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REVIEW 4 major objections 5 minor 138 references

Static dark-fluid thin shells around a black hole are radially stable only when the exterior mass exceeds the interior mass (m_+/m_->1); the paper maps the stability windows and shows the shells enlarge the black-hole shadow for distant obs

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2026-08-02 20:45 UTC pith:GSAA6VGS

load-bearing objection Solid Israel-formalism stability map for two-SdS thin shells, but the headline claim 'stable only for m+/m−>1' is contradicted by the paper's own λ=w0 scan; needs a corrected abstract before it can be trusted. the 4 major comments →

arxiv 2602.22141 v2 pith:GSAA6VGS submitted 2026-02-25 gr-qc

Static Dark Fluid Thin Shells in Schwarzschild-de Sitter Spacetimes: Stability and Black Hole Shadows

Dimitrios Efstratiou , Evangelos Achilleas Paraskevas , Leandros Perivolaropoulos This is my paper
classification gr-qc MSC 83C57 PACS 04.20.-q04.70.-s
keywords thin-shell stabilitySchwarzschild–de Sitter spacetimesdark fluidlinear barotropic equation of stateblack hole shadoweffective potentialcosmological constant
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 asks when a thin spherical shell of "dark fluid" can sit stably between two spherically symmetric vacuum spacetimes with a cosmological constant. Matching the two sides with the standard junction conditions, it derives an effective potential for the shell's radial motion and finds that stable static shells with positive surface density and a physically reasonable sound speed exist only when the exterior mass parameter exceeds the interior one. Adopting the measured value of the cosmological constant and a supermassive black-hole mass, it maps three stability windows in the pressure-to-density ratio w0, corresponding to shells near the photon sphere, near the static-radius scale, and near the cosmological horizon. It then computes how such a transparent shell bends light, predicting an enlarged black-hole shadow for observers outside the shell — a deviation of order 10^-3 for positive-pressure shells near the photon sphere, below current sensitivity but within projected future shadow observations.

Core claim

With the linear barotropic law p=λ(σ−σ1)c², the sound speed c_s²=λc² is independent of the equilibrium ratio w0=p0/(σ0c²), so tension shells can be stable with a real sound speed. Solving Veff(R0)=V'eff(R0)=0 with V''eff(R0)>0 and σ0>0, the authors find stable static shells only for m_+/m_->1. At λ=1 the stability windows are −3/7≲w0≲1/2 (Λ+=Λ−), −2/3≲w0≲1/2 (Λ+>Λ−), and 0≲w0≲1 (Λ+<Λ−). Positive-pressure shells sit near the photon sphere; negative-pressure shells near the static radius or cosmological horizon. A transparent shell refracts light, enlarging the shadow for outside observers; deviations reach ~10^−3 for positive-pressure shells near the photon sphere.

What carries the argument

The central object is the effective potential Veff(R) for the shell's radial motion, obtained by squaring the junction condition β_− − β_+ = κ(R). The load-bearing parameterization is the linear equation of state p=λ(σ−σ1)c², which splits the sound-speed parameter λ from the equilibrium pressure ratio w0=p0/(σ0c²); this is what allows negative-pressure (tension) shells to be stable without an imaginary sound speed. Stability is decided by V''eff(R0)>0, and the paper solves Veff=0=V'eff together with that inequality to construct the (w0,λ) stability maps and the shadow formula sin²Θcrit that follows from matching impact parameters across the shell.

Load-bearing premise

Everything rests on the phenomenological linear response p=λ(σ−σ1)c² with constant λ and σ1, plus the neglect of back-reaction on the backgrounds; if the shell's real pressure response differs, the stability windows move.

What would settle it

Set m_+/m_−=0.75 with Λ+=Λ− at the measured vacuum density and scan −1<w0<1, 0<λ≤1 using the paper's equilibrium equations; the claim is that no σ0>0, V''eff>0 solution exists. Any such static equilibrium would refute the central existence condition. A cheaper check: at λ=w0=0 the paper states stability is lost, so confirming that the dust-shell limit has no local minimum of the effective potential corroborates the mechanism.

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

If this is right

  • Stable static shells with positive surface density and 0<λ≤1 are confined to m_+/m_->1; the three windows at λ=1 provide a complete classification for the linear equation of state.
  • Positive-pressure shells approach the photon sphere asymptotically as w0→1/2, while negative-pressure shells sit at the static-radius scale or the cosmological horizon depending on Λ+/Λ−.
  • A transparent shell changes the photon impact parameter by a factor sqrt(f_−(R)/f_+(R)), so an outside observer sees an enlarged shadow; deviations reach ~10^−3 for positive-pressure shells near the photon sphere, below current sensitivity but within projected future capabilities.
  • For the worked example (m_+/m_−=Λ_+/Λ_−=1.001, m_−=10^10 M⊙), stable oscillations have period ~10^15 s, and perturbations above a critical energy send the shell to the cosmological horizon rather than into the black hole.
  • The stability classification is qualitative: the windows depend only on whether m_+/m_− and Λ_+/Λ_− are greater, equal, or less than unity, not on their precise values.

