Constraints on a charge in the Reissner-Nordström metric for the black hole at the Galactic Center

Using an algebraic condition of vanishing discriminant for multiple roots of fourth-degree polynomials, we derive an analytical expression of a shadow size as a function of a charge in the Reissner-Nordstr\"om (RN) metric [1,2]. We consider shadows for negative tidal charges and charges corresponding to naked singularities $q={\mathcal{Q}}^{2}/{M}^{2}>1$, where $\mathcal{Q}$ and $M$ are black hole charge and mass, respectively, with the derived expression. An introduction of a negative tidal charge $q$ can describe black hole solutions in theories with extra dimensions, so following the approach we consider an opportunity to extend the RN metric to negative ${\mathcal{Q}}^{2}$, while for the standard RN metric ${\mathcal{Q}}^{2}$ is always non-negative. We found that for $q>9/8$, black hole shadows disappear. Significant tidal charges $q=\ensuremath{-}6.4$ (suggested by Bin-Nun [3--5]) are not consistent with observations of a minimal spot size at the Galactic Center observed in mm-band; moreover, these observations demonstrate that a Reissner-Nordstr\"om black hole with a significant charge $q\ensuremath{\approx}1$ provides a better fit of recent observational data for the black hole at the Galactic Center in comparison with the Schwarzschild black hole.

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