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This paper argues that solar neutrino oscillations carry a measurable imprint of symmetric teleparallel gravity, with non-metricity couplings a2 and a4 altering the oscillation phase only when neutrino mass eigenstates have opposite spin or

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2026-08-02 23:45 UTC pith:X2FCIZ75

load-bearing objection First STPG neutrino-oscillation calculation, but the central derivation rests on a degenerate tetrad and an unshown Hamiltonian approximation; needs major revision before I would trust the phase formulas. the 4 major comments →

arxiv 2602.13355 v3 pith:X2FCIZ75 submitted 2026-02-13 gr-qc hep-phhep-th

Symmetric teleparallel gravitational effects on solar neutrino oscillations

Aysel Cetinkaya , Muzaffer Adak , Ozcan Sert , Caglar Pala This is my paper
classification gr-qc hep-phhep-th MSC 83D0583C6081V15 PACS 04.50.Kd14.60.Pq
keywords symmetric teleparallel gravitynon-metricityneutrino oscillationssolar neutrinosDirac equationKerr metriccoincident gaugemass eigenstates
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 seeks to establish that solar neutrino oscillations register the geometry of symmetric teleparallel gravity, a theory in which gravity comes entirely from non-metricity, with zero curvature and torsion. Working with the reduced Kerr metric of a slowly rotating Sun in the coincident gauge, the authors derive a Dirac Hamiltonian from a generalized Dirac equation and compute the phase each neutrino mass eigenstate accumulates. They find that the couplings a2 and a4 alter the oscillation phase only when the two mass eigenstates have opposite spin orientations, while b3 shifts the effective mass-squared difference as Δm² → Δm² − 8 b3 m_p Δm for all spin configurations. Comparison with solar neutrino data yields bounds |a1|~|a3|≲1, |a2|~|a4|≲10³, and |b3|~|b4|≲10⁻³³. If correct, neutrino oscillations become a new observational handle on non-metricity, and any theory predicting larger b3 or b4 would already be excluded.

Core claim

The central claim is that in symmetric teleparallel spacetime, where curvature and torsion vanish but non-metricity does not, the phase difference between neutrino mass eigenstates acquires non-metricity corrections that are absent in general relativity. Using the generalized Dirac equation with the spinor connection Ω = ½σ_ab ω^[ab] + (a1 I + a2 γ5)Q + (a3 I + a4 γ5)P + (b3 γa + b4 γa γ5)e^a, the authors compute the accumulated phases Φ↑,↓_1,2 for radial propagation from the solar surface to Earth. The resulting phase differences show that b3 acts as a universal shift of the mass-squared difference, Δm² → Δm² − 8 b3 m_p Δm, while a2 and a4 contribute only when the two mass eigenstates have

What carries the argument

The engine of the calculation is the generalized spinor connection of Eq. (6), the rule determining how a neutrino spinor responds to the flat-but-non-metric connection of symmetric teleparallel gravity; it contains the six free couplings a1, a2, a3, a4, b3, b4 attached to the non-metricity trace forms Q and P and to the coframe. Combined with the coincident-gauge connection ω^a_b = h^a_α d h^α_b and the reduced Kerr tetrad, it produces the Dirac Hamiltonian whose eigenvalues E↑,↓_± enter the dynamical-phase integrals. The key manipulations are block-diagonalizing the Hamiltonian, extracting the 2×2 upper block, and diagonalizing it to first order in the Sun's rotation parameter a, after whi

Load-bearing premise

The load-bearing premise is the assumed spinor connection of Eq. (6), the rule connecting neutrino spin to non-metricity, which is imported from the authors' companion work rather than derived here — the paper itself concedes the physical meaning of its constants remains an open problem — and the coincident-gauge tetrad used for the Sun is singular (det h = 0), so if that parameterization is incomplete the phase formulas and bounds collapse.

