Pith. sign in

REVIEW 4 major objections 6 minor 73 references

Gravitational Waves beyond the Linear Approximation and Gravitational Wave Reflection

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read Metric perturbations obey a massive Klein-Gordon equation, so gravitons are massive, black holes scatter gravitational waves, and density interfaces reflect them.

desk verdict A clear, ambitious paper that fails at its first technical step: Eq. (6) is not the variation of the Ricci tensor, so the massive Klein-Gordon field and all its consequences are unsupported. read the letter →

arxiv 2502.01687 v2 pith:ICAGY5P7 submitted 2025-02-02 physics.gen-ph

Victor Atanasov , Avadh Saxena This is my paper · ORCID
classification physics.gen-ph MSC 83C3583C5783C25 PACS 04.30.-w04.70.-s
keywords gravitationalwavesmassivegravitonKlein-Gordonequationblackholescatteringwavereflectionmirrorindexofrefractionpropulsion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to describe gravitational waves beyond the weak-field, flat-background approximation of general relativity and claims that the trace of the metric perturbation, $\psi = g^{\sigma\nu}\delta g_{\sigma\nu}$, is a genuine scalar field obeying the massive Klein-Gordon equation $\Box\psi + m^2\psi = 0$, with $m^2 = \Lambda - \tfrac{1}{2}\kappa T$ fixed by the cosmological constant and local matter content. If this holds, gravitons are massive, gravitational waves disperse with a minimum frequency, black holes reflect a portion of incoming gravitational radiation through a potential barrier outside the horizon, and sharp density boundaries behave like refractive interfaces for gravitational waves. A sympathetic reader should care because the consequences are concrete and testable: black-hole scattering would be an unambiguous signature of black-hole existence, and layered stacks of alternating dense and sparse materials would form gravitational-wave mirrors.

What carries the argument

The load-bearing object is the scalar trace field $\psi = g^{\sigma\nu}\delta g_{\sigma\nu}$, the metric-contracted amplitude of the perturbation, together with the contraction identity $g^{\sigma\nu}\delta R_{\sigma\nu} = -\Box\,\psi$ (Eq. 7) that turns the tensor perturbation equations into a single scalar equation, and the mass term $m^2 = \Lambda - \tfrac{1}{2}\kappa T$ that emerges from varying the material-content tensor under the dust assumption. An argument that $\psi$ is invariant under coordinate transformations is what prevents it from being gauged away in the manner of the transverse-traceless gauge of linearized gravity, so the scalar channel cannot be discarded and carries the whole wave dynamics. From there the machinery is mechanical: on Schwarzschild space-time the ansatz $\Psi(x) = N\,\mathrm{e}^{-\ln(x\sqrt{F})}\chi(x)$ eliminates the first-derivative term and produces the effective Schrödinger problem with the potential $V_{\mathrm{eff}}(l,x)$ of Eq. (41), whose positive barrier outside the horizon is the reflection mechanism; in the Newtonian limit replacing the curved d'Alembertian by the weak-field operator $(-n^2\partial_{ct}^2 + \Delta + \mu^2)$ produces the index of refraction $n = 1 - 2\Phi/c^2$ and the Fresnel-style reflection coefficient $R = |n_1 - n_2|/(n_1 + n_2)$.

What would settle it

Recompute $g^{\sigma\nu}\delta R_{\sigma\nu}$ from the full Palatini identity, keeping the cross terms $\nabla_\alpha\nabla_\sigma\delta g^\alpha{}_\nu + \nabla_\alpha\nabla_\nu\delta g^\alpha{}_\sigma$ and the background-curvature terms that Eq. (6) drops; wherever these do not vanish, the contracted identity (7) fails, and with it the massive Klein-Gordon equation (25), the black-hole barrier, and the mirror predictions. The paper's own binary experiment supplies the observational half: a stack with an even number of sharp density interfaces reflects a nonzero fraction of a gravitational wave under the Fresnel-style boundary conditions but exactly zero under the quantum-gluing conditions, so a single measurement decides which boundary condition — and hence which theory — is right.

