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REVIEW 4 major objections 5 minor 77 references

A generalized geometric flow turns a black hole horizon into a traversable wormhole throat when the surrounding fluid enters the dark-energy era.

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2026-07-31 23:51 UTC pith:BMBG7YLD

load-bearing objection A correct but mostly algebraic soliton calculation is dressed up as an endogenous black-hole-to-wormhole transition; the transition claim fails on inspection. the 4 major comments →

arxiv 2607.23282 v1 pith:BMBG7YLD submitted 2026-07-25 gr-qc math.DG

Traversable Wormhole De-singularization: Almost η-Ricci-Yamabe Solitons in Static Spherically Symmetric Imperfect Fluid Spacetimes

Jay Prakash Singh , Jaswant This is my paper
classification gr-qc math.DG MSC 53C4453C2583C1583C57 PACS 04.20.-q04.70.-s
keywords traversable wormholesalmost η-Ricci-Yamabe solitonsimperfect fluidstatic spherically symmetric spacetimeReissner-Nordström-de SitterNull Energy ConditionHawking temperaturegeometric flow
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 the almost η-Ricci-Yamabe soliton — a geometric flow with radially varying couplings — can act as an endogenous regulator that removes the coordinate singularity at a black hole's apparent horizon. When the imperfect fluid accreting onto the black hole reaches the dark-energy equation of state (γ = −1) and violates the null energy condition, the flow's scaling factor ω(r) becomes strictly positive at the horizon, which forces the standard wormhole flare-out condition b′(r_H) < 1 on the shape function b(r) and keeps the time coordinate finite. The authors derive this transition directly from the flow equation, without assuming a wormhole shape function a priori, and they connect the flow parameters to the Hawking temperature. If correct, the result offers a purely geometric route from black hole to traversable wormhole in classical general relativity, though it still requires the surrounding fluid to supply NEC-violating stress.

Core claim

On its own terms, the paper's central discovery is that the almost η-Ricci-Yamabe soliton equation, written with smoothly varying radial parameters α(r), β(r), λ(r), ω(r) and the radial vector field ξ = ∂_r, implies a definite horizon behavior: the scaling parameter satisfies ω(r) = f′(r) − α(r)[S_tt + f(r)²S_rr], where S_tt and S_rr are the time-time and radial-radial components of the Ricci tensor, and this reduces at the horizon to ω(r_H) = f′(r_H) > 0 for a non-extremal static black hole. The paper then uses the identification f(r) = 1 − b(r)/r between the black hole lapse and the wormhole shape function to convert this into the flare-out requirement b′(r_H) < 1. Simultaneously, requirin

What carries the argument

The central object is the almost η-Ricci-Yamabe soliton — a geometric flow equation (1/2)L_ξ g_{μν} + α(r)S_{μν} + (λ(r) − β(r)R/2) g_{μν} + ω(r)η_μη_ν = 0 along the radial vector field ξ = ∂_r, with all coupling parameters promoted to smooth radial functions. Its role is to encode the geometry's response to the imperfect fluid; in particular, the scaling factor ω(r) emerges as the difference between the metric slope f′(r) and the α-weighted Ricci trace, giving a purely geometric expression for the wormhole flare-out condition. The second key identity is the map between the black hole lapse f(r) and the wormhole shape function b(r) via f(r) = 1 − b(r)/r, which turns the positivity of ω at th

Load-bearing premise

The load-bearing premise is that the soliton can regularize the time coordinate at the horizon while keeping the background metric fixed; the explicit Reissner-Nordström-de Sitter example used in the paper violates the required condition α(r_H)S_tt(r_H) = f′(r_H)/2 because S_tt vanishes at the horizon, so the proof would need an extremal horizon or a different metric to go through.

What would settle it

Direct evaluation for the paper's own example (a charged black hole with M = 1, Q = 0.6, Λ = 0.05 at r_H ≈ 1.9346): compute the Ricci component S_tt and the metric slope f′ at the apparent horizon. One finds S_tt(r_H) = 0 and f′(r_H) > 0, which makes λ(r) from the derived formula diverge at the horizon; a single consistent calculation exhibiting that divergence — or, alternatively, a solution of the flow with the proposed ω(r) that yields a finite λ — would settle the temporal-regularization claim.

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

If this is right

  • A static black hole geometry can, in principle, be de-singularized into a traversable wormhole by the geometric flow itself, with no shape function assumed in advance.
  • The transition is triggered by the accreting fluid entering the dark-energy era (γ = −1) and violating the null energy condition; the sign of ω(r_H) is the operational indicator of whether the throat opens.
  • The coupling α(r_H)S_tt(r_H) = 2πT_H ties the geometric flow's parameters to the Hawking temperature, connecting a purely geometric construction to black hole thermodynamics.
  • Perturbations around the wormhole obey a damped wave equation with damping proportional to 1/α(r), so the flow introduces a local dissipative 'geometric drag' that bounds gravitational wave amplitudes and supports linear stability.
  • Asymptotically, the soliton's scaling factor recovers the underlying spacetime's expansion (Minkowski or de Sitter), so the wormhole modification stays localized at the throat.

Where Pith is reading between the lines

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

  • A natural next step, not taken in the paper, is to promote the soliton equation to the underlying Ricci-Yamabe flow and let the metric itself evolve; that would test whether the temporal regularization is a genuine dynamical smoothing or a coordinate redefinition.
  • Because the damping coefficient in the perturbed wave equation is 1/α(r), the ratio of damping time to oscillation period at the throat can be estimated from the flow's parameters; extending the same calculation to rotating spacetimes would predict an observable extra damping in quasinormal ringing.
  • The framework suggests a search strategy for wormhole candidates: look for accreting black holes whose surrounding medium has an effective equation of state crossing γ = −1; the sign of the geometric scaling factor at the horizon then serves as a local diagnostic for whether the throat can flare open.

