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REVIEW 3 major objections 3 minor 113 references

The paper proves a dual generalized second law at future null infinity: the unavoidable growth is not of entropy but of a thermodynamic potential — free energy or grand potential — built from the Bondi mass.

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2026-08-04 06:22 UTC pith:USLWGF3W

load-bearing objection A real extension of Wall's modular-logic to null infinity, but the thermodynamic identification is conditional on a slow-variation assumption that becomes arbitrarily restrictive for the soft vacuum's high-l modes. the 3 major comments →

arxiv 2601.03353 v2 pith:USLWGF3W submitted 2026-01-06 hep-th gr-qc

Black hole thermodynamics at null infinity. Part 1: Dual Generalized Second Law

Antoine Rignon-Bret , Matthieu Vilatte This is my paper
classification hep-th gr-qc PACS 04.70.-s04.62.-v
keywords black hole thermodynamicsgeneralized second lawnull infinityrelative entropymodular HamiltonianBondi massalgebraic quantum field theorythermodynamic potentials
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that the second law of black hole thermodynamics can be formulated from the perspective of observers at future null infinity, who never see the horizon area. Using algebraic quantum field theory and the monotonicity of relative entropy under restriction of observable algebras, it shows that the monotone quantity is not the generalized entropy but a thermodynamic potential built from the Bondi mass. For the Hartle–Hawking state regularized with a hard angular-momentum cutoff, the potential is the free energy M − T_H S; for the soft regularization with mode-dependent temperatures, it is a grand potential with chemical potentials. These inequalities define a 'dual generalized second law' that can be combined with the standard horizon GSL to yield a decreasing generalized free energy M − T_H(S + A/4G). If correct, this gives asymptotic observers a well-defined irreversible law and predicts that work can be extracted from the non-thermal radiation.

Core claim

The central claim is that the monotonicity of relative entropy, applied to nested algebras of observables on future null infinity, yields the dual generalized second law: ΔM − T_H ΔS ≤ 0 for the hard-regularized vacuum and ΔM − Σ μ Δn − T_H ΔS ≤ 0 for the soft-regularized vacuum. The key step is relating the one-sided modular Hamiltonian of the chosen vacuum state to the integrated normal-ordered stress tensor, then using the semiclassical Bondi mass-loss equation together with a slow-variation assumption to identify the modular boost-energy variation with the Bondi mass variation. The result is that the spontaneous evolution of quantum fields toward the vacuum is governed by the decrease of

What carries the argument

The central object is the one-sided modular Hamiltonian of a vacuum state restricted to the algebra of observables in a region of future null infinity, written as an integral of the normal-ordered stress tensor. The proof uses the monotonicity of relative entropy under algebra restriction and the fact that the Hartle–Hawking state and its hard and soft regularizations are thermal (KMS) states for the geometric boost flow. The slow-variation assumption κ^{-1} ∂_u ⟨T_uu⟩ ≪ ⟨T_uu⟩ is the bridge that converts the Bondi mass-loss equation into a relation between mass variation and modular boost energy.

Load-bearing premise

The proof assumes the radiation flux at null infinity varies slowly: κ^{-1} ∂_u ⟨T_uu⟩ must be much smaller than ⟨T_uu⟩ itself. If the flux changes rapidly, the Bondi mass variation can no longer be identified with the modular boost-energy variation, and the inequalities break down.

What would settle it

Compute the relative entropy for a quantum state on two nested algebras of null infinity in which the stress tensor has a sudden, order-one change over a timescale shorter than κ^{-1}. If the difference of modular Hamiltonians does not equal the Bondi mass difference, the dual GSL as stated fails. Alternatively, find a physically allowed Hadamard state with rapidly varying flux whose free energy M − T_H S increases between two cuts.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Asymptotic observers, who lack access to the horizon area, can still formulate an irreversible law; the monotone quantity is the free energy or grand potential, not the generalized entropy.
  • Combining the dual GSL with the horizon GSL gives a decreasing generalized free energy F_gen = M − T_H(S + A/4G), extending the usual GSL to slices that end at different cuts of null infinity.
  • The form of the thermodynamic potential depends on the vacuum: hard angular-momentum regularization yields a free energy, soft mode-dependent regularization yields a grand potential with chemical potentials μ_{ωl} = ω(1 − κ/κ_l).
  • For states satisfying the slow-variation assumption, the dual GSL holds for a large class of semiclassical states representing local excitations of the vacuum; violations would require rapidly varying fluxes.