Watch

Extended reading notes

Core claim

On the author's own terms, the central discovery is that contracting the varied Einstein equations produces a free, massive, Lorentz-covariant wave equation for the scalar amplitude $\psi = g^{\sigma\nu}\delta g_{\sigma\nu}$ on any curved background: $\Box\psi + m^2\psi = 0$, where $\Box$ is the curved-space d'Alembertian and $m^2 = \Lambda - \tfrac{1}{2}\kappa T$. Because $\Lambda$ is positive, the effective mass $m_g = (\hbar/c)\sqrt{\Lambda - \tfrac{1}{2}\kappa T}$ stays real in vacuum, so gravitational waves remain stable and oscillatory, and the paper assigns dark energy the dynamical role of keeping the graviton mass real where matter is absent. Solving the equation on a Schwarzschild background and removing the first-derivative term by a change of dependent variable yields an effective Schrödinger equation whose geometric potential is purely attractive for zero angular momentum (bound standing waves) but develops a positive barrier just outside the event horizon for $l \geq 1$; the paper reads this barrier as a scattering potential, so black holes reflect part of any incoming gravitational radiation, which it presents as an unambiguous testable prediction of black-hole existence. In the Newtonian limit the same equation gives gravitational waves an index of refraction $n = 1 - 2\Phi/c^2$, so propagation slows in gravitational potentials and an interface between regions of different density reflects a small fraction of the power; a hemispherical stack of alternating high- and low-density layers can then break the symmetry of a symmetric emitter and produce directed thrust without violating Newton's third law.

Load-bearing premise

The derivation assumes that when the Ricci tensor is perturbed, only two terms survive — one that becomes the wave operator acting on $\psi$ and one that vanishes on contraction — with all other pieces of the standard variation dropping out; if those extra pieces do not vanish on a curved background, the massive wave equation for $\psi$ and every later prediction fails to follow.

Editorial extensions

If this is right

  • Gravitons acquire a mass $m_g = (\hbar/c)\sqrt{\Lambda - \tfrac{1}{2}\kappa T}$, making gravitational waves dispersive with a minimal frequency $\nu_{\mathrm{min}} = (c/2\pi)\sqrt{\Lambda - \tfrac{1}{2}\kappa T}$ below which no radiation propagates.
  • Black holes scatter gravitational waves: incoming waves with $l \geq 1$ encounter a positive potential barrier just outside the horizon and are partially reflected, giving a direct and unambiguous observational test of black-hole existence.
  • In the Newtonian limit gravitational waves travel slower than light with index of refraction $n = 1 - 2\Phi/c^2$, and each sharp density interface reflects a fraction $R \approx V|\rho_1 - \rho_2|/(r c^2)$ of the incident power.
  • Stacking many alternating dense and sparse layers (for example iridium and aluminum) adds these small reflections nearly linearly, producing a gravitational-wave mirror; a hemispherical mirror around a symmetric quadrupole emitter converts emission into directed thrust while conserving momentum, so Newton's third law is never violated.
  • Dark matter is re-interpreted as an apparent effect of the non-vanishing graviton mass rather than a new particle species, with the $\Lambda$CDM cosmological model recovered, and the cosmological term is assigned the role of keeping the graviton mass real in empty space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: the same barrier that reflects incoming waves should also produce gravitational-wave echoes — delayed, weaker replicas of a merger's ringdown — because part of the wave is temporarily trapped between the barrier and the horizon; the paper does not compute the delay, but the potential in Eq. (41) implies it.
  • Extension: since $m^2 = \Lambda - \tfrac{1}{2}\kappa T$ depends on local density, the effective graviton mass varies with environment, so gravitational waves passing through different matter columns should accumulate a density-dependent phase; comparing the dispersion of a single event along different lines of sight would probe this, a consequence the paper leaves implicit.
  • Extension: if $\psi$ is a real gauge-invariant radiation channel, then detector analyses that model only the two transverse-traceless tensor polarizations are missing a scalar signal; re-analyzing existing merger and ringdown waveforms for a scalar-polarization component is a direct, data-only test of the paper's premise.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper derives an equation for the trace of the metric perturbation ψ = g^{σν}δg_{σν} by varying the Einstein field equations. The authors claim this yields a massive Klein-Gordon equation □ψ + m²ψ = 0 with m² = Λ − (1/2)κT, implying a density-dependent graviton mass. They then solve this equation on a Schwarzschild background, finding an effective potential that binds l=0 modes and gives a repulsive barrier for l≥1, which they interpret as gravitational wave reflection off black holes. In the Newtonian limit they obtain an index of refraction n = 1 − 2Φ/c² and propose layered-density gravitational wave mirrors and a propulsion scheme. The paper also suggests that the massive graviton could account for dark matter.

