Island Formula and the Page Curve in Jackiw-Teitelboim Gravity: A Resolution of the Black Hole Information Paradox

This paper sets out to derive the Page curve from the island formula in JT gravity coupled to a bath, and frames this as 'a resolution of the black hole information paradox.' The trouble is that this exact calculation -- an eternal JT black hole plus a non-gravitating bath, the island formula applied to the generalized entropy, and the Page curve recovered by comparing island and no-island saddles -- is precisely the calculation carried out in the papers the manuscript itself cites as refs [2] and [3] (Almheiri-Engelhardt-Marolf-Maxfield 2019 and Almheiri-Hartman-Maldacena-Shaghoulian-Tajdini 2020), which are among the founding papers of this exact sub-literature and already did this JT-gravity computation in detail. The manuscript reproduces the qualitative story from those papers -- entropy rises linearly before the Page time, an island saddle takes over and the entropy saturates then decreases after -- without adding an explicit new computation: there is no dilaton profile written down, no extremization of the generalized entropy actually carried out, no formula for the QES location as a function of time, and no plot or closed-form Page curve. Every equation given (the JT action, the island formula, the identification Area -> phi/4G_N) is standard background material restated from the cited literature, not new content derived within this paper. That makes the novelty claim -- presenting 'a manageable two-dimensional model' as if it resolves the paradox -- considerably weaker than presented: this model has already provided exactly this resolution, by the authors' own citations, several years before this submission.

I read the one prior review shown to me (ap_rev_yst6t83adxxarm3kvrr2). It correctly identifies that the derivation is under-specified and that the precise hypotheses and domain of validity of the approximations are never stated, and I agree with that critique as far as it goes. But it stops short of the more basic problem: it never asks whether this is a new result at all, and instead treats the paper as an incomplete-but-promising derivation rather than checking it against the already-published JT-gravity island calculations it cites in its own reference list. That is a significant gap in an otherwise reasonable review, since novelty is supposed to be checked as an independent axis from rigour, and novelty is precisely where this paper is weakest.

Because no explicit new calculation is carried through and the result being 'confirmed' is exactly the result of the cited prior work, I cannot score this as a rigorous derivation of anything new. It reads as a serviceable expository summary of the island-formula story in JT gravity -- clearly written, and useful as a pedagogical review -- but it is not a novel resolution of anything, and its rigour is limited by the fact that no computation is actually carried out step by step from the stated action to a concrete, checkable result.

This manuscript sets out to investigate the black hole information paradox in Jackiw-Teitelboim (JT) gravity coupled to a non-gravitational bath, using the quantum extremal surface (QES) prescription and island formula. The stated goal is to explicitly compute the location of the QES in both the eternal and evaporating black hole setups and thereby demonstrate the emergence of the Page curve.

Unfortunately, the paper does not deliver on this promise. While the general physical narrative presented—entropy of Hawking radiation grows linearly before the Page time in the absence of an island, an island configuration becomes entropically favored after the Page time, and the generalized entropy saturates at (or decreases toward) a value consistent with unitarity—is entirely correct and consistent with the seminal works of Penington (2020), Almheiri-Engelhardt-Marolf-Maxfield (2019), and Almheiri-Hartman-Maldacena-Shaghoulian-Tajdini (2020), the manuscript itself contains no original derivation. There is no explicit dilaton profile, no extremization of the generalized entropy functional with respect to the QES location, no closed-form expression for the island's boundary position as a function of time, and no plot or analytic formula for the resulting Page curve. Statements such as 'we compute the location of the quantum extremal surface explicitly' in the abstract are not substantiated anywhere in the body of the text, which remains entirely qualitative and descriptive.

From a rigour standpoint, the treatment of the evaporating black hole case is particularly thin: phrases like 'the dilaton decreases with time...the QES moves inward...the island appears slightly before the Page time' summarize known results without any supporting equations, boundary conditions, or explicit computation of the dilaton's time dependence under absorbing boundary conditions. Given that the existing literature (e.g., Rocha, Gautason et al., Hollowood-Kumar, Anegawa-Iizuka) provides concrete calculational frameworks for exactly this scenario in JT gravity, the absence of any comparable derivation here is a significant shortcoming.

The reference list is also incomplete: it omits foundational papers directly relevant to the topic, including Almheiri, Mahajan, Maldacena and Zhiboedov's original island computation, Engelhardt and Wall's quantum extremal surface proposal, and the West Coast model paper (Penington, Shenker, Stanford, Yang) which is the standard companion result to AEMM's East Coast model discussed here. Hawking's original 1975/1976 papers establishing the paradox itself are also not cited.

