  The introduction of coordinates representing the points of view of various
observers results in the possibility of horizons when acceleration and
gravitation are included. A horizon is a surface of possible light beams in a
region of space of finite distance from the observer, which means that since
nothing travels faster than light, events on the far side of a horizon cannot
influence those on the causal side. A black hole has such a horizon, where some
radially outgoing light beams can never reach a distant (or even nearby)
observer. However, since one suspects that black holes can swallow energy, and
even evaporate by Hawking radiation, such horizons must take on a time
dependency. A naive introduction of temporal dependency results in infinities
(singularities) in energy densities, suggesting in such descriptions that an
in-falling observer would encounter a hard surface at the horizon. However, if
coordinates representing space-time as analogous to a "flowing river" are used
to describe the dynamics of a black hole, no such singularities are
encountered. Such a parameterization of time dependent horizons will be offered
in this presentation. A Penrose space-time diagram (which represents the entire
space-time on a finite diagram with light beams always moving at a 45 degree
angle to vertical) describing the growth and evaporation of an example black
hole, along with the resulting coordinate anomaly, will be constructed.
