Within current observational uncertainties, almost all observed gravitational phenomena are consistent with General Relativity (GR), which postulates “Local position invariance” (LPI). However, the violation of LPI has been implied by observations of quasars, and various modified theories of gravity predict the violation of LPI in extreme regimes, such as in a cosmological scale and strong gravity. Recently, Do et al. (2019) and Saida et al. (2019) have 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 around the Galactic SMBH. The key quantity of the LPI test is the difference of the gravitational potential of S24 from 2023 to its pericenter passage and thereafter, 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 null test around the SMBH. Therefore this will be the most precise LPI null test ever performed, in a strong gravity regime.
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