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Timelike Holographic Complexity

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Timelike subregion complexity stays real even when the extremal surface crosses a black hole horizon.

desk verdict First timelike extension of subregion CV complexity; clean pure-AdS results, but the reality is built into the definition and the quoted formulas have a factor-of-2 error. read the letter →

arxiv 2510.25700 v4 pith:A4MQOCYH submitted 2025-10-29 hep-th

Mohsen Alishahiha This is my paper · ORCID
classification hep-th
keywords timelikesubregioncomplexityComplexity=Volumepseudo-entropyentanglemententropyAdS/CFTextremalsurfacesSblackbranes
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

The paper extends the holographic 'Complexity=Volume' proposal to Lorentzian (timelike) boundary intervals. It claims that, despite the Lorentzian embedding and the appearance of a timelike branch of extremal surfaces, the enclosed bulk volume is purely real. For hyperbolic intervals in pure AdS and for strips in AdS black branes—including surfaces that penetrate the horizon—the complexity has the same universal UV divergences as the spacelike case and receives no imaginary part. This contrasts with timelike entanglement entropy and pseudo-entropy, which acquire imaginary contributions. A sympathetic reader would care because it suggests holographic complexity is a real, geometric quantity tied to causal reach rather than to analytic continuation and phase structure.

What carries the argument

The key object is the subregion 'Complexity=Volume' (CV) functional, V/G, with V defined in Eq. (7) as the volume enclosed between the two extremal branches for a timelike interval: the spacelike branch ξ² = r² + T²/4 and the timelike branch ξ² = r² − T²/4, the latter existing only for r ≥ T/2. The argument hinges on the cancellation of apparent IR and near-horizon divergences between the branches, leaving real, finite, near-boundary UV terms. This choice of 'enclosed volume between branches' is what carries the reality; the paper admits that a boundary-field-theory rationale for this precise volume is lacking.

What would settle it

A concrete check: compute timelike subregion complexity directly in a 1+1D boundary theory using a non-unitary circuit or tensor network for a transition matrix, and compare with the AdS3 prediction C_T = R T/(2G ϵ). If the boundary quantity is complex, or its leading divergence is not linear in T/ϵ, the central reality and divergence claims collapse. Alternatively, evaluate the complexified extremal-surface volume used in the timelike-entanglement prescription the paper cites for the black-brane strip: if its real part differs from Eq. (24) in any finite term, the identification of the 'real

Watch

Extended reading notes

Core claim

The central claim is that the natural CV-type quantity for a timelike subregion—defined as the bulk volume enclosed between the spacelike and timelike extremal surfaces anchored on the interval—is real. The paper computes this volume for hyperbolic regions in AdS_{d+2} and for strips in AdS black branes. In the black brane case the extremal surface can either be a constant-time slice ending at the horizon or a horizon-penetrating surface with turning point behind the horizon; both give a real volume, with apparent divergent contributions from the two sides of the horizon cancelling. The author argues that, unlike timelike entanglement entropy, no analytic continuation of the volume functiona

Load-bearing premise

The load-bearing premise is that the correct holographic entry for timelike subregion complexity is the real bulk volume enclosed between the spacelike and timelike extremal surfaces (Eq. 7): the paper gives no boundary-field-theory derivation for this identification, and for the black-brane strip it admits that the setup 'lacks a clear interpretation of bulk extremal surface'—if the correct prescription required analytically continuing the CV functional (as is done for timel

Editorial extensions

If this is right

  • Timelike subregion complexity in pure AdS and AdS black branes has the same UV divergence structure as spacelike subregion complexity, so the leading divergence is set by boundary volume.
  • Since the result is real, the CV functional under Lorentzian continuation does not require complexification, in contrast to the analytic continuation used for timelike entanglement entropy.
  • In black brane geometries, horizon-penetrating extremal surfaces are admissible and give finite, real complexity; horizon-crossing divergences cancel.
  • For odd-dimensional AdS, timelike complexity lacks the universal constant term of the spacelike case, so it is not a Wick rotation of the spatial result.
  • A possible transition between constant-time and horizon-penetrating branches could produce a geometric phase transition in timelike complexity, as the paper suggests as plausible.

