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Timelike and gravitational anomalous entanglement from the inner horizon

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Pith's one-line read The paper argues that the inner horizon of Rindler AdS3 maps to a bulk geodesic—the inner RT surface—that simultaneously reproduces the real part of holographic timelike entanglement entropy and the Chern-Simons correction to holographic…

desk verdict A genuinely new geometric object (the inner RT surface) with real explanatory payoff, but the key algebraic bridge to the twist description and the replica interpretation are asserted rather than proven. read the letter →

arxiv 2412.21058 v3 pith:YAOQTPUI submitted 2024-12-30 hep-th

Qiang Wen , Mingshuai Xu , Haocheng Zhong This is my paper · ORCID
classification hep-th
keywords innerhorizonRTsurfacetimelikeentanglemententropyRindlerAdS3topologicallymassivegravitygravitationalanomalywedgecrosssectionpartial
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

This paper tries to show that a single bulk object, the pre-image of the inner horizon of Rindler AdS3—called the inner RT surface—carries two quantum-information meanings that were previously described separately. For any spacelike boundary interval, its causal development has two tips; the timelike interval joining those tips has a holographic timelike entanglement entropy whose real part is a spacelike geodesic, and the paper identifies that geodesic with the inner RT surface. In topologically massive gravity, the Chern-Simons correction to holographic entanglement entropy has been computed by a twist along the RT surface; the paper claims this correction is simply the regulated length of a geodesic chord on the inner RT surface, and that the anomalous part of the balanced partial entanglement entropy is the length of a saddle geodesic connecting two pieces of that surface, the inner entanglement wedge cross section. If correct, anomaly corrections become purely geometric lengths rather than normal-frame-dependent worldline data, and timelike entanglement acquires a bulk replica interpretation.

What carries the argument

The load-bearing object is the inner RT surface $\widehat E$, defined as the inverse Rindler image of the inner horizon $\tilde\rho=-T_{\tilde U}T_{\tilde V}$ of Rindler $\widetilde{\mathrm{AdS}}_3$. It is an extremal surface, the fixed-point set of the modular momentum flow $k_t^{\beta,\text{bulk}}$, and its length parameter $\hat\tau$ organizes geodesic chords whose lengths reproduce partial entanglement entropies. The argument works by mapping the interval to a thermal state, computing thermal entropy on the outer and inner horizons, and then mapping back; because the inner horizon length is already known to give the Chern-Simons entropy correction in topologically massive gravity, the IRT chord length inherits that role for arbitrary boundary intervals.

What would settle it

Compute the anomalous part of the entanglement entropy for a boundary interval using the normal-frame worldline action (100) under two different but equally valid smooth normal-frame configurations along the RT surface; if the result is frame-dependent and not equal to the regulated inner-RT chord length (113), the claimed equivalence between twist and IRT descriptions fails. A second check would be to test whether the inner-EWCS saddle (148) still equals the twist-based correction when the balance conditions (116) are violated; the equality should break precisely there.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the pre-image of the inner horizon of the Rindler $\widetilde{\mathrm{AdS}}_3$ in the original Poincaré AdS3, denoted $\widehat E$, is exactly the spacelike geodesic used in the holographic description of the timelike entanglement entropy of the partner interval, and moreover that with a suitable cutoff this same surface computes the Chern-Simons correction to entanglement entropy. In formulas, for an interval $A: (-l_U/2,-l_V/2)\to(l_U/2,l_V/2)$, the inner RT surface is $\widehat E:\ \rho=-2l_V/(l_U(l_V^2-4V^2)),\ U=-(l_U/l_V)V$, anchored at the two tips of the causal development $D_A$. The regulated chord on $\widehat E$ gives the anomalous entropy $S^a_A=\frac{1}{4\mu G}\log(l_U\varepsilon_V/(l_V\varepsilon_U))$, matching both the replica-method result and the twist-description result. The paper further claims that the mixed-state correlation dual to the entanglement wedge cross section receives an anomalous part equal to the length of a saddle geodesic connecting the two components of the inner RT surface of the mixed state, called the inner EWCS, and that the twist description of [50] and [51] is equivalent to this purely geometric description.

