Across-horizon propagation and infinite time-dilation: An independent approach to Hawking radiation

The conformal scalar propagator as an explicit function of the Schwarzschild black-hole space-time has been established in the preceding three projects. The present project examines the precise relation which that propagator has to the amplitude the Schwarzschild black hole emit a scalar particle in a particular mode of positive energy. It is thereby established that the effect of infinite time-dilation on the event horizon results in both, the thermal radiation in the background of a static exterior space-time geometry and the detraction from thermality if energy-conservation is properly imposed as a constraint on scalar propagation. The derivation of the latter signifies an equivalent approach to the result of Parikh and Wilczek from the semi-classical perspective of the distant observer. From that perspective for that matter, Hawking radiation emerges in both contexts as the exclusive consequence of infinite time-dilation on the event horizon. These results are consistent with Hawking radiation as the exclusive consequence of the causal structure of the Schwarzschild black-hole space-time both, in a static exterior geometry and in such dynamical exterior geometry as energy-conservation signifies. The extension of the thermal case to other spin-fields and black-hole geometries is discussed.