Constraints on a charge in the Reissner--Nordstr\"om metric for the black hole at the Galactic Center

Abstract

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 \cite{Reissner_16,Nordstrom_18}. 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 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 $q=-6.4$ (suggested by Bin-Nun (2010)) 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 $q \approx 1$ provides a better fit of recent observational data for the black hole at the Galactic Center in comparison with the Schwarzschild black hole.