This paper supplies missing proofs for two technical but important results about gravitational perturbations of Kerr black holes. The results were stated without proof in a recent landmark work that completes the Kerr stability conjecture. Here the author derives the precise form of the Teukolsky wave–transport system in a perturbed Kerr spacetime and proves the needed energy–Morawetz estimates for that system.

Teukolsky equations are the linear equations that govern certain components of the spacetime curvature that carry gravitational radiation. In a null frame they describe the so-called spin ±2 curvature components (often written A and "+A" in complex form). The paper works in a tensorial formalism adapted to a perturbed principal null pair and writes these curvature components as a family of tensors with spin weight s=±2.

The main technical construction is a coupled wave/transport hierarchy. The authors define tensors psi_s^(p) for p=0,1,2 and show that they satisfy transport equations along the null directions (schematically like e3 or e4 derivatives) and coupled wave equations. The wave operator that appears contains lower-order coupling terms. Concretely, the wave equations include rotation-dependent terms (for example a term proportional to the rotation parameter a times a time derivative) and linear couplings whose coefficients decay like powers of the radius r (schematically r^-3 plus smaller terms). The transport equations relate successive psi_s^(p) and contain source terms that are controlled by the geometry.

The second main result in the paper is a proof of energy–Morawetz estimates for these Teukolsky tensors on perturbations of Kerr with |a|<m (the subextremal range where the black hole has nondegenerate horizons). An energy–Morawetz estimate is an inequality that controls both the energy on slices and an integrated spacetime norm (a Morawetz or ‘‘local decay’’ term) of solutions. These estimates are a central analytic tool used in the recent global nonlinear stability proofs for Kerr, and the paper provides the detailed justification of the Teukolsky part of that analysis.