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

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 the existence of sufficient amount of dark matters around Sgr A*. 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.
Comment: 43 pages, 4 figures, Accepted for Publication in Prog.Theor.Exp.Phys