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arxiv: 2510.21673 · v2 · submitted 2025-10-24 · ✦ hep-th · gr-qc· hep-ph

Imprint of the black hole singularity on thermal two-point functions

Nima Afkhami-Jeddi , Simon Caron-Huot , Joydeep Chakravarty , Alexander Maloney This is my paper

Pith reviewed 2026-05-18 04:15 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-ph
keywords holographic theoriesthermal two-point functionsblack hole singularityoperator product expansionnull geodesicsnonperturbative correctionsWKB approximationAdS black holes
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The pith

Thermal two-point functions receive exponentially small corrections from black hole singularity geodesics

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

The paper studies two-point correlation functions of light operators in holographic theories at finite temperature and large real frequencies. It establishes that the standard high-frequency expansion coming from the operator product expansion is modified by nonperturbative terms that are exponentially suppressed. These terms are determined by the behavior of null geodesics in the two-sided eternal black hole that reflect off the central singularity. The authors provide a bulk WKB analysis to describe these geodesics and to compute the reflection coefficients needed for the corrections.

Core claim

We show that the high-frequency expansion obtained from the Operator Product Expansion receives exponentially small nonperturbative corrections, which are controlled by null geodesics bouncing off the black hole singularity in the two-sided eternal black hole geometry. We develop a bulk WKB description of these bouncing geodesics and explain how to calculate reflection coefficients at the singularity.

What carries the argument

Null geodesics bouncing off the black hole singularity in the two-sided eternal black hole geometry, described via bulk WKB methods to obtain reflection coefficients

If this is right

  • The perturbative OPE series for high-frequency thermal correlators must be supplemented by these nonperturbative terms.
  • Boundary two-point functions encode information about the interior geometry including the singularity.
  • Reflection coefficients at the singularity become accessible through boundary observables.
  • Similar nonperturbative effects controlled by interior geodesics appear in other high-frequency correlation functions.

Where Pith is reading between the lines

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

  • This mechanism suggests boundary correlators can indirectly probe properties of the black hole interior singularity.
  • The WKB treatment of bouncing geodesics may extend to higher-point functions or different black hole backgrounds.
  • Similar exponentially small corrections could appear when complexified geodesics reach other spacetime singularities.

Load-bearing premise

The two-point functions in the thermal system dual to a single-sided AdS black hole are controlled by geodesics propagating in the two-sided eternal black hole geometry.

What would settle it

An exact high-frequency asymptotic expansion of a two-point function computed in a solvable model such as BTZ that shows no exponentially small terms matching the predicted singularity contributions would falsify the claim.

Figures

Figures reproduced from arXiv: 2510.21673 by Alexander Maloney, Joydeep Chakravarty, Nima Afkhami-Jeddi, Simon Caron-Huot.

Figure 2
Figure 2. Figure 2: FIG. 2. Nonperturbative contributions [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The original real-time contour (in red) for the Fourier [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Stoke’s lines (in blue, with phases [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The spectral density [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Stokes lines (blue, at angles 0 and [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

We consider two-point functions of light fields at finite temperature and large real frequencies in holographic theories. The thermal system is dual to a single-sided AdS black hole. We show that the high-frequency expansion obtained from the Operator Product Expansion receives exponentially small nonperturbative corrections, which are controlled by null geodesics bouncing off the black hole singularity in the two-sided eternal black hole geometry. We develop a bulk WKB description of these bouncing geodesics and explain how to calculate reflection coefficients at the singularity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies thermal two-point functions of light fields in holographic theories at large real frequencies. It claims that the high-frequency expansion obtained from the operator product expansion receives exponentially small nonperturbative corrections controlled by null geodesics that traverse the Einstein-Rosen bridge and bounce off the black hole singularity in the two-sided eternal AdS black hole geometry. The authors develop a bulk WKB description of these geodesics and explain the calculation of reflection coefficients at the singularity.

