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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-Maxwellequations around a Reissner-Nordstr\"om (A)dS black hole were analyzed. Here wecomplete the proof of the nonmodal linear stability of this spacetime byanalyzing the even sector of the linear perturbations. We show that all thegauge invariant information in the metric and Maxwell field even perturbationsis encoded in two spacetime scalars: ${\mathcal S}$, which is a gauge invariantcombination 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 Weyltensor, $F_{\alpha \beta}$ the Maxwell field and $\delta$ means first ordervariation. We prove that $\mathcal{S}$ and $\mathcal{T}$ are are in one-onecorrespondence with gauge classes of even linear perturbations, and that thelinearized Einstein-Maxwell equations imply that these scalar fields arepointwise bounded on the outer static region.

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