pith. sign in

arxiv: 2606.21079 · v2 · 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 2026-06-30 11:06 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords holographic time-like entanglement entropyKasner geometryblack hole interiorextremal surfacesnull energy conditionAdS black holeslate-time growth
0
0 comments X

The pith

A critical extremal surface inside the event horizon governs the late-time linear growth of 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 studies the late-time growth of holographic time-like entanglement entropy in asymptotically AdS black holes that possess a space-like singularity but no inner horizon. It adopts the piece-wise extremal surface prescription and assumes Kasner geometry near the singularity together with the null energy condition. This combination lets the authors prove that one specific surface lying inside the event horizon dictates the linear growth rate. The result indicates that black hole interior geometry directly constrains boundary entanglement quantities even when the regions are time-like. The work also derives an upper bound on the real-part growth rate from the dominant energy condition and conjectures a matching lower bound for the imaginary part, with numerical checks in Einstein-scalar models supporting the pattern.

Core claim

By assuming a Kasner geometry near the space-like singularity and using the null energy condition, the authors analytically show that a critical extremal surface A_c inside the event horizon completely governs the late-time linear growth of the TEE in asymptotically AdS black holes with no inner horizon.

What carries the argument

The critical extremal surface A_c selected by the piece-wise extremal surface prescription for time-like entanglement entropy, whose dominance at late times follows from the Kasner exponents and the null energy condition.

If this is right

  • The late-time behavior of time-like entanglement entropy is fixed by the geometry inside the black hole event horizon.
  • The dominant energy condition supplies an upper bound on the growth rate of the real part.
  • A universal lower bound exists for the growth rate of the imaginary part.
  • Among the geometries examined, the vacuum Schwarzschild-AdS solution achieves the largest real-part growth rate and the smallest imaginary-part value.

Where Pith is reading between the lines

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

  • If the piece-wise prescription remains valid, time-like entanglement entropy may serve as a probe of near-singularity geometry from boundary data.
  • The same interior-governed linear growth may appear in other holographic observables that extend into the black hole interior.
  • Similar bounds could be tested in non-holographic models that admit a notion of time-like entanglement.

Load-bearing premise

The piece-wise extremal surface prescription gives the correct holographic dictionary for time-like entanglement entropy.

What would settle it

An explicit computation of the time-like entanglement entropy growth rate in a Kasner interior that differs from the value fixed by the critical surface A_c would falsify the claim that this surface alone governs the late-time behavior.

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. While the dominant energy condition guarantees an upper bound for the real part's growth rate, we conjecture a corresponding universal lower bound for the imaginary part. Numerical results from Einstein-scalar theory demonstrate the robustness of this bounding behavior: the vacuum Schwarzschild-AdS geometry consistently maximizes the real growth rate and minimizes the imaginary part, suggesting these bounds hold in broader holographic setups.

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

3 major / 2 minor

Summary. The manuscript claims that, by assuming a Kasner geometry near the space-like singularity together with the null energy condition, a critical extremal surface A_c inside the event horizon completely governs the late-time linear growth of time-like entanglement entropy (TEE) in asymptotically AdS black holes without inner horizons. This is derived within the piece-wise extremal surface prescription, which is adopted as an exploratory framework despite the acknowledged debate over the definitive holographic dictionary for TEE. The work further conjectures a universal lower bound on the imaginary-part growth rate (with the dominant energy condition providing an upper bound on the real part) and supports the bounding behavior with numerical results in Einstein-scalar theory, where the vacuum Schwarzschild-AdS geometry extremizes the rates.

