Black hole shadows as a source for testing extended theories

The first images of black holes have opened up new possibilities for testing extended theories of gravity. In this paper, we propose and develop a method for constructing the background of the shadow of spherically symmetric non-rotating black holes for the case of 𝑔11 ̸= -𝑔-1 00 . The results of the analysis for the extended theories of gravity are compared with the predictions of the general theory of relativity (GRT) when taking into account the data of the Event Horizon Telescope (the global network of radio telescopes on which the first direct image of the BH was obtained). After the publication of the article [1], the Event Horizon Telescope (HET) project obtained the first direct image of a black hole in the center of our galaxy: Sagittarius A* [2]. The results obtained in [1] are in full agreement with the results of observations of Sagittarius A* and M87* for the following models: the Horndeski model with the Gauss-Bonnet invariant, loop quantum gravity, Bambelby scalar model, Gauss-Bonnet model. In conformal gravity, large values of 𝑚2 and should be excluded. In 𝑓(𝑄) gravity, observations of Sgr A* additionally limit the values of the parameter 𝛼: -0.025