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arxiv: 2602.11297 · v2 · submitted 2026-02-11 · 🌀 gr-qc

Dynamical systems approach to stellar modelling in f(G, B) gravity

Sudan Hansraj , Christian G. Boehmer , Ndumiso Buthelezi This is my paper

Pith reviewed 2026-05-16 05:10 UTC · model grok-4.3

classification 🌀 gr-qc
keywords modified gravitystellar modelsdynamical systemsf(G,B) gravityisotropy conditioninvariant submanifoldsstability analysis
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The pith

Stellar isotropy in f(G, B) gravity yields an autonomous equation whose invariant submanifolds are stable attractors.

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

The paper splits the Ricci scalar into bulk and boundary terms to build a modified gravity theory for stellar interiors. Only the bulk functional enters the dynamics, so the resulting field equations stay second order and free of ghosts. For the pure quadratic case the isotropy condition becomes autonomous, a property the authors exploit by choosing a gauge that converts it into an autonomous dynamical system. Phase-space analysis then shows that the invariant submanifolds are stable, with nearby trajectories approaching them.

Core claim

The isotropy equation for stellar models in f(G, B) gravity is autonomous. After a gauge choice that splits the equation into a first-order system, the phase portrait reveals that the invariant submanifolds are generally stable fixed structures, with solution curves from nearby initial conditions converging toward them. Two distinct vacuum metrics arise when the energy-momentum tensor is set to zero.

What carries the argument

The autonomous isotropy equation, converted by gauge choice into a dynamical system whose phase portrait determines the stability of its invariant submanifolds.

Load-bearing premise

The chosen gauge keeps the isotropy equation autonomous while preserving the physical content of the stellar interior solutions.

What would settle it

A phase-portrait integration or numerical solution of the autonomous system in which trajectories diverge from the invariant submanifolds rather than approach them would show the claimed stability does not hold.

Figures

Figures reproduced from arXiv: 2602.11297 by Christian G. Boehmer, Ndumiso Buthelezi, Sudan Hansraj.

Figure 1
Figure 1. Figure 1: • On the conic 3U 2 + 4UV + 8U − V 2 + 8V = 0 the non-zero eigenvalue have the form λ2 = A1F1 + B1F2 A2 1 + B2 1 where A1 = 2(2V 2 + 6UV + 3U 2 ), B1 = 2U(3U + 4V ) and F1 = ∂F ∂U = −12U 3 − 30U 2V − 24U 2 − 14UV 2 − 48UV + 2V 3 − 16V 2 , F2 = ∂F ∂V = 2U(−5U 2 − 7UV − 12U + 3V 2 − 16V ) where F = −U(U + 2V )(3U 2 + 4UV + 8U − V 2 + 8V ) is the common parts of the numerators of U˙ and V˙ . The sign of λ2 de… view at source ↗
Figure 1
Figure 1. Figure 1: Phase space portrait of U, ˙ V˙ overlaid with invariant submanifolds U = 0, U + 2V = 0, and Q = 0 in dashed lines. The fixed point (0, 0), (0, 8), and (−16/3, 8/3) are in red. • unstable for U > − 16 3 . Additionally it may be observed that the conic Q(U, V ) = 0 is a one-dimensional set of equilibria (a shifted hyperbola). Its transverse stability λ⊥ varies by segment; it changes sign precisely at the int… view at source ↗
Figure 2
Figure 2. Figure 2: Global phase space portrait. The black dots denote critical points at infinity, red dots are the fixed [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
read the original abstract

The novel proposal to invoke the split of the Ricci scalar into bulk and boundary terms in the gravitational action, opens up a new avenue of investigation into stellar dynamics. The Lagrangian contains functional forms of the bulk term while the boundary term do not contribute to the dynamics. The advantage of the proposition is that the stellar structure equations are up to order two, thus the theory is not haunted by ghosts. We obtain explicitly the defining equations for the thermodynamical variables and the geometry for the pure quadratic case, since the linear case amounts to general relativity. In trying to establish the vacuum geometry associated with the theory it turns out that two possible metrics emerge through the vanishing of the energy-momentum tensor. Next, we analyse the isotropy equation and make the observation that it is autonomous. It is rare that this happens in astrophysical modelling. This behaviour prompted the use of dynamical systems to understand the stability properties of fixed points or invariant submanifolds. It was necessary to choose a gauge in order to split the autonomous equation into a system from which we could plot a phase portrait and deduce the stability of solution trajectories. We find that the invariant submanifolds were generally stable with nearby paths approaching them.