Where Pith is reading between the lines

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

  • Beyond the paper's argument, the same junction-condition rendering with modified f± (e.g., a charged or modified-gravity exterior) would produce a shadow formula of identical structure, so the shell lensing signature is a general probe of compact dark layers.
  • The authors discuss opacity only qualitatively; quantifying the opacity threshold at which absorption overtakes refraction is a natural next step that would connect their idealized transparent-shell picture to radiative-transfer observations.
  • The discrepancy between the abstract's (1−√13)/6 and the conclusion's −3/7 for the Λ+=Λ− lower bound is worth resolving; re-deriving the analytic test-shell bound would tell which value is correct.
  • The existence condition m_+/m_->1 could serve as a sharp test: any claimed stable thin-shell equilibrium near a black hole in this fluid family must hide a mass jump, which gravitational-gradient measurements might in principle probe.

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

4 major / 5 minor

Summary. The paper studies static, spherically symmetric thin shells joining two Schwarzschild–de Sitter spacetimes with possibly different masses m± and cosmological constants Λ±. Using the Israel junction formalism and a linear barotropic surface equation of state p = λ(σ−σ1)c² (Eq. 21), it derives an effective potential (Eq. 34), solves the equilibrium and stability conditions (Eqs. 38–40), and numerically maps stable equilibria over (m±, Λ±, λ, w0). The central claim is that, for σ0>0 and 0<λ≤1, stable static shells exist only when m+/m−>1, occurring at three scales: photon sphere, SdS static radius, and cosmological horizon. For λ=1 the abstract quotes windows (1−√13)/6≲w0≲1/2, −2/3≲w0≲1/2, and 0≲w0 for Λ+=Λ−, Λ+>Λ−, and Λ+<Λ− respectively. The paper also analyzes small perturbations and bounded excursions (Sec. III), and computes the shell's effect on the SdS black-hole shadow for a static observer (Sec. IV), finding deviations of order 10⁻³ for positive-pressure shells near the photon sphere and much smaller deviations for shells near the static radius.

Significance. If the stability classification were correct, the paper would provide a systematic map of parameter space for a phenomenological dark-fluid thin-shell model, with an explicit prediction for shadow modification. The derivation is largely explicit: the effective potential, equilibrium equations, and stability conditions are written in full, and the shadow formula (Eq. 56) is a concrete, falsifiable prediction. The manuscript also makes the figure-generating code publicly available, which is a positive feature. However, the headline claim is overstated relative to the paper's own results, and the numerical exploration is not accompanied by convergence checks or a supplied code listing, so the quantitative stability windows rest on representative scans rather than a fully reproducible or analytically complete classification.

