Relativistic Self-similar Dynamic Collapses of Black Holes in General Polytropic Spherical Clouds

We study the hydrodynamic self-similar mass collapses of general polytropic (GP) spherical clouds to central Schwarzschild black holes and void evolution with or without shocks. In order to grossly capture characteristic effects of general relativity (GR) outside yet close to the event horizon of a Schwarzschild black hole and to avoid mathematical complexity, we adopt the approximation of the Paczynski-Wiita gravity to replace the simple Newtonian gravity in our model formulation. A new dimensionless parameter s appears with the physical meaning of the square of the ratio of the sound speed to the speed of light $c$. Various self-similar dynamic solutions are constructed for a polytropic index $\gamma>4/3$. Two (for small enough $s<1$) or no (for large enough $s<1$) expansion-wave collapse solutions (EWCSs) with central event horizons exist when $\gamma>4/3$, representing the collapse of static singular GP spheres towards the central singularity of spacetime. Such GP spherical dynamic mass collapse is shown to be highly efficient for the rapid formation of supermassive black holes (SMBHs; mass range of $10^6-10^{10}M_{\odot}$) in the early universe or even hypermassive black holes (HMBHs; mass range of $10^{10}-10^{12}M_{\odot}$) if extremely massive mass reservoirs could be sustained for a sufficiently long time, which may evolve into hard X-ray/gamma ray sources or quasars according to their surroundings. Self-similar dynamic solutions of a GP gas are also proposed for the stellar mass black hole formation during the violent supernova explosion of a massive progenitor star, the timescale of which is estimated of $10^{-3}$ seconds. Rebound shocks travelling in supernovae are also discussed based on our self-similar shock expansion solutions.