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arxiv: 2606.19879 · v1 · pith:SK7WVWCPnew · submitted 2026-06-18 · 🌀 gr-qc

Covariant Tolman-Oppenheimer-Volkoff equations in Energy-Momentum Squared Gravity

Eduardo Bittencourt , Mariam Campbell , Peter K. S. Dunsby , Sergio E. Jor\'as This is my paper

Pith reviewed 2026-06-26 16:56 UTC · model grok-4.3

classification 🌀 gr-qc
keywords energy-momentum squared gravitytolman-oppenheimer-volkoff equationstellar structurecovariant formalismeffective fluidmodified gravityspherical symmetrydynamical systems
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The pith

In Energy-Momentum Squared Gravity the stellar equilibrium equations retain the standard Tolman-Oppenheimer-Volkoff form when the nonlinear corrections are absorbed into effective fluid variables.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies static spherically symmetric stellar configurations in an extended class of Energy-Momentum Squared Gravity using the covariant 1+1+2 semi-tetrad formalism. For perfect physical fluids the nonlinear matter corrections are shown to act exactly like the stress-energy of an additional perfect fluid. The structure equations therefore keep the familiar Tolman-Oppenheimer-Volkoff shape once written in terms of effective density and pressure. The equations are given both in metric and dimensionless variables and reduce to an autonomous planar dynamical system whenever an effective closure relation exists. Specializing to linear equations of state recovers the general-relativity benchmark in some sectors while identifying others, particularly dust, where the planar reduction fails.

Core claim

For perfect physical fluids the nonlinear matter corrections arising from the Energy-Momentum Squared Gravity action can be reinterpreted as the stress-energy of an effective perfect fluid. This allows the covariant structure equations to retain the standard Tolman-Oppenheimer-Volkoff form when expressed in effective variables. The exterior spacetime remains Schwarzschild, yet the surface matching condition becomes the vanishing of the effective pressure rather than the physical pressure.

What carries the argument

Reinterpretation of the nonlinear matter corrections as the stress-energy tensor of an effective perfect fluid inside the covariant 1+1+2 semi-tetrad formalism for static spherical symmetry.

If this is right

  • The stellar equilibrium equations reduce to the standard Tolman-Oppenheimer-Volkoff form in effective variables.
  • For linear equations of state some sectors are exactly, asymptotically or piecewise equivalent to general relativity.
  • Dust configurations require the full three-dimensional covariant flow rather than a planar dynamical system.
  • The natural surface matching condition is the vanishing of effective pressure, which need not coincide with vanishing physical pressure for self-bound matter.
  • The phase space of stellar configurations is described globally by finite and asymptotic critical points of the reduced dynamical system.

Where Pith is reading between the lines

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

  • Numerical integration routines already written for the general-relativity Tolman-Oppenheimer-Volkoff equation could be reused after a simple redefinition of the input density and pressure.
  • Self-bound stars might possess a nonzero physical pressure at the surface while still matching smoothly to the exterior Schwarzschild geometry.
  • The breakdown of the planar reduction for dust indicates that pressureless matter may produce qualitatively different interior solutions than in general relativity.
  • The same effective-fluid rewriting may apply to other modified-gravity models whose field equations contain quadratic or higher powers of the matter tensor.

Load-bearing premise

The nonlinear matter corrections from the modified action can be exactly rewritten as the energy-momentum tensor of a perfect fluid when the spacetime is static and spherically symmetric.

What would settle it

A direct substitution of the effective density and pressure into the hydrostatic equilibrium equation that fails to hold identically would disprove the effective-fluid reinterpretation.

Figures

Figures reproduced from arXiv: 2606.19879 by Eduardo Bittencourt, Mariam Campbell, Peter K. S. Dunsby, Sergio E. Jor\'as.

