One of the most interesting astronomical objects is the Galactic Center. It is a subject of intensive astronomical observations in different spectral bands in recent years. We concentrate our discussion on a theoretical analysis of observational data of bright stars in the IR-band obtained with large telescopes. We also discuss the importance of VLBI observations of bright structures which could characterize the shadow at the Galactic Center. If we adopt general relativity (GR), there are a number of theoretical models for the Galactic Center, such as a cluster of neutron stars, boson stars, neutrino balls, etc. Some of these models were rejected or the range of their parameters is significantly constrained with consequent observations and theoretical analysis. In recent years, a number of alternative theories of gravity have been proposed because there are dark matter (DM) and dark energy (DE) problems. An alternative theory of gravity may be considered as one possible solution for such problems. Some of these theories have black hole solutions, while other theories have no such solutions. There are attempts to describe the Galactic Center with alternative theories of gravity and in this case one can constrain parameters of such theories with observational data for the Galactic Center. In particular, theories of massive gravity are intensively developing and theorists have overcome pathologies presented in the initial versions of these theories. In theories of massive gravity, a graviton is massive in contrast with GR where a graviton is massless. Now these theories are considered as an alternative to GR. For example, the LIGO–Virgo collaboration obtained the graviton mass constraint of about [Formula: see text] eV in their first publication about the discovery of the first gravitational wave detection event that resulted of the merger of two massive black holes. Surprisingly, one could obtain a consistent and comparable constraint of graviton mass at a level around [Formula: see text][Formula: see text]eV from the analysis of observational data on the trajectory of the star S2 near the Galactic Center. Therefore, observations of bright stars with existing and forthcoming telescopes such as the European extremely large telescope (E-ELT) and the thirty meter telescope (TMT) are extremely useful for investigating the structure of the Galactic Center in the framework of GR, but these observations also give a tool to confirm, rule out or constrain alternative theories of gravity. As we noted earlier, VLBI observations with current and forthcoming global networks (like the Event Horizon Telescope) are used to check the hypothesis about the presence of a supermassive black hole at the Galactic Center.
On the Black Holes in Alternative Theories of Gravity: The Case of Non-linear Massive Gravity
