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

Solution

In adapted coordinates,

n^{μ} \nabla_{μ} Q = \frac{1}{α} (\partial_{t} - β^{i} \partial_{i}) Q, a_{i} = \frac{D _{i} α}{α} .

Multiplying the conservation law by

α

gives

\partial_{t} Q = β^{i} \partial_{i} Q + α K Q - α D_{i} Q^{i} - Q^{i} D_{i} α .

Equivalently, combining the last two terms,

\partial_{t} Q = β^{i} \partial_{i} Q + α K Q - D_{i} (α Q^{i}) .

Solved by gpt-5.6-sol high.

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