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arXiv:2509.03997 · Progress of Theoretical and Experimental Physics

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

Yuuta Saito

Published Dec 3, 2025• 1 citation
arXivHTMLPDF

Abstract

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 multiple energy injections. We derive a solution of the model, by J. G. Russo, L. Susskind and L. Thorlacius, for a general number n of 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.

DOI
10.1093/ptep/ptaf174

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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 multiple energy injections. We derive a solution of the model, by J. G. Russo, L. Susskind and L. Thorlacius, for a general number n of total injections, and discuss the entropy only for the case n=2n=2n=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.
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Abstract We study a thermofield double type entangled state on two disjoint universes A and B, where one of the universes is asymptotically flat containing a black hole. As we increase the entanglement temperature, this black hole receives back-reaction from the stress–energy tensor of the state. This results in lengthening of the wormhole region in the black hole interior, and decreasing of its horizon area, both of which are key features of an evaporating black hole. We then compute the entanglement entropy in universe A through the island formula, and argue that it naturally follows the Page curve of an evaporating black hole in flat space. We also study the effects of local operations in the gravitating universe with the black hole. We find that they accelerate the evaporation of the black hole, thereby disrupting the entanglement between the two universes. Furthermore, we observe that, depending on whether the operation can be regarded as a local operation and classical communication or not, the behavior of the entanglement entropy changes. In particular, when the operation is made neither in the entanglement wedge of the radiation system nor that of the black hole, the transition between the island phase and the no-island phase can happen multiple times.
We study thermodynamic properties and the entanglement island of a black hole in asymptotically safe quantum gravity, analyzing key thermodynamic quantities such as the Hawking temperature, heat capacity, and entropy, as well as the mass-horizon radius relation. Unlike Schwarzschild black holes, the temperature decreases with mass near the evaporation endpoint, signaling a phase transition and possible stable remnant. The entanglement entropy of Hawking radiation is obtained both with and without island contributions. Without islands, the radiation entropy grows linearly indefinitely, leading to the information paradox. By including island contributions and extremizing the generalized entropy functional, we resolve this paradox. At late times, the radiation entropy saturates at the Bekenstein-Hawking entropy, confirming unitary evolution. From this, we derive the Page time and scrambling time by equating early- and late-time entanglement entropies. The result of this study establishes the finiteness of the radiation entropy and consistency with quantum mechanics.
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We compute the capacity of entanglement in two dimensional dilaton gravity in a setting where Hawking radiation, backreaction, and islands can be treated analytically. Our focus is the eternal black hole of the Russo Susskind Thorlacius model coupled to N conformal matter fields. Unlike previous gravitational computations, which were mostly carried out in JT gravity, the RST model forces one to deal with a genuinely dynamical conformal factor and with the global constraints of the replica construction. The main technical step is therefore to solve the replica deformation on the orbifold globally at first order near n=1, including the homogeneous sector fixed by single valuedness and by the requirement of a fixed microcanonical state. For a single interval we obtain a time independent generalized capacity, parallel to the generalized entropy. For two intervals, even in the late time factorization regime, the global solution generates an interaction term between replica fixed points; after Lorentzian continuation this produces a time dependent capacity on the two QES saddle, despite the corresponding entropy plateau. We discuss the regime of validity of the resulting expressions and explain how the large size of the two QES capacity implies a highly non uniform saddle competition near n=1, providing a concrete mechanism for sharp features of the capacity at the Page transition.