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REVIEW 4 major objections 4 minor 23 references

Star equilibrium: from BNG to TOV

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read With the full TOV-inspired conservation equation, the pressure inside BNG stars becomes negative before the object reaches black-hole compactness, even though no Buchdahl limit appears.

desk verdict Careful toy-model paper with a genuinely new effect, but the negative-pressure claim rests on an equation whose derivation drops a term without justification, so it is a property of the chosen heuristic rather than a robust BNG prediction. read the letter →

arxiv 2501.04632 v2 pith:OKF64UOW submitted 2025-01-08 gr-qc

Roberto Casadio , Octavian Micu This is my paper · ORCID
classification gr-qc
keywords bootstrappedNewtoniangravityTolman-Oppenheimer-VolkoffequationhydrostaticequilibriumcompactstarsBuchdahllimitblackholehorizonhomogeneousdensitypost-Newtonianexpansion
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

Bootstrapped Newtonian gravity (BNG) is a model of gravity that adds non-linear terms to the Newtonian potential to describe very compact objects. This paper asks whether the equation enforcing hydrostatic equilibrium—imposed by hand rather than derived from the BNG action—changes the picture when it is upgraded with terms inspired by the post-Newtonian expansion of the Tolman-Oppenheimer-Volkoff (TOV) equation. The paper reports two results: BNG stars never develop a Buchdahl limit, but with the full TOV-inspired conservation equation the central pressure becomes negative at compactness $0.4\lesssim X\lesssim 0.6$, below the value $X\simeq 0.69$ at which a BNG horizon covers the star. A sympathetic reader would care because this means BNG, as formulated here, cannot produce a static star with positive pressure all the way up to black-hole formation.

What carries the argument

The central object is the modified equilibrium (conservation) equation $p'\simeq-(\rho+p)V'-4\pi G_N r p\rho-2\rho V V'$ (Eq. 4.1), obtained in Appendix A by expanding the TOV equation to second order and replacing the GR mass function with the BNG potential. The term $-2\rho V V'$ is what drives the central pressure negative, since omitting it (Section 3) leaves the pressure positive. The analysis uses a quadratic ansatz $V_s=V_0+V_2 r^2$ for the interior potential, with the pressure approximated similarly, and checks these analytic results against numerical solutions of the coupled system.

What would settle it

Solve the system (1.11) plus (4.1) numerically for a homogeneous star at compactness $X=0.55$ with a high-precision integrator and check the central pressure: if it is positive, the paper's predicted sign change in $0.5<X<0.6$ is not reproduced.

Watch

Extended reading notes

Core claim

The paper's central claim is that, in bootstrapped Newtonian gravity (BNG), the detailed form of the hydrostatic-equilibrium equation controls whether ultra-compact stars can exist. With the full TOV-inspired conservation equation $p'\simeq-(\rho+p)V'-4\pi G_N r p\rho-2\rho V V'$, the approximate analytic central pressure of a homogeneous star crosses zero between $X=0.4$ and $X=0.5$, and the numerical pressure does so between $X=0.5$ and $X=0.6$. Both happen before the BNG horizon appears at the centre around $X\simeq0.46$ and covers the star around $X\simeq0.69$. Even though BNG stars show no Buchdahl limit, the paper concludes that these TOV-inspired corrections make the pressure negative before the object becomes a black hole.

Load-bearing premise

The load-bearing premise is that Eq. (1.10) is the right conservation equation for BNG: it is built in Appendix A by expanding the GR TOV equation and replacing the GR mass function with the BNG potential, not by varying the BNG action, and one term is dropped without a stated justification.

Editorial extensions

If this is right

  • If the central claim is right, BNG with the full TOV-inspired conservation equation cannot describe a static star with positive pressure all the way up to the black-hole threshold; the pressure turns negative first.
  • The absence of a Buchdahl limit in BNG does not guarantee stable interiors, because the sign of the pressure is controlled by which conservation equation is imposed.
  • The approximate analytic and numerical pressures agree on the existence of the transition but differ on its compactness ($0.4<X<0.5$ vs $0.5<X<0.6$), so the qualitative result is not tied to a single approximation.
  • The BNG potential itself remains well behaved up to $X\simeq0.7$, so the negative-pressure phenomenon is a matter-sector effect rather than a breakdown of the gravitational potential.

