Yes I am saying that we ought to have conservation of energy and Hawking radiation in the Schwarzchild case and so by extension Hawking radiation should affect the mass of the black holein order to satisfy this conservation of energy. But I’m also saying the negative energy flux in to the black hole is due to the global symmetry of the spacetime ratehr than the local symmetry.
btw in between the first and the second sentence of the above paragraph I read what Wald has to say on this as he does cover it in his book General relativity, needless to say his arguments on why there has to be a negative nergy flux in to the black hole (and he offers approximations as to what this flux should be) are considerably more robust than what I’ve offered so far
