We construct from first principles a general relativistic approach to study Schwarzschild black hole (BH) rotation curves and estimate the mass-to-distance ratio of the active galactic nucleus of NGC 4258 in terms of astrophysical observable quantities. The presented method allows one to clearly distinguish and quantify the general and special relativistic contributions to the total redshift expression. The total relativistic redshift/blueshift comprises three components: the gravitational redshift due to the spacetime curvature generated by the mass of the BH in its vicinity, the kinematic shift, originated by the photons’ local Doppler effect, and the redshift due to a special relativistic boost that describes the motion of a galaxy from a distant observer. We apply our method to the largest data set of highly redshifted water megamaser measurements on the accretion disk of the NGC 4258 active galaxy and use this general relativistic method to estimate its BH mass-to-distance ratio: M/D = (0.5326 ± 0.00022) × 107 M ⊙ Mpc−1.

We derive closed formulas for the mass M and spin a parameters of a Kerr black hole in terms of a minimal quantity of observational data: the red-/blue-shifts of photons emitted by massive particles (stars) moving on geodesics around the black hole and their respective orbital radius. It turns out that for a single star orbiting the black hole, we need a minimal set of four observational measurements to analytically determine both parameters. Indeed, given a set of two (three) stars revolving around the black hole, these formulas involve just eight (twelve) observational data.

In this work we perform a Bayesian statistical fit to estimate the mass-to-distance ratio and the recessional redshift of 10 different black holes hosted at the centre of active galactic nuclei, namely the galaxies NGC 5765b, NGC 6323, UGC 3789, CGCG 074-064, ESO 558-G009, NGC 2960, NGC 6264, NGC 4388, J0437+2456 and NGC 2273. Our general relativistic method makes use of the positions in the sky and frequency shift observations of water megamasers circularly orbiting the central black hole on their accretion disks. This approach also allows us to quantify the gravitational redshift which is not considered in a Newtonian analysis. The gravitational redshift of the megamasers closest to the black hole is found to be within the range 1-6 km/s. The order of the fitted black hole masses corresponds to supermassive black holes and lies on the range 106 − 107M/odot;.