Abstract
Hawking's model of black hole evaporation is not unitary and leads to a mixed density matrix for the emitted radiation, while the Page model describes a unitary evaporation process in which the density matrix evolves from an almost thermal state to a pure state. We compare a recently proposed model of semiclassical black hole evaporation to the two established models. In particular, we study the density matrix of the outgoing radiation and determine how the magnitude of the off-diagonal corrections differs for the three frameworks. For Hawking's model, we find power-law corrections to the two-point functions that induce exponentially suppressed corrections to the off-diagonal elements of the full density matrix. This verifies that the Hawking result is correct to all orders in perturbation theory and also allows one to express the full density matrix in terms of the single-particle density matrix. We then consider the semiclassical theory for which the corrections, being non-perturbative from an effective field-theory perspective, are much less suppressed and grow monotonically in time. In this case, the R\'enyi entropy for the outgoing radiation is shown to grow linearly at early times; but this growth slows down and the entropy eventually starts to decrease at the Page time. In addition to comparing models, we emphasize the distinction between the state of the radiation emitted from a black hole, which is highly quantum, and that of the radiation emitted from a typical classical black body at the same temperature.
Highlights
In Hawking’s model, the process of BH evaporation is not unitary [2]
Hawking argued that the correlation functions of the emitted radiation are diagonal in mode-occupancy number, frequency and emission time and, from this, deduced that the density matrix for the radiation is diagonal in the same quantities
Having established that the Renyi entropy of the radiation in the semiclassical model starts to decrease after the Page time, let us speculate on a viable form for its functional dependence on the number of emitted particles
Summary
Let us start here by recalling the original Hawking description of BH evaporation, which completely dismisses the back-reaction and time-dependence effects. The initial vacuum contains no particles with respect to the creation and annihilation operators {a+i , ai} as defined at past null infinity, ai|0− = 0. We will be interested in the expectation values of the observables at future infinity These will be composed only from the outgoing creation and annihilation operators {b+i , bi} and so can be written as O = Oout ⊗ Iin, where Iin denotes the identity operator in the Hilbert space of the ingoing states. The result of Hawking indicates that the density matrix for the outgoing radiation is thermal and is similar to that of a maximally mixed state. The full density matrix ρ is completely defined by ρij = βikβjk, with the normalization factor Z expressed as. It is much easier to calculate and provides an accurate measure of the entanglement in the state of the BH radiation.
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