Significance. If the central claim is established, the result supplies a holographic mechanism linking the black hole singularity to nonperturbative corrections in boundary thermal correlators. The explicit WKB treatment of bouncing geodesics and the associated reflection coefficients constitute a concrete technical contribution that may be reusable in other bulk calculations involving complexified geodesics.

major comments (1)
  1. [Abstract and bulk WKB development] Abstract, paragraph 2 and the bulk WKB section: the mapping of single-sided thermal correlators to geodesics that cross into the two-sided geometry and reflect off the singularity rests on an implicit contour deformation in the complex frequency plane. An explicit demonstration is needed that this contour avoids other saddles or branch-cut contributions that could dominate the exponentially small corrections; without it the nonperturbative control asserted in the central claim remains unsecured.
minor comments (2)
  1. [WKB analysis] Clarify the precise matching conditions between the WKB solutions on either side of the singularity and the boundary conditions used for the reflection coefficient.
  2. [High-frequency expansion] Add a short discussion of how the result reduces to the known OPE expansion in the limit where the nonperturbative corrections are neglected.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. The concern about securing the nonperturbative control via explicit contour analysis is well taken, and we address it directly below.

read point-by-point responses

  1. Referee: [Abstract and bulk WKB development] Abstract, paragraph 2 and the bulk WKB section: the mapping of single-sided thermal correlators to geodesics that cross into the two-sided geometry and reflect off the singularity rests on an implicit contour deformation in the complex frequency plane. An explicit demonstration is needed that this contour avoids other saddles or branch-cut contributions that could dominate the exponentially small corrections; without it the nonperturbative control asserted in the central claim remains unsecured.

    Authors: We agree that the present text leaves the contour deformation implicit and that an explicit verification is required to fully establish dominance of the bouncing-geodesic saddles. In the revised version we will add a new subsection to the bulk WKB development that performs the contour deformation in the complex frequency plane explicitly. This subsection will (i) identify the relevant analytic structure of the integrand, (ii) show that the deformed contour can be chosen to avoid other saddles and branch cuts whose exponential suppression is weaker than or comparable to the singularity-bouncing contribution, and (iii) confirm that no additional nonperturbative terms enter at the same order. We believe this addition will secure the central claim without altering the overall conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from standard holographic bulk WKB to boundary OPE corrections

full rationale

The paper's central claim derives exponentially small corrections to the high-frequency OPE expansion of thermal two-point functions from null geodesics in the two-sided eternal black hole geometry via a bulk WKB analysis. This is a first-principles calculation within the AdS/CFT framework, with no fitted parameters renamed as predictions, no self-definitional loops in the equations, and no load-bearing self-citations that reduce the result to unverified inputs. The single-to-two-sided geometry mapping is presented as a standard holographic duality step rather than a derived output that collapses back to the input. The derivation chain remains independent and externally falsifiable through bulk geodesic calculations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the AdS/CFT correspondence and the validity of the WKB approximation for null geodesics near the singularity; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The thermal system is dual to a single-sided AdS black hole whose correlators are controlled by the two-sided eternal geometry
    Stated in abstract paragraph 2; this is the standard holographic dictionary for finite-temperature states.

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Forward citations

Cited by 6 Pith papers

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  2. Bouncing singularities in Schwarzschild: a geometric origin of the QNM convergence region

    gr-qc 2026-05 unverdicted novelty 7.0

    Bouncing singularities from null geodesics off the black hole singularity set the convergence region of the QNM expansion for the Schwarzschild retarded Green's function.

  3. Bulk-cone singularities and echoes from AdS exotic compact objects

    hep-th 2025-12 unverdicted novelty 7.0

    AdS exotic compact objects imprint bulk-cone singularities from null geodesics and echoes from trapped waves on CFT Green functions, signaling no horizon.

  4. Bouncing singularities and thermal correlators on line defects

    hep-th 2026-03 unverdicted novelty 6.0

    Retarded correlators of displacement operators on line defects in holographic thermal CFTs exhibit bouncing singularities that match between interior-sensitive WKB and boundary-only OPE analyses.

  5. Complex Geodesics in the Nariai Geometry

    hep-th 2026-04 unverdicted novelty 5.0

    Two-point functions in Nariai geometry are sums over complex geodesics whose phases must be retained to eliminate artificial singularities.

  6. Complex Geodesics in the Nariai Geometry

    hep-th 2026-04 unverdicted novelty 5.0

    Obtains the two-point correlator in Nariai geometry as a sum over complex geodesics via heat kernel approximation on sphere products followed by analytic continuation, extending de Sitter results.

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