Significance. If the central derivation were correct, the paper would offer a striking unification: an effective graviton mass emerging directly from the cosmological constant and local matter density, with testable predictions for gravitational wave reflection by black holes and by laboratory density interfaces. The manuscript is clearly written in parts and engages with the massive-gravity literature. However, the main result depends on an incorrect expression for the first-order variation of the Ricci tensor, and the subsequent gauge-invariance claim and boundary-condition discussion contain additional serious flaws. Because these errors occur at the very first step and propagate through all applications, the paper's predictions are unsupported. The work does not provide numerical simulations, machine-checked derivations, or new observational analyses that could rescue the conclusions.

major comments (4)
  1. [§2.1, Eq. (6)-(7)] The expression for the first-order variation of the Ricci tensor is incomplete. Using the Palatini identity (3) and the standard Christoffel variation (5), one obtains δR_{σν} = (1/2)(∇^β∇_νδg_{σβ} + ∇^β∇_σδg_{βν} − □δg_{σν} − ∇_σ∇_νψ). Equation (6) omits the first two cross terms. Their metric trace contributes ∇^α∇^βδg_{αβ}, so Eq. (7) should read g^{σν}δR_{σν} = ∇^α∇^βδg_{αβ} − □ψ, not −□ψ. Consequently the contracted field equation (22) and the massive Klein-Gordon equation (25) are missing the term ∇^α∇^βδg_{αβ}; this term cannot be discarded unless the perturbation is transverse, a condition that is incompatible with the paper's claim that ψ is coordinate-invariant. Since Eqs. (29)-(41) and (52)-(55) all follow from Eq. (25), the central predictions are unsupported.
  2. [§2.2, Eq. (13)-(14)] The variation of the dust stress-energy tensor is not correctly evaluated. From T_{σν}=ρu_σu_ν, the trace variation is δT = δ(g^{σν}T_{σν}) = T_{σν}δg^{σν} + g^{σν}δT_{σν}. Even if g^{σν}δT_{σν}=0 as claimed in Eq. (13), the first term equals −ρ u^αu^βδg_{αβ} (up to the sign convention for δg), which is generically nonzero. The statement in Eq. (14) that δT=0 because the equation of state does not depend on the metric is therefore invalid: the trace is a metric-dependent contraction. Since δT appears in the source J in Eq. (23), the derivation of Eq. (22) and the subsequent mass formula (24) are not established.
  3. [§2.3] The proof that ψ = g^{σν}δg_{σν} is gauge invariant does not apply to the standard gauge freedom of metric perturbation theory. In linearized gravity an infinitesimal coordinate change x^μ→x^μ+ξ^μ sends the perturbation to h_{μν}+∇_μξ_ν+∇_νξ_μ, so the trace transforms to ψ+2∇·ξ, which is not invariant. The paper instead treats δg_{σν} as an arbitrary tensor field under an active diffeomorphism and subtracts the Lie derivative of the background metric; this is a different operation and does not correspond to the gauge freedom used in the TT gauge. The claim that ψ 'cannot be made to vanish by the choice of the gauge condition' is therefore incorrect, and the interpretation of ψ as a coordinate-invariant physical scalar field is unsupported.
  4. [§5.1] The derivation of the gravitational wave reflection coefficient is internally inconsistent. The 'quantum mechanical gluing' conditions (62)-(63) lead to R_{12} = (k_1−k_2)/(k_1+k_2) and, by the same argument, R_{21} = −R_{12} in Eqs. (66) and (69). The paper then replaces this antisymmetric result by the symmetric Fresnel-like formula (72) with R_{12}=R_{21}=|n_1−n_2|/(n_1+n_2), based on an unproved phase-shift assumption. The paper itself acknowledges this is an open question, but Sections 5 and 6 nevertheless assume the symmetric form to construct the mirror and the propulsion device. Without a physical derivation of the correct boundary conditions, the reflection and propulsion predictions are not established.
minor comments (6)
  1. [Abstract and Section 1] The title and Section 1 describe the work as 'beyond the linear approximation', but the derivation keeps only terms first order in δg_{σν}; no quadratic or higher-order terms are retained. The paper should temper this claim or include the nonlinear terms.
  2. [Throughout] There are numerous typographical errors, e.g., 'd'Alambert' for 'd'Alembert' (after Eq. 7), 'insted' for 'instead' (Section 5.1), and 'Schwatzschild' for 'Schwarzschild' (Section 1).
  3. [References] Reference [8] appears to misspell the author name ('John T. Goblin Jr.' should likely be 'John T. Giblin Jr.').
  4. [Section 5] The notation switches from ψ to ϕ to χ without explanation; using a single symbol for the perturbation amplitude would improve readability.
  5. [Section 4] The claim that gravitational wave scattering off a potential barrier provides an 'unambiguous testable prediction of black hole existence' (Section 4) overstates the case, since any sufficiently compact mass distribution would produce a similar effective potential; the paper does not provide a background-independent observable or an amplitude estimate.
  6. [§2.5, Eq. (24)] Equation (24) says the effective mass is 'defined as a square root of (20)', but only m^2 is defined; define m = sqrt(M) explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central scalar wave equation is derived algebraically from stated inputs, and the resulting predictions are not imposed by construction.

