A revisit of rotating black hole shadow with astrometric observables

The first image of black hole in M87 galaxy taken by Event Horizon Telescope shows that directed observation of supermassive black holes would be a promising way to test general relativity in strong gravity field regime. In order to calculate shadow of rotating black holes with respect to observers located at finite distance, orthonormal tetrads have been introduced in previous works. However, it is noticed that different choice of tetrads does not lead to the same shape of shadow for observers in near regions. In this paper, we alternatively use formula of astrometric observables for calculating the shadow of a general rotating black hole with respect to these observers. For the sake of intuitive, we also consider Kerr-de Sitter black holes as a representative example. In this space-time, size and shape of Kerr-de Sitter black hole shadows are expressed as functions of distance between the black hole and observer. It is forecasted that the distortion of shadow would increase with distance.