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Notes on the islands in the CGHS black holes

Takanori Anegawa, Norihiro Iizuka

Published Apr 3, 2020• 0 citations
OpenAlex

Abstract

We study the islands and the Page curve in the 1+1-dimensional eternal CGHS 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.

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We study the islands and the Page curve in the 1+1-dimensional eternal CGHS 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.
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.
We compute the entanglement entropy of Hawking radiation in a bath attached to a deformed eternal AdS black hole. This black hole is dual to the two identical strongly coupled large- N c thermal field theories, where each theory is backreacted (deformed) by the presence of a uniform static distribution of heavy fundamental quarks. In our observation we find that the entanglement entropy of Hawking radiation increases in a quadratic manner for an early time and linearly for the late time. The large time expression for the entanglement entropy of Hawking radiation is used to find the Page curve and Page time. After the Page time, the entanglement entropy saturates to a constant value due to the appearance of an island. We observe that introducing deformation (backreaction) delays the appearance of island and shifts the Page curve to a later time. Subsequently, the computation of the scrambling time reveals an increase with the backreaction parameter, suggesting a longer duration for information retrieval in the presence of deformation. Moreover, our analysis of the mutual information between the radiation subsystems shows that it vanishes at a critical time which increases with the deformation before the Page time. After the Page time, the appearance of island leads to the vanishing of mutual information between black hole subsystems and gives the time difference of the order of scrambling time.
Abstract The role played by the mutual information of subsystems on the Page curve is explored in this paper. With the total system consisting of the black hole and radiation, together with the inclusion of island, we observe that the vanishing of mutual information between B+B_+B+​ B + and B−B_-B−​ B - which in turn means the disconnected phase of the entanglement wedge corresponding to B+∪B−B_+\cup B_-B+​∪B−​ B + ∪ B - , yields a time scale of the order of scrambling time. This results in a time independent expression for the fine grained entropy of Hawking radiation consistent with the correct Page curve. We also find corrections to this entropy and Page time which are logarithmic and inverse power law in form.
In this paper, we consider the entanglement entropy and entanglement islands in the two-sided generalization of the Schwarzschild black hole in a cavity. We find that entanglement entropy saturates at some constant value thus for some values avoiding information paradox (in the Page curve formulation). Also we find, that inclusion of the entanglement islands leads to a universal effect induced by the boundary presence, which we call ``blinking island". For some time the entanglement island inevitably disappears, thus leading to a short-time information paradox.
Analyses suggest that the quantum RT construction can be effectively extended beyond the asymptotically AdS spacetime for the calculation of the entanglement entropy. We study the entanglement entropy of a one-parameter family of exactly solvable gravities in the 2-dimensional asymptotically-flat space, and investigate the evaporating and the bath-removed black hole solutions including both the RST and the BPP black holes as two special cases. We generalize the results on the Page curves of evaporating (dynamical) black holes to this family of gravities and compute the corresponding entanglement entropy explicitly. The Page curves and the configuration of islands directly follow from the calculation. In the case of evaporating black holes formed by collapsing matter shells, the Page time is around a third of the black hole evaporating time and the island lies inside the horizon. In the bath-removed black hole case, the position of the islands is discontinuous across two different spacetime geometries and two transition points are witnessed in this model. We analyze the island configurations and the Page curves in the large-N limit in both cases and restore the unitary Page curves.
The Island Formula for Quantum Extremal Surfaces of large AdS black holes, which are made to evaporate by coupling the CFT of the AdS/CFT correspondence to an external system, demonstrates that the space time picture can reproduce the quantum mechanically required Page curve of the CFT/bath system if a region in the interior of the black hole has become entangled with the external radiation by the Page time. In an effective quantum field theory (EFT) description in ``nice slice" coordinates, this appears to imply paradoxical entanglement between degrees of freedom that have been causally disconnected since long before the Page time. We will give two arguments that all of the quantum information in the island is in fact concentrated in a tiny space-like portion of the island very close to the black hole horizon at the Page time. The first applies the covariant entropy bound\cite{fsb} (CEB) to the causal diamonds of geodesics originating on the space-like region that becomes the island, but on a time slice just before the Page time. All points on the island except those in a stretched horizon region very close to the horizon have entropy bounds that are tiny fractions of the black hole entropy. The second argument uses standard quantum statistical mechanics combined with the tensor network renormalization group regularization of CFT to argue that the wave function of a typical pure state in the microcanonical evaporating black hole always has most of its probability concentrated in a small range of wavelengths corresponding to a small range of radii in the dual tensor network/AdS slice.