pith. sign in

arxiv: 2509.05700 · v3 · pith:SUKYIO5Vnew · submitted 2025-09-06 · ✦ hep-th · gr-qc

Entanglement Entropy and Thermodynamics of Dynamical Black Holes

Weizhen Jia , Qiongyu Qi , Christina Gao This is my paper

Pith reviewed 2026-05-21 23:07 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords dynamical black holesentanglement entropyf(R) gravityWald entropyreplica trickapparent horizongeneralized second lawphysical process first law
0
0 comments X

The pith

In f(R) theories the dynamical black hole entropy equals Wald entropy on the generalized apparent horizon at first order.

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

The paper shows that for dynamical black holes the thermodynamic entropy defined by Hollands-Wald-Zhang coincides with the Wald entropy computed on the generalized apparent horizon when only first-order perturbations are considered. Using the replica method to compute entanglement entropy, the authors demonstrate that selecting the apparent horizon as the entangling surface is the only choice that recovers this entropy and obeys the physical-process first law. They further reinterpret the generalized second law in terms of matter entanglement entropy across the apparent horizon, allowing the total entropy to be written as a renormalized generalized entropy at the leading area-law term.

Core claim

Under first-order perturbations, the dynamical black hole entropy in any f(R) theory equals the Wald entropy evaluated on the generalized apparent horizon. Only the apparent-horizon prescription in the replica method reproduces this entropy and satisfies the physical-process first law.

What carries the argument

The generalized apparent horizon used as the entangling surface in the replica trick for computing gravitational entropy in dynamical spacetimes.

If this is right

  • The apparent horizon must be used instead of the event horizon to obtain consistent entropy in evolving black holes.
  • The modified von Neumann entropy represents matter entanglement across the apparent horizon.
  • The total entropy reduces to the renormalized generalized entropy on the apparent horizon at leading order.

Where Pith is reading between the lines

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

  • This choice of surface may generalize to other higher-curvature theories beyond f(R).
  • Numerical simulations of perturbed black holes could test whether apparent-horizon replica entropy matches thermodynamic expectations.
  • Similar horizon prescriptions might apply to other dynamical systems like collapsing stars or cosmological horizons.

Load-bearing premise

The replica trick remains valid when applied to a dynamical spacetime with the apparent horizon chosen as the entangling surface and still satisfies the first law to linear order.

What would settle it

An explicit computation in a specific f(R) model where the event-horizon replica entropy fails to match the dynamical black hole entropy or violates the first law at linear order.

read the original abstract

We explore the thermodynamic and entanglement properties of dynamical black holes based on the recently proposed dynamical black hole entropy by Hollands-Wald-Zhang. We first provide direct proof that, under first-order perturbations, the dynamical black hole entropy in any $f(R)$ theory equals the Wald entropy evaluated on the generalized apparent horizon. Then, we compute the gravitational entropy explicitly from the replica method using both the event horizon and the apparent horizon as the entangling surfaces, and we show that only the apparent horizon prescription reproduces the correct dynamical black hole entropy satisfying the physical process first law. Furthermore, we reinterpret the generalized second law by identifying the modified von Neumann entropy as the matter entanglement across the apparent horizon. This allows us to express the total entropy as the renormalized generalized entropy evaluated on this surface at the level of the leading local area-law term.

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

1 major / 1 minor

Summary. The paper claims to provide a direct proof that, under first-order perturbations, the dynamical black hole entropy of Hollands-Wald-Zhang equals the Wald entropy evaluated on the generalized apparent horizon in any f(R) theory. It then computes gravitational entropy via the replica method, comparing event-horizon and apparent-horizon choices as entangling surfaces, and shows that only the apparent-horizon prescription reproduces the dynamical entropy while satisfying the physical-process first law at linear order. It further reinterprets the generalized second law by equating the modified von Neumann entropy to matter entanglement across the apparent horizon, expressing the total entropy as the renormalized generalized entropy on that surface at leading order.