major comments (4)
  1. [Abstract, Conclusion, Sec. II.B.2, Fig. 2] The central assertion that stable shells with σ0>0 and 0<λ≤1 exist only when m+/m−>1 is contradicted by the paper's own Fig. 2 and the accompanying text in Sec. II.B.2. There, for λ=w0 (the same EoS Eq. 21), stable configurations are reported for m+/m−<1 and Λ+/Λ−>1 when 0≲w0≲1, and also for w0≲−1.5. The claim that this regime is 'already contained within the broader case where w0≠λ' is not substantiated and is inconsistent with the earlier statement that the broader case has no stable m+/m−<1 solutions. Unless an explicit exclusion λ≠w0 is added to the abstract and conclusion, the 'only if' statement is false as written. This is a load-bearing issue: it is the paper's central result.
  2. [Abstract, Sec. II.B.1, Conclusion] The abstract quotes the λ=1, Λ+=Λ− stability window as (1−√13)/6 ≲ w0 ≲ 1/2, but the body and conclusion quote −3/7 ≲ w0 ≲ 1/2. These lower bounds are not equal: (1−√13)/6 ≈ −0.434, while −3/7 ≈ −0.429. The discrepancy is not explained. The abstract's version appears to be the exact analytic bound from the test-shell limit, whereas the body's version is presumably the numerical window; the relationship between them must be stated, and the numbers reconciled.
  3. [Sec. II.B.2, footnote 2] Footnote 2 states that static stable configurations satisfying m+/m−<1 'exist but require λ<0, which leads to instability through an imaginary...' — the sentence is incomplete and internally confusing. A negative λ gives an imaginary sound speed, but the phrase 'which leads to instability' seems to be describing the consequence, yet then the claim about existence is unclear. This footnote is directly relevant to the central claim and should be removed, corrected, or expanded, since as written it appears to concede a counterexample to the 'only if' statement.
  4. [Sec. II.B, Eqs. (38)–(40), Figs. 1–2] The stability windows and the m+/m−>1 claim are based on numerical scans for representative parameters (m−=10^10 M⊙, ρ_Λ+=ρ_crit). The paper does not provide convergence criteria, grid resolution, or a direct verification that the reported windows are exhaustive in (λ,w0,Λ+/Λ−) at fixed masses. Given that the central claim is an 'only if' statement, a more systematic or analytic exclusion is needed. The promised public code will help, but it is not yet available at the time of review. Please either supply the code and convergence checks or temper the exhaustive claim.
minor comments (5)
  1. [Sec. IV, Eq. (56)] The shadow formula in Eq. (56) uses sin²Θcrit on the left, but the text refers to Θcrit. Since sin² is not one-to-one on [0,π/2], it is worth stating explicitly that the physical angular radius is arcsin of the right-hand side, and that only the branch θcrit∈[0,π/2] is considered.
  2. [Abstract and body] The three bullets in the conclusion each begin with 'For Λ+=Λ−' etc., but the abstract condenses them into a single sentence with commas. Consider aligning the notation for the windows, using consistent symbols for the inequalities (≲ vs. ≤) and for the Λ+<Λ− window (abstract: '0≲w0', conclusion: '0≤w0≤1').
  3. [Throughout] The manuscript contains several typos and minor inconsistencies, e.g., 'Schwarzschild' vs. 'Schwarzschild–de Sitter' in the title and text, and the formatting of the footnote marker in Sec. II.B.2. A careful proofreading pass is advised.
  4. [Sec. III, Fig. 5] The caption of Fig. 5(a) refers to '(Veff−ε²)/c²' but the variable ε is later called a velocity and has units of m/s. It may help to define ε explicitly as the energy parameter with the same units as V_eff (or as an initial velocity with a factor √2).
  5. [Sec. IV, Fig. 6] The Fig. 6 caption says 'for negative-pressure shells, as w0 decreases from 0 to −1/2 the deviation initially decreases, reaching a minimum at w0=−1/2, and then increases again as w0 approaches −2/3.' This statement is in the caption but is not derived in the text; either provide the derivation or move it to the main text for clarity.

Circularity Check

0 steps flagged

No circularity: stability windows and shadow predictions are solved outputs from the junction equations and scanned parameters, not fitted inputs; self-citations are contextual.

full rationale

The central derivation is self-contained. V_eff follows from the Israel junction conditions (Eqs. 17-18, 27-32) with the linear EoS in Eq. (21); no parameter is fitted to the stability windows or shadow predictions. Planck [86] fixes Lambda_+ externally, m_- is a representative input, and (lambda, w0, m_+/m_-, Lambda_+/Lambda_-) are scanned independent variables. Eq. (41) is a derived sigma0>0 condition, not an imposed m_+/m_->1 constraint. The stability regions are obtained by solving Eqs. (38)-(39) with V''_eff>0 and sigma0>0, so the lambda=1 windows are numerical outputs. The shadow formula Eq. (56) follows from geodesic equations and the derived impact-parameter ratio Eq. (53); the deviations in Fig. 6 are novel outputs. Self-citations (e.g., [4], [82], [102], [120], [121]) are contextual and not load-bearing; the existence/stability result does not depend on them. Acknowledged limitations—transparent shell, neglected back-reaction, toy-model status—are caveats, not circular reductions. The internal tension between the abstract's m_+/m_->1 "only if" claim and Fig. 2's stable lambda=w0, m_+/m_-<1 branch is a consistency/correctness issue, not a circularity; likewise the abstract's (1-sqrt(13))/6 versus the body's -3/7 lower-bound discrepancy is a numerical inconsistency, not a circular step.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 1 invented entities

The central claim rests on standard GR junction theory plus a phenomenological linear shell EoS with parameters λ and w0 chosen by hand. No data are fitted to the target result, but the EoS itself is an unverified modeling assumption, and the numerical conclusions rely on representative choices of mass and cosmological-constant ratios.

