The Teukolsky–Starobinsky identities in type D vacuum backgrounds with cosmological constant

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Abstract
It is shown that in all the type D solutions to the Einstein vacuum field equations with cosmological constant, the one-variable functions appearing in the separable maximal spin-weight components of the neutrino, electromagnetic, and gravitational perturbations are related by certain differential operators and the corresponding proportionality constants are obtained. It is also shown that analogous relations hold in the case of perturbations by a Rarita–Schwinger field if the cosmological constant vanishes.