Significance. If the central claims hold, the work is significant because it supplies an explicit bridge between the replica-trick definition of entanglement entropy and the Hollands-Wald-Zhang dynamical entropy in modified gravity, thereby furnishing evidence that the apparent horizon is the physically preferred surface for thermodynamic relations in non-stationary spacetimes. The direct proof of equality to Wald entropy and the first-law match at linear order constitute concrete, falsifiable statements that can be checked against existing constructions.

major comments (1)
  1. [Abstract, paragraph 2] Abstract, paragraph 2 and the replica-method derivation: the claim that selecting the apparent horizon as the entangling surface allows a consistent n-fold cover, analytic continuation, and extraction of the linear entropy variation requires explicit verification that the conical singularity produces no O(ε²) surface terms or stress-tensor contributions that would invalidate the first-law match. Because the apparent horizon is not a Killing horizon, the standard Euclidean replica construction does not apply directly; the manuscript must demonstrate that all higher-order corrections vanish at the order needed for the physical-process first law.
minor comments (1)
  1. [Notation and definitions] Clarify the precise definition of the generalized apparent horizon used in the Wald-entropy comparison and ensure it is stated identically in the proof and in the replica calculation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive major comment. The point concerning the justification of the replica construction on the apparent horizon is well taken, and we will revise the paper to supply the requested explicit verification at linear order.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] Abstract, paragraph 2 and the replica-method derivation: the claim that selecting the apparent horizon as the entangling surface allows a consistent n-fold cover, analytic continuation, and extraction of the linear entropy variation requires explicit verification that the conical singularity produces no O(ε²) surface terms or stress-tensor contributions that would invalidate the first-law match. Because the apparent horizon is not a Killing horizon, the standard Euclidean replica construction does not apply directly; the manuscript must demonstrate that all higher-order corrections vanish at the order needed for the physical-process first law.

    Authors: We agree that the apparent horizon is not a Killing horizon and that the standard Euclidean replica construction therefore requires additional care in the dynamical case. Our derivation is performed entirely within a first-order perturbative expansion around a stationary background, in which the apparent horizon coincides with the event horizon at zeroth order. Within this framework the n-fold cover is constructed perturbatively, the conical singularity is introduced at linear order in the replica parameter, and the entropy variation is extracted from the coefficient of the linear term. We will revise the manuscript to add an explicit perturbative calculation (in a new subsection or appendix) demonstrating that all O(ε²) contributions from the conical singularity to the surface terms and to the stress-tensor expectation value vanish identically at the order relevant for the physical-process first law. This will make the linear-order match with the Hollands-Wald-Zhang entropy fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained against external benchmarks

full rationale

The paper builds directly on the external Hollands-Wald-Zhang dynamical black hole entropy definition (cited as prior independent work) and performs explicit first-order perturbation proofs for f(R) theories plus replica-method comparisons between event and apparent horizons. No load-bearing step reduces by construction to a self-definition, fitted input renamed as prediction, or self-citation chain from the current authors. The replica-trick application to dynamical spacetimes is treated as a calculational choice whose validity is checked against the external entropy and first-law condition rather than assumed to force the outcome. This is the most common honest finding for papers that reference external constructions while adding independent computations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the validity of the replica trick in dynamical spacetimes, the definition of dynamical black hole entropy from Hollands-Wald-Zhang, and standard assumptions of f(R) gravity at linear order.

axioms (2)
  • domain assumption The replica trick yields the correct gravitational entropy when the entangling surface is chosen as the apparent horizon.
    Invoked when the authors compute entropy on the apparent horizon and compare to the dynamical definition.
  • domain assumption First-order perturbation theory is sufficient to compare dynamical entropy with Wald entropy.
    Stated in the first result of the abstract.

pith-pipeline@v0.9.0 · 5669 in / 1454 out tokens · 36487 ms · 2026-05-21T23:07:31.903623+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · 39 internal anchors

  1. [1]

    Bekenstein,Black Holes and the Second Law,Lett

    J.D. Bekenstein,Black Holes and the Second Law,Lett. Nuovo Cim.4(1972) 737

    work page 1972

  2. [2]

    Bekenstein,Black holes and entropy,Phys

    J.D. Bekenstein,Black holes and entropy,Phys. Rev. D7(1973) 2333

    work page 1973

  3. [3]

    Hawking,Particle Creation by Black Holes,Commun

    S.W. Hawking,Particle Creation by Black Holes,Commun. Math. Phys.43(1975) 199

    work page 1975

  4. [4]

    Bardeen, B

    J.M. Bardeen, B. Carter and S.W. Hawking,The Four Laws of Black Hole Mechanics, Commun. Math. Phys.31(1973) 161

    work page 1973

  5. [5]

    Black Hole Entropy is Noether Charge

    R.M. Wald,Black Hole Entropy is the Noether Charge,Phys. Rev. D48(1993) R3427 [gr-qc/9307038]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  6. [6]

    Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy

    V. Iyer and R.M. Wald,Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy,Phys. Rev. D50(1994) 846 [gr-qc/9403028]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  7. [7]

