Non Linear Perturbations of Kerr Spacetime in External Regions and the Peeling Decay

We prove, outside the influence region of a ball of radius R ₀ centred in the origin of the initial data hypersurface, Σ₀, the existence of global solutions near to Kerr spacetime, provided that the initial data are sufficiently near to those of Kerr. This external region is the “far” part of the outer region of the perturbed Kerr spacetime. Moreover, if we assume that the corrections to the Kerr metric decay sufficiently fast, o(r ⁻³), we prove that the various null components of the Riemann tensor decay in agreement with the “Peeling theorem”.