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arxiv: 2606.30797 · v1 · pith:LAIJOZPUnew · submitted 2026-06-29 · ✦ hep-th · gr-qc

On Black Holes Surrounded by Radiation II: Thermodynamics

Marcos Riojas , Matthew J. Strassler This is my paper

Pith reviewed 2026-07-01 01:36 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords black holesthermodynamicsradiation oceanhillingar black holethermal equilibriumentropytemperatureAdS space
0
0 comments X

The pith

A black hole enveloped by a radiation ocean shares the temperature and entropy of an ordinary black hole of the same mass when thermal equilibrium is assumed.

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

The paper studies a Schwarzschild black hole of mass m wrapped in a thick ocean of massless particles that reaches out to a finite depth, forming a hillingar black hole whose total ADM mass is M. Under the assumption that the black hole and ocean reach thermal equilibrium, the composite system obeys the same temperature and entropy formulas as an ordinary black hole of mass M. The result is confirmed by several independent calculations. The same mimicry fails in anti-de Sitter space, where the solutions instead display a more complex structure. The authors also observe that an HBH placed in a cavity whose radius is at least 3M can evaporate while remaining in equilibrium.

Core claim

The formal assumption of thermal equilibrium implies the system has the same temperature and entropy as an ordinary black hole of mass M. Multiple independent methods confirm this thermodynamic mimicry for the hillingar black hole in flat space. In AdS the mimicry is absent and the solutions exhibit richer thermodynamic behavior. Assuming equilibrium can be maintained, an HBH inside a cavity of radius at least 3M is able to evaporate, which may place the information puzzle inside a small finite volume.

What carries the argument

The assumption of thermal equilibrium between the central black hole and the surrounding radiation ocean, which permits standard thermodynamic relations to be applied directly to the composite ADM mass M.

If this is right

  • The hillingar black hole can be assigned the Hawking temperature and Bekenstein-Hawking entropy of an ordinary black hole of mass M.
  • An HBH inside a reflecting cavity of radius at least 3M can lose mass through evaporation while remaining in equilibrium.
  • The information puzzle can be posed inside a finite cavity whose size is only a few times the black-hole radius.
  • In anti-de Sitter space the thermodynamic mimicry disappears and the solutions display additional branches and phase structure.

Where Pith is reading between the lines

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

  • The thermodynamic identity may survive even if the detailed distribution of the radiation ocean is altered, provided equilibrium is preserved.
  • The contrast between flat-space mimicry and AdS non-mimicry suggests that the result depends on the choice of asymptotic boundary conditions.
  • Numerical evolution of an HBH in a cavity could test whether evaporation proceeds while the ocean remains in equilibrium with the hole.

Load-bearing premise

Thermal equilibrium can be established and maintained between the black hole and the radiation ocean long enough for the usual thermodynamic relations to hold for the whole system.

What would settle it

An explicit calculation of the temperature or entropy for a concrete hillingar black hole configuration that yields a value different from the corresponding ordinary black hole of mass M, while still satisfying the equilibrium assumption.

read the original abstract

In a companion paper we considered a Schwarzschild black hole of mass $m$ enveloped by a thick "ocean'' of massless particles that extends the black hole's photon sphere into a region of finite depth. There we showed that this "hillingar black hole'', of ADM mass $M$, optically mimics an ordinary black hole of the same mass. Here we find it also mimics the black hole thermodynamically: the formal assumption of thermal equilibrium implies the system has the same temperature and entropy as an ordinary black hole of mass $M$. We check this result carefully using multiple methods; a further method and indications of metastability are given by one of us in a companion paper. In AdS space, the mimicry does not hold, and the solutions have a richer structure. While it is far from clear that these systems are models for more realistic ones, we note possible connections with black hole evolution. In particular, assuming thermal equilibrium can be established and maintained, an HBH in a cavity of radius $\geq 3M$ can evaporate, potentially posing the information puzzle in a small finite volume.

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 manuscript claims that a hillingar black hole (central Schwarzschild black hole of mass m enveloped by a thick ocean of massless particles yielding ADM mass M) thermodynamically mimics an ordinary Schwarzschild black hole of mass M: under the formal assumption of thermal equilibrium the composite system has identical temperature and entropy. The result is checked via multiple methods, with one additional method and indications of metastability deferred to a companion paper. In AdS the mimicry fails and the solutions exhibit richer structure. Possible implications for black-hole evaporation in a cavity of radius ≥3M are noted.

