Parametrized-Post-Newtonian Test of Black Hole Spacetime for Galactic Center Massive Black Hole Sgr A*: Formulation and $\chi^2$ Fitting

TL;DR

This study uses a parametrized post-Newtonian model to analyze the spacetime around Sgr A* with star S0-2 data, detecting potential lensing effects and constraining deviations from general relativity.

Contribution

It introduces a PPN framework for black hole spacetime testing using observational data and reports initial constraints on deviations from Schwarzschild geometry.

Findings

Detectability of gravitational lensing effect if PPN parameter exceeds GR value by an order.

Black hole spin effects on S0-2's orbit are not detectable with current data.

Current data do not statistically confirm deviations from Schwarzschild metric.

Abstract

We have performed a parametrized post-Newtonian (PPN) test of a black hole spacetime using observational data of the star S0-2/S2 orbiting the massive black hole at our galactic center Sgr A*. After introducing our PPN model of black hole spacetime, we report the result of $\chi^2$ fitting of the PPN model with the observational data. A new finding through our PPN model is a detectability of the gravitational lens effect on the null geodesics connecting S0-2 and observer under the present observational uncertainties, if a PPN parameter is about one order larger than the value for general relativity case. On the other hand, the effect of black hole spin on the S0-2's motion is not detectable. Thus our present PPN test is performed with spherically symmetric vacuum black hole spacetime. The resultant value of the PPN parameter, which corresponds to the minimum $\chi^2$, implies that the gravitational field of Sgr A* is not of Schwarzschild metric or that there exists a sufficient amount of dark matters around Sgr A* to be detected by present telescopes. However, the difference between the minimum $\chi^2$ and the $\chi^2$ of Schwarzschild case is not large enough to ensure a statistical significance of non-Schwarzschild result. A more precise statistical analysis than $\chi^2$ statistics is necessary to extract a statistically significant information of the gravitational field of Sgr A* from present observational data. We will report a result by a Bayesian analysis in next paper.