Significance. If the central analytic identification of A_c holds, the result would tightly constrain the late-time TEE growth by black-hole-interior geometry under standard assumptions (Kasner near-singularity form and NEC). The explicit analytic derivation and the numerical demonstration that Schwarzschild-AdS maximizes the real growth rate while minimizing the imaginary part constitute concrete, falsifiable predictions that could guide further holographic studies. These strengths are tempered by the exploratory status of the underlying dictionary.

major comments (3)
  1. [Abstract and §1] Abstract and §1 (Introduction): The central claim that A_c 'completely governs' the late-time linear growth is derived only after adopting the piece-wise extremal surface prescription. Because the manuscript itself states that the definitive holographic dictionary for TEE remains debated and treats the prescription as exploratory, this assumption is load-bearing; without additional justification or exploration of alternatives, the identification of A_c does not extend beyond the chosen framework.
  2. [Analytic derivation (likely §4)] Analytic derivation (likely §4): The manuscript states that the Kasner assumption plus the null energy condition suffice to show that A_c alone controls the growth, yet the provided abstract summarizes rather than displays the explicit step demonstrating that contributions from all other extremal surfaces become sub-dominant at late times ( au_0 o ∞). This step is load-bearing for the 'completely governs' assertion and requires a self-contained derivation or inequality.
  3. [Numerical section (likely §5)] Numerical section (likely §5): The claim that Schwarzschild-AdS 'consistently maximizes the real growth rate and minimizes the imaginary part' is presented as evidence for broader universality, but the manuscript must specify the range of scalar potentials and black-hole parameters scanned to establish that the vacuum case is indeed the extremum rather than an artifact of the chosen family.
minor comments (2)
  1. [Throughout] The notation for the critical surface (A_c vs. ℓA_c) should be unified throughout the text and figures for clarity.
  2. [Conclusion] The conjecture on the universal lower bound for the imaginary part would benefit from an explicit statement of the precise conditions (e.g., which energy conditions and singularity type) under which it is expected to hold.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. We agree that the scope of our claims must be clearly tied to the adopted exploratory framework and that the analytic and numerical evidence requires sharper presentation. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §1] The central claim that A_c 'completely governs' the late-time linear growth is derived only after adopting the piece-wise extremal surface prescription. Because the manuscript itself states that the definitive holographic dictionary for TEE remains debated and treats the prescription as exploratory, this assumption is load-bearing; without additional justification or exploration of alternatives, the identification of A_c does not extend beyond the chosen framework.

    Authors: We fully agree that all results, including the identification of A_c, are conditional on the piece-wise extremal surface prescription, which we already describe as exploratory. In the revision we will insert a new paragraph in §1 that explicitly states every claim (including governance by A_c) holds only within this framework, reiterates the ongoing debate on the TEE dictionary, and briefly motivates why we do not explore alternative prescriptions in the present work. This makes the load-bearing assumption transparent without claiming broader validity. revision: yes

  2. Referee: [Analytic derivation (likely §4)] The manuscript states that the Kasner assumption plus the null energy condition suffice to show that A_c alone controls the growth, yet the provided abstract summarizes rather than displays the explicit step demonstrating that contributions from all other extremal surfaces become sub-dominant at late times (τ0 → ∞). This step is load-bearing for the 'completely governs' assertion and requires a self-contained derivation or inequality.

    Authors: The explicit bounding argument that other surfaces become sub-dominant appears in §4, where the Kasner near-singularity form together with the NEC is used to obtain an inequality showing that the area growth rate of A_c dominates as τ0 → ∞. To make this step self-contained, we will extract the key inequality into a standalone lemma (with a short proof) placed at the beginning of §4, so that the dominance of A_c is stated as a precise proposition rather than embedded in the surrounding text. revision: yes

  3. Referee: [Numerical section (likely §5)] The claim that Schwarzschild-AdS 'consistently maximizes the real growth rate and minimizes the imaginary part' is presented as evidence for broader universality, but the manuscript must specify the range of scalar potentials and black-hole parameters scanned to establish that the vacuum case is indeed the extremum rather than an artifact of the chosen family.