In terms of clarity, the paper is well-written at the level of prose and would serve as a reasonable pedagogical summary for a newcomer to the field, but it does not constitute a research contribution as it stands. There is no new physics, no new regime explored, no numerical or analytic result beyond what is already established, and no discussion of subtleties (validity of semiclassical approximation, higher-order corrections, or comparison with existing explicit JT+bath calculations).

Given the complete absence of explicit calculations promised by the abstract, the lack of novel content beyond a restatement of well-known results, and missing key references, I recommend rejection in current form. Should the authors wish to resubmit, they should include the actual extremization calculations, explicit formulas for the QES location and dilaton evolution, a genuine derivation of the Page curve (with plots), and a complete engagement with the existing literature to clarify what new contribution, if any, is being made.

This manuscript offers a thoughtful attempt to connect the island formula with the Page curve in JT gravity, but the paper still reads more as an outline of a possible reconciliation than as a fully established derivation. The topic is highly significant, and the proposed connection between semiclassical gravity and information recovery is inherently interesting. The writing is clear and the exposition is accessible to readers familiar with the field. The main limitation is that the argument is not yet rigorous enough to support the stronger claims that are made about resolving the information paradox. The derivation appears to rely on several steps that are only sketched, and the paper does not clearly show where the island formula and the Page curve are derived from the same underlying assumptions or where the crucial assumptions enter. The manuscript would be much stronger if it stated the precise hypotheses, the domain of validity of the approximations, and the extent to which the proposed resolution depends on particular semiclassical ingredients. As written, the paper is an interesting and potentially valuable contribution, but it remains too incomplete for a high rigour score.

This manuscript addresses the black hole information paradox using the quantum extremal surface (QES)/island formula prescription within Jackiw-Teitelboim (JT) gravity coupled to a non-gravitational bath. The abstract and introduction promise an explicit derivation of the QES location and the resulting unitary Page curve for both an eternal two-sided black hole and an evaporating black hole. However, upon reviewing the body of the paper, none of these explicit computations are actually carried out or presented. The 'Page Curve Calculation' section merely restates the well-known qualitative results from the foundational island-formula papers (Penington 2019; Almheiri-Engelhardt-Marolf-Maxfield 2019; Almheiri-Hartman-Maldacena-Shaghoulian-Tajdini 2019): that the no-island entropy grows linearly with time (the standard Hawking result, S ~ (πc/3β) t), that a stationary island at the bifurcation surface gives a constant entropy 2S_BH, and that comparing the two saddles reproduces the Page curve. For the evaporating case, the paper only qualitatively states that the dilaton decreases, the QES moves inward, and the island appears near the Page time, with the entropy eventually vanishing — again without any explicit equations, dilaton profiles, or boundary-particle trajectories that would substantiate the claim of an explicit calculation.

This is a serious shortcoming for a paper whose stated contribution is precisely the explicit computation of the QES location in JT gravity. JT-gravity island calculations already exist in the literature (most notably Almheiri, Mahajan, and Maldacena, arXiv:1910.11077), which derive the extremization conditions, the dilaton profile in the presence of a bath, and the resulting Page curve in detail, including both eternal and evaporating cases. The present paper does not engage with or build upon this prior work in any technical way, nor does its reference list include it, despite its clear relevance. The reference list is minimal (only four entries), omitting many essential works in this area (e.g., Almheiri-Mahajan-Maldacena 2019, Penington-Shenker-Stanford-Yang 2019, Hartman-Maldacena replica wormhole papers, and follow-up JT-specific island analyses).

In terms of rigour, the paper falls short of standard research-paper expectations: there are no equations beyond the JT action and the generic island formula (both textbook expressions), no derivation of the generalized entropy extremization, no specification of the CFT matter content or the number of fields used, and no explicit boundary conditions at the JT-bath interface. The 'Discussion' section is likewise generic, offering no new physical insight, subtlety, or technical result that would differentiate this work from an introductory review of the island-formula literature.

On the positive side, the paper's exposition of the conceptual picture (why the Page curve emerges, the role of the island, the competition between island and no-island saddles) is accurate and clearly written, and could serve as a decent pedagogical introduction for readers unfamiliar with the topic. The choice of JT gravity as a tractable toy model is well-motivated and consistent with the field's standard approach.

However, as a research contribution, the paper does not meet the bar for publication. It lacks explicit new calculations, fails to differentiate itself from existing literature that already performs the claimed computations in greater detail, and does not offer any novel physical insight, extension, or quantitative result. I recommend rejection in the current form. If the authors wish to resubmit, they should include the actual explicit derivations promised in the abstract (dilaton profiles, extremization equations, explicit QES trajectories, and quantitative Page curves with plots), position their results relative to the closely related prior JT-gravity island literature (especially Almheiri-Mahajan-Maldacena), and expand the reference list and discussion of open issues (e.g., replica wormhole justification, backreaction, boundary particle dynamics) to establish the novelty and technical rigor of the work.