A differential-geometry approach to black hole characterization of megamaser systems in static spherically symmetric spacetimes

Abstract

We develop a geometry-first model that maps measured thin-disk water megamaser observables--sky angles, frequency shifts, their secular drifts and the angular redshift rate--to the black hole parameters in a generic static, spherically symmetric (SSS) spacetime written in the Schwarzschild gauge. The core of the approach is local: dot-product relations in the equatorial curved geometry relate the conserved light-deflection parameter to the observed detector angle at finite distance, providing a connection between sky positions and photon constants of motion. These local identities feed a closed model for the frequency shift of photons traveling between a maser clump circularly orbiting a black hole and a finite-distance detector, making explicit the dependence on the metric at emission and detection radii. We also apply the Gauss-Bonnet theorem to this construction on the equatorial two-manifold as an intrinsic cross-check. This theorem provides a global consistency relation between the local emission and detection angles, helping to validate sign conventions and angle branch choices in the local setup. In this sense, the local and global perspectives on the megamaser system support each other. To supplement the instantaneous information contained in frequency shifts, we incorporate the time-domain general relativistic invariant, the redshift rapidity. We further introduce a prospective angular-domain observable, the angular redshift rate, and give its analytic expression in the SSS framework. The results are formulated for generic SSS backgrounds, providing closed relations suited for likelihood-based inference from VLBI positions and spectral monitoring. In particular, for a Schwarzschild background, the black hole mass, its distance to Earth and megamaser orbital radius are fully constrained in the language of astrophysical observables.

Figures (2)

• Figure 1: Local equatorial geometry of a null ray connecting a maser clump at radius $r_e$ to a detector at distance $D$. The photon wave--vector $\vec{k}$ is decomposed into radial and azimuthal components $\vec{k}_r$ and $\vec{k}_\varphi$, and the angles $\alpha$ (local emission angle), $\Theta$ (local detection angle), and $\varphi_e$ (azimuthal separation between the maser and the line of sight, LOS) are defined in a local orthonormal basis adapted to the equatorial two--manifold $(N,\tilde{g})$. These conventions are used throughout the Gauss--Bonnet construction.
• Figure 2: Geometric configuration of the Gauss--Bonnet triangle $\triangle P_1P_2P_3$ on the equatorial two--manifold $(N,\tilde{g})$. The vertices are $P_1=(r_e,0)$, where the maser circle intersects the LOS, $P_3=(r_e,\varphi_e)$, the actual maser position, and $P_2=(D,0)$, the detector. The boundary consists of the maser circular arc $\overset{\frown}{P_3P_1}$, the projected null ray $\overset{\frown}{P_2P_3}$, and the radial LOS segment $\overset{\frown}{P_1P_2}$. The interior angles are $\gamma_1=\pi/2$ at $P_1$, $\gamma_2=\Theta$ at $P_2$, and $\gamma_3=\pi/2-\alpha$ at $P_3$.

Theorems & Definitions (1)

• Theorem 1: Gauss--Bonnet