full rationale

The paper's central equation (22) is obtained by contracting the perturbed Einstein equation (19) with the metric; the effective mass m^2 = Lambda - (1/2) kappa T is a combination of the cosmological term and the stress-energy trace, not a parameter fitted to the black-hole or reflection outcomes. The Schwarzschild analysis, effective potential (41), index of refraction (52), and reflection coefficients (55)/(66) are all derived algebraically from that equation and standard separation and gluing assumptions; none of these predictions is defined in terms of the input parameters in a way that makes the prediction tautological. The self-citations ([12], [72], [73]) occur in peripheral contexts (exotic equations of state, electrical driving of a proposed emitter) and are not used to justify the main derivation or to forbid alternatives. The paper itself flags open issues, such as the absence of initial or boundary conditions for quantization in Section 4 and the unresolved boundary-condition choice in Section 5.1; these are limitations, not circular reductions. A separate, independent mathematical concern is that Eq. (6) drops the standard cross terms involving gradients of delta g, so the traced relation (7) and hence the Klein-Gordon equation (22) may be incorrect; that is a correctness risk, not an equivalence of inputs and outputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper uses no free parameters fitted by hand; Lambda and rho are external inputs. The axioms, however, carry the load: the incorrect Ricci variation formula (Eq. 6) is framed as a standard identity but is not one; the vanishing trace variation for dust (Eq. 14) is an ad hoc simplification; the gauge-invariance of psi (Section 2.3) is asserted against standard TT-gauge practice; and the Fresnel boundary conditions (Section 5.1) are assumed with the paper itself noting an alternative choice predicts no net reflection.

assumptions (5)
  • standard math Einstein field equations in the form R_{sigma nu} = kappa(T_{sigma nu} - (1/2) g_{sigma nu} T) + g_{sigma nu} Lambda (Eq. 1)
    Starting point of the derivation; standard general relativity.
  • domain assumption Metric perturbation delta g_{sigma nu} is small and treated to first order in the expansion around the background metric
    Section 2.1 defines the perturbative framework used throughout the paper.
  • ad hoc to paper The variation of the stress-energy trace for dust is zero, delta T = 0, because the equation of state does not depend on the metric (Eq. 14)
    This neglects density perturbations delta rho, which are nonzero in standard perturbation theory; the trace variation is actually -c^2 delta rho.
  • ad hoc to paper The trace psi = g^{sigma nu} delta g_{sigma nu} is a coordinate-invariant scalar that cannot be gauged away (Section 2.3)
    Asserts a property that conflicts with the transverse-traceless gauge practice for linearized gravitational waves; the monopole and dipole perturbation modes are gauge artifacts in standard GR.
  • ad hoc to paper Reflection at material interfaces obeys Fresnel-like boundary conditions with R = |n1 - n2| / (n1 + n2) (Eq. 55)
    Section 5.1 assumes these boundary conditions, while the paper itself acknowledges that an alternative quantum-mechanical gluing prescription predicts zero net reflection for an even number of interfaces.
invented entities (1)
  • Massive graviton with density-dependent mass m_g = (hbar/c) sqrt(Lambda - (1/2) kappa T)
    purpose: Account for dark matter as an apparent effect and provide a minimal gravitational wave frequency
    The mass is defined through the already-known cosmological constant and matter density; no new falsifiable signature is derived beyond existing Yukawa/massive-gravity proposals cited in Refs. 47-49.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gravitational Waves beyond the Linear Approximation and Gravitational Wave Reflection." pith.science (2026). https://pith.science/paper/ICAGY5P7

