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arxiv: 2606.21079 · v1 · pith:PHPS7VUDnew · submitted 2026-06-19 · ✦ hep-th · gr-qc

Linear Growth of Holographic Time-like Entanglement Entropy and Kasner exponents

Zi-Hao Li , Run-Qiu Yang This is my paper

Pith reviewed 2026-06-26 14:06 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords holographic time-like entanglement entropyextremal surfacesKasner geometryblack hole interiorsAdS black holesnull energy condition
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0 comments X

The pith

A critical extremal surface inside the event horizon governs the late-time linear growth of holographic time-like entanglement entropy.

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

The paper examines the late-time growth of time-like entanglement entropy in asymptotically AdS black holes that have a space-like singularity. Using the piece-wise extremal surface prescription and assuming Kasner geometry near the singularity together with the null energy condition, it demonstrates that one particular critical surface inside the horizon fully determines the linear growth rate. Numerical checks in Einstein-scalar theory and analytic proofs under the dominant energy condition in planar static cases further show that the Schwarzschild-AdS solution supplies an upper bound on the real-part growth rate and a lower bound on the imaginary-part growth rate.

Core claim

By assuming Kasner geometry near the space-like singularity and using the null energy condition, a critical extremal surface A_c inside the event horizon completely governs the late-time linear growth of the TEE. Numerical results indicate that the vacuum Schwarzschild-AdS sets an upper bound on the real part growth rate and lower bound on the imaginary part, proven under dominant energy condition in static planar symmetric cases.

What carries the argument

The critical extremal surface A_c inside the event horizon that sets the coefficient of the linear growth term in the piece-wise extremal surface prescription for TEE.

If this is right

  • The late-time behavior of TEE is tightly constrained by the geometry of black hole interiors.
  • The real-part growth rate is bounded above by the Schwarzschild-AdS value in static planar symmetric spacetimes under the dominant energy condition.
  • The imaginary-part growth rate is bounded below by the Schwarzschild-AdS value in the same class of spacetimes.
  • The bounds are conjectured to hold more generally beyond the proven planar static case.

Where Pith is reading between the lines

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

  • Kasner exponents may enter the explicit linear coefficient of TEE growth once the interior geometry is fixed.
  • If the prescription remains valid, similar interior surfaces could control late-time growth for other time-like boundary observables.
  • The result suggests that time-like entanglement may serve as a probe of near-singularity physics even when direct access is impossible.
  • Violations of the null or dominant energy conditions could alter the dominance of A_c and therefore change the growth bounds.

Load-bearing premise

The piece-wise extremal surfaces prescription correctly computes holographic time-like entanglement entropy and the near-singularity region is described by Kasner geometry.

What would settle it

A counter-example black hole spacetime in which the late-time TEE growth rate deviates from the value fixed by any single critical surface inside the horizon would falsify the governing role of A_c.

read the original abstract

The holographic time-like entanglement entropy (TEE) extends entanglement to time-like boundary subregions. While its definitive holographic dictionary remains debated, one concrete proposal utilizes piece-wise extremal surfaces. In this work, we adopt this geometric prescription as an exploratory framework to holographically investigate the late-time ($\tau_0\to \infty$) growth of TEE in asymptotically AdS black holes with a space-like singularity and no inner horizon. By assuming a Kasner geometry near the space-like singularity and using null energy condition, we analytically show that a critical extremal surface $\mathcal{A}_c$ inside the event horizon completely governs the late-time linear growth of the TEE. This result suggests that the late-time behavior of TEE is tightly constrained by the geometry of black hole interiors. Using numerical results from Einstein-scalar theory, we find a robust behavior: the vacuum Schwarzschild-AdS geometry sets an upper bound on the growth rate of the real part and a lower bound on the imaginary part. We prove these bounds in static planar symmetric case under dominant energy condition and conjecture that it should be true in more general cases.

