The quantum stress-energy tensor of a massless scalar field propagating in the two-dimensional Vaidya-de Sitter metric, which describes a classical model spacetime for a dynamical evaporating black hole in an inflationary universe, is analysed. The author presents a possible way to obtain the Hawking radiation terms for the model with arbitrary functions of mass. It is used to see how the expansion of universe will affect the dynamical process of black-hole evaporation. The results show that the cosmological inflation has an inclination to depress the black hole evaporation. However, if the cosmological constant is sufficiently large then the back-reaction effect has the inclination to increase the black-hole evaporation. He also presents a simple method to show that it will always produce a divergent flux of outgoing radiation along the Cauchy horizon where the curvature is a finite value. This means that the Hawking radiation will be very large in there and shall modify classical spacetime drastically. Therefore black hole evaporation cannot be discussed self-consistently on the classical Vaidya-type spacetime. The method can also be applied to analyse the quantum stress-energy tensor in the more general Vaidya-type spacetimes.
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