Gravitomagnetic London Moment and the Graviton Mass inside a Superconductor

Clovis Jacinto de Matos; Martin Tajmar

REVIEW

Gravitomagnetic London Moment and the Graviton Mass inside a Superconductor

Superconductivity · Submitted 24 Feb 2006 · 1 reference

Pith's verdict: The paper's graviton mass is a re-parameterization of the Tate anomaly through an unverified B_g conjecture; only the EM Proca reformulation is sound.

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arXiv IDcond-mat/0602591 (version 1)

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CitationClovis Jacinto de Matos, Martin Tajmar. "Gravitomagnetic London Moment and the Graviton Mass inside a Superconductor." arXiv:cond-mat/0602591 (2006). https://pith.science/paper/NYJUOKEB

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Editor's note

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Two things you should know. The first half of the paper, the electromagnetic derivation of the London moment from Proca equations, is a tidy reformulation that would be fine in a pedagogy review. The second half, the graviton mass i x 4.61 x 10^-55 kg, is not supported by the evidence in the paper. The number is a restatement of the Tate anomaly through the authors' own unverified conjecture, not a measurement or a derivation from first principles.

The good part: the authors are explicit about the conditional nature of their conclusion. They write 'If our conjecture is correct...' before Eq. (12), and they do not claim to have measured B_g. The EM section correctly shows that in a massive-photon treatment with photon wavelength equal to the London depth, you recover the London moment. That's a known result (Sternberg, Ryder), but it is cleanly presented. The comparison to Modanese's estimate at least gives the reader a sense of scale.

The soft spots: the load-bearing premise, B_g = -2 omega (Delta m*/m*), is imported from the authors' own earlier papers. This paper does not derive it, does not measure it, and does not link it to a falsifiable prediction. Eq. (12) then adjusts the graviton mass so that the Proca gravitomagnetic equation reproduces the same B_g. That is a one-parameter fit; if the anomaly were any other value, you would get a different m_g, and the 'explanation' would still look consistent. There is also the issue that the mass is imaginary. The authors cite Novello's de-Sitter result for complex graviton masses, but that is a high-level speculation, not a parameter-free constraint. Modanese's Pb value of i x 10^-62 kg is seven orders away from their Nb value, which they acknowledge; that gap does not constitute confirmation. The paper's own caveat undercuts the abstract's stronger phrasing.

I don't think this paper is a waste of time to read for someone in the gravity/superconductivity phenomenology niche. It is a serious attempt to give a Proca-based formalism to a speculative idea. But it does not rise to a testable claim. I would not cite it, and I would not bring it to a reading group as a substantive result. A serious journal can desk-reject this with a clear explanation; sending it to review would likely consume referee time without a testable kernel. If the authors can produce a prediction that distinguishes B_g from zero, that would be a different and much more interesting paper.

Referee report

Major objections (4)

[The Gravitomagnetic London Moment and the Graviton Mass, Eq. (7) and Eq. (12)] The load-bearing premise is the existence of the gravitomagnetic London field B_g = -2ω(Δm*/m*) in Eq. (7). This field is imported from refs. 13-14 and is not derived from the Proca equations of Eq. (9); no measurement or independent derivation is presented. Equation (12) then uses Tate's anomaly to fix m_g by equating the hypothesized B_g with the gravitomagnetic London solution of Eq. (11). The argument is therefore circular: the graviton mass is defined as whatever value makes the conjecture reproduce the anomaly, and the 'explanation' of the anomaly is the assumption itself.
[Discussion, comparison with Modanese's estimate] The claimed support from Modanese's estimate is not quantitative: his value for a Pb superconductor is i·10^-62 kg, which differs from the paper's Nb value of i·4.61×10^-55 kg by about seven orders of magnitude. The manuscript offers no mechanism for this large difference, so the comparison cannot be taken as corroboration of Eq. (12).
[Conclusion] The conclusion states that 'we can calculate the mass of the graviton needed to solve the anomaly', but the calculation is explicitly conditional on a conjecture that is not tested, as acknowledged in the sentence 'If our conjecture is correct' preceding Eq. (12). The manuscript establishes only that a suitable complex graviton mass could make the conjectured B_g consistent with Tate's data; it does not establish that such a graviton mass or such a B_g field actually exists.
[The Gravitomagnetic London Moment and the Graviton Mass, Eq. (10)] The complex gravitomagnetic London penetration depth is invoked to argue that gravitomagnetic fields penetrate the superconducting bulk with an oscillating signature, but no experimental signature, quantitative consequence, or practical detection strategy is derived. As presented, the mechanism is unfalsifiable within the scope of the paper.

