Scalar curvature of an umbilic vacuum hypersurface (${}^{(3)}R=6\epsilon {\lambda }^2$)

For a timelike unit normal with squared norm epsilon and vanishing ambient Ricci tensor, contraction of the Gauss equation for a nonnull hypersurface gives the displayed formula. In signature $(+---)$ the timelike normal has epsilon plus one, so the scalar of the induced negative-definite metric is nonnegative. In signature $(-+++)$ the positive-definite spatial metric instead has scalar $-6{\lambda }^2$. Replacing the induced metric by its negative reverses the scalar, so the inequality must not be transferred between these conventions without adjustment.

// Target: general-relativity.bigb