3+1 Noether-current conservation law
= 3+1 Noether-current conservation law
For $J^\mu=Qn^\mu+Q^\mu$ with spatial $Q^\mu$, current conservation gives $n^\mu\nabla_\mu Q=KQ-D_\mu Q^\mu-Q^\mu D_\mu\log\alpha$ under the stated extrinsic-curvature convention.