Gauss–Codazzi equations for a spatial hypersurface

With signature $(-,+,+,+)$, future unit normal $n$ and $K_{ab}=-P^c{}_a{P}^d{}_b{\nabla }_c{n}_d$, the Gauss–Codazzi equations become $(\perp R{)}_{abcd}={\mathcal{R}}_{abcd}+K_{ac}{K}_{db}-K_{ad}{K}_{cb}$ and $(\perp R{)}_{abcd}{n}^d=-D_a{K}_{bc}+D_b{K}_{ac}$, where in the second expression only the first three indices are projected. The intrinsic curvature and spatial covariant derivative use the spatial induced metric. The normal is timelike, so the normal-sign convention differs from a spacelike normal in a positive-definite ambient metric.

For a Ricci-flat spacetime with the stated Gauss–Codazzi equations for a spatial hypersurface and induced orientation ${\widetilde{ϵ}}_{abc}=n^d{ϵ}_{dabc}$, the electric part of the Weyl tensor and magnetic part of the Weyl tensor satisfyThe electric relation comes from the vanishing spatial projection of the Ricci tensor. The magnetic relation combines the normal curvature projection with the sign ${ϵ}_{acde}{n}^c=-{\widetilde{ϵ}}_{ade}$.