Where Pith is reading between the lines

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

  • If the dual GSL holds for evaporating black holes, it suggests that the black hole–radiation system behaves like an open quantum system coupled to a thermal reservoir; the slow-variation assumption is then the Markovian condition, and the chemical potentials in the soft regularization could be interpreted as work terms extractable from the non-thermal radiation.
  • The framework may extend to the Unruh vacuum and to rotating black holes by using mode-dependent effective temperatures and transmission coefficients; the chemical potentials would then be fixed by gray-body factors rather than by an ad hoc choice of κ_l.
  • A direct numerical test would be to simulate a collapsing shell and check whether F = M − T_H S decreases between two cuts of future null infinity; the paper leaves the explicit Unruh-state calculation to future work.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper develops an algebraic proof of a 'dual generalized second law' at future null infinity for a massless scalar field on an eternal Schwarzschild background. The central claim is that monotonicity of Araki relative entropy under restriction to nested algebras on I^+_R, with the Hartle–Hawking state or its hard/soft regularizations as reference, yields a monotone thermodynamic potential built from the Bondi mass: a free energy ΔM − T_H ΔS ≤ 0 for the L-vacuum, and a grand potential ΔM − Σ μ_{ωl} Δn_{ωl} − T_H ΔS ≤ 0 for the κ_l-vacuum. The paper further combines the dual law with the horizon generalized second law to obtain the decrease of F_gen = M − T_H(S + A/4G). The proof chain is: construct the algebra and GNS Hilbert space at I_R, identify the one-sided modular Hamiltonian with an integral of the normal-ordered stress tensor (Appendices D–E), relate it to the Bondi mass via the semiclassical BMS mass-loss equation under a slow-variation assumption (2.27), and apply relative-entropy monotonicity.

Significance. If the central result is correct, it gives asymptotic observers a genuine irreversible thermodynamic law based on the Bondi mass rather than the horizon area, and it substantially extends the earlier heuristic treatment [52] with an algebraic-QFT framework. The paper's strengths include a careful GNS construction for several vacuum states, an explicit account of the relation between modular Hamiltonians and stress-energy operators, and a clear separation of covariant versus normal-ordered stress tensors. The regularized vacua are physically motivated. However, the proof is conditional on a slow-variation assumption that is not derived or quantified, and there is a sign inconsistency in the horizon-area step of Section 4.4. These are load-bearing issues; they are likely fixable, but the manuscript in its current form does not fully establish the advertised law in the stated generality.

