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=Q²/M²>1, where 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 Q², while for the standard RN metric Q² is always non-negative. We found that for $q>9/8$, black hole shadows disappear. Significant tidal charges q=-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≈1 provides a better fit of recent observational data for the black hole at the Galactic Center in comparison with the Schwarzschild black hole.