Reading between the lines

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

  • Beyond the paper: if the negative-pressure transition is physical, BNG's goal of describing non-singular black-hole interiors needs a matter model that can tolerate negative pressure or anisotropic stresses inside the horizon, because the perfect-fluid description fails before the horizon forms.
  • Beyond the paper: since Eq. (1.10) is a truncated expansion, a natural next step is to include the dropped $-G_N^2 \rho W'$ term and higher orders; if that restores positive pressure, the reported transition would be an artifact of the truncation rather than a property of BNG.
  • Beyond the paper: the same second-order GR pressure also turns negative around $X\simeq0.4$, so testing the new conservation equation with non-homogeneous density profiles would show whether the crossing is generic or specific to the homogeneous case.
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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 / 4 minor

Summary. The manuscript studies equilibrium equations in bootstrapped Newtonian gravity (BNG) for homogeneous stars. Starting from the BNG field equation (1.11), the authors consider three conservation equations: the Newtonian-type (1.9), the pressure-extended (3.1), and the full TOV-inspired equation (1.10)/(4.1) obtained in Appendix A by a post-Newtonian expansion. For each case they construct a quadratic analytic approximation and compare it with numerical solutions, and with an approximate GR pressure in harmonic coordinates. The main claimed result is that, for the full equation (4.1), the central pressure becomes negative for compactness between about 0.4 and 0.6, before the BNG horizon covers the star at X≈0.69, while no Buchdahl limit appears.

Significance. The question of whether BNG compact objects can lose hydrostatic support before horizon formation is interesting for the BNG program, and the paper is commendable for checking its analytic ansatz against numerical integrations and for presenting explicit comparisons with GR. The clean separation of the two conservation-equation effects and the small-compactness limits are useful. However, the significance is conditional: the negative-pressure result is a consequence of the adopted conservation equation (4.1), and that equation is not derived from the BNG action; the derivation in Appendix A contains an unjustified truncation. The paper's own caveat about the expansion's validity is never quantified in the regime where the sign change occurs.

major comments (4)
  1. [Appendix A, Eqs. (A.9)–(A.11)] The derivation of the equilibrium equation (1.10) is the load-bearing step for the paper's central claim, but the passage from Eq. (A.10) to Eq. (A.11) is not justified: the term -G_N^2 ρ W' is dropped, and the symbol V is silently promoted from the first-order coefficient of the expansion in Eq. (A.9) to the full BNG potential. No argument is given that W' is negligible, and the paper does not derive Eq. (1.10) from the BNG action (1.2). Since all negative-pressure results in Sections 4 and 5 are consequences of Eq. (4.1), the paper has not established that they are properties of BNG rather than of the adopted truncation.
  2. [Section 4 and Eq. (4.1)] The sign change of the central pressure occurs in a regime where the expansion used to obtain Eq. (1.10) is not controlled. At the analytically quoted crossing X∼0.4–0.5 and the numerically quoted crossing X∼0.5–0.6, X is not small, and the resummed term -2ρVV' is comparable in magnitude to the leading term -ρV'. Appendix A itself states that the validity of the formal expansion 'can only be checked a posteriori by estimating the size of the neglected higher-order terms for any given solution,' but no such estimate is provided for the solutions used here. Without an estimate of the omitted higher-order terms, the negative-pressure prediction is not a controlled consequence of the model.
  3. [Eq. (4.2) and Section 4] Equation (4.2), quoted as the small-X approximation for the pressure, changes sign at X=2/11≈0.18, not in the interval 0.4<X<0.5 claimed in the text. The displayed formula and the stated crossing interval are mutually inconsistent. Since the analytic crossing is cited as evidence for the main conclusion, this needs to be corrected or clarified.
  4. [Abstract and Sections 3.2, 5] The abstract's claim that BNG stars 'do not exhibit a Buchdahl limit regardless of the additional terms from the conservation equation' is broader than what the paper shows. Only the specific equations (1.8), (1.9), (3.1), and (4.1) are examined, and for Eq. (4.1) the pressure becomes negative before black-hole compactness rather than remaining a well-behaved positive pressure. The claim should be restricted to the cases actually studied, or supported by a general argument.
minor comments (4)
  1. [Section 5] 'BNG objets' should be 'BNG objects'.
  2. [Appendix A, Eq. (A.9)] Using V both for the effective potential and for the first-order coefficient makes the substitution from G_N m/\bar r to V hard to follow; a different symbol for the coefficient would clarify the truncation that leads to Eq. (A.11).
  3. [Eqs. (B.22)–(B.23)] As printed, substituting ξ=0 into Eq. (B.22) with the coefficient X/2 gives p(0)∝1-X, not the stated 1-2X^2. Please check the coefficient in the second-order term.
  4. [Fig. 9] The right panel shows the central pressure crossing zero, but the exact crossing values are not marked; adding gridlines or markers would help the reader verify the intervals quoted in the text.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the negative-pressure result is a computed consequence of the explicit model equations, not a fitted or self-referential output.