major comments (3)
  1. [§2.3.2, Eq. (2.27); §4.3, Eq. (4.64)] The slow-variation/Markovian assumption is load-bearing but is not given a precise domain of validity. The passage from (2.26) to (2.28) drops the term −κ^{-1}∂_u⟨T_uu⟩; if this term is not small, Eq. (2.33) — ΔM = κ(⟨K_{U2}⟩−⟨K_{U1}⟩) — fails, and with it the bridge from relative-entropy monotonicity to ΔM − T_H ΔS ≤ 0 in (4.46) and (4.73). The condition is not implied by membership in the GNS Hilbert space, and no norm bound or small parameter is supplied. For the κ_l-vacuum the per-mode condition (4.64) is even more restrictive: convergence (4.60) forces κ_l → 0, so κ_l^{-1} becomes enormous and even extremely slow high-l transients violate (4.64). Thus the grand-potential form is not established for the high-l sector outside the strictly stationary limit. The authors should state a precise condition (e.g. a bound on ∥∂_u T^l_uu∥ relative to κ_l) and discuss states that violate it.
  2. [§4.4, Eq. (4.84), compared with §2.3.1, Eqs. (2.21), (2.23)] There is a sign inconsistency in the horizon-area step. With K^{A^H_i} defined in (4.83) as 2π∫_{D^H_i}(Ṽ−Ṽ_i) T_{ṼṼ}, Eq. (2.21) gives ⟨K_i⟩ = (A|∞ − A_i)/4, so the difference is Δ⟨K^{A^H}⟩ = −(A_2−A_1)/4. Eq. (4.84), which states Δ(A/4G) = Δ⟨K^{A^H}⟩, has the opposite sign. The correct relation is Δ(A/4G) = −Δ⟨K^{A^H}⟩. Since the monotonicity inequality (4.7) contains −Δ⟨K_tot⟩, this sign changes the area contribution in the derived thermodynamic potential. The displayed sign must be corrected before (4.85)–(4.86) follows from the stated modular-Hamiltonian argument.
  3. [§4.1–4.3, Eqs. (4.17), (4.45), (4.68)–(4.69)] The conversion of the reference-vacuum energy variation ΔM_Ω into an entropy variation ΔS_Ω via the Clausius relation is an additional physical input, not a consequence of relative-entropy monotonicity. In the Hartle–Hawking case (4.17) this involves formally infinite quantities and a cancellation; in the L- and κ_l-vacua it assumes the reference radiation is an equilibrium thermal reservoir satisfying the first law. The final free-energy/grand-potential inequality depends on this identification. This should be stated explicitly as an assumption or derived from a microscopic definition of the reference entropy, rather than presented as part of the algebraic proof.
minor comments (3)
  1. [§4, Eq. (4.7)] Eq. (4.7) is written for the total algebra A^B_i but displays only Δ⟨K^{A^I_i}_Ω⟩. If the horizon modular term is intended to be dropped, the text should say so; if it is included, the sign conventions need to be aligned with Eq. (4.84).
  2. [§4.2, Eq. (4.39)] The statement that (4.39) is 'true for most states if L≫1' is not a precise theorem. The condition should be formulated as a restriction on the high-l flux of the state |Ψ⟩, with a norm or energy bound, so that the domain of validity of the L-vacuum inequality is clear.
  3. [General] There are several typographical and notational inconsistencies: e.g. 'with shall consider' in §4.2; the vacuum state |Ω^L_H⟩ is sometimes written |Ω_L⟩; and the symbol ΔS is redefined in (4.19), (4.47), and (4.72) with slightly different content. A careful editing pass would improve readability.

Circularity Check

0 steps flagged

No circular derivation: the dual GSL follows from external relative-entropy monotonicity plus explicitly stated semiclassical assumptions; the cited prior work is motivational, not load-bearing.

full rationale

The derivation chain is self-contained. The central input is Araki's monotonicity of relative entropy (D.28), used as an external information-theoretic theorem, combined with the semiclassical Einstein equations (2.25)–(2.33) and the explicitly constructed vacua and modular Hamiltonians of Section 3 and Appendix E. The dual GSL inequalities (4.46), (4.73), and (4.85)–(4.86) are obtained by algebraically rewriting the fundamental inequality (4.7); no step redefines the target inequality as an input. The slow-variation/Markovian condition (2.27) and its per-mode analogue (4.64) are stated as domain-of-validity assumptions before the identification ΔM = κΔK (2.33), and the paper restricts its claims to states satisfying them; this is a scope limitation, not a circular fit. The chemical potentials μωl = ω(1 − κ/κ_l) are defined from the chosen κ_l-vacuum sequence (4.77), not fitted to make the inequality hold, and the inequality is derived from relative-entropy monotonicity. The cited prior work [52] by one of the authors is used for motivation and context, and the paper explicitly says it improves [52] by providing rigorous algebraic foundations; all load-bearing modular-Hamiltonian and stress-tensor relations are proven in the text or appendices. Thus no prediction reduces by construction to an input, and no self-citation is load-bearing.

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 0 invented entities

The inequalities are derived for families of auxiliary reference vacua. L and κ_l are not fixed by the theory: L is an arbitrary finite cutoff, while κ_l is a free positive sequence with κ_l ≤ κ and Σ(2l+1)κ_l < ∞. The form of the potential (free versus grand) and the chemical potentials depend on these choices. No parameters are fitted to data; the dependence is explicit. No new physical entities (particles, forces, dimensions) are introduced.

