Discussion on event horizon and quantum ergosphere of evaporating black holes in a tunnelling framework

In this paper, with the Parikh-Wilczek tunnelling framework the positions of the event horizon of the Vaidya black hole and the Vaidya-Bonner black hole are calculated, respectively. We find that the event horizon and the apparent horizon of these two black holes correspond, respectively, to the two turning points of the Hawking radiation tunnelling barrier. That is, the quantum ergosphere coincides with the tunnelling barrier. Our calculation also implies that the Hawking radiation comes from the apparent horizon.