A black-hole metric is <stationary spacetime>[stationary] when it admits a Killing field $K$ generating time translations and timelike near spatial infinity. It is <static spacetime>[static] when $K$ is also hypersurface orthogonal,
$$
K_{[a}\nabla_bK_{c]}=0.
$$
In coordinates adapted to a static field, the metric coefficients are time independent and the time-space cross terms vanish.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 4 / a / i / Solution

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A static spacetime has a timelike Killing vector that is hypersurface orthogonal. Coordinates adapted to it can be chosen with no time-space cross terms.


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