Reading between the lines

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

  • Editorial: If this reality persists under a fully complexified prescription, it suggests pseudo-entropy's imaginary part encodes phase information that complexity is blind to; a boundary definition of timelike complexity should therefore be real-valued and could be probed by tensor networks built from non-Hermitian transition matrices.
  • Editorial: The same volume-between-branches construction could be applied to the Complexity=Action proposal on a Wheeler–DeWitt patch anchored to a timelike interval; if reality holds there, the geometric interpretation extends beyond CV.
  • Editorial: The paper notes that the horizon-penetrating solution coincides with one branch of the complex extremal surfaces used for timelike entanglement entropy; a testable question it leaves open is whether the real volume generally equals the real part of that complex volume for all dimensions.
  • Editorial: The admitted lack of a clear bulk-surface interpretation for the strip suggests the cleanest test is in AdS3, where the result C_T = R T/(2G ϵ) can be compared with a direct tensor-network computation for a timelike interval.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper proposes a timelike version of holographic subregion complexity in the CV framework. For a timelike hyperbolic boundary interval in AdS_{d+2}, the author takes the bulk volume enclosed between the spacelike and timelike extremal surfaces (Eq. 7) as the definition, C_T=V/G, and evaluates it explicitly for d=1,...,4. The result is claimed to be purely real, in contrast to complex pseudo-entropy, with the same hierarchy of UV divergences as the spacelike CV proposal. The construction is then extended to AdS black branes, where the extremal surface is allowed to penetrate the horizon; Eq. (24) is quoted as a real, finite expression. The paper concludes that timelike complexity retains a geometric, real character, although a boundary field-theoretic definition is left open.

Significance. If the proposed identification is accepted, the paper is a natural and mostly explicit extension of [25] to Lorentzian boundary subregions, with no free parameters and with calculations that are straightforward to verify. Its main conceptual claim — reality of timelike complexity — would distinguish complexity from pseudo-entropy and would support a geometric rather than analytic-continuation interpretation of the CV functional. However, the significance is conditional: Eq. (7) is a new prescription, not a derived holographic entry, and the paper's own caveats (footnote 1, Sec. IV) leave its bulk/boundary meaning open. The factor-of-2 mismatch with the quoted results and the uncomputed horizon-crossing integral must be resolved before the quantitative claims can be trusted.

major comments (3)
  1. [Eq. (7); Eqs. (9)-(12)] The volume formula (7) contains an overall factor 2 that is not consistent with the quoted results. For d=1, direct evaluation of (7) gives a leading divergence proportional to T/ε (with the transverse volume factors used in the paper), whereas (9) states C_T = (R/G)(T/2ε), exactly half. The same factor of 2 appears in the d=2,3,4 expressions. If the factor 2 is meant to count the ξ<0 and ξ>0 parts of the enclosed region, this doubling must be stated explicitly and applied consistently to the r<T/2 cap as well; if not, (7) should be corrected. In either case the reported numerical coefficients in (9)-(12) do not follow from (7) as written.
  2. [Sec. II, Eqs. (6)-(8); Sec. IV] The central claim that timelike subregion complexity is purely real is built into the definition rather than derived. Eq. (7) integrates only over r≥T/2 on the timelike branch, discarding the segment 0<r<T/2 where ξ^2=r^2−T^2/4 is negative. A complexified evaluation of the same volume functional, in the spirit of the timelike entanglement entropy prescriptions of [34,35], would produce an imaginary contribution from this segment. The paper gives no boundary circuit construction and no bulk principle that selects the real-branch volume; indeed footnote 1 questions whether the causal volume exists, Sec. III states the strip setup 'lacks a clear interpretation of bulk extremal surface', and Sec. IV concedes the field-theoretic definition is open. As it stands, the reality property is a consequence of choosing the integration contour in (7), not a falsifiable prediction.
  3. [Sec. III, Eqs. (22)-(24), footnote 2] The horizon-penetrating calculation is incomplete. Eq. (24) is the final expression for C_T, but the integral is not evaluated. Footnote 2 asserts that the potentially divergent contributions from the two sides of the horizon cancel, but no such cancellation is shown, and the integral as written crosses the pole f(r)=0 at r=r_h. Since the finiteness and reality of the black-brane result are load-bearing for the paper's conclusions, this step requires an explicit evaluation or a controlled principal-value/regularization argument. In addition, the paper does not provide a criterion for selecting the horizon-penetrating branch over the constant-time branch; the statement that a transition is 'plausible' is not sufficient to establish which surface computes the CV volume.
minor comments (5)
  1. [Section II, Eq. (9)] The d=1 result in Eq. (9) omits the transverse volume factor H_0 that appears in the general formula (7); please clarify the normalization (e.g. H_0=1) for consistency with (10)-(12).
  2. [Sec. II, text after Eq. (7)] The statement that both integrals in (7) individually have IR divergences that cancel is not demonstrated by the displayed formulas; a short explanation of the cancellation for general d would help.
  3. [Fig. 2 caption] The caption says the figure 'essentially reproduces Fig. 3 of [8]'; if this is a reproduction, the source should be acknowledged explicitly in the caption (beyond the reference), and the overlap with [8] should not be presented as new.
  4. [Sec. III, discussion after Eq. (24)] The possible transition between the t'=0 and horizon-penetrating branches is mentioned twice, but no condition for the transition is derived. Please mark this as speculative or provide a criterion.
  5. [Sec. IV] The discussion of [36,37] is disconnected from the rest of the paper; consider shortening to a brief pointer or moving to a separate paragraph.