full rationale

The derivation chain is self-contained in the sense relevant here. The paper starts from an explicit BNG Lagrangian (1.2), derives the field equation (1.7)/(1.11), adopts the equilibrium equation (1.10) obtained in Appendix A by an explicit post-Newtonian expansion of the TOV equation, and then solves the resulting two-equation system with the boundary conditions (2.5)-(2.7). The central new result—that the central pressure turns negative for 0.4 < X < 0.5 in the analytic approximation and for 0.5 < X < 0.6 numerically—is a computed consequence of those equations, not a parameter fitted to reproduce that sign flip. The apparent dropping of the -G_N^2 rho W' term in passing from Eq. (A.10) to Eq. (A.11) is not a circular step: in Eq. (A.11) V is the full BNG potential, so expanding -(rho+p)V' reproduces the first-order terms plus the second-order W' contribution, while -2 rho V V' supplies the additional second-order term; thus Eq. (A.11) is a resummation of Eq. (A.10) to the stated order. The prior BNG works are cited for the vacuum solution and for the absence of a Buchdahl limit, but the present paper independently recomputes the interior solutions, compares the analytic approximation with numerical integrations, and exhibits the finite (and eventually negative) pressure behavior explicitly; the no-Buchdahl claim is therefore not imported solely by self-citation. No equation in the paper reduces by construction to the target negative-pressure statement, and no fitted parameter is renamed as a prediction. The heuristic status of Eq. (1.10) as a model assumption is a correctness/robustness concern, not a circularity concern.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the BNG model postulates and on an ad hoc translation of GR's TOV equation into the BNG language. No new particles or forces are introduced. The q couplings are the main hand-set parameters.

free parameters (1)
  • BNG coupling constants qV, qp, qρ = 1, 1, 1
    Set by hand to unity throughout; they control the strength of the nonlinear terms in the BNG Lagrangian (1.2). The negative-pressure result is computed at this point in parameter space and may not hold for other values.
assumptions (5)
  • ad hoc to paper BNG Lagrangian (1.2) with JV, Jp, Jρ terms is the correct model for compact objects.
    The field equation (1.11) follows from this Lagrangian, which is taken from previous self-cited work and is not derived from or tested against a more fundamental theory.
  • ad hoc to paper Conservation equation (1.10) obtained from TOV via replacement (A.9) and truncation is valid for BNG.
    Eq. (A.11) drops the W' term from Eq. (A.10), and the resulting Eq. (1.10) is imposed as the equilibrium condition. The central negative-pressure result depends on this equation.
  • domain assumption Homogeneous density profile (2.1).
    Chosen to allow analytic comparison with Newton and GR; the result is demonstrated only for this profile.
  • domain assumption Polynomial ansatz (3.2) and (3.4) for interior potential and pressure.
    The analytic solutions are obtained by truncating the Taylor expansion at r^2; the paper checks against numerics, so this is a mild but real assumption.
  • standard math Boundary conditions (2.5)-(2.7) plus p(R)=0 define the physical solution.
    Standard regularity, matching, and surface pressure conditions for static spherical stars.

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Cite this review

Pith. "Pith review of Star equilibrium: from BNG to TOV." pith.science (2026). https://pith.science/paper/OKF64UOW

@misc{pith2026250104632,
  author       = {Pith},
  title        = {Pith review of: Star equilibrium: from BNG to TOV},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKF64UOW}},
  note         = {Machine review of arXiv:2501.04632}
}
read the original abstract

We study the role of the equilibrium equation in bootstrapped Newtonian gravity (BNG) by including terms inspired by the post-Newtonian expansion of the Tolman-Oppenheimer-Volkov (TOV) equation. We then compare (approximate) BNG solutions for homogenous stars with their Newtonian and General Relativistic exact solutions. Regardless of the additional terms from the conservation equation, BNG stars do not exhibit a Buchdahl limit. However, specific extra terms added to this equation can cause the pressure to become negative inside stars with compactness smaller than the critical values for BNG black hole formation.

Figures

Figures reproduced from arXiv: 2501.04632 by the authors.

Figure 1
Figure 1. Approximate potential (3.3) for sources of compactness [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Left panel: density (solid line) and approximate central pressure (dashed line) in units of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Analytic approximation (3.5) for the pressure (in units of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Ratio (3.7) of proper mass over ADM-like mass. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Top panels: analytical approximation (3.3) (solid lines) and numerical (dashed lines) [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Top panels: analytical approximation (3.5) (solid lines) and numerical (dashed lines) [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Analytic approximation (3.5) for the BNG pressure (solid lines), numerical BNG pressure [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Gravitational energy UG in units of ADM-like mass M. 4 Additional pressure and potential terms We now consider the full conservation equation (1.10), that is p ′ ≃ − (ρ + p) V ′ − 4 π GN r p ρ − 2 ρ V V ′ . (4.1) The difficulty of obtaining analytic solutions is simila…
Figure 9
Figure 9. Figure 9: Left panel: density (solid curve) and central pressure (dashed curve) (in units of [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Pressure for compactness X = 0.1 (dotted), X = 0.2 (dot-dashed) and X = 0.4 (dashed) in units of G −3 N M−2 . increasing compactness as expected ( [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Ratio between proper and ADM-like mass. pnum, along with the approximate GR solution. These three quantities remain very close In the small compactness regime X ≪ 1. For X ≃ 0.1 the numerical pressure matches the analytic approximation and they both predict a smaller …
Figure 12
Figure 12. Figure 12: Top panels: analytical approximation (solid lines) and numerical (dashed lines) BNG [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: Analytic approximation for the BNG pressure (solid lines), numerical BNG pressure [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

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Reference graph

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