Slow rotation black hole perturbation theory

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