  A spherically symmetric hydrodynamic stellar core collapse under gravity is
time-dependent and may become unstable once disturbed. Specifically for a
homologously collapse of stellar core characterized by a polytropic exponent
\Gamma=4/3, we examine oscillations and/or instabilities of three dimensional
(3D) general polytropic perturbations. For compressible 3D perturbations, the
polytropic index \gamma of perturbations can differ from \Gamma=4/3 of the
general polytropic hydrodynamic background flow. Our model formulation here is
more general and allows the existence of internal gravity g(-)-modes and/or
g(+)-modes. Eigenvalues and eigen-functions of various oscillatory and unstable
perturbation modes are computed. As studied in several specialized cases of
Goldreich & Weber and of Cao & Lou, we further confirm that acoustic p-modes
and surface f-modes remain stable in the current more general situations. In
comparison, g(-)-modes and sufficiently high radial order g(+)-modes are
unstable, leading to inevitable convective motions within the collapsing
stellar interior. Unstable growths of 3D g-mode disturbances are governed
dominantly by the angular momentum conservation and modified by the gas
pressure restoring force. We note in particular that unstable temporal growths
of 3D vortical perturbations exist even when the specific entropy distribution
becomes uniform and \gamma=\Gamma=4/3. Conceptually, unstable g-modes might
bear conceivable physical consequences on supernova explosions, the initial
kicks of nascent proto-neutron stars (PNSs) and break-ups of the collapsing
core, while unstable growths of vortical perturbations can lead to fast spins
of compact objects and other relevant results.