Minor objections (4)

[Introduction] There are typographical errors, including 'Ochselfeld' for 'Ochsenfeld', 'at el' for 'et al.', and 'don not' for 'do not'.
[Discussion, Eq. (25)] The equation numbering is inconsistent: after Eq. (13) the text refers to a displayed equation numbered (25), with Eqs. (14)-(24) absent from the manuscript.
[References] Ref. 28 (Podkletnov and Modanese) is cited as supporting 'gravity waves produced by HV discharges', a claim that is highly controversial and not generally accepted; the text would benefit from a more cautious framing.
[Throughout] The notation alternates between 'Equ.' and 'Eq.', and the graviton mass is interchangeably called 'complex' and then used to infer a 'real part of this wavelength'; the exposition would be clearer with a consistent treatment of the complex nature of m_g.

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The paper

The paper claims

This paper claims that a massive graviton inside a rotating superconductor generates a gravitomagnetic London field $B_g=-2\omega(\Delta m^*/m^*)$ that accounts for the measured Cooper-pair mass anomaly, and uses the anomaly to calculate…

The reading

This paper argues that the same massive-photon (Proca) description that produces the magnetic London moment in a rotating superconductor has a gravitational counterpart: a massive graviton produces a gravitomagnetic London field. The authors use that field, $B_g=-2\omega(\Delta m^*/m^*)$, to explain a 1989 measurement in which the Cooper-pair mass in niobium came out heavier than the theoretical prediction. From that anomaly they calculate the graviton mass inside the superconductor as $m_g=i\,4.61\times10^{-55}$ kg, about fourteen orders of magnitude above the accepted free-space upper limit, and they note that a complex graviton mass is consistent with a de Sitter background. If the conjecture is right, a rotating superconductor becomes a laboratory-scale source of gravitomagnetic fields far stronger than classical Lense-Thirring estimates.

What carries the argument

The central machinery is the Proca equation for electromagnetism (Maxwell's equations with a photon mass term) and its gravitomagnetic analogue with a graviton mass term. Taking the curl of the fourth Proca equation yields a screened field equation; in the electromagnetic case the photon wavelength is identified with the London penetration depth, so the solution contains both the Meissner-Ochsenfeld expulsion and the London moment. The same step in the gravitational sector introduces a complex gravitomagnetic London penetration depth $\lambda_{Lg}$, so gravitomagnetic fields are not confined to a surface layer but penetrate the bulk with an oscillating signature. The graviton wavelength $\lambda_g=\hbar/(m_g c)$ is deliberately not set equal to $\lambda_{Lg}$, and this free ratio is what lets the gravitomagnetic London moment match $-2\omega(\Delta m^*/m^*)$, fixing the graviton mass at $m_g=i\,4.61\times10^{-55}$ kg.

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Proca-equation treatment of superconductivity, in which the photon acquires mass through symmetry breaking, works in parallel for gravitomagnetism and yields a gravitomagnetic London moment $B_g=-2\omega(\Delta m^*/m^*)$. Combined with conservation of the full canonical momentum, this field shifts the effective Cooper-pair mass inside a rotating niobium superconductor so that the measured value $m^*/2m_e=1.000084(21)$ is no longer an anomaly. Solving the gravitomagnetic Proca equation for the graviton mass that produces this field gives $m_g=i\,4.61\times10^{-55}$ kg and a corresponding wavelength $\lambda_g=7.63\times10^{11}$ m. The imaginary value is presented as consistent with a complex graviton mass in a de Sitter background, and the paper notes that the resulting graviton mass is about fourteen orders of magnitude larger than the free-space upper limit set by the cosmological constant, a gap it attributes to the same mechanism that makes the photon mass enormous inside a superconductor.

