A Note on Black Hole Entropy and Wormhole Instabilities

E. Velasco-Aja; J.L.F. Barbon

REVIEW 1 major objections 5 minor 2 cited by

A Note on Black Hole Entropy and Wormhole Instabilities

T0 review · 1 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper argues that a microcanonical formulation resolves the Jeans and GPY instabilities of Euclidean wormhole saddle points, so the black hole entropy overlap calculation yields a real, positive result.

desk verdict A legitimate robustness check of the wormhole-overlap entropy program; the main sign cancellation is asserted rather than derived, but the gap is fixable and the paper deserves refereeing. read the letter →

arxiv 2502.00769 v2 pith:DEYW5AHG submitted 2025-02-02 hep-th

J.L.F. Barbon , E. Velasco-Aja This is my paper · ORCID

classification hep-th

keywords blackholeentropyEuclideanwormholesmicrocanonicalensembleGPYinstabilityJeansGrammatrixrankstateoverlapsAdS/CFT

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Recent proposals compute black hole entropy from Euclidean wormhole saddle points that estimate overlaps of specially prepared microstates; these saddle points can develop Jeans and GPY instabilities whose one-loop determinants contribute imaginary factors. This paper argues that passing to a microcanonical ensemble, by inverse Laplace transform from temperature to energy, resolves the problem. The normalized cyclic overlap ratio becomes $1/(\bar N_\Gamma(E_L)\bar N_\Gamma(E_R))^{n-1}$, so the Gram matrix has the expected rank, and the semiclassical overlap is real and positive because the GPY factor $1/i$, the measure factor $1/i$, and a contour-orientation factor $-1$ for negative heat capacity multiply to $+1$. If correct, the Euclidean wormhole program for black hole entropy survives its apparent instabilities and reproduces $\exp(S(E_L)+S(E_R))$.

What carries the argument

The machinery is the microcanonical inverse Laplace transform of partially entangled thermal states, applied to the universality-limit factorization of $n$-replica wormhole partition functions. An orthogonal rotation $J$ isolates the combination $\sum_j\beta_j$ as a single longitudinal variable, producing delta functions that localize the band integrals and yield Eq. (35). The stability argument turns on the steepest-descent contour for the inverse Laplace integral around a saddle with negative heat capacity: the contour is tangential to the real axis and oriented toward negative real values, giving the compensating $-1$, while the branch cut of the analytically continued black hole actions $I_\pm(\beta)$ at $\bar\beta$ is placed along the negative imaginary axis. The product $(-1)\cdot(1/i)\cdot(1/i)=+1$ is what converts the GPY negative mode, a negative eigenvalue localized near a small black hole horizon, into a harmless phase.

What would settle it

Evaluate the microcanonical overlap integral (40) for a concrete model, say a small AdS black hole with explicit one-loop determinants or a random-matrix ensemble with the same band structure, without prescribing the contour orientation or the branch-cut placement; if the steepest-descent flow through the negative-heat-capacity saddle yields a phase opposite to the paper's $-1$, the cancellation fails and the overlap is not real and positive.

Watch

Extended reading notes

Core claim

The central claim is that the apparent failure of the Euclidean path integral as a machine for positive-definite inner products, signalled by imaginary factors from Jeans and GPY negative modes, is an artifact of the canonical fixed-temperature framing. In the microcanonical framing, the negative mode contributes $1/i$, the inverse Laplace measure contributes $1/i$, and the steepest-descent contour around a negative-heat-capacity saddle contributes $-1$; the product is $+1$. Consequently the semiclassical overlap in Eq. (45) is real and positive, and the normalized cyclic overlap in Eq. (35) equals $1/(\bar N_\Gamma(E_L)\bar N_\Gamma(E_R))^{n-1}$, so the Gram matrix has rank $\bar N_\Gamma(E_L)\bar N_\Gamma(E_R)$ and the entropy estimate is $\exp(S(E_L)+S(E_R))$ in the universality limit.

Load-bearing premise

The load-bearing step is the choice of steepest-descent contour for the inverse Laplace integral at negative heat capacity, together with the placement of the branch cut of the black hole action along the negative imaginary axis; the paper calls this placement convenient rather than deriving it from the path integral, and a different contour would turn the compensating $-1$ into $+1$, leaving the overlap imaginary.