Figure 1
Figure 1. Figure 1: Radiation case in EMSG with Lm = p and n = 1 2 . For this value of n, the effective closure is defined branchwise according to σ. On each branch, the system is topologically equivalent to the GR linear-EoS case under the replacement λGR → λσ, with λσ given by Eq. (66); consequently, the finite fixed points and the asymptotic directions follow the GR classification, with the stability at infinity determined… view at source ↗
Figure 2
Figure 2. Figure 2: Radiation case in EMSG with Lm = p and n = 1. This sector is dynamically equivalent to the GR radiation when written in terms of the effective variables, but the full effective (K, ˜ p˜)-plane is displayed because p˜ and µ˜ are not restricted a priori. Left panel: nullclines K˜ ′ = 0 and p˜ ′ = 0, shown as solid and dashed curves, respectively. Their intersections give P0, P1/4, PK0, and Pint. The point PK… view at source ↗
Figure 3
Figure 3. Figure 3: Dust sector in EMSG for Lm = p and n = 1, with η = 1.0. Top: representative integral curves of the full 3D covariant system in the physically relevant sector (ϕ, A, ρ) > 0, showing orbits that connect a regular central asymptotic regime (indicated by the dots) to the vacuum boundary ρ = 0. The semitransparent coordinate planes are included only as guides to the eye. Bottom: projections of the same trajecto… view at source ↗
read the original abstract

We study static, spherically symmetric stellar configurations in an extended class of Energy--Momentum Squared Gravity using the covariant \(1+1+2\) semi-tetrad formalism. For perfect physical fluids, we show that the nonlinear matter corrections can be reinterpreted as an effective perfect fluid, so that the stellar equilibrium equations retain the standard Tolman--Oppenheimer--Volkoff form when written in terms of effective variables. The resulting covariant structure equations are formulated in both metric and dimensionless variables and, whenever an effective closure relation exists, reduce to an autonomous planar dynamical system. This provides a global qualitative description of the stellar phase space in terms of finite and asymptotic critical points. Specializing to linear physical equations of state, we recover the general relativistic benchmark and identify sectors that are exactly, asymptotically, or piecewise equivalent to general relativity, as well as sectors -- particularly dust configurations -- for which the planar reduction breaks down and the full three-dimensional covariant flow must be considered. We further recover the standard metric Tolman--Oppenheimer--Volkoff equation in terms of effective variables and show that, although the exterior spacetime remains Schwarzschild, the natural matching condition at the stellar surface is \(p_{\rm eff}(R)=0\), which need not coincide with \(p(R)=0\) for self-bound matter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript investigates static spherically symmetric stellar models in Energy-Momentum Squared Gravity using the covariant 1+1+2 semi-tetrad formalism. For perfect fluids it shows that nonlinear matter corrections can be reinterpreted as an effective perfect fluid, so the stellar equilibrium equations retain the standard Tolman-Oppenheimer-Volkoff form when expressed in effective variables. The covariant structure equations are given in both metric and dimensionless variables and reduce to an autonomous planar dynamical system whenever an effective closure exists. Specializing to linear physical equations of state recovers the GR benchmark, identifies sectors of exact/asymptotic/piecewise equivalence to GR, and notes that dust configurations require the full three-dimensional flow. The exterior remains Schwarzschild, but the surface matching condition is p_eff(R)=0.

Significance. If the effective-fluid reinterpretation is established by direct substitution into the field equations under the static spherical symmetry and perfect-fluid ansatz, the paper supplies a covariant route to stellar structure that reuses familiar TOV machinery while exposing the modified-gravity corrections explicitly. The dynamical-systems reduction supplies a global qualitative description of the phase space via critical points, which is a genuine methodological advance for this class of theories. Recovery of GR limits and the explicit caveat for dust are appropriately cautious and strengthen the comparative value of the results.

