A <stationary axisymmetric spacetime> additionally admits an axial Killing field $m$ with closed orbits. The two symmetry generators commute,
$$
[K,m]=0,
$$
and preserve the black-hole exterior and horizon. Coordinates adapted to them make the metric independent of both $t$ and $\phi$.

Solution

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

A stationary axisymmetric spacetime has commuting stationary and axial Killing vector fields, with the axial field having closed spacelike orbits.


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