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The paper argues that a specific analytic continuation of a thermal four-point correlator turns one-sided boundary data into local flat-space scattering amplitudes about a bulk point inside the black hole, making the interior readable from

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2026-08-03 16:58 UTC pith:IHXY3WL4

load-bearing objection A candid interior-scattering proposal that extends the bulk-point program, but the load-bearing analytic continuation through the horizon is asserted rather than proven. the 3 major comments →

arxiv 2512.10912 v2 pith:IHXY3WL4 submitted 2025-12-11 hep-th gr-qchep-ph

Imprint of the black hole interior on thermal four-point correlators

Joydeep Chakravarty This is my paper
classification hep-th gr-qchep-ph PACS 04.70.-s11.25.Tq04.60.-m
keywords black hole interiorthermal correlatorsAdS/CFTbulk point singularityflat-space scatteringboundary hyperboloidsanalytic continuationholography
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.

The paper proposes that boundary thermal correlators of a single-sided planar AdS black hole, smeared into directed wavepackets and analytically continued through an integral transform, can be read as local flat-space scattering amplitudes about a bulk point inside the horizon. The central formula (4.15) factorizes the continued correlator into a 3→1 amplitude with a bulk momentum-conserving delta function, with the left-boundary operators created by the transform (4.7). If this dictionary holds, interior physics—including the approach to the singularity—is not hidden from exterior observers but is extractable from four-point thermal data at an exponential-in-frequency cost that can be boosted away. The paper argues this extends the exterior bulk-point program to the interior, with boundary hyperboloids flattening and the signal dimming as the singularity is approached.

Core claim

The claim is that a particular analytic continuation of a thermal four-point function—taking late-time right-boundary operators to left-boundary operators via t_R → −t_L − iβ/2 with a smearing kernel (4.7)—turns a one-sided exterior experiment into a two-sided interior one. The continued operators create local right-moving modes behind the horizon, and the correlator factorizes as in Eq. (4.15) into an integral over bulk momenta of a flat-space-like scattering amplitude iM about an interior bulk point X. This provides a state-dependent operator dictionary (Table 1) mapping left/right smeared boundary operators to interior creation/annihilation modes. The paper also derives the boundary hyper

What carries the argument

The carrying object is the smearing transform (4.7), a Fourier-integral kernel f_β with poles at δt′ = −δt ± iβ/2 that implements the light-cone-coordinate analytic continuation t_R → −t_L − iβ/2 and maps right-boundary late-time operators to left-boundary operators with an O(e^{−βω/2}) suppression that is boosted away. Around this sits the WKB bulk-to-boundary propagator whose geodesic phase function S(p,X) localizes the smeared operators to plane waves about a bulk point and supplies the dictionary between boundary operators and local creation/annihilation modes. The factorization formula (4.15) is the output of this machinery.

Load-bearing premise

The load-bearing premise is that the analytic continuation specified in (4.3)–(4.4) and implemented by the transform (4.7) genuinely carries the WKB phase and the mode dictionary to a real local bulk point in the interior; the paper asserts this continuation without a proof that it is the correct saddle, and it notes that the resulting complex contour violates standard positivity criteria for viable path-integral geometries.

What would settle it

Locate an exact or numerically controlled thermal four-point correlator in a solvable planar black hole (e.g., the three-dimensional case) and evaluate it on the continued contour; if the result does not develop the predicted momentum-conserving peak of a 3→1 amplitude, or if continuing the contour slightly differently changes the amplitude, then Eq. (4.15) is not the physical saddle the paper claims.

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

If this is right

  • Interior local scattering amplitudes (3→1 and 2→2) are encoded in one-sided thermal four-point correlators, so no second boundary is needed to probe behind the horizon.
  • The boundary signature of interior bulk points is carried by boundary hyperboloids; as the point approaches the singularity, hyperboloids flatten and the correlator intensity falls as z^{−2d+4} for d≠2,3, with the d=2 case giving a constant.
  • Experiments must be timed between β/2 and the scrambling time; outside this window two-point saddles contaminate the four-point signal.
  • The dictionary is state-dependent and comes with an exponential suppression e^{−βω/2}, which the boosted kernel removes to make the measurement O(1).
  • The continuation violates standard positivity criteria for viable complex path-integral geometries, a fact the author acknowledges while arguing the concrete setup still yields physically meaningful conclusions.

