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In this paper, we present a detailed analysis of first-order perturbations ofthe Kerr metric in the slow-rotation limit. We perform the calculation byperturbing the Schwarzschild metric plus up to second-order corrections in thespin in the Regge-Wheeler gauge. The apparent coupling between differentangular momentum axial-led and polar-led modes can be removed by suitablycombining the perturbation equations and projecting them onto spin-weightedspherical harmonics. In this way, we derive the corrections to theRegge-Wheeler and the Zerilli equations up to second-order in the spin. We showthat the two potentials remain isospectral as in the non-rotating limit.However, it is easy to demonstrate it only for a precise choice of the tortoisecoordinate. The isospectrality with slow-rotating Teukolsky equation is alsoverified. We discuss the main implication of this result for the problem ofvacuum metric reconstruction, providing the transformation rule betweenslow-spinning Teukolsky variables and metric perturbations. The existence ofthis relation leaves us with the conjecture that a resummation of the expansionin the spin is possible, leading to two decoupled differential equations forperturbations of the Kerr metric.

Slow rotation black hole perturbation theory