 3+1 Noether-current conservation law

For $J^{\mu }=Q{n}^{\mu }+Q^{\mu }$ 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.