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Perihelion precession of planetary orbits solved from quantum field theory

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper derives the perihelion precession of planetary orbits from a quantum field theory that treats gravity as a gauge field in flat spacetime, without invoking the curved metric of general relativity.

desk verdict A load-bearing algebraic error inverts a key factor, and an unjustified substitution does the rest: the claimed derivation of the GR perihelion shift from unified gravity fails on its own equations. read the letter →

arxiv 2506.14447 v1 pith:NX7J6DON submitted 2025-06-17 gr-qc

Mikko Partanen , Jukka Tulkki This is my paper · ORCID
classification gr-qc MSC 83C1083C2570F1581Q05 PACS 04.20.-q03.65.Pm04.60.-m95.10.Ce
keywords perihelionprecessionunifiedgravityDiracequationFoldy-WouthuysentransformationBinetquantumgaugetheorygeneralrelativityclassicallimit
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

This paper claims that the perihelion precession of planetary orbits, a classic test of general relativity, can be derived from a quantum field theory in which gravity is a gauge field rather than spacetime curvature. Starting from a Dirac equation extended by a gravity gauge field, the authors diagonalize the Hamiltonian, take a classical wavepacket limit, and obtain an orbit equation. Through a specific change of variables, that equation reduces to the standard relativistic Binet equation, whose perturbative solution gives the famous perihelion shift $6\pi GM/[c^2 a(1-e^2)]$. If correct, this means a key general-relativistic prediction can emerge from quantum field theory without a curved metric or other geometric concepts.

What carries the argument

The central object is the Dirac equation of unified gravity, a theory in which gravity is carried by a gauge field $H^{\mu\nu}$ in flat Minkowski spacetime rather than by a curved metric. The carrying identity is Eq. (15), the change of variables $r(\phi)=1/u(\phi)-GM/c^2$, which shifts the radial variable by the gravitational radius and, when combined with a small-$GM/c^2$ expansion, transforms the orbit equation into the Binet form with the $3GMu^2/c^2$ correction. The other essential machinery is the Foldy-Wouthuysen transformation that diagonalizes the Dirac Hamiltonian, and the classical wavepacket of Eq. (12) that justifies replacing operators with expectation values.

What would settle it

Use the standard radial variable $u=1/r$ in Eq. (14) while keeping the Foldy-Wouthuysen Hamiltonian; if the resulting first-order equation is not $d^2u/d\phi^2+u=GM/J^2+3GMu^2/c^2$, the claimed result is an artifact of the substitution in Eq. (15).

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Extended reading notes

Core claim

The central claim is that the Dirac equation of unified gravity, applied to a localized electron wavepacket in the classical limit, yields orbital dynamics whose perihelion precession matches the general-relativistic result at lowest order. After the Foldy-Wouthuysen diagonalization and the substitution $r(\phi)=1/u(\phi)-GM/c^2$, the orbit equation becomes $d^2u/d\phi^2+u=GM/J^2+3GMu^2/c^2$, exactly the Binet equation with the general-relativistic correction term. Its perturbative solution gives the perihelion shift of Eq. (23), and the paper states that this was derived without a curved metric or any other concept of general relativity.

Load-bearing premise

The load-bearing premise is the substitution $r(\phi)=1/u(\phi)-GM/c^2$; if this shift is not physically justified, the expansion does not yield the general-relativistic Binet equation.

Editorial extensions

If this is right

  • If the derivation is sound, the perihelion advance is no longer a unique signature of curved spacetime; a gauge-field extension of the Standard Model reproduces it at lowest order.
  • The result is independent of the orbiting body's mass, so it applies to planets as well as to the electron state used in the calculation.
  • The full orbit equation (20) contains higher-order terms in $GM/c^2$; the paper states these will differ between unified gravity and general relativity, giving a route to experimentally distinguish the theories.
  • The same quantum field theory approach is being applied to light deflection in a companion preprint, suggesting that the full set of classical gravitational tests may be reproducible from gauge theory.