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

2 major / 2 minor

Summary. The manuscript adopts the piece-wise extremal surface prescription for holographic time-like entanglement entropy (TEE) as an exploratory framework and analytically demonstrates that a critical extremal surface A_c inside the event horizon governs the late-time linear growth of TEE in asymptotically AdS black holes with space-like singularities. This is shown by assuming Kasner geometry near the singularity together with the null energy condition. Numerical solutions in Einstein-scalar theory indicate that the vacuum Schwarzschild-AdS geometry supplies an upper bound on the real-part growth rate and a lower bound on the imaginary-part growth rate; these bounds are proven for static planar symmetric cases under the dominant energy condition, with a conjecture for generality.

Significance. If the adopted TEE prescription holds, the result supplies analytic control linking black-hole interior geometry (via Kasner exponents) to late-time TEE growth rates, together with concrete bounds that are at least partially proven. The combination of analytic derivation under stated assumptions and numerical checks in a concrete theory constitutes a clear strength, though the exploratory framing limits the immediate scope.

major comments (2)
  1. [Abstract] Abstract and § on analytic derivation: the identification of A_c as completely governing late-time growth, and the derived bounds on Re/Im rates, are obtained entirely within the piece-wise extremal surface ansatz. The abstract explicitly flags this dictionary as debated and adopted only exploratorily; if an alternative prescription applies, the central claim does not transfer.
  2. [Proof of bounds (static planar case)] Section containing the DEC proof and conjecture: the upper/lower bounds on growth rates are proven only in the static planar symmetric case under the dominant energy condition. The extension to more general cases is stated as a conjecture without an outline of a proof strategy or additional evidence, which is load-bearing for the 'robust behavior' claim.
minor comments (2)
  1. [Analytic section] The definition of the critical surface A_c and its relation to the Kasner exponents should be stated with an explicit equation or diagram for clarity.
  2. [Introduction] A brief reference list entry or footnote on the status of the piece-wise TEE prescription would help readers locate the debate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and § on analytic derivation: the identification of A_c as completely governing late-time growth, and the derived bounds on Re/Im rates, are obtained entirely within the piece-wise extremal surface ansatz. The abstract explicitly flags this dictionary as debated and adopted only exploratorily; if an alternative prescription applies, the central claim does not transfer.

    Authors: We agree that the identification of the critical surface A_c and the derived bounds are obtained within the piece-wise extremal surface prescription. The abstract and introduction already emphasize that this is an exploratory framework adopted because the definitive holographic dictionary for TEE remains debated. All central claims are therefore conditional on this prescription, as stated. revision: no

  2. Referee: [Proof of bounds (static planar case)] Section containing the DEC proof and conjecture: the upper/lower bounds on growth rates are proven only in the static planar symmetric case under the dominant energy condition. The extension to more general cases is stated as a conjecture without an outline of a proof strategy or additional evidence, which is load-bearing for the 'robust behavior' claim.

    Authors: The referee correctly notes that the analytic proof applies only to the static planar symmetric case under the dominant energy condition. The conjecture for generality is supported by the numerical evidence in Einstein-scalar theory, but we acknowledge that no explicit proof strategy for the general case is outlined. This remains a conjecture. revision: partial

Circularity Check

0 steps flagged

No significant circularity; central claims derived from external conditions and stated assumptions

full rationale

The paper adopts the piece-wise extremal surface prescription explicitly as an 'exploratory framework' because the dictionary 'remains debated,' rather than deriving or fitting it internally. Analytic control of late-time TEE growth by A_c follows from the Kasner near-singularity assumption plus the null energy condition (NEC); bounds on growth rates are proven under the dominant energy condition (DEC) for the static planar case and verified numerically in Einstein-scalar theory. No derivation step reduces by construction to a fitted input, self-citation chain, or redefinition of the target quantity. The result is therefore self-contained against the stated external benchmarks and assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Kasner near-singularity assumption, the debated piece-wise extremal surface prescription for TEE, and the null/dominant energy conditions. No free parameters or invented entities are introduced in the abstract. Based on abstract only.

axioms (2)
  • domain assumption Null energy condition
    Invoked to show analytically that the critical surface A_c governs late-time linear growth.
  • domain assumption Dominant energy condition
    Used to prove the Schwarzschild-AdS bounds in the static planar symmetric case.

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discussion (0)

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Reference graph

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