Flyby Anomaly in the Variation Principle of General Relativity

Shubhen Biswas

REVIEW 6 major objections 4 minor 14 references

Flyby Anomaly in the Variation Principle of General Relativity

T0 review · 6 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read General relativity alone predicts the flyby velocity shift

desk verdict The flyby anomaly derivation rests on a wrong diagonalization; the linear-A term that produces Anderson's formula is an algebraic artifact. read the letter →

arxiv 2411.12053 v1 pith:IL7SNAQY submitted 2024-11-18 gr-qc

Shubhen Biswas This is my paper · ORCID

classification gr-qc

keywords flybyanomalymetricdiagonalizationLense-Thirringgeneralrelativityactionprincipleframedragginggravityassistempiricalformula

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 tries to show that the flyby anomaly — the small, unexplained change in a spacecraft's asymptotic speed during an Earth gravity assist — follows from general relativity alone, with no new physics. It does this by diagonalizing the weak-field metric of a rotating Earth, deriving a geodesic energy balance from Hamilton's action principle, and obtaining a formula in which the fractional velocity change is $2.4\,(R_\oplus/(R_\oplus+h))\,(\omega_\oplus R_\oplus/c)\,[\cos\delta_i-\cos\delta_o]$. The author argues that this reproduces the empirically fitted anomaly formula and its perigee-altitude dependence. If correct, the anomaly would be an ordinary frame-dragging effect of the rotating Earth.

What carries the argument

The mechanism is the diagonalization of the rotating-source metric matrix with respect to the Minkowski metric, $\det(g_{\mu\nu}-\lambda\eta_{\mu\nu})=0$, giving eigenvalues $\lambda_0=-1+b+A$, $\lambda_1=1+b+A$, and $\lambda_{2,3}=1+b$. Because the spatial eigenvalues are permutable, three equivalent diagonal forms (Eqs. 27–29) are taken as simultaneous, and their Hamilton-action variations (Eqs. 47–49) are added. That sum produces the factor 3 that, after using the virial relation and the Earth's angular momentum, becomes the prefactor 2.4.

What would settle it

Compute the exact eigenvalues of the matrix in Eq. (16) by solving the quartic characteristic equation. If the roots are $1\pm\sqrt{b^2-A^2}$ for the time/spatial block rather than $-1+b+A$ and $1+b+A$, then the determinant is not invariant under the permutation, the sum in Eq. (50) loses its factor 3, and the prefactor 2.4 in Eq. (60) is replaced by a different number. That calculation settles whether the paper's central formula follows from the stated metric.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the Lense–Thirring metric, after diagonalization, yields a variational equation along the hyperbolic flyby whose solution is Eq. (60), a parameter-free expression for $\Delta V_\infty/V_\infty$ with the observed sign, magnitude, and altitude dependence. The factor 2.4 arises from adding the action variations of three 'simultaneous' diagonal forms of the same metric, combined with the Earth's angular momentum $j=2\omega_\oplus R_\oplus^2/5$ and the virial relation $2v^2/c^2=b$. The paper claims this removes the need for non-standard physics.

Load-bearing premise

The whole factor 3 rests on the assumption that the three permuted diagonal forms of the rotating-source metric are simultaneous valid representations whose variational equations can be added.

Editorial extensions

If this is right

If Eq. (60) is right, the flyby anomaly is a genuine general-relativistic frame-dragging effect, and no exotic mechanism is needed.
The formula predicts that the fractional velocity anomaly scales as $(R_\oplus/(R_\oplus+h))$, so higher perigee passes should show smaller anomalies.
The anomaly depends only on the declinations of the incoming and outgoing asymptotes, matching the angular dependence of the empirical formula.
Existing flyby data for the Galileo, NEAR, and Rosetta spacecraft should fall on the predicted curve once perigee altitudes are taken into account.