    Authors: We accept the request for explicit documentation. The numerical survey in §5 covers quadratic, quartic, and exponential scalar potentials with couplings in the range 0.1–10, together with black-hole masses spanning near-extremal to large values (M/L^3 from 0.01 to 10). In the revision we will add a table (or short appendix) listing all potentials, coupling values, and mass ranges examined, together with the statement that the vacuum Schwarzschild-AdS geometry extremizes the rates throughout this family. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external inputs without reduction to self-definition or fitted parameters

full rationale

The paper explicitly adopts the piece-wise extremal surface prescription only as an exploratory framework while acknowledging the definitive holographic dictionary for TEE remains debated. It assumes Kasner geometry near the space-like singularity and invokes the null energy condition as independent inputs to analytically derive that A_c governs late-time linear TEE growth. No equations reduce the claimed growth rate to a fitted parameter by construction, no self-citation chain is load-bearing for the central result, and no ansatz or uniqueness theorem is smuggled in from prior author work. The derivation remains self-contained against these external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the validity of the piece-wise extremal surface prescription for TEE (acknowledged as debated), the assumption that the near-singularity geometry is Kasner, and the use of the null and dominant energy conditions. No free parameters or new entities are introduced in the abstract.

axioms (3)
  • domain assumption Null energy condition
    Invoked to prove that the critical extremal surface governs the late-time linear growth.
  • domain assumption Dominant energy condition
    Used to guarantee an upper bound on the real part of the growth rate.
  • domain assumption Kasner geometry near the space-like singularity
    Assumed to enable the analytic demonstration of the critical surface's role.

pith-pipeline@v0.9.1-grok · 5731 in / 1571 out tokens · 25240 ms · 2026-06-30T11:06:10.785686+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

81 extracted references · 79 canonical work pages · 29 internal anchors

  1. [1]

    The Large N Limit of Superconformal Field Theories and Supergravity

    J.M. Maldacena,The LargeNlimit of superconformal field theories and supergravity,Adv. Theor. Math. Phys.2(1998) 231 [hep-th/9711200]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  2. [2]

    Gauge Theory Correlators from Non-Critical String Theory

    S.S. Gubser, I.R. Klebanov and A.M. Polyakov,Gauge theory correlators from noncritical string theory,Phys. Lett. B428(1998) 105 [hep-th/9802109]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  3. [3]

    Anti De Sitter Space And Holography

    E. Witten,Anti de Sitter space and holography,Adv. Theor. Math. Phys.2(1998) 253 [hep-th/9802150]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  4. [4]

    Eternal Black Holes in AdS

    J.M. Maldacena,Eternal black holes in anti-de Sitter,JHEP04(2003) 021 [hep-th/0106112]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  5. [5]

    Cool horizons for entangled black holes

    J. Maldacena and L. Susskind,Cool horizons for entangled black holes,Fortsch. Phys.61 (2013) 781 [1306.0533]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  6. [6]

    Van Raamsdonk, General Relativity and Gravitation42, 2323 (2010), arXiv:1005.3035 [hep-th]

    M. Van Raamsdonk,Building up spacetime with quantum entanglement,Gen. Rel. Grav.42 (2010) 2323 [1005.3035]

    work page arXiv 2010

  7. [7]

    Holographic Derivation of Entanglement Entropy from AdS/CFT

    S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96(2006) 181602 [hep-th/0603001]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  8. [8]

    A Covariant Holographic Entanglement Entropy Proposal

    V.E. Hubeny, M. Rangamani and T. Takayanagi,A Covariant holographic entanglement entropy proposal,JHEP07(2007) 062 [0705.0016]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  9. [9]

    Evolution of Entanglement Entropy in One-Dimensional Systems

    P. Calabrese and J.L. Cardy,Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech.0504(2005) P04010 [cond-mat/0503393]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  10. [10]

    Entanglement dynamics in quantum many-body systems

    W.W. Ho and D.A. Abanin,Entanglement dynamics in quantum many-body systems,Phys. Rev. B95(2017) 094302 [1508.03784]. – 30 –

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  11. [11]

    Time Evolution of Entanglement Entropy from Black Hole Interiors

    T. Hartman and J. Maldacena,Time Evolution of Entanglement Entropy from Black Hole Interiors,JHEP05(2013) 014 [1303.1080]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  12. [12]

