), but probably this is because it's a linearization.

The spin connection is usually solved by the constraint that the vielbein curvature (the "torsion") vanishes. In the first order formalism this is done by varying the action wrt the spin connection.