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 – Nordström ( 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 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 ( 3 ; 4 ; 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 – Nordström 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 .