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Timelike-separated boundary points in anti-de Sitter space can be connected by a composite geodesic whose complex length matches the length read off from the CFT two-point function at timelike separation.

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2026-08-03 19:49 UTC pith:67NG4JAE

load-bearing objection A useful refinement of the composite-geodesic recipe for timelike separations, with clean global/BTZ extremization but a Poincaré section that doesn't actually compute the divergent spacelike legs and a degeneracy-fixing step that still imports the CFT answer. the 3 major comments →

arxiv 2511.22168 v1 pith:67NG4JAE submitted 2025-11-27 hep-th gr-qcquant-ph

Composite AdS geodesics for CFT correlators and timelike entanglement entropy

Hardik Bohra , Allic Sivaramakrishnan This is my paper
classification hep-th gr-qcquant-ph
keywords AdS/CFT correspondencecomposite geodesicstimelike separationcomplex geodesic lengthtimelike entanglement entropyBTZ black holeanalytic continuationbulk reconstruction
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 a bulk prescription for timelike-separated boundary points in AdS/CFT, where ordinary spacelike geodesics do not reach the boundary. It claims that extremizing the sum of a spacelike leg length and a timelike leg proper time over the bulk join points, and then fixing the remaining ambiguity by matching the time-ordered boundary two-point function's analytic continuation, determines a unique composite geodesic with complex length ℓ = ℓs + iℓt. The paper verifies the equality between this complex length and the correlator length in the planar slicing of AdS, in global AdS for generic points, and in the non-rotating BTZ black hole, where the timelike segment runs behind the horizon. If correct, this provides a real-geometry bulk dual for timelike boundary separations and a sharper derivation of timelike entanglement entropy in AdS3/CFT2.

Core claim

The central claim is that the complex length ℓ(x1,x2) = ℓs + iℓt extracted from the time-ordered two-point function at timelike separation is realized as the length of a real, piecewise-geodesic curve in the original, uncomplexified AdS geometry. The curve consists of two spacelike geodesics running from the boundary points to bulk join points, joined by one timelike geodesic; extremizing the total ℓs and ℓt over the join points leaves a discrete family, and the paper selects the branch by requiring ℓs + iℓt to reproduce the analytically continued correlator length with a specific +iπ prescription. In global AdS this selects the n = 1 solution with timelike proper time π; in BTZ it forces th

What carries the argument

The composite geodesic: a curve made of two boundary-anchored spacelike geodesic segments and one interior timelike geodesic, assigned the complex length ℓs + iℓt. The extremization procedure varies the two bulk join points to make the total ℓs + ℓt stationary; the residual degeneracy is resolved by matching ℓs + iℓt to the length obtained from the time-ordered CFT two-point function continued to timelike separation with a fixed iε prescription. In BTZ this matching fixes the endpoint product u₁² = 1/(V₁V₂), placing both join points on the singularity and giving a continuous family of timelike segments of equal length π, of which only one crosses the horizon.

Load-bearing premise

The whole construction rests on taking the +iπ analytic-continuation branch of the time-ordered boundary two-point function as the rule that fixes which composite geodesic to select; if a different iε convention were used, the selected branch would change and the claimed equality would fail.

What would settle it

Compute the timelike two-point function length using the opposite analytic continuation (ℓ = log|...| − iπ) and check whether any extremal composite geodesic reproduces it, or find generic boundary points in global AdS where the extremization equations admit no solution with Im(ℓs) = 0. Either outcome would show the prescription does not uniquely determine the composite geodesic.

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

If this is right

  • For the planar, global, and BTZ backgrounds, the composite geodesic length ℓs + iℓt equals the CFT correlator length at timelike separation, so the prescription reproduces the boundary data without complexifying the bulk geometry.
  • In BTZ the timelike segment can lie behind the horizon, giving a concrete real-geometry realization of probing the black hole interior from boundary correlators.
  • The method derives previously assumed conditions—selecting n = 1 in global AdS and placing endpoints on the singularity in BTZ—from extremization plus the iε matching rule.
  • Applied to the twist operators whose correlation functions define entanglement entropy, the same prescription recovers known timelike entanglement entropy in AdS3/CFT2.
  • Because the spacelike legs solve extremality with real lengths, the construction works in the original Lorentzian geometry, unlike earlier complex-extremal-surface approaches.