Significance. If the central claim is robust, the work extends optical mimicry to thermodynamics and supplies a concrete example in which standard thermodynamic relations apply to a composite system whose exterior is indistinguishable from a vacuum black hole. The explicit use of multiple independent methods is a positive feature. The metastability caveat, however, directly touches the load-bearing equilibrium assumption.

major comments (1)
  1. [Abstract] Abstract: the thermodynamic mimicry is obtained only after imposing thermal equilibrium between the central black hole and the radiation ocean. The same abstract states that a companion paper finds 'indications of metastability.' If the equilibrium configuration is merely metastable, the uniform temperature required for the first law and entropy maximization may not be maintained on relevant timescales; this issue is load-bearing for the claim that T and S are identical to those of a Schwarzschild black hole of mass M and must be addressed quantitatively in the present manuscript rather than deferred.
minor comments (1)
  1. The distinction between the central mass m and the ADM mass M should be introduced with an explicit equation in the opening section.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their detailed review and valuable feedback on our manuscript. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the thermodynamic mimicry is obtained only after imposing thermal equilibrium between the central black hole and the radiation ocean. The same abstract states that a companion paper finds 'indications of metastability.' If the equilibrium configuration is merely metastable, the uniform temperature required for the first law and entropy maximization may not be maintained on relevant timescales; this issue is load-bearing for the claim that T and S are identical to those of a Schwarzschild black hole of mass M and must be addressed quantitatively in the present manuscript rather than deferred.

    Authors: We acknowledge that the metastability indicated in the companion paper raises important questions about the stability of the thermal equilibrium assumption, which is indeed central to our thermodynamic claims. The present manuscript focuses on deriving the thermodynamic properties under the formal assumption of thermal equilibrium, with the metastability analysis deferred to the companion paper as it employs distinct methods. To address the referee's concern, we will revise the manuscript to include a more explicit discussion in the introduction and conclusions sections highlighting the assumption and its potential limitations, while maintaining the cross-reference to the companion paper for the quantitative metastability study. We believe this provides sufficient context without requiring a full reproduction of the companion paper's results here. revision: partial

standing simulated objections not resolved
  • The full quantitative analysis of metastability and associated timescales is developed in the companion paper using additional methods and cannot be quantitatively addressed within the scope of the current manuscript.

Circularity Check

0 steps flagged

No significant circularity; claim is explicitly conditional on stated assumption

full rationale

The paper's thermodynamic mimicry result is presented as following directly from the formal assumption of thermal equilibrium, with the temperature and entropy then matching those of a standard Schwarzschild black hole of ADM mass M. This is checked via multiple methods as stated in the abstract. The companion paper is cited only for an additional method and metastability indications, which does not bear the load of the central claim here. No equations or steps reduce the result to a self-definition, fitted parameter renamed as prediction, or self-citation chain; the derivation applies standard thermodynamic relations to the composite system under the explicit assumption and remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Abstract-only review prevents full enumeration; the claim rests on standard general-relativity assumptions plus the thermal-equilibrium premise for the composite system.

axioms (2)
  • domain assumption Standard black-hole thermodynamics (Hawking temperature and Bekenstein-Hawking entropy) applies to the ADM mass M of the composite system
    Invoked when stating that the hillingar black hole has the same T and S as an ordinary black hole of mass M
  • ad hoc to paper Thermal equilibrium can be established and maintained between the central black hole and the surrounding radiation ocean
    Explicitly stated as the formal assumption that implies the thermodynamic mimicry
invented entities (1)
  • Hillingar black hole (thick ocean of massless particles around Schwarzschild black hole) no independent evidence
    purpose: Model that optically and thermodynamically mimics an ordinary black hole
    Introduced in companion paper; no independent evidence provided beyond the model definition

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discussion (0)

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Reference graph

Works this paper leans on

79 extracted references · 27 linked inside Pith

  1. [1]

    On black holes surrounded by radiation: Classical considerations

    M. Riojas and M.J. Strassler, “On black holes surrounded by radiation: Classical considerations. ” 2026

    2026

  2. [2]

    As Cold as a Black Hole: Extended Photon Spheres

    M. Riojas, “As Cold as a Black Hole: Extended Photon Spheres. ” 2026

    2026

  3. [3]