@misc{pith2026250201687,
  author       = {Pith},
  title        = {Pith review of: Gravitational Waves beyond the Linear Approximation and Gravitational Wave Reflection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICAGY5P7}},
  note         = {Machine review of arXiv:2502.01687}
}
read the original abstract

We derive a relativistic field equation for the trace of the metric perturbation beyond the weak field approximation to the Einstein field equations. The dynamics is governed by a massive Klein-Gordon equation on curved space-time, where the effective mass of the field is associated with the material and the dark energy content via the cosmological term. We solve the equation in the case of a Schwarzschild black hole and show that it can be cast into an effective Schr\"odinger form with an effective geometric potential which binds the zero angular momentum states. The non-zero angular momentum states experience a positive potential peak before the event horizon pointing to gravitational waves scattering. Black holes scatter gravitational waves and thus we provide an unambiguous testable prediction of black hole existence. The Newtonian limit for this equation points to the possibility of reflecting gravitational waves at interfaces with sharp density boundary, thus opening up gravitational wave propulsion physics. We discuss this type of propulsion in the light of Newton's third law of Mechanics. Compelling questions such as the existence of quanta of this field which may account for the dark matter content are also addressed.

Figures

Figures reproduced from arXiv: 2502.01687 by the authors.

Figure 1
Figure 1. The effective potential for the radial part of the equation for gravitational waves on Schwarzschild metric. The blue line corresponds to l = 0 orbital momentum and is strictly negative (bound states). The red line corresponds to l = 1 orbital momentum. The positive energy peak for l ≥ 1 states right after the event horizon at x = 1 points to gravitational waves coming in towards the singularity at x = 0 from infini… view at source ↗
Figure 2
Figure 2. The solution to the radial part (37) as dictated by the ansatz (39) in the exterior region x ≥ 1. The blue line is the real part and the red line is the imaginary part. Note the typical oscillatory behavior for large x. Such a behavior is expected due to Vef f (41). Here r 2 sµ 2 = 0.0001, r 2 sω 2/c2 = 0.5 p −2r 2 sµ2 + 2l 2 + 2l + 2 and l = 1. of the energy spectrum. This remains a research topic in progress [PIT… view at source ↗
Figure 3
Figure 3. The solution to the radial part (37) as dictated by the ansatz (39) in the interior of the black hole. The blue line is the real part and the red line is the imaginary part. Since inside the black hole the signature of the metric changes, we are no longer certain of the interpretation of (37) as the radial part of the equation of motion. Here the parameters are the same as those in [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The propulsion scheme proposed here consists of a GWE, i.e. a quadrupole mass oscillating device hence the opposing arrow directions, placed in the center of a hemisphere shaped gravitational wave deflector made of layered materials with large density differential such…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

73 extracted references · 65 canonical work pages

  1. [1]

    Albert Einstein, Jahrbuch der Radioaktivit¨ at,4 411 (1907)

    work page 1907

  2. [2]

    Document 31: Ideas and Methods. II. The theory of General Relativity,

    Albert Einstein, “Document 31: Ideas and Methods. II. The theory of General Relativity,” in Michel Janssen et al. (eds.), The Collected Papers of Albert Einstein. Volume 7: The Berlin Years: Writings, 1918–1921 (Princeton University Press, 2002)

    work page 1918

  3. [3]

    Albert Einstein, Feldgleichungen der Gravitation , Preussische Akademie der Wissenschaften, Sitzungsberichte, 844–847 (1915)

    work page 1915

  4. [4]

    Everitt et al., The Gravity Probe B test of General Relativity , Classical and Quantum Gravity 32, 224001 (2015)

    C.W.F. Everitt et al., The Gravity Probe B test of General Relativity , Classical and Quantum Gravity 32, 224001 (2015)

    work page 2015

  5. [5]

    B. P. Abbott et al., Phys. Rev. Lett. 116 (2016) 061102

    work page 2016

  6. [6]

    Misner, Kip S

    Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, Gravitation (Princeton Univ. Press, 2017)

    work page 2017

  7. [7]

    Sean Carroll, Spacetime and Geometry: An Introduction to General Relativity , Addison-Wesley (2003)

    work page 2003

  8. [8]

