Late time approach to Hawking radiation: Terms beyond leading order

Abstract

Black hole evaporation is studied using wave packets for the modes. These allow for approximate frequency and time resolution. The leading order late time behavior gives the well known Hawking radiation that is independent of how the black hole formed. The focus here is on the higher order terms and the rate at which they damp at late times. Some of these terms carry information about how the black hole formed. A general argument is given which shows that the damping is significantly slower (power law) than what might be naively expected from a stationary phase approximation (exponential). This result is verified by numerical calculations in the cases of 2D and 4D black holes that form from the collapse of a null shell.

How Not to Extract Information From Black Holes: Cosmic Censorship as a Guiding Principle

“…Nevertheless such an optimization to release information as soon as possible is consistent with the notion that after the Page time, quantum information that falls into the black hole is quickly re-emitted in a timescale set by T −1 log S, the "scrambling time" [5][6][7][8][9][10][11][12], where S denotes the Bekenstein-Hawking entropy of the black hole (a quarter of the horizon area). This is also consistent with recent findings that terms beyond leading order in the Hawking temperature can have pronounced effects at late time [56]. In any case, since we are only interested in the final moments of the evaporation, we shall simply work with the Kim-Wen model.…”

Moving mirror model for quasithermal radiation fields

“…Time evolution can be resolved with the use of wave packets [12] β ϵ jn that pick out frequencies near jϵ at retarded time u ¼ 2πnϵ −1 with width 2πϵ −1 . This localization of the global beta coefficients corresponds to the sensitivity response of a particle detector at a given time, frequency, and bandwidth [22,34]. With large g ≫ κ, and good time resolution (i.e., large ϵ; the particles pile up in the single j ¼ 0 bin) one observes a flattening plateau, Fig.…”

Method to compute the stress-energy tensor for a quantized scalar field when a black hole forms from the collapse of a null shell

“…In this paper we present a method to compute the renormalized stress-energy tensor, hinjT ab jini, for a massless minimally coupled scalar field in the case that a black hole forms from the collapse of a spherically symmetric null shell. This model has been previously used to derive the Hawking effect [19,20], investigate how the stress-energy tensor is affected by the production of a pair of particles due to the Hawking effect [21], study some details of how the spectrum and number of produced particles changes in time during and after the collapse [22,23], and in 2D to compute the stress-energy tensor for a massless minimally coupled scalar field [20,24]. While this is not a realistic model for collapse because the shell begins with an infinite size, this is probably the simplest model to work with that involves collapse in 4D to form a black hole.…”