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 considers a static, spherically symmetric spacetime (1.2) coupled to an imperfect fluid and assumes it admits an almost η-Ricci-Yamabe soliton along the radial vector field. It derives expressions for the soliton parameters λ(r) and ω(r) (Theorem 1), claims that finiteness of λ at the horizon imposes the Hawking-temperature relation α(r_H)S_tt(r_H)=2πT_H (Corollary 1), and argues that NEC violation makes ω(r_H)>0 and enforces the Morris-Thorne flare-out condition b'(r_H)<1 (Theorem 3). Theorem 4 asserts that these conditions produce a topological transition to a traversable wormhole, and Theorem 5 claims that linear perturbations satisfy a damped wave equation. An explicit RNdS example is given in §4.

Significance. If the main claims were correct, the paper would offer a novel mechanism for converting a black hole metric into a traversable wormhole without prescribing a shape function a priori. The component computations in §3 are transparent, and the idea of promoting the soliton parameters to radial functions is reasonable. However, the central ‘transition’ is not actually proved: the key positivity condition reduces to a property of the background lapse function, and the temporal regularization is inconsistent with the very metric used in the example. The paper therefore does not establish its advertised result and, in its current form, contains a load-bearing contradiction.

major comments (4)
  1. [Theorem 3, Eqs. (3.38)–(3.39)] The claim that the soliton ‘generates’ ω(r_H)>0 from NEC violation is not a dynamical result. Combining Eq. (3.12) with Eq. (3.31) gives ω(r)=f'(r)-α(r)f(r)S_μν k^μ k^ν. Since f(r_H)=0, the second term vanishes at the horizon, so ω(r_H)=f'(r_H) identically. Thus the positivity is an input property of a non-extremal lapse function, not a consequence of the soliton flow or of the NEC. Similarly, Eq. (3.37) makes b'(r_H)<1 algebraically equivalent to f'(r_H)>0. The theorem therefore restates the coordinate relation f=1-b/r rather than proving an endogenous geometric transition.
  2. [Corollary 1 and §4, Eqs. (3.15), (4.8)] The temporal regularization condition contradicts the explicit metric. For (1.2), direct computation gives S_tt = f(r)(f''(r)/2 + f'(r)/r), so S_tt(r_H)=0 whenever f(r_H)=0 with finite derivatives. Corollary 1 requires α(r_H)S_tt(r_H)=f'(r_H)/2, which then forces f'(r_H)=0, i.e. an extremal horizon. The RNdS example in §4 has r_H≈1.9346 and f'(r_H)≈0.3705>0, while S_tt(r_H)=0. Hence Eq. (4.8) is numerically false, and the asserted Hawking-temperature relation (3.19) is not satisfied by the model.
  3. [Theorem 4, Eqs. (3.52)–(3.55)] The temporal de-singularization is asserted rather than derived. The proof substitutes the Morris-Thorne form g_tt=-e^{2Φ} into the flow equation, but the background being evolved is g_tt=-f(r). The paper never shows that the soliton replaces f(r) with e^{2Φ}; it merely writes Φ'(r) and declares it finite. Since g_tt=-f(r_H)=0 remains the actual component of the metric (1.2), the coordinate singularity is not removed. Moreover, if one sets -e^{2Φ}=-f, then Φ(r_H)=-∞, so finiteness of Φ' alone does not imply finiteness of Φ.
  4. [Theorem 5, Eqs. (3.70)–(3.74)] There is an internal factor inconsistency in the perturbation derivation. Eq. (3.70) writes the perturbed soliton as 1/2 L_ξ g̃ + 2α(r) S̃ + (...)g̃ + 2ω(r)η⊗η=0, but the unperturbed equation (2.1) has α and ω without the factor 2. The subsequent expansion (3.72) silently reverts to α and ω. This invalidates the stated reduction to the damped wave equation as written, at least without a corrected derivation.
minor comments (5)
  1. [Eq. (2.2)] The heat-flux terms appear as q_μ u_ν + q_ν u_ν; the second should presumably be q_ν u_μ (or symmetrized) to be consistent with the imperfect-fluid stress tensor.
  2. [Throughout] ‘Morris & Throne’ should be ‘Morris & Thorne’.
  3. [Fig. 1 caption] The caption states Q=60 while the text and Table 1 use Q=0.60; this is clearly a typo but should be corrected.
  4. [Table 1] The entry for λ(r) lists two different functions (“0.1+0.05 sin(r)” and “2.0+0.2 sin(r)”) without specifying which is used in which figure or derivation.
  5. [§3, around Eq. (3.43)] The symbol T(r) is introduced as a ‘geometric matter-trace equivalent’ and defined as S_μν k^μ k^ν, which is different from the trace T=g^{μν}T_{μν} used earlier. The notation should be disambiguated.

Circularity Check

4 steps flagged

Core 'black hole to wormhole transition' reduces to algebraic restatements: ω(rH)>0 and flare-out b'(rH)<1 are just f'(rH)>0 after defining b=r(1−f), while the temporal regularization is internally inconsistent with the explicit RNdS example.

specific steps
  1. self definitional [Theorem 3, Eqs. (3.32)–(3.39), p.12–13]
    "Equating equations (3.32) and (3.33), we obtain the relation: f(r)=1−b(r)/r ... f′(rH)=1−b′(rH)/rH ... f′(rH)>0 ... 1−b′(rH)>0 =⇒ b′(rH)<1."

