An active stage of relativistic astrophysics started in 1963 since in this year, quasars were discovered, Kerr solution had 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 (NSs) had been proposed, meanwhile 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. NSs were observed in different spectral band of electromagnetic radiation. In addition, a neutrino signal had 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. At the moment, only few astronomical signatures for strong gravitational field are found, including a shape of relativistic iron [Formula: see text] line, size and shape of shadows near black holes at the Galactic Center (GC) and M87, trajectories of bright stars near the GC. After two observational runs, the LIGO–Virgo collaboration provided a confirmation for a presence of mergers for 10 binary black holes and one binary NS system where gravitational wave signals were found. In addition, in the last years, a remarkable progress has been reached in a development of observational facilities to investigate a gravitational potential, for instance, the number of telescopes operating in the Event Horizon Telescope network is increasing and accuracy of a shadow reconstruction near the GC is improving, meanwhile largest VLT, Keck telescopes with adaptive optics and especially GRAVITY facilities observe bright IR stars at the GC with perfect accuracy. More options for precision observations of bright stars will be available with creating extremely large telescopes Thirty Meter Telescope (TMT) and E-ELT. It is clear that the GC (Sgr [Formula: see text]) is a specific object for observations. Our solar system is located at a distance around 8 kpc from the GC. Earlier, theorists proposed a number of different models including exotic ones for GC such as boson star, fermion ball, neutrino ball, a cluster of NSs. Later, some of these models were ruled out or essentially constrained with consequent observations and theoretical considerations. Currently, a supermassive black hole with mass around [Formula: see text] is the most natural model for GC. Using results of observations for trajectories of bright stars in paper [A. F. Zakharov, P. Jovanović, D. Borka and V. B. Jovanović, J. Cosmol. Astropart. Phys. 05 (2016) 045] the authors got a graviton mass constraint which is comparable and consistent with constraints obtained recently by the LIGO–Virgo collaboration. Later, we consider opportunities to improve current graviton mass constraints with future observations of bright stars [A. F. Zakharov, P. Jovanović, D. Borka and V. B. Jovanović, J. Cosmol. Astropart. Phys. 04 (2018) 050]. Similarly, from an analysis of bright star trajectories, one could constrain a tidal charge which was predicted by a gravity theory with an additional dimension [A. F. Zakharov, Eur. Phys. J. C 78 (2018) 689].
A note on the linear stability of black holes in quadratic gravity