Circularity Check

2 steps flagged · score 6.0 of 10

The advertised 'purely real' conclusion is built into the definition: C_T is defined as the volume of a real bulk region, so the absence of an imaginary part is imposed by Eq. (7), not derived from the CV proposal.

  1. self definitional [Sec. II, Eqs. (7)-(9)]
    "Following [25], the timelike subregion complexity is defined as C_T=V/(GR)... The second integral starts at r=T/2 since the timelike branch exists only for r≥T/2... in the complexity-volume relation the integration is performed over a strictly Lorentzian region with r≥T/2, yielding a real-valued geometric volume."

    The reality claim is the defining content of C_T: Eq. (8) sets C_T equal to V/(GR), where V in Eq. (7) is by construction an integral over real r and real branch functions, with the timelike integral artificially cut off at r=T/2. That cutoff is exactly what discards the imaginary contribution that would arise from analytically continuing xi^2=r^2-T^2/4 into r<T/2, the continuation used for timelike entanglement entropy in [8,9,34,35]. Thus 'no imaginary contribution' is not a holographic prediction; it is imposed by the choice of observable. The UV-divergence computation is genuine, but the headline reality result reduces to the definition.

  2. self definitional [Sec. III, Eqs. (21)-(24)]
    "Solving for t'(r) gives [Eq. (21)]... Importantly, the total timelike complexity remains real, even when the extremal surface penetrates the horizon."

    The black-brane version of the reality claim is likewise put in by hand. The paper notes that 'in general no globally real solution exists unless the extremal surface extends beyond the horizon,' and then defines C_T through the real integral (22)-(23) using the real horizon-penetrating branch (21). The complex extremal surfaces of [34,35] are mentioned but not used to define C_T. So the realness is an assumption about which branch represents complexity, not a consequence derived from the CV proposal.

full rationale

The computation itself is self-contained: the extremal surfaces come from independent references [8,9], no parameter is fitted, and the UV-divergence structure follows from near-boundary AdS kinematics. The self-citation to [25] is only used to import the standard CV subregion-complexity definition, which is not itself circular here. However, the paper's central advertised finding, that timelike subregion complexity is 'purely real,' is true by construction: C_T is defined as the volume of a real bulk region, with the timelike branch truncated at r=T/2, so any imaginary contribution is excluded at the level of the definition rather than excluded by a boundary derivation or by an independent analytic-continuation principle. The paper's own caveats (Sec. IV: no field-theoretic definition; footnote 1: 'it remains unclear whether such a causal volume genuinely exists') reinforce that the reality statement is a property of the chosen observable. I therefore score 6: one or more central claims reduce by construction, while the volume integrals and divergence computations retain independent content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the CV conjecture, on extremal-surface profiles imported from the timelike-entanglement literature, and on an ad hoc definition of which bulk volume is the 'timelike complexity.' The black-brane extension further assumes horizon-penetrating saddles and an unproven cancellation of divergences. No free parameters are fitted; the only new entity is the observable C_T itself, which lacks an independent definition.