    The shape function b(r) is not independently derived from the flow; it is introduced by the coordinate identity b(r)=r(1−f(r)) applied to the original black hole metric. Then b(rH)=rH and f'(rH)>0 are standard properties of any non-extremal static, spherically symmetric horizon, e.g., Schwarzschild has b=2M and b'=0. The 'flare-out condition' b'(rH)<1 is therefore algebraically equivalent to f'(rH)>0 under this definition and holds for the input black hole before any flow. It cannot certify a topological transition.

  2. fitted input called prediction [Theorem 3, Eqs. (3.38)–(3.39); Section 4, Eq. (4.6)]
    "Also for a standard, non-extremal static, spherically symmetric black hole, the lapse function f(r) is negative inside the horizon and positive outside. Then the slope of the function at r=rH must point strictly upward, ensuring an inherently positive metric gradient f′(rH)>0. Therefore, equation (3.38) mathematically guarantees that the geometric flow expands at the boundary: ω(rH)=f′(rH)>0 ... Because the scaling factor is strictly positive ω(rH)=0.3705>0, from theorem 3 that successfully generates the exact outward geometric repulsion required to open and maintain the Morris-Thorne throat."

    At f(rH)=0, the matter term α(r)f(r)T(r) in ω(r)=f'(r)−α(r)f(r)T(r) vanishes identically, so ω(rH)=f'(rH) is forced by the soliton equation itself. The 'strictly positive scaling factor' is just the input lapse slope at the horizon, not a consequence of NEC violation or dark energy. Section 4 then presents the number ω(rH)=0.3705, which is exactly f'(rH) from the chosen RNdS metric, as 'outward geometric repulsion'; this is a fitted metric input renamed as a soliton prediction. The exponentially decaying exotic matter T(r) does not even enter at the throat.

  3. renaming known result [Corollary 1, Eqs. (3.13)–(3.19), p.9]
    "To prevent a geometric singularity, the numerator 2α(rH)Stt(rH)−f′(rH) must simultaneously vanish at the exact coordinate rH ... α(rH)Stt(rH)=1/2 f′(rH) ... we obtain the exact thermogeometric equivalence: α(rH)Stt(rH)=2πTH."

    The 'thermogeometric equivalence' is not an independent law; it is exactly the finiteness condition for λ(rH) at f(rH)=0, re-expressed after substituting the standard surface-gravity and Hawking-temperature definitions. Equation (3.19) is just equation (3.15) relabeled. Moreover, for the metric (1.2) a direct computation gives S_tt=f(f''/2+f'/r), hence S_tt(rH)=0 at any horizon; the condition would force f'(rH)=0, i.e., an extremal horizon. The explicit RNdS example has f'(rH)≈0.3705>0, so the claimed temporal regularization is not satisfied by the model used to demonstrate it.

  4. self definitional [Theorem 4, Eqs. (3.52)–(3.55), p.16–17]
    "Now, by using the standard Morris-Thorne redshift metric, gtt=−e2Φ(r). Taking the Lie derivative of this metric component along the radial vector yields: (Lξg)tt=∂r(−e2Φ(r))=−2Φ′(r)e2Φ(r)."

    The soliton equation was solved for the original black hole metric (1.2), whose tt component is g_tt=−f(r) and remains zero at rH. In the proof of temporal de-singularization, this background component is silently replaced by the Morris-Thorne component −e^{2Φ} in the same flow equation. That substitution assumes the very regularized wormhole temporal metric that Theorem 4 is supposed to derive. No flow equation or dynamical step converts f(r) into e^{2Φ}; thus the conclusion g_tt≠0 is inserted as an input rather than produced by the geometry.

full rationale

The paper's central transition claim reduces to its own inputs by construction. Theorem 1 defines ω(r) from the soliton equation, and at the horizon it collapses to ω(rH)=f'(rH); Theorem 3 then obtains ω(rH)>0 and the Morris-Thorne flare-out condition b'(rH)<1 by defining b(r)=r(1−f(r)) and using the standard non-extremal horizon property f'(rH)>0. Every static, spherically symmetric black hole with a non-extremal horizon satisfies these inequalities automatically, so they do not establish an endogenous wormhole transition. The NEC-violating matter term T(r) is fitted as an exponential negative function but vanishes at the throat, so it does no work in the decisive inequality. Corollary 1's 'Hawking temperature coupling' is the same λ-finiteness condition restated; for the explicit RNdS metric it would require S_tt(rH)=0 to imply f'(rH)=0, contradicting the example's f'(rH)=0.3705. Theorem 4's temporal de-singularization substitutes the Morris-Thorne metric into the flow equations, assuming the conclusion, and no metric evolution is shown. These are not merely missing rigor: the 'predictions' are definitionally or algebraically identical to the input metric properties. No load-bearing self-citation chain is involved; the circularity is internal to the equations.

