Reconstruction of a star motion in the vicinity of a Schwarzschild black hole by redshift of the spectrum

TL;DR

This paper presents an exact relativistic method to reconstruct a star's motion near a Schwarzschild black hole using redshift data, combining nonlinear equations and least squares for improved accuracy.

Contribution

It introduces a novel two-step approach for reconstructing stellar motion from redshift measurements, including solving nonlinear equations and applying least squares in a relativistic context.

Findings

Exact solution for redshift evolution near Schwarzschild black hole

Method for reconstructing star motion from redshift data

Numerical validation using Galactic Center star model

Abstract

We investigate the time evolution of the redshift of the spectrum of electromagnetic radiation that is emitted by the star that moves in the vicinity of supermassive black hole. For the case of Schwarzschild black hole we solve this problem exactly in General Theory of Relativity. Also we formulate an approach that gives possibilities to solve the inverse problem: reconstruction of motion of a star in the vicinity of black hole using the redshift data. This approach consists of two steps. The first step gives possibilities to find unique solution for the integrals of motion of the star by solution of a system of non-linear equations, rather than the direct application of standard statistical methods. For this purpose we consider properties of congruences of isotropic geodesics that connect of the worldline of the source and the worldline of the observer. The second step is the application of standard least squares method in order to obtain more accurate solution. This approach is used on the numerical model of the star in the vicinity of the Center of our Galaxy.