Mass, charge, and distance to Reissner-Nordstr\"{o}m black hole in terms of directly measurable quantities

Abstract

In this paper, we employ a general relativistic formalism and develop new theoretical tools that allow us to analytically express the mass and electric charge of the Reissner-Nordstr\"{o}m black hole as well as its distance to a distant observer in terms of few directly observable quantities, such as the total frequency shift, aperture angle of the telescope, and redshift rapidity. Our analytic and concise formulas are valid on the midline, and the redshift rapidity is a relativistic invariant observable that represents the evolution of the frequency shift with respect to the proper time in the Reissner-Nordstr\"{o}m spacetime. This procedure is applicable for particles undergoing circular motion around a spherically symmetric and electrically charged black hole, which is the case for accretion disks orbiting supermassive black holes hosted at the core of active galactic nuclei. Although this type of black hole is expected to be electrically neutral, our results provide a novel method to measure the electric charge of the Reissner-Nordstr\"{o}m black hole, hence can shed some light on this claim. Besides, these results allow us to measure the mass of the black hole and its distance from the Earth, and we can employ the general formulas in black hole parameter estimation studies.