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

1 major / 1 minor

Summary. The paper proposes splitting the Ricci scalar into bulk (G) and boundary (B) terms in f(G,B) gravity for stellar modeling, yielding second-order structure equations free of ghosts. For the pure quadratic case it derives explicit equations for thermodynamic variables and geometry, identifies two vacuum metrics from vanishing energy-momentum tensor, observes that the isotropy equation is autonomous, imposes a gauge to obtain an explicit first-order system, and uses dynamical-systems methods to conclude that the invariant submanifolds are generally stable with nearby trajectories approaching them.

Significance. If the gauge choice is shown to preserve equivalence with the original field equations, the work opens a new route to stellar equilibria in modified gravity by exploiting the rare autonomy of the isotropy equation and applying phase-portrait analysis to establish stability of invariant submanifolds.

major comments (1)
  1. [dynamical systems analysis] In the dynamical-systems section where the gauge is chosen to split the autonomous isotropy equation: the stability result for the invariant submanifolds rests on trajectories in the reduced phase space. No explicit verification is supplied that every solution of the gauged system satisfies the unmodified f(G,B) field equations together with the stellar interior matching conditions; an auxiliary constraint not implied by the original equations could render the apparent stability unphysical or gauge-dependent.
minor comments (1)
  1. [abstract] Abstract, sentence on boundary term: 'the boundary term do not contribute' should read 'does not contribute'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address the concern directly below and commit to revisions that strengthen the presentation of the dynamical-systems analysis.

read point-by-point responses

  1. Referee: In the dynamical-systems section where the gauge is chosen to split the autonomous isotropy equation: the stability result for the invariant submanifolds rests on trajectories in the reduced phase space. No explicit verification is supplied that every solution of the gauged system satisfies the unmodified f(G,B) field equations together with the stellar interior matching conditions; an auxiliary constraint not implied by the original equations could render the apparent stability unphysical or gauge-dependent.

    Authors: We appreciate the referee highlighting the need for explicit equivalence between the gauged system and the original field equations. The gauge is introduced solely to reduce the autonomous isotropy equation to an explicit first-order dynamical system for phase-portrait analysis; it is chosen so that it is identically satisfied along the solution trajectories of the unmodified equations and does not impose an independent auxiliary constraint. Nevertheless, we acknowledge that the manuscript does not contain a direct back-substitution check confirming that all reduced trajectories, once lifted, satisfy the full f(G,B) field equations and the interior-exterior matching conditions. In the revised manuscript we will add a short subsection that performs this verification explicitly, thereby confirming that the reported stability of the invariant submanifolds is not an artifact of the gauge choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper starts from the f(G,B) action with Ricci split, derives the stellar field equations and isotropy condition explicitly, observes autonomy, introduces a gauge to obtain an explicit first-order autonomous system, and reads stability from the resulting phase portrait. No equation reduces to a fitted parameter renamed as prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled via prior work. The stability conclusion is a direct consequence of the constructed dynamical system and does not collapse to the input assumptions by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central construction rests on the validity of splitting the Ricci scalar and on the assumption that only the bulk term enters the field equations. No explicit free parameters or new entities are named in the abstract.

axioms (2)
  • domain assumption The Ricci scalar admits a split into bulk G and boundary B such that only G contributes to the dynamics.
    This is the novel proposal stated in the abstract that enables second-order equations.
  • domain assumption The isotropy equation can be rendered autonomous by a suitable gauge choice without loss of physical content.
    Required to convert the single equation into a dynamical system for phase-portrait analysis.

pith-pipeline@v0.9.0 · 5517 in / 1345 out tokens · 54992 ms · 2026-05-16T05:10:31.383653+00:00 · methodology

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