What would settle it

Evaluate the coincident-gauge connection with a non-singular tetrad for the Kerr metric and check whether the phase integrals survive; a direct polarized-neutrino solar experiment could also look for the predicted opposite-helicity phase asymmetry of the a2/a4 contribution — a null result at the predicted level would falsify that part of the effect.

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

If this is right

  • If correct, existing solar neutrino oscillation data already rule out any symmetric-teleparallel model with |b3| or |b4| above about 10⁻³³, because the universal mass shift would be visible.
  • The a2 and a4 couplings change the oscillation probability only for opposite-spin mass eigenstate pairs, so the effect is invisible in standard unpolarized solar neutrino measurements and would require polarization-sensitive detection.
  • Solar rotation has no effect on the oscillation phase at first order in a/r, so frame-dragging corrections are negligible for solar neutrinos.
  • The phase formulas give a concrete target: a measurable wavelength shift Δm²_eff = Δm² − 8 b3 m_p Δm for all spin configurations, with additional geometric terms proportional to GMΔr/(c² r_A r_B) for the opposite-spin channels.
  • The same Hamiltonian method provides a template for testing other metric-affine geometries; comparing with torsion-based Einstein-Cartan results would isolate which geometric sector affects neutrino phases.

Where Pith is reading between the lines

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

  • Editorial extension: the same phase-integral machinery could be applied to atmospheric and accelerator neutrinos crossing Earth's gravitational field; shorter baselines and higher energies might sharpen or exclude the a2/a4 spin-dependent term.
  • Editorial extension: because the a2/a4 contributions average to zero for an unpolarized beam, the realistic observational probe is a helicity-correlation asymmetry; a null result would push these couplings toward zero rather than disproving non-metricity.
  • Editorial extension: the bound |b3| ≲ 10⁻³³ is so stringent that it effectively forces b3 to vanish in any viable theory, so the more interesting phenomenological window is the much weaker (≲10³) a2/a4 sector.
  • Editorial extension: recomputing the connection with a non-singular tetrad for the Kerr background (the paper's coincident-gauge tetrad has det h = 0 and e² = 0) would be the cleanest check on whether the phase formulas survive a fully well-defined connection.

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 / 4 minor

Summary. The paper analyzes solar neutrino oscillations in symmetric teleparallel gravity (STPG). Using a generalized Dirac equation that couples the spinor to non-metricity via six coupling constants (a1...a4, b3, b4), the authors model the solar gravitational field as the reduced Kerr metric in coincident gauge, restrict to the equatorial plane, and derive a Dirac Hamiltonian in block form. They compute dynamical phases for spin-up and spin-down mass neutrinos, obtaining phase differences (Eqs. 30–33) in which b3 shifts the effective mass-squared difference and the a2/a4 terms appear only for opposite spin configurations. They also report phenomenological upper bounds on the coupling constants from solar neutrino data. The central claims are that non-metricity modifies neutrino oscillation phases and that solar neutrino data can constrain STPG parameters.

Significance. If the derivation were sound, this would be the first analysis of neutrino oscillations in symmetric teleparallel geometry, a previously unexplored regime, and would open a new observational window on non-metricity. The paper makes the useful observation that certain non-metricity couplings introduce spin-dependent geometric phases, and that the b3 coupling acts as a universal mass-shift. It also provides explicit, order-of-magnitude bounds that are in principle checkable. However, the calculation as written is not reproducible because a central mathematical input—the equatorial coframe—is degenerate, and because the reduction of the Hamiltonian to the two-by-two block H+ is not shown. These issues are load-bearing; without them the phase formulas and bounds lack a valid derivation.

