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Under a horizon-preserving radial perturbation, the variation of a thermal one-point function is controlled by the variation of the boundary-to-horizon geodesic length, in the large-dimension limit.

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2026-08-03 10:36 UTC pith:QJCNTS7B

load-bearing objection A plausible robustness check of Kraus-Maloney that currently fails its own quantitative test: the m-scaling is inconsistent across (5.7), (5.8), and (5.13). the 3 major comments →

arxiv 2601.09397 v4 pith:QJCNTS7B submitted 2026-01-14 hep-th gr-qc

Geodesics, One Point Functions and Black Hole Perturbations

Parijat Dey , Arundhati Goldar , Nirmalya Kajuri This is my paper
classification hep-th gr-qc
keywords AdS/CFT correspondenceBTZ black holethermal one-point functiongeodesic lengthWKB approximationblack hole perturbationslarge conformal dimensionholographic correlators
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tests whether the exponential relation between thermal one-point functions and boundary-to-horizon geodesic lengths, previously established for the Euclidean BTZ black hole, survives small metric perturbations. It considers a radial deformation f(r)→f(r)+εδf(r) that preserves the horizon and computes both sides to first order in ε. The main result is that, in the large conformal-dimension limit, the variation of the one-point function is proportional to e^{-mℓ_hor} δℓ_hor, with δℓ_hor the first-order change in the renormalized geodesic length. The derivation relies on WKB approximations for the Green function and bulk-boundary propagator, and these approximations are shown to match the exact large-dimension hypergeometric expressions. If correct, the result shows that heavy operators remain faithful probes of bulk geometry even in slightly deformed black hole backgrounds.

Core claim

The central claim is that the one-point function of a heavy primary operator, dual to a bulk scalar of mass m, obeys δ⟨O⟩ = δ(e^{-mℓ_hor}) ∝ e^{-mℓ_hor} δℓ_hor at first order in a radial, horizon-preserving perturbation of Euclidean BTZ. The paper proves this by computing the first-order change in the zero-mode bulk-boundary propagator, δK0(r) = -m K0(r) δℓ(r), using WKB forms for the propagator and Green function, and then verifying these WKB forms are nothing but the large conformal-dimension limits of the exact hypergeometric solutions. The result is confirmed for a concrete power-law perturbation H(r)=r^p, where the saddle-point evaluation of the one-point function reproduces the geodesi

What carries the argument

The key object is the zero-mode bulk-boundary propagator K0(r) and its first-order variation δK0(r) under the metric perturbation. The computation proceeds via WKB approximations: the Green function takes the form e^{-m|ℓ(r)-ℓ(r')|}/(2m[rr'√(f(r)f(r'))]^{1/2}) and the propagator behaves as e^{-mℓ(r')}/√(r'f(r')^{1/4}), where ℓ(r) is the radial geodesic length. The dominant large-m contribution to δK0 collapses to -m K0 δℓ(r), which directly links the propagator correction to the geodesic length variation. The paper also derives the large-h asymptotics of the hypergeometric functions appearing in the exact zero-mode Green function and propagator, showing they reproduce the WKB expressions, th

Load-bearing premise

The perturbation is required to be purely radial with no change to the redshift function (δχ=0), which is sourced only by matter with T^t_t = T^r_r; if the perturbation has time dependence or off-diagonal components, the simple proportionality between δ⟨O⟩ and δℓ_hor could fail.

