Black hole nonmodal linear stability: even perturbations in the Reissner-Nordstr\"om case

This paper is a companion of [Phys. Rev. D 95, 124041 (2017)] in which, following a program on black hole nonmodal linear stability initiated in Phys. Rev. Lett. 112 (2014) 191101, odd perturbations of the Einstein-Maxwell equations around a Reissner-Nordstr\"om (A)dS black hole were analyzed. Here we complete the proof of the nonmodal linear stability of this spacetime by analyzing the even sector of the linear perturbations. We show that all the gauge invariant information in the metric and Maxwell field even perturbations is encoded in two spacetime scalars: ${\mathcal S}$, which is a gauge invariant combination of $\delta (C_{\alpha \beta \gamma \epsilon}C^{\alpha \beta \gamma \epsilon})$ and $\delta (C_{\alpha \beta \gamma \delta} F_{\alpha \beta} F^{\gamma \delta})$, and ${\mathcal T}$, a gauge invariant combination of $\delta ( \nabla _\mu F_{\alpha \beta} \nabla^\mu F^{\alpha \beta })$ and $\delta ( \nabla_{\mu} C_{\alpha \beta \gamma \delta} \nabla^{\mu}C^{\alpha \beta \gamma \delta})$. Here $C_{\alpha \beta \gamma \delta}$ is the Weyl tensor, $F_{\alpha \beta}$ the Maxwell field and $\delta$ means first order variation. We prove that $\mathcal{S}$ and $\mathcal{T}$ are are in one-one correspondence with gauge classes of even linear perturbations, and that the linearized Einstein-Maxwell equations imply that these scalar fields are pointwise bounded on the outer static region.