= Spatial projector derivative identity
{title2=$P^\mu{}_\rho P^\nu{}_\sigma\nabla_\nu P^\lambda{}_\mu=-n^\lambda K_{\sigma\rho}$}
Differentiating the <spatial projection tensor> and projecting its derivative and covariant indices leaves $-n^\lambda K_{\sigma\rho}$, where $K$ is the <extrinsic curvature of a spatial hypersurface>. Contracting $\lambda$ with $\rho$ gives zero because $K$ is transverse. This distinction separates the informative tensor identity, with a free normal index, from its vanishing trace. <Hypersurface orthogonality implies symmetric extrinsic curvature>.
Back to article page