Load-bearing premise

The argument assumes that the conjectured gravitomagnetic London field $B_g=-2\omega(\Delta m^*/m^*)$ actually exists in a rotating superconductor; that field is imported from earlier work by the authors and is not independently derived or measured in this paper, so if it is absent the graviton-mass estimate loses its anchor.

Editorial extensions

If this is right

The 1989 niobium Cooper-pair mass measurement would be reinterpreted as evidence for a gravitomagnetic London moment, and the theoretical prediction for the Cooper-pair mass would need a large gravitational correction.
Rotating superconductors would produce gravitomagnetic fields about 31 orders of magnitude larger than the classical Lense-Thirring field of a rotating ring, making laboratory detection potentially feasible.
The graviton mass inside a superconductor would depend on material composition, connecting anomalies in London-moment slopes, such as the 25% deviation seen in one high-temperature superconductor, to the photon-to-graviton mass ratio.
If the cosmological constant inside a superconductor is set by the condensate, as the paper quotes for lead, then different superconductors should show different internal graviton masses, giving a testable material dependence.

Reading between the lines

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

Editorial inference: a rotating superconducting ring would be a tunable gravitomagnetic source, so a null search for gravitomagnetic fields near such rings at the level $B_g\simeq -2\omega(\Delta m^*/m^*)$ would directly test the conjecture.
Editorial inference: the complex penetration depth predicts an oscillatory spatial profile for gravitomagnetic fields inside the bulk, a concrete experimental fingerprint that differs from the simple exponential decay seen in the magnetic case.
Editorial inference: a dedicated run of the niobium experiment at several angular velocities would test the linear scaling of the Cooper-pair mass anomaly with $\omega$, which the paper's Eq. (7) predicts.

Original abstract

Using Proca electromagnetic and gravitoelectromagnetic equations the magnetic and gravitomagnetic properties of a rotating superconductor are respectively derived. Perfect diamagnetism, and the magnetic London moment are deduced from the photon mass in the superconductor. Similarly, it is shown that the conjecture proposed by the authors to resolve the cooper pair mass anomaly reported by Tate, can be explained by a graviton mass in the superconductor different with respect to its expected cosmological value.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,990 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.02 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.46 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Using Proca electromagnetic and gravitoelectromagnetic equations the magnetic and gravitomagnetic properties of a rotating superconductor are respectively derived. Perfect diamagnetism, and the magnetic London moment are deduced from the photon mass in the superconductor. Similarly, it is shown that the conjecture proposed by the authors to resolve the cooper pair mass anomaly reported by Tate, can be explained by a graviton mass in the superconductor different with respect to its expected cosmological value.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Using Proca electromagnetic and gravitoelectromagnetic equations the magnetic and gravitomagnetic properties of a rotating superconductor are respectively derived.	method 0.58	0.90	0.12	asserted 1.00
s1	Perfect diamagnetism, and the magnetic London moment are deduced from the photon mass in the superconductor.	result 0.98	0.86	0.21	asserted 1.00
s2	Similarly, it is shown that the conjecture proposed by the authors to resolve the cooper pair mass anomaly reported by Tate, can be explained by a graviton mass in the superconductor different with respect to its expected cosmological value.	result 0.57	0.70	0.54	asserted 0.92

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

How the literature uses it

2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Discussion

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[1] [1]

1Becker, R., Heller, G., and Sauter, F., "Über die Stromverteilung in einer Supraleitenden Kugel", Z. Physik, 85, 1933, pp. 772-787 2Rystephanick, R.G., "On the London Mom ent in Rotating Superconducting Cylinders", Can. J. Phys., 51, 1973, 789-794 3Hildebrandt, A.F., "Magnetic Field of a Rotating Superconductor", Physical Review Letters, 12(8), 1964, pp....

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