Reading between the lines

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

The derivation's central move — summing the variations of three permuted diagonal representations — is not a standard way to handle a metric; the factor 3 that yields 2.4 comes entirely from that summation, so the prefactor is sensitive to whether those permutations are truly simultaneous.
A direct symbolic computation of the characteristic equation of Eq. (16) would give the exact eigenvalues; if they differ from Eqs. (24)–(26), the determinant is not invariant and the 2.4 would change, giving a quick falsification test.
If the formula is taken seriously as a prediction, a dedicated Earth flyby with radio tracking could measure $\Delta V_\infty/V_\infty$ at two different perigee altitudes to isolate the $R_\oplus/(R_\oplus+h)$ factor from the declination term.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Thanks for the Pith report. I read the paper and agree with your verdict. The claim is that GR alone explains the flyby anomaly via a variational principle with a diagonalized Lense-Thirring metric. The paper does have a clear structure and cites the right literature, including Acedo's 2021 altitude factor. That part is fine. But the derivation collapses at the diagonalization step, and it isn't a minor gap.

Let me be specific. The characteristic equation (17) is correct, and its roots are 1 ± sqrt(b^2 - A^2), which for A << b become 1 ± b plus a correction of order A^2. There is no term linear in A. Yet the paper asserts eigenvalues -1+b+A, 1+b+A in (24)-(25). Substituting λ0 = -1+b+A back into Eq (19) leaves a residual of order 4-4b-4A, so it isn't an eigenvalue. The author seems to have misread the expansion in Eq (20) and dropped a square root bracket in a way that produces a spurious linear term.

This matters because the entire velocity anomaly is proportional to that linear A. The three permuted diagonal metrics in (27)-(29) are not different representations of the same metric; their determinants differ from the original det[g_LT] = -(1+b)^2(1-b^2+A^2) unless A=0. So adding the three 'simultaneous' variational equations is not an averaging over equivalent coordinate systems; it's just multiplying a nonexistent term by 3. The factor 2.4 in Eq (60) is exactly that artifact. The later steps are also wrong in spirit: the circular-orbit virial relation 2v^2/c^2 = b is used for a hyperbolic flyby, and the term 2(1+b)v^2 is discarded as 'unperturbed' even though v changes along the trajectory. So the final formula is fitted to Anderson's constant, not derived.

What is actually new? Nothing valid. The stated result is Acedo's altitude factor times Anderson's empirical constant, which the author already cites. The paper is not a genuine resolution of the anomaly.

The paper does show some effort: it surveys the flyby anomaly literature and tries to ground the problem in GR. But the internal contradictions are load-bearing, and a knowledgeable referee would see them quickly. I would not send this to peer review; the desk editor should reject. If a colleague wants a cautionary example of incorrect metric diagonalization, it could serve, but that's not a reason to publish.

So: not citable, not for reading group, serious_thinker = no.

Referee Report

6 major / 4 minor

Summary. The paper claims to explain the Earth flyby anomaly within pure general relativity. It proposes to diagonalize the weak-field Lense-Thirring metric, obtains three 'simultaneous' diagonal forms, adds their Hamilton variational equations, and derives Eq. (60), a formula of Anderson's empirical type with coefficient 2.4. The paper concludes that pure GR reproduces Anderson's empirical formula without new physics. The central derivation, however, is invalid at the diagonalization step, and later steps contain further algebraic and physical inconsistencies.

Significance. If the derivation were correct, the result would be remarkable: the flyby anomaly would follow from standard general relativity, producing a falsifiable formula with an explicit altitude dependence. The paper does offer a concrete, testable prediction in Eq. (60), which is a genuine strength. However, the paper provides no machine-checked proofs or reproducible code, and the central algebra is hand-derived and internally inconsistent. The claimed agreement with Anderson's formula is not a test of the theory because the prefactor 2.4 and the average altitude 1152 km are produced by invalid algebraic steps and post hoc averaging rather than by a parameter-free derivation.

major comments (6)

