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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 3 / g / Solution

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.


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 Incoming links: 3+1 Noether-current conservation law