Slow rotation black hole perturbation theory

In this paper, we present a detailed analysis of first-order perturbations of the Kerr metric in the slow-rotation limit. We perform the calculation by perturbing the Schwarzschild metric plus up to second-order corrections in the spin in the Regge-Wheeler gauge. The apparent coupling between different angular momentum axial-led and polar-led modes can be removed by suitably combining the perturbation equations and projecting them onto spin-weighted spherical harmonics. In this way, we derive the corrections to the Regge-Wheeler and the Zerilli equations up to second-order in the spin. We show that 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 tortoise coordinate. The isospectrality with slow-rotating Teukolsky equation is also verified. We discuss the main implication of this result for the problem of vacuum metric reconstruction, providing the transformation rule between slow-spinning Teukolsky variables and metric perturbations. The existence of this relation leaves us with the conjecture that a resummation of the expansion in the spin is possible, leading to two decoupled differential equations for perturbations of the Kerr metric.