Separation of gravitational perturbations of singly spinning Myers-Perry black holes

Gravitational perturbations of rotating black holes in higher dimensions generally obey coupled partial differential equations. We show that scalar gravitational perturbations invariant under rotations of the transverse sphere of a singly spinning Myers-Perry black hole separate, despite the obstruction to a decoupled Teukolsky equation. Two commuting, second-order angular operators organise the three gauge-invariant curvature components into joint eigenfunctions. The physical radial problem is a $2\times 2$ first-order system, equivalent to a single second-order equation. This gives a separation of the sector containing both the bar-mode and axisymmetric ultraspinning instabilities. Using only ordinary differential equations, we compute the bar-mode quasinormal spectrum in $d=5,6,7$ and find instability onsets at $a\mathrm{/}{r}_{+}=0.90498$ and $0.83228$ in $d=6,7$. We also compute the growing axisymmetric modes and their onset rotations in $d=6,\dots ,9$. The construction shows how separability can survive the coupling between gravitational polarisations.