free parameters (2)
  • Angular momentum cutoff L = free integer, chosen large but finite; L=0 gives Minkowski vacuum, L=∞ gives Hartle–Hawking
    Introduced in eq. (3.63) to make the energy flux at null infinity finite by keeping only modes l < L thermal. The resulting dual GSL (4.46) has the same free-energy form for any finite L, but L is not fixed by the theory.
  • Effective surface-gravity sequence {κ_l} = free positive sequence with κ_l ≤ κ and Σ_{l≥0}(2l+1)κ_l < ∞; a suggested form is κ_l ~ e^{-α l(l+1)} with α>0 (footnote
    Defines the soft regularized vacuum (3.69). The chemical potentials μ_{ωl} = ω(1 − κ/κ_l) in eq. (4.77) depend on this arbitrary sequence, so the grand-potential form (4.73) is parameter-dependent.
axioms (7)
  • standard math Standard results of algebraic QFT and modular theory: GNS construction, Tomita–Takesaki theory, monotonicity of Araki relative entropy under restriction of algebras.
    Invoked throughout Section 4 and Appendices C–D; the GSL proof is built on these theorems.
  • domain assumption Semiclassical Einstein equations: Raychaudhuri–Damour equation at the horizon (2.15–2.21) and BMS mass-loss equation at null infinity (2.24–2.33), with renormalized stress tensor.
    Used to relate modular-Hamiltonian vevs to area and Bondi-mass variations.
  • domain assumption Slow-variation/Markovianity assumption (2.27) and its per-mode version (4.64).
    Needed to neglect ∂_u terms and identify ΔM with the boost-energy difference; not proven, only argued for small excitations of the chosen vacua.
  • domain assumption Clausius relation ΔS_Ω = ΔM_Ω/T_H applied to the reference-vacuum radiation.
    Converts vacuum energy variation into entropy variation; assumes thermal equilibrium and zero extractable work. The paper itself notes that the multi-reservoir κ_l vacuum may in principle allow work extraction.
  • domain assumption Hadamard condition and existence of a renormalized stress tensor for the states in the GNS Hilbert spaces.
    Required for finite energy densities and for the normal-ordered stress tensor used in the modular Hamiltonian (Appendix B.3, Section 3.3).
  • domain assumption Reeh–Schlieder-type cyclicity/separating property for vacuua restricted to subalgebras A^I_i at null infinity.
    Needed so that modular Hamiltonians and relative entropies are well defined; stated as a straightforward extension in Section 4 and Appendix D.
  • domain assumption Tensor-product factorization A^B_i = A^H ⊗ A^I_i and product reference states |Ω_H⟩⊗|Ω⟩.
    Used in the complete proof, eqs. (4.6), (4.81), to combine horizon and null-infinity contributions.

pith-pipeline@v1.3.0-alltime-deepseek · 75848 in / 13402 out tokens · 135040 ms · 2026-08-04T06:22:42.052567+00:00 · methodology

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The generalized second law (GSL) of black hole thermodynamics asserts the monotonic increase of the generalized entropy combining the black hole area and the entropy of quantum fields outside the horizon. Modern proofs of the GSL rely on information theoretic methods and are typically formulated using algebras of observables defined on the event horizon together with a vacuum state invariant under horizon symmetries, inducing a geometric modular flow. In this work, we formulate a dual version of the generalized second law from the perspective of asymptotic observers at future null infinity, who do not have access to the black hole area. Our approach exploits the dependence of the second law on the choice of algebra of observables and of a reference state invariant under suitable symmetries, in close analogy with open quantum thermodynamics. Using algebraic quantum field theory and modular theory, we analyze several physically motivated vacuum states, including the Hartle Hawking state and two classes of regularized vacua. We show that, at null infinity, the monotonic quantity governing an irreversible evolution is no longer the generalized entropy, but rather a thermodynamic potential constructed from asymptotic observables. Depending on the chosen vacuum, this potential takes the form of the free energy or of a generalized grand potential built from the Bondi mass and additional (angular) mode dependent chemical potentials. The resulting inequalities define a dual generalized second law at future null infinity, which can be consistently combined with the standard GSL involving variations of the black hole area.

Figures

Figures reproduced from arXiv: 2601.03353 by Antoine Rignon-Bret, Matthieu Vilatte.