Load-bearing premise

The argument depends on the known result that the Chern-Simons correction to black hole entropy in topologically massive gravity equals the inner-horizon length, together with the assumption that this relation transfers, via the Rindler map, to arbitrary boundary intervals; it also assumes the replica argument applies to the inner RT surface so that timelike entanglement entropy has a von Neumann entropy interpretation.

Editorial extensions

If this is right

  • The real part of holographic timelike entanglement entropy is given by the length of the inner RT surface, and the imaginary part by the timelike geodesic at the boundary of the extended entanglement wedge.
  • The Chern-Simons correction to entanglement entropy in TMG/CFT with gravitational anomaly is a regulated geodesic length on the inner RT surface, not a normal-frame-dependent quantity.
  • The anomalous part of the balanced partial entanglement entropy equals the length of the inner EWCS, a saddle geodesic connecting pieces of the inner RT surface.
  • The twist description and the IRT description agree because the normal-frame boundary data along the RT surface encode the same point-to-point partnership as the modular momentum slices.
  • In the flat limit, the IRT picture reduces to the swing-surface picture of holographic entanglement entropy in flat-space holography.

Reading between the lines

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

  • The paper does not draw this, but if the replica argument for the inner RT surface is valid, timelike entanglement entropy would be a genuine von Neumann entropy generated by the modular momentum, not merely an analytic continuation of the spacelike formula.
  • The paper leaves implicit that the equivalence between twist and inner-RT length turns the twist observable in pure AdS3 into a timelike-entanglement diagnostic; a direct test would be to compare twist fluctuations along the RT surface with fluctuations of IRT chord lengths under boundary perturbations.
  • The construction is specific to three bulk dimensions and locally AdS3 spacetimes; a speculative extension would ask whether an inner-horizon pre-image plays a similar role for gravitational-anomaly corrections in higher-dimensional holography, where no chord-length formula is currently known.
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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 / 4 minor

Summary. The paper studies the pre-image of the inner horizon of the Rindler AdS3 obtained by the bulk Rindler transformation for a spacelike interval A, calling it the inner RT (IRT) surface. It argues that the IRT surface is exactly the spacelike geodesic representing the real part of the holographic timelike entanglement entropy for the partner timelike interval, with the timelike geodesic at the boundary of the extended entanglement wedge representing the imaginary part. In the context of topologically massive gravity, it proposes that the Chern-Simons correction to the holographic entanglement entropy is the regulated length of a geodesic chord on the IRT surface, and that the anomalous part of the balanced partial entanglement entropy is the length of a saddle geodesic chord (the "inner EWCS") connecting the two pieces of the IRT surface. Section 6 attempts to prove the equivalence between the twist description of [50] and the IRT length via Eq. (156).

Significance. If established, the paper's central claim would be a genuinely useful result: the anomalous correction to holographic entanglement entropy, previously encoded in a normal-frame "twist" along the RT surface, would become the length of a purely geometric geodesic chord on the inner RT surface, and the inner EWCS would give a geometric picture for the anomalous mixed-state correlation. The paper contains many explicit analytic computations, including the Rindler mapping, the fine structure of modular momentum slices, the explicit IRT surface equations, and a flat-limit comparison with the swing surface prescription. The derivation in Section 6 is a genuine attempt to connect the twist and IRT descriptions, and the appendices provide useful technical detail. However, the central equivalence rests on an unproved algebraic identity, and the transfer of the BTZ inner-horizon entropy formula to Rindler intervals is assumed; these gaps must be closed before the main claim can be regarded as established.

