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arxiv: 1907.07661 · v1 · pith:DMVTVOK6new · submitted 2019-07-17 · 🌀 gr-qc

Extra dimensions' influence on the equilibrium and radial stability of strange quark stars

Jos\'e D. V. Arba\~nil , Geanderson A. Carvalho , Ronaldo V. Lobato , Rubens M. Marinho Jr. , Manuel Malheiro This is my paper

Pith reviewed 2026-05-24 20:13 UTC · model grok-4.3

classification 🌀 gr-qc
keywords strange quark starsextra dimensionsradial stabilityMIT bag modelSchwarzschild-Tangherlini metricstellar structure equationshigher-dimensional gravityradial perturbations
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0 comments X

The pith

Strange quark stars gain more radial stability as spacetime dimensions increase for fixed normalized mass and density ranges.

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

In higher-dimensional gravity the equilibrium and radial oscillation equations are solved for strange quark stars using an extended MIT bag equation of state. Within intervals of central energy density and total mass the configurations become more stable against small radial perturbations when the number of dimensions rises. The maximum of the normalized mass M G_d/(d-3) occurs at the same central density where the fundamental oscillation frequency reaches zero, so the sign of dM/dρ_cd continues to separate stable from unstable models. Radius and mass values depend on the chosen normalization of constants in the higher-dimensional theory. In the Newtonian limit the adiabatic index must satisfy Γ1 ≥ 2(d-2)/(d-1) for stability.

Core claim

For an interval of central energy densities ρ_cd G_d and total masses M G_d/(d-3), the stars gain more stability when the dimension is increased. The maximum value of M G_d/(d-3) and the zero eigenfrequency of oscillation are found with the same value of ρ_cd G_d; the peak value of M G_d/(d-3) marks the onset of instability. This indicates that the necessary and sufficient conditions to recognize regions constructed by stable and unstable equilibrium configurations against radial perturbations are, respectively, dM/dρ_cd>0 and dM/dρ_cd<0. Some physical parameters of the compact object depend on the normalization. Within the Newtonian framework, compact objects with adiabatic index Γ1 ≥ 2(d-2

What carries the argument

d-dimensional stellar structure and radial perturbation equations integrated with the MIT bag model equation of state and Schwarzschild-Tangherlini exterior metric.

If this is right

  • The turning-point criterion dM/dρ_cd >0 for stability remains valid for all d greater than or equal to 4.
  • Stable configurations exist at higher values of the normalized mass M G_d/(d-3) when d is increased.
  • The Newtonian stability bound on the adiabatic index rises with dimension as 2(d-2)/(d-1).
  • Both radius and total mass of equilibrium models depend on the normalization chosen for G_d and other constants.

Where Pith is reading between the lines

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

  • If the normalization dependence persists in other equations of state, direct comparison of masses across dimensions would require a consistent choice of units before observational constraints on extra dimensions can be drawn.
  • The result suggests that searches for stable compact objects above the four-dimensional maximum mass could be reinterpreted as possible signatures of higher-dimensional effects rather than new physics in four dimensions.
  • The same numerical setup could be applied to rotating or magnetized models to test whether the dimension dependence of the stability boundary survives those extensions.

Load-bearing premise

The MIT bag equation of state retains its functional form when extended to d greater than or equal to 4 and the exterior geometry is exactly the Schwarzschild-Tangherlini metric for every d.

What would settle it

A numerical integration showing that the fundamental radial frequency does not reach zero at the central density where M G_d/(d-3) reaches its maximum would falsify the claimed coincidence of turning point and stability boundary.

Figures

Figures reproduced from arXiv: 1907.07661 by Geanderson A. Carvalho, Jos\'e D. V. Arba\~nil, Manuel Malheiro, Ronaldo V. Lobato, Rubens M. Marinho Jr..

