Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Islands in Schwarzschild black holes

Koji Hashimoto, Norihiro Iizuka, Yoshinori Matsuo

TL;DR

The paper demonstrates that the Page curve for an eternal Schwarzschild black hole in asymptotically flat spacetime is reproduced by the island rule, extending the quantum extremal surface/replica wormhole framework beyond two dimensions. Without islands, the entanglement entropy grows linearly and would exceed the finite BH entropy, but a single island near the horizon yields a time-independent entropy of order the Bekenstein-Hawking value, effectively saturating the Page curve. This behavior holds in four and higher dimensions, with the Page time t_{\rm Page} scaling universally as t_{\rm Page} = \frac{3}{\pi} \frac{S_{\rm BH}}{c T_{\rm H}}, and the scrambling time given by t_{\rm scr} \simeq \frac{1}{2\pi T_{\rm H}} \log S_{\rm BH}; evaporation backreaction only enters at subleading order. Together, these results support a semiclassical resolution of the information paradox for eternal BHs and reveal a universal island-induced mechanism for entropy saturation across dimensions.

Abstract

We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole.

Islands in Schwarzschild black holes

TL;DR

The paper demonstrates that the Page curve for an eternal Schwarzschild black hole in asymptotically flat spacetime is reproduced by the island rule, extending the quantum extremal surface/replica wormhole framework beyond two dimensions. Without islands, the entanglement entropy grows linearly and would exceed the finite BH entropy, but a single island near the horizon yields a time-independent entropy of order the Bekenstein-Hawking value, effectively saturating the Page curve. This behavior holds in four and higher dimensions, with the Page time t_{\rm Page} scaling universally as t_{\rm Page} = \frac{3}{\pi} \frac{S_{\rm BH}}{c T_{\rm H}}, and the scrambling time given by t_{\rm scr} \simeq \frac{1}{2\pi T_{\rm H}} \log S_{\rm BH}; evaporation backreaction only enters at subleading order. Together, these results support a semiclassical resolution of the information paradox for eternal BHs and reveal a universal island-induced mechanism for entropy saturation across dimensions.

Abstract

We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole.

Paper Structure

This paper contains 9 sections, 71 equations, 2 figures.

Table of Contents

  1. Introduction and our strategy
  2. No island, no entropy bound
  3. Island saves the entropy bound
  4. Close look at the black hole
  5. View from a distance
  6. Higher dimensions
  7. Page time and scrambling time
  8. Early time growth of the entropy
  9. Geodesic distance and extremal volume

Figures (2)

  • Figure 1: Penrose diagram of the static Schwarzschild spacetime without island (left) and that with an island $I$ (right). The region $R$ whose states are identified with the Hawking radiation has two parts $R_+$ and $R_-$, which are located in the right and the left wedge, respectively. The boundary surfaces of $R_+$ and $R_-$ are indicated as $b_+$ and $b_-$, respectively. The island extends between the right wedge and the left wedge. The boundaries of $I$ are located at $a_+$ and $a_-$. At late times, the distance between the right wedge and the left wedge is very large.
  • Figure 2: The Page curve for the eternal Schwarzschild black hole. In this plot we ignore terms of higher order in $c \, G_{\rm N}/r_{\rm h}^{D-2}$, which are small compared to $t_{\rm Page}$ or $S_{\rm BH}$.