Reading between the lines

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

  • A direct check of the derivation would be to redo the classical limit using the ordinary substitution $u=1/r$ in Eq. (14); the resulting first-order correction changes, so the paper's result is sensitive to the choice in Eq. (15).
  • The Foldy-Wouthuysen step drops the commutators $[\hat{p},C_1]$ and $[\hat{p},C_2]$; if these are kept and the classical limit is taken later, residual quantum terms could survive and modify the orbit equation.
  • Because the paper only compares to the lowest-order general-relativistic result, the theory is not yet tested at second order; extracting a quantitative second-order prediction from Eq. (20) for Mercury would make the claim falsifiable.
  • If the second-order deviation predicted by unified gravity lies within reach of proposed laser astrometric missions, those missions could adjudicate between the geometric and gauge pictures of gravity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper claims to derive the perihelion precession of planetary orbits from the Dirac equation of a gauge theory of gravity ('unified gravity', UG), without introducing a curved spacetime metric. The authors model a gravitational bound state of an electron, take a classical 'planetary state' limit, and obtain a Binet-type orbit equation whose perturbative solution gives the standard GR perihelion shift Δφ = 6πGM/(c^2 a(1-e^2)). The derivation proceeds through a Foldy-Wouthuysen transformed Hamiltonian, a change of variables r = 1/u - GM/c^2, and a Taylor expansion of the resulting orbit equation.

Significance. If valid, the result would be significant: it would demonstrate that a quantum field theory built on gauge symmetries, without curved spacetime, can reproduce a classic GR test. The manuscript is clearly structured and uses standard techniques (Foldy-Wouthuysen transformation, classical wave-packet limit). However, the central derivation contains algebraic errors and an unjustified variable substitution. The claimed Binet equation does not follow from the preceding equations, and the final GR correction term is not independently derived; it is obtained only after the problematic substitution. The paper does not currently support its central claim.

major comments (4)
  1. [Planetary orbit dynamics, Eq. (16)] The chain-rule expression for dr/dt is incorrect. With r(φ)=1/u(φ)-GM/c^2, we have u^2 r^2 = (1 - GM u/c^2)^2, so dr/dt = J (1 - GM u/c^2)^{-2} du/dφ, not J (1 - GM u/c^2)^2 du/dφ as printed. The factor is inverted, and this error propagates into Eq. (19).
  2. [Planetary orbit dynamics, Eq. (18)] The rewriting of J^2/r^2 is also in error. Since r = (1 - GM u/c^2)/u, one has J^2/r^2 = J^2 u^2 (1 - GM u/c^2)^{-2}, not J^2 u^2 (1 - GM u/c^2)^2 as written. The inverted exponent is essential for the subsequent simplification; with the correct expression, Eq. (19) takes a different form.
  3. [Planetary orbit dynamics, Eqs. (19)-(20)] Equation (20) does not follow from differentiating Eq. (19). Even if one accepts Eq. (19) as printed, differentiating with respect to φ and applying the stated multiplier yields a different equation. For example, setting k = GM/c^2, the differentiation of Eq. (19) leads to an equation containing (1+ku)(1-ku)^2 u'' - 2k(1-ku)(u')^2 + u = k c^2/J^2 after rearrangement, not the displayed Eq. (20). Therefore Eq. (21) is not derived from the preceding equations.
  4. [Planetary orbit dynamics, Eq. (15)] The substitution r = 1/u - GM/c^2 is introduced without physical justification. It is not derived from the UG field equations, and it changes the relationship between u and the physical radius by a constant shift GM/c^2. This shift is load-bearing: with the standard u = 1/r, the first-order expansion of the same orbit equation (14) gives only the Newtonian Binet equation u'' + u = constant, with no u^2 correction term. Thus the advertised 3GM u^2/c^2 term in Eq. (21) is manufactured by the choice of substitution rather than derived from the theory.
minor comments (3)
  1. [Planetary orbit dynamics, Eq. (20)] Equation (20) as typeset appears dimensionally inconsistent: the term (1 - GM u/c^2)^3 is dimensionless while d^2u/dφ^2 has dimension inverse length. The intended form is likely (1 - GM u/c^2)^3 u, but this should be stated explicitly and corrected.
  2. [Planetary orbit dynamics, Eq. (22)] The solution in Eq. (22) and the associated shift in Eq. (23) are introduced from prior literature. Since these are central to the final claim, the authors should at least outline the standard perturbative solution procedure or provide an explicit derivation for completeness.
  3. [Introduction and Conclusion] The manuscript repeatedly emphasizes that the result is obtained 'without a curved metric or other concepts of GR.' Given the algebraic issues in the derivation, this claim should be carefully re-evaluated and softened or supported by a corrected calculation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the coordinate change in Eq. (15) is an allowed reparametrization, and the perihelion formula is obtained by solving the resulting Binet equation rather than assumed.

