Boundedness and decay for the Teukolsky equation of spin ±1 on Reissner–Nordström spacetime: the spherical mode

Abstract

We prove boundedness and polynomial decay statements for solutions to the spin 1 Teukolsky-type equation projected to the spherical harmonic on Reissner–Nordström 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 (Dafermos et al 2019 Acta Math. 222 1–214). The derived quantity verifies a Fackerell–Ipser-type equation, with right hand side vanishing at the spherical harmonics. The boundedness and decay for the projection to the spherical harmonics are implied by the boundedness and decay for the Teukolsky system of spin 2 obtained in Giorgi (2018 (arXiv:1811.03526)).The spin 1 Teukolsky-type equation is verified by the curvature and electromagnetic components of a gravitational and electromagnetic perturbation of the Reissner–Nordström spacetime. Consequently, together with the estimates obtained in Giorgi (2018 (arXiv:1811.03526)), these bounds allow to prove the full linear stability of Reissner–Nordström metric for small charge to coupled gravitational and electromagnetic perturbations.