Where Pith is reading between the lines

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

  • The recurring value ℓt = π in all three examples looks like a robust feature of the matching condition; if it holds beyond these cases, it would make the proper time of the timelike segment a universal, diffeomorphism-invariant bulk clock ticking at a fixed rate for boundary two-point data.
  • The reliance on the time-ordered (rather than anti-time-ordered) iε prescription suggests that different Lorentzian orderings of boundary insertions may select different composite geodesics, potentially connecting this construction to pseudo-entropy or modular-flow reconstructions.
  • A natural test is to push the same extremization into higher dimensions or non-constant-φ configurations; the paper states the procedure is dimension-agnostic, so a null result there would localize where the prescription breaks.

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 / 4 minor

Summary. The paper proposes a three-step bulk prescription for connecting timelike-separated boundary points in AdS/CFT by composite curves made of two spacelike geodesic segments and one timelike geodesic segment. The total complex length ℓ = ℓs + iℓt is extremized over the bulk joining points, and any residual degeneracy is fixed by matching to the analytically continued time-ordered CFT two-point function. The authors test this in Poincaré AdS, in global AdS for generic boundary points, and in the BTZ black hole. In the BTZ case the timelike segment lies behind the horizon, and the paper also connects the construction to known results on timelike entanglement entropy.

Significance. If the central claim holds, composite geodesics provide a real-geometry bulk dual for timelike CFT separations and a toy model for reconstructing bulk observer worldlines. The BTZ section is the strongest part: it gives an analytic construction of a timelike segment behind the horizon with proper time π, and Appendix B provides an explicit family of such curves. The paper also carefully positions itself relative to earlier work [36,37,39,40]. However, the Poincaré demonstration is incomplete, and the prescription's branch-fixing step imports the boundary answer, so the bulk extremization does not determine the composite geodesic as strongly as the abstract suggests.

major comments (3)
  1. [Section II, Eqs. (3)-(6)] The paper states 'We can in fact show that ℓ = iℓt + ℓs is the length of a bulk curve', but the spacelike contribution is never computed from the bulk geometry. For Q²<0 the spacelike geodesics described by Eq. (2) have no turning point and reach the Poincaré horizon z=∞; the proper length integral ∫ dz/(z√(1−Q²z²)) diverges logarithmically. The finite real part 2 log(Δτ/δ) in Eq. (3) is obtained by analytically continuing the length formula from spacelike separation, i.e. by importing the boundary answer. Thus the Poincaré 'demonstration' of Eq. (6) is not a bulk geodesic calculation of ℓs; it presupposes the equality. This is load-bearing because Poincaré is the first advertised example. The later global and BTZ sections do compute ℓs independently, but Section II should either regulate the divergence explicitly or be presented as a check rather than a derivation.
  2. [Section I, Step 3; Sections III and IV] The residual degeneracy in the extremization is fixed by comparing ℓs+iℓt to the analytically continued boundary length. Concretely, in global AdS the n=±1 branch is chosen 'Comparison to (13)' after Eq. (16), and in BTZ the sign u1=1/√(V1V2) is selected as the 'time-ordered configuration' after Eq. (31). With the opposite iε prescription (−iπ), the selected branches would change and the claimed equality would fail. Step 3 is explicit, so this is not a hidden circularity; however, it means the bulk extremization alone does not determine a unique composite geodesic, and the final agreement is partly imported from the CFT. The abstract's claim that the procedure 'determines' the geodesic should be qualified to say that the boundary iε rule is part of the input. The free parameters pt (Section II) and a (Appendix B) are additional illustrations that the bulk equations leave a family.
  3. [Section III, Eq. (19)] The generic global-AdS case is not proven analytically. After deriving the extremization conditions (19), the authors state 'We have checked numerically for a variety of (Ti,Ri)' and report agreement. The special symmetric configuration R1=R2=0 is solved in Eqs. (20)-(21), but the general claim of the abstract — agreement for generic boundary points — rests on this numerical check. Please either supply an analytic solution of (19) or explicitly label the generic case as a numerical conjecture.
minor comments (4)
  1. [Title page] Title contains a typo: 'en tanglement' should be 'entanglement'.
  2. [Section II, Eq. (3)] 'For reasons that will become clear' should be explained at that point; the iε prescription is only introduced later. Also, ℓs is used in Eq. (6) before being defined as the total spacelike length; define it when first used.
  3. [Section IV, around Eq. (31)] The treatment of the s1=s2=−1 solution (uv=−1, i.e. the boundary) is not explained. Since these are candidates for bulk endpoints, clarify why only the singularity branch is kept.
  4. [Appendix B, Eq. (B11)] The free parameter a is said to remain within |a|<1/v+; it would be useful to state explicitly that the total length is independent of a, so the family does not affect the central result.

