Exact solutions for shells collapsing towards a pre-existing black hole
  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. This paradox has been discussed intensively and seems to have been
solved in the comoving-like coordinates. However, to a real astrophysical
observer within a finite time, this problem should be discussed in the point of
view of the distant rest-observer, which is the main purpose of this paper.
Following the seminal work of Oppenheimer and Snyder (1939), we present the
exact solution for one or two dust shells collapsing towards a pre-existing
black hole. We find that the metric of the inner region of the shell is
time-dependent and the clock inside the shell becomes slower as the shell
collapses towards the pre-existing black hole. This means the inner region of
the shell is influenced by the property of the shell, which is contrary to the
result in Newtonian theory. It does not contradict the Birkhoff's theorem,
since in our case we cannot arbitrarily select the clock inside the shell in
order to ensure the continuity of the metric. This result in principle may be
tested experimentally if a beam of light travels across the shell, which will
take a longer time than without the shell. It can be considered as the
generalized Shapiro effect, because this effect is due to the mass outside, but
not inside as the case of the standard Shapiro effect. We also found that in
real astrophysical settings matter can indeed cross a black hole's horizon
according to the clock of an external observer and will not accumulate around
the event horizon of a black hole, i.e., no "frozen star" is formed for an
external observer as matter falls towards a black hole. (abridged)
