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Phases of information release during black hole evaporation

Ram Brustein, A. J. M. Medved

TL;DR

This work reframes black hole evaporation as a time-dependent, semiclassical process in which background-geometry fluctuations generate small but pivotal off-diagonal correlations in the Hawking radiation density matrix. The authors introduce a coherence time $t_{coh}$ (and its Schwarzschild-time counterpart $t_{coh}=R_S^2/l_p$) that dictates when information begins to emerge ($t_{1bit}=t_{coh}$) and when the emission becomes fully information-purifying ($t_{trans}$, just before evaporation ends). They show that Page-time language splits into two distinct scales and that late-stage purification is driven by a growing coherence scale $N_{coh}$ and back-reaction, with entanglement between early and late radiation becoming parametrically maximal near the end. The results offer a unitary-consistent narrative for information release, quantify the information flow rate, and illuminate potential implications for the firewall paradox by shifting the critical timing away from the original Page time.

Abstract

In a recent article, we have shown how quantum fluctuations of the background geometry modify Hawking's density matrix for black hole (BH) radiation. Hawking's diagonal matrix picks up small off-diagonal elements whose influence becomes larger with the number of emitted particles. We have calculated the "time-of-first-bit", when the first bit of information comes out of the BH, and the "transparency time", when the rate of information release becomes order unity. We have found that the transparency time is equal to the "Page time", when the BH has lost half of its initial entropy to the radiation, in agreement with Page's results. Here, we improve our previous calculation by keeping track of the time of emission of the Hawking particles and their back-reaction on the BH. Our analysis reveals a new time scale, the radiation "coherence time", which is equal to the geometric mean of the evaporation time and the light crossing time. We find, as for our previous treatment, that the time-of-first-bit is equal to the coherence time, which is much shorter than the Page time. But the transparency time is now much later than the Page time, just one coherence time before the end of evaporation. Close to the end, when the BH is parametrically of Planckian dimensions but still large, the coherence time becomes parametrically equal to the evaporation time, thus allowing the radiation to purify. We also determine the time dependence of the entanglement entropy of the early and late-emitted radiation. This entropy is small during most of the lifetime of the BH, but our qualitative analysis suggests that it becomes parametrically maximal near the end of evaporation.

Phases of information release during black hole evaporation

TL;DR

This work reframes black hole evaporation as a time-dependent, semiclassical process in which background-geometry fluctuations generate small but pivotal off-diagonal correlations in the Hawking radiation density matrix. The authors introduce a coherence time (and its Schwarzschild-time counterpart ) that dictates when information begins to emerge () and when the emission becomes fully information-purifying (, just before evaporation ends). They show that Page-time language splits into two distinct scales and that late-stage purification is driven by a growing coherence scale and back-reaction, with entanglement between early and late radiation becoming parametrically maximal near the end. The results offer a unitary-consistent narrative for information release, quantify the information flow rate, and illuminate potential implications for the firewall paradox by shifting the critical timing away from the original Page time.

Abstract

In a recent article, we have shown how quantum fluctuations of the background geometry modify Hawking's density matrix for black hole (BH) radiation. Hawking's diagonal matrix picks up small off-diagonal elements whose influence becomes larger with the number of emitted particles. We have calculated the "time-of-first-bit", when the first bit of information comes out of the BH, and the "transparency time", when the rate of information release becomes order unity. We have found that the transparency time is equal to the "Page time", when the BH has lost half of its initial entropy to the radiation, in agreement with Page's results. Here, we improve our previous calculation by keeping track of the time of emission of the Hawking particles and their back-reaction on the BH. Our analysis reveals a new time scale, the radiation "coherence time", which is equal to the geometric mean of the evaporation time and the light crossing time. We find, as for our previous treatment, that the time-of-first-bit is equal to the coherence time, which is much shorter than the Page time. But the transparency time is now much later than the Page time, just one coherence time before the end of evaporation. Close to the end, when the BH is parametrically of Planckian dimensions but still large, the coherence time becomes parametrically equal to the evaporation time, thus allowing the radiation to purify. We also determine the time dependence of the entanglement entropy of the early and late-emitted radiation. This entropy is small during most of the lifetime of the BH, but our qualitative analysis suggests that it becomes parametrically maximal near the end of evaporation.

Paper Structure

This paper contains 19 sections, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of previous results on semiclassical corrections to Hawking radiation
  3. Conventions
  4. Semiclassical density matrix
  5. Entropy and information
  6. Time-dependent radiation density matrix
  7. A model of time-dependent emission
  8. The time-dependent density matrix
  9. The coherence time
  10. A simplified qualitative description of BH evaporation
  11. Time dependence of information release
  12. Time-dependence of the purity
  13. The rate of information transfer
  14. Qualitative discussion of the final purification
  15. Early-Late Entanglement
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Information released as a function of the number $N_T$ of emitted Hawking particles shown for the Page model (blue) and our model (solid and dashed purple). Here, $N$ denotes the maximal value of $N_T$ which is approximately $S_{BH}(0)$. The lower dashed line depicts the block picture of $\S$3.4. The upper dashed line is for $N_{coh} C_{BH}=1$, so it estimates the derivative at the transparency time correctly and corresponds to the approximation in Eq. (\ref{['inforate2']}). The solid purple line depicts the approximation in Eq. (\ref{['inforate3']}).