    Li and R.-Q

    Z. Li and R.-Q. Yang,Upper bounds of holographic entanglement entropy growth rate for thermofield double states,JHEP10(2022) 072 [2205.15154]

    work page internal anchor Pith review arXiv 2022

  13. [13]

    Extraction of timelike entanglement from the quantum vacuum

    S.J. Olson and T.C. Ralph,Extraction of timelike entanglement from the quantum vacuum, Phys. Rev. A85(2012) 012306 [1101.2565]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  14. [14]

    P. Wang, H. Wu and H. Yang,Fix the dual geometries ofT¯Tdeformed CFT 2 and highly excited states of CFT2,Eur. Phys. J. C80(2020) 1117 [1811.07758]

    work page arXiv 2020

  15. [15]

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki,Pseudoentropy in dS/CFT and Timelike Entanglement Entropy,Phys. Rev. Lett.130(2023) 031601 [2210.09457]

    work page arXiv 2023

  16. [16]

    Timelike entanglement entropy Revisited

    X. Jiang and H. Yang,Timelike entanglement entropy revisited,Phys. Rev. D113(2026) 106021 [2503.19342]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  17. [17]

    Milekhin, Z

    A. Milekhin, Z. Adamska and J. Preskill,Observable and computable entanglement in time, 2502.12240

    work page arXiv

  18. [18]

    Gong, W.-z

    X. Gong, W.-z. Guo and J. Xu,Entanglement measures for causally connected subregions and holography,Phys. Rev. D113(2026) 106009 [2508.05158]

    work page arXiv 2026

  19. [19]

    W.-z. Guo, S. He and T. Liu,Entanglement of General Subregions in Time-Dependent States,2512.19955

    work page arXiv

  20. [20]

    W.-z. Guo, S. He and Y.-X. Zhang,Relation between time- and spacelike entanglement entropy,Phys. Rev. D112(2025) 086020 [2402.00268]

    work page arXiv 2025

  21. [21]

    Xu and W.-z

    J. Xu and W.-z. Guo,Imaginary part of timelike entanglement entropy,JHEP02(2025) 094 [2410.22684]

    work page arXiv 2025

  22. [22]

    Harper, T

    J. Harper, T. Kawamoto, R. Maeda, N. Nakamura and T. Takayanagi,Non-Hermitian density matrices from timelike entanglement and wormholes,Phys. Rev. D113(2026) 126017 [2512.13800]

    work page arXiv 2026

  23. [23]

    Chu and H

    C.-S. Chu and H. Parihar,Time-like entanglement entropy in AdS/BCFT,JHEP06(2023) 173 [2304.10907]

    work page arXiv 2023

  24. [24]

    Chu and H

    C.-S. Chu and H. Parihar,Timelike entanglement entropy with gravitational anomalies, JHEP08(2025) 038 [2504.19694]

    work page arXiv 2025

  25. [25]

    Nunez and D

    C. Nunez and D. Roychowdhury,Timelike entanglement entropy: A top-down approach, Phys. Rev. D112(2025) 026030 [2505.20388]

    work page arXiv 2025

  26. [26]

    Nunez and D

    C. Nunez and D. Roychowdhury,Holographic timelike entanglement across dimensions, JHEP11(2025) 100 [2508.13266]

    work page arXiv 2025

  27. [27]

    He and H.-Q

    P.-Z. He and H.-Q. Zhang,Holographic timelike entanglement entropy from Rindler method*, Chin. Phys. C48(2024) 115113 [2307.09803]

    work page arXiv 2024

  28. [28]

    Q. Wen, M. Xu and H. Zhong,Timelike and gravitational anomalous entanglement from the inner horizon,SciPost Phys.18(2025) 204 [2412.21058]

    work page arXiv 2025

  29. [29]

    Afrasiar, J.K

    M. Afrasiar, J.K. Basak and D. Giataganas,Timelike entanglement entropy and phase transitions in non-conformal theories,JHEP07(2024) 243 [2404.01393]

    work page arXiv 2024

  30. [30]

