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Mathematically, GR equations have singular solutions. The metric of a single spherical mass has two features: at zero (r = 0) and at the event horizon (r = rg). A material particle crossing the horizon falls fatally into the center of the sphere, i.e. RG predicts the existence of a gravitational collapse. The collapse of massive stars with (M > 3 M0) is also predicted by the theory of stellar evolution from thermodynamic and statistical considerations. The singularity at zero is explained by the unsuitability of the RG theory on quantum scales. However, the singularity of the event horizon is associated with an unfortunate choice of reference system (Schwarzschild). The Lemaitre reference system is known, in which there is no singularity at the event horizon, i.e. it is not physical. The problem (mystery) of the event horizon requires the use of data from astrophysical observations. It was shown by Zeldovich Ya.B. and Salpiter E.E. non-spherical accretion must be accompanied by X-ray emission. In this case, the released energy can significantly (by an order of magnitude) exceed the energy of thermonuclear reactions. Thus, the detection of black holes is associated with the search for astrophysical objects with powerful X-ray luminosity. The study of X-ray binaries showed that their relativistic NS and BH objects differ not only in masses, but also in observational manifestations. In most cases, when a compact object with M< 3Mo shows signs of an observed surface, it is an NS: - X-ray pulsar, radio-pulsar or X-ray burster of the first type - requiring the presence of a strong magnetic field. On the contrary, none of the three dozen massive (M > 3 Mo ) compact objects (candidates for black holes) shows signs of an observable surface in accordance with the GR prediction. Instead, there are only conceivable boundaries - event horizons - light surfaces in space-time (which depend on the choice of an appropriate frame of reference). The statistics of relativistic stellar mass objects is already sufficient to accept this significant result - experimental evidence of the absence of observable surfaces in stellar BHs (or the presence of an event horizon). To prove the existence and study the properties of the event horizon, it is necessary to observe the effects generated by its changes. Such an opportunity is provided by the observation of the merging of binary BHs, using the GW waves emitted. In this case, the resulting BH with a new event horizon is formed from two initial BHs with their own horizons. Experimentally, the task is to detect and study the mode oscillations of the newly formed remnant after the merger and the process of damping these oscillations (phase - ring-down). In general relativity, isolated BHs are invisible due to the infinitely large redshift of photons propagating from the event horizon to a distant observer. However, the dark shadow (silhouette) of a black hole can be seen against the background of matter radiation lensed by the gravitational field of black holes. The shadow of a black hole is a projection of the cross section of the celestial sphere when a photon is captured by a black hole. The event horizon of any black hole is a surface formed by geodesic photon trajectories that do not end at spatial infinity. Direct evidence was first presented in April 2019, when the international EHT (Event Horizon Telescope) collaboration imaged a supermassive black hole in the M87 galaxy with a record high angular resolution of the order of several microarcseconds. For a mass of 6×10^9 Mo, the event horizon of this black hole has exactly this angular size. The image qualitatively consists of a bright asymmetric ring, interpreted as the glow of an accretion disk, and a dark central part, interpreted as the observed silhouette of a black hole. A photo of the shadow of a black hole at the center of our Galaxy, the Milky Way, was published by a research group of astronomers from the Event Horizon Telescope project. The image was obtained using eight radio telescopes in the world. Provides compelling evidence for the existence of a supermassive black hole at the center of the galaxy (this object, known as Sagittarius A* (Sgr A*) at the center of our galaxy, is about 27,000 light-years from Earth)