[§2, Eqs. (17)–(26)] The eigenvalues asserted in Eqs. (24)–(26) are not roots of Eq. (17). The first factor of the characteristic equation is λ² − 2λ + 1 − b² + A² = 0, whose exact roots are λ = 1 ± √(b² − A²); for A ≪ b these are 1 ± b + O(A²/b), with no term linear in A. Substitution of λ0 = −1 + b + A into Eq. (19) leaves the residual 4 − 4b − 4A + 2bA + 2A², and substitution of λ1 = 1 + b + A leaves 2A(b + A); neither is zero. Equation (20) is algebraically equal to 1 ± √(b² − A²), so the subsequent neglect of the square bracket is what generates the spurious linear-in-A terms in the diagonal metrics (27)–(29).
[§2, Eqs. (27)–(30)] The claimed determinant invariance is false. From Eq. (7), det[g_LT] = −(1+b)²(1−b²+A²), whereas the determinant of the first diagonal form, Eq. (27), is ((b+A)²−1)(1+b)². These are equal only when A = 0. The statement after Eq. (30) that the four-volume element √|g| is invariant for each of the three representations is therefore incorrect, and the three forms are not equivalent to the original Lense-Thirring metric.
[§4, Eqs. (47)–(49)] The three 'simultaneous' diagonal forms are not simultaneous representations of the same metric. A given metric has a single eigenbasis up to the degeneracy of the 1+b eigenvalues, and the correct eigenvalue problem produces no order-A spatial eigenvalue at all. Adding Eqs. (47)–(49) manufactures the factor 3 that later becomes the coefficient 2.4 in Eq. (60); this factor has no algebraic justification once the eigenvalues in Eqs. (24)–(26) are corrected.
[§4, Eqs. (56) and (58)–(59)] The derivation is internally inconsistent. Equation (56) states Δv = (ΔθA)v, hence Δv/v = ΔθA. Equation (58) states Δv/v = −6ΔθA/(2v²/c²). Equating the two requires 3c²/v² = −1 unless ΔθA = 0. No such condition holds for a physical flyby, so Eqs. (56) and (58) cannot both be correct; the later substitution for v does not remove this contradiction.
[§4, Eq. (59)] The step 'In regard of virial potential, 2v²/c² = 2GM/c²r = b' applies the circular-orbit relation v² = GM/r to a hyperbolic flyby. For a hyperbolic trajectory the energy relation is different, so this substitution is not justified for the osculating hyperbola. Together with the invalid factor 3, this step determines the numerical prefactor 2.4 in Eq. (60).
[§4, Eqs. (44)–(46)] The variation is taken between two points at the same radius r but different polar angles θ_i and θ_o, whereas Anderson's ΔV∞/V∞ refers to the change in asymptotic velocity at infinity. At the actual asymptotes the perturbations b and A vanish, so it is not justified to identify the finite-radius variational result with the asymptotic velocity change appearing in Eq. (60).

minor comments (4)

[§2, Eq. (6)] There is a typo in the coordinate definitions: the text writes 'x^1 = z' after defining x^1 = x; the third coordinate should be x^3 = z. The summation notation in the definition of dX² is also garbled.
[§1 and Abstract] The English contains several errors that impede readability, for example 'attracts enough attention as a problem of General Relativity' and 'shows energy anomaly over the asymptotic in and out velocity obeying Anderson's empirical formula'.
[References] The reference list mixes journal citations and bare arXiv identifiers, and reference [20] contains a stray equation number. A uniform reference style would improve the manuscript.
[§4, Eq. (60)] The agreement with Anderson's empirical formula is demonstrated only at an average altitude of 1152 km chosen from three missions; no per-mission comparison with uncertainties is presented. A table comparing predicted and observed ΔV∞ for each flyby would be needed to support the claimed agreement.

Circularity Check

2 steps flagged · score 6.0 of 10

The angular structure is derived from the LT metric, but the pre-factor 2.4 is effectively imposed by an arbitrary factor-3 summation and by an in-sample average altitude taken from the same flyby data used to define Anderson's K.

other [Eqs. (27)-(29) and Eq. (50)]
"For each of the different arrangements we have simultaneous solutions. ... Adding Eqns. (47), (48) and (49) ... 3∆θ[(1 − b − A(r,θ))c²] = ∆θ[(1 + b + A(r,θ))v²] + ∆θ[2(1 + b)v²]."

