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The paper numerically computes, for the first time, the exact strong-field post-Newtonian parameters of scalar-tensor stars and finds they differ from the standard weak-field values by up to tens of percent.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 00:00 UTC pith:72IUFFA7

load-bearing objection First numerical computation of the exact PPN parameters, with a usable codebase, but the Entangled Relativity numbers do not survive contact with the paper's own equations — likely a bug in that section. the 3 major comments →

arxiv 2602.11811 v2 pith:72IUFFA7 submitted 2026-02-12 gr-qc

Numerical evaluation of the exact post-Newtonian parameters in Brans-Dicke and entangled relativity theories

Thomas Chehab , Olivier Minazzoli This is my paper
classification gr-qc MSC 83C2583D0585A15 PACS 04.25.Nx04.50.Kd04.30.-w
keywords exact post-Newtonian parametersBrans-Dicke theoryEntangled RelativityTolman-Oppenheimer-Volkoff equationsJanis-Newman-Winicour solutiondipolar gravitational wavesscalar-tensor theoriesneutron stars
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.

Scalar-tensor theories ordinarily predict post-Newtonian parameters from the theory alone, but a recent analytical result says that in strong gravity the effective parameters should depend on the star's pressure and density. This paper makes that concrete by integrating the Tolman–Oppenheimer–Volkoff equations for Brans–Dicke stars and for Entangled Relativity, using two independent methods that agree to better than a percent. It finds that the exact γ and δ can deviate from the standard perturbative values by tens of percent for neutron stars, and in Brans–Dicke δ_exact can differ from unity by as much as 83.7%. For Entangled Relativity with L_m = −ρ, the predicted neutron-star deviations are large enough that the implied dipolar gravitational-wave emission from PSR J1738+0333 overshoots the observed orbital-period constraint by two orders of magnitude, which the authors argue likely excludes that branch of the theory. If L_m = T instead, the theory matches General Relativity for ordinary matter and the same test does not apply.

Core claim

The paper establishes that the 'exact' post-Newtonian parameters γ_exact, δ_exact (and β_exact, which stays 1 for these theories) are numerically accessible from standard TOV integration, and that they are genuinely body-dependent strong-field quantities. In Brans–Dicke theory the deviation of γ_exact from unity reaches 44.1%, δ_exact deviates from unity by up to 83.7%, and the relative difference between exact and standard perturbative parameters reaches 44.7% for γ and 46.3% for δ. A cross-check fits the Janis–Newman–Winicour parameter to the exterior of the same numerical star; the two methods agree to 0.07% for γ and 0.3% for δ, validating the analytical expressions. The same machinery g

What carries the argument

The central object is the exact post-Newtonian parameter set (γ_exact, β_exact, δ_exact) introduced analytically in previous work: one reads these parameters from the Taylor expansion of the Janis–Newman–Winicour exterior metric matched to a static spherical star, which makes them explicit functionals of the star's integrated energy E*, pressure P*, and, for Entangled Relativity, the integrated matter Lagrangian L*_m. The paper's numerical machinery is the Tolman–Oppenheimer–Volkoff integration of the Brans–Dicke and Entangled Relativity field equations, with a polytropic (and, for the pulsar test, a realistic piecewise) equation of state; the radial-coordinate invariance of the integrals ju

Load-bearing premise

The whole computation rests on identifying a star's exact post-Newtonian parameters with the coefficients read off the Janis–Newman–Winicour exterior solution matched to the TOV interior — for the Entangled Relativity exclusion, together with the assumption L_m = −ρ.

What would settle it

Compute the exterior field of a rotating or magnetized neutron star in Entangled Relativity by full numerical relativity or by matching to a different exact solution family, and read off the scalar charge independently: if γ_exact and α_DEF for a 1.4 M☉ star no longer imply Ṗ_D more than an order of magnitude above the observed PSR J1738+0333 limit, the two-order-of-magnitude exclusion would be an artifact of the JNW matching. Alternatively, a high-precision measurement of Ṗ_D in a white-dwarf–pulsar system, fitted within Entangled Relativity, that remains consistent with the General Relativit