    Witten,A Mini-Introduction To Information Theory,Riv

    E. Witten,A Mini-Introduction To Information Theory,Riv. Nuovo Cim.43(2020) 187 [1805.11965]

    work page arXiv 2020

  8. [8]

    Notes on Some Entanglement Properties of Quantum Field Theory

    E. Witten,APS Medal for Exceptional Achievement in Research: Invited Article on Entanglement Properties of Quantum Field Theory,Rev. Mod. Phys.90(2018) 045003 [1803.04993]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  9. [9]

    On Geometric Entropy

    C.G. Callan, Jr. and F. Wilczek,On Geometric Entropy,Phys. Lett. B333(1994) 55 [hep-th/9401072]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  10. [10]

    Geometric and Renormalized Entropy in Conformal Field Theory

    C. Holzhey, F. Larsen and F. Wilczek,Geometric and Renormalized Entropy in Conformal Field Theory,Nucl. Phys. B424(1994) 443 [hep-th/9403108]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  11. [11]

    Entanglement Entropy and Quantum Field Theory

    P. Calabrese and J.L. Cardy,Entanglement Entropy and Quantum Field Theory,J. Stat. Mech.0406(2004) P06002 [hep-th/0405152]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  12. [12]

    Generalized gravitational entropy

    A. Lewkowycz and J. Maldacena,Generalized Gravitational Entropy,JHEP08(2013) 090 [1304.4926]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  13. [13]

    Dimensional Reduction in Quantum Gravity

    G. ’t Hooft,Dimensional Reduction in Quantum Gravity,Conf. Proc. C930308(1993) 284 [gr-qc/9310026]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  14. [14]

    The World as a Hologram

    L. Susskind,The World as a Hologram,J. Math. Phys.36(1995) 6377 [hep-th/9409089]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  15. [15]

    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]. – 29 –

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  16. [16]

    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

  17. [17]

    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

  18. [18]

    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

  19. [19]

    Holographic Entanglement Entropy: An Overview

    T. Nishioka, S. Ryu and T. Takayanagi,Holographic Entanglement Entropy: An Overview,J. Phys. A42(2009) 504008 [0905.0932]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  20. [20]

    1983 paper on entanglement entropy: "On the Entropy of the Vacuum outside a Horizon"

    R.D. Sorkin,1983 Paper on Entanglement Entropy: ”On the Entropy of the Vacuum Outside a Horizon”, in10th International Conference on General Relativity and Gravitation, vol. 2, pp. 734–736, 1984 [1402.3589]

    work page internal anchor Pith review Pith/arXiv arXiv 1983

  21. [21]

    Bombelli, R.K

    L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin,A Quantum Source of Entropy for Black Holes,Phys. Rev. D34(1986) 373

    work page 1986

  22. [22]

    Entropy and Area

    M. Srednicki,Entropy and Area,Phys. Rev. Lett.71(1993) 666 [hep-th/9303048]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  23. [23]

    Frolov and I

    V.P. Frolov and I. Novikov,Dynamical Origin of the Entropy of a Black Hole,Phys. Rev. D 48(1993) 4545 [gr-qc/9309001]

    work page arXiv 1993

  24. [24]

    Black Hole Entropy and Induced Gravity

    T. Jacobson,Black Hole Entropy and Induced Gravity,gr-qc/9404039

    work page internal anchor Pith review Pith/arXiv arXiv

  25. [25]

    The Conical Singularity And Quantum Corrections To Entropy Of Black Hole

    S.N. Solodukhin,The Conical Singularity and Quantum Corrections to Entropy of Black Hole,Phys. Rev. D51(1995) 609 [hep-th/9407001]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  26. [26]

    On the Description of the Riemannian Geometry in the Presence of Conical Defects

    D.V. Fursaev and S.N. Solodukhin,On the Description of the Riemannian Geometry in the Presence of Conical Defects,Phys. Rev. D52(1995) 2133 [hep-th/9501127]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  27. [27]

    On one-loop renormalization of black-hole entropy

    D.V. Fursaev and S.N. Solodukhin,On one Loop Renormalization of Black Hole Entropy, Phys. Lett. B365(1996) 51 [hep-th/9412020]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  28. [28]

    Black Hole Entropy without Brick Walls

    J.-G. Demers, R. Lafrance and R.C. Myers,Black Hole Entropy Without Brick Walls,Phys. Rev. D52(1995) 2245 [gr-qc/9503003]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  29. [29]

