Radiations From Geometries Using Painlev\'e-Gullstrand Coordinates

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.