Circularity Check

2 steps flagged

Branch/degeneracy selection uses the boundary length being matched; Poincaré real part is taken from analytic continuation rather than an explicit bulk computation.

specific steps
  1. self definitional [Section II, Eqs. (3)-(6)]
    "Now, we analytically continue ℓ in x1, x2 to reach the timelike configuration (x1 − x2)^2 < 0. ... ℓ = iπ + 2 log(√((t1 − t2)^2 − (w1 − w2)^2)/δ). (3) ... Analytically continuing to timelike separations gives ℓ = log(|x1 − x2|^2) + iπ = ℓs + iℓt, (6), in agreement with the geodesic calculation we provided."

    The text never evaluates the proper length of the spacelike legs described just before; it only notes that they reach the Poincaré horizons. The real part assigned to ℓs comes from Eq. (3), which is the analytic continuation of the spacelike boundary two-point length, and Eq. (6) then labels that same value ℓs. So in the Poincaré demonstration the real part of the 'geodesic calculation' is imported from the boundary answer rather than obtained from the stated extremization of ℓs over y1,y2.

  2. fitted input called prediction [Section III after Eq. (16); Section IV after Eqs. (27)-(31)]
    "Comparison to (13) indicates we should choose the n = ±1 solution, as explained in Appendix A. ... Comparing to (26) instructs us to exclude the coincident-point solution. ... The time-ordered configuration is where (u1, v1) lies on the future singularity, or u1 = 1/√(V1V2)."

    In both the global and BTZ cases, extremization leaves discrete degeneracy (n ∈ Z in global; the coincident solution and the u1 = ±1/√(V1V2) branch in BTZ). The paper selects among these branches by comparing to the target boundary length ℓ from Eqs. (13)/(26). Thus the final equality ℓs + iℓt = ℓ is partly enforced by the selection rule. This is partial rather than total circularity: the bulk-extremized spacelike lengths are genuine, and the remaining orientation choice does not change the computed length.

full rationale

The paper is open about using the boundary value as a selector: its Step 3 says 'to fix any remaining degeneracy, compare ℓs+iℓt to ℓ(x1,x2) obtained by analytically continuing...' This makes part of the advertised agreement a selection rule rather than an independent prediction. The clearest definitional step is in Poincaré AdS, where the real part ℓs is taken from Eq. (3)/(6), the analytically continued boundary length, rather than being computed from the proper lengths of the bulk spacelike segments. In global AdS and BTZ, the sections do contain genuine bulk extremization of spacelike segments and ℓt=π, and the discrete branch choice (n=±1, time-ordered u1) does not change the final length, so the central construction has nontrivial content. I find no load-bearing self-citation: the prior timelike-entanglement references [36,37] are by Doi et al., not by the present authors, and the authors' own earlier works cited here are background, not the justification of the composite-geodesic prescription. Overall, the circularity is partial and openly stated: score 4.0 rather than higher because the real part in global/BTZ is not reduced to a fit, and the paper's own multiplicity caveat in Future Directions further confirms that uniqueness is not being claimed from extremization alone.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central claim rests on the standard AdS/CFT geodesic dictionary plus the paper's own extremization prescription. No new physical entities are introduced. The main free inputs are discrete analytic-continuation/branch choices imported from the CFT to fix degeneracies, plus continuous family parameters that cancel in the final length.