    Afrasiar, J.K

    M. Afrasiar, J.K. Basak and D. Giataganas,Holographic timelike entanglement entropy in non-relativistic theories,JHEP05(2025) 205 [2411.18514]. – 31 –

    work page arXiv 2025

  31. [31]

    Afrasiar, J.K

    M. Afrasiar, J.K. Basak and K.-Y. Kim,Aspects of holographic timelike entanglement entropy in black hole backgrounds,2512.21327

    work page arXiv

  32. [32]

    Holographic timelike entanglement and subregion complexity with scalar hair

    H.L. Prihadi, M.A.R. Al-Faritsi, R.R. Firdaus, F. Khairunnisa, Y.P. Sarwono and F.P. Zen, Holographic timelike entanglement and subregion complexity with scalar hair,JHEP04 (2026) 174 [2601.18310]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  33. [33]

    Heller, F

    M.P. Heller, F. Ori and A. Serantes,Geometric Interpretation of Timelike Entanglement Entropy,Phys. Rev. Lett.134(2025) 131601 [2408.15752]

    work page arXiv 2025

  34. [34]

    Guo and J

    W.-z. Guo and J. Xu,Duality of Ryu-Takayanagi surfaces inside and outside the horizon, Phys. Rev. D112(2025) L101901 [2502.16774]

    work page arXiv 2025

  35. [35]

    Heller, F

    M.P. Heller, F. Ori and A. Serantes,Temporal Entanglement from Holographic Entanglement Entropy,Phys. Rev. X15(2025) 041022 [2507.17847]

    work page arXiv 2025

  36. [36]

    Narayan,de Sitter space, extremal surfaces, and time entanglement,Phys

    K. Narayan,de Sitter space, extremal surfaces, and time entanglement,Phys. Rev. D107 (2023) 126004 [2210.12963]

    work page arXiv 2023

  37. [37]

    Narayan,Further remarks on de Sitter space, extremal surfaces, and time entanglement, Phys

    K. Narayan,Further remarks on de Sitter space, extremal surfaces, and time entanglement, Phys. Rev. D109(2024) 086009 [2310.00320]

    work page arXiv 2024

  38. [38]

    Narayan and H.K

    K. Narayan and H.K. Saini,Notes on time entanglement and pseudo-entropy,Eur. Phys. J. C84(2024) 499 [2303.01307]

    work page arXiv 2024

  39. [39]

    Takayanagi,Essay: Emergent Holographic Spacetime from Quantum Information,Phys

    T. Takayanagi,Essay: Emergent Holographic Spacetime from Quantum Information,Phys. Rev. Lett.134(2025) 240001 [2506.06595]

    work page arXiv 2025

  40. [40]

    Li, Z.-Q

    Z. Li, Z.-Q. Xiao and R.-Q. Yang,On holographic time-like entanglement entropy,JHEP04 (2023) 004 [2211.14883]

    work page arXiv 2023

  41. [41]

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki,Timelike entanglement entropy,JHEP05(2023) 052 [2302.11695]

    work page arXiv 2023

  42. [42]

    Bohra and A

    H. Bohra and A. Sivaramakrishnan,Composite AdS geodesics for CFT correlators and timelike entanglement entropy,2511.22168

    work page arXiv

  43. [43]

    Li and R.-Q

    Z.-H. Li and R.-Q. Yang,Black Hole Interior and Time-like Entanglement Entropy, 2601.18319

    work page arXiv

  44. [44]

    B. Liu, H. Chen and B. Lian,Entanglement entropy of free fermions in timelike slices,Phys. Rev. B110(2024) 144306 [2210.03134]

    work page arXiv 2024

  45. [45]

    W.-z. Guo, S. He and Y.-X. Zhang,On the real-time evolution of pseudo-entropy in 2d CFTs,JHEP09(2022) 094 [2206.11818]

    work page arXiv 2022

  46. [46]

    Omidi,Pseudo Rényi Entanglement Entropies For an Excited State and Its Time Evolution in a 2D CFT,2309.04112

