Solution
= Solution
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.