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

TL;DR

This paper derives an analytical expression for the shadow size of a Reissner--Nordström black hole as a function of charge, considering negative tidal charges and comparing with observational data from the Galactic Center.

Contribution

It introduces an analytical formula for black hole shadow size depending on charge, extending the Reissner--Nordström metric to negative charges and testing against observational data.

Findings

Black hole shadows vanish for charge q > 9/8.

Negative tidal charges like q=-6.4 are inconsistent with observations.

A charged black hole with q ≈ 1 fits Galactic Center data better than Schwarzschild.

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.