free parameters (5)
  • λ (sound-speed parameter) = scanned 0<λ≤1
    Chosen by hand; fixes c_s²=λc² in the linear EoS and controls the restoring pressure response.
  • w0 (equilibrium pressure-to-density ratio) = scanned, e.g. −2/3 to 1
    Chosen by hand; sets p0=w0 σ0 c² at equilibrium and labels the stable branches.
  • m+/m− mass ratio = examples 0.75, 1.001, 1.25
    Chosen by hand; central to the claim that stable shells need m+>m−.
  • Λ+/Λ− cosmological-constant ratio = examples 0.75, 0.999, 1.0, 1.001, 1.25
    Chosen by hand; separates the three stability branches (Λ+=Λ−, Λ+>Λ−, Λ+<Λ−).
  • m− black-hole mass = 10^10 M⊙
    Representative astrophysical scale chosen for the numerical maps; the paper claims the shape of stability regions is insensitive to its value.
axioms (7)
  • standard math Israel junction conditions and the Einstein equations govern the thin-shell matching.
    Section II uses the standard junction formalism and SdS vacuum solutions.
  • standard math The spacetime on each side is Schwarzschild–de Sitter with metric (1)–(2).
    Invoked in Section II as the background geometry.
  • ad hoc to paper The shell obeys the linear barotropic EoS p=λ(σ−σ1)c² with constant λ and σ1.
    Introduced in Eq. (21); it decouples sound speed from w0 and is the main modeling input. No microphysical derivation is given.
  • domain assumption Perturbations of the shell do not back-react on the background spacetimes.
    Stated in Section III; used to treat Veff as fixed and to integrate Eq. (33) without metric feedback.
  • domain assumption The dark-fluid shell is transparent to light.
    Stated in Section IV; the shadow computation is purely geometric and neglects emission, absorption, or scattering by the shell.
  • domain assumption Λ+ is fixed by Planck 2018 so that its vacuum energy density equals the critical density.
    Physical input taken from [86]; sets the numerical scale for Λ+ in the maps.
  • standard math A local minimum of the effective potential (Veff''>0) is the correct radial-stability criterion.
    Standard mechanics of the one-dimensional effective potential, used throughout the stability analysis.
invented entities (1)
  • Dark fluid thin shell no independent evidence
    purpose: Localized stress-energy surface at the junction between two SdS spacetimes; source of the shadow distortion.
    The paper labels it a 'theoretical toy model'; the predicted shadow deviations are internal outputs of the model, not independent external evidence for the entity.

pith-pipeline@v1.3.0-alltime-deepseek · 27921 in / 13341 out tokens · 128159 ms · 2026-08-02T20:45:02.790550+00:00 · methodology

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read the original abstract

We study the existence and radial stability of static, spherically symmetric thin shells joining two Schwarzschild--de~Sitter (SdS) spacetimes $(m_\pm,\Lambda_\pm)$. Using the Israel junction formalism, we map the stable equilibria ($V_{\mathrm{eff}}''>0$) of the effective potential. Near the equilibrium radius $R_0$ the shell's surface density $\sigma$ and pressure $p$ obey the linearized barotropic law $p=p_0+c_s^2(\sigma-\sigma_0)$, with sound speed $c_s^2=\lambda c^2$. Since $c_s^2$ is independent of the equilibrium ratio $w_0\equiv p_0/(\sigma_0 c^2)$, tension shells ($w_0<0$) stay radially stable with real $c_s$. Fixing $\Lambda_+$ so that its vacuum energy density equals the critical density (Planck~2018), and taking $m_-$ representative of astrophysical black holes, we systematically map the stable equilibria $(R_0,\sigma_0)$ over $(m_\pm,\Lambda_\pm,\lambda,w_0)$ and find that stable shells with $\sigma_0>0$ and $0<\lambda\le1$ exist only for $m_+/m_->1$, at three scales -- the photon sphere, the SdS static radius, and the cosmological horizon. At $\lambda=1$ the numerical windows, checked against the analytic test-shell bounds, are $(1-\sqrt{13})/6\lesssim w_0\lesssim 1/2$ ($\Lambda_+=\Lambda_-$), $-2/3\lesssim w_0\lesssim 1/2$ ($\Lambda_+>\Lambda_-$), and $0\lesssim w_0$ ($\Lambda_+<\Lambda_-$). Positive-pressure shells ($0\lesssim w_0\lesssim 1/2$) sit near the photon sphere and those with $w_0\gtrsim1/2$ near the static radius scale, while tension shells reach the cosmological horizon scale for $\Lambda_+=\Lambda_-$, only the static radius scale for $\Lambda_+>\Lambda_-$, and are absent for $\Lambda_+<\Lambda_-$. Finally, we compute the dark fluid shell's imprint on the SdS black-hole shadow seen by a static observer at varying radial distance.

