  A simple ordinary differential equation is derived governing the red-shifts
of wave-fronts propagating through a non-stationary spherically symmetric
space-time. Approach to an event horizon corresponds to approach to a fixed
point; in general, the phase portrait of the equation illuminates the
qualitative features of the geometry. In particular, the asymptotics of the
red-shift as a horizon is approached, a critical ingredient of Hawking's
prediction of radiation from black holes, are easily brought out. This
asympotic behavior has elements in common with the universal behavior near
phase transitions in statistical physics. The validity of the Unruh vacuum for
the Hawking process can be understood in terms of this universality. The
concept of surface gravity is extended to to non-stationary spherically
symmetric black holes. Finally, it is shown that in the non-stationary case,
Hawking's predicted flux of radiation from a black hole would be modified.
