  The gravitational collapse of a star is an important issue both for general
relativity and astrophysics, which is related to the well known "frozen star"
paradox. Following the seminal work of Oppenheimer and Schneider (1939), we
present the exact solution for two dust shells collapsing towards a
pre-existing black hole. We find that the inner region of the shell is
influenced by the property of the shell, which is contrary to the result in
Newtonian theory and and the clock inside the shell becomes slower as the shell
collapses towards the pre-existing black hole. This result in principle may be
tested experimentally if a beam of light travels across the shell. We conclude
that the concept of the "frozen star" should be abandoned, since matter can
indeed cross a black hole's horizon according to the clock of an external
observer. Since matter will not accumulate around the event horizon of a black
hole, we predict that only gravitational wave radiation can be produced in the
final stage of the merging process of two coalescing black holes. Our results
also indicate that for the clock of an external observer, matter, after
crossing the event horizon, will never arrive at the "singularity" (i.e. the
exact center of the black hole.