major comments (4)
  1. [§4, Eqs. (10)–(13)] The orthonormal coframe is defined with e^2 ≃ 0, so the matrix h^a_α in Eq. (11) has a zero row and det(h^a_α)=0. The inverse tetrad h^α_a therefore does not exist, and the coincident-gauge connection ω^a_b = h^a_α d h^α_b in Eq. (3) is not well-defined. All subsequent quantities—Q_ab in Eq. (13), the traces Q and P in Eq. (14), the Hamiltonian (24), and the phase differences (30)–(33)—are computed from this ill-defined connection. A valid treatment must work in the full four-dimensional Kerr frame and restrict to θ=π/2 only after computing the spin connection, or use an induced 3D tetrad/spinor formalism. As written, this invalidates the central derivation.
  2. [§4, Eqs. (22)–(23)] The reduction from the four-dimensional Hamiltonian H to the two-by-two block H+ is announced with 'under certain assumptions based on the observational data to be discussed later,' but those assumptions are never displayed. The blocks H11, H12, H21, H22 are not given, and the approximations leading to A, B, C in Eq. (24) are not shown. Since every phase formula and bound in the paper follows from H+, the calculation is not reproducible. The authors should present the full Hamiltonian and the explicit truncation/disposal criteria, or make an ancillary derivation available.
  3. [§5 and Abstract] The abstract and the concluding list state that solar neutrino data place upper bounds |a1| ~ |a3| ≲ 1. In the main text (§5), however, the authors write 'we anticipated the upper bound |a1| ≈ |a3| ≲ 1 and carried out our calculations accordingly.' This bound is an input, not an output of the data analysis. Moreover, the paper later argues that a1 and a3 are suppressed by ℏ/(pr) ≪ 1 and do not affect oscillation phases, so solar data do not actually test them. The claim that these bounds follow from consistency with solar data is therefore overstated and should be corrected.
  4. [§3, Eq. (6)] The generalized spinor connection Ω, including the five additional non-metricity terms with couplings a1...a4, b3, b4, is imported from the authors' companion preprint [29] without derivation. The paper itself notes the physical interpretation is open. Because all phase results and bounds depend on the precise form and on the reality conditions (7), the authors should at least outline the derivation of Eq. (6) in the STPG sector, or state which independent results in [29] justify it. Otherwise the central results are conditional on an unverified parameterization.
minor comments (4)
  1. [Eq. (10)] The notation e^2 ≃ 0 should be flagged as a post-restriction condition, not a coframe component; if a 1-form vanishes identically, the coframe is not a basis of the cotangent space. This is related to the major concern but should be clarified in the geometry setup.
  2. [Eq. (24a)] The typesetting of A is garbled: 'M2 + 16b2 4 4 2p' should likely be (M^2 + 16 b_4^2)/(2p). Please correct the equation for readability.
  3. [§4 notation] The symbols H±, H+, and E↑+ etc. are used inconsistently; the submatrices H11, H12, H21, H22 are introduced but never defined explicitly. Defining the block structure of the four-dimensional Hamiltonian would improve clarity.
  4. [§5] The argument for |b3| ≲ 10^-33 uses m_p ~ 10^19 GeV, Δm ~ 10^-2 eV, and δ(Δm^2) ~ 10^-6 eV^2; these orders of magnitude are plausible but should be given with more explicit unit conversion, and the statement that the bound reflects 'Planck-scale suppression' should be phrased as an assumption about the origin of b3, not a consequence.

Circularity Check

2 steps flagged

The a1/a3 bounds are anticipated inputs later reported as data-driven constraints, and the central phase formulas inherit the authors' own ansatz spinor connection (6) from [29]; the non-metricity contributions are therefore built in rather than independently predicted.

specific steps
  1. fitted input called prediction [Section 5 (Phenomenological bounds); Section 1 item 4; Abstract]
    "On the other hand, since the measured momenta of solar neutrinos are of the order of (5–15) MeV/c, we anticipated the upper bound |a1| ≈ |a3| ≲ 1 and carried out our calculations accordingly. ... Consistency with solar neutrino data places upper bounds |a1| ∼ |a3|≲1, |a2| ∼ |a4|≲10^3 and |b3| ∼ |b4|≲10^{-33}."