    York, Jr.,Black hole thermodynamics and the Euclidean Einstein action,Phys

    J.W. York, Jr.,Black hole thermodynamics and the Euclidean Einstein action,Phys. Rev. D 33(1986) 2092

    1986

  4. [4]

    Maeda, V

    K.-i. Maeda, V. Cardoso and A. Wang,Einstein cluster as central spiky distribution of galactic dark matter,Phys. Rev. D111(2025) 044060 [2410.04175]. – 39 –

    arXiv 2025

  5. [5]

    Cardoso, K

    V. Cardoso, K. Destounis, F. Duque, R.P. Macedo and A. Maselli,Black holes in galaxies: Environmental impact on gravitational-wave generation and propagation,Phys. Rev. D105 (2022) L061501 [2109.00005]

    arXiv 2022

  6. [6]

    Jusufi,Black holes surrounded by Einstein clusters as models of dark matter fluid,Eur

    K. Jusufi,Black holes surrounded by Einstein clusters as models of dark matter fluid,Eur. Phys. J. C83(2023) 103 [2202.00010]

    arXiv 2023

  7. [7]

    Gibbons and S.W

    G.W. Gibbons and S.W. Hawking,Action Integrals and Partition Functions in Quantum Gravity,Phys. Rev. D15(1977) 2752

    1977

  8. [8]

    Brady, J

    P.R. Brady, J. Louko and E. Poisson,Stability of a shell around a black hole,Phys. Rev. D 44(1991) 1891

    1991

  9. [9]

    In preparation

    M. Riojas and M.J. Strassler, “In preparation. ” 2026

    2026

  10. [10]

    Sorkin, R.M

    R.D. Sorkin, R.M. Wald and Z.J. Zhang,Entropy of selfgravitating radiation,Gen. Rel. Grav.13(1981) 1127

    1981

  11. [11]

    Tolman,Static solutions of Einstein’s field equations for spheres of fluid,Phys

    R.C. Tolman,Static solutions of Einstein’s field equations for spheres of fluid,Phys. Rev.55 (1939) 364

    1939

  12. [12]

    Oppenheimer and G.M

    J.R. Oppenheimer and G.M. Volkoff,On massive neutron cores,Phys. Rev.55(1939) 374

    1939

  13. [13]

    Bondi,Spherically symmetrical models in general relativity,Mon

    H. Bondi,Spherically symmetrical models in general relativity,Mon. Not. Roy. Astron. Soc. 107(1947) 410

    1947

  14. [14]

    Buchdahl,General Relativistic Fluid Spheres,Phys

    H.A. Buchdahl,General Relativistic Fluid Spheres,Phys. Rev.116(1959) 1027

    1959

  15. [15]

    Misner and D.H

    C.W. Misner and D.H. Sharp,Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse,Physical Review136(1964) 571

    1964

  16. [16]

    Bowers and E.P.T

    R.L. Bowers and E.P.T. Liang,Anisotropic Spheres in General Relativity,Astrophys. J.188 (1974) 657

    1974

  17. [17]

    Herrera and N.O

    L. Herrera and N.O. Santos,Local anisotropy in self-gravitating systems,Phys. Rept.286 (1997) 53

    1997

  18. [18]

    Andreasson,Sharp bounds on 2m/r of general spherically symmetric static objects,J

    H. Andreasson,Sharp bounds on 2m/r of general spherically symmetric static objects,J. Diff. Eq.245(2008) 2243 [gr-qc/0702137]

    Pith/arXiv arXiv 2008

  19. [19]

    Gao,A general maximum entropy principle for self-gravitating perfect fluid,Phys

    S. Gao,A general maximum entropy principle for self-gravitating perfect fluid,Phys. Rev. D 84(2011) 104023 [1109.2804]

    Pith/arXiv arXiv 2011

  20. [20]

    Kim and Y

    H.-C. Kim and Y. Lee,Entropy of self-gravitating anisotropic matter,Eur. Phys. J. C79 (2019) 679 [1901.03148]

    arXiv 2019

  21. [21]

    Cardoso and P

    V. Cardoso and P. Pani,Testing the nature of dark compact objects: a status report,Living Rev. Rel.22(2019) 4 [1904.05363]

    Pith/arXiv arXiv 2019

  22. [22]

    Bambi et al.,Black hole mimickers: from theory to observation, 5, 2025 [2505.09014]

    C. Bambi et al.,Black hole mimickers: from theory to observation, 5, 2025 [2505.09014]

    arXiv 2025

  23. [23]