    Black Hole Entropy and Entropy of Entanglement

    D.N. Kabat,Black Hole Entropy and Entropy of Entanglement,Nucl. Phys. B453(1995) 281 [hep-th/9503016]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  30. [30]

    Renormalization of Black Hole Entropy and of the Gravitational Coupling Constant

    F. Larsen and F. Wilczek,Renormalization of Black Hole Entropy and of the Gravitational Coupling Constant,Nucl. Phys. B458(1996) 249 [hep-th/9506066]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  31. [31]

    Cones, Spins and Heat Kernels

    D.V. Fursaev and G. Miele,Cones, Spins and Heat Kernels,Nucl. Phys. B484(1997) 697 [hep-th/9605153]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  32. [32]

    Wald,Entropy and Black-Hole Thermodynamics,Phys

    R.M. Wald,Entropy and Black-Hole Thermodynamics,Phys. Rev. D20(1979) 1271

    work page 1979

  33. [33]

    Unruh and R.M

    W.G. Unruh and R.M. Wald,Acceleration radiation and the generalized second law of thermodynamics,Phys. Rev. D25(1982) 942

    work page 1982

  34. [34]

    Unruh and R.M

    W.G. Unruh and R.M. Wald,Entropy Bounds, Acceleration Radiation, and the Generalized Second Law,Phys. Rev. D27(1983) 2271

    work page 1983

  35. [35]

    A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices

    A.C. Wall,A Proof of the Generalized Second Law for Rapidly Changing Fields and Arbitrary Horizon Slices,Phys. Rev. D85(2012) 104049 [1105.3445]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  36. [36]

    Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory

    L. Susskind and J. Uglum,Black hole entropy in canonical quantum gravity and superstring theory,Phys. Rev. D50(1994) 2700 [hep-th/9401070]. – 30 –

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  37. [37]

    Renormalization of Entanglement Entropy and the Gravitational Effective Action

    J.H. Cooperman and M.A. Luty,Renormalization of Entanglement Entropy and the Gravitational Effective Action,JHEP12(2014) 045 [1302.1878]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  38. [38]

    A Second Law for Higher Curvature Gravity

    A.C. Wall,A Second Law for Higher Curvature Gravity,Int. J. Mod. Phys. D24(2015) 1544014 [1504.08040]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  39. [39]

    Wald,Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics, Chicago Lectures in Physics, University of Chicago Press, Chicago, IL (1995)

    R.M. Wald,Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics, Chicago Lectures in Physics, University of Chicago Press, Chicago, IL (1995)

    work page 1995

  40. [40]

    The "physical process" version of the first law and the generalized second law for charged and rotating black holes

    S. Gao and R.M. Wald,The ’Physical Process’ Version of the First Law and the Generalized Second Law for Charged and Rotating Black Holes,Phys. Rev. D64(2001) 084020 [gr-qc/0106071]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  41. [41]

    Hollands, R

    S. Hollands, R.M. Wald and V.G. Zhang,Entropy of Dynamical Black Holes,Phys. Rev. D 110(2024) 024070 [2402.00818]

    work page arXiv 2024

  42. [42]

    Visser and Z

    M.R. Visser and Z. Yan,Properties of Dynamical Black Hole Entropy,JHEP10(2024) 029 [2403.07140]

    work page arXiv 2024

  43. [43]

    Rignon-Bret,Second law from the Noether current on null hypersurfaces,Phys

    A. Rignon-Bret,Second Law from the Noether Current on Null Hypersurfaces,Phys. Rev. D 108(2023) 044069 [2303.07262]

    work page arXiv 2023

  44. [44]

    D. Kong, Y. Tian, H. Zhang and J. Zhao,Dynamical Black Hole Entropy Beyond General Relativity from the Einstein Frame,Phys. Rev. D111(2025) 084005 [2412.00647]

    work page arXiv 2025

  45. [45]

    Holographic Entanglement Entropy for General Higher Derivative Gravity

    X. Dong,Holographic Entanglement Entropy for General Higher Derivative Gravity,JHEP 01(2014) 044 [1310.5713]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  46. [46]

    Hollands, A.D

    S. Hollands, A.D. Kov´ acs and H.S. Reall,The Second Law of Black Hole Mechanics in Effective Field Theory,JHEP08(2022) 258 [2205.15341]

    work page arXiv 2022

  47. [47]

    Bhattacharyya, P

    S. Bhattacharyya, P. Dhivakar, A. Dinda, N. Kundu, M. Patra and S. Roy,An Entropy Current and the Second Law in Higher Derivative Theories of Gravity,JHEP09(2021) 169 [2105.06455]

    work page arXiv 2021

  48. [48]