Where Pith is reading between the lines

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

  • A natural testable extension is to derive the same factorization from a Lorentzian AdS/CFT computation without the complex contour; if the contour cannot be deformed through a positivity-admissible region, the claim that the continued correlator is physically meaningful needs a separate positivity-based justification.
  • The dictionary suggests a new diagnostic for state dependence: a localized perturbation deforming the thermofield-double state should produce a computable phase shift in the interior scattering amplitude, which could be checked in a two-sided setup.
  • Because the singularity signal is a smooth dimming rather than a sharp jump, a cleaner smoking gun would be a sign change or cusp in the phase of the amplitude, not just its intensity; the author notes this concern about focal points explicitly.
  • One could test the contour by asking whether the continued correlator satisfies the expected thermal periodicity in the left time; the transform is engineered to do so, and a failure would indicate a missed branch cut in the analytic continuation.

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 / 3 minor

Summary. The paper proposes that the interior of a single-sided planar AdS black hole leaves an imprint on thermal four-point correlators. The construction starts from the exterior dictionary of Caron-Huot–Chakravarty–Namjou, in which boundary operators smeared with directed wavepackets factorize to flat-space scattering amplitudes about a bulk point. The author extends this to an interior bulk point by an analytic continuation of late-time right-boundary operators to left-boundary operators, implemented through the integral transform (4.7) with the kernel (4.6), exploiting poles at t' = −t ± iβ/2. This maps late-time absorption/emission operators on the right boundary to left-boundary operators, with an accompanying O(e^{−βω/2}) suppression that is boost-compensated. The claimed output is a dictionary (Table 1) and a factorization formula (4.15): a radar-type interior correlator ⟨Ψ|O_{x4} O_{L,y3}^† O_{x2}^† O_{x1}^†|Ψ⟩ reduces to a 3→1 flat-space amplitude with a momentum-conserving delta function about an interior bulk point. A similar 2→2 formula (4.20) is given for the two-timefold correlator. The paper also analyzes boundary hyperboloids for interior points, finding that they flatten near the singularity, and gives a 2d CFT/BTZ example in §3.5. The central physics claim is that local interior scattering — including approach to the singularity — is imprinted on and extractable from boundary thermal correlators.

Significance. If the central claim holds, this would be a significant step: a concrete, perturbatively calculable observable in a single-sided thermal CFT that encodes local flat-space physics in the black hole interior, going beyond the two-point-function probes of [24] and providing a state-dependent dictionary of the Papadodimas–Raju type. The paper is explicit and honest about its limitations: it acknowledges in §6.1 and the Fig. 2 caption that the geometric continuation violates KSW positivity criteria, and in §4.3.1 that the WKB description breaks down near the singularity. It also includes concrete 2d CFT formulas (3.40)–(3.41) that could in principle be checked numerically or analytically. The main strength is that the proposal is specific enough to be tested: an exact BTZ saddle-point computation, or a direct evaluation of the continued WKB propagator across the horizon, would either confirm or falsify the dictionary. However, as it stands, the load-bearing analytic continuation is asserted rather than proven, and the factorization (4.15) is proposed rather than derived. These are fixable within the manuscript's scope, which is why I recommend major revision rather than rejection.

major comments (3)
  1. [§4.2, Eq. (4.7), Table 1, Fig. 2 caption, §6.1] The central step of the paper is the analytic continuation of right-boundary future operators to left-boundary operators via the poles of f_β in (4.6) and the Kruskal continuations (4.3)–(4.4), t_R → −t_L − iβ/2. It is asserted that this continuation takes the WKB bulk-to-boundary propagator (A.2) and its Hamilton–Jacobi phase S(p,X) to a genuine interior bulk point, so that the continued operators O_L create local right-moving modes behind the horizon. But the WKB solution and its subleading corrections (Appendix B) are derived only in the exterior region z < z_h; their analytic continuation through the branch point at z_h is not controlled. The paper explicitly acknowledges in the Fig. 2 caption and §6.1 that this operation 'violates path integral rules' and violates KSW positivity, yet claims it is 'no surprise that we still obtain physically meaningful conclusions.' This is a load-be
  2. [Eq. (4.15) and §4.3.2] The factorization formula (4.15) is stated as an approximation, but it is not derived from the boundary correlator in the way that the exterior factorization (2.15) is derived in §2.1.4. In the exterior, the factorization follows from the bulk-point singularity (3.10), the Hamilton–Jacobi phase relation (2.8), and the localization of wavepackets to a unique bulk point. In the interior, the corresponding steps are assumed: the continued phase S(p,X) is assumed to select a unique interior bulk point, and the wavepacket overlap is assumed to produce the flat-space delta function δ^{d+1}(P1+P2−P4−Q3). The manuscript would need to show this explicitly — for example, by inserting the continued version of (2.5) into the left-right correlator and performing the wavepacket integrals — to elevate (4.15) from an ansatz to a consequence. As written, the factorization is plausible but unproven.
  3. [§4.3.1 and §5.2.1] The domain-of-validity discussion introduces a lower bound β/2 < t_f < O(t_scr) because 'the WKB description breaks down near the black hole singularities' and because competing saddles (4.11) can dominate at intermediate times. However, the paper's advertised 'imprint' includes the approach to the singularity: the singularity signal (5.10) is computed from the same WKB description that the paper admits is invalid there. Moreover, the competing-saddle criterion (4.11) shows that for generic wavepacket parameters the claimed four-point saddle may not be the dominant contribution, but no systematic analysis of the regime where it is dominant is given. Thus the extractability claim for the interior, especially near the singularity, is not quantitatively supported.
minor comments (3)
  1. [§4.1, Eq. (4.4)] The notation 'Im z_* ~ 0' is imprecise; the text says 'net zero imaginary part of the z_* coordinates' but the equation could be misread. Please clarify whether the Tortoise coordinate returns to a real value after the full R→F→L sequence, and state the branch choices explicitly.
  2. [§3.5.3 and §5.1] The equations for the boundary hyperboloids (3.44) and (3.45) switch between arccoth and arctanh for the exterior and interior, but the derivation of this switch from the analytic continuation is not shown. A short derivation would help the reader trust these formulas, which are used later for the singularity analysis.
  3. [Throughout] There are several places where 'right' and 'left' CFT operators are used with slightly different sign conventions (e.g., (4.5) versus (4.7)). A summary table of the conventions for t_L, the sign of the frequency modes, and the location of the poles would improve readability.

