Boundedness and Decay for the Teukolsky System in Kerr-Newman Spacetime I: The Case $|a|, |Q| \ll M$

We prove boundedness and polynomial decay statements for solutions of the Teukolsky system for electromagnetic-gravitational perturbations of a Kerr-Newman exterior background, with parameters satisfying $|a|, |Q| \ll M$. The identification and analysis of the Teukolsky system in Kerr-Newman has long been problematic due to the long-standing problem of coupling and failure of separability of the equations. Here, we analyze a system satisfied by novel gauge-invariant quantities representing gravitational and electromagnetic radiations for coupled perturbations of a Kerr-Newman black hole. The bounds are obtained by making use of a generalization of the Chandrasekhar transformation into a system of coupled generalized Regge-Wheeler equations derived in previous work. Crucial in our resolution is the use of a combined energy-momentum tensor for the system which exploits its symmetric structure, performing an effective decoupling of the perturbations. As for other black hole solutions, such bounds on the Teukolsky system provide the first step in proving the non linear stability of the Kerr-Newman metric to gravitational and electromagnetic perturbations.