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.
We present our first numerical results of axisymmetric magnetohydrodynamic simulations for neutrino-cooled accretion tori around rotating black holes in general relativity. We consider tori of mass $sim 0.1$--0.4$M_{odot}$ around a black hole of mass
$M=4M_{odot}$ and spin $a=0$--$0.9M$; such systems are candidates for the central engines of gamma-ray bursts (GRBs) formed after the collapse of massive rotating stellar cores and the merger of a black hole and a neutron star. In this paper, we consider the short-term evolution of a torus for a duration of $approx 60$ ms, focusing on short-hard GRBs. Simulations were performed with a plausible microphysical equation of state that takes into account neutronization, the nuclear statistical equilibrium of a gas of free nucleons and $alpha$-particles, black body radiation, and a relativistic Fermi gas (neutrinos, electrons, and positrons). Neutrino-emission processes, such as $e^{pm}$ capture onto free nucleons, $e^{pm}$ pair annihilation, plasmon decay, and nucleon-nucleon bremsstrahlung are taken into account as cooling processes. Magnetic braking and the magnetorotational instability in the accretion tori play a role in angular momentum redistribution, which causes turbulent motion, resultant shock heating, and mass accretion onto the black hole. The mass accretion rate is found to be $dot M_* sim 1$--$10 M_{odot}$/s, and the shock heating increases the temperature to $sim 10^{11}$ K. This results in a maximum neutrino emission rate of $L_{ u}=$ several $times 10^{53}$ ergs/s and a conversion efficiency $L_{ u}/dot M_* c^2$ on the order of a few percent for tori with mass $M_{rm t} approx 0.1$--0.4$M_{odot}$ and for moderately high black hole spins.
