Kähler decoupling for Kerr perturbations

The Euclidean Kerr metric is conformal, in two distinct ways, to a Kahler metric, with conformal factors determined by the repeated eigenvalues of the two chiral halves of the Weyl curvature. An analogous statement holds in Lorentzian signature, where the conformally related metrics are complex but retain some key properties of the Euclidean Kahler geometry. In this paper we explain how this hidden geometry is related to the decoupling of fields on Kerr. The essential geometric ingredient is the decomposition of the bundle of self-dual 2-forms into three line subbundles preserved by the reduced Levi-Civita connection. The essential geometric ingredient is the decomposition of the bundle of self-dual 2-forms into three line subbundles preserved by the reduced Levi-Civita connection. Differential operators compatible with this structure preserve the three components rather than mix them. We make this relation precise in two complementary ways. First, for arbitrary spin weight k, we show by direct calculation on Kerr that the Teukolsky operator is related, by a similarity transformation and an overall rescaling, to a natural Laplace-type operator associated with the complex Kahler metric. This is an operator identity and does not by itself derive the corresponding equation from the field equations for an arbitrary spin. Second, for Maxwell fields we establish the complete geometric mechanism. Conformal invariance allows Maxwell's equations to be expressed using the Kahler metric, where they imply a Hodge-Laplacian equation for the self-dual field strength. The parallel Kahler decomposition makes this equation diagonal: its middle component gives the FI equation, while its two extremal components agree with the spin-1 Teukolsky equations. Thus Kahler geometry supplies the mechanism preventing the mixing of components when the physical field operator is compatible with the parallel decomposition; separability of the extremal equations is an additional property of the Teukolsky operators.