Vacuum Weyl tensors from hypersurface data ($E_{ab},B_{ab}$)

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 ${ϵ}_{acde}{n}^c=-{\widetilde{ϵ}}_{ade}$.