Axiom & Free-Parameter Ledger

8 free parameters · 5 axioms · 0 invented entities

The central derivation relies on the assumed existence of the soliton and on coordinate mappings between black hole and wormhole descriptions. The free parameters are all chosen by hand for the explicit example, and the key inequalities are obtained by tuning these functions. No independent evidence is supplied for the dark-energy fluid or its NEC-violating trace. No new particles or forces are posited; the 'geometric drag' is a labeling of an existing term.

free parameters (8)
  • Central mass M = 1.00
    Chosen for the explicit RNdS example; no derivation.
  • Electric charge Q = 0.60
    Chosen sub-extremal to have a horizon.
  • Cosmological constant Λ = 0.05
    Chosen to model de Sitter expansion.
  • Ricci coupling α(r) = 1.5+0.1r
    Ad hoc linear function; picked to keep α(r_H)>0 and satisfy inequality in Corollary 3.
  • Yamabe coupling β(r) = 1.5+0.1cos(r)
    Ad hoc oscillating function.
  • Solitonic expansion λ(r) = 0.1+0.05 sin(r) or 2.0+0.2 sin(r)
    Ad hoc oscillating functions to mimic ripples; chosen to make A²(r)>0.
  • Exotic matter trace T(r) = -0.5 e^{-(r-r_H)}
    Exponential decay chosen so that ω'(r_H)>f''(r_H) in Corollary 3; not derived from the fluid model.
  • EoS parameter γ = -1
    Assumed to enter dark energy era; not derived.
axioms (5)
  • domain assumption The spacetime admits an almost η-Ricci-Yamabe soliton along the radial vector field ξ=∂_r with smooth functional parameters.
    Definition 3; this is the core mathematical setup, not derived.
  • domain assumption The fluid satisfies the barotropic equation of state ρ=γσ.
    Used in Theorem 2 to classify eras; restricts the fluid.
  • ad hoc to paper The apparent horizon r_H is identified with the wormhole throat and the metric is rewritten as f(r)=1-b(r)/r.
    This mapping in Theorem 3 assumes a one-sided wormhole interpretation; no two-sided throat is constructed.
  • ad hoc to paper The soliton flow leaves the background metric unchanged.
    All proofs treat g as fixed; the claimed regularization of g_tt is never reflected in a modified line element.
  • ad hoc to paper The perturbations of η, ω, and R are neglected; only h and δS are kept in first order.
    In Theorem 5, δg = ϵh, but η depends on g and R depends on S and g, so these should perturb as well.

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read the original abstract

In this paper, we investigate the almost $\eta$-Ricci-Yamabe soliton as a fundamental geometric regulator for a static, spherically symmetric black hole coupled to an imperfect fluid. We have shown that the scaling parameter $\omega(r)$ is governed by thermodynamic friction along the radial vector field, and the geometric coupling with the Hawking temperature: $\alpha(r_H) S_{tt} = 2\pi T_H$ at the horizon. We also derive the Poisson equation along the gradient vector field of the soliton and prove that the flow's kinematic expansion is explicitly dependent on the fluid's equation of state $\rho = \gamma \sigma$. Diverging from traditional methodologies that assume a geometric shape function apriori, we analytically proved the geometric flow endogenously transitions the black hole geometry into a traversable wormhole throat by regularizing of temporal coordinate and satisfying spatial flare-out condition. This transition occurs when fluid enters the dark energy era at $\gamma = -1$ and violates the Null Energy Condition $\rho + \sigma < 0$, with the soliton strictly dominating the local curvature gradient $\omega^{\prime}(r_H) > f^{\prime\prime}(r_H)$, to keep the throat open. Moreover, by smoothly attenuating at spatial infinity, the soliton preserves the exact cosmological spacetime. Finally, through tensorial perturbation analysis, we demonstrate that the geometric flow introduces a localized dissipative mechanism, that the perturbation evolution reduces to damped wave equation, imposing geometric drag on the manifold.

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Reference graph

Works this paper leans on

77 extracted references · 15 canonical work pages · 2 internal anchors

  1. [1]

    arXiv preprint gr-qc/9704082 (1997) https://doi.org/10.1103/PhysRevD.56.4745

    Hochberg, D., Visser, M.: Geometric structure of the generic st atic traversable wormhole throat. arXiv preprint gr-qc/9704082 (1997) https://doi.org/10.1103/PhysRevD.56.4745

    Pith/arXiv arXiv 1997

  2. [2]

    arXiv pre print gr-qc/9710001 (1997) https://doi.org/10.48550/arXiv.gr-qc/9710001

    Visser, M., Hochberg, D.: Geometric wormhole throats. arXiv pre print gr-qc/9710001 (1997) https://doi.org/10.48550/arXiv.gr-qc/9710001

    Pith/arXiv arXiv doi:10.48550/arxiv.gr-qc/9710001 1997

  3. [3]

    The European Physical Journal C 84(10), 1063 (2024) https://doi.org/10.1140/epjc/s10052-024-13401-5 22

    Nath, P.P., Sarma, D.: A new class of traversable wormhole metrics . The European Physical Journal C 84(10), 1063 (2024) https://doi.org/10.1140/epjc/s10052-024-13401-5 22

    work page doi:10.1140/epjc/s10052-024-13401-5 2024

  4. [4]

    American Journal of Physics 56(5), 395–412 (1988) https://doi

    Morris, M.S., Thorne, K.S.: Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity. American Journal of Physics 56(5), 395–412 (1988) https://doi. org/10.1119/1.15620

    doi:10.1119/1.15620 1988

  5. [5]

    Sitzungsbericht e der Køniglich Preußischen Akademie der Wissenschaften, 844–847 (1915)

    Einstein, A.: Die feldgleichungen der gravitation. Sitzungsbericht e der Køniglich Preußischen Akademie der Wissenschaften, 844–847 (1915)

    1915

  6. [6]

    Cambridge university press, Cambridge (2023)

    Hawking, S.W., Ellis, G.F.: The Large Scale Structure of Space-time. Cambridge university press, Cambridge (2023)

    2023

  7. [7]

    University of Chicago press, Chicag o (2010)