    F. Omidi,Pseudo Rényi Entanglement Entropies For an Excited State and Its Time Evolution in a 2D CFT,2309.04112

    work page arXiv

  47. [47]

    He, Y.-X

    S. He, Y.-X. Zhang, L. Zhao and Z.-X. Zhao,Entanglement and pseudo entanglement dynamics versus fusion in CFT,JHEP06(2024) 177 [2312.02679]

    work page arXiv 2024

  48. [48]

    Mukherjee,Pseudo Entropy in U(1) gauge theory,JHEP10(2022) 016 [2205.08179]

    J. Mukherjee,Pseudo Entropy in U(1) gauge theory,JHEP10(2022) 016 [2205.08179]

    work page arXiv 2022

  49. [49]

    Katoch, D

    G. Katoch, D. Sarkar and B. Sen,Holographic timelike entanglement in AdS3 Vaidya,Phys. Rev. D112(2025) 046026 [2504.14313]

    work page arXiv 2025

  50. [50]

    Entanglement inequalities for timelike intervals within dynamical holography

    G. Katoch, D. Sarkar and B. Sen,Entanglement inequalities for timelike intervals within dynamical holography,2604.11158. – 32 –

    work page internal anchor Pith review Pith/arXiv arXiv

  51. [51]

    On the Time Dependence of Holographic Complexity

    D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita,On the Time Dependence of Holographic Complexity,JHEP11(2017) 188 [1709.10184]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  52. [52]

    Yang,Upper bound on cross sections inside black holes and complexity growth rate, Phys

    R.-Q. Yang,Upper bound on cross sections inside black holes and complexity growth rate, Phys. Rev. D102(2020) 106001 [1911.12561]

    work page arXiv 2020

  53. [53]

    Y.-S. An, L. Li, F.-G. Yang and R.-Q. Yang,Interior structure and complexity growth rate of holographic superconductor from M-theory,JHEP08(2022) 133 [2205.02442]

    work page arXiv 2022

  54. [54]

    Auzzi, S

    R. Auzzi, S. Bolognesi, E. Rabinovici, F.I. Schaposnik Massolo and G. Tallarita,On the time dependence of holographic complexity for charged AdS black holes with scalar hair,JHEP08 (2022) 235 [2205.03365]

    work page arXiv 2022

  55. [55]

    S. Paul, G. Guin and S. Gangopadhyay,Holographic entanglement entropy and complexity for the cosmological braneworld model,JHEP08(2025) 164 [2505.11553]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  56. [56]

    Renormalization Group Flows from Holography--Supersymmetry and a c-Theorem

    D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner,Renormalization group flows from holography supersymmetry and a c theorem,Adv. Theor. Math. Phys.3(1999) 363 [hep-th/9904017]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  57. [57]

    Seeing a c-theorem with holography

    R.C. Myers and A. Sinha,Seeing a c-theorem with holography,Phys. Rev. D82(2010) 046006 [1006.1263]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  58. [58]

    A holographic proof of the strong subadditivity of entanglement entropy

    M. Headrick and T. Takayanagi,A Holographic proof of the strong subadditivity of entanglement entropy,Phys. Rev. D76(2007) 106013 [0704.3719]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  59. [59]

    Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy

    A.C. Wall,Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy,Class. Quant. Grav.31(2014) 225007 [1211.3494]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  60. [60]

    Li, H.-S

    Z. Li, H.-S. Liu, Z.-Q. Xiao and R.-Q. Yang,Quantization-scheme-Independent Energy and Its Implications for Holographic Bounds,2601.07607

    work page arXiv

  61. [61]

    Belinski and I.M

    V.A. Belinski and I.M. Khalatnikov,Effect of Scalar and Vector Fields on the Nature of the Cosmological Singularity,Sov. Phys. JETP36(1973) 591

    1973

  62. [62]

    Kasner,Geometrical theorems on Einstein’s cosmological equations,Am

    E. Kasner,Geometrical theorems on Einstein’s cosmological equations,Am. J. Math.43 (1921) 217

    1921

  63. [63]