Figure 1
Figure 1. Figure 1: Penrose diagram of the maximal extension of the Schwarschild solution. The solid blue (reps. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Conformal extension of the black hole spacetime centered on spacelike infinity. Regions [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The trajectory in red is generated by a vector [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Setup in which the hypersurfaces Σ1 and Σ2 both starts at the horizon bifurcation surface B and end at different cuts 𝑈 = 𝑈1 and 𝑈 = 𝑈2 respectively at I+ 𝑅 . Are also depicted the regions DH and DI 𝑖 . as both the horizon and null infinity have topology R×𝑆 2 . If we show the dual GSL between DB 1 and DB 2 , then unitarity ensures that the proof extends between Σ1 and Σ2. At the algebraic level our notati… view at source ↗
Figure 5
Figure 5. Figure 5: The two spacelike hypersurfaces Σ1 and Σ1 we consider in this proof, as well as the regions on the horizon and null infinity on which the algebras are defined. 50Strictly speaking, in order to be spacelike, we should add to these regions the part between the bifurcation surface Band the cut 𝑆𝑖 together with the part between 𝜄 0 and the cut C𝑖 , see footnote 35. 50 [PITH_FULL_IMAGE:figures/full_fig_p051_5.png] view at source ↗

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Works this paper leans on

113 extracted references · 57 linked inside Pith

  1. [1]

    Springer,1978

    M.Planck,Über das gesetz der energieverteilung im normalspektrum,inVon Kirchhoff bis Planck: Theorie der Wärmestrahlung in historisch-kritischer Darstellung,pp.178–191. Springer,1978

    1978

  2. [2]

    A.Einstein,Über einem die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt,Annalenderphysik4(1905)

    1905

  3. [3]

    J.D.Bekenstein,Black holes and entropy,PhysicalReviewD7(1973),no.8,2333

    1973

  4. [4]

    S.W.Hawking,Black hole explosions,Nature248(1974)30–31

    1974

  5. [5]

    World Scientific,1975

    S.W.Hawking,Particle creation by black holes,inEuclidean quantum gravity,pp.167–188. World Scientific,1975

    1975

  6. [6]

    S.W.Hawking,Gravitational radiation from colliding black holes,Phys.Rev.Lett.26(1971) 1344–1346

    1971

  7. [7]

    J.M.Bardeen,B.CarterandS.W.Hawking,The four laws of black hole mechanics, Communicationsinmathematicalphysics31(1973),no.2,161–170

    1973

  8. [8]

    J.D.Bekenstein,Generalized second law of thermodynamics in black-hole physics,PhysicalReview D9(1974),no.12,3292

    1974

  9. [9]

    WorldScientific,2020

    J.D.Bekenstein,Black holes and entropy,inJACOB BEKENSTEIN: The Conservative Revolutionary,pp.307–320. WorldScientific,2020

    2020

  10. [10]

    R.D.Sorkin,The statistical mechanics of black hole thermodynamics,Blackholesandrelativistic stars(1998)177–194,gr-qc/9705006

    Pith/arXiv arXiv 1998

  11. [11]

    A.C.Wall,Ten Proofs of the Generalized Second Law,JHEP06(2009)021,0901.3865

    Pith/arXiv arXiv 2009

  12. [12]

    A.C.Wall,A Proof of the generalized second law for rapidly-evolving Rindler horizons,Phys.Rev.D 82(2010)124019,1007.1493

    Pith/arXiv arXiv 2010

  13. [13]

    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,[Erratum: Phys.Rev.D87,069904(2013)]

    Pith/arXiv arXiv 2012

  14. [14]

    A.C.Wall,A Second Law for Higher Curvature Gravity,Int.J.Mod.Phys.D24(2015),no.12, 1544014,1504.08040

    Pith/arXiv arXiv 2015

  15. [15]

    T.FaulknerandA.J.Speranza,Gravitational algebras and the generalized second law,JHEP11 (2024)099,2405.00847

    Pith/arXiv arXiv 2024

  16. [16]

    J.Kirklin,Generalised second law beyond the semiclassical regime,JHEP07(2025)192,2412.01903 89

    Pith/arXiv arXiv 2025

  17. [17]

    G.Lindblad,Completely positive maps and entropy inequalities,CommunicationsinMathematical Physics40(1975)147–151

    1975

  18. [18]

    A.Uhlmann,Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory,CommunicationsinMathematicalPhysics54(1977),no.1,21–32

    1977

  19. [19]

