Boundedness and decay for the Teukolsky equation of spin $\pm1$ on Reissner-Nordstr\"om spacetime: the $\ell=1$ spherical mode

We prove boundedness and polynomial decay statements for solutions to the spin $\pm1$ Teukolsky-type equation projected to the $\ell=1$ spherical harmonic on Reissner-Nordstr\"om spacetime. The equation is verified by a gauge-invariant quantity which we identify and which involves the electromagnetic and curvature tensor. This gives a first description in physical space of gauge-invariant quantities transporting the electromagnetic radiation in perturbations of a charged black hole. The proof is based on the use of derived quantities, introduced in previous works on linear stability of Schwarzschild by Dafermos-Holzegel-Rodnianski. The derived quantity verifies a Fackerell-Ipser-type equation, with right hand side vanishing at the $\ell=1$ spherical harmonics. The boundedness and decay for the projection to the $\ell\geq 2$ spherical harmonics are implied by the boundedness and decay for the Teukolsky system of spin $\pm2$ obtained in our previous work. The spin $\pm1$ Teukolsky-type equation is verified by the curvature and electromagnetic components of a gravitational and electromagnetic perturbation of the Reissner-Nordstr\"om spacetime. Consequently, together with the estimates obtained in our previous work, these bounds allow to prove the full linear stability of Reissner-Nordstr\"om metric for small charge to coupled gravitational and electromagnetic perturbations.