It is well-known that Reissner-Nordström black holes cannot support spatially regular configurations of minimally coupled massive scalar fields in their exterior regions. But can they do so in their interior regions? In the present compact paper we explicitly prove that the answer to this physically interesting question is ‘yes’. In particular, it is proved that the interior spacetimes of extremal Reissner-Nordström black holes can support static (zero-frequency) bound-state configurations of linearized massive scalar fields whose characteristic eigenfunctions are regular throughout the entire interval $r\in [0,r_{\mathrm{H}}]$. Interestingly, solving analytically the Klein-Gordon wave equation for the massive scalar fields in the interior region of a maximally charged eternal black hole, we identify a discrete set $\{{\mu }_n(r_{\mathrm{H}},l){\}}_{n=0}^{n=\infty }$ of field masses that characterize antiresonant modes of the composed black-hole-field system, a unique family of massive scalar fields that cannot be supported inside the extremal black hole (here $r_{\mathrm{H}}$ is the horizon radius of the black hole and $\{\mu ,l\}$ are, respectively, the proper mass of the scalar field and its angular harmonic index).

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• Gravitation