Time dependent Schroedinger equation for black holes: No information loss

Introducing a black hole's e ective temperature, we obtained an Honorable Mention in the 2012 Gravity Research Foundation Essay Competition interpreting black hole's quasi-normal modes naturally in terms of quantum levels. This permitted us to show that quantities which are fundamental to realize the underlying quantum gravity theory, like Bekenstein-Hawking entropy, its sub-leading corrections and the number of microstates, are function of the black hole's excited state, i.e. of the black hole's quantum level, which is symbolized by the quantum overtone number n that denotes the countable sequence of quasi-normal modes. Here, we improve the analysis by nding a fundamental equation that directly connects the probability of emission of an Hawking quantum with the two emission levels which are involved in the transition. That equation permits us to interpret the correspondence between Hawking radiation and black hole quasi-normal modes in terms of a time dependent Schroedinger system. In such a system, the quasi-normal modes energies, which are also the total energies emitted by the black hole in correspondence of the various quantum levels, represent the eigenvalues of the unperturbed Hamiltonian of the system and the Hawking quanta represent the energy transitions among the eigenvalues, which correspond to a perturbed Hamiltonian ∝ δ(t). In this way, we explicitly write down a time dependent Schroedinger equation for the system composed by Hawking radiation and black hole quasi-normal modes. The states of the correspondent Schroedinger wave-function can be written in terms of a unitary evolution matrix instead of a density matrix. Thus, they result to be pure states instead of mixed ones. Hence, we conclude with the non-trivial consequence that information comes out in black hole's evaporation. This issue is also a con rmation of the assumption by 't Hooft that Schroedinger equations can be used universally for all dynamics in the universe, further endorsing the conclusion that black hole evaporation must be information preserving.