    Wald, R.M.: General Relativity. University of Chicago press, Chicag o (2010)

    2010

  8. [9]

    https://doi.org/https://arxiv.org/abs/ 0908.2006

    Cao, H.-D.: Recent Progress on Ricci Solitons (2009). https://doi.org/https://arxiv.org/abs/ 0908.2006

    Pith/arXiv arXiv 2009

  9. [10]

    https://doi.org/ https://arxiv.org/abs/0903.3927

    Cao, H.-D.: Geometry of Complete Gradient Shrinking Ricci Soliton s (2009). https://doi.org/ https://arxiv.org/abs/0903.3927

    Pith/arXiv arXiv 2009

  10. [12]

    In: Ellip tic and Parabolic Methods in Geometry, pp

    Cao, H.-D.: Existence of gradient k¨ ahler-ricci solitons. In: Ellip tic and Parabolic Methods in Geometry, pp. 1–16. AK Peters/CRC Press, New York (1996)

    1996

  11. [13]

    Yamabe, H.: On a deformation of riemannian structures on comp act manifolds (1960)

    1960

  12. [14]

    unpublished manusc ript (1989)

    Hamilton, R.S.: Lectures on geometric flows. unpublished manusc ript (1989)

    1989

  13. [15]

    PhD thesis, MZU (2022)

    Khatri, M.: A study on certain almost contact manifolds and invar iant submanifolds. PhD thesis, MZU (2022)

    2022

  14. [16]

    arXiv preprint math/0211159 (2002) https://doi.org/10.48550/arXiv.math/0211159

    Perelman, G.: The entropy formula for the ricci flow and its geom etric applications. arXiv preprint math/0211159 (2002) https://doi.org/10.48550/arXiv.math/0211159

    Pith/arXiv arXiv doi:10.48550/arxiv.math/0211159 2002

  15. [17]

    Turkish Journal of Mathematics 43(5), 2631–2641 (2019)

    G¨ uler, S., Crasmareanu, M.: Ricci-yamabe maps for riemannian flows and their volume variation and volume entropy. Turkish Journal of Mathematics 43(5), 2631–2641 (2019)

    2019

  16. [18]

    Almost Kenmotsu metric as Ricci-Yamabe soliton

    Dey, D.: Almost kenmotsu metric as ricci-yamabe soliton. arXiv pr eprint arXiv:2005.02322 (2020) https://doi.org/10.48550/arXiv.math/0303109

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.math/0303109 2005

  17. [19]

    Nonlinear Analy sis 132, 66–94 (2016)

    Catino, G., Mazzieri, L.: Gradient einstein solitons. Nonlinear Analy sis 132, 66–94 (2016)

    2016

  18. [20]

    Canadian Mathematical Bulletin 64(3), 591–604 (2021)

    Dwivedi, S.: Some results on ricci-bourguignon solitons and almost solitons. Canadian Mathematical Bulletin 64(3), 591–604 (2021)

    2021

  19. [21]

    ANNALI DELL’UNIVERSITA’DI FERRARA 65(2), 375–388 (2019)

    Venkatesha, V., Aruna Kumara, H.: Gradient ρ-einstein soliton on almost kenmotsu manifolds. ANNALI DELL’UNIVERSITA’DI FERRARA 65(2), 375–388 (2019)

    2019

  20. [22]

    arXiv preprint arXiv:14 02.0223 (2014)

    Blaga, A.M.: η-ricci solitons on para-kenmotsu manifolds. arXiv preprint arXiv:14 02.0223 (2014)

    2014

  21. [23]

    Tohoku Mathematical Journal, Second Series 61(2), 205–212 (2009)

    Cho, J.T., Kimura, M.: Ricci solitons and real hypersurfaces in a c omplex space form. Tohoku Mathematical Journal, Second Series 61(2), 205–212 (2009)

    2009

  22. [24]

    Afrika Matematika 32(7), 1645–1656 (2021) https://doi.org/10.1007/s13370-021- 00925-2

    Singh, J.P., Khatri, M.: On ricci–yamabe soliton and geometrical st ructure in a perfect fluid spacetime. Afrika Matematika 32(7), 1645–1656 (2021) https://doi.org/10.1007/s13370-021- 00925-2

    work page doi:10.1007/s13370-021- 2021

  23. [25]

    : η-ricci-yamabe soliton on riemannian submersions from riemannian manifolds

    Siddiqi, M., Akyol, M.A., et al. : η-ricci-yamabe soliton on riemannian submersions from riemannian manifolds. arXiv preprint arXiv:2004.14124 (2020) https://doi.org/10.48550/arXiv. 2004.14124

    Pith/arXiv arXiv doi:10.48550/arxiv.2004.14124 2004

  24. [26]

    Nuclear Physics B, 11 7360 (2026) https://doi.org/10

    Shaikh, A.A., et al.: Symmetry and pseudosymmetry properties w ith ricci soliton of the reissner-nordstr¨ om-de sitter spacetime. Nuclear Physics B, 11 7360 (2026) https://doi.org/10. 1016/j.nuclphysb.2026.117360

    arXiv 2026

  25. [27]

    Advances in Mathem atical Physics 2021(1), 2485804 (2021) https://doi.org/10.1155/2021/2485804 23

    Alkhaldi, A.H., Siddiqi, M.D., Khan, M.A., Alqahtani, L.S.: Imperfect fluid generalized robertson walker spacetime admitting ricci-yamabe metric. Advances in Mathem atical Physics 2021(1), 2485804 (2021) https://doi.org/10.1155/2021/2485804 23

    work page doi:10.1155/2021/2485804 2021

  26. [28]