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

If this is right

  • Neutron-star tests of scalar-tensor gravity must use body-dependent strong-field parameters, not the universal weak-field PPN values, because the two can differ by tens of percent.
  • In Brans–Dicke theory the exact parameters approach General Relativity for the most compact stars, because the scalar-field source ∝(−ρ + 3P) nearly cancels when P → ρ/3; this partially offsets the naive expectation of larger deviations at higher density.
  • In Entangled Relativity with L_m = −ρ, the Sun's γ_exact − 1 ≈ 7 × 10⁻⁶ is just below current solar-system bounds, at a level where a full c⁻⁴ light-propagation treatment becomes necessary to test it.
  • The predicted Ṗ_D for PSR J1738+0333 in Entangled Relativity with L_m = −ρ is two orders of magnitude above the observed constraint, so existing timing data likely exclude that branch once system parameters are re-fit.
  • If L_m = T, Entangled Relativity reduces to General Relativity for ordinary matter, leaving only strongly magnetized compact objects (magnetars) as prospective laboratories, with expected deviations about seven orders of magnitude smaller than the L_m = −ρ case.

Where Pith is reading between the lines

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

  • Because the two cross-checking methods both inherit the same Janis–Newman–Winicour matching premise, a genuinely independent strong-field calculation — e.g., full numerical relativity for rotating or magnetized neutron stars in these theories — would be the natural next test of whether the exact parameters are unique, not just self-consistent.
  • The Sun's γ_exact − 1 ≈ 7 × 10⁻⁶ prediction suggests a near-term experimental target: improving solar-system γ measurements by an order of magnitude, with the required c⁻⁴ light-time formalism, would begin to probe Entangled Relativity with L_m = −ρ.
  • The same algebraic link that yields α_DEF from γ_exact could translate future measurements of scalar charges in binary pulsars into direct constraints on the exact parameters, bypassing the weak-field PPN framework entirely.
  • If the L_m = −ρ exclusion survives re-fitting, the theory's viability shifts to the L_m = T branch, where magnetars and extreme electromagnetic fields become the only observable window; forecasting that branch would require TOV models with strong magnetic fields, not included here.

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

3 major / 3 minor

Summary. The paper computes, by numerical TOV integration, the so-called exact (non-perturbative) post-Newtonian parameters introduced by Chauvineau [16], for Brans-Dicke theory and for Entangled Relativity. In the Brans-Dicke case it develops two methods: a direct integration of the [16] formulas over the stellar interior, and an external fit to the Janis-Newman-Winicour (JNW) solution; the two agree at the sub-0.1--0.3% level. It then derives an analogous ER exterior solution, evaluates the exact gamma and delta for the Sun, Earth and neutron stars, and uses the Damour-Esposito-Farese parameter alpha_DEF to estimate dipolar gravitational-wave emission from PSR J1738+0333, concluding that the L_m=-rho branch of ER is heavily constrained, if not ruled out.

Significance. If correct, this would be the first numerical evaluation of the exact PPN parameters and a useful strong-field tool for scalar-tensor theories. The Brans-Dicke section is self-contained, internally consistent, cross-checked by two methods, accompanied by public code, and validated by a first-integral check at 10^-11 (Appendix A). The ER application is potentially very interesting because it connects the exact parameters to a sharp binary-pulsar constraint. However, the central ER equations as printed are mathematically inconsistent with the derivation that precedes them, and the reported ER numbers cannot all follow from the printed formulas. The ER results and the resulting pulsar constraint therefore need major correction before the paper's central claims can be accepted.

