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 χ ² fitting of the PPN model with the observational data. A new finding through our PPN model is the detectability of the gravitational lens effect on the null geodesics connecting S0-2 and the observer under the present observational uncertainties, if a PPN parameter is about one order larger than the value for the general relativity case. On the other hand, the effect of black hole spin on 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 χ ², implies that the gravitational field of Sgr A\, is not of the 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 χ ² and the χ ² of the Schwarzschild case is not large enough to ensure the statistical significance of the non-Schwarzschild result. A more precise statistical analysis than χ ² statistics is necessary to extract statistically significant information on the gravitational field of Sgr A\, from the present observational data. We will report a result by a Bayesian analysis in our next paper.