    H.Araki,Relative entropy of states of von Neumann algebras,PublicationsoftheResearchInstitute forMathematicalSciences11(1975),no.3,809–833

    1975

  20. [20]

    D.Petz,Monotonicity of quantum relative entropy revisited,ReviewsinMathematicalPhysics15 (2003),no.01,79–91,quant-ph/0209053

    Pith/arXiv arXiv 2003

  21. [21]

    SpringerScience&BusinessMedia,2004

    M.OhyaandD.Petz,Quantum entropy and its use. SpringerScience&BusinessMedia,2004

    2004

  22. [22]

    H.Umegaki,Conditional expectation in an operator algebra, IV (entropy and information),inKodai Mathematical Seminar Reports,vol.14,pp.59–85,DepartmentofMathematics,TokyoInstituteof Technology. 1962

    1962

  23. [23]

    H.Spohn,Entropy production for quantum dynamical semigroups,JournalofMathematicalPhysics 19(1978),no.5,1227–1230

    1978

  24. [24]

    B.SchumacherandM.D.Westmoreland,Relative entropy in quantum information theory, quant-ph/0004045

    Pith/arXiv arXiv

  25. [25]

    OUPOxford,2002

    H.-P.BreuerandF.Petruccione,The theory of open quantum systems. OUPOxford,2002

    2002

  26. [26]

    V.Vedral,The role of relative entropy in quantum information theory,ReviewsofModernPhysics 74(2002),no.1,197,quant-ph/0102094

    Pith/arXiv arXiv 2002

  27. [27]

    J.Goold,M.Huber,A.Riera,L.DelRioandP.Skrzypczyk,The role of quantum information in thermodynamics—a topical review,JournalofPhysicsA:MathematicalandTheoretical49(2016), no.14,143001,1505.07835

    Pith/arXiv arXiv 2016

  28. [28]

    Springer,2019.1801.08314

    R.AlickiandR.Kosloff,Introduction to quantum thermodynamics: History and prospects,in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions,pp.1–33. Springer,2019.1801.08314

    arXiv 2019

  29. [29]

    G.T.LandiandM.Paternostro,Irreversibleentropyproduction: Fromclassicaltoquantum,Reviews ofModernPhysics93(2021),no.3,035008,2009.07668

    Pith/arXiv arXiv 2021

  30. [30]

    H.Casini,Relative entropy and the Bekenstein bound,ClassicalandQuantumGravity25(2008), no.20,205021,0804.2182

    Pith/arXiv arXiv 2008

  31. [31]

    Rev.D99(2019),no.12,125020,1903.00109 90

    H.Casini,S.GrilloandD.Pontello,Relative entropy for coherent states from Araki formula,Phys. Rev.D99(2019),no.12,125020,1903.00109 90

    Pith/arXiv arXiv 2019

  32. [32]

    J.Kudler-Flam,S.Leutheusser,A.A.Rahman,G.SatishchandranandA.J.Speranza,Covariant regulator for entanglement entropy: Proofs of the Bekenstein bound and the quantum null energy condition,Phys.Rev.D111(2025),no.10,105001,2312.07646

    Pith/arXiv arXiv 2025

  33. [33]

    R.Bousso,Z.Fisher,J.Koeller,S.LeichenauerandA.C.Wall,Proof of the Quantum Null Energy Condition,Phys.Rev.D93(2016),no.2,024017,1509.02542

    Pith/arXiv arXiv 2016

  34. [34]

    Lett.118(2017),no.15,151601,1701.03196

    A.C.Wall,Lower Bound on the Energy Density in Classical and Quantum Field Theories,Phys.Rev. Lett.118(2017),no.15,151601,1701.03196

    Pith/arXiv arXiv 2017

  35. [35]

    F.CeyhanandT.Faulkner,Recovering the QNEC from the ANEC,Communicationsin MathematicalPhysics377(2020),no.2,999–1045,1812.04683

    Pith/arXiv arXiv 2020

  36. [36]

    S.HollandsandR.Longo,A New Proof of the QNEC,Commun.Math.Phys.406(2025),no.11, 269,2503.04651

    arXiv 2025

  37. [37]

    S.Hollands,Modular Time Evolution and the QNEC,2503.21385

    Pith/arXiv arXiv

  38. [38]