minor comments (2)
  1. The abstract states that the planar reduction holds 'whenever an effective closure relation exists'; an explicit example of such a closure (or the condition under which it is guaranteed for linear EOS) should be stated early in the introduction or in the section deriving the reduced system.
  2. Notation for physical versus effective quantities (p vs p_eff, ho vs ho_eff) must be introduced once and used consistently; any place where the effective variables are first defined should include a short table or list of the redefinitions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of our results on the effective-fluid reinterpretation, the dynamical-systems reduction, and the caveats for dust configurations. The recommendation for minor revision is noted. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper begins from the field equations of Energy-Momentum Squared Gravity and applies the covariant 1+1+2 formalism to perfect fluids under static spherical symmetry. It demonstrates that nonlinear corrections can be exactly recast as the stress-energy of an effective perfect fluid (same 4-velocity, isotropic pressure), allowing the structure equations to retain TOV form in effective variables. This is a direct algebraic re-expression from the given action and symmetries, with no fitted parameters renamed as predictions, no load-bearing self-citations, and no ansatz or uniqueness theorem imported from prior author work. The central claim is therefore independent of its inputs and self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The Energy-Momentum Squared Gravity action is presumed to contain at least one free coupling parameter, but its value and fitting procedure are not stated. The derivation rests on the domain assumptions listed below.

axioms (2)
  • domain assumption The spacetime is static and spherically symmetric
    Invoked to apply the 1+1+2 semi-tetrad formalism to stellar configurations.
  • domain assumption Matter is described by a perfect fluid
    Stated explicitly for the fluids for which the effective-fluid reinterpretation is shown.

pith-pipeline@v0.9.1-grok · 5782 in / 1406 out tokens · 39178 ms · 2026-06-26T16:56:00.518787+00:00 · methodology

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Works this paper leans on

34 extracted references · 1 canonical work pages

  1. [1]

    Roshan M and Shojai F 2016Phys. Rev. D94044002 (Preprint1607.06049)

    Pith/arXiv arXiv

  2. [2]

    Katırcı N and Kavuk M 2014Eur. Phys. J. Plus129163 (Preprint1302.4300)

    Pith/arXiv arXiv

  3. [3]

    Board C V R and Barrow J D 2017Phys. Rev. D96123517 (Preprint1709.09501)

    Pith/arXiv arXiv

  4. [4]

    Akarsu Ö, Katırcı N and Kumar S 2018Phys. Rev. D97024011 (Preprint1709.02367)

    Pith/arXiv arXiv

  5. [5]

    Akarsu Ö, Barrow J D, Çikıntoğlu S, Ekşi K Y and Katırcı N 2018Phys. Rev. D97124017 (Preprint1802.02093)

    Pith/arXiv arXiv

  6. [6]

    Nazari E, Roshan M and De Martino I 2022Phys. Rev. D105044014 (Preprint2201.08578)

    arXiv

  7. [7]

    Cipriano R A C, Ganiyeva N, Harko T, Lobo F S N, Pinto M A S and Rosa J L 2024Universe10 339 (Preprint2408.14106)

    arXiv

  8. [8]

    Dunsby P K S, Caldis M A and Bittencourt E 2025 Cosmological perturbations in energy- momentum squared gravity (Preprint2511.05166) URLhttps://arxiv.org/abs/2511.05166

    Pith/arXiv arXiv 2025

  9. [9]

    Rev.55364–373

    Tolman R C 1939Phys. Rev.55364–373

  10. [10]

    Rev.55374–381

    Oppenheimer J R and Volkoff G M 1939Phys. Rev.55374–381

  11. [11]

    Nari N and Roshan M 2018Phys. Rev. D98024031 (Preprint1802.02399)

    Pith/arXiv arXiv

  12. [12]

    Alam N, Pal S, Rahmansyah A and Sulaksono A 2024Phys. Rev. D109083007 (Preprint 2309.06022)

    arXiv

  13. [13]

    Banerjee A, İzzet Sakallı, Rayimbaev J, Ibragimov I and Muminov S 2025Physics of the Dark Universe48101866 ISSN 2212-6864