    Classical and Quantum Gravity 19(5), 935–952 (2002) https://doi.org/10.1088/0264-9381/19/5/307

    Rahman, S., Visser, M.: Spacetime geometry of static fluid spher es. Classical and Quantum Gravity 19(5), 935–952 (2002) https://doi.org/10.1088/0264-9381/19/5/307

    doi:10.1088/0264-9381/19/5/307 2002

  27. [29]

    Physical Review D 112(10), 104034 (2025) https://doi.org/10.1103/djx3-f3ht

    Livine, E.R., Yokokura, Y.: Effective dynamics of spherically symme tric static spacetime. Physical Review D 112(10), 104034 (2025) https://doi.org/10.1103/djx3-f3ht

    work page doi:10.1103/djx3-f3ht 2025

  28. [30]

    Annals of Physics 151(2), 466–496 (1983)

    Hiscock, W.A., Lindblom, L.: Stability and causality in dissipative relat ivistic fluids. Annals of Physics 151(2), 466–496 (1983)

    1983

  29. [31]

    Cambridge university press, Cambridge (2 009)

    Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E .: Exact Solutions of Einstein’s Field Equations. Cambridge university press, Cambridge (2 009)

  30. [32]

    Phys ical Review D—Particles, Fields, Gravitation, and Cosmology 71(4), 043520 (2005)

    Sushkov, S.: Wormholes supported by a phantom energy. Phys ical Review D—Particles, Fields, Gravitation, and Cosmology 71(4), 043520 (2005)

    2005

  31. [33]

    General Relativity and Gravitation 35(2), 285–305 (2003)

    Ali, M.H.: Spinning particles in reissner-nordstr¨ om-de sitter spa cetime. General Relativity and Gravitation 35(2), 285–305 (2003)

    2003

  32. [34]

    Physical Review D 100(12), 124001 (2019)

    Gim, Y., Gwak, B.: Charged particle and strong cosmic censorship in reissner–nordstr¨ om–de sitter black holes. Physical Review D 100(12), 124001 (2019)

    2019

  33. [35]

    Living Reviews in Relativity 7(1), 10 (2004) https://doi.org/10.12942/lrr-2004-10

    Ashtekar, A., Krishnan, B.: Isolated and dynamical horizons an d their applications. Living Reviews in Relativity 7(1), 10 (2004) https://doi.org/10.12942/lrr-2004-10

    doi:10.12942/lrr-2004-10 2004

  34. [36]

    Physical Review Letters 14(3), 57 (1965) https://doi.org/10.1103/PhysRevLett.14.57

    Penrose, R.: Gravitational collapse and space-time singularities . Physical Review Letters 14(3), 57 (1965) https://doi.org/10.1103/PhysRevLett.14.57

    doi:10.1103/physrevlett.14.57 1965

  35. [37]

    Dynamical behavior of black-hole spacetimes

    Destounis, K.: Dynamical behavior of black-hole spacetimes. ar Xiv preprint arXiv:1909.08597 (2019) https://doi.org/10.48550/arXiv.1909.08597

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1909.08597 1909

  36. [38]

    PhD thesis, Universidade de S˜ ao Paulo

    Jimenez, A.F.C.: Thermodynamics and phase transitions of black h oles. PhD thesis, Universidade de S˜ ao Paulo

  37. [39]

    O’neill, B.: Semi-Riemannian Geometry with Applications to Relativity v ol. 103. Academic press, California (1983)

    1983

  38. [40]

    Living Reviews in Relativity 16(1), 1 (2013)

    Abramowicz, M.A., Fragile, P.C.: Foundations of black hole accretio n disk theory. Living Reviews in Relativity 16(1), 1 (2013)

    2013

  39. [41]

    Annals of Physics 118(2), 341–372 (1979)

    Israel, W., Stewart, J.M.: Transient relativistic thermodynamics and kinetic theory. Annals of Physics 118(2), 341–372 (1979)

    1979

  40. [42]

    arXiv prepr int astro-ph/9609119 (1996)

    Maartens, R.: Causal thermodynamics in relativity. arXiv prepr int astro-ph/9609119 (1996)

    Pith/arXiv arXiv 1996

  41. [43]

    obse rvational appearance

    Shakura, N.I., Sunyaev, R.A.: Black holes in binary systems. obse rvational appearance. Astronomy and Astrophysics, Vol. 24, p. 337-355 24, 337–355 (1973)

    1973

  42. [44]

    Physics Reports 509(4-5), 167–321 (2011)

    Capozziello, S., De Laurentis, M.: Extended theories of gravity. Physics Reports 509(4-5), 167–321 (2011)

    2011

  43. [45]

    Spacetime an d geometry 101, 102 (2004)

    Carroll, S.M.: An introduction to general relativity. Spacetime an d geometry 101, 102 (2004)

    2004

  44. [46]

    In: Wormholes, Warp Drives and Energy Conditions, pp

    Mart ´ ın–Moruno, P., Visser, M.: Classical and semi-classical ene rgy conditions. In: Wormholes, Warp Drives and Energy Conditions, pp. 193–213. Springer, Chem ( 2017). https://doi.org/10. 1007/978-3-319-55182-1 9

    2017

  45. [47]

    from einstein to hawking

    Visser, M.: Lorentzian wormholes. from einstein to hawking. Woo dbury (1995)

    1995

  46. [48]

    Classical and Quantum Gravity 25(22), 222002 (2008)

    Husain, V., Seahra, S.S.: Ricci flows, wormholes and critical phen omena. Classical and Quantum Gravity 25(22), 222002 (2008)