What would settle it

Compute the first-order variation of the one-point function in the large-dimension limit for a perturbation that includes a nonzero δχ(r), such as a metric with e^{2εχ(r)}(f+εδf(r))dt² + ...; if the result differs from e^{-mℓ_hor}δℓ_hor at leading order, the relation is confined to the δχ=0 class.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The exponential one-point function/geodesic-length relation is stable under the class of horizon-preserving radial perturbations, so heavy operators can be used to measure small metric deformations from the boundary.
  • For any static perturbation of this type, the first-order correction to a thermal one-point function can be computed directly from δℓ_hor without solving the full bulk wave equation.
  • The structure of the derivation suggests the same relation holds for any static black hole geometry where the WKB approximation for the propagator is valid, not just BTZ.
  • The Euclidean result carries over to Lorentzian static time-slices, so similar geodesic-imprint statements apply to real-time thermal correlators (as the paper notes).
  • The dominant contribution to δ⟨O⟩ comes solely from δK, while variations of the measure and of ⟨χ²⟩ are subleading at large m, simplifying future computations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The restriction to δχ=0 means the result does not cover perturbations that alter the redshift function; if such a deformation were included, the one-point function variation might acquire an additional contribution not captured by δℓ_hor.
  • One could test the universality of δK0 = -m K0 δℓ(r) by applying the same WKB procedure to Schwarzschild-AdS or higher-dimensional black holes, where exact hypergeometric verification is harder but the leading-order relation should still hold.
  • The saddle-point structure hints that time-dependent perturbations could be probed by inserting the perturbation's Fourier modes; the zero-mode dominance might be lost, potentially yielding corrections beyond δℓ_hor that encode the time dependence.
  • Because the relation is parameter-free and background-independent within the WKB regime, it may be a general property of holographic one-point functions for heavy operators, possibly extending to non-black-hole geometries with a natural 'horizon' cutoff.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper tests whether the holographic relation between thermal one-point functions and the boundary-to-horizon geodesic length, previously established for Euclidean BTZ, survives infinitesimal radial perturbations of the metric. For a perturbation of the form f(r) -> f(r)+epsilon delta f(r), the authors compute the first-order variation of the bulk-boundary propagator using WKB methods, compare it with exact hypergeometric asymptotics, and evaluate the one-point function by saddle-point integration. They claim the central relation delta <O> proportional to delta(e^{-m l_hor}) holds at large mass. The physical setup is clear and the goal is well motivated, but the quantitative verification contains several unresolved algebraic inconsistencies.

Significance. If established, the result would be a useful robustness statement: heavy scalar operators would remain faithful probes of bulk geometry even when the BTZ background is perturbed by a restricted class of matter sources. The paper includes a physically motivated derivation of the allowed perturbation from Einstein equations (Appendix A), a WKB treatment of the Sturm-Liouville problem, and an attempt to justify the WKB approximation from exact large-h hypergeometric expansions. These are useful technical elements. However, the central quantitative verification currently fails: three expressions for the same saddle-point result disagree in the power of m and in the exponential factor, and the exact asymptotic formulas used in the matching are internally inconsistent. The result may be true, but the paper as written does not establish it.

major comments (3)
  1. [Section 5, Eq. (5.8) and Appendix C] The saddle-point evaluation in Appendix C gives I(m)=sqrt(pi)/2 r_+^{1-m} delta_l(r_+) m^{-3/2} (Eq. C.2). Substituting this into Eq. (5.6) yields delta<O> = sqrt(pi) m^{-1/2} r_+^{-m} e^{-2 pi r_+} delta_l(r_+), not the m^{3/2} quoted in Eq. (5.8). The worked example in Eq. (5.13) quotes m^{-3/2} and replaces e^{-m l_hor} by e^{-m r_+}, again without derivation. These three expressions cannot all be correct. Since the central claim (1.4) is precisely that delta<O> is governed by delta(e^{-m l_hor}), the m-scaling determines whether the geodesic-exponent relation holds at leading order. As written, the result is quantitatively unverified.
  2. [Eq. (3.4) vs Eqs. (4.19)/(5.5)] The first-order geodesic variation is defined in (3.4) as an integral of H(r)/sqrt(r_+^2 - r^2) from r_+ to infinity, but Eq. (4.19) and Eq. (5.5) define delta_l(r) = (1/2) integral from infinity to r of H(r')/sqrt(f(r')) dr'. Apart from the missing factor 1/2, the integrand in (3.4) is imaginary for r > r_+ (likely a typo for sqrt(r^2-r_+^2)), and the orientation of the integration limits differs. Since (1.4) equates delta<O> with delta_l_hor, these discrepancies are load-bearing. The sign structure of the preceding perturbation theory also needs checking: from delta K = - G delta_square K, Eq. (3.17) should contain an overall minus sign relative to what is written.
  3. [Section 4.2, Eqs. (4.29)-(4.30) and Appendix B] The large-h asymptotic formulas used for the exact matching are inconsistent with each other. Eq. (4.30) has a prefactor 1/(2 sqrt(pi) h x^{1/4}), while the derivation in Appendix B, Eq. (B.5), gives 1/(4 sqrt(2 pi h) x^{1/4}); these differ by a factor of order sqrt(h). The paper states that substituting these asymptotic forms into the exact Green function and bulk-boundary propagator recovers the WKB expressions (4.31)-(4.33), but with different h-dependent prefactors this does not follow automatically. The cancellation must be shown explicitly. This matters because Section 4.2 is the justification that the WKB approximation is valid; as written, that justification is not established.
minor comments (5)
  1. [Section 5, near Eq. (5.13)] The line 'H(r)=r^p, p<0 1' appears to contain a stray '1' from the footnote marker; please clean up the typesetting and separate the footnote.
  2. [Section 2, Eq. (2.16)] The renormalized length l_hor is used before it is properly defined. It would improve readability to state the cutoff prescription, e.g., l(r)=arccosh(r/r_+) minus log(2r), before taking the large-m limit.
  3. [Section 6] The sentence claiming the result 'should straightforwardly extend to the Lorentzian continuation' is not demonstrated. Either provide a short argument or remove the unsupported claim.
  4. [Appendix B] The derivation of Eq. (B.5) should be reconciled with the expression used in Eq. (4.30); the two differ by a non-negligible h-dependent factor (see major comment).
  5. [References] Reference [13] is given only as an arXiv identifier with no publication status; consider updating if a journal version exists.