assumptions (5)
  • domain assumption Complexity=Volume: subregion complexity is the bulk volume enclosed by extremal surfaces (C = V/GR)
    The entire computation is conditional on the CV conjecture. Invoked in Sec. II, Eq. (8); no independent boundary definition of timelike complexity is given.
  • domain assumption Extremal-surface profiles xi^2 = r^2 ± T^2/4 from [8,9] are the correct surfaces for the timelike interval
    Adopted from the timelike entanglement entropy literature (Refs. [8,9]); the profiles are not re-derived and their extension to the volume functional is assumed. Used in Sec. II, Eq. (5).
  • ad hoc to paper The relevant volume in timelike subregion complexity is the region bounded by the spacelike branch and timelike branch, with the integration range and factor 2 of Eq. (7)
    This defines the new quantity C_T; there is no field-theoretic or independent holographic derivation that this region — rather than, e.g., a complexified volume — is the complexity. The ambiguity is apparent in Eq. (7) vs. Fig. 1.
  • ad hoc to paper For the black brane, the extremal surface may penetrate the horizon with turning point r0 > r_h, and this branch is the relevant one
    Introduced in Sec. III after Eq. (18): 'Under this assumption...'; the surface is claimed to parallel Vaidya geodesics, but no proof of dominance or boundary interpretation is supplied.
  • ad hoc to paper Divergent contributions to the volume integral across the horizon cancel, yielding a finite result
    Footnote 2 asserts cancellation without a principal-value definition or explicit evaluation; the integral in Eq. (24) is not computed.
invented entities (1)
  • Timelike subregion complexity C_T
    purpose: Extends the CV subregion complexity to Lorentzian boundary intervals; the paper's central observable.
    The paper introduces C_T as the CV volume enclosed by timelike extremal surfaces; no boundary dual, no independent falsifiable prediction is provided. Its reality and UV-divergence structure are properties of the chosen geometric definition.

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Cite this review

Pith. "Pith review of Timelike Holographic Complexity." pith.science (2026). https://pith.science/paper/A4MQOCYH

@misc{pith2026251025700,
  author       = {Pith},
  title        = {Pith review of: Timelike Holographic Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4MQOCYH}},
  note         = {Machine review of arXiv:2510.25700}
}
read the original abstract

Motivated by the pseudo-entropy program, we study timelike subregion complexity within the holographic Complexity-equal-Volume framework, extending previous spatial constructions to Lorentzian boundary intervals. For hyperbolic timelike regions in pure AdS, we compute the enclosed bulk volume and show that, despite the Lorentzian embedding, the resulting complexity is purely real. We generalize the analysis to AdS black brane geometries, where extremal surfaces may either remain entirely outside the horizon or penetrate it, placing their timelike branch inside the black brane interior. In both configurations, the complexity exhibits the same universal UV divergences as the spacelike case, yet it receives no imaginary contribution, highlighting its causal and geometric origin. This reality stands in sharp contrast to the complex-valued pseudo-entropy and indicates that holographic complexity retains a genuinely geometric, real character even under Lorentzian continuation.

Figures

Figures reproduced from arXiv: 2510.25700 by the authors.

Figure 1
Figure 1. The colored region between the spacelike (green) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Extremal surfaces for a timelike entangling region [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Holographic timelike complexity for de Sitter

    hep-th 2026-07 conditional novelty 6.0 of 10

    Timelike subregion volume complexity in de Sitter grows exponentially early and diverges hyperfast at a maximal duration; near the SdS black hole horizon the divergence is replaced by slower, claimed-nonlinear growth.

Reference graph

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