    P.DorauandA.Much,From Quantum Relative Entropy to the Semiclassical Einstein Equations, 2510.24491

    arXiv

  39. [39]

    S.LeutheusserandH.Liu,Causal connectability between quantum systems and the black hole interior in holographic duality,PhysicalReviewD108(2023),no.8,086019,2110.05497

    Pith/arXiv arXiv 2023

  40. [40]

    E.Witten,Gravity and the crossed product,JHEP10(2022)008,2112.12828

    Pith/arXiv arXiv 2022

  41. [41]

    V.Chandrasekaran,R.Longo,G.PeningtonandE.Witten,An algebra of observables for de Sitter space,JHEP02(2023)082,2206.10780

    Pith/arXiv arXiv 2023

  42. [42]

    V.Chandrasekaran,G.PeningtonandE.Witten,Large N algebras and generalized entropy,JHEP 04(2023)009,2209.10454

    Pith/arXiv arXiv 2023

  43. [43]

    K.Jensen,J.SorceandA.J.Speranza,Generalized entropy for general subregions in quantum gravity,JHEP12(2023)020,2306.01837

    Pith/arXiv arXiv 2023

  44. [44]

    J.Kudler-Flam,S.LeutheusserandG.Satishchandran,Generalized Black Hole Entropy is von Neumann Entropy,2309.15897

    Pith/arXiv arXiv

  45. [45]

    S.AliAhmadandR.Jefferson,Crossed product algebras and generalized entropy for subregions, SciPostPhysicsCore7(2024),no.2,020,2306.07323

    Pith/arXiv arXiv 2024

  46. [46]

    J.DeVuyst,S.Eccles,P.A.HoehnandJ.Kirklin,Gravitational entropy is observer-dependent,JHEP 07(2025)146,2405.00114

    Pith/arXiv arXiv 2025

  47. [47]

    J.DeVuyst,S.Eccles,P.A.HoehnandJ.Kirklin,Linearization (in)stabilities and crossed products, JHEP05(2025)211,2411.19931 91

    Pith/arXiv arXiv 2025

  48. [48]

    J.Kudler-Flam,S.LeutheusserandG.Satishchandran,Algebraic Observational Cosmology, 2406.01669

    Pith/arXiv arXiv

  49. [49]

    J.DeVuyst,S.Eccles,P.A.HoehnandJ.Kirklin,Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy,JHEP07(2025)063,2412.15502,[Erratum: JHEP10,234(2025)]

    Pith/arXiv arXiv 2025

  50. [50]

    C.-H.ChenandG.Penington,A clock is just a way to tell the time: gravitational algebras in cosmological spacetimes,2406.02116

    Pith/arXiv arXiv

  51. [51]

    A.J.Speranza,An intrinsic cosmological observer,Class.Quant.Grav.42(2025),no.21,215023, 2504.07630

    Pith/arXiv arXiv 2025

  52. [52]

    A.Rignon-Bret,Black hole thermodynamic potentials for asymptotic observers,Phys.Rev.D110 (2024),no.12,124002,2406.15843

    Pith/arXiv arXiv 2024

  53. [53]

    D.Kapec,A.-M.RaclariuandA.Strominger,Area, Entanglement Entropy and Supertranslations at Null Infinity,Class.Quant.Grav.34(2017),no.16,165007,1603.07706

    Pith/arXiv arXiv 2017

  54. [54]

    R.Bousso,Asymptotic Entropy Bounds,Phys.Rev.D94(2016),no.2,024018,1606.02297

    Pith/arXiv arXiv 2016

  55. [55]

    R.Bousso,I.HalpernandJ.Koeller,Information Content of Gravitational Radiation and the Vacuum,Phys.Rev.D94(2016),no.6,064047,1607.03122

    Pith/arXiv arXiv 2016

  56. [56]

    W.Israel,Thermo-field dynamics of black holes,PhysicsLettersA57(1976),no.2,107–110

    1976

  57. [57]

    J.B.HartleandS.W.Hawking,Path-integral derivation of black-hole radiance,PhysicalReviewD 13(1976),no.8,2188

    1976

  58. [58]

    W.G.Unruh,Notes on black hole evaporation,Phys.Rev.D14(1976)870

    1976

  59. [59]

