Mass and spin of Kerr black holes in terms of observational quantities: The dragging effect on the redshift

TL;DR

This paper develops a formalism to determine Kerr black hole mass and spin from observational redshift data, accounting for gravitational dragging effects and orbital geometry, enabling improved parameter estimation in astrophysical systems.

Contribution

It provides closed-form formulas linking observable redshifts and orbital parameters to Kerr black hole characteristics, including effects of black hole rotation and motion.

Findings

Derived explicit formulas for mass and spin from redshift observations.

Incorporated gravitational dragging effects into redshift and light bending calculations.

Applicable to astrophysical systems like megamaser disks around supermassive black holes.

Abstract

In this work, we elaborate on the development of a general relativistic formalism that allows one to analytically express the mass and spin parameters of the Kerr black hole in terms of observational data: the total redshift and blueshift of photons emitted by geodesic massive particles revolving the black hole and their orbital parameters. Thus, we present concise closed formulas for the mass and spin parameters of the Kerr black hole in terms of few directly observed quantities in the case of equatorial circular orbits either when the black hole is static or is moving with respect to a distant observer. Furthermore, we incorporate the gravitational dragging effect generated by the rotating nature of the Kerr black hole into the analysis and elucidate its non-trivial contribution to the expression for the light bending parameter and the frequency shifts of photons emitted by orbiting particles that renders simple symmetric expressions for the kinematic redshift and blueshift. We also incorporate the dependency of the frequency shift on the azimuthal angle, a fact that allows one to express the total redshift/blueshift along any point of the orbit of the revolving particle for the cases when the black hole is both static or moving with respect to us. These formulas allow one to compute the Kerr black hole parameters by applying this general relativistic formalism to astrophysical systems like the megamaser accretion disks orbiting supermassive black holes at the core of active galactic nuclei. Our results open a new window to implement parameter estimation studies to constrain black hole variables, and they can be generalized to black hole solutions beyond Einstein gravity.