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An Uneventful Horizon in Two Dimensions

Ahmed Almheiri, James Sully

TL;DR

This work investigates the firewall paradox within the $1+1$-dimensional CGHS dilaton gravity model. By leveraging Ashtekar et al.'s mean-field quantum backreaction, it shows the horizon remains smooth and the Hawking radiation is in a mixed state entangled with a remnant beyond the last ray, challenging a key black hole complementarity postulate. The authors derive the renormalized entanglement entropy for Hawking radiation in CGHS via $1+1$D CFT methods and the moving-mirror analogy, finding that the entropy scales with the ADM mass and is purified by degrees of freedom behind the last ray. They further discuss how uplift to higher dimensions preserves the remnant picture but reveals tensions with holography, and conclude that while the CGHS model avoids firewalls, it highlights limitations of 2D toy models in capturing higher-dimensional gravity and holographic constraints.

Abstract

We investigate the possibility of firewalls in the Einstein-dilaton gravity model of CGHS. We use the results of the numerical simulation carried out by Ashtekar et al. to demonstrate that firewalls are absent and the horizon is drama free. We show that the lack of a firewall is consistent because the model does not satisfy one of the postulates of black hole complementarity. In particular, we show that the Hawking radiation is not pure, and is completely entangled with a long-lived remnant beyond the last ray.

An Uneventful Horizon in Two Dimensions

TL;DR

This work investigates the firewall paradox within the -dimensional CGHS dilaton gravity model. By leveraging Ashtekar et al.'s mean-field quantum backreaction, it shows the horizon remains smooth and the Hawking radiation is in a mixed state entangled with a remnant beyond the last ray, challenging a key black hole complementarity postulate. The authors derive the renormalized entanglement entropy for Hawking radiation in CGHS via D CFT methods and the moving-mirror analogy, finding that the entropy scales with the ADM mass and is purified by degrees of freedom behind the last ray. They further discuss how uplift to higher dimensions preserves the remnant picture but reveals tensions with holography, and conclude that while the CGHS model avoids firewalls, it highlights limitations of 2D toy models in capturing higher-dimensional gravity and holographic constraints.

Abstract

We investigate the possibility of firewalls in the Einstein-dilaton gravity model of CGHS. We use the results of the numerical simulation carried out by Ashtekar et al. to demonstrate that firewalls are absent and the horizon is drama free. We show that the lack of a firewall is consistent because the model does not satisfy one of the postulates of black hole complementarity. In particular, we show that the Hawking radiation is not pure, and is completely entangled with a long-lived remnant beyond the last ray.

Paper Structure

This paper contains 14 sections, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A Review of the CGHS Model
  3. The Classical Solution
  4. The Mean Field Approximation
  5. Numerical Results
  6. Entanglement Entropy in 2D CFTs
  7. Entanglement Entropy of Hawking Radiation in the CGHS model
  8. Entanglement Entropy of radiation from Moving Mirrors
  9. CGHS Remnants
  10. Identifying Entangled Modes
  11. Long-Range Behaviour
  12. Discussion and Conclusions
  13. Lifting to Higher Dimensions
  14. Conclusions

Figures (4)

  • Figure 1: The geometry of an evaporating black hole in the mean field approximation of the CGHS model, as found by Ashtekar2008.
  • Figure 2: A set of right-moving modes on an interval of $\mathcal{I}^+_R$ can be mapped back to a set of left-moving modes in the vacuum on $\mathcal{I}^-_R$. The same interval on $\mathcal{I}^+_R$, with different mirror configurations, maps back to different length intervals on $\mathcal{I}^-_R$. In this specific example, we see the intervals on $\mathcal{I}^-_R$ differ, at leading order, only by the length of the cutoff at the future bondary.
  • Figure 3: The radiated modes in the Hawking-like region are exactly the localized modes described in Giddings1992b. They are entangled with partner modes across the last ray of black hole singularity.
  • Figure 4: (a) When the interval is much longer to the past of the kink than to the future, there are many modes whose entangled partner is within the future boundary in the vacuum, but outside the boundary in the kinked state. (b) When the interval is much longer to the future of the kink than to the past, there are many modes who entangled partner is outside the past boundary in the vacuum, but inside the boundary in the kinked state.