Figures

Figures reproduced from arXiv: 2602.22141 by Dimitrios Efstratiou, Evangelos Achilleas Paraskevas, Leandros Perivolaropoulos.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: c (where Λ− > Λ) we again consider shells near the static radius; here the shadow initially deviates and becomes smaller for static observers located inside R0, and then increases beyond the shell — the overall ampli￾tude of the effect is comparable to that in Fig. 6b but the detailed behaviour is distinct. Note that in the case of Fig. 6b, for static observers located outside the shell and far from it, an… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

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Reference graph

Works this paper leans on

138 extracted references · 1 canonical work pages

  1. [1]

    This corresponds to match- ing two Schwarzschild–de Sitter spacetimes with distinct mass parametersm + ̸=m −

    Static stable thin shell in the case whereΛ+ = Λ− A particularly physically realistic class of static and stable dark-fluid thin shells—relevant, at least, for a late- time accelerating universe—emerges in the case Λ + = Λ− ≡Λ, where the positive cosmological constant den- sity satisfiesρ Λ =ρ crit [86]. This corresponds to match- ing two Schwarzschild–de...

  2. [2]

    This is accomplished by solving the cou- pled system defined by Eqs

    Static stable thin shell in the case whereΛ+ ̸= Λ− In the more general case with distinct cosmological constants (Λ + ̸= Λ −), the system of equations to be solved becomes: Veff(R0) =− G(m − +m +) 2R0 − c2 12 −6 +R 2 0(Λ− + Λ+) − 6cG(m− −m +) +c 3R3 0(Λ− −Λ +) 2 1152G 2π2R4 0σ2 0 − 2G2π2R2 0σ2 0 c2 = 0 (38) V ′ eff(R0) =− 1 576c 2G2π2R5 0σ2 0 " 6c6G(m− −m...

  3. [3]

    Enhancement and Sup- port of the Operational, Research, and Educational Ac- tivities of the University of Ioannina

    × 10-6 ProperTimeτ(×1015 s) (R(τ) - R0) / R0 0 5.0 × 1014 1.0 × 1015 1.5 × 1015 0.0 0.2 0.4 0.6 0.8 1.0 Proper Timeτ (s) R(τ) / Rch (b)Full Nonlinear Dynamics FIG. 5:Stability analysis of the shell for the case wherem +/m− = 1.001 and Λ +/Λ− = 1.001.(a)The effective potential shifted by the perturbation energy, (V eff(R)−ϵ 2)/c2. The shell is confined bet...

  4. [4]

    Christoffel symbols The nonvanishing Christoffel symbols associated with metric in Eq. (B1) are Γt tr = f ′(r) 2f(r) ,Γ r tt = 1 2 f ′(r)f(r),(B3) Γr rr =− f ′(r) 2f(r) ,Γ r θθ =−rf(r),(B4) Γr ϕϕ =−rf(r) sin 2 θ,Γ θ ϕϕ =−sinθcosθ,(B5) Γϕ rϕ = Γθ rθ = 1 r ,Γ ϕ θϕ = cotθ.(B6)

  5. [5]

    gravitational forces

    The static radiusR st A key feature of the SdS geometry is the existence of a unique radius at which the “gravitational forces” from the black hole and the cosmological constant exactly balance [129, 130]. To identify this radius, we examine the geodesic equation for a radially moving test particle. We parameterize the timelike geodesic using an affine pa...

  6. [6]

    Lightlike trajectories in Schwarzchild-de Sitter spacetime We parameterize the null geodesic using an affine pa- rameterλ. For motion confined to the equatorial plane θ=π/2, the equations of motion are derived from the geodesic equation and the null conditionds 2 = 0 as [131]: ˙t= k f(r) ,(B13) ˙ϕ= h r2 ,(B14) c2k2 = ˙r2 + h2 r2 f(r),(B15) where . ≡d/dλan...

  7. [7]

    Shadow of the SdS black hole for a static observer We determine the angular radius of the shadow for a static observer in Schwarzschild–de Sitter (SdS) space- time by introducing an orthonormal tetrad adapted to the observer’s local rest frame [93]. For a static ob- serverO, atθ O =π/2, the four-velocity has only a t−component, [uµ O] = (ut O,0,0,0).(B23)...

