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Remarks on self-interaction correction to black hole radiation

In the work [P. Kraus and F. Wilczek, \textit{Self-interaction correction to black hole radiation, Nucl. Phys.} B433 (1995) 403], it has been pointed out that the self-gravitation interaction would modify the black hole radiation so that it is no longer thermal, where it is, however, corrected in an approximate way and therefore is not established its relationship with the underlying unitary theory in quantum theory. In this paper, we revisit the self-gravitation interaction to Hawking radiation of the general spherically symmetric black hole, and find that the precisely derived spectrum is not only deviated from the purely thermal spectrum, but most importantly, is related to the change of the Bekenstein-Hawking entropy and consistent with an underlying unitary theory.