    The |a1|≈|a3|≲1 bound is stated in Section 5 to have been 'anticipated' before the calculation, yet Section 1 (item 4) and the Abstract report it as an upper bound imposed by consistency with solar neutrino data. Thus the constraint is an input assumption used to simplify Section 4, not a consequence of the data; presenting it as an inferred bound is a fitted-input-called-prediction.

  2. ansatz smuggled in via citation [Section 3, Eq. (6) and its preamble]
    "Recently, its generalized form was established in Ref. [29]. For the symmetric teleparallel geometric sector with vanishing torsion, the covariant exterior derivative of spinor takes the form Dψ= dψ+ Ωψ, where Ω = ... (b3γa + b4γaγ5)e^a. ... if one requires that all basis elements of the Clifford algebra be present in the spinor connection in a metric-affine spacetime, this combination plus the term (b1I + b2γ5)T emerge, see [29]."

    This Ω is the single load-bearing input of the paper: Eqs. (24)-(25) and the phase differences (30)-(33) are expansions of it for the Kerr coframe. It is not re-derived here; it is imported from the same authors' companion preprint [29], whose 'all Clifford-algebra elements' motivation is an ansatz. Consequently the headline results—that a2/a4 contribute to the phase and b3 shifts Δm²—are guaranteed by the assumed form of the spinor connection rather than independently derived. The paper itself adds that 'the physical interpretation of these constants remains an open problem,' confirming the input is unverified.

full rationale

Two circularity/self-citation defects are present, but the paper is not wholly circular. First, the a1/a3 bound is an anticipated assumption (Section 5) that is then advertised as a data-derived constraint (Section 1, Abstract). Second, the generalized Dirac equation used for the central calculation is lifted from the authors' own preprint [29]; the spinor connection (6) is an ansatz (all Clifford basis elements present), and all phase formulas (26)-(33) are algebraic consequences of that ansatz, so the non-metricity contributions are inserted rather than predicted. The b3/b4 and a2/a4 bounds in Section 5 are genuine order-of-magnitude comparisons with external solar-neutrino data, and the spin-dependent structure of Eqs. (32)-(33) is nontrivial algebra; those parts give the paper independent content. Separately, the equatorial coframe (10)-(11) is singular (e^2=0, det h=0), so the coincident-gauge connection (3) is not well-defined as written; this is a mathematical-validity defect to be repaired, and it is not itself a circularity.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central result rests on: the assumed spinor connection (Eq. 6) from the authors' own [29] with six arbitrary couplings; reality constraints (Eq. 7) also from [29]; the coincident-gauge Kerr background (Eqs. 9-13) as a model of the Sun, with the singular equatorial tetrad; the ultra-relativistic radial-propagation/vacuum-oscillation idealization (Eqs. 21, 29); and standard solar oscillation parameters (Eq. 34). The six coupling constants are the free parameters. The §5 bounds are inferences from data for a2/a4/b3, assumptions for a1/a3, and vacuous for b4. No new particles, forces, or geometric entities are introduced.