    Elsevier,2013

    L.D.LandauandE.M.Lifshitz,Statistical Physics: Volume 5,vol.5. Elsevier,2013

    2013

  60. [60]

    R.Kubo,Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems,JournalofthephysicalsocietyofJapan12(1957), no.6,570–586

    1957

  61. [61]

    I,PhysicalReview115(1959),no.6, 1342

    P.C.MartinandJ.Schwinger,Theory of many-particle systems. I,PhysicalReview115(1959),no.6, 1342

    1959

  62. [62]

    R.M.Wald,On Particle Creation by Black Holes,Commun.Math.Phys.45(1975)9–34

    1975

  63. [63]

    R.H.Price,Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations,Phys.Rev.D5(1972)2419–2438

    1972

  64. [64]

    R.H.Price,Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields,Phys.Rev.D5(1972)2439–2454 92

    1972

  65. [65]

    Oxforduniversitypress,1998

    S.Chandrasekhar,The mathematical theory of black holes,vol.69. Oxforduniversitypress,1998

    1998

  66. [66]

    M.DafermosandI.Rodnianski,A proof of Price’s law for the collapse of a self-gravitating scalar field,Inventionesmathematicae162(2005),no.2,381–457,gr-qc/0309115v1

    Pith/arXiv arXiv 2005

  67. [67]

    I,Phys.Rev.98(May,1955)1123–1126

    A.Raychaudhuri,Relativistic Cosmology. I,Phys.Rev.98(May,1955)1123–1126

    1955

  68. [68]

    PhDthesis,ParisU.,VI-VII,1979

    T.Damour,Quelques proprietes mecaniques, electromagnet iques, thermodynamiques et quantiques des trous noir. PhDthesis,ParisU.,VI-VII,1979

    1979

  69. [69]

    A.Rignon-Bret,Second law from the Noether current on null hypersurfaces,Phys.Rev.D108(2023), no.4,044069,2303.07262

    Pith/arXiv arXiv 2023

  70. [70]

    S.Hollands,R.M.WaldandV.G.Zhang,Entropy of dynamical black holes,Phys.Rev.D110(2024), no.2,024070,2402.00818

    Pith/arXiv arXiv 2024

  71. [71]

    M.R.VisserandZ.Yan,Properties of dynamical black hole entropy,JHEP10(2024)029, 2403.07140

    Pith/arXiv arXiv 2024

  72. [72]

    G.Odak,A.Rignon-BretandS.Speziale,General gravitational charges on null hypersurfaces,JHEP 12(2023)038,2309.03854

    Pith/arXiv arXiv 2023

  73. [73]

    L.Ciambelli,L.FreidelandR.G.Leigh,Null Raychaudhuri: canonical structure and the dressing time,JHEP01(2024)166,2309.03932

    Pith/arXiv arXiv 2024

  74. [74]

    A.Rignon-Bret,Note on the physical process first law of black hole mechanics,Phys.Rev.D108 (2023),no.2,024005,2303.06731

    Pith/arXiv arXiv 2023

  75. [75]

    H.Bondi,Gravitational Waves in General Relativity,Nature186(1960),no.4724,535–535

    1960

  76. [76]

    R.K.Sachs,Asymptotic symmetries in gravitational theory,PhysicalReview128(1962),no.6,2851

    1962

  77. [77]

    R.Penrose,Quasi-local mass and angular momentum in general relativity,ProceedingsoftheRoyal SocietyofLondon.A.MathematicalandPhysicalSciences381(1982),no.1780,53–63

    1982

  78. [78]

    Part 2: Open systems, Markovian dynamics and work extraction from non-rotating black holes,arXiv2601:???? (2026)

    A.Rignon-BretandM.Vilatte,Black hole thermodynamics at null infinity. Part 2: Open systems, Markovian dynamics and work extraction from non-rotating black holes,arXiv2601:???? (2026)

    2026

  79. [79]

    I.GelfandandM.Neumark,On the imbedding of normed rings into the ring of operators in Hilbert space,Rec.Math.12(1943),no.2,197–217

    1943

  80. [80]

    I.E.Segal,Irreducible representations of operator algebras,Bull.Amer.Math.Soc.53(February, 1947)73–88

    1947

Showing first 80 references.