  8. [8]

    LeMaitre and E

    P. LeMaitre and E. Poisson, Am. J. Phys.87, 961 (2019), arXiv:1909.06253 [gr-qc]

    Pith/arXiv arXiv 2019

  9. [9]

    Vilenkin, Phys

    A. Vilenkin, Phys. Rev. D23, 852 (1981)

    1981

  10. [10]

    Perivolaropoulos, Phys

    L. Perivolaropoulos, Phys. Rev. D97, 124035 (2018), arXiv:1804.08098 [gr-qc]

    Pith/arXiv arXiv 2018

  11. [11]

    Alestas, G

    G. Alestas, G. V. Kraniotis, and L. Perivolaropoulos, Phys. Rev. D102, 104015 (2020), arXiv:2005.11702 [gr- qc]

    Pith/arXiv arXiv 2020

  12. [12]

    Antoniou, D

    I. Antoniou, D. Kazanas, D. Papadopoulos, and L. Perivolaropoulos, Int. J. Mod. Phys. D31, 2250064 (2022), arXiv:2204.14003 [gr-qc]

    Pith/arXiv arXiv 2022

  13. [13]

    Alestas and L

    G. Alestas and L. Perivolaropoulos, Phys. Rev. D99, 064026 (2019), arXiv:1901.06659 [gr-qc]

    Pith/arXiv arXiv 2019

  14. [14]

    Randall and R

    L. Randall and R. Sundrum, Phys. Rev. Lett.83, 3370 (1999)

    1999

  15. [15]

    Antoniadis, N

    I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B436, 257 (1998), arXiv:hep- ph/9804398

    arXiv 1998

  16. [16]

    Arkani-Hamed, S

    N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B429, 263 (1998), arXiv:hep-ph/9803315

    Pith/arXiv arXiv 1998

  17. [17]

    Israel, Nuovo Cim

    W. Israel, Nuovo Cim. B44S10, 1 (1966), [Erratum: Nuovo Cim.B 48, 463 (1967)]

    1966

  18. [18]

    Kijowski, G

    J. Kijowski, G. Magli, and D. Malafarina, General Rel- ativity and Gravitation38, 1697 (2006)

    2006

  19. [19]

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

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

    2009

  20. [20]

    O. y. Grøn and S. Hervik,Einstein ’s General Theory of Relativity: With Modern Applications in Cosmology, 1st ed. (Springer, New York, 2007)

    2007

  21. [21]

    Frauendiener, C

    J. Frauendiener, C. Hoenselaers, and W. Konrad, Class. Quant. Grav.7, 585 (1990)

    1990

  22. [22]

    P. R. Brady, J. Louko, and E. Poisson, Phys. Rev. D 44, 1891 (1991)

    1991

  23. [23]

    B. G. Schmidt, Phys. Rev. D59, 024005 (1999)

    1999

  24. [24]

    K. G. Zloshchastiev, Int. J. Mod. Phys. D8, 549 (1999), arXiv:gr-qc/9802041

    Pith/arXiv arXiv 1999

  25. [25]

    R. Chan, M. F. A. da Silva, and P. Rocha, JCAP12, 017, arXiv:0910.2054 [gr-qc]

    Pith/arXiv arXiv 2054

  26. [26]

    J. P. Pereira, J. G. Coelho, and J. A. Rueda, Phys. Rev. D90, 123011 (2014), arXiv:1412.1848 [gr-qc]

    Pith/arXiv arXiv 2014

  27. [27]

    Pradhan, D

    S. Pradhan, D. Mohanty, and P. K. Sahoo, Chin. Phys. 21 C47, 095104 (2023), arXiv:2306.17435 [gr-qc]

    Pith/arXiv arXiv 2023

  28. [28]

    Visser and D

    M. Visser and D. L. Wiltshire, Class. Quant. Grav.21, 1135 (2004), arXiv:gr-qc/0310107

    Pith/arXiv arXiv 2004

  29. [29]

    P. Pani, E. Berti, V. Cardoso, Y. Chen, and R. Norte, Phys. Rev. D80, 124047 (2009), arXiv:0909.0287 [gr- qc]

    Pith/arXiv arXiv 2009

  30. [30]

    Berezin, V

    V. Berezin, V. Kuzmin, and I. Tkachev, Physics Letters B120, 91 (1983)

    1983

  31. [31]

    Laguna-Castillo and R

    P. Laguna-Castillo and R. A. Matzner, Phys. Rev. D 34, 2913 (1986)

    1986

  32. [32]

    V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys. Rev. D36, 2919 (1987)

    1987

  33. [33]

    Ishak and K

    M. Ishak and K. Lake, Phys. Rev. D65, 044011 (2002), arXiv:gr-qc/0108058

    Pith/arXiv arXiv 2002

  34. [34]

    S. M. C. V. Goncalves, Phys. Rev. D66, 084021 (2002), arXiv:gr-qc/0212124

    Pith/arXiv arXiv 2002

  35. [35]

    F. S. N. Lobo and P. Crawford, Class. Quant. Grav.22, 4869 (2005), arXiv:gr-qc/0507063

    Pith/arXiv arXiv 2005

  36. [36]

    Habib Mazharimousavi, M

    S. Habib Mazharimousavi, M. Halilsoy, and S. N. Hamad Amen, Int. J. Mod. Phys. D26, 1750158 (2017), arXiv:1708.04588 [gr-qc]