Editorial extensions

If this is right

The Euclidean wormhole method for black hole entropy survives the Jeans and GPY instabilities: the microcanonical overlap is real and positive, and the Gram matrix has rank $\bar N_\Gamma(E_L)\bar N_\Gamma(E_R)$.
The microcanonical projection lets the calculation zoom into the energy band where small AdS black holes dominate, precisely the band carrying the GPY instability, and still obtain a positive-definite inner product.
The Jeans instability never afflicts the dominant saddle manifolds in the universality limit, and the GPY instability is confined to an intermediate energy window $E_C<E<E_\ell$ where the phase product neutralizes it.
In the universality limit the exponential part of the overlap becomes $\exp(S(E_L)+S(E_R))$, matching the expected black hole entropy of the two-sided system.
The final entropy estimate is insensitive to the dust-shell details, since the dust worldvolume renormalizes the ADM mass and contributes only subleading corrections to the action.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same sign-cancellation pattern may apply to other Euclidean saddle computations with negative heat capacity, such as gravitational sphalerons in flat space, suggesting that negative modes need not invalidate norm computations once the microcanonical contour is fixed.
A direct microcanonical Euclidean path integral, defined without inverse Laplace transforms, should reproduce the same real, positive overlap; such a check would confirm whether the branch-cut placement here is a consistent definition of the analytic continuation rather than an ad hoc fix.
Because the universality limit must be taken before any flat-space limit, the method as stated does not transfer to asymptotically flat black holes; the dust worldvolume's Euclidean time extent grows with the turning-point radius in flat space, so a different construction would be needed.
The equality between operator averaging and GPI coarse-graining is assumed in the universality limit; testing it at finite dust-shell mass would show whether the microscopic Gram rank emerges from geometry alone or relies on the random-matrix ansatz.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a useful, honest robustness check, and my read is the central calculation holds up. The genuinely new piece is the instability census in the universality limit. Barbon and Velasco-Aja point out that the factorized saddles are thermal AdS spaces with known problems—Jeans modes in the radiation on both X0 and Xmu-, and the GPY negative mode on the small black hole Xmu-—and then show that a microcanonical Laplace transform can assemble the signs so that the cyclic overlap in Eq. (45) is real and positive. The derivation of Eq. (35), with the J rotation that turns the band projection into a one-dimensional integral over the diagonal, is clean. The stability analysis in Section 3 is physically sensible and uses standard results properly. Credit where due: the paper is clear about what it is doing and does not oversell the universality limit.

Now the soft spot. The minus sign that makes the GPY 1/i harmless comes from the orientation of the steepest-descent contour through a negative-heat-capacity saddle, and that orientation is stated as "towards the negative direction of the real axis as shown in Figure 6", not derived from the Bromwich contour in Eq. (20). The branch cut at beta-bar is also placed by convenience. The stress-test note is right that this is the decisive step: if the orientation is wrong, the overlap flips sign and the resolution of the instability collapses. I do not think this is fatal, because the authors flag it openly and point to the direct microcanonical GPI of Marolf-Santos as the natural place to settle it. But it is a real gap, and it is exactly the kind of thing a referee should ask to be closed before publication—either by deriving the contour from the original inversion integral or by matching the Marolf-Santos prescription.

For whom: people working on the Balasubramanian et al. microstate program and on semiclassical Euclidean gravity will want this on hand. It is a robustness check, not a breakthrough; nothing here opens new territory. The citation pattern is normal and the paper is not padded. I would send it to a serious referee. I would want the contour question addressed before fully trusting Eq. (45), but the paper deserves refereeing rather than desk rejection.

Referee Report

1 major / 5 minor

Summary. The paper studies recent Euclidean wormhole proposals for computing black hole entropy through overlaps of partially entangled thermal states (PETS), specifically the construction of Balasubramanian, Lawrence, Magan and Sasieta. It observes that the relevant Euclidean saddle manifolds can exhibit Jeans and GPY instabilities in the universality limit, which threaten the interpretation of the GPI as a positive-definite inner product. The paper's central claim is that a microcanonical inverse-Laplace projection resolves these puzzles. The main results are Eq. (35), giving the normalized cyclic overlap ratio as (1/(Nbar_Gamma(E_L) Nbar_Gamma(E_R)))^{n-1} and hence the Gram rank, and Eq. (45), in which the semiclassical microcanonical overlap is real and positive because the GPY 1/i factor, the Laplace-measure 1/i, and a contour-orientation factor -1 multiply to +1. The paper concludes that the Euclidean wormhole overlap method survives both the Jeans and GPY instabilities and yields the expected microcanonical entropy exp(S(E_L)+S(E_R)).