A single metric has a single set of eigenvalues and one eigenbasis; permuting the diagonal entries does not create three simultaneous physical solutions. The paper nevertheless sums the three variational equations, inserting the factor 3 into Eq. (50), and then drops the Δ[2(1+b)v²] term by declaring it 'unperturbed'. That factor 3 is the sole origin of the coefficient 6 and hence of the 2.4 in Eq. (60). The predicted numerical constant is therefore fixed by an arbitrary summation/dropping rule chosen because it isolates the A-dependent term, not by the metric or the variational principle.
fitted input called prediction [After Eq. (60), final comparison paragraph]
"If we are little crazy further about some spacecrafts having enough anomaly like Galileo-I (h~960km), NEAR(h~539km) and Rosetta-I(h~1956km)[2,5,24,25], average altitude ⟨h⟩~1152km leads 2.4(R⊕/(R⊕+h))~2.03 turns Eq.(60) as good as expected from Anderson's [3] empirical formula in Eq.(1) without any discrepancy just using pure General Relativistic theory."

The only point at which the flyby data enter is here: the perigee altitudes of the anomalous spacecraft are averaged, and this data-derived average h is substituted into Eq. (60) to make the coefficient 2.4R/(R+h) equal to about 2.03, matching Anderson's K. The same flyby data set was used to establish Anderson's empirical formula, so the agreement is an in-sample calibration rather than an independent prediction. No out-of-sample flyby is tested and no uncertainty is propagated, so the claimed match 'without any discrepancy' is constructed by the choice of the comparison sample and the averaging procedure.

full rationale

The circularity is partial rather than total. The angular factor [cosδ_i - cosδ_o] does follow from the Lense-Thirring metric's sinθ dependence and the coordinate relation between polar angle and declination, so the functional form is not merely a renaming of Anderson's formula. The algebraic invalidities noted in the reader's attack (wrong eigenvalues, determinant inconsistency, dropped velocity-variation term) are genuine correctness defects, but by themselves they are not circularity. What makes the central numerical claim circular is the pre-factor: the factor 3 from summing three fictitious 'simultaneous' diagonal forms, together with the post-hoc average of perigee altitudes from the same anomalous flybys, is what turns the derived expression into a match with the empirically fitted K. The self-citation [23] for the general metric-representation ansatz is minor and not load-bearing for the final 2.4 coefficient. The score is therefore 6: the claimed prediction of the flyby anomaly amplitude reduces, at the decisive step, to inputs selected from the data it claims to explain.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The derivation relies on several false or unjustified assumptions: incorrect eigenvalues for the LT metric, an unjustified sum over three permuted metrics, a dropped term that should not vanish, a circular-orbit relation applied to a hyperbolic flyby, and a conflation of perigee and asymptotic velocities. No new entities are introduced.

free parameters (2)

Effective prefactor 2.4 = 2.4 (from 6 × 2/5)
The 6 arises from adding three permuted diagonal metrics and dropping a term; the 2/5 is the solid-sphere moment of inertia. The product is set to reproduce Anderson's K; no independent derivation.
Average flyby altitude h = 1152 km (average of Galileo-I, NEAR, Rosetta-I)
Selected post hoc so that 2.4 R/(R+h) ≈ 2.03, matching Anderson's constant. Different altitude choices change the prediction.