Circularity Check

0 steps flagged

No significant circularity: WKB exponential forms are independently matched to exact hypergeometric asymptotics; central claim has independent content.

full rationale

The central derivation is not circular. The WKB replacements K0(r') = e^{-mℓ(r')}/√(r' f(r')^{1/4}) (Eq. 4.9) and G0(r,r') = e^{-m|ℓ(r)-ℓ(r')|}/(2m[rr'√(f(r)f(r'))]^{1/2}) (Eq. 4.8) are not imposed by the target relation; they are checked in Sec. 4.2 against the large-h asymptotics of the exact hypergeometric expressions (Eqs. 4.29-4.33), which independently reproduce the same exponential forms. The leading result δK0 = -mK0δℓ (Eq. 4.26) follows from a Laplace/saddle evaluation of the perturbed Klein-Gordon integral (Eqs. 4.14-4.25), not simply from differentiating the claimed final relation. The one-point function integral (5.2)-(5.7) is then evaluated by saddle point in Appendix C; no parameter is fitted to a target output and no prediction is renamed from a fit. There is no load-bearing self-citation: the unperturbed relation (2.16) is quoted from Kraus-Maloney [5] and the propagator from David-Kumar [9], both external prior work, and the perturbation result is derived within the paper. The apparent m-power discrepancy among (5.7), (5.8), and (5.13) is an internal algebraic consistency issue affecting verification, not a circularity; nothing in it shows the output is equivalent to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters are fitted to data: H(r) is an arbitrary radial profile, and the example H=r^p with p<0 is a test choice. The axioms are standard holographic assumptions plus the paper-specific restriction to radial, δχ=0 perturbations. No new entities are introduced.

axioms (4)
  • domain assumption AdS/CFT correspondence and the scalar field setup: two interacting massive scalars dual to primary operators with one large conformal dimension.
    The entire derivation relies on the holographic dictionary converting bulk scalar fields to boundary operators, introduced in §2.1.
  • ad hoc to paper The metric perturbation is purely radial and horizon-preserving: g_tt = f(r)+εδf(r), g_rr = 1/(f(r)+εδf(r)), and the redshift function δχ=0.
    This restricted form is stated in Eq (1.3) and justified in Appendix A only for matter with T^t_t = T^r_r; it excludes more general perturbations.
  • domain assumption Large conformal dimension (large m) justifies WKB and saddle-point approximations for propagators and integrals.
    Used throughout §4 and §5; the paper partially justifies WKB by exact hypergeometric asymptotics in §4.2.
  • standard math The exact hypergeometric asymptotic expansions in Appendix B are valid in the integration regimes (0 < x < 1, large h).
    These saddle-point evaluations of hypergeometric functions are standard mathematical results, invoked in §4.2.

pith-pipeline@v1.3.0-alltime-deepseek · 11818 in / 21198 out tokens · 180719 ms · 2026-08-03T10:36:41.948465+00:00 · methodology

0 comments
read the original abstract

Holographic black holes exhibit a striking relation between thermal boundary one-point functions and bulk geodesic lengths. In the large conformal-dimension limit, the one-point function of a primary operator is given by the exponential of the geodesic length from its boundary insertion point to the horizon. We test the robustness of this relation under perturbations by considering a class of deformations of an Euclidean BTZ black hole and working to first order in the perturbation.We find that, at leading order in the large conformal-dimension limit and to first order in the radial horizon-preserving perturbation, the logarithmic variation of the one-point function is governed by the variation of the renormalized boundary-to-horizon geodesic length. The result is established using WKB and saddle-point methods, and WKB expressions at large conformal dimension are checked against the exact Green function and bulk-boundary propagator.

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Forward citations

Cited by 1 Pith paper

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