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

A black-hole metric is stationary when it admits a Killing field $K$ generating time translations and timelike near spatial infinity. It is static when $K$ is also hypersurface orthogonal,In coordinates adapted to a static field, the metric coefficients are time independent and the time-space cross terms vanish.

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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 $\varphi$.

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The Kerr black hole coefficients are independent of $t$ and $\varphi$, so ${\partial }_t$ and ${\partial }_{\varphi }$ are commuting Killing fields; the latter has closed circular orbits. Kerr is therefore stationary and axisymmetric.

Expanding the metric gives the nonzero cross componentFor $a\ne 0$ this cannot be removed throughout the exterior by a constant redefinition of $\varphi$, and the asymptotically timelike stationary Killing field has nonzero twist. It is not hypersurface orthogonal, so Kerr is not static.

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