free parameters (4)
  • Analytic continuation branch (iε sign) = +iπ (positive imaginary part, time-ordered)
    The paper defines the timelike complex length by continuing the spacelike formula with a specific iε prescription chosen to match the time-ordered CFT correlator. This choice is an input used to fix degeneracies, not derived from bulk principles. Location: Section I item 3; Section II Eqs. (3)-(6).
  • Global AdS branch n = ±1 = n = 1 (time-ordered)
    After extremizing the timelike distance, solutions satisfy t1−t2 = nπ. The paper selects n = ±1 by comparing to the boundary length (13), and chooses n = 1 as the time-ordered configuration. This is a discrete input from the CFT. Location: Section III near Eq. (17).
  • Poincare timelike segment family parameter pt = undetermined (arbitrary real)
    In Poincare AdS, the timelike geodesic segment has ℓt = π independent of pt, and the paper notes a family of distinct geodesics whose pt is not fixed by boundary conditions. The length is unique, but the composite curve is not. Location: Section II, after Eq. (4).
  • BTZ timelike segment family parameter a = arbitrary with |a| < 1/v+; future-directed choice a = 0
    Appendix B shows a continuous family of timelike curves connecting the singularity endpoints, all with the same length iπ. Extremization of the spacelike legs fixes v+ but not a; the future-directed choice a = 0 is selected by convention. Location: Section IV and Appendix B.
axioms (4)
  • domain assumption Geodesic/WKB dictionary: ⟨O(x1)O(x2)⟩ ∼ e^{−mℓ} for spacelike separations, continued to timelike separations.
    The paper relies on the standard holographic relation between CFT two-point functions and bulk geodesic lengths, then analytically continues it to define ℓ for timelike separations. This is standard but unproved in the paper. Location: Section I, second paragraph.
  • domain assumption The iε prescription of the CFT correlator directly specifies the analytic continuation of the bulk geodesic length.
    The paper assumes the sign of the imaginary part of the analytically continued length is fixed by the time-ordered correlator's iε choice, and uses this to define the bulk prescription. This is a nontrivial assumption connecting boundary and bulk analytic structure. Location: Section II Eqs. (5)-(6); Section I item 3.
  • ad hoc to paper The extremization prescription (varying y1,y2 to extremize both ℓs and ℓt) is the correct bulk rule for timelike separations.
    The paper proposes this as a new prescription and uses it to derive results; it is not derived from a deeper principle such as string theory or a variational principle for the full correlator. Location: Section I items 1-3.
  • domain assumption In BTZ, the timelike segment endpoints may lie on the singularity.
    The geometry constraint −1 ≤ s1,s2 ≤ 1 forces the endpoints to s1 = s2 = 1, i.e., on the singularity. The paper treats this as an admissible endpoint for the geodesic segments. Location: Section IV after Eq. (31).

pith-pipeline@v1.3.0-alltime-deepseek · 12007 in / 16925 out tokens · 143339 ms · 2026-08-03T19:49:56.468508+00:00 · methodology

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read the original abstract

We study how to recover timelike worldlines in AdS from CFT data as a toy model for holographically reconstructing realistic observers. We give a bulk extremization procedure that determines composite timelike-spacelike geodesics that connect timelike-separated boundary points. The total geodesic length matches the length extracted from CFT correlators at the timelike-separated points. We show agreement in Poincar\'e AdS, for generic boundary points in global AdS, and also for the BTZ solution, in which the timelike segment probes behind the horizon. We refine related methods to compute timelike entanglement entropy in AdS$_3$/CFT$_2$ and recover known results.

Figures

Figures reproduced from arXiv: 2511.22168 by Allic Sivaramakrishnan, Hardik Bohra.

Figure 1
Figure 1. Figure 1: FIG. 1. For timelike-separated boundary points [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

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