Boundedness and decay for the Teukolsky system of spin $\pm2$ on Reissner-Nordstr\"om spacetime: the case $|Q| \ll M$

We prove boundedness and polynomial decay statements for solutions to the spin $\pm2$ generalized Teukolsky system on a Reissner-Nordstr\"om background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $\alpha$. The second equation, coupled with the first one, is a new equation for a new gauge-invariant quantity involving the electromagnetic curvature components. The proof is based on the use of derived quantities, introduced in previous works on linear stability of Schwarzschild. These quantities verify a generalized coupled Regge-Wheeler system. These equations are the ones verified by the extreme null curvature and electromagnetic components of a gravitational and electromagnetic perturbation of the Reissner-Nordstr\"om spacetime. Consequently, as in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of Reissner-Nordstr\"om metric for small charge to coupled gravitational and electromagnetic perturbations.