Significance. If correct, the note removes a potential obstruction to using Euclidean wormhole saddles for black hole state overlaps: a naive GPY negative mode would otherwise make the GPI norm indefinite. The treatment of the i-factor bookkeeping is explicit, the derivation of Eq. (35) via an orthogonal change of variables is clean, and the paper correctly invokes standard results (GPY, Hawking-Page, Wishart/Marchenko-Pastur). The paper is also honest in citing the direct microcanonical GPI of [24] as a formulation that would avoid these issues from the start. The main weakness is that the central sign cancellation in Eq. (45) depends on a contour-orientation and branch-cut prescription that is stated rather than derived, so the central claim is currently conditional. A derivation or independent justification of that prescription would make the argument substantially more robust.

major comments (1)

[Section 3, Eqs. (40)-(45), Fig. 6] The claimed cancellation of unwelcome i factors depends on the sign of the steepest-descent contour through the negative-heat-capacity saddle and on the placement of the branch cut emanating from beta-bar. The text states that the contour is oriented 'towards the negative direction of the real axis' and that placing the cut along the negative imaginary axis is 'a convenient way', but it does not derive these choices from the original Bromwich contours gamma_s in Eq. (20) or from a first-principles definition of the analytically continued GPI. If the contour orientation is reversed, the factor -1 in Eq. (45) becomes +1 and the semiclassical overlap acquires an overall -1; a different cut placement can change which sheet the saddle lies on and hence the sign of sqrt(lambda_GPY). This is the load-bearing step that makes the microcanonical overlap real and positive. I request either a derivation of the contour from Eq. (20) with the branch cut tracked explicitly, or an independent check via the direct microcanonical GPI of [24].

minor comments (5)

[Section 3, Eq. (35)] Because Nbar_Gamma(E) in Eq. (36) is defined as the integral of the very same GPI density of states used in the calculation, Eq. (35) is best described as a consistency check between the GPI cyclic product and the ETH/Gaussian operator-averaging ansatz (28)-(30), rather than as an independent derivation of the black hole entropy. The text makes this reasonably clear, but stating it explicitly would prevent over-reading.
[Figure 6] Figure 6 should explicitly label the branch cut and the arrows indicating the orientations of gamma_+ and gamma_-; the sign of the -1 factor in Eq. (45) is determined by this orientation, so the figure should make the direction unambiguous.
[Section 3, Eq. (40)] The replacement of the smearing function f_Gamma(beta) by its saddle-point value Gamma is justified only if Re(beta_s) Gamma << 1, but the saddle points of the analytically continued integrand can be complex; a brief comment on the validity of this approximation along the deformed contour would be helpful.
[Section 3, final paragraph] The text notes that the direct microcanonical GPI of [24] would avoid the GPY instabilities; adding a sentence on whether the contour prescription of Fig. 6 agrees with the one derived in [24] would significantly strengthen the central argument.
[Section 2, Eq. (11)] The overline notation is used both for GPI evaluation (e.g., Eq. (9)) and for the microscopic operator average (Eq. (11)), with a much later parenthetical clarification; using distinct symbols would reduce the risk of confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main results are conditional consistency checks whose sign and rank follow from stated GPI inputs, with the contour-orientation caveat being a correctness risk rather than a circular reduction.

full rationale

The paper's central derivation is a consistency check. The annealed overlap ratio (35) is obtained from the universality-limit factorization (15) — an input taken from the earlier proposal [4], not from the authors' own prior work — followed by exact changes of variables in the inverse Laplace transform (Eqs. (31)-(36)). The paper itself labels the narrow-band version of this argument a 'mere consistency check' (Section 2). The rank result rank(G)=Nbar_Gamma(E_L)Nbar_Gamma(E_R) is the GPI one-sided density of states integrated over the chosen band: the overlap moments and the band dimension are computed from the same GPI partition function, so the paper is verifying internal consistency of the wormhole-overlap method, not claiming an independent derivation of exp(S). This is not a circular reduction because Nbar_Gamma is not defined as the output rank; the moment computation could in principle have produced a different functional form. The stability argument culminating in (45) does rely on a heavy assumption: the steepest-descent contour through the negative-heat-capacity saddle is asserted to be 'oriented towards the negative direction of the real axis' and the branch cut is placed as 'a convenient way', rather than derived from the Bromwich contour of (20). This is a missing justification and a genuine correctness risk — if the orientation or branch choice differs, the product of i-factors would not be +1. However, a conditional derivation is not circular: the sign of the overlap is a consequence of the stated contour/cut prescription, not the definition of that prescription. The self-citations ([17], [23]) are background references for optical volumes and string thresholds and do not carry the argument. Overall, no claim in the paper reduces to its inputs by construction; the appropriate verdict is no significant circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The microcanonical stability result does not introduce fitted parameters. It relies on the universality-limit factorization taken from [4], on standard results for the GPY and Jeans instabilities in thermal AdS, and on an explicit but not fully derived contour-prescription choice. The band window parameters (center energy and width) are physical choices, not adjusted to obtain the answer; the result is independent of them. There are no invented entities.

