= Vacuum Weyl tensors from hypersurface data
{title2=$E_{ab},\ B_{ab}$}
For a Ricci-flat spacetime with the stated <Gauss–Codazzi equations for a spatial hypersurface> and induced orientation $\tilde\epsilon_{abc}=n^d\epsilon_{dabc}$, the <electric part of the Weyl tensor> and <magnetic part of the Weyl tensor> satisfy
$$
E_{ab}=\mathcal R_{ab}+KK_{ab}-K_a{}^cK_{cb},\qquad B_{ab}=\tilde\epsilon_a{}^{cd}D_cK_{db}.
$$
The electric relation comes from the vanishing spatial projection of the <Ricci tensor>. The magnetic relation combines the normal curvature projection with the sign $\epsilon_{acde}n^c=-\tilde\epsilon_{ade}$.
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