    Goblin Jr., James B

    John T. Goblin Jr., James B. Mertens, Glenn D. Starkman, and Chi Tian, Limited accuracy of linearized gravity., Phys. Rev. D 99, 023527 (2019)

    work page 2019

Show all 73 references
  1. [9]

    E. T. Newman and R. Penrose. An Approach to Gravitational Radiation by a Method of Spin Coefficients. J. Math. Phys, 3:566, 1962

    1962

  2. [10]

    Teukolsky, Perturbations of a Rotating Black Hole

    S. Teukolsky, Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations, Astrophysical Journal, Vol. 185, pp. 635-648 (1973)

    1973

  3. [11]

    Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton

    A. Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rend. Circ. Matem. Palermo 43, 203 (1919)

    1919

  4. [12]

    Atanasov and A

    V. Atanasov and A. Saxena, Class. Quantum Grav. 41 075015, (2024)

    2024

  5. [13]

    D. J. Rowan and G. Stephenson, Solutions of the time-dependent Klein-Gordon equation in a Schwarzschild background space., J. Phys. A: Math. Gen. 9, 1631 (1976). Grav. Waves beyond the Linear Approximation and Grav. Wave Reflection 23

    1976

  6. [14]

    D. J. Rowan and G. Stephenson, The Klein-Gordon equation in a Kerr-Newman background space., J. Phys. A: Math. Gen. 10, 15 (1977)

    1977

  7. [15]

    R. D. Lehn, S. S. Chabysheva, and J. R. Hiller, Klein–Gordon equation in curved space-time., Eur. J. Phys., 39 045405 (2018)

    2018

  8. [16]

    Elizalde, Exact solutions of the massive Klein-Gordon-Schwarzschild equation

    E. Elizalde, Exact solutions of the massive Klein-Gordon-Schwarzschild equation. , Phys. Rev. D 37, 2127 (1988)

    1988

  9. [17]

    Elizalde, Series solutions for the Klein-Gordon equation in Schwarzschild space-time

    E. Elizalde, Series solutions for the Klein-Gordon equation in Schwarzschild space-time. , Phys. Rev. D 36, 1269 (1987)

    1987

  10. [18]

    V. B. Bezerra, H. S. Vieira and A. A. Costa, The Klein–Gordon equation in the spacetime of a charged and rotating black hole , Class. Quantum Grav. 31, 045003 (2014)

    2014

  11. [19]

    Derezi´ nski and D

    J. Derezi´ nski and D. Siemssen, An evolution equation approach to the Klein-Gordon operator on curved spacetime., Pure and Applied Analysis, 1, 215 (2019). doi: 10.2140/paa.2019.1.215

    2019 doi

  12. [20]

    Englert, and R

    F. Englert, and R. Brout, Phys. Rev. Lett. 13, 321 (1964)

    1964

  13. [21]

    Higgs, Phys

    P.W. Higgs, Phys. Rev. Lett. 13, 508 (1964)

    1964

  14. [22]

    G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964)

    1964

  15. [23]

    Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC , Phys

    S. Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC , Phys. Lett. B, 716, 30-61 (2012)

    2012

  16. [24]

    Aad et al., Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC , Phys

    G. Aad et al., Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC , Phys. Lett. B, 716, 1-29 (2012)

    2012

  17. [25]

    Bargmann and E

    V. Bargmann and E. P. Wigner, Group Theoretical Discussion of Relativistic Wave Equations , PNAS 34, 211 (1948)

    1948

  18. [26]

    Weinberg and E

    S. Weinberg and E. Witten, Limits on massless particles , Phys. Lett. B, 96, 59 (1980). doi: 10.1016/0370-2693(80)90212-9

    1980 doi

  19. [27]

    A. H. Chamseddine and V. Mukhanov, JHEP 8, 011 (2010)

    2010

  20. [28]

    Ellithi, Modified gravity/entropic gravity correspondence due to graviton mass, Annals of Physics, 169717 (2024)

    Kimet Jusufi, Ahmed Farag Ali, Abdelrahman Yasser, Nader Inan, A.Y. Ellithi, Modified gravity/entropic gravity correspondence due to graviton mass, Annals of Physics, 169717 (2024). doi.org/10.1016/j.aop.2024.169717

    2024

  21. [29]

