Einstein General Relativity Theory is usually used for the interpretation of the experimental cosmological data in the Friedman-Robertso-Worker framework. Meanwhile, the General Relativity Theory is verified only in the Solar system and in the near cosmological environments.

Observations of the black hole images open the unique possibility for this verification in the strong field limit when Newtonian gravitational potential is of the squared light velocity order and when gravitation dominates over astrophysical factors.

Experimental verification of the Einstein General Relativity Theory and different Modified Gravity theory by observations of detailed images of supermassive black holes with the use of projected Space Millimetron Observatory with nano-arcsecond angular resolution. These observations are very crucial for physical interpretation of astrophysical and cosmological data on the Universe and for understanding, in part, the physical origin of enigmatic dark matter and dark energy. Scrutinizing the different gravity theories would be possible in the nearest future after construction of the Space Millimetron Observatory with nano-arcsecond angular resolution.

The very promising astrophysical supermassive black holes for these verifications are the SgrA* at the center of our native Milky Way Galaxy and supermassive black hole M87* at the center of the giant elliptical galaxy M87, which itself is placed in the Virgo galactic cluster. The fist images of these black holes were observed recently by the Event Horizon Telescope collaboration with milli-arcsecond resolution.

It must be accented that the explicit point of this research is the verification of Einstein General Relativity and different modifications of gravity theory in the strong field limit.