    Banks, W

    T. Banks, W. Fischler, A. Kashani-Poor, R. McNees and S. Paban,Entropy of the stiffest stars,Class. Quant. Grav.19(2002) 4717 [hep-th/0206096]

    Pith/arXiv arXiv 2002

  24. [24]

    Zel’dovich,The Equation of State at Ultrahigh Densities and Its Relativistic Limitations,Zh

    Y.B. Zel’dovich,The Equation of State at Ultrahigh Densities and Its Relativistic Limitations,Zh. Eksp. Teor. Fiz.41(1961) 1609

    1961

  25. [25]

    Brustein and A.J.M

    R. Brustein and A.J.M. Medved,Resisting collapse: How matter inside a black hole can withstand gravity,Phys. Rev. D99(2019) 064019 [1805.11667]. – 40 –

    Pith/arXiv arXiv 2019

  26. [26]

    Brustein, A.J.M

    R. Brustein, A.J.M. Medved and T. Simhon,Black holes as frozen stars,Phys. Rev. D105 (2022) 024019 [2109.10017]

    arXiv 2022

  27. [27]

    Brustein, A.J.M

    R. Brustein, A.J.M. Medved and T. Simhon,Thermodynamics of frozen stars,Phys. Rev. D 110(2024) 024066 [2310.11572]

    arXiv 2024

  28. [28]

    Martinez,Fundamental thermodynamical equation of a selfgravitating system,Phys

    E.A. Martinez,Fundamental thermodynamical equation of a selfgravitating system,Phys. Rev. D53(1996) 7062 [gr-qc/9601037]

    Pith/arXiv arXiv 1996

  29. [29]

    Andr´ e, J.P.S

    R. Andr´ e, J.P.S. Lemos and G.M. Quinta,Thermodynamics and entropy of self-gravitating matter shells and black holes inddimensions,Phys. Rev. D99(2019) 125013 [1905.05239]

    Pith/arXiv arXiv 2019

  30. [30]

    Andr´ e and J.P.S

    R. Andr´ e and J.P.S. Lemos,Thermodynamics ofd-dimensional Schwarzschild black holes in the canonical ensemble,Phys. Rev. D103(2021) 064069 [2101.11010]

    arXiv 2021

  31. [31]

    Lemos and O.B

    J.P.S. Lemos and O.B. Zaslavskii,Black holes and hot shells in the Euclidean path integral approach to quantum gravity,Class. Quant. Grav.40(2023) 235012 [2304.06740]

    arXiv 2023

  32. [32]

    S. Kim, S. Kundu, E. Lee, J. Lee, S. Minwalla and C. Patel,Grey Galaxies’ as an endpoint of the Kerr-AdS superradiant instability,JHEP11(2023) 024 [2305.08922]

    arXiv 2023

  33. [33]

    Wheeler,Geons,Phys

    J.A. Wheeler,Geons,Phys. Rev.97(1955) 511

    1955

  34. [34]

    Misner, K.S

    C.W. Misner, K.S. Thorne and J.A. Wheeler,Gravitation, W. H. Freeman, San Francisco (1973)

    1973

  35. [35]

    Martinez and J.W

    E.A. Martinez and J.W. York, Jr.,Additivity of the entropies of black holes and matter in equilibrium,Phys. Rev. D40(1989) 2124

    1989

  36. [36]

    Anderson,On boundary value problems for Einstein metrics,arXiv Mathematics e-prints(2006) math/0612647 [math/0612647]

    M.T. Anderson,On boundary value problems for Einstein metrics,arXiv Mathematics e-prints(2006) math/0612647 [math/0612647]

    Pith/arXiv arXiv 2006

  37. [37]

    An and M.T

    Z. An and M.T. Anderson,The initial boundary value problem and quasi-local Hamiltonians in General Relativity,2103.15673

    arXiv

  38. [38]

    Witten,A note on boundary conditions in Euclidean gravity,Rev

    E. Witten,A note on boundary conditions in Euclidean gravity,Rev. Math. Phys.33(2021) 2140004 [1805.11559]

    arXiv 2021

  39. [39]

    Odak and S

    G. Odak and S. Speziale,Brown-York charges with mixed boundary conditions,JHEP11 (2021) 224 [2109.02883]

    arXiv 2021

  40. [40]