assumptions (8)

domain assumption Thin-shell dust model with conserved Newtonian mass m and junction conditions.
The wormhole geometries are constructed by gluing two AdS black holes along a spherical dust shell; Eq. (1) and Fig. 1 in Section 2 inherit this from [4].
domain assumption AdS/CFT dictionary: gravitational path integral equals CFT partition function; GPI with wormhole boundary conditions computes coarse-grained state overlaps.
Used throughout to identify Z[X] with Tr O^dagger e^{-beta H} O e^{-beta H}, Eq. (4), and the GPI overlaps (13).
domain assumption d > 2 so that small AdS black holes exist with beta(mu) below the maximum.
Stated explicitly in Section 2: 'requires d >2, a condition that we shall assume throughout this paper.'
domain assumption Universality limit factorization (15): Z_{i1...in} -> Z(n beta_L) Z(n beta_R) prod C_j as m -> infinity.
Taken from [4] (with refinements in [6]); it is the key simplification that turns the n-replica wormhole into one-sided thermal partition functions and underlies Eqs. (16), (33)-(35).
domain assumption GPY negative mode for small AdS black holes, with lambda_GPY ~ -1/beta^2, and the Jeans instability of high-temperature thermal AdS (beta << ell).
Imported from [11], [19]; the paper does not recompute these determinants, it only uses their scaling and localization.
domain assumption The analytic continuation of the one-loop determinant picks the branch of sqrt(lambda_GPY) that migrates from positive to negative as mu passes the maximum of beta(mu).
Section 3, 'Semiclassical Stability': 'we are taking the positive branch of the square root, consistent with the fact that the GPY negative eigenvalue migrates from an ordinary positive eigenvalue'. This is a convention.
ad hoc to paper The saddle-point dominance phase diagram (Fig. 5) based on the competition of X0, X_mu-, X_mu+ and the mild effect of dust shells on the fluctuation spectrum.
Constructed from estimates in Section 3; it identifies the energy window E_C < E < E_ell where the GPY-unstable manifold dominates, which is essential for the contour argument. The text calls it 'a crude description'.
ad hoc to paper The contour-orientation and branch-cut prescription in Fig. 6 produces the correct analytic continuation.
The -1 sign that cancels the GPY factor comes from choosing the steepest-descent contour toward negative real values and placing the cut along the negative imaginary axis; this is stated rather than derived.

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Cite this review

Pith. "Pith review of A Note on Black Hole Entropy and Wormhole Instabilities." pith.science (2026). https://pith.science/paper/DEYW5AHG

@misc{pith2026250200769,
  author       = {Pith},
  title        = {Pith review of: A Note on Black Hole Entropy and Wormhole Instabilities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DEYW5AHG}},
  note         = {Machine review of arXiv:2502.00769}
}

read the original abstract

We discuss recent approaches to the computation of black hole entropies through semiclassical estimates of appropriate state overlaps, saturated by Euclidean wormhole configurations. We notice that the relevant saddle-point manifolds may exhibit instabilities, thereby compromising the interpretation of the Euclidean path integral as a tool for computing positive-definite inner products. We show that a proper treatment using a microcanonical formulation effectively addresses the puzzles posed by these instabilities.

Figures

Figures reproduced from arXiv: 2502.00769 by the authors.

**Figure 1.** The global structure of eternal AdS black holes with horizons H (µL) and H (µR) featuring long Lorentzian wormholes in their shared interior, supported by a shell of dust with worldvolume Wm. The simplest matter model (and in many ways the more convenient one), is a fluid of dust with energy density σ. Einstein’s equations in the thin-shell approximation imply that the Newtonian mass of the shell m = σ(r) r d−1 Vol(… view at source ↗

**Figure 2.** The Euclidean manifold XµL,m,µR entering the GPI for the norm-squared of the PETS state. The crosses denote the left-right Euclidean horizons In order to infer the form of the CFT microstates, it is useful to switch to Euclidean signature. In [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** The (r, τ ) section of a 4-replica wormhole made of two copies of the Xµ manifold. An identical copy hides in the back, glued through the dust worldvolumes Wi , and a S d−1 lies at each point. Similar geometries can be constructed replacing the Xµ manifold by the vacuum X0 manifold. In the universality limit m → ∞ the Euclidean time extent of the dust ∆τWi shrinks to zero as ℓ 2/r¯ and the path integral (13) factori… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: An Euclidean ‘flat box’ corresponding to a small Euclidean black hole which [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: The bulk inverse temperature βE for thermal states on the manifolds X0 and Xµ, as a function of the microcanonical energy E. Regions of local instability are indicated by discontinuous graph curves. The Jeans instability (dotted line) occurs for β < βJ on both the Xµ− …

**Figure 6.** Figure 6: Contours γ± around a saddle β(Es) in the complex βs plane, for saddles of positive (γ+) and negative (γ−) specific heat, avoiding the branch cut emanating from β¯, the maximal inverse temperature of the black hole manifolds. a behavior Y s=L,R Γ e −Es P W ∆τW e β(Es)Es…

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Pith tools

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