    2008

  47. [49]

    Lobo, F.S.: Wormholes, Warp Drives and Energy Conditions vol. 18 9. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-55182-1

    doi:10.1007/978-3-319-55182-1 2017

  48. [50]

    Chandrasekhar, S.: The Mathematical Theory of Black Holes vo l. 69. Oxford university press, Oxford (1998)

    1998

  49. [51]

    Quaestiones Mathematicae 45(1), 143–163 (2022) https://doi.org/10.2989/ 16073606.2020.1850538

    Blaga, A.M., Ozgur, C.: Almost η-ricci and almost η-yamabe solitons with torse-forming potential vector field. Quaestiones Mathematicae 45(1), 143–163 (2022) https://doi.org/10.2989/ 16073606.2020.1850538

    arXiv 2022

  50. [52]

    Bulletin of the Belgian Mathematical 24 Society-Simon Stevin 25(5), 641–653 (2018) https://doi.org/10.36045/bbms/1547780426

    Blaga, A.M.: Almost η-ricci solitons in ( lcs)n-manifolds. Bulletin of the Belgian Mathematical 24 Society-Simon Stevin 25(5), 641–653 (2018) https://doi.org/10.36045/bbms/1547780426

    arXiv 2018

  51. [53]

    Arab Journal of Mathematical Sciences 31(1), 118–129 (2025) https://doi.org/10.1108/AJMS-07-2022-0171

    Khatri, M., Singh, J.P.: Almost ricci-yamabe soliton on contact met ric manifold. Arab Journal of Mathematical Sciences 31(1), 118–129 (2025) https://doi.org/10.1108/AJMS-07-2022-0171

    work page doi:10.1108/ajms-07-2022-0171 2025

  52. [54]

    Journal of AppliedMath 2(2), 231–231 (2024) https:// doi.org/10.59400/jam.v2i2.231

    Mert, T., At¸ ceken, M.: Pseudosymmetric normal paracontac t metric space forms admitting ( α, β)- type almost η- ricci-yamabe solitons. Journal of AppliedMath 2(2), 231–231 (2024) https:// doi.org/10.59400/jam.v2i2.231

    work page doi:10.59400/jam.v2i2.231 2024

  53. [55]

    Acta Mathematica Universitatis Comenianae 93(3), 171–183 (2024)

    Pandey, S., Mert, T., At¸ ceken, M.: ( α, β)-type almost η-ricci-yamabe solitons in perfect fluid spacetime. Acta Mathematica Universitatis Comenianae 93(3), 171–183 (2024)

    2024

  54. [56]

    International Journal of Theoretical Physics 65(5), 131 (2026) https://doi.org/10.1007/s10773- 026-06340-2

    Jafari, M., Azami, S., Chand De, U.: Generalized almost η-ricci solitons on spacetimes. International Journal of Theoretical Physics 65(5), 131 (2026) https://doi.org/10.1007/s10773- 026-06340-2

    doi:10.1007/s10773- 2026

  55. [57]

    Physical Review D 67(2), 024035 (2003) https://doi.org/10.1103/PhysRevD.67

    Carr, B.J., Gundlach, C.: Spacetime structure of self-similar sph erically symmetric perfect fluid solutions. Physical Review D 67(2), 024035 (2003) https://doi.org/10.1103/PhysRevD.67. 024035

    doi:10.1103/physrevd.67 2003

  56. [59]

    Physical review letters 90(20), 201102 (2003) https://doi.org/10.1103/PhysRevLett

    Visser, M., Kar, S., Dadhich, N.: Traversable wormholes with arbit rarily small energy condition violations. Physical review letters 90(20), 201102 (2003) https://doi.org/10.1103/PhysRevLett. 90.201102

    doi:10.1103/physrevlett 2003

  57. [60]

    Journal of Cosmology and Astroparticle Physics 2019(02), 042–042 (2019) https://doi.org/10.1088/1475-7516/2019/02/ 042

    Simpson, A., Visser, M.: Black-bounce to traversable wormhole. Journal of Cosmology and Astroparticle Physics 2019(02), 042–042 (2019) https://doi.org/10.1088/1475-7516/2019/02/ 042

    work page doi:10.1088/1475-7516/2019/02/ 2019

  58. [61]

    Physical Review Letters 128(9), 091104 (2022) https://doi.org/10.1103/PhysRevLett.128.091104

    Konoplya, R., Zhidenko, A.: Traversable wormholes in general re lativity. Physical Review Letters 128(9), 091104 (2022) https://doi.org/10.1103/PhysRevLett.128.091104

    doi:10.1103/physrevlett.128.091104 2022

  59. [62]

    The European Physical Journal C 77(11), 748 (2017) https://doi.org/10.1140/epjc/s10052-017-5332-5

    Cataldo, M., Liempi, L., Rodriguez, P.: Traversable schwarzschild -like wormholes. The European Physical Journal C 77(11), 748 (2017) https://doi.org/10.1140/epjc/s10052-017-5332-5

    work page doi:10.1140/epjc/s10052-017-5332-5 2017

  60. [63]

    F ortschritte der Physik 69(8-9), 2100048 (2021) https://doi.org/10.1002/prop.202100048

    Mustafa, G., Ahmad, M., ¨Ovg¨ un, A., Farasat Shamir, M., Hussain, I.: Traversable wormholesin the extended teleparallel theory of gravity with matter coupling. F ortschritte der Physik 69(8-9), 2100048 (2021) https://doi.org/10.1002/prop.202100048

    work page doi:10.1002/prop.202100048 2021

  61. [64]