major comments (3)
  1. [Section 6, Eq. (156)] The equivalence between the twist description and the IRT length rests entirely on the identity log((q - \tilde{q})|_H \cdot n|_H) = \hat{\tau}_{\hat{H}}, stated in Eq. (156). This identity is asserted without derivation, and it is the bridge between the twist integral in Eqs. (154)-(155) and the IRT length parameter defined in Eq. (32). Since Eq. (157) and the claimed equivalence in Section 6, as well as the appendix-G explanation of Eq. (125), all depend on this identity, the paper should provide a proof. If the identity fails for a generic interval, the central anomaly-reproduction claim loses its derivation, and the earlier argument via Eq. (93) remains only an analogy from compact BTZ black holes to Rindler intervals.
  2. [Section 4.3, Eqs. (93), (95), (107)] The derivation of the anomalous holographic entanglement entropy (95) starts from the known BTZ inner-horizon entropy formula (93) and the modular Hamiltonian correction (107), and then transfers this result through the Rindler mapping to arbitrary boundary intervals. The text explicitly calls these "our starting points." This transfer is load-bearing: without it, the claim that the IRT geodesic chord computes the anomalous part of the entropy for general intervals is not established. The authors should either justify the transfer from compact BTZ horizons to the non-compact Rindler black string with the regulated cutoffs, or clearly state this step as an assumption whose failure would leave the reproduction claim heuristic.
  3. [Section 2.2, around Eqs. (36)-(40)] The paper proposes that the timelike entanglement entropy can be interpreted as a holographic von Neumann entropy by applying the Lewkowycz-Maldacena replica prescription to the IRT surface. The text itself later acknowledges that this replica interpretation is incomplete, stating that the role of the timelike geodesic in the analog replica story is "unclear" and that the point will be revisited in the future. Because the abstract and summary present the timelike-entanglement interpretation as one of the paper's main results, this limitation should be stated more prominently. If this replica interpretation is not needed for the anomaly-reproduction claim, the paper should say so explicitly; if it is part of the claim, it needs a concrete derivation or a clear relegation to conjecture.
minor comments (4)
  1. [Section 2.1, text after Eq. (10)] The sentence "where T~U and T~U are the parameters" should read "T~U and T~V"; as printed, the second symbol repeats the first.
  2. [Section 5.1] "One the other hand" should be "On the other hand".
  3. [Section 5.2.2] "were we also take L(...)" should be "where we also take L(...)".
  4. [Sections 2.2 and 3.2.2] The phrase "interaction line" should be "intersection line" in the sentences following Eqs. (30) and (31), and in the related discussion of M_\pm and the IRT surface.

Circularity Check

1 steps flagged · score 4.0 of 10

The anomalous-entropy 'reproduction' via the inner RT surface re-expresses the known TMG inner-horizon formula (93) in new coordinates, but the paper's independent geometric identifications and the twist/IRT bridge are not fitted; overall partial, not pervasive, circularity.

  1. renaming known result [Sec. 2.2 (definition of bE, Eq. (31)) and Sec. 4.3 (Eqs. (93)-(95), (113))]
    "their interaction line is bE which we refer to as the inner Ryu-Takayanagi (IRT) surface ... the correction of the thermal entropy of a BTZ black hole from the CS term is proportional to the length of the inner horizon. Then it is natural to think that, the CS correction to the holographic entanglement entropy for intervals should be represented via the inner horizon in the Rindler AdS3, as well as its pre-image, the IRT surface bE."

    The IRT surface is defined as the pre-image of the inner horizon under the inverse Rindler map, so its regulated length coincides with the inner-horizon length appearing in Eq. (93). Eq. (93) already states that the CS correction to the outer-horizon entropy is proportional to that inner-horizon length, and substituting the Rindler interval lengths (94) into (93) yields the anomalous part (95). Eq. (113) then re-labels the same quantity as Length(bE_reg)/(4Gµ). The 'reproduction' is therefore the input formula (93) expressed in the new bE coordinates, rather than an independent derivation of the CS correction.

full rationale

Most of the paper is a self-contained coordinate/geometric analysis: the IRT surface is constructed as the pre-image of the Rindler inner horizon; the HTEE identification is an equality between the IRT surface and the known spacelike geodesic of timelike entanglement; the modular-momentum slicing and partner-point construction are explicit; and the IEWCS length is computed directly and matched to an independent ALC evaluation of BPEa. No parameters are fitted, and the main external benchmarks ([50] and the replica result (84)) are genuinely independent of the paper's own fitted values. The one circular-looking move is the anomaly 'reproduction' in Sec. 4.3: because bE is defined as the pre-image of the inner horizon, Length(bE_reg) is the inner-horizon length of Eq. (93) by construction, so Eq. (113) restates the input TMG formula in new coordinates. This reduces the novelty of the anomaly-reproduction claim, but it does not undermine the HTEE identification or the IEWCS construction. Separately, the twist/IRT equivalence in Sec. 6 rests on the unproved algebraic identity (156), which equates the normal-frame integral primitive with the IRT length parameter; this is a completeness or correctness risk rather than a circularity, since the identity, if verified, would be a genuine algebraic bridge rather than a restatement of the conclusion.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

