Hawking Radiation and Energy Conservation in an Evaporating Black Hole

We define the Bondi energy for two-dimensional dilatonic gravity theories by generalizing the known expression of the ADM energy. We show that our definition of the Bondi energy is exactly the ADM energy minus the radiation energy at null infinity. An explicit calculation is done for the evaporating black hole in the RST model with the Strominger's ghost decoupling term. It is shown that the infalling matter energy is completely recovered through the Hawking radiation and the thunderpop.