COLLAPSE OF SINGULAR ISOTHERMAL SPHERES TO BLACK HOLES

We study the gravitational collapse of a relativistic singular isothermal sphere that is initially in unstable equilibrium. In the subsequent collapse, the dynamic spacetime is self-similar. The infall proceeds in an inside-out fashion, mimicking its Newtonian counterpart in star formation. A spherical expansion wave propagates outward at the speed of sound, initiating an inward collapse relative to local static observers. Outside of the expansion wave front, matter remains in local equilibrium. Inside, fluid elements are accelerated from rest toward the expanding black hole event horizon. When the singular isothermal sphere is initially threaded by a uniform but weak magnetic field, the frozen-in field lines accumulate above the horizon according to a distant observer, while assuming a split-monopole configuration on a larger scale. When the magnetized system also possesses rotation, such a configuration may naturally develop a vigorous outflow in the simultaneous presence of an accretion inflow. We speculate that such a process underlies the well-known relationship between mass and bulge velocity dispersion of super-massive black holes in the nuclei of galaxies.