    Pith/arXiv arXiv 2017

  37. [37]

    A. D. D. Masa, E. S. de Oliveira, and V. T. Zanchin, Phys. Rev. D103, 104051 (2021), arXiv:2009.10948 [gr- qc]

    Pith/arXiv arXiv 2021

  38. [38]

    Cataldo, A

    M. Cataldo, A. Cid, and P. Labra˜ na, Eur. Phys. J. C 85, 1461 (2025), arXiv:2512.13575 [gr-qc]

    arXiv 2025

  39. [39]

    Crisostomo, S

    J. Crisostomo, S. del Campo, and J. Saavedra, Phys. Rev. D70, 064034 (2004), arXiv:hep-th/0311259

    Pith/arXiv arXiv 2004

  40. [40]

    Crisostomo and R

    J. Crisostomo and R. Olea, Phys. Rev. D69, 104023 (2004), arXiv:hep-th/0311054

    Pith/arXiv arXiv 2004

  41. [41]

    Bueno, P

    P. Bueno, P. A. Cano, R. A. Hennigar, and´A. J. Murcia, Phys. Rev. D111, 104009 (2025), arXiv:2412.02740 [gr- qc]

    Pith/arXiv arXiv 2025

  42. [42]

    A. G. Schubert, Entropy28, 10.3390/e28010096 (2026)

    work page doi:10.3390/e28010096 2026

  43. [43]

    Gravanis and S

    E. Gravanis and S. Willison, Phys. Rev. D75, 084025 (2007), arXiv:gr-qc/0701152

    Pith/arXiv arXiv 2007

  44. [44]

    Javed, J

    F. Javed, J. Lin, G. Mustafa, and F. M. O. Tawfiq, Fortsch. Phys.72, 2300081 (2024)

    2024

  45. [45]

    Beauchesne and A

    H. Beauchesne and A. Edery, Phys. Rev. D85, 044056 (2012), arXiv:1108.0449 [gr-qc]

    Pith/arXiv arXiv 2012

  46. [46]

    Javed, Eur

    F. Javed, Eur. Phys. J. C83, 513 (2023)

    2023

  47. [47]

    Hussain, F

    A. Hussain, F. Javed, G. Fatima, F. Tchier, S. Nozima, and G. Mustafa, Int. J. Geom. Meth. Mod. Phys.22, 2450297 (2025)

    2025

  48. [48]

    Musgrave and K

    P. Musgrave and K. Lake, Class. Quant. Grav.13, 1885 (1996), arXiv:gr-qc/9510052

    Pith/arXiv arXiv 1996

  49. [49]

    R. J. Gleiser and M. A. Ramirez, Class. Quant. Grav. 26, 045006 (2009), arXiv:0807.4728 [gr-qc]

    Pith/arXiv arXiv 2009

  50. [50]

    E. F. Eiroa and C. Simeone, Phys. Rev. D83, 104009 (2011), arXiv:1102.1683 [gr-qc]

    Pith/arXiv arXiv 2011

  51. [51]

    L. M. Reyes, M. Chiapparini, and S. E. P. Bergliaffa, Eur. Phys. J. C82, 151 (2022)

    2022

  52. [52]

    Habib Mazharimousavi and M

    S. Habib Mazharimousavi and M. Halilsoy, Eur. Phys. J. C73, 2527 (2013), arXiv:1305.2909 [gr-qc]

    Pith/arXiv arXiv 2013

  53. [53]

    S. H. Mazharimousavi and M. Halilsoy, Eur. Phys. J. C 75, 334 (2015), arXiv:1503.05587 [gr-qc]

    Pith/arXiv arXiv 2015

  54. [54]

    J. P. S. Lemos and G. M. Quinta, Phys. Rev. D88, 067501 (2013), arXiv:1309.1478 [gr-qc]

    Pith/arXiv arXiv 2013

  55. [55]

    J. P. S. Lemos and G. M. Quinta, Phys. Rev. D89, 084051 (2014), arXiv:1403.0579 [gr-qc]

    Pith/arXiv arXiv 2014

  56. [56]

    E. F. Eiroa, E. Rub ´ ın de Celis, and C. Simeone, Eur. Phys. J. C76, 546 (2016), arXiv:1608.04729 [gr-qc]

    Pith/arXiv arXiv 2016

  57. [57]

    Poisson and M

    E. Poisson and M. Visser, Phys. Rev. D52, 7318 (1995), arXiv:gr-qc/9506083

    Pith/arXiv arXiv 1995

  58. [58]

