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An asymptotically flat spacetime is a stationary spacetime if it admits a complete stationary Killing vector field $K$, with ${\mathcal{L}}_{K}g=0$, timelike near infinity and normalized there to a unit time translation. It need not remain timelike in an ergoregion of a stationary spacetime. It is axisymmetric if it admits an axial Killing vector field $\Phi$, whose flow is a rotational $U(1)$ action: its spacelike orbits away from the axis are closed with period $2\pi$, and it vanishes on the regular rotation axis. For a stationary axisymmetric spacetime, the generators can be chosen to commute, $[K,\Phi ]=0$. These are geometric symmetries, not merely time-independent or angle-independent coordinate labels.

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