    Journal of Cosmology and Astropar ticle Physics 2023(10), 038 (2023) https://doi.org/10.1088/1475-7516/2023/10/038

    Ilyas, M., Bamba, K.: Traversable wormholes with static spherica l symmetry and their stability in higher-curvature gravity. Journal of Cosmology and Astropar ticle Physics 2023(10), 038 (2023) https://doi.org/10.1088/1475-7516/2023/10/038

    work page doi:10.1088/1475-7516/2023/10/038 2023

  62. [65]

    International Journal of Modern Physics A 37(05), 2250010 (2022) https://doi.org/10.1142/ S0217751X22500105

    Mishra, B., Agrawal, A., Tripathy, S., Ray, S.: Traversable wormh ole models in f (r) gravity. International Journal of Modern Physics A 37(05), 2250010 (2022) https://doi.org/10.1142/ S0217751X22500105

    2022

  63. [66]

    International Journal of Modern Physics D 30(13), 2150100 (2021) https://doi.org/10

    Sahoo, P., Moraes, P., Lapola, M.M., Sahoo, P.: Traversable worm holes in the traceless f (r, t) gravity. International Journal of Modern Physics D 30(13), 2150100 (2021) https://doi.org/10. 1142/S0218271821501005

    2021

  64. [67]

    JANOLI International Journal of Physics 2(1) (2026) https://doi.org/10.64758/wy7gn443

    Ashraf, S.: Traversable wormhole solutions and f (r) gravity mo dels. JANOLI International Journal of Physics 2(1) (2026) https://doi.org/10.64758/wy7gn443

    work page doi:10.64758/wy7gn443 2026

  65. [68]

    The European Physical Journal C 83(6), 522 (2023) https://doi.org/10.1140/epjc/s10052-023-11704-7

    Malik, A., Naz, T., Qadeer, A., Shamir, M.F., Yousaf, Z.: Investigat ion of traversable wormhole solutions in modified f (r) gravity with scalar potential. The European Physical Journal C 83(6), 522 (2023) https://doi.org/10.1140/epjc/s10052-023-11704-7

    work page doi:10.1140/epjc/s10052-023-11704-7 2023

  66. [69]

    The European Physical Journal C 80(12), 1102 (2020) https://doi.org/10.1140/epjc/s10052- 020-08689-y

    Shamir, M.F., Fayyaz, I.: Traversable wormhole solutions in f (r) g ravity via karmarkar condition. The European Physical Journal C 80(12), 1102 (2020) https://doi.org/10.1140/epjc/s10052- 020-08689-y

    doi:10.1140/epjc/s10052- 2020

  67. [70]

    The European Physical Journal C 82(4), 280 (2022) https://doi.org/10.1140/epjc/ s10052-022-10249-5

    Sokoliuk, O., Mandal, S., Sahoo, P., Baransky, A.: Generalised ellis– bronnikov wormholes in f 25 (r) gravity. The European Physical Journal C 82(4), 280 (2022) https://doi.org/10.1140/epjc/ s10052-022-10249-5

    doi:10.1140/epjc/ 2022

  68. [71]

    Anna li della Scuola normale superiore di Pisa-Classe di scienze 10(4), 757–799 (2011)

    Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Anna li della Scuola normale superiore di Pisa-Classe di scienze 10(4), 757–799 (2011)

    2011

  69. [72]

    Oxford Univ ersity Press, Oxford (2013)

    Rezzolla, L., Zanotti, O.: Relativistic Hydrodynamics. Oxford Univ ersity Press, Oxford (2013)

    2013

  70. [73]

    Classical and Quantum Gravity 39(19), 195002 (2022) https://doi.org/10.1088/1361-6382/ac8861

    Maeda, H., Harada, T.: Criteria for energy conditions. Classical and Quantum Gravity 39(19), 195002 (2022) https://doi.org/10.1088/1361-6382/ac8861

    work page doi:10.1088/1361-6382/ac8861 2022

  71. [74]

    Adamiak, J.P.: Static and dynamic traversable wormholes. In: Th e Eleventh Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (In 3 Volumes), pp. 2187–2 189 (2008). https://doi. org/10.1142/9789812834300 0357 . World Scientific

    doi:10.1142/9789812834300 2008

  72. [75]

    Alpha Sc ience International Limited, Oxford (2007)

    De, U.C., Shaikh, A.A.: Differential Geometry of Manifolds. Alpha Sc ience International Limited, Oxford (2007)

    2007

  73. [76]

    Cambridge university press, Cambridge (2004)

    Poisson, E.: A Relativist’s Toolkit: the Mathematics of Black-hole M echanics. Cambridge university press, Cambridge (2004)

    2004

  74. [77]

    Communications in m athematical physics 43(3), 199–220 (1975)

    Hawking, S.W.: Particle creation by black holes. Communications in m athematical physics 43(3), 199–220 (1975)

    1975

  75. [78]

    Petersen, P.: Riemannian Geometry, p. 499. Springer, Cham (2 016). https://doi.org/10.1007/ 978-3-319-26654-1

  76. [79]

    Rendiconti del Circolo Matematico di Palermo (1884-1940) 43(1), 203–212 (1919)

    Palatini, A.: Deduzione invariantiva delle equazioni gravitazionali d al principio di hamilton. Rendiconti del Circolo Matematico di Palermo (1884-1940) 43(1), 203–212 (1919)

    1940

  77. [80]

    Maggiore, M.: 1: Theory and Experiments vol. 1. Oxford Univers ity Press, Oxford (2008) 26

    2008