The Dynamical Stability of Tori around a Schwarzschild Black Hole

Recently, geometrically thick disks around black holes have been studied by many authors. I) These models may exist in the inner region of accretion disks and explain the generation of high luminosity and colliminated jets in quasers and active galactic nuclei. In such tori, there is no reason for the rotation law to be Keplerian as pressure force plays a role in the equilibrium structure. These authors use the rotation laws which satisfy the stability criterion for local axisymmetric perturbation!) However, it is not sufficient. These exist unstable modes for global, nonaxisymmetric perturbation, although the tori are marginally stable for local, axisymmetric one.) Some properties of these global unstable modes have been discussed by many authors.')-7l The growth rates of unstable modes in the tori with constant specific angular momentum have also been obtained numerically in the Newtonian theory.7) It is necessary to study the stability of the tori in general relativity, because the tori near a black hole are of special interest. We must take account of the general relativistic effects at least in the inner part of the tori. The relativistic criterion for local stability will be identical in form to Newtonian one, if we consider it in the local proper frame. As for global stability, however, the criterion is not necessarily the same, a priori, because such a global phenomenon is related to collective motion of entire system. In this short note, we consider an. isentropic perfect flow rotating around a Schwarzschild black hole. The total mass of the torus is so small compared with M (the mass of the central black hole) that the background metric is always fixed as *) Present address: Research Institute for Fundamental Physics, Kyoto University. ds2=:-(1;)dt2+(1;rdr2 + r2de2+ r2sin2ed¢2 .