No fitted parameters beyond the UV regulator scheme; mu, G, cL, cR are theory inputs from prior literature. The central calculations rest on the TMG inner-horizon entropy formula and on the worldline action of [50], both treated as established. The new objects (inner RT surface, inner EWCS, extended entanglement wedge) are geometric constructs, not new physical entities. The TEE replica interpretation is an added conjecture, listed as an ad hoc axiom.

free parameters (1)
  • UV cutoffs epsilon_U and epsilon_V = not fitted; infinitesimal regulators, often set equal
    Introduced in Eq (16) to regulate divergent entanglement entropies. The ratio epsilon_U/epsilon_V controls the finite anomalous term, so the regulator scheme affects the expression, though it is not fitted to data.
assumptions (7)
  • domain assumption AdS3/CFT2 duality and the Ryu-Takayanagi formula are valid for the geometries considered.
    The entire calculation uses the holographic dictionary, including RT surfaces, entanglement wedges, and the Rindler method; see Secs 1 and 2.1.
  • domain assumption The Rindler transformation maps the causal development of any boundary interval to a thermal CFT and the entanglement wedge to Rindler AdS3, preserving entropies.
    The Rindler method is used throughout; the explicit map (10) and its boundary limit (12) are taken from [16,59].
  • domain assumption In topologically massive gravity, the Chern-Simons correction to black hole entropy equals the inner horizon length divided by 4 mu G (Eq 93), and the modular Hamiltonian receives the correction (107).
    This is the stated starting point for the anomaly-correction part in Sec 4.3, based on [28-31,86].
  • domain assumption The worldline action with normal frame (100) from [50] correctly describes holographic entanglement entropy with gravitational anomaly.
    The twist description is used as the benchmark in Secs 4.2 and 6; the paper aims to reproduce it geometrically.
  • domain assumption The balanced partial entanglement entropy (BPE) is dual to the entanglement wedge cross section, with balance conditions (117) applied separately to normal and anomalous parts.
    Sec 5 relies on the BPE proposal from [51,75,76] to define the anomalous EWCS.
  • domain assumption Timelike entanglement entropy is defined by analytical continuation of the spacelike entropy formula (Eq 35), and its geometric picture involves spacelike and timelike geodesics.
    The paper takes the TEE geometric picture from [37,38] as given and then identifies the spacelike part with the IRT surface.
  • ad hoc to paper The Lewkowycz-Maldacena replica prescription can be applied to the IRT surface for a timelike interval, making timelike entanglement entropy a holographic von Neumann entropy.
    Argued in Sec 2.2 by analogy with the spacelike RT story; no Lorentzian replica derivation is given.

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Pith. "Pith review of Timelike and gravitational anomalous entanglement from the inner horizon." pith.science (2026). https://pith.science/paper/YAOQTPUI

@misc{pith2026241221058,
  author       = {Pith},
  title        = {Pith review of: Timelike and gravitational anomalous entanglement from the inner horizon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YAOQTPUI}},
  note         = {Machine review of arXiv:2412.21058}
}
abstract

