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

Abstract

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 red and blue shifts of photons that are emitted by geodesic massive particles (stars and galactic gas) and travel along null geodesics towards a distant observer, and ii) the radius of these star/gas 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 red/blue shifts in terms of the metric parameters (mass and angular momentum per unit mass) and the orbital radii of both the emitter star (and/or galactic gas) and the distant observer.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 red/blue shifts 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 discs of rotating black holes, binary systems and active galactic nuclei, among others, to which this formalism can be applied. Keywords: Kerr black hole, black holes rotation curves, red/blue shifts, determination of mass, angular momentum and distance to the black hole.

Reconstruction of relative motion of a binary star in the vicinity of black hole by its redshift

“…The problem of the extracting of mentioned parameters from the function z 0 (t) is analogous to the problem of the extracting parameters of motion of a single source in external gravitational field. Some authors have proposed methods for the solution of such problem (see, e. g., [7], [1]). In the present work we consider only the inverse problem for the relative motion that is determined by function z r (t).…”

“…Here B (α) defined by (7). It must be calculated at arbitrary time τ that is belong to It is possible to find also orbital inclination:…”

“…Moreover, underlying mathematical models that are used in the works [1][2][3][4][5][6] enable to investigate the effects of strong gravitational field on the motion of the source and propagation of its radiation. The inverse problem for the source moving in the equatorial plane of a Kerr black hole was solved in [7] using the redshift of the source only. The case of more general motion of the source in gravitational field of non rotating supermassive black hole is considered in [3], but in the cited paper the redshift data as well as the radiation intensity both are used for the solution of inverse problem.…”

Null geodesics and observables around the Kerr–Sen black hole

“…For this purpose we have followed the approach as opted in [18] and [19]. The generalised expression for the frequency of a photon measured by an observer with proper 12) where the index C refers to the emission (e) and/or detection (d) at the corresponding spacetime point P C .…”

“…Various important issues such as properties of shadow recasted by KSBH [10], cosmic censorship conjecture in KSBH [11], Hawking-temperature and entropy of KSBH [12,13]are discussed in recent past. Further, the increasing amount of evidence that various galaxies contain a supermassive BH at their center [14][15][16][17], motivated researchers to develop a theoretical approach to obtain the mass and rotation parameter of a rotating BH in terms of the redshift z red and blueshift z blue of photons emitted by massive test particles travelling around them along geodesics and the radius of their orbits [18,19].…”