    Banihashemi, E

    B. Banihashemi, E. Shaghoulian and S. Shashi,Flat space gravity at finite cutoff,Class. Quant. Grav.42(2025) 035010 [2409.07643]

    arXiv 2025

  41. [41]

    Frauendiener, C

    J. Frauendiener, C. Hoenselaers and W. Konrad,A shell around a black hole,Class. Quant. Grav.7(1990) 585

    1990

  42. [42]

    Israel,Singular hypersurfaces and thin shells in general relativity,Nuovo Cim

    W. Israel,Singular hypersurfaces and thin shells in general relativity,Nuovo Cim. B44S10 (1966) 1

    1966

  43. [43]

    Andr´ easson,Existence of Steady States of the Massless Einstein–Vlasov System Surrounding a Schwarzschild Black Hole,Annales Henri Poincare22(2021) 4271 [2102.08170]

    H. Andr´ easson,Existence of Steady States of the Massless Einstein–Vlasov System Surrounding a Schwarzschild Black Hole,Annales Henri Poincare22(2021) 4271 [2102.08170]

    arXiv 2021

  44. [44]

    Andr´ easson, D

    H. Andr´ easson, D. Fajman and M. Thaller,Models for Self-Gravitating Photon Shells and Geons,Annales Henri Poincare18(2017) 681 [1511.01290]

    arXiv 2017

  45. [45]

    Sorkin,A Criterion for the onset of instability at a turning point,Astrophys

    R. Sorkin,A Criterion for the onset of instability at a turning point,Astrophys. J.249 (1981) 254. – 41 –

    1981

  46. [46]

    Witten,Anti de Sitter space and holography,Adv

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

    Pith/arXiv arXiv 1998

  47. [47]

    Hawking and D.N

    S.W. Hawking and D.N. Page,Thermodynamics of Black Holes in anti-De Sitter Space, Commun. Math. Phys.87(1983) 577

    1983

  48. [48]

    Banks, M.R

    T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec,AdS dynamics from conformal field theory,hep-th/9808016

    Pith/arXiv arXiv

  49. [49]

    Horowitz,Comments on black holes in string theory,Class

    G.T. Horowitz,Comments on black holes in string theory,Class. Quant. Grav.17(2000) 1107 [hep-th/9910082]

    Pith/arXiv arXiv 2000

  50. [50]

    Christensen and S.A

    S.M. Christensen and S.A. Fulling,Trace Anomalies and the Hawking Effect,Phys. Rev. D 15(1977) 2088

    1977

  51. [51]

    Aharony, J

    O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk,The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories,Adv. Theor. Math. Phys.8(2004) 603 [hep-th/0310285]

    Pith/arXiv arXiv 2004

  52. [52]

    Aharony, J

    O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk,A First order deconfinement transition in large N Yang-Mills theory on a small S**3,Phys. Rev. D 71(2005) 125018 [hep-th/0502149]

    Pith/arXiv arXiv 2005

  53. [53]

    Susskind and J

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

    Pith/arXiv arXiv 1994

  54. [54]

    Dvali,Black Holes and Large N Species Solution to the Hierarchy Problem,Fortsch

    G. Dvali,Black Holes and Large N Species Solution to the Hierarchy Problem,Fortsch. Phys. 58(2010) 528 [0706.2050]

    Pith/arXiv arXiv 2010

  55. [55]

    Dvali and M

    G. Dvali and M. Redi,Black Hole Bound on the Number of Species and Quantum Gravity at LHC,Phys. Rev. D77(2008) 045027 [0710.4344]

    Pith/arXiv arXiv 2008

  56. [56]

    Mathur,The Information paradox: A Pedagogical introduction,Class

    S.D. Mathur,The Information paradox: A Pedagogical introduction,Class. Quant. Grav.26 (2009) 224001 [0909.1038]

    Pith/arXiv arXiv 2009

  57. [57]

    Mathur,A Proposal to resolve the black hole information paradox,Int

    S.D. Mathur,A Proposal to resolve the black hole information paradox,Int. J. Mod. Phys. D 11(2002) 1537 [hep-th/0205192]

    Pith/arXiv arXiv 2002

  58. [58]

    Mathur,Fuzzballs and the information paradox: A Summary and conjectures, 0810.4525

    S.D. Mathur,Fuzzballs and the information paradox: A Summary and conjectures, 0810.4525

    Pith/arXiv arXiv

  59. [59]

    Polchinski,The black hole information problem., inTheoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp

