Toward the Gravitational Redshift Detection in NGC 4258 and the Estimation of Its Black Hole Mass-to-distance Ratio

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.