Within current observational uncertainties, almost all observed gravitational phenomena are consistent with General Relativity (GR), which postulates “Local position invariance” (LPI). However, various modified theories of gravity predict the violation of LPI in extreme regimes, such as in strong gravity. Recently, Do et al. (2019) has shown that a spectroscopic monitoring of stars in the vicinity of a supermassive black hole (SMBH) is a powerful tool to study GR in a strong gravity. If the LPI violation is found around a SMBH, it will be an unambiguous signal of new physics. In order to test LPI in the strong gravity, we propose spectroscopic monitoring observations of a star S24 orbiting the Galactic SMBH. Observations of S24 in 2024 is crucial for this study, because it will pass the bottom of the gravitational potential well in 2024. The key quantity of the LPI test is the difference of the gravitational potential of S24 from 2024 (pericenter passage) and beyond, which is 6 orders of magnitude larger than that accessible in ground experiments. The expected uncertainty in our LPI measurements is ∼ 10−4, more than 100 times better than the previous tests around the SMBH. Therefore this will be the most precise LPI test ever performed, in a strong gravity regime.
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