    F. S. N. Lobo and P. Crawford, Class. Quant. Grav.21, 391 (2004), arXiv:gr-qc/0311002

    Pith/arXiv arXiv 2004

  59. [59]

    F. S. N. Lobo, inAPCTP Winter School and Work- shop on Quantum Gravity, Black Holes and Wormholes (2004) arXiv:gr-qc/0401083

    Pith/arXiv arXiv 2004

  60. [60]

    F. S. N. Lobo, Class. Quant. Grav.21, 4811 (2004), arXiv:gr-qc/0409018

    Pith/arXiv arXiv 2004

  61. [61]

    E. F. Eiroa and C. Simeone, Phys. Rev. D76, 024021 (2007), arXiv:0704.1136 [gr-qc]

    Pith/arXiv arXiv 2007

  62. [62]

    J. P. S. Lemos and F. S. N. Lobo, Phys. Rev. D78, 044030 (2008), arXiv:0806.4459 [gr-qc]

    Pith/arXiv arXiv 2008

  63. [63]

    G. A. S. Dias and J. P. S. Lemos, Phys. Rev. D82, 084023 (2010), arXiv:1008.3376 [gr-qc]

    Pith/arXiv arXiv 2010

  64. [64]

    N. M. Garcia, F. S. N. Lobo, and M. Visser, Phys. Rev. D86, 044026 (2012), arXiv:1112.2057 [gr-qc]

    Pith/arXiv arXiv 2012

  65. [65]

    Nakao, T

    K.-i. Nakao, T. Uno, and S. Kinoshita, Phys. Rev. D 88, 044036 (2013), arXiv:1306.6917 [gr-qc]

    Pith/arXiv arXiv 2013

  66. [66]

    Sharif and M

    M. Sharif and M. Azam, JCAP04, 023, arXiv:1305.4441 [gr-qc]

    Pith/arXiv arXiv

  67. [67]

    Tsukamoto and T

    N. Tsukamoto and T. Kokubu, Phys. Rev. D98, 044026 (2018), arXiv:1807.01528 [gr-qc]

    Pith/arXiv arXiv 2018

  68. [68]

    Li, W.-L

    A.-C. Li, W.-L. Xu, and D.-F. Zeng, JCAP03, 016, arXiv:1812.07224 [hep-th]

    Pith/arXiv arXiv

  69. [69]

    Amirabi, Eur

    Z. Amirabi, Eur. Phys. J. C79, 410 (2019)

    2019

  70. [70]

    Berry, F

    T. Berry, F. S. N. Lobo, A. Simpson, and M. Visser, Phys. Rev. D102, 064054 (2020), arXiv:2008.07046 [gr- qc]

    Pith/arXiv arXiv 2020

  71. [71]

    F. S. N. Lobo, G. J. Olmo, E. Orazi, D. Rubiera-Garcia, and A. Rustam, Phys. Rev. D102, 104012 (2020), arXiv:2009.10997 [gr-qc]

    Pith/arXiv arXiv 2020

  72. [72]

    F. S. N. Lobo, A. Simpson, and M. Visser, Phys. Rev. D101, 124035 (2020), arXiv:2003.09419 [gr-qc]

    Pith/arXiv arXiv 2020

  73. [73]

    Sharif and F

    M. Sharif and F. Javed, Phys. Scripta96, 055003 (2021)

    2021

  74. [74]

    J. L. Rosa, R. Andr´ e, and J. P. S. Lemos, Gen. Rel. Grav.55, 65 (2023), arXiv:2305.06829 [gr-qc]

    Pith/arXiv arXiv 2023

  75. [75]

    A. Eid, A. Alkaoud, M. M. Khader, and M. A. Bakry, Sci. Rep.14, 12696 (2024)

    2024

  76. [76]

    Eid, New Astron.98, 101934 (2023)

    A. Eid, New Astron.98, 101934 (2023)

    2023

  77. [77]

    F. S. N. Lobo, Universe11, 270 (2025), arXiv:2508.17823 [gr-qc]

    Pith/arXiv arXiv 2025

  78. [78]

    T. S. Rippentrop, A. Bera, and M. Ishak, Phys. Rev. D 112, 124069 (2025), arXiv:2507.00315 [gr-qc]

    arXiv 2025

  79. [79]

    Antoniou, A

    G. Antoniou, A. Papageorgiou, and P. Kanti, Universe 9, 147 (2023), arXiv:2210.17533 [gr-qc]

    Pith/arXiv arXiv 2023

  80. [80]

    Sakai, H

    N. Sakai, H. Saida, and T. Tamaki, Phys. Rev. D90, 104013 (2014), arXiv:1408.6929 [gr-qc]

    Pith/arXiv arXiv 2014

Showing first 80 references.