# Reference For This Tetrad Form of the Einstein-Hilbert Action?

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Is there a name of, theory for, and/or reference for the derivation of, the following form of the Einstein-Hilbert action in terms of a tetrad $e_{\mu} \, ^a$:
\begin{align}
S &= \int d^D x \, e R(e_{\mu} \, ^a, \omega_{\mu a} \, ^b (e)) \\
&= \int d^D x \, e (T_{ca,} \, ^a T^{cb,} \, _{b} - \frac{1}{2} T_{ab,c} T^{ac,b} - \frac{1}{4} T_{ab,c} C^{ab,c}),
\end{align}
where
$$T_{\mu \nu} \, ^a = \partial_{\mu} e_{\nu} \, ^a - \partial_{\nu} e_{\mu} \, ^a$$

and the spin connection is

$$\omega_{\mu b c} = \frac{1}{2}(e^{\rho} \, _b \partial_{\mu} e_{\rho c} - e^{\rho} \, _c \partial_{\mu} e_{\rho b}) - \frac{1}{2}(e^{\rho} \, _b \partial_{\rho} e_{\mu c} - e^{\rho} \, _c \partial_{\rho} e_{\mu b} ) \\ - \frac{1}{2}(e^{\lambda} \, _b e^{\rho} \, _c \partial_{\lambda} e_{\rho a} - e^{\lambda} \, _c e^{\rho} \, _b \partial_{\lambda} e_{\rho a})e_{\mu} \, ^a ?$$

Thank you.

Closed by author request
asked Jan 24, 2018 in Q&A
closed Jan 24, 2018