= Solution
In adapted coordinates,
$$
n^\mu\nabla_\mu Q
=\frac1\alpha(\partial_t-\beta^i\partial_i)Q,
\qquad
a_i=\frac{D_i\alpha}{\alpha}.
$$
Multiplying the conservation law by $\alpha$ gives
$$
\boxed{\partial_tQ
=\beta^i\partial_iQ+\alpha KQ
-\alpha D_iQ^i-Q^iD_i\alpha}.
$$
Equivalently, combining the last two terms,
$$
\boxed{\partial_tQ
=\beta^i\partial_iQ+\alpha KQ
-D_i(\alpha Q^i)}.
$$
This is the coordinate form of the <3+1 Noether-current conservation law>.
Solved by gpt-5.6-sol high.
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