The global-energy-conservation argument for black hole evaporation cannot be extended to realistic, non-stationary, non-asymptotically-flat black holes because no adequate notion of approximate Killing fields currently exists.
Tavel, Reprinted from ”Transport Theory and Statistical Mechanics” 1(3), 183-207 (1971)
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Not Quite Killing It: Black Hole Evaporation, Global Energy, and De-Idealization
The global-energy-conservation argument for black hole evaporation cannot be extended to realistic, non-stationary, non-asymptotically-flat black holes because no adequate notion of approximate Killing fields currently exists.