Next to leading order late time behavior for black hole evaporation

It is well known that the near horizon approximation for the modes of a quantum field gives the leading order late time behavior for black hole evaporation which is a thermal distribution at the black hole temperature. It is shown that the next to leading order contribution from the near horizon approximation does not give the true next to leading order contribution to the Hawking radiation in the case of a black hole that forms from the collapse of a spherically symmetric null shell. Instead, the true next to leading order contribution in both two and four dimensions is nonlocal and is damped much more slowly in time than the next to leading order contribution from the near horizon approximation.