Stability of stationary, spherical accretion onto a Schwarzschild black hole

The general problem of perturbing a (non-self-gravitating) perfect fluid potential flow in an arbitrary background gravitational field is formulated. The special case of perturbing stationary, spherical accretion onto a Schwarzschild black hole is then considered, and the following stability results are derived: (1) no unstable normal modes exist which extend outside the sound horizon of the background flow; and (2) there are no unstable modes which represent a standing shock at the sound horizon. The high-frequency (JWKB) approximation to traveling wave perturbations is also derived and it is shown that these approximate solutions are regular across the black hole's event horizon.