Kerr black hole parameters in terms of the redshift/blueshift of photons emitted by geodesic particles

We are motivated by the recently reported dynamical evidence of stars with short orbital periods moving around the center of the Milky Way and the corresponding hypothesis about the existence of a supermassive black hole hosted at its center. In this paper we show how the mass and rotation parameters of a Kerr black hole (assuming that the putative supermassive black hole is of this type), as well as the distance that separates the black hole from the Earth, can be estimated in a relativistic way in terms of (i) the redshift and blueshift of photons that are emitted by geodesic massive particles (stars) and travel along null geodesics towards a distant observer (located at a finite distance), and (ii) the radius of these star orbits. As a concrete example and as a first step towards a full relativistic analysis of the above-mentioned star orbits around the center of our Galaxy, we consider stable equatorial circular orbits of stars and express their corresponding redshift/blueshift in terms of the metric parameters (mass and angular momentum per unit mass) and the orbital radii of both the emitter star and the distant observer. These radii are linked through the constants of motion along the null geodesics followed by photons since their emission until their detection, allowing us to get a closed expression for the orbital radius of the observer in terms of the emitter orbital radius, which is known from observations, and the black hole parameters M and a. In principle, these expressions allow one to statistically estimate the mass and rotation parameters of the Kerr black hole, and the radius of our orbit, through a Bayesian fitting, i.e., with the aid of observational data: the redshift/blueshift measured at certain points of stars' orbits and their radii, with their respective errors, a task that we hope to perform in the near future. We also point to several astrophysical phenomena, like accretion disks of rotating black holes, binary systems and active galactic nuclei, among others, to which this formalism can be applied.