    J. Polchinski,The black hole information problem., inTheoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 353–397, 2017, DOI [1609.04036]

    Pith/arXiv arXiv 2017

  60. [60]

    Almheiri, D

    A. Almheiri, D. Marolf, J. Polchinski and J. Sully,Black Holes: Complementarity or Firewalls?,JHEP02(2013) 062 [1207.3123]

    Pith/arXiv arXiv 2013

  61. [61]

    Almheiri, D

    A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully,An Apologia for Firewalls, JHEP09(2013) 018 [1304.6483]

    Pith/arXiv arXiv 2013

  62. [62]

    Penington,Entanglement Wedge Reconstruction and the Information Paradox,JHEP09 (2020) 002 [1905.08255]

    G. Penington,Entanglement Wedge Reconstruction and the Information Paradox,JHEP09 (2020) 002 [1905.08255]

    Pith/arXiv arXiv 2020

  63. [63]

    Almheiri, N

    A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield,The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,JHEP12(2019) 063 [1905.08762]

    Pith/arXiv arXiv 2019

  64. [64]

    Almheiri, R

    A. Almheiri, R. Mahajan and J. Maldacena,Islands outside the horizon,1910.11077. – 42 –

    arXiv 1910

  65. [65]

    Penington, S.H

    G. Penington, S.H. Shenker, D. Stanford and Z. Yang,Replica wormholes and the black hole interior,JHEP03(2022) 205 [1911.11977]

    Pith/arXiv arXiv 2022

  66. [66]

    Almheiri, R

    A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao,The Page curve of Hawking radiation from semiclassical geometry,JHEP03(2020) 149 [1908.10996]

    Pith/arXiv arXiv 2020

  67. [67]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini,Replica Wormholes and the Entropy of Hawking Radiation,JHEP05(2020) 013 [1911.12333]

    Pith/arXiv arXiv 2020

  68. [68]

    Raju,Lessons from the information paradox,Phys

    S. Raju,Lessons from the information paradox,Phys. Rept.943(2022) 1 [2012.05770]

    arXiv 2022

  69. [69]

    Giddings and G.J

    S.B. Giddings and G.J. Turiaci,Wormhole calculus, replicas, and entropies,JHEP09(2020) 194 [2004.02900]

    arXiv 2020

  70. [70]

    Geng and A

    H. Geng and A. Karch,Massive islands,JHEP09(2020) 121 [2006.02438]

    arXiv 2020

  71. [71]

    H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas et al.,Information Transfer with a Gravitating Bath,SciPost Phys.10(2021) 103 [2012.04671]

    arXiv 2021

  72. [72]

    H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas et al.,Inconsistency of islands in theories with long-range gravity,JHEP01(2022) 182 [2107.03390]

    arXiv 2022

  73. [73]

    Antonini, C.-H

    S. Antonini, C.-H. Chen, H. Maxfield and G. Penington,An apologia for islands,JHEP10 (2025) 034 [2506.04311]

    arXiv 2025

  74. [74]

    H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall and M. Riojas,Seeing Page Curves and Islands with Blinders On,2602.06543

    arXiv

  75. [75]

    Matsuo,Islands and stretched horizon,JHEP07(2021) 051 [2011.08814]

    Y. Matsuo,Islands and stretched horizon,JHEP07(2021) 051 [2011.08814]

    arXiv 2021

  76. [76]

    Bousso and G

    R. Bousso and G. Penington,Islands far outside the horizon,JHEP11(2024) 164 [2312.03078]

    arXiv 2024

  77. [77]

    Karch, C

    A. Karch, C. Perez-Pardavila, M. Riojas and M. Youssef,Subregion entropy for the doubly-holographic global black string,JHEP05(2023) 195 [2303.09571]

    arXiv 2023

  78. [78]

    diffgeo.m: A package for doing GR-type tensor algebra and calculus

    M. Headrick, “diffgeo.m: A package for doing GR-type tensor algebra and calculus. ” https://sites.google.com/view/matthew-headrick/mathematica

  79. [79]

    classical

    B. Shoshany,OGRe: An Object-Oriented General Relativity Package for Mathematica,J. Open Source Softw.6(2021) 3416 [2109.04193]. A The HBH forΛ̸= 0and Higher Dimensional Spaces As shown below, HBH solutions exist in alld≥4 with any cosmological constant, and have similar unique properties in each dimension. For other choices ofP i/ρ, linear self-similar so...

    arXiv 2021