In the context of the AdS$_3$/CFT$_2$, the boundary causal development and the entanglement wedge of any boundary spacelike interval can be mapped to a thermal CFT$_2$ and a Rindler $\widetilde{\text{AdS}_3}$ respectively via certain boundary and bulk Rindler transformations. Nevertheless, the Rindler mapping is not confined in the entanglement wedges. While the outer horizon of the Rindler $\widetilde{\text{AdS}_3}$ is mapped to the RT surface, we also identify the pre-image of the inner horizon in the original AdS$_3$, which we call the inner RT surface. In this paper we give some new physical interpretation for the inner RT surface. First, the inner RT surface breaks into two pieces which anchor on the two tips of the causal development. Furthermore, we can take the two tips as the endpoints of a certain timelike interval and the inner RT surface is exactly the spacelike geodesic that represents the real part of the so-called holographic timelike entanglement entropy (HTEE). We also identify a timelike geodesic at boundary of the extended entanglement wedge, which represents the imaginary part of the HTEE. Second, in the duality between the topological massive gravity (TMG) and gravitational anomalous CFT$_2$, the entanglement entropy and the mixed state correlation that is dual to the entanglement wedge cross section (EWCS) receive correction from the Chern-Simons term in the TMG. We find that, the correction to the holographic entanglement entropy can be reproduced by the area of the inner RT surface with a proper regulation, while the mixed state correlation can be represented by the saddle geodesic chord connecting the two pieces of the inner RT surface of the mixed state we consider, which we call the inner EWCS. The equivalence between the twist on the RT surface and the length of inner RT surface is also discussed.

Figures

Figures reproduced from arXiv: 2412.21058 by the authors.

Figure 1
Figure 1. The figure is extracted from [59]. The pink surfaces represent the null hypersurfaces N±, and the blue curve represents the RT surface E. where TU˜ and TU˜ are the parameters characterizing the size of a thermal circle of the outer horizon, outer horizon: (U˜, V˜) ∼ (U˜ + i π TU˜ , V˜ − i π TV˜ ), (11) The bulk Rindler transformation (10) reduces to the boundary Rindler transformation at the asymptotic boundary ρ → … view at source ↗
Figure 2
Figure 2. The left and the right figures show the modular flow lines of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The right figure is extracted from [61]. The left figure: the blue curve is the IRT surface that is the intersection line between the two null hypersurfaces M±. The right figure: the green solid line represents the real part of the timelike entanglement entropy, and the red solid line represents the imaginary part, which correspond to the spacelike geodesic and the timelike geodesic respectively. If we consider a ca… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: The left figure shows the modular Hamiltonian slices in the Rindler [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: The figure is extracted from [75]. The PEEs sA (Ai ) are captured by the length of Ei . The purple lines are the saddle geodesics that are normal to the RT surface E. Although the figure looks static, we should consider it to be a covariant configuration. see Fig.5 for…
Figure 6
Figure 6. Figure 6: The gray blue surfaces are the spacetime slices with fixed [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: The orange curve is the boundary modular momentum flow line [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: In the left figure, the blue surface is the modular momentum slice [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: The figures are extracted from [75]. The region A ′B ′ serves as an auxiliary system to purify the bipartite system ρAB. The left figure represents the adjacent case. While the right figure represents the non-adjacent case, where we need to make a further separation, A…
Figure 10
Figure 10. Figure 10: Extracted from [51]. Illustration for the EWCS in the non-adjacent case. The extension of the EWCS ΣAB intersects with the asymptotic boundary at the two points Q1 and Q2 . Without loss of generality, we suppose that A have a smaller size than that of B, i.e. U4 − U3 …
Figure 11
Figure 11. Figure 11: Illustration for the anomalous part E a W (A, B) in the non-adjacent case. The dashed line represents the boundary of the causal development. The purple lines γ1 and γ2 represent the null geodesics. The IEWCS ΣbAB (the red solid line) and its extension (the red dashed…
Figure 12
Figure 12. Figure 12: The blue lines represent the boundary of the causal development [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]
Figure 13
Figure 13. Figure 13: Illustration for the APEE of the subinterval [PITH_FULL_IMAGE:figures/full_fig_p037_13.png]
Figure 14
Figure 14. Figure 14: The surfaces enclosed by the dash curve represent the null infinities of the [PITH_FULL_IMAGE:figures/full_fig_p042_14.png]
Figure 15
Figure 15. Figure 15: The yellow surface represents the null hypersurface [PITH_FULL_IMAGE:figures/full_fig_p045_15.png]
Figure 16
Figure 16. Figure 16: The black and the purple lines represent the interval [PITH_FULL_IMAGE:figures/full_fig_p047_16.png]

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