The four independent <Killing vector fields> are the stationary field $\partial_t$ and the three generators of spatial rotations on the two-spheres. In the Schwarzschild interior, $f(r)<0$, so
$$
g(\partial_t,\partial_t)=-f(r)>0.
$$
Every rotational Killing field is tangent to the positive-definite round-sphere metric. All four fields are tangent to a surface of constant $r$, whose induced metric
$$
(-f)dt^2+r^2d\Omega^2
$$
is positive definite. Consequently every nonzero linear combination of the Killing fields is spacelike wherever it does not vanish. A pure rotational Killing field can vanish on its rotation axis, but it is nowhere timelike.

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Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 3 / a / i / Solution

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