Circularity Check

2 steps flagged

Interior amplitude is defined by the same continuation that is said to 'predict' it; Eq. (4.15) restates the proposed dictionary.

specific steps
  1. self definitional [§4.3.2 and Fig. 3 caption; construction in §4.2.1, Eq. (4.7)]
    "The analytic continuation can be thought of as pushing the bulk point X from a well-defined exterior scattering experiment on single timefold in Fig 1 to an interior scattering experiment with bulk point X′."

    The interior scattering experiment is defined as the image of the exterior experiment under the same analytic continuation (4.3)–(4.4) that underlies the transform (4.7). The paper then presents Eq. (4.15) as a derived factorization into an interior amplitude, but the interior bulk point X′ and the left-boundary insertions O_L are manufactured by that continuation. The claimed 'imprint' is therefore a restatement of how the observable was constructed, not an independent extraction from the thermal correlator.

  2. self definitional [§4.2.2, Eq. (4.8) and Table 1; used in Eq. (4.15)]
    "Here O_{L,p,σ} are operators obtained by acting upon O†_{p,σ} with the integral transform (4.7), which in turn introduces local right-moving modes near X. This generalizes the exterior dictionary of [25,26] to the interior, where we showed its plane wave limit in Table 1."

    The dictionary is the assumption that the continued left operator O_L creates the interior oscillator \tilde a†_{X,P}. Substituting this dictionary into the four-point function yields the factorization formula (4.15) by one algebraic step; no independent solution of the interior bulk-to-boundary problem is supplied, and the Hamilton-Jacobi function is only 'extend[ed] ... using the continuation' from the right exterior. Thus Eq. (4.15) carries no information beyond the proposed dictionary itself.

full rationale

The exterior machinery is largely self-contained: the WKB propagator is re-derived in Appendices A and B, and the flat-space factorization template (2.15) follows standard bulk-point kinematics. The self-citations to [24–26] supply the exterior dictionary and the two-point-singularity timescale, but those are not the target result and are not invoked as uniqueness theorems; they are not the basis for the circularity score. The circularity is definitional and located in the interior step. The paper constructs the left-boundary operators O_L via the integral transform (4.7), which is built on the same analytic continuation (4.3)–(4.4) said to move the bulk point from X to an interior X′ (Fig. 3). It then assigns these operators to interior oscillators \tilde a† via Eq. (4.8)/Table 1. Eq. (4.15) is obtained by substituting this dictionary into the four-point function. Thus the 'prediction' that the thermal correlator factorizes to an interior scattering amplitude is a restatement of the proposed dictionary rather than an independent consequence. The paper is candid that this is a proposal and that the continuation violates KSW positivity criteria (Fig. 2 caption, §6.1), and it explicitly leaves the WKB behavior behind the horizon as an extension ('we can extend the Hamilton-Jacobi function S(p,X) using the continuation'). These caveats are weighed as correctness risks rather than additional circularity. An exact saddle-point check (e.g., in BTZ) would be needed to break the definitional loop; absent that, the central claim is coextensive with its dictionary input.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The central claim rests on the WKB/dictionary formalism inherited from [25,26], plus the specific analytic continuation (4.7)–(4.4) proposed here. No new physical entities are introduced; the 'interior oscillators' ã are standard bulk modes in the Kruskal extension. The main ad hoc element is the KSW-violating continuation and the chosen boost factor.

