Parametrized-Post-Newtonian Test of Black Hole Spacetime for Galactic Center Massive Black Hole Sgr A* : Formulation and χ2 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 χ2 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 χ2, 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 χ2 and the χ2 of Schwarzschild case is not large enough to ensure a statistical significance of non-Schwarzschild result. A more precise statistical analysis than χ2 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.