Tests of gravity theories with Galactic Center observations

An active stage of relativistic astrophysics started in 1963 since in this year, quasars were discovered, Kerr solution has been found and the first Texas Symposium on Relativistic Astrophysics was organized in Dallas. Five years later, in 1967--1968 pulsars were discovered and their model as rotating neutron stars has been proposed, meanwhile J. A. Wheeler claimed that Kerr and Schwarzschild vacuum solutions of Einstein equations provide an efficient approach for astronomical objects with different masses. Wheeler suggested to call these objects black holes. Neutron stars were observed in different spectral band of electromagnetic radiation. In addition, a neutrino signal has been found for SN1987A. Therefore, multi-messenger astronomy demonstrated its efficiency for decades even before observations of the first gravitational radiation sources. However, usually, one has only manifestations of black holes in a weak gravitational field limit and sometimes a model with a black hole could be substituted with an alternative approach which very often looks much less natural, however, it is necessary to find observational evidences to reject such an alternative model. After two observational runs the LIGO-- Virgo collaboration provided a confirmation for an presence of mergers for ten binary black holes and one binary neutron star system where gravitational wave signals were found. In addition, in last years a remarkable progress has been reached in a development of observational facilities to investigate a gravitational potential, for instance, a number of telescopes operating in the Event Horizon Telescope network is increasing and accuracy of a shadow reconstruction near the Galactic Center is improving, meanwhile largest VLT, Keck telescopes with adaptive optics and especially, GRAVITY facilities observe bright IR stars at the Galactic Center with a perfecting accuracy.