Operator weak values and black hole complementarity

In conventional field theories, the emission of Hawking radiation in the background of a collapsing star requires transplanckian energy fluctuations. These fluctuations are encoded in the weak values of the energy-momentum operator constructed from matrix elements between both -in and -out states. It is argued that taming of these weak values by back-reaction may lead to geometrical backgrounds which are also build from weak values of the gravitational field operators. This leads to different causal histories of the black hole as reconstructed by observers crossing the horizon at different times but reduces, in accordance with the equivalence principle, to the classical description of the collapse for the proper history of the star as recorded by an observer comoving with it. For observers never crossing the horizon, the evaporation would be interpreted within a topologically trivial ``achronon geometry" void of horizon and singularity: after the initial ignition of the radiation from pair creation out of the vacuum of the collapsing star of mass M, as in the conventional theory, the source of the thermal radiation would shift gradually to the star itself in a time at least of order $4M\ln 2M$. The burning of the star could be consistent with a quantum unitary evolution along the lines suggested by 't Hooft. A provisional formal expression of general black hole complementarity is proposed and its possible relevance for testing features of a theory of quantum gravity is suggested.