free parameters (6)
  • a1 = -1/2 + iA1 (imaginary part A1) = |a1| ≲ 1 (assumed, not derived)
    Appears in Eq. (6); §5 states the bound was 'anticipated' to simplify the calculation, and a1 drops out of the final phase differences.
  • a2 (real A2) = |2a2 + a4| ≲ 10^3
    Drives the helicity-dependent phase in Eqs. (32)-(33); bounded in §5 by requiring 2(2a2+a4)MGΔr/(c²rArB) ~ 10^-5(2a2+a4) to stay below the ~10^-2 phase uncertainty.
  • a3 = +1/2 + iA3 (imaginary part A3) = |a3| ≲ 1 (assumed, not derived)
    Drops out of the phases; the 'bound' in §5 is anticipated, not inferred from data.
  • a4 (real A4) = |2a2 + a4| ≲ 10^3
    Appears together with a2 in the opposite-helicity phase (Eqs. 32-33); only the combination 2a2+a4 is bounded.
  • b3 (real B3) = |b3| ≲ 10^-33
    Produces the universal shift Δm² → Δm² − 8b3·m_p·Δm (Eqs. 30-33); bound from |8b3·m_p·Δm| ≲ δ(Δm²) ~ 10^-6 eV² with Δm ~ 10^-2 eV and m_p ~ 10^19 GeV.
  • b4 (real B4) = |b4| ≲ 10^-33 (claimed; vacuous)
    Cancels identically from all four phase differences (Eqs. 30-33); the §5 bound '16b4²m_p² ≲ m²' compares a quantity that no oscillation observable measures, so it is not supported by solar data.
axioms (6)
  • domain assumption Generalized Dirac equation with spinor connection Ω = ½σ_abω^[ab] + (a1I + a2γ5)Q + (a3I + a4γ5)P + (b3γa + b4γaγ5)e^a (Eq. 6), with six arbitrary couplings, imported from Ref. [29].
    The entire phase calculation depends on this assumed coupling structure; its derivation is deferred to the authors' own companion preprint [29], and §3 explicitly says the physical interpretation of the constants 'remains an open problem.'
  • domain assumption Reality constraints on couplings, Eq. (7): a1=-1/2+iA1, a2=A2, a3=+1/2+iA3, a4=A4, b3=B3, b4=B4, required for consistency between variational and canonical Dirac equations.
    Stated as a consistency condition inherited from [29]; not derived or checked in this paper.
  • domain assumption The Sun's exterior is described by the reduced Kerr metric (Eq. 9) at first order in M/r and a/r.
    Standard background choice; the paper never writes an STPG action or field equations, so whether Kerr is a stationary solution of the underlying symmetric teleparallel dynamics is not addressed (it is consistent for the STEGR subclass, but this is not stated).
  • ad hoc to paper The equatorial-plane coframe (Eq. 10) with e² = 0 forms a valid 4D tetrad for use in Eq. (3).
    The displayed tetrad matrices (11) have a zero row, det h = 0, so h^a_α dh^α_b is not well-defined as written; the paper proceeds as if the θ-sector decouples, which is plausible but not demonstrated.
  • domain assumption Ultra-relativistic approximation p ≈ E, dt ≈ dr with radial propagation, and the vacuum two-flavor oscillation probability (29) apply to solar neutrinos.
    Used to obtain the phases (21) and probability (29); solar neutrinos are in fact MSW-dominated, so the vacuum formula with parameters (34) is an idealization that is never flagged.
  • standard math Solar oscillation parameters Δm²21 ≈ 7.5e-5 eV² and sin²(2θ12) ≈ 0.307 are inputs (Eq. 34).
    Standard PDG/KamLAND measured inputs used to calibrate the bounds in §5; external data, not derived here.

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Neutrino oscillations probe the quantum gravity interface in unique ways. While gravitational effects on neutrinos are well studied in general relativity and torsion based geometries, the symmetric teleparallel regime where gravity stems solely from non-metricity, with zero curvature and torsion has remained uncharted. In this work, we perform the first analysis of neutrino oscillations in such a spacetime. Using the reduced Kerr metric in coincident gauge for the slowly rotating and weakly gravitating spherical Sun, we derive the Dirac Hamiltonian from the generalized Dirac equation and compute the accumulated phase of neutrino mass eigenstates. There are six free coupling constants in our model. Based on certain observational inputs, we inferred upper bounds on our arbitrary coupling constants. This allowed us to simplify the otherwise cumbersome calculations to some extent. Ultimately, we computed the phase differences that play a crucial role in solar neutrino oscillations and analyzed the contributions arising from our arbitrary coupling constants. Our results establish neutrino oscillations as a novel probe of non-metricity and open a new avenue for testing symmetric teleparallel gravity through astrophysical observations.

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