Back

This page is from following chapter 3 of 2013 Rovelli, Vidotto "Covariant Loop Quantum Gravity"

More on $e_I = {e^a}_I e_a$ for $e_a$ torsion-free coordinate basis (commutation-free)
...such that $e_I \cdot e_J = {e^a}_I {e^b}_J g_{ab} = \eta_{IJ}$
...so $(\Gamma_\eta)_{IJK} = -(\Gamma_\eta)_{KJI}$
...and $\omega_{IK} = (\Gamma_\eta)_{IJK} e^J$
...so $\omega_{IJ} = -\omega_{JI}$

Next comes ${\Omega^I}_J = d {\omega^I}_J + {\omega^I}_K \wedge {\omega^K}_J$
So long as we restriction to torsion-free, we have only one possible connection, the Levi-Civita connection, which we are re-parameterizing to be a Minkowski metric basis.

Next show that the Riemann curvature tensor ${R^a}_{bcd} = {e^a}_I {e_b}^J {\Omega^I}_{Jcd}$
(Seems intuitive if the connection of the reparameterized basis $\omega_{IJ} = (\Gamma_\eta)_{IaJ} e^a$ is just a linear transformation of the coordinate basis connection ${\Gamma^c}_{de})$
From there, $\Omega^{IJ} = \frac{1}{2} {e_a}^I {e_b}^J {R^{ab}}_{cd} e^c \wedge e^d$ (book doesn't have $\frac{1}{2}$)

$e^I \wedge e^J \wedge e^K \wedge e^L$
$= {e_a}^I dx^a \wedge {e_b}^J dx^b \wedge {e_c}^K dx^c \wedge {e_d}^L dx^d$
$= {e_a}^I {e_b}^J {e_c}^K {e_d}^L dx^a \wedge dx^b \wedge dx^c \wedge dx^d$
$= {e_a}^I {e_b}^J {e_c}^K {e_d}^L \cdot 4! \cdot dx^{[a} \otimes dx^b \otimes dx^c \otimes dx^{d]}$
$= {e_a}^I {e_b}^J {e_c}^K {e_d}^L \delta^{abcd}_{0123} dx^0 \otimes dx^1 \otimes dx^2 \otimes dx^3$
$= {e_a}^I {e_b}^J {e_c}^K {e_d}^L \epsilon^{abcd} dx^0 \otimes dx^1 \otimes dx^2 \otimes dx^3$
$= s |det(e)| dV$

Next we rewrite the Einstein-Hilbert action:
$S = \int R \sqrt{-g} dx^4$
... $S = \int e^I \wedge e^J \wedge F^{KL} \epsilon_{IJKL}$
$S = \int {e_a}^I e^a \wedge {e_b}^J e^b \wedge {F^{KL}}_{cd} e^c \wedge e^d \epsilon_{IJKL}$
$S = \int {e_a}^I {e_b}^J {F^{KL}}_{cd} \epsilon_{IJKL} e^a \wedge e^b \wedge e^c \wedge e^d$
$S = \int {e_a}^I {e_b}^J {F^{KL}}_{cd} \epsilon_{IJKL} dx^a \wedge dx^b \wedge dx^c \wedge dx^d$
$S = \int {R^{ef}}_{cd} \epsilon_{abef} dx^a \wedge dx^b \wedge dx^c \wedge dx^d$

Back