Positive-expansion convention for extrinsic curvature ($K_{\mu \nu }=P^{\rho }{}_{\mu }{P}^{\sigma }{}_{\nu }{\nabla }_{\rho }{n}_{\sigma }$)

With a future unit timelike normal in signature $(-,+,+,+)$, this convention makes the extrinsic curvature trace equal to the normal congruence's expansion, $K={\nabla }_{\mu }{n}^{\mu }$. An expanding flat Friedmann-Lemaître-Robertson-Walker metric has $K_{ij}=a^{\prime }{\delta }_{ij}$ and $K=3a^{\prime }/a^2$. This is the opposite sign to the negative-expansion convention used in many 3+1 decomposition of spacetime formulas. Projection identities and momentum constraints must use the same convention throughout.