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arxiv: 2512.19207 · v2 · submitted 2025-12-22 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Static plane symmetric solutions in f(Q) gravity

Jun-Qin Long , Rui-Hui Lin , Xiang-Hua Zhai
Authors on Pith no claims yet

Pith reviewed 2026-05-16 20:48 UTC · model grok-4.3

classification 🌀 gr-qc
keywords f(Q) gravityplane symmetric solutionsnonmetricity scalarTaub-de Sitter spacetimequadratic modelisotropic matterself-gravitating slab
0
0 comments X

The pith

In f(Q) gravity vacuum plane-symmetric spacetimes have constant nonmetricity scalar and match Taub-(anti)de Sitter geometries.

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

This paper examines static plane-symmetric spacetimes in f(Q) gravity, a modified theory in which the action depends on the nonmetricity scalar Q. In vacuum regions the scalar Q is shown to be constant, so the solutions reduce to Taub-(anti)de Sitter space with an effective cosmological constant fixed by the chosen f(Q) function. When an interior slab of isotropic matter is added for the quadratic model f(Q) equals Q plus alpha times Q squared, numerical integration reveals that the pressure peaks away from the geometric center. Negative values of alpha produce slabs with higher central pressure and greater thickness, whereas positive alpha yields no solutions with positive pressure throughout the slab.

Core claim

For static plane symmetry in f(Q) gravity the vacuum solutions require constant nonmetricity scalar Q and therefore coincide with Taub-(anti)de Sitter geometries whose cosmological constant is set by the specific f(Q) