Pith. sign in

REVIEW 3 major objections 4 minor 65 references

For black holes whose trapped region forms in finite time for distant observers, this paper argues that the apparent horizon behaves as a two-dimensional viscous membrane, and that its redshifted acceleration recovers the standard surface g

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 23:49 UTC pith:JEAJD7Z7

load-bearing objection A plausible membrane description of timelike apparent horizons, but the printed closed-form formulas have algebra errors and the surface-gravity recovery is partly self-calibrated. the 3 major comments →

arxiv 2511.03959 v2 pith:JEAJD7Z7 submitted 2025-11-06 gr-qc math-phmath.MP

Apparent horizon as a membrane

Daniel R. Terno This is my paper
classification gr-qc math-phmath.MP MSC 83C5783C75 PACS 04.70.-s04.20.-q04.62.+v
keywords apparent horizonphysical black holemembrane paradigmsurface gravitynear-horizon geometryspherical symmetrynull energy conditionblack hole evaporation
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.

This paper argues that a real ('physical') black hole — one whose light-trapping region forms in finite time according to a distant observer — has an apparent horizon that is timelike rather than null, and that this horizon can be treated as a membrane. Working in spherical symmetry, the author derives closed-form expressions for the horizon's redshift (α_v² = 2|r'_+|), proper acceleration, and extrinsic curvature, and assigns it a two-dimensional viscous-fluid stress tensor through junction conditions. The central payoff is an identity: the acceleration of a comoving observer at the horizon, redshifted to infinity, approaches (1−w1)/(2r_+), the same value as the invariant dynamical surface gravity; with the expansion coefficient w1 set to zero this reduces to the Schwarzschild value 1/(2r_g). If correct, this gives astrophysical models of finite-time black holes a concrete set of boundary data for computing quasinormal modes, light rings, and possible echoes, and it identifies which dynamical definitions of surface gravity survive.

Core claim

The central claim is that the timelike apparent horizon of a physical black hole — a spherically symmetric trapped region that forms in finite distant-observer time — is geometrically rich enough to carry a full membrane description. The near-horizon geometry belongs to a class in which the metric function f behaves as a constant times √(r−r_g) and the redshift function h diverges logarithmically, a consequence of all three effective energy-momentum components approaching the same negative constant −Υ² at the horizon. On this background the paper computes, in closed form, the redshift α_v²=2|r'_+|, the proper acceleration g_v, and the extrinsic curvature diag(g_v, α_v/(2r_+), α_v/(2r_+)). Ap

What carries the argument

The load-bearing object is the 'k=0' near-horizon solution class for spherically symmetric physical black holes: the effective energy-momentum components τ_t, τ_r, τ^r_t all scale as f^0 and approach the same negative constant −Υ² at the horizon, producing the metric behavior f≈α_{1/2}√x and h≈−(1/2)ln(x/ξ). This special near-horizon form makes the apparent horizon timelike and gives the redshift α_v²=2|r'_+| that enters every membrane quantity. The membrane construction itself proceeds through standard junction conditions: the horizon hypersurface is assigned a surface stress tensor of a two-dimensional dissipative fluid, with shear/bulk viscosity inherited from the usual membrane choice (η

Load-bearing premise

The load-bearing premise is that a physical matter source can actually realize the k=0 near-horizon class — where all effective energy-momentum components approach the same negative constant at the horizon and the trapped region forms in finite distant-observer time — and that the auxiliary choice w1=0 (which fixes the free parameter to match Schwarzschild surface gravity from the start) is warranted; if either fails, the membrane formulas and the recovered surface gravity do

What would settle it

Take any concrete spherically symmetric dynamical collapse solution of the semiclassical Einstein equations that forms a trapped region in finite distant-observer time and expand the metric near the apparent horizon: if f does not behave like (constant)·√(r−r_g) but like (constant)·(r−r_g), or if the three effective energy-momentum components do not share a common limit, the k=0 class (and hence the membrane construction) is not realized. Alternatively, measure the product α_v g_v in such a solution and check whether it equals (1−w1)/(2r_+); a different limit would invalidate the surface-gravi

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

If this is right

  • The apparent horizon of a physical black hole can replace the stretched horizon of the standard membrane picture, so a distant observer can ignore the interior and still reproduce the exterior phenomenology.
  • The closed-form membrane quantities parameterize dissipation and reflectivity; they can be fed into quasinormal-mode and echo calculations for any rate of horizon dynamics, not just slow evolution.
  • Among dynamical definitions of surface gravity, only the redshifted-acceleration definition survives for these geometries; it coincides with the invariant dynamical surface gravity and reduces to the Schwarzschild value when the first mass-expansion coefficient vanishes.
  • The near-horizon metric can be 'frozen' at fixed evaporation rate, giving an explicit modified metric whose deviations from Schwarzschild are confined to a narrow band of width |r'_g| r_g, so infall times into the apparent horizon remain of order r_g.
  • The separatrix — a nearby hypersurface that approximates the event-horizon generators in absence of a true horizon — has approximately zero redshift, providing a simple accelerated-observer description for studying thermal effects on these backgrounds.

Where Pith is reading between the lines

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

  • Editorial: the membrane's speed of sound, computed from ρ and p, exceeds unity (c_s≈1/(2α_v²)≫1), so the membrane fluid is an effective description rather than a physical medium; this suggests the viscous-fluid parameters should be read as boundary data, not as matter properties.
  • Editorial: the explicit values of ρ and p depend on the w1=0 choice and on the standard evaporation law; if those assumptions fail — for instance, if a dynamical collapse produces w1≠0 — the membrane stress tensor changes and the recovered surface gravity deviates from the Schwarzschild value, which a future gravitational-wave measurement of ring-down frequencies could in principle probe.
  • Editorial: the membrane description is derived in spherical symmetry; the same k=0 classification does not yet exist for rotating horizons, so whether a timelike apparent horizon of a spinning physical black hole admits an analogous viscous-membrane description remains open.
  • Editorial: the paper's frozen near-horizon metric provides a concrete starting point for computing quasinormal-mode spectra and light rings, and comparing them against the standard Schwarzschild predictions could yield the first observational discriminant between physical black holes and eternal-horizon black holes.

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 / 4 minor

Summary. The paper develops a membrane description for the timelike apparent horizon of spherically symmetric 'physical black holes' in the k=0 near-horizon class. It derives closed-form results for the horizon redshift α_v, the proper acceleration g_v of a comoving observer, the extrinsic curvature, and a two-dimensional viscous-fluid stress tensor obtained via Israel junction conditions. It then argues that the redshifted membrane acceleration recovers the Kodama surface gravity κ_K=(1−w_1)/(2r_+) in the slowly evolving limit, and reduces to the Schwarzschild value when w_1=0. The paper also discusses the static limit, the York–Frolov separatrix, and the relation between Rindler and near-horizon geometries. The central claims are the membrane data of Section III and the surface-gravity recovery of Eq. (68).

Significance. If the construction is correct, it provides a concrete, observation-facing framework for computing membrane properties, quasinormal-mode boundary conditions, and possible echoes for black holes that form in finite time for distant observers. The paper is explicit about its assumptions — the k=0 solution class, Page-law identification, and w_1=0 — and it carries out the Israel junction-condition calculation in detail. This is a strength: the membrane data are not formal but tied to a specific metric class. The viscosity-dependent reflectivity, sound speed, and surface-gravity relation are falsifiable predictions. However, the printed closed-form formulas contain internal inconsistencies that affect the central claims and must be corrected before the results can be relied upon.

major comments (3)
  1. [Sec. III A, Eq. (45) vs (44), (68)] Equation (45) does not follow from Eq. (44) as printed. From Eq. (44), using ζ_1∼|r'_+|/r_+ (Eq. 30) and the slow-evolution/Page relations, the leading small-|r'_+| term is g_v ≈ (1−w_1)/(2√2 r_+ √|r'_+|) = (1−w_1)/(2 r_+ α_v), with α_v²=2|r'_+|. If Eq. (45) is meant as (2√(2|r'_+|) r_+)^{-1}, it is only the w_1=0 version and the w_1 dependence is lost; if it is meant as (2√(2|r'_+| r_+))^{-1}, then α_v g_v ≈ 1/(2√r_+), not (1−w_1)/(2r_+) of Eq. (68). Either way, the displayed Eq. (45) does not support the central surface-gravity recovery as written. Please rewrite Eq. (45) with the restored (1−w_1) factor and disambiguate the radical.
  2. [Sec. III A, Eq. (50)–(52)] The approximate pressure in Eq. (52) is inconsistent with the stated leading form of g_v. Substituting g_v = 1/(2 r_+ α_v) (the w_1=0 limit) into p = (1/8π)(g_v + 3α_v/(2r_+)) gives p = (1/(16π r_+))(1/α_v + 3α_v), not (1/(6π r_+))(1/α_v + 3α_v). The coefficient 1/(6π) is a factor 8/3 too large. In addition, Eq. (50) states σ_ab = −(ϑ/2)γ_ab, but this shear tensor is not trace-free: γ_ab has trace 2, so Tr σ = −ϑ ≠ 0. For a round sphere with u = ∂_τ the shear actually vanishes. The viscous contribution to the Israel junction condition therefore needs to be recomputed, and Eq. (52) and the sound speed Eq. (53) revised accordingly.
  3. [Sec. IV, Eq. (68); Sec. II B, Eqs. (27)–(28)] The presentation of the surface-gravity recovery is partly circular as written. The paper fixes w_1=0 and uses the Page evaporation law to identify the free coefficients (Eqs. 27–28), and then Eq. (68) returns (1−w_1)/(2r_+), which was already identified as the Kodama surface gravity in Eq. (63). This is a consistency check, not an independent derivation from the membrane data. The identity α_v g_v → (1−w_1)/(2r_+) actually follows from Eq. (44) without the Page-law identification, and the paper should present it that way, then note that w_1=0 gives the Schwarzschild value. As it stands, the abstract's claim of 'recovering' the intuitive surface gravity overstates the logical status.
minor comments (4)
  1. [Sec. III B, Eq. (57)] Equation (57) has y_sep ∼ 2r_+(1+w_1)r'_+, while Appendix B, Eq. (B1), gives the leading term as 2r_+(1−w_1)r'_+. The sign of w_1 should be corrected.
  2. [Throughout] There are numerous typographical errors and OCR artifacts: 'anlysis', 'Relativisitc', 'fom', 'witζ', 'Enstein', 'coordin tes', and inconsistent notation such as α2 for α². A careful proofreading pass is needed.
  3. [Sec. II A, Eq. (25)] The relation |r'_g|/|r'_+| = α√(2|r'_+|) appears dimensionally and algebraically inconsistent with Eq. (26) and with the surrounding text; it is likely meant to be α/√(2|r'_+|). Please clarify.
  4. [Sec. III A, Eq. (40)–(41)] The membrane paradigm conventions (η=−ζ=1/(16π)) are stated, but the sign conventions in Eq. (49) should be reconciled explicitly with the Israel junction form used in Eq. (52), especially after the shear correction.

Circularity Check

0 steps flagged

No significant circularity: membrane quantities are computed from the k=0 metric and junction conditions, and the surface-gravity matching is a derived consistency relation.

full rationale

The paper's derivation is not circular in the sense prohibited here. The k=0 near-horizon solution class (Eqs. 14–17) is imported from prior work, but it is an independent mathematical classification with stated assumptions; it does not encode the membrane results and is externally checkable. The membrane data are computed rather than assumed: α_v^2=2|r'_+| follows directly from dτ^2=α_v^2 dv^2 on the horizon (Eq. 42); g_v is the four-acceleration magnitude obtained from the metric (Eq. 44); the static-limit expression (Eq. 45) is the slow-evolution leading term of Eq. (44); and Eqs. (46)–(52) follow from the Israel junction conditions with the explicitly stated K^-=0 and η=-ζ=1/(16π) conventions. None of these definitions presuppose the surface-gravity value. The surface-gravity section defines κ_K=(1-w_1)/(2r_+) geometrically via the Kodama vector (Eq. 63), then Eq. (68) evaluates α_v g_v and obtains the same combination; this is a derived equality, not an identity imposed by construction. The choice w_1=0 is an explicit parameter choice made to match the standard Schwarzschild value, stated before the recovery rather than fitted to it. The self-citations supplying the k=0 classification are load-bearing but are not circular: they are stated as inputs with derivations elsewhere and do not assume the membrane or surface-gravity conclusions of this paper. No specific reduction of a claimed result to its own input was found.

Axiom & Free-Parameter Ledger

7 free parameters · 7 axioms · 1 invented entities

The central claim rests almost entirely on the authors' previous classification of k=0 solutions (self-cited), on the physical premise of finite-time horizon formation (which requires NEC violation), and on auxiliary choices (w1=0, Page law) that fix the free functions Υ and ξ. The membrane stress tensor is then computed from that metric rather than predicted. The 'recovery' of surface gravity from the membrane acceleration is a consistency identity that presupposes the same standard κ used to set the constants. The approximate near-horizon metric (37)-(38) contains additional non-unique regularization constants b,d. No new physical entity is postulated beyond the fictitious membrane.

free parameters (7)
  • Υ(t) = Υ² = A/(8π r_g⁴)
    Horizon energy-density scale in the k=0 EMT scaling τ→-Υ² (Eq. 14). Fixed via the Page evaporation law and w1=0 (Eq. 28); determines α_{1/2}=4√π r_g Υ.
  • ξ(t) = ξ = A/(2r_g)
    Redshift scale in h=-(1/2)ln(x/ξ). Fixed by Eq. (28); leads to α²=4ξ/r_g=2|r'_g| and the membrane redshift.
  • w1(v) = 0
    First mass-expansion coefficient in C+(v,r)=r+ + w1 y + ...; set to zero 'unless it is assumed otherwise' to make Kodama surface gravity equal 1/(2r_g). Nonzero w1 rescales κK and all membrane formulas.
  • ζ1(v) = ζ1 ∼ |r'_+|/r_+
    h+ expansion coefficient; adopted from Bardeen's steady-state argument (Eq. 30), making h+≈0 and the metric approximately Vaidya.
  • b, d = b=1/4, d=1/2
    Regularization constants in the approximate e^h=√(b²+ξ/x)+d (Eq. 38), chosen in App. A3 to match the h_{1/2} expansion and the static limit e^h→1. The approximate e^h is one of many possible forms.
  • c1(t), h_{1/2}(t) = c1→w1, h_{1/2}≈1/(4√(π r_g³) Υ)
    Subleading metric coefficients tied to EMT coefficients e12,p12 (Eqs. A5-A6); enter Eq. (23) for the redshift but are mostly absorbed after the w1=0/Page-law choices.
  • A (Page constant) = A (from literature)
    Evaporation constant in r'_g=-A/r_g², r'_+=-A/r_+²; imported from Hawking evaporation, not derived in this paper.
axioms (7)
  • domain assumption Semiclassical Einstein equations with an effective EMT (Eq. 1) govern the spacetime.
    The entire ΦBH framework assumes quantum corrections enter through a renormalized EMT; no specific matter model or quantum state is used (Sec. II A).
  • domain assumption Weak cosmic censorship: curvature scalars G and G_2 finite at the apparent horizon.
    Used to restrict admissible solutions (Sec. II A); not proved here.
  • domain assumption A trapped domain forms in finite time for a distant observer (scenario (iii) of Sec. I).
    This defines ΦBHs and makes the apparent horizon timelike; if false, the membrane formalism for finite-time horizons is moot.
  • ad hoc to paper For k=0 solutions the EMT components scale as τt,τr,-τ^t_r→Υ² and the metric is f=α_{1/2}√x+O(x), h=-(1/2)ln(x/ξ)+O(√x) (Eqs. 14-17).
    Imported from the authors' earlier classification [10,25]; not re-derived in this paper. All membrane and surface-gravity results rest on it.
  • ad hoc to paper w1=0 and the Page evaporation law r'_g=-A/r_g², r'_+=-A/r_+² hold in both coordinate systems (Eqs. 27-28).
    Fixes Υ and ξ; presupposes the standard Schwarzschild surface gravity and Hawking temperature that the paper later 'recovers.'
  • domain assumption The York–Frolov separatrix equation (56) approximates the event horizon / D-geodesic.
    Used for the Rindler relation (Sec. III B); cited to [46-49] without derivation.
  • domain assumption The static limit is regularized by e^h→1 and f→f(r) (Eq. 35), and the approximate e^h is chosen to satisfy this plus matching of h_{1/2}.
    The k=0 static limit is singular (ξ→0); the paper specifies asymptotic requirements rather than an exact limiting solution (Sec. II C, App. A3).
invented entities (1)
  • 2D viscous-fluid membrane on the timelike apparent horizon no independent evidence
    purpose: Represents the horizon's exterior effect for distant observers; carries the effective stress tensor S_ab with density, pressure, and viscosities via Israel junction conditions.
    It is a fictitious construct ('formal postulating discontinuity', Sec. III A), not a new physical substance; no independent detection handle beyond future QNM/echo predictions.

pith-pipeline@v1.3.0-alltime-deepseek · 15643 in / 28583 out tokens · 246584 ms · 2026-08-03T23:49:08.666424+00:00 · methodology

0 comments
read the original abstract

The requirement that a trapped spacetime domain forms in finite time for distant observers is logically possible and sometimes unavoidable, but its consequences are not yet fully understood. In spherical symmetry, the characterization of the near-horizon geometry of these physical black holes is complete and shows marked differences from their eternal counterparts. Whether these differences lead to observable signatures remains unclear. We construct an approximate near-horizon metric that encapsulates them and is suitable for modeling. The timelike apparent horizon of physical black holes provides a natural surface for a consistent membrane description: we obtain closed-form expressions for the redshift, proper acceleration, and extrinsic curvature, and assign a two-dimensional viscous-fluid stress tensor via junction conditions. These results also provide an additional perspective on the relation between Rindler and near-horizon geometries. Among dynamical generalizations of surface gravity, only a subset applies to these models. We complete their analysis and recover the intuitive definition of surface gravity -- the acceleration in the frame of a near-horizon observer, redshifted to infinity -- directly from the membrane acceleration.

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

65 extracted references · 4 linked inside Pith

  1. [1]

    We also quote the coeffi- cientℓ 1 of Eq

    EMT structure and the Einstein equations The effective EMT components ofk= 0 ΦBHs outside the Schwarzschild radiusr g has the form τt =−Υ 2 +e 12(t)√x+e 1(t)x+O(x 3/2),(A1) τ r t =−Υ 2 +ϕ 12(t)√x+ϕ 1(t)x+O(x 3/2),(A2) τ r =−Υ 2 +p 12(t)√x+p 1(t)x+O(x 3/2),(A3) where ϕ12 = 1 2 (e12 +p 12).(A4) The two subleading metric terms have the coefficients c1 = 1 3 ...

  2. [2]

    As we found that∂ t¯t≈1in its vicinity (see Appendix C), this version of the surface gravity is also untenable for aΦBH

    that κPG2 = ∂¯t ¯C 2rg r=rg ,(65) where ∂¯t ¯C=∂ tC∂ ¯tt|r.(66) 8 Thus ∂¯t ¯C≈ r′ g ∂t¯t  1 + 2 q πr3g Υ √r−r g   ,(67) and the behavior of the function ¯t(t, r)near the apparent hori- zon determines the limit. As we found that∂ t¯t≈1in its vicinity (see Appendix C), this version of the surface gravity is also untenable for aΦBH. On the other hand, if...

  3. [3]

    (38) eh ≈ r b2 + ξ x +d,(A30) allows to match the expansion of Eq

    Limits The approximation ofe h via Eq. (38) eh ≈ r b2 + ξ x +d,(A30) allows to match the expansion of Eq. (34) up toh 12 term for all finite values ofr ′ g (and, therefore,ξandΥ). The results of Section II B set the constraint |r′ g| |r′ +| = (1−w 1)√ξ 2 q πr3g Υ .(A31) Assumingw 1 = 0, Page’s evaporation law and the static limit, c1 →w 1, we find using t...

  4. [4]

    (A11) in a series aroundr + and the RHS aroundr g, after making use of Eq

    Expansion coefficients and gap functions Expanding the LHS of Eq. (A11) in a series aroundr + and the RHS aroundr g, after making use of Eq. (A9), and com- paring order-by-order, one arrives at the following relation for w1(v): w1(v) = e12 −p 12 Υ √πr3/2 g (A13) The conditione 12(t) =p 12(t)is therefore equivalent to w1(v) = 0. Using Eqs. (23),(A5) and (A...

  5. [5]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time(Cambridge University Press, Cambridge, Eng- land, 1973)

    1973

  6. [6]

    LIGO Scientific Collaboration, Virgo Collaboration, and KA- GRA Collaboration, Phys. Rev. X13, 041039 (2023)

    2023

  7. [7]

    Event Horizon Telescope Collaboration, Astrophys. J. Lett. 930, L16 (2022)

    2022

  8. [8]

    Cardoso and P

    V . Cardoso and P. Pani, Living Rev. Relativ.22, 4 (2019)

    2019

  9. [9]

    Berti, V

    E. Berti, V . Cardoso, G. Carullo, (eds.), arXiv:2505.23895 (2025)

    Pith/arXiv arXiv 2025

  10. [10]

    R. B. Mann, S. Murk, and D. R. Terno, Int. J. Mod. Phys. D31, 2230015 (2022)

    2022

  11. [11]

    V . P. Frolov and I. D. Novikov,Black Hole Physics: Basic Con- cepts and New Developments(Kluwer, Dordrecht, 1998)

    1998

  12. [12]

    Faraoni,Cosmological and Black Hole Apparent Horizons (Springer, Heidelberg, 2015)

    V . Faraoni,Cosmological and Black Hole Apparent Horizons (Springer, Heidelberg, 2015)

    2015

  13. [13]

    Visser, Phys

    M. Visser, Phys. Rev. D90, 127502 (2014)

    2014

  14. [14]

    Bambi,Black Holes: A Laboratory for Testing Strong Grav- ity(Springer Nature, Singapore, 2017)

    C. Bambi,Black Holes: A Laboratory for Testing Strong Grav- ity(Springer Nature, Singapore, 2017)

    2017

  15. [15]

    Murk, Int

    S. Murk, Int. J. Mod. Phys. D32, 2342012 (2023)

    2023

  16. [16]

    N. D. Birrel and P. C. W. Davies,Quantum Fields in Curved Space(Cambridge University Press, Cambridge, 1984)

    1984

  17. [17]

    Brout, S

    R. Brout, S. Massar, R. Parentani, and P. Spindel, Phys. Rep. 260, 329 (1995)

    1995

  18. [18]

    V . P. Frolov, arXiv:1411.6981 (2014)

    Pith/arXiv arXiv 2014

  19. [19]

    Rezzolla and O

    L. Rezzolla and O. Zanotti,Relativisitc Hydrodynamics(Ox- ford University Press, Oxford, England, 2013)

    2013

  20. [20]

    ´E. ´E. Flanagan and R. M. Wald, Phys. Rev. D54, 6233 (1996)

    1996

  21. [21]

    H. A. Buchdahl, Phys. Rev.116, 1027 (1959)

    1959

  22. [22]

    Soranidia and D

    I. Soranidia and D. R. Terno, arXiv:2505.09189 (2025)

    arXiv 2025

  23. [23]

    Franzin, S

    E. Franzin, S. Liberati and V . Vellucci, JCAP2024, 020 (2024)

    2024

  24. [24]

    P. K. Dahal, S. Maharana, F. Simovic, I. Soranidis, and D. R. Terno, Phys. Rev. D110, 044032 (2024)

    2024

  25. [25]

    Maharana and R

    S. Maharana and R. Vadapalli, arXiv:2509.11578 (2025)

    arXiv 2025

  26. [26]

    Kiefer,Quantum Gravity(Oxford University Press, 2007)

    C. Kiefer,Quantum Gravity(Oxford University Press, 2007)

    2007

  27. [27]

    Hu and E

    B.-L. Hu and E. Verdaguer,Semiclassical and Stochastic Grav- ity: Quantum Field Effects on Curved Spacetime(Cambridge University Press, Cambridge, England, 2020)

    2020

  28. [28]

    P. K. Dahal, F. Simovic, I. Soranidis and D. R. Terno, Phys. Rev. D108, 104014 (2023)

    2023

  29. [29]

    Faraoni, G

    V . Faraoni, G. F. R. Ellis, J. T. Firouzjaee, A. Helou, and I. Musco, Phys. Rev. D95, 024008 (2017)

    2017

  30. [30]

    J. M. Bardeen, Phys. Rev. Lett.46, 382 (1981)

    1981

  31. [31]

    Kontou and K

    E.-A. Kontou and K. Sanders, Class. Quantum Gravity37, 193001 (2020)

    2020

  32. [32]

    P. K. Dahal, I. Soranidis, and D. R. Terno, Phys. Rev. D106, 124048 (2022)

    2022

  33. [33]

    Maharana, F

    S. Maharana, F. Simovic, I. Soranidia, and D. R. Terno, Phys. Rev. D111, 104063 (2025)

    2025

  34. [34]

    Page, New J

    D. Page, New J. Phys.7, 203 (2005)

    2005

  35. [35]

    Ireland, S

    A. Ireland, S. Profumo and J. Scharnhorst Phys. Rev. D107, 104021 (2023)

    2023

  36. [36]

    Relativ.25, 4 (2022)

    LISA consortium, Living Rev. Relativ.25, 4 (2022)

    2022

  37. [37]

    Carr and J

    B. Carr and J. Silk, MNRAS478, 3756 (2018)

    2018

  38. [38]

    Yoo, Galaxies10, 112 (2022)

    C.-M. Yoo, Galaxies10, 112 (2022)

    2022

  39. [39]

    Auffinger, Prog

    J. Auffinger, Prog. Part. Nucl. Phys. 131 104040 (2023)

    2023

  40. [40]

    Chakraborty, E

    S. Chakraborty, E. Maggio, A. Mazumdar, and P. Pani, Phys. Rev. D106, 024041 (2022)

    2022

  41. [41]

    K. S. Thorne, R. Price, and D. MacDonald (eds.),Black Holes: The Membrane Paradigm(Yale University, New Haven, CT, 12 1986)

    1986

  42. [42]

    Poisson,A Relativist’s Toolkit: The Mathematics of Black- Hole Mechanics(Cambridge University Press, Cambridge, England, 2004)

    E. Poisson,A Relativist’s Toolkit: The Mathematics of Black- Hole Mechanics(Cambridge University Press, Cambridge, England, 2004)

    2004

  43. [43]

    Abedi, N

    A. Abedi, N. Afshordi, N Oshita, and Q Wang, Universe6, 43 (2020)

    2020

  44. [44]

    Maggio, L

    E. Maggio, L. Buoninfante, A. Mazumdar, and P. Pani, Phys. Rev. D102, 064053 (2020)

    2020

  45. [45]

    Padmanabhan, Rep

    T. Padmanabhan, Rep. Prog. Phys.73, 046901 (2010)

    2010

  46. [46]

    Silvestrini, E

    M. Silvestrini, E. Maggio, S. Chakraborty, and P. Pani, arXiv: 2506.16516 (2025)

    Pith/arXiv arXiv 2025

  47. [47]

    R. B. Mann, I. Nagle, and D. R. Terno, Nucl. Phys. B936, 19 (2018)

    2018

  48. [48]

    Jacobson, Phys

    T. Jacobson, Phys. Rev. Lett.75, 1260 (1995)

    1995

  49. [49]

    Padmanabhan, Class

    T. Padmanabhan, Class. Quantum Grav.1953879 (2002)

    2002

  50. [50]

    Ashtekar and B

    A. Ashtekar and B. Krishnan, Living Rev. Relativ.7, 10 (2004)

    2004

  51. [51]

    P. K. Dahal and F. Simovic, arXiv:2304.11833 (2023)

    Pith/arXiv arXiv 2023

  52. [52]

    J. W. York, Jr., Phys. Rev. D28, 2929 (1983)

    1983

  53. [53]

    V . P. Frolov, Phys. Rev. D94, 104056 (2016)

    2016

  54. [54]

    Bin ´etruy, A

    P. Bin ´etruy, A. Helou, and F. Lamy, Phys. Rev. D98, 064058 (2018)

    2018

  55. [55]

    S. A. Hayward, Class. Quantum Gravity15, 3147 (1998)

    1998

  56. [56]

    It was shown

    was introduced using flat slice Painlev´e–Gullstrand coor- dinates( ¯t, r)[53] (whose relevant properties are summarized in Appendix C), κPG2 := 1 2rg (1−∂ r ¯C+∂ ¯t ¯C) r=rg ,(64) where ¯C(¯t, r) = 2M t(¯t, r), r is the MSH mass expressed in the Painlev ´e–Gullstrand (PG) coordinates. It was shown

  57. [57]

    Vanzo, G

    L. Vanzo, G. Acquaviva, and R. Di Criscienzo, Class. Quantum Gravity28, 183001 (2011)

    2011

  58. [58]

    Cropp, S

    B. Cropp, S. Liberati, and M. Visser, Class. Quantum Gravity 30, 125001 (2013)

    2013

  59. [59]

    A. B. Nielsen and M. Visser, Class. Quantum Gravity23, 4637 (2006)

    2006

  60. [60]

    Kodama, Prog

    H. Kodama, Prog. Theor. Phys.63, 1217 (1980)

    1980

  61. [61]

    R. B. Mann, S. Murk, and D. R. Terno, Phys. Rev. D105, 124032 (2022)

    2022

  62. [62]

    Martel and E

    K. Martel and E. Poisson, Am. J. Phys.69, 476 (2001)

    2001

  63. [63]

    Rezzolla and A

    L. Rezzolla and A. Zhidenko, Phys. Rev. D90, 084009 (2014)

    2014

  64. [64]

    Konoplya, L

    R. Konoplya, L. Rezzolla, and A. Zhidenko, Phys. Rev. D93, 064015 (2016)

    2016

  65. [65]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D105, 104032 (2022)

    2022