Anisotropic radiation field and trapped photons around the Kerr black hole

Aims. In order to understand the anisotropic properties of local radiation field in the curved spacetime around a rotating black hole, we investigate the appearance of a black hole seen by an observer located near the black hole. When the black hole is in front of a source of illumination the black hole cast shadow in the illumination. Accordingly, the appearance of the black hole is called the black hole shadow. Methods. We first analytically describe the shape of the shadow in terms of constants of motion for a photon seen by the observer in the locally non-rotating reference frame (LNRF). Then, we newly derive the useful equation for the solid angle of the shadow. In a third step, we can easily plot the apparent image of the black hole shadow. Finally, we also calculate the ratio of the photon trapped by the hole and the escape photon to the distant region for photons emitted near the black hole. Results. From the shape and the size of the black hole shadow, we can understand the signatures of the curved spacetime; i.e., the mass and spin of the black hole. Our equations for the solid angle of the shadow has technical advantages in calculating the photon trapping ratio. That is, this equation is computationally very easy, and gives extremely precise results. This is because this equation is described by the one-parameter integration with given values of the spin and location for the black hole considered. After this, the solid angle can be obtained without numerical calculations of the null geodesics for photons.