General relativity (GR) was tested in many experiments and observations and always its predictions were confirmed. However, in a majority of tests GR was checked
and confirmed in a weak gravitational field regime.
In 2005 it was predicted that a shadow formed near a supermassive black hole at the Galactic Center could be reconstructed from observations
of ground based global VLBI system or ground -- space interferometer acting in mm or sub-mm bands. In 2022 the Event Horizon Telescope (EHT) collaboration reconstructed the shadow for the black hole at the Galactic Center (GC), therefore
our prediction was confirmed.
In 2019 the EHT collaboration presented the first image reconstruction around the shadow for the supermassive black hole in M87*. In 2021 the EHT collaboration constrained parameters (“charges”) of spherical symmetrical metrics of black holes from an allowed interval for shadow radius. Earlier, we obtained analytical expressions for the shadow radius as a function of charge (including a tidal one) in the case of Reissner –Nordstr\"om metric. Based on results of the shadow size evaluation for M87* done by the EHT team we constrain a tidal charge. Similarly we constrained a tidal charge for the black hole at the Galactic Center based on shadow reconstruction done by the EHT collaboration in 2022. We discussed opportunities to use shadows to test alternative theories of gravity and alternative theories for galactic centers. We used also observational data for trajectories of bright stars near the Galactic Center to test gravity theories and theoretical models for the Galactic Center.