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Adopting the notation of Teukolsky and Press, we derive the connectionrelation for asymptotic solutions of the massless Dirac equation on a Kerrbackground. We show that, unlike bosonic fields, the connection relation formassless Dirac fields (neutrino) provides a rigorous proof of mode stability.The same relation also implies that every incoming mode can be absorbed by theblack hole or there is no superradiance. Recent works on overspinning blackholes have shown that this can lead to the formation of naked singularities. Weargue that the fact that both the mode stability of the black hole underneutrino perturbations and the instability of the event horizon (therefore theinstability of the black hole) can be derived from the same connection relationleads to a paradox. In other words mode, stability implies event horizoninstability as far as neutrino perturbations are concerned.

Is there a mode stability paradox for neutrino perturbations of Kerr black holes?