    Hawking and J.B

    S.W. Hawking and J.B. Hartle,Energy and Angular Momentum Flow Into a Black Hole, Commun. Math. Phys.27(1972) 283

    work page 1972

  49. [49]

    Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge University Press (12, 2009), 10.1017/CBO9780511606601

    E. Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge University Press (12, 2009), 10.1017/CBO9780511606601

    work page doi:10.1017/cbo9780511606601 2009

  50. [50]

    Matsuda and S

    T. Matsuda and S. Mukohyama,Covariant Entropy Bound Beyond General Relativity,Phys. Rev. D103(2021) 024002 [2007.14015]

    work page arXiv 2021

  51. [51]

    Nishioka,Entanglement ntropy: Holography and Renormalization Group,Rev

    T. Nishioka,Entanglement ntropy: Holography and Renormalization Group,Rev. Mod. Phys. 90(2018) 035007 [1801.10352]

    work page arXiv 2018

  52. [52]

    A Quantum Focussing Conjecture

    R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall,Quantum Focusing Conjecture,Phys. Rev. D93(2016) 064044 [1506.02669]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  53. [53]

    Furugori, K

    H. Furugori, K. Nishii, D. Yoshida and K. Yoshimura,Apparent Horizons Associated with Dynamical Black Hole Entropy,2507.14105

    work page arXiv

  54. [54]

    Isolated Horizons: Hamiltonian Evolution and the First Law

    A. Ashtekar, S. Fairhurst and B. Krishnan,Isolated Horizons: Hamiltonian Evolution and the First Law,Phys. Rev. D62(2000) 104025 [gr-qc/0005083]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  55. [55]

    Generic Isolated Horizons and their Applications

    A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krishnan, J. Lewandowski et al.,Isolated Horizons and Their Applications,Phys. Rev. Lett.85(2000) 3564 [gr-qc/0006006]. – 31 –

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  56. [56]

    Geometry of Generic Isolated Horizons

    A. Ashtekar, C. Beetle and J. Lewandowski,Geometry of Generic Isolated Horizons,Class. Quant. Grav.19(2002) 1195 [gr-qc/0111067]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  57. [57]

    Dynamical Horizons and their Properties

    A. Ashtekar and B. Krishnan,Dynamical Horizons and Their Properties,Phys. Rev. D68 (2003) 104030 [gr-qc/0308033]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  58. [58]

    Isolated and dynamical horizons and their applications

    A. Ashtekar and B. Krishnan,Isolated and Dynamical Horizons and Their Applications, Living Rev. Rel.7(2004) 10 [gr-qc/0407042]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  59. [59]

    Ashtekar and B

    A. Ashtekar and B. Krishnan,Quasi-Local Black Hole Horizons: Recent Advances, 2502.11825

    work page arXiv

  60. [60]

    Ciambelli, S

    L. Ciambelli, S. Pasterski and E. Tabor,Radiation in holography,JHEP09(2024) 124 [2404.02146]

    work page arXiv 2024

  61. [61]

    Fern´ andez-´Alvarez and J

    F. Fern´ andez-´Alvarez and J.M.M. Senovilla,Gravitational radiation at infinity with negative cosmological constant and AdS4 holography,2507.05826

    work page arXiv

  62. [62]

    Radiation in Fluid/Gravity and the Flat Limit

    G. Arenas-Henriquez, L. Ciambelli, F. Diaz, W. Jia and D. Rivera-Betancour,Radiation in Fluid/Gravity and the Flat Limit,2508.01446

    work page internal anchor Pith review Pith/arXiv arXiv

  63. [63]

    Decoding the Apparent Horizon: A Coarse-Grained Holographic Entropy

    N. Engelhardt and A.C. Wall,Decoding the Apparent Horizon: Coarse-Grained Holographic Entropy,Phys. Rev. Lett.121(2018) 211301 [1706.02038]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  64. [64]

    Coarse Graining Holographic Black Holes

    N. Engelhardt and A.C. Wall,Coarse Graining Holographic Black Holes,JHEP05(2019) 160 [1806.01281]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  65. [65]

    Wall and Z

    A.C. Wall and Z. Yan,Linearized Second Law for Higher Curvature Gravity and Nonminimally Coupled Vector Fields,Phys. Rev. D110(2024) 084005 [2402.05411]. – 32 –

    work page arXiv 2024