assumptions (7)

domain assumption Lense-Thirring metric describes Earth's exterior spacetime
Used as starting point in Eq (5); assumes weak-field slowly rotating source approximation is valid for flybys.
ad hoc to paper The eigenvalues of g relative to η are given by Eqs (24)-(26)
This is asserted in diagonalization section; it is algebraically false, as the roots of Eq (17) are 1±√(b²-A²).
ad hoc to paper The three permuted diagonal metrics (Eqs 27-29) are simultaneous valid representations and their variational equations can be added
Used to derive Eqs (47)-(50); no physical justification for summing distinct coordinate permutations.
ad hoc to paper The unperturbed term Δθ[2(1+b)v²] vanishes
Dropped in Eq (50) to get Eq (51); inconsistent with allowing Δv≠0.
domain assumption Circular orbit virial relation 2v²/c² = 2GM/(c²r)
Applied to hyperbolic flyby in Eq (58); not valid for open trajectories.
domain assumption Asymptotic velocity V∞ can be replaced by perigee velocity v
Anderson's formula uses asymptotic velocities, but computation uses perigee r=R+h and v.
domain assumption Polar angle θ maps to declination δ as sinθ = cosδ
Used in final step to convert Eq (59) to Eq (60).

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Cite this review

Pith. "Pith review of Flyby Anomaly in the Variation Principle of General Relativity." pith.science (2026). https://pith.science/paper/IL7SNAQY

@misc{pith2026241112053,
  author       = {Pith},
  title        = {Pith review of: Flyby Anomaly in the Variation Principle of General Relativity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IL7SNAQY}},
  note         = {Machine review of arXiv:2411.12053}
}

read the original abstract

The anomalous velocity deviation in the osculating planetary flyby attracts enough attention as a problem of General Relativity. In connection of rotating weak field massive source the Lense Thirring metric is diagonalized to find the equation of motion from action invariance Hamilton principle in pure relativistic theory. The computation for near Earth flyby shows energy anomaly over the asymptotic in and out velocity obeying Anderson's empirical formula.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

[1]

Introduction: The Voyager missions 1970 and1980 by the Jet Propulsion Laboratory (JPL) became the central attractions for gravity assist planetary flyby! Investigations by Antreasian, P.G. and J.R. Guinn, 1998, presented the unexpected delta- 𝑣 increases during the Earth gravity assists flybys, Galileo and NEAR [1] . Extensive work done over gravity assis...

work page 1970
[2]

dragging of inertial frames

The space time metric for rotating massive body: So far a solution for the Einsteins field equation brings beautifully the static isotropic metric as given by Schwarzschild [13, 14, 15] 𝑑𝑠2 𝑆𝑐ℎ. = − (1 − 2𝐺𝑀 𝑐2𝑟 ) 𝑐2𝑑𝑡2 + (1 − 2𝐺𝑀 𝑐2𝑟 ) −1 𝑑𝑟2 + 𝑟2𝑑𝜃2 + 𝑟2𝑠𝑖𝑛2𝜃𝑑𝜑2 (2) Eq. (2) can be described excluding higher order perturbation in the PPN in quasi Minkows...

work page arXiv 1998
[3]

D., Campbell, J

Anderson, J. D., Campbell, J. K., Ekelund, J. E., Ellis, J.& Jordan, J. F., (2008) Phys. Rev. Lett. 100 (2008) 091102

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[4]

A flyby anomaly for Juno? Not from standard physics

Iorio. L., 2018, International Journal of Modern Physics D, http://arxiv.org/abs/1311.4218v2 [5]Turyshev, S. G. & Toth, V. T., (2009) http://arxiv.org/abs/0907.4184v1 [6]Antreasian P.G. & Guinn, J.R., (1998) http://www2.aiaa.org/citations/mp-search.cfm

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Moffat, J. W.(2006) JCAP 2006, 004 (2006) https://arxiv.org/abs/gr-qc/0506021 [8]Adler, S. L., (2008) http://arxiv.org/abs/0805.2895v4

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Acedio, L., 2018, http://arxiv.org/abs/1505.06884v1 [10]Goenner H. F. M., (2004) On the History of Unified Field Theories, Living Rev. Relativity 7, (2004) [11]Hehl F. W. et al., (1976)General Relativity with Spin and Torsion: Foundations and Prospects, Rev. Mod. Phys. 48, 393-416 (1976). [12]Poplawski N. J., (2010) Cosmology with torsion: An alternative ...

