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 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 $q=-6.4$ (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-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.

Astronomy & Astrophysics (A&A) is an international journal which publishes papers on all aspects of astronomy and astrophysics

Recently, gravitational microlensing has been investigated in the framework of the weak field limit of fourth order gravity theory. However, solar system data (i.e. planetary periods and light bending) can be used to put strong constraints on the parameters of this class of gravity theories. We find that these parameters must be very close to those corresponding to the Newtonian limit of the theory.