• averaged energy conditions
• black hole evaporation
• Black holes
• energy conditions
• Hawking's area theorem
• index forms

In 1971 Hawking published one of his most famous results: the area of the horizon of a black hole should never shrink. A few years later, in "Particle creation by Black Holes" (Hawking, 1975), we can find the extraordinary prediction of a black body radiation, expected to be emitted by any black hole.
This prediction left the scientific world astonished, not only because it was burning down the image of a black hole as an eternal prison, but also because would force them to evaporate, and indeed break that very law from which we started from.
In order to resolve this tension it is necessary to carefully evaluate the hypothesis taken for the deduction of the area theorem: it had already been pointed out by Epstein, Glaser and Jaffe that any point-wise energy conditions shall be violated in the presence of quantum fields; this matches with the intrinsic quantum nature of black hole evaporation in suggesting that the violation of the black hole area theorem shall be ascribed to a failure of the null energy condition in presence of quantum effects.
In this work we generalise the black hole area theorem to the case where the null energy condition is substituted by an averaged quantum energy inequality; we archive this result by means of index forms methods, reviewed at the beginning of the project. The new statement is in the form of "the area of the horizon cannot decrease faster than some bound V", and finally we proceed to compare this bound with the expected rate of evaporation of black holes.