Figure 1
Figure 1. Figure 1: FIG. 1: Pressure versus energy density for a few different [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Total mass of the object against the total radius for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Ratio ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Fundamental mode eigenfrequency versus the mass [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Change of the fundamental mode eigenfrequency of [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Fundamental mode eigenfrequency against the central [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

We analyze the influence of extra dimensions on the static equilibrium configurations and stability against radial perturbations. For this purpose, we solve stellar structure equations and radial perturbation equations, both modified for a $d$-dimensional spacetime ($d\geq4$) considering that spacetime outside the object is described by a Schwarzschild-Tangherlini metric. These equations are integrated considering a MIT bag model equation of state extended for $d\geq4$. We show that the spacetime dimension influences both the structure and stability of compact objects. For an interval of central energy densities $\rho_{cd}\,G_d$ and total masses $MG_d/(d-3)$, we show that the stars gain more stability when the dimension is increased. In addition, the maximum value of $M{G_d}/(d-3)$ and the zero eigenfrequency of oscillation are found with the same value of $\rho_{cd}\,G_d$; i.e., the peak value of $M{G_d}/(d-3)$ marks the onset of instability. This indicates that the necessary and sufficient conditions to recognize regions constructed by stable and unstable equilibrium configurations against radial perturbations are, respectively, $dM/d\rho_{cd}>0$ and $dM/d\rho_{cd}<0$. We obtain that some physical parameter of the compact object in a $d$-dimensional spacetime, such as the radius and the mass, depend of the normalization. Finally, within the Newtonian framework, the results show that compact objects with adiabatic index $\Gamma_1\geq2(d-2)/(d-1)$ are stable against small radial perturbations.

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

3 major / 2 minor

Summary. The paper solves the d-dimensional stellar structure equations and radial perturbation equations for strange quark stars with an extended MIT bag EOS P=(ρ-4B)/3, assuming a Schwarzschild-Tangherlini exterior metric. It reports that, in normalized variables ρ_cd G_d and M G_d/(d-3), the range of stable configurations widens with increasing d, that the maximum mass coincides with the zero-eigenfrequency point, and that dM/dρ_cd>0 (respectively <0) is necessary and sufficient for stability against radial perturbations. A separate Newtonian analysis yields a d-dependent adiabatic-index stability bound.

Significance. If the EOS extension and metric assumptions are valid, the work supplies a concrete extension of the turning-point criterion to d>4 and quantifies how dimensionality enlarges the stable domain in normalized mass-central-density space. The independent integration of the perturbation equations (rather than assuming the turning-point result a priori) is a methodological strength that allows the coincidence of mass maximum and zero frequency to be an output.

major comments (3)
  1. [stellar structure equations / EOS paragraph] Section on stellar structure equations and the paragraph introducing the EOS: the MIT bag model is extended to d≥4 by retaining the identical linear form P=(ρ-4B)/3 without d-dependent corrections arising from higher-dimensional Fermi-gas thermodynamics or bag energy density. Because the reported gain in stability interval with d and the location of the mass peak rest directly on the slope and adiabatic index implied by this EOS, any d-dependent modification would shift both the turning-point location and the eigenfrequency zero-crossing, undermining the central claim.
  2. [results / normalization discussion] Results section (normalized quantities): the stability conclusions are presented in terms of ρ_cd G_d and M G_d/(d-3), yet the manuscript states that radius and mass themselves depend on the choice of normalization. It is not shown whether the reported widening of the stable interval with d survives a change in normalization convention or is an artifact of the particular scaling adopted.
  3. [Newtonian framework paragraph] Newtonian framework paragraph: the derived bound Γ_1 ≥ 2(d-2)/(d-1) is stated without explicit derivation or connection to the relativistic eigenfrequency calculation; for d=4 it recovers the familiar Γ_1≥4/3, but the consistency of this Newtonian limit with the relativistic d-dimensional perturbation equations is not demonstrated.
minor comments (2)
  1. [stellar structure equations] Notation for the gravitational constant is written G_d; its explicit d-dependence and relation to the usual 4D G should be stated once in the stellar-structure section.
  2. [abstract / stability discussion] The abstract claims the turning-point criterion is 'necessary and sufficient'; the text shows only that the two diagnostics coincide numerically, which is weaker than a general proof.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: Section on stellar structure equations and the paragraph introducing the EOS: the MIT bag model is extended to d≥4 by retaining the identical linear form P=(ρ-4B)/3 without d-dependent corrections arising from higher-dimensional Fermi-gas thermodynamics or bag energy density. Because the reported gain in stability interval with d and the location of the mass peak rest directly on the slope and adiabatic index implied by this EOS, any d-dependent modification would shift both the turning-point location and the eigenfrequency zero-crossing, undermining the central claim.

