  We consider spherically-symmetric black holes in semiclassical gravity. For a
collapsing radiating thin shell we derive a sufficient condition on the
exterior geometry that ensures that a black hole is not formed. This is also a
sufficient condition for an infalling test particle to avoid the apparent
horizon of an existing black hole and approach it only within a certain minimal
distance. Taking the presence of a trapped region and its outer boundary ---
the apparent horizon--- as the defining feature of black holes, we explore the
consequences of their finite time of formation according to a distant observer.
Assuming regularity of the apparent horizon we obtain the limiting form of the
metric and the energy-momentum tensor in its vicinity that violates the null
energy condition (NEC). The metric does not satisfy the sufficient condition
for horizon avoidance: a thin shell collapses to form a black hole and test
particles (unless too slow) cross into it in finite time. However, there may be
difficulty in maintaining the expected range of the NEC violation, and
stability against perturbations is not assured.

General Relativity and Quantum Cosmology