  14. [14]

    Dark Univ.31 100774 (Preprint2007.00455)

    Singh K N, Banerjee A, Maurya S K, Rahaman F and Pradhan A 2021Phys. Dark Univ.31 100774 (Preprint2007.00455)

    arXiv

  15. [15]

    Phys.458169440 (Preprint2308.05197)

    Pretel J M Z, Tangphati T and Banerjee A 2023Ann. Phys.458169440 (Preprint2308.05197)

    arXiv

  16. [16]

    Nasir M M M, Bhatti M Z and Yousaf Z 2023Int. J. Mod. Phys. D322350009

  17. [17]

    Naz S and Sharif M 2022Universe8142

  18. [18]

    Quantum Grav.203855–3884 (Preprintgr-qc/ 0209051)

    Clarkson C A and Barrett R K 2003Class. Quantum Grav.203855–3884 (Preprintgr-qc/ 0209051)

  19. [19]

    Carloni S and Vernieri D 2018Phys. Rev. D97124056 (Preprint1709.02818)

    Pith/arXiv arXiv

  20. [20]

    Naidu N F, Carloni S and Dunsby P K S 2021Phys. Rev. D104044014 (Preprint2104.09595)

    arXiv

  21. [21]

    Nzioki A M, Carloni S, Goswami R and Dunsby P K S 2010Phys. Rev. D81084028 (Preprint 0908.3333)

    Pith/arXiv arXiv

  22. [22]

    Quantum Grav.42085014 (Preprint2403.00070)

    Campbell M, Carloni S, Dunsby P K S and Naidu N F 2025Class. Quantum Grav.42085014 (Preprint2403.00070)

    arXiv

  23. [23]

    Harko T, Lobo F S N, Nojiri S and Odintsov S D 2011Phys. Rev. D84024020 (Preprint 1104.2669)

    Pith/arXiv arXiv

  24. [24]

    Akarsu Ö, Bouhmadi-López M, Katırcı N, Nazari E, Roshan M and Uzun N M 2024Phys. Rev. D109104055 (Preprint2305.07506)

    arXiv

  25. [25]

    Clarkson C 2007Phys. Rev. D76(10) 104034 URLhttps://link.aps.org/doi/10.1103/ PhysRevD.76.104034

  26. [26]

    Bittencourt E, Campbell M, Dunsby P K S and Jorás S E 2026 Covariant dynamical systems formulation of the tolman-oppenheimer-volkoff equations (Preprint2605.26187) URLhttps: //arxiv.org/abs/2605.26187

    Pith/arXiv arXiv 2026

  27. [27]

    Collins C B 1977Journal of Mathematical Physics181374–1377 ISSN 0022-2488 URLhttps: //doi.org/10.1063/1.523431

    work page doi:10.1063/1.523431

  28. [28]

    Collins C B 1985Journal of Mathematical Physics262268–2274 ISSN 0022-2488

  29. [29]

    Nilsson U S and Uggla C 2000Annals of Physics286278–291

  30. [30]

    Nilsson U S and Uggla C 2000Annals of Physics286292–319

  31. [31]

    Heinzle J M, Röhr N and Uggla C 2003Classical and Quantum Gravity204567–4586 (Preprint gr-qc/0304012) Covariant TOV equations in EMSG26

    Pith/arXiv arXiv

  32. [32]

    Wainwright J and Ellis G F R (eds) 1997Dynamical Systems in Cosmology(Cambridge: Cambridge University Press) ISBN 0521554578

  33. [33]

    Bahamonde S, Marciu M and Rudra P 2019Phys. Rev. D100083511 (Preprint1906.00027)

    arXiv

  34. [34]

    Dumortier F, Llibre J and Artés J C 2006Qualitative Theory of Planar Differential Systems Universitext (Berlin Heidelberg: Springer) ISBN 978-3-540-32893-3