free parameters (2)
  • Wavepacket widths σ_x, σ_y = chosen freely; σ > β/2 in (4.5)
    The smearing widths in (2.2)/(4.5) are hand-picked experimental resolutions; the derivation's regime is σ > β/2, and the width appears in the exponential damping e^{-ν²σ₀²/2} in (3.40). Results are not fitted to data.
  • Boost factor e^{βω/2} in (4.6) = exactly e^{βω/2}
    Introduced ad hoc in the smearing kernel f_β(δt,δt') to cancel the O(e^{-βω/2}) suppression of the analytic continuation; it is a normalization that reverses a computed suppression rather than a fitted constant.
axioms (6)
  • domain assumption AdS/CFT duality: boundary CFT thermal state dual to planar AdS black hole (2.3)
    Invoked throughout as the setting; standard but unproved within the paper.
  • domain assumption WKB/geometric-optics approximation for bulk-to-boundary propagator at βω ≫ 1 (Eq. (A.2), §2.1.1)
    Central; the entire phase-factor dictionary rests on the WKB phase S(p,X) and its first-order variation (2.8).
  • domain assumption KMS/thermal identification O_R(−t − iβ/2) = O_L(t) and its use behind the horizon (Eq. (4.7), (4.3)–(4.4))
    Standard in TFD, but its extension into the interior (and cancellation of e^{−βω/2}) is the paper's key assumption; not derived from the bulk equation of motion.
  • domain assumption Universality of the flat-space factorization formula (2.15) for any bulk point, including the interior (Eq. (4.15))
    The paper assumes the exterior factorization of [25] continues to hold behind the horizon without explicit proof.
  • ad hoc to paper Validity of the KSW-violating analytic continuation as a physical saddle (§6.1, Fig. 2 caption)
    The paper acknowledges the violation of Kontsevich-Segal-Witten criteria but asserts the result is still meaningful; no supporting argument is provided.
  • domain assumption Validity of the interior description for k ≪ N insertions (footnote 2)
    The paper bounds the domain of validity to k ≪ N up to O(e^{-N}) corrections but does not derive this bound.

pith-pipeline@v1.3.0-alltime-deepseek · 38552 in / 14460 out tokens · 132757 ms · 2026-08-03T16:58:40.247635+00:00 · methodology

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read the original abstract

We consider correlators smeared against directed wavepackets over a thermal state dual to a single-sided planar AdS black hole. In the large frequency limit, our measurement is simplified using a bulk WKB description. We propose a dictionary that maps the action of smeared boundary operators to flat-space oscillators near an interior bulk point on the thermal state, by analytically continuing late-time operators from the right to the left boundary via an integral transform. Using the dictionary the smeared correlator factorizes to a flat-space like scattering amplitude about the interior event. Our transformed correlators describe local physics in the two-sided black hole interior, while incurring a suppression of $\mathcal{O}(e^{-\beta \omega / 2})$. These measurements necessitate a non-trivial time ordering of operators living on boundary hyperboloids which are causally connected to the past light cone of the bulk point, as well as on a corresponding future branch.

Figures

Figures reproduced from arXiv: 2512.10912 by Joydeep Chakravarty.

Figure 1
Figure 1. Figure 1: (a) Radar scattering process ⟨Ψ| Ox4,p4,σ4 Oy3,q3,σ3 O † x2,p2,σ2 O † x1,p1,σ1 |Ψ⟩ with a bulk point in black hole exterior as in [25,26], see equation (2.1) for the definition of smeared operators. The dashed line is trace over all final states that may fall behind the horizon. The process has three early-time (two emission and one absorption) operators and a single late-time (absorption) operator. (b) Ou… view at source ↗
Figure 2
Figure 2. Figure 2: (a) and (b) On a Schwinger Keldysh (SK) fold with a thermal identification, the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The analytic continuation can be thought of as pushing the bulk point [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Going from the Euclidean to the radar configuration. Crossing these light cones [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Analytic continuation along the complexified paths [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The path of the cross ratios for the holographic cameras configuration over the com [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Different regions using lightcone coordinates [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Exterior experiment with two timefolds with the contour ( [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The transformation in boundary time to a deformed boundary contour which takes us [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Boundary hyperboloids for the AdS5 black hole corresponding to a radially infalling geodesic T = z∗, with X = 0, plotted over a fixed range of shooting angle. The red line is just before the infalling geodesic crosses the horizon. equation for the boundary hyperboloids takes the form: −ω 2 coth2 t − T zh + p2 coth2 y − X zh = 0, (5.5) while the corresponding equation with an interior bulk point takes the … view at source ↗
Figure 11
Figure 11. Figure 11: The hyperboloids for the BTZ black hole corresponding to a radially infalling geodesic [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗

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