    Capozziello, A

    S. Capozziello, A. Stabile, and A. Troisi, Phys. Rev. D 76, 104019 (2007), arXiv:0708.0723 [gr-qc]

    2007 arXiv

  22. [30]

    Capozziello, G

    S. Capozziello, G. Lambiase, M. Sakellariadou, and A. Stabile, Phys. Rev. D 91, 044012 (2015)

    2015

  23. [31]

    De Martino and M

    I. De Martino and M. De Laurentis, Phys. Lett. B 770, 440 (2017)

    2017

  24. [32]

    Benisty and S

    D. Benisty and S. Capozziello, Phys. Dark Univ. 39, 101175 (2023)

    2023

  25. [33]

    Fierz and W

    M. Fierz and W. Pauli, On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field, Proc. Roy. Soc. Lond. A 173, 211 (1939)

    1939

  26. [34]

    Massive and mass- less Yang-Mills and gravitational fields,

    H. van Dam and M. J. G. Veltman, “Massive and mass- less Yang-Mills and gravitational fields,” Nucl. Phys. B 22, 397 (1970); V. I. Zakharov, “Linearized gravita- tion theory and the graviton mass,” JETP Lett. 12, 312 (1970) [Pisma Zh. Eksp. Teor. Fiz. 12, 447 (1970)]

    1970

  27. [35]

    To the problem of nonvanishing grav- itation mass,

    A. I. Vainshtein, “To the problem of nonvanishing grav- itation mass,” Phys. Lett. B 39, 393 (1972)

    1972

  28. [36]

    Can gravitation have a finite range?

    D. G. Boulware and S. Deser, “Can gravitation have a finite range?”, Phys. Rev. D 6, 3368 (1972)

    1972

  29. [37]

    de Rham, G

    C. de Rham, G. Gabadadze and A. J. Tolley, Phys.Lett. B 711, 190 (2012)

    2012

  30. [38]

    Bimetric Gravity from Ghost-free Massive Gravity,

    S. F. Hassan and R. A. Rosen, “Bimetric Gravity from Ghost-free Massive Gravity,” JHEP 2, 126 (2012)

    2012

  31. [39]

    Theoretical Aspects of Massive Gravity

    K. Hinterbichler, “Theoretical Aspects of Massive Gravity”, Rev. Mod. Phys. 84, 671 (2012)

    2012

  32. [40]

    Jusufi, E

    K. Jusufi, E. Gonz´ alez, and G. Leon, (2024), arXiv:2402.02512 [astro-ph.CO]

    2024 arXiv

  33. [41]

    De Felice, S

    A. De Felice, S. Mukohyama, and J.-P. Uzan, Gen. Rel. Grav. 50, 21 (2018)

    2018

  34. [42]

    De Felice, S

    A. De Felice, S. Mukohyama, M. Oliosi, and Y. Watanabe, Phys. Rev. D 97, 024050 (2018)

    2018

  35. [43]

    Aoki and S

    K. Aoki and S. Mukohyama, Phys. Rev. D 96, 104039 (2017)

    2017

  36. [44]

    Agazie et al

    G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 951, L8 (2023)

    2023

  37. [45]

    Wu, Z.-C

    Y.-M. Wu, Z.-C. Chen, Y.-C. Bi, and Q.-G. Huang, (2023), arXiv:2310.07469 [astro-ph.CO]

    2023 arXiv

  38. [46]

    Hellings and G.S

    R.W. Hellings and G.S. Downs, Upper limits on the isotropic gravitational radiation background from pulsar timing analysis The Astrophysical Journal. 265, L39 (1983). Grav. Waves beyond the Linear Approximation and Grav. Wave Reflection 24

    1983

  39. [47]

    Dark Universe phenomenology from Yukawa potential?,

    K. Jusufi, G. Leon, and A. D. Millano, “Dark Universe phenomenology from Yukawa potential?,” Phys. Dark Univ. 42, 101318 (2023)

    2023

  40. [48]

    Observational constraints on Yukawa cosmology and con- nection with black hole shadows,

    E. Gonz´ alez, K. Jusufi, G. Leon and E. N. Saridakis, “Observational constraints on Yukawa cosmology and con- nection with black hole shadows,” Phys. Dark Univ. 42, 101304 (2023)

    2023

  41. [49]

