are local coordinates near $p$ with $x^3 \ge 0$ and $x^3 = 0$ on $\partial {\cal S} = \Sigma$. The fields $e_A$, $A = 1, 2$, are tangential to ${\cal S}_c = {\cal S} \cap \{x^3 = c =const. \ge 0\}$.

On ${\cal M} \sim [0, T[ \times {\cal S}$ the time-like unit vector field $e_0$ is future directed, tangential to ${\cal T} = [0, T[ \times \partial {\cal S}$, and orthogonal to $S_c$. The coordinate $x^0$ is a natural parameter on the integral curves of $e_0$ and the other coordinates are dragged along with $e_0$ so that $e_0(x^{\mu}) = \delta^{\mu}\,_0$. The fields $e_A$ are $D$-Fermi-transported in direction of $e_0$ on ${\cal T}_c = \{x^3 = c\}$, where $D$ denotes the covariant derivative defined by the metric induced on ${\cal T}_c$. The field $e_3$ is normal to ${\cal T}_c$ and inward pointing on ${\cal T} = {\cal T}_0$.

In this gauge we have $$e^{\mu}\,_0 = \delta^{\mu}\,_0, \quad 
D_a\,e_b = \Gamma_a\,^c\,_b\,e_c, \quad
\chi_{ab} \equiv \Gamma_a\,^3\,_b = \Gamma_b\,^3\,_a, \quad
\chi \equiv g^{ab}\,\chi_{ab}, \quad a, b, c = 0, 1, 2,$$ with $\chi_{ab}$ the second fundamental form and $\chi$ the mean extrinsic curvature on ${\cal T}_c$, $$\label{e0-eA-transport}
\Gamma_0\,^A\,_B = 0, \quad 
D_{e_0}\,e_0 = 