Beyond the semi-classical description of black hole evaporation

Abstract

In the semi-classical treatment, i.e. in a classical black hole geometry, Hawking quanta emerge from trans-Planckian configurations because of scale invariance. There is indeed no scale to stop the blue-shifting effect encountered in the backward propagation towards the event horizon. On the contrary, when taking into account the gravitational interactions neglected in the semi-classical treatment, a UV scale stopping the blue-shift could be dynamically engendered. To show that this is the case, we use a non-perturbative treatment based on the large-N limit, where $N$ is the number of matter fields. In this limit, the semi-classical treatment is the leading contribution. Non-linear gravitational effects appear in the next orders and in the first of these, the effects are governed by the two-point correlation function of the energy-momentum tensor evaluated in the vacuum. Taking this correlator into account, backward propagated modes are dissipated at a distance from the horizon $\propto G\kappa$ when measured in a freely falling frame. ($G$ is Newton's constant and $\kappa$ the surface gravity.) This result can be also obtained by considering light propagation in a stochastic ensemble of metrics whose fluctuations are determined by the above correlator.