    Testing Yukawa cosmology at the Milky Way and M31 galactic scales,

    R. D’Agostino, K. Jusufi and S. Capozziello, “Testing Yukawa cosmology at the Milky Way and M31 galactic scales,” Eur. Phys. J. C 84, 386 (2024)

    2024

  42. [50]

    P. J. E. Peebles, Astrophys. J. 277, 470 (1984)

    1984

  43. [51]

    Trimble, Ann

    V. Trimble, Ann. Rev. Astron. Astrophys. 25, 425 (1987)

    1987

  44. [52]

    Weinberg, Rev

    S. Weinberg, Rev. Mod. Phys. 61, 1 (1989)

    1989

  45. [53]

    Zlatev, L.-M

    I. Zlatev, L.-M. Wang, and P. J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999)

    1999

  46. [54]

    Padmanabhan, Phys

    T. Padmanabhan, Phys. Rept. 380, 235 (2003)

    2003

  47. [55]

    Aghanim et al

    N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron. Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    2020 arXiv

  48. [56]

    D. G. Boulware and S. Deser, Can gravitation have a finite range? , Phys. Rev. D 6, 3368 (1972)

    1972

  49. [57]

    de Rham, G

    C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Lett. B 711, 190 (2012)

    2012

  50. [58]

    S. F. Hassan and R. A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity, JHEP 2 (2012), 126

    2012

  51. [59]

    Hinterbichler, Theoretical Aspects of Massive Gravity , Rev

    K. Hinterbichler, Theoretical Aspects of Massive Gravity , Rev. Mod. Phys. 84, 671 (2012)

    2012

  52. [60]

    Rickles, and C´ ecile M

    D. Rickles, and C´ ecile M. DeWitt, (2011). An Expanded Version of the Remarks by R.P. Feynman on the Reality of Gravitational Waves. In: The Role of Gravitation in Physics: Report from the 1957 Chapel Hill Conference. doi: 10.34663/9783945561294-34

    2011 doi

  53. [61]

    Weinberg, Gravitation and Cosmology , Wiley, New York, (1972)

    S. Weinberg, Gravitation and Cosmology , Wiley, New York, (1972)

    1972

  54. [62]

    Minter, K

    S.J. Minter, K. Wegter-McNelly, and R.Y. Chiao,Do mirrors for gravitational waves exist? Physica E 42, 234 (2010). doi:10.1016/j.physe.2009.06.056

    2010 doi

  55. [63]

    Inan, J.J

    N.A. Inan, J.J. Thompson, and R.Y. Chiao, Interaction of gravitational waves with superconductors. Fortschritte Der Physik, 65, 1600066 (2016). doi:10.1002/prop.201600066

    2016 doi

  56. [64]

    Event Horizon Telescope Collaboration et al., ApJL 930 L12 (2022)

    2022

  57. [65]

    Abramowitz and I.A

    M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables., Dover Publications (1964)

    1964

  58. [66]

    Arfken and H.J

    G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists. Elsevier Academic Press (2005)

    2005

  59. [67]

    Motte, and N

    Isaac Newton, A. Motte, and N. W. Chittenden, Newton ’s Principia. The mathematical principles of natural philosophy , New-York, D. Adee, (1848)

  60. [68]

    Die Grundlagen der Physik

    David Hilbert, “Die Grundlagen der Physik” [Foundations of Physics], Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen – Mathematisch-Physikalische Klasse (in German), 3, 395 (1915)

    1915

  61. [69]

    Feynman, Feynman Lectures on Gravitation

    Richard P. Feynman, Feynman Lectures on Gravitation. Addison-Wesley (1995)

    1995

  62. [70]

    Introduction to solid state physics (8th ed.) ch.22

    Kittel, Charles (2005). Introduction to solid state physics (8th ed.) ch.22. New York: Wiley. ISBN 978-0-471-68057-4

    2005

  63. [71]

    Shabalin, I.L. (2014). Iridium. In: Ultra-High Temperature Materials I. Springer, Dordrecht

    2014

  64. [72]

    Atanasov and A

    V. Atanasov and A. Saxena, Mass Oscillations in Superconducting Junctions for Gravitational Wave Emission and Detection , Annals of Physics (in press)

  65. [73]

    Atanasov, Gravitational wave emission from quadrupole Josephson junction device , Phys

    V. Atanasov, Gravitational wave emission from quadrupole Josephson junction device , Phys. Lett. A 384, 126042 (2020)

    2020

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.