Shadow of a spinning black hole in an expanding universe

We study the influence of the cosmic expansion on the size of the shadow of a spinning black hole observed by a comoving observer. We first consider that the expansion is driven by a cosmological constant only and build the connection between the Kerr-de Sitter metric and the Friedmann--Lemaître--Robertson--Walker metric. We clarify that the notion of a comoving observer is well defined in the spacetime of a spinning black hole only in the sense of being asymptotic. The angular size of the shadow for a comoving observer is calculated. Significantly we find that the angular size approaches a nonzero finite value for a distant comoving observer, while it vanishes for a distant static observer. Furthermore, by adopting the approximate method proposed in [G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Phys. Rev. D 98, 084020 (2018)., 10.1103/PhysRevD.98.084020] we extend the study to the general multicomponent universe case. The results show that the difference between the horizontal and vertical angular size changes a lot, while their ratio, i.e., the oblateness, keeps unchanged when the supermassive spinning black hole is at a high redshift, due to the common amplification factor exerted by the cosmic expansion. In addition, when a =0 , our results agree with the previous studies in [V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D 97, 104062 (2018)., 10.1103/PhysRevD.97.104062, G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Phys. Rev. D 98, 084020 (2018)., 10.1103/PhysRevD.98.084020].