major comments (3)
  1. [Sec. V.C, Eq. (65)] Equation (75b) as printed is not the formula that follows from the preceding derivation. For L_m=-rho, inserting Eq. (74) into Eq. (11c) with omega0=0 gives gamma_exact = (1+Theta - Theta/sqrt(3))/(1+Theta + Theta/sqrt(3)) < 1 for Theta>0, whereas the printed Eq. (75b) is (3+4Theta)/(3+2Theta) > 1. The reported values are mutually inconsistent with one of these: Eq. (79a) reports gamma_sun-1 about +7e-6 (the sign of the printed formula), while Fig. 5 reports 1-gamma up to 16.1%, which cannot be obtained from the printed Eq. (75b) with the stated polytrope, since Theta <= max(P/rho) <= 1/5 would bound |1-gamma| <= 11.8%. The claimed 0.01% cross-check with Eq. (11) in Sec. VI B cannot hold for both interpretations. Please correct Eq. (75b), state the sign convention explicitly, and recompute/verify all ER numerical results and the Pdot estimates in Sec. VII.
  2. [Sec. IV.B/IV.D/VII] The metric function A is written with a minus sign: A = -1/Phi (1-k/r)^d. Together with ds^2 = -A dt^2 + ..., this gives g_tt = +1/Phi (1-k/r)^d, i.e. a non-Lorentzian signature. Since the subsequent integrals and the sign of d in Eqs. (69)-(74) depend on the metric convention, please correct this sign and re-derive the ER formulas if necessary.
  3. [Sec. V.C, Eq. (65)] The two numerical methods used for the 'exact' parameters are not independent tests of the physical identification of the JNW-matching quantities with strong-field PPN parameters: both methods share the static, spherical JNW ansatz. The dipolar-radiation constraint in Sec. VII additionally assumes that this JNW scalar charge is the DEF charge of a real, non-spherical, rotating neutron star. The authors acknowledge some of these limitations, but the central conclusion -- that the L_m=-rho branch of ER is likely ruled out -- inherits this unidentified matching step. Please state this more prominently, or provide a concrete test (e.g. a non-spherical or slowly rotating model) if the stronger claim is intended.
minor comments (3)
  1. [Eq. (75b) formatting] If Eq. (75b) is corrected to the form that follows from Eq. (11c), the sign of gamma-1 for the Sun and Earth changes; the order of magnitude may be unaffected, but the printed signs should be checked.
  2. [Throughout] Many displayed equations have lost fraction bars in the typesetting (e.g. Eq. (75b) is ambiguous as printed). Please carefully re-typeset the equations; this is not merely cosmetic because ambiguities in signs and denominators are load-bearing.
  3. [Throughout] The paper consistently writes 'Tolman-Oppeheimer-Volkoff' in prose; the standard spelling is 'Tolman-Oppenheimer-Volkoff'. Also, the first-integral constant in Eq. (83) has a dimensionality issue that should be clarified (arguments of logarithms).

Circularity Check

1 steps flagged

No substantial circularity in the main derivation; one overclaimed 'independent test' is a self-consistency check that shares the same JNW ansatz.

specific steps
  1. other [Sec. IV C (Cross-checking results), echoed in Sec. VIII]
    "This notably is an independent test of the validity of the analytical results presented in [16]."

    The cross-check is not independent of [16]. Method A evaluates the [16] interior integrals (Eqs. 29-30) while method B fits the JNW parameter α to the exterior of the same TOV solution and then uses Eqs. (37)/(46), which are precisely the [16] relations, to map α to γ and δ. Both legs therefore assume the JNW exterior and the [16] identification of the exact PPN parameters as Taylor coefficients of that exterior. The <0.3% agreement checks numerical consistency and the matching, but cannot independently validate the [16] formulas; the claimed validation is equivalent to using the same ansatz in both methods. This is a minor overclaim, not a derivation-level circularity, and the central numerical values do not depend on this 'independence' being genuine.

full rationale

The central quantities γ_exact, β_exact, δ_exact, and α_DEF are computed by direct numerical integration of analytical expressions imported from Chauvineau [16] (Eqs. 29 and 75) or derived in-paper from the same JNW matching (Sec. V C), not by fitting to the observational data later compared. The TOV integration is checked with a conserved first integral to ~10^-11 and the code is publicly available, which is independent support. The main assumptions — the JNW exterior as the strong-field generalization of PPN parameters, and L_m=-ρ in Entangled Relativity — are stated explicitly and inherited from prior work; they are modeling premises and potential correctness risks rather than circular reasoning, because the target results are not fed back into those premises. The only near-circular element is the Sec. IV C claim that the two-method agreement is an 'independent test' of [16]; since both methods share the [16]/JNW identification, that agreement is a self-consistency check. That overstatement is not load-bearing for the reported numbers, so the circularity score is low: the derivation is self-contained once the imported premises are granted, and no prediction reduces by construction to a fitted input.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The paper's inputs are: the imported [16] formulas, the TOV integration machinery from [17-19], the chosen EOS (K, Γ), the scan range and cutoffs (central density, c_s<c/√3, M>1M_⊙), and the matter-Lagrangian choice L_m=−ρ for ER. No new particles, forces or fields are invented; the scalar degree of freedom of BD/ER comes from prior theory. The load-bearing assumptions are the JNW-matched definition of exact parameters and the L_m=−ρ choice.

