  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.
