  The principle of equivalence is used to examine covariant descriptions of
quantum phenomena within the global exterior of geometries described using
Painlev\'e- Gullstrand coordinates, which are everywhere non-singular away from
their center. The differences between the descriptions of observers stationary
within the curvilinear geometry and those in a locally-flat freely-falling
system that is momentarily at rest are found to depend only on local proper
accelerations, becoming vanishingly small for increasingly distant observers.
Away from the horizon, the local detections of outgoing massless radiations
from the ground state of the freely-falling observers are found to be
consistent with those of Rindler observers accelerating in flat space-time.
