We show that perturbatively small higher-derivative corrections to the Einstein-Hilbert action can lead to order-one modifications of the quasinormal mode spectrum of near-extremal Kerr black holes. The spectrum of such black holes contains zero-damping modes (ZDMs) and damped modes (DMs), with the latter only existing when the ratio $\mu =m/(l+1/2)$ is below a critical value ${\overline{\mu }}_{\mathrm{cr}}\approx 0.744$. Thus, this value represents a “phase boundary” that separates a region with both ZDMs and DMs and a region with only ZDMs. We find that the modes lying close to the phase boundary are very sensitive to modifications of general relativity, as their lifetimes receive corrections inversely proportional to their distance to the boundary. We link this growth of the corrections to a modification of the critical point ${\overline{\mu }}_{\mathrm{cr}}$, which can lead to a change in the number of DMs and produce order-one effects in the spectrum. We show that these large effects can take place in a regime in which the higher-derivative expansion remains under control. We also perform an exact analysis of the modification of the phase boundary for lower $(l,m)$ modes and pinpoint those that are most sensitive to corrections. Our results indicate that spectroscopy of highly rotating black holes is by far the most powerful way to search for new physics in ringdown signals.

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• Alternative gravity theories
• Classical black holes
• General relativity
• Gravitational waves