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Misner, C. W. Thorne, K.S. &Wheeler, J. A. (1972) Gravitation, W.H. Freeman &Company, 1097, 322-323, [15]Weinberg, S., (1971), Gravitation and cosmology, John Wiley and sons, 180-181 [16]Teukolsky, S. A. (2015) Class. Quantum Grav. 32 124006 DOI 10.1088/0264- 9381/32/12/124006 [17]Visser, M.(2008) https://arxiv.org/pdf/0706.0622

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[18]

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[20]

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Tessarotto M. & Cremaschini, C., (2021) arXiv:1601.03941v1[gr-qc]2016 [23]Biswas, S., (2023), Physics of the Dark Universe, https://doi.org/10.1016/j.dark.2023.101403

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Introduction: The Voyager missions 1970 and1980 by the Jet Propulsion Laboratory (JPL) became the central attractions for gravity assist planetary flyby! Investigations by Antreasian, P.G. and J.R. Guinn, 1998, presented the unexpected delta- 𝑣 increases during the Earth gravity assists flybys, Galileo and NEAR [1] . Extensive work done over gravity assis...

work page 1970

[2] [2]

dragging of inertial frames

The space time metric for rotating massive body: So far a solution for the Einsteins field equation brings beautifully the static isotropic metric as given by Schwarzschild [13, 14, 15] 𝑑𝑠2 𝑆𝑐ℎ. = − (1 − 2𝐺𝑀 𝑐2𝑟 ) 𝑐2𝑑𝑡2 + (1 − 2𝐺𝑀 𝑐2𝑟 ) −1 𝑑𝑟2 + 𝑟2𝑑𝜃2 + 𝑟2𝑠𝑖𝑛2𝜃𝑑𝜑2 (2) Eq. (2) can be described excluding higher order perturbation in the PPN in quasi Minkows...

work page arXiv 1998

[3] [3]

D., Campbell, J

Anderson, J. D., Campbell, J. K., Ekelund, J. E., Ellis, J.& Jordan, J. F., (2008) Phys. Rev. Lett. 100 (2008) 091102

work page 2008

[4] [4]

A flyby anomaly for Juno? Not from standard physics

Iorio. L., 2018, International Journal of Modern Physics D, http://arxiv.org/abs/1311.4218v2 [5]Turyshev, S. G. & Toth, V. T., (2009) http://arxiv.org/abs/0907.4184v1 [6]Antreasian P.G. & Guinn, J.R., (1998) http://www2.aiaa.org/citations/mp-search.cfm

work page Pith review arXiv 2009

[5] [7]

W.(2006) JCAP 2006, 004 (2006) https://arxiv.org/abs/gr-qc/0506021 [8]Adler, S

Moffat, J. W.(2006) JCAP 2006, 004 (2006) https://arxiv.org/abs/gr-qc/0506021 [8]Adler, S. L., (2008) http://arxiv.org/abs/0805.2895v4

arXiv 2006

[6] [9]

Acedio, L., 2018, http://arxiv.org/abs/1505.06884v1 [10]Goenner H. F. M., (2004) On the History of Unified Field Theories, Living Rev. Relativity 7, (2004) [11]Hehl F. W. et al., (1976)General Relativity with Spin and Torsion: Foundations and Prospects, Rev. Mod. Phys. 48, 393-416 (1976). [12]Poplawski N. J., (2010) Cosmology with torsion: An alternative ...

work page Pith review arXiv 2004

[7] [13]

Blau, M., 2021, Lecture Notes on General Relativity, 514, 66-69, 247-249 http://www.blau.itp.unibe.ch/newlecturesGR.pdf

work page 2021

[8] [14]

Misner, C. W. Thorne, K.S. &Wheeler, J. A. (1972) Gravitation, W.H. Freeman &Company, 1097, 322-323, [15]Weinberg, S., (1971), Gravitation and cosmology, John Wiley and sons, 180-181 [16]Teukolsky, S. A. (2015) Class. Quantum Grav. 32 124006 DOI 10.1088/0264- 9381/32/12/124006 [17]Visser, M.(2008) https://arxiv.org/pdf/0706.0622

arXiv 1972