full rationale

The derivation chain is not circular in the sense defined here. Eq. (15) is an invertible change of variables for the orbital radius; a coordinate reparametrization cannot by itself create or destroy the physical perihelion advance, so the final precession, if correctly computed, is a property of the UG orbital dynamics rather than an input. The paper does not fit GM/c^2 or any other parameter to the target formula; the shift in Eq. (15) is fixed by the known gravitational constant and the selected coordinate variable. The solution of the resulting Binet equation, Eqs. (22)-(23), is taken from standard literature, but that is a mathematical solution step after the equation is obtained, not an assumption of the final result. The self-citations to the authors' prior UG papers (Refs. [30] and [33]) supply the starting Dirac equation and the UG framework, but the perihelion formula is not assumed there; the present paper derives a new consequence from those equations. There are legitimate correctness concerns: the chain-rule factor in Eq. (16) appears inverted, Eq. (19) does not follow from Eq. (16) as written, and the specific choice of the coordinate shift in Eq. (15) is not physically justified. These issues bear on the validity of the derivation, but they do not make the claimed result equivalent to its inputs by construction, and no fitted parameter is being relabeled as a prediction. Accordingly, no circular step is identified.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The central claim rests on the authors' own UG formalism (field equation, Dirac coupling) and on ad hoc mathematical choices (variable shift, commutator neglect). No free parameters are fit to data, but the substitution in Eq. (15) is a free choice that determines the final form of the Binet equation.

free parameters (1)
  • GM/c^2 shift in the u-substitution = GM/c^2 (a fixed constant, not fitted to data)
    Introduced ad hoc in Eq. (15) to make the perturbative expansion produce the standard GR Binet equation. Without this shift, the first-order correction differs from GR.
assumptions (3)
  • ad hoc to paper Dirac equation of UG (Eq. 3) correctly couples the electron to the gravity gauge field.
    Taken from the authors' own theory [30]; not independently validated.
  • ad hoc to paper Gravity field equation of UG (Eq. 1) and its point-mass solution (Eq. 2) describe the Sun's gravitational field.
    This is a linearized Einstein equation with a Newtonian-potential solution; central input.
  • domain assumption The commutators [p,C1] and [p,C2] can be set to zero and expectation values factorize in the wavepacket state of Eq. (12).
    This classical-limit approximation is asserted but not rigorously justified for a 1/r potential.
invented entities (1)
  • Gravity gauge field Hμν of unified gravity (UG)
    purpose: Mediates gravitational interaction in flat Minkowski spacetime, replacing the curved metric of GR.
    The field is defined in the authors' prior paper [30]. This paper provides no independent falsifiable evidence; the only derived prediction is the already-known GR precession.

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Pith. "Pith review of Perihelion precession of planetary orbits solved from quantum field theory." pith.science (2026). https://pith.science/paper/NX7J6DON

@misc{pith2026250614447,
  author       = {Pith},
  title        = {Pith review of: Perihelion precession of planetary orbits solved from quantum field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NX7J6DON}},
  note         = {Machine review of arXiv:2506.14447}
}
read the original abstract

We derive the perihelion precession of planetary orbits using quantum field theory extending the Standard Model to include gravity. Modeling the gravitational bound state of an electron via the Dirac equation of unified gravity [Rep. Prog. Phys. 88, 057802 (2025)], and taking the classical planetary state limit, we obtain orbital dynamics exhibiting a precession in agreement with general relativity. This demonstrates that key general relativistic effects in planetary motion can emerge directly from quantum field theory without invoking the geometric framework of general relativity.

Figures

Figures reproduced from arXiv: 2506.14447 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of the perihelion precession of an object [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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    A new interaction term is added to the unified gravity Lagrangian to couple gravitons to their own stress-energy-momentum tensor, yielding a nonlinear, gauge-invariant field equation.

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Reference graph

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