  In this work, we have studied the dynamical evolution of the shock front in a
neutron star. The shock wave is expected to possess enough strength to ignite
the nuclear matter thereby converting it to quark matter. The conversion of
nuclear to quark matter is assumed to take place at the shock discontinuity.
The density and pressure discontinuity is studied both spatially and temporally
as it starts near the center of the star and moves towards the surface.
Polytropic equations of state which mimics real original nuclear matter and
quark matter equations of state are used to study such dynamical phase
transition. Solving relativistic hydrodynamic equations for a spherically
symmetric star we have studied the phase transition assuming a considerable
density discontinuity near the center. We find that as the shock wave
propagates outwards, its intensity decreases with time, however the shock
velocity peaks up and reaches a value close to that of light. Such fast shock
velocity indicates rapid phase transition in neutron star taking place on a
timescale of some tens of microseconds. Such a result is quite an interesting
one, and it differs from previous calculations that the phase transition in
neutron stars takes at least some tens of milliseconds. Rapid phase transition
can have significant observational significance because such fast phase
transition would imply quite strong gravitational wave signals but very short
lived. Such short-lived gravitational wave signals would be accompanied with
short-lived gamma-ray bursts and neutrino signals originating from the neutrino
and gamma-ray generation from the phase transition of nuclear matter to quark
matter.
