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Notes on islands in asymptotically flat 2d dilaton black holes

Takanori Anegawa, Norihiro Iizuka

TL;DR

This work extends the island paradigm to eternal 1+1D dilaton black holes in asymptotically flat spacetime, showing that without islands the radiation entropy grows linearly with time, while including an island leads to a Page-like saturation. Using the quantum-extremal-surface framework, the authors compute the generalized entropy $S_R$ for the CGHS-type 2d dilaton gravity, analyze the no-island and island phases, and show that the island boundary sits near the outer region ($a\approx b$) at late times, driving $S_R$ toward $S_R\approx 2S_{BH}$. The result demonstrates a late-time transition where the island dominates and the entanglement entropy of Hawking radiation halts its growth, reproducing the Page curve in a non-AdS, asymptotically flat 2d setting. The findings support the universality of the island mechanism and motivate explorations across broader parameter regimes and higher-dimensional generalizations.

Abstract

We study the islands and the Page curve in the 1+1-dimensional eternal dilaton black hole models. Without islands, the entanglement entropy of the radiation grows linearly at late time. However with an island, its growth stops at the value of almost twice of the black hole entropy. Therefore an island emerges at the late time, and the entanglement entropy of the radiation shows the Page curve.

Notes on islands in asymptotically flat 2d dilaton black holes

TL;DR

This work extends the island paradigm to eternal 1+1D dilaton black holes in asymptotically flat spacetime, showing that without islands the radiation entropy grows linearly with time, while including an island leads to a Page-like saturation. Using the quantum-extremal-surface framework, the authors compute the generalized entropy for the CGHS-type 2d dilaton gravity, analyze the no-island and island phases, and show that the island boundary sits near the outer region () at late times, driving toward . The result demonstrates a late-time transition where the island dominates and the entanglement entropy of Hawking radiation halts its growth, reproducing the Page curve in a non-AdS, asymptotically flat 2d setting. The findings support the universality of the island mechanism and motivate explorations across broader parameter regimes and higher-dimensional generalizations.

Abstract

We study the islands and the Page curve in the 1+1-dimensional eternal dilaton black hole models. Without islands, the entanglement entropy of the radiation grows linearly at late time. However with an island, its growth stops at the value of almost twice of the black hole entropy. Therefore an island emerges at the late time, and the entanglement entropy of the radiation shows the Page curve.

Paper Structure

This paper contains 8 sections, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Islands in the 2d dilaton black holes
  3. Classical eternal 2d dilaton black holes
  4. Entanglement entropy without islands
  5. Entanglement entropy with islands
  6. Short summary
  7. Review of 2d dilaton black holes
  8. Approximation of the entropy formula

Figures (3)

  • Figure 1: We consider the entropy of the interval $[-\infty,-b]_L\cup[b,\infty]_R$. This entropy, which is the entanglement entropy between pair created particles near the horizon, increases linearly and eternally in time at the late time. If unitarity is hold, the fine-grained entropy cannot increase eternally and it must to reach to a fixed value at the late time, which is contradicted with the linear growth.
  • Figure 2: We add the interval $[a_L,-a_R]$ in $\sigma$ coordinates. This island region lies on the non-zero time $t'$ slice. After the Page time $t_{\rm Page}$, this entropy becomes less than the entropy of no-island configuration. Thus, a phase transition occurs.
  • Figure 3: The Page curve. The blue line is the entropy without islands, which increases eternally. After the Page time $t_{\rm Page} \simeq 6 \beta S_{BH}/2 \pi c$, the one without islands becomes greater than the one with an island. Thus phase transition occurs, and the entropy reaches an constant value, which is approximately twice of the black hole entropy.