free parameters (4)
  • Polytropic EOS stiffness K = 1.475×10⁻³ (fm³/MeV)^(2/3)
    Input model parameter P=Kρ^(5/3) from [17,22]; not fitted in this paper. Sets stellar structure and hence all exact parameters.
  • Adiabatic index Γ = 5/3
    Fixed polytropic index for BD and ER neutron-star models (Sec. III B, VI B).
  • Central-density scan with sound-speed cutoff = 100–1570 (BD) / 100–1578 (ER) MeV/fm³
    Scan range plus the c_s<c/√3 selection [25] determines the most compact configurations; maximum deviations occur at the cutoff. The two sections quote different cutoff densities for the same EOS.
  • Brans–Dicke coupling ω (scanned range) = 10⁻¹ to 10⁶
    Theory parameter scanned in Figs. 1-4; headline deviations are reported at scan extremes, and the Fig. 2/4 relative differences depend on this range.
axioms (6)
  • domain assumption The exact PPN parameter framework of Chauvineau [16]: PPN parameters of a star are read from the Taylor expansion of a Janis–Newman–Winicour exterior matched to the star, with scalar charge d given by volume integrals (8)-(9)
    Sec. II A (Eqs. 8-12) imports this without re-derivation; both numerical methods and the ER analogue (Sec. V C, Eq. 75) inherit it. If the JNW matching is not the correct strong-field generalization, all reported numbers change.
  • domain assumption Static, spherically symmetric perfect-fluid stars with a JNW exterior describe BD/ER neutron stars
    Rotation, magnetic fields and anisotropy are neglected (Sec. III B, VI B); the exterior-fit cross-check (Sec. IV B) assumes the numerical solution's exterior is JNW.
  • domain assumption On-shell matter Lagrangian L_m = −ρ for Entangled Relativity
    Sec. V B (Eq. 56) and Sec. VI B; all ER numbers and the dipolar-GW estimate depend on it. The authors state L_m = T gives no deviation (Sec. VII B, VIII).
  • domain assumption Configurations with c_s < c/√3 are the physically allowed ones [25]
    This cut sets the maximal central density (quoted as 1570 in Sec. IV A and 1578 in Sec. VI B) and hence the maximum reported deviations.
  • domain assumption The polytrope P = Kρ^(5/3) (plus a piecewise EOS in Sec. VII) captures the broad behaviour of neutron-star matter
    Sec. III B and VI B state the goal is broad behaviour; the Ṗ_D estimates are EOS-dependent (Fig. 7).
  • standard math Standard BD/ER field equations and the Jordan–Einstein frame conformal transformation
    Eqs. (13)-(15) and (33)-(42); standard background from [1,17-19] used to build both exterior solutions.

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read the original abstract

In context of Brans-Dicke scalar-tensor theories of gravity, it has recently been obtained that the post-Newtonian parameters should be generalized in the context of strongly gravitating bodies, and that its generalization -- the so-called $\textit{exact parameters}$ -- actually depends on the pressure and energy density of a considered celestial body. Here we develop two new methods to numerically obtain the $\textit{exact parameters}$ by means of usual Tolman-Oppenheimer-Volkoff computation, and find that the difference with the value of standard post-Newtonian parameters can be more than 80% in some situations. We also provide the connection with the Damour-Esposito Far\`ese non-pertubative parameter $\alpha_{DEF}$. We then apply the methodology to the case of Entangled Relativity, and derive these exact parameters for the Sun and the Earth, as well as for neutron stars. We argue that current and foreseeable experiments are likely able to constrain the theory under the assumption that $\mathcal{L}_m=-\rho$, where $\rho$ is the total energy density. If $\mathcal{L}_m=T$ instead, as often advocated in the literature, then there is no deviation with respect to General Relativity and the prospects of testing Entangled Relativity become much more remote in time, as only compact objects with extreme electric or magnetic fields could lead to some deviation from General Relativity.

Figures

Figures reproduced from arXiv: 2602.11811 by Olivier Minazzoli, Thomas Chehab.

Figure 1
Figure 1. Figure 1: FIG. 1: Plot of 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Plot of the relative difference between the exact [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Plot of the relative difference between the exact [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: —it seems likely that Entangled Relativity could be heavily constrained under this assumption from observations of binary pulsars [63], white dwarf–pulsar systems [33], or triple star systems [21]. Indeed, one would first expect a significant deviation of the Shapiro delay, which is taken into account when fitting the system parameters to pulsar timing data [63]. More importantly, the non-negligible numeri… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Plot of 1 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Plot of [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Plot of the relative variation of the constant [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗

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Forward citations

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