Page curves and island's delays in asymptotically flat 2d spacetimes with injections

TL;DR

This work analyzes entanglement entropy and Page curves in asymptotically flat 2D spacetimes with multiple energy injections using the Russo–Susskind–Thorlacius (RST) model. By deriving the general solution for $n$ injections and focusing on $n=2$, it demonstrates that the island endpoint experiences a delay relative to the injection times, creating an intermediate state where the island lies between injections. Analytically and numerically, the paper shows that this island-delay mechanism yields a continuous entropy evolution across injections and can reproduce a two-step Page curve with a second Page time, highlighting the importance of the intermediate state for a consistent information-recovery picture. These results deepen the understanding of how islands regulate information flow in dynamical, non-stationary black-hole spacetimes within two-dimensional dilaton gravity and suggest avenues for extending the framework to more general geometries and injection histories.

Abstract

We explore spacetimes with multiple energy injections in asymptotically flat two-dimensional black hole and analyze the entanglement entropy in such spacetimes. This work is an extension of the setup of the single-injection case, by F. F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, to include the multiple energy injections. We derive the solution of the model, by J. G. Russo, L. Susskind and L. Thorlacius, for a general number $n$ of the total injections, and discuss the entropy only for the case $n=2$. The essential point of this work is in the delay of the island. This delay makes the intermediate state necessary, where the island's endpoint lies between the 1st and the 2nd injections while the observer is located after the 2nd injection. The intermediate state makes the entanglement entropy evolve continuously across the 2nd injection time.

Figures (6)

• Figure 1: The Page curve for the single-injection case,Gautason:2020tmk. Here, in the numerical computation, we use the following parameters: $M=3$, $\epsilon = 1/15$, $x_1 =1$, $\sigma_{\rm A}=2$. The increasing and decreasing line correspond to the cases without and with the island respectively. The red line expresses the minimum value at each time, so is the Page curve.
• Figure 2: The Penrose diagram illustrating a two-injection setup. The red line indicates the anchor curve. The blue thin lines correspond to the energy injections. The dashed line represents the event horizon, and the blue thick line denotes the apparent horizon. This case corresponds to the short period between the 1st and the 2nd injections, which means that the first appearance of the island's endpoint is located after the 2nd injection. Therefore, after the 2nd injection, the case without the island is initially favored, and then the case with the island is selected after the Page time. The situation is similar to the single-injection case, in that after all injections we only need to consider two possibilities: the no-island case and the case with the island.
• Figure 3: The Penrose diagram illustrating another two-injection setup. The interpretation of the diagram is the same as in figure \ref{['pd1']}. In this case, the 2nd injection occurs after the island appears. Therefore, we consider 2 cases; (i) the island's endpoint exists in the region between the 1st injection and the 2nd injection— this will be referred to as the intermediate state in the main text — and (ii) in the region after the 2nd injection.
• Figure 4: The Page curve with two injections. The parameters are chosen as follows: $E_1=E_2=3$, $t_2=4.2$, and the other parameters are the same as in figure \ref{['1p']}. The red dashed line indicates the time of the 2nd injection. The monotonically increasing behavior corresponds to the no-island case. The decreasing behavior before the 2nd injection is the case that both points A and I lie between the injections. For the later argument, we indicate it by writing $(1,1)$. The curve with the local minimum after the 2nd injection corresponds to the intermediate state. The other decreasing behavior after the 2nd injection is the case that both points A and I lie after the 2nd injection. We indicate it by writing $(2,2)$. The red solid line in this graph represents the minimum value, i.e., the Page curve.
• Figure 5: The Page curve with two injections using an alternative choice of the parameters: $E_1=3$, $E_2=11.1$, $\sigma_{\rm A}=1$, and the other parameters are the same as in figure \ref{['1p']}. Using these parameters, the entropy of the intermediate state monotonically increases.