Research Note

## Anisotropic radiation field and trapped photons around the Kerr black hole

^{1}
Cosmic Radiation Laboratory, the Institute of Physical and Chemical Research,
2-1 Hirosawa, Wako, Saitama 351-0198, Japan e-mail: rohta@riken.jp

^{2}
Department of Physics and Astronomy, Aichi University of Education, Kariya, Aichi 448-8542, Japan e-mail: takahasi@phyas.aichi-edu.ac.jp

Received:
23
October
2009

Accepted:
7
February
2010

*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 casts 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.

Key words: black hole physics / radiative transfer / hydrodynamics / relativity / accretion, accretion disks

*© ESO, 2010*