    Authors: We agree that the MIT bag EOS is extended phenomenologically by retaining P=(ρ-4B)/3 without incorporating d-dependent corrections from higher-dimensional Fermi-gas thermodynamics. This choice isolates the geometric effects of extra dimensions on equilibrium and stability while treating the EOS as an effective model. The reported trends hold specifically for this EOS; we will add an explicit statement of this assumption and its implications in the revised manuscript. revision: partial

  2. Referee: Results section (normalized quantities): the stability conclusions are presented in terms of ρ_cd G_d and M G_d/(d-3), yet the manuscript states that radius and mass themselves depend on the choice of normalization. It is not shown whether the reported widening of the stable interval with d survives a change in normalization convention or is an artifact of the particular scaling adopted.

    Authors: The normalization ρ_cd G_d and M G_d/(d-3) is the natural one arising from the d-dimensional Einstein equations and the Schwarzschild-Tangherlini metric, where the mass parameter scales as G_d M/(d-3). The manuscript already notes that physical parameters such as radius and mass depend on normalization. The widening of the stable interval with d originates from the explicit d-dependence in the hydrostatic equilibrium and perturbation equations; we will add a short discussion confirming that the qualitative trend persists under alternative scalings. revision: yes

  3. Referee: Newtonian framework paragraph: the derived bound Γ_1 ≥ 2(d-2)/(d-1) is stated without explicit derivation or connection to the relativistic eigenfrequency calculation; for d=4 it recovers the familiar Γ_1≥4/3, but the consistency of this Newtonian limit with the relativistic d-dimensional perturbation equations is not demonstrated.

    Authors: We will include an explicit derivation of the Newtonian bound Γ_1 ≥ 2(d-2)/(d-1) in an appendix of the revised manuscript, obtained from the d-dimensional Newtonian hydrostatic equilibrium and the condition for marginal stability against radial perturbations. We will also note its consistency with the relativistic d-dimensional equations, recovering the standard Γ_1 ≥ 4/3 result for d=4. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper numerically integrates the d-dimensional hydrostatic equilibrium and radial perturbation equations using the MIT bag EOS extended by direct substitution of the same linear form. The reported coincidence between the maximum of M G_d/(d-3) and the zero eigenfrequency is an output of that independent integration, not an input definition or fitted parameter renamed as a prediction. The turning-point criterion (dM/dρ_cd >0 for stability) is verified rather than presupposed. No self-citation load-bearing steps, ansatz smuggling, or self-definitional reductions appear in the derivation. The result is self-contained against the solved differential equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions: the exterior metric is Schwarzschild-Tangherlini for any d, and the MIT bag EOS extends directly to d dimensions without additional terms. No free parameters are fitted inside the derivation itself; normalization dependence is noted but not quantified as a fitted constant.

axioms (2)
  • domain assumption Exterior spacetime is described by the Schwarzschild-Tangherlini metric for d≥4
    Invoked for the matching condition at the stellar surface.
  • domain assumption MIT bag model equation of state extends to d-dimensional spacetime with the same functional form
    Used to close the stellar structure equations.

pith-pipeline@v0.9.0 · 5846 in / 1491 out tokens · 21729 ms · 2026-05-24T20:13:09.219808+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    We solve stellar structure equations and radial perturbation equations, both modified for a d-dimensional spacetime (d≥4) considering that spacetime outside the object is described by a Schwarzschild-Tangherlini metric. These equations are integrated considering a MIT bag model equation of state extended for d≥4.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean SphereAdmitsCircleLinking D ↔ D = 3 contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    For an interval of central energy densities ρ_cd G_d and total masses M G_d/(d-3), the stars gain more stability when the dimension is increased.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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