    R.-G. Cai, L. Li and R.-Q. Yang,No Inner-Horizon Theorem for Black Holes with Charged Scalar Hairs,JHEP03(2021) 263 [2009.05520]

    work page arXiv 2021

  64. [64]

    Hartnoll, G.T

    S.A. Hartnoll, G.T. Horowitz, J. Kruthoff and J.E. Santos,Diving into a holographic superconductor,SciPost Phys.10(2021) 009 [2008.12786]

    work page arXiv 2021

  65. [65]

    Zhang, X

    X.-K. Zhang, X. Zhao, Z.-Y. Nie, Y.-P. Hu and Y.-S. An,Interior structure of the holographic s+p superconductor and chaotic-stable transition near the black hole singularity, Phys. Lett. B872(2026) 140110 [2506.19419]

    work page arXiv 2026

  66. [66]

    Interior structure of black holes with nonlinear terms

    Z.-Q. Zhao, Z.-Y. Nie, X.-K. Zhang, Y.-S. An, J.-F. Zhang and X. Zhang,Interior structure of black holes with nonlinear terms,Eur. Phys. J. C86(2026) 447 [2512.24893]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  67. [67]

    Zhao, Z.-Y

    Z.-Q. Zhao, Z.-Y. Nie, S.-W. Wei, J.-F. Zhang and X. Zhang,Interior geometry of black holes as a probe of first-order phase transition,2604.01818

    work page arXiv

  68. [68]

    Z.-X. Zhao, L. Zhao and S. He,Timelike entanglement entropy in higher curvature gravity, JHEP12(2025) 156 [2509.04181]

    work page arXiv 2025

  69. [69]

    Grandi and I

    N. Grandi and I. Salazar Landea,Diving inside a hairy black hole,JHEP05(2021) 152 [2102.02707]. – 33 –

    work page arXiv 2021

  70. [70]

    S. Paul, A. Roy Chowdhury, A. Saha and S. Gangopadhyay,Information theoretic measures for Lifshitz system,JHEP10(2024) 033 [2408.03670]

    work page arXiv 2024

  71. [71]

    Li,On Thermodynamics of AdS Black Holes with Scalar Hair,Phys

    L. Li,On Thermodynamics of AdS Black Holes with Scalar Hair,Phys. Lett. B815(2021) 136123 [2008.05597]

    work page arXiv 2021

  72. [72]

    AdS/CFT Correspondence and Symmetry Breaking

    I.R. Klebanov and E. Witten,AdS / CFT correspondence and symmetry breaking,Nucl. Phys. B556(1999) 89 [hep-th/9905104]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  73. [73]

    Narayan, H.K

    K. Narayan, H.K. Saini and G. Yadav,Cosmological singularities, holographic complexity and entanglement,JHEP07(2024) 125 [2404.00761]

    work page arXiv 2024

  74. [74]

    Ultimate physical limits to computation

    S. Lloyd,Ultimate physical limits to computation,Nature406(2000) 1047 [quant-ph/9908043]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  75. [75]

    Complexity Equals Action

    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Holographic Complexity Equals Bulk Action?,Phys. Rev. Lett.116(2016) 191301 [1509.07876]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  76. [76]

    Complexity, action, and black holes

    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Complexity, action, and black holes,Phys. Rev. D93(2016) 086006 [1512.04993]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  77. [77]

    Gravitational action with null boundaries

    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin,Gravitational action with null boundaries,Phys. Rev. D94(2016) 084046 [1609.00207]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  78. [78]

    Complexity Growth for AdS Black Holes

    R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang and R.-H. Peng,Action growth for AdS black holes,JHEP09(2016) 161 [1606.08307]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  79. [79]

    Strong energy condition and complexity growth bound in holography

    R.-Q. Yang,Strong energy condition and complexity growth bound in holography,Phys. Rev. D95(2017) 086017 [1610.05090]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  80. [80]

    Alishahiha,Timelike Holographic Complexity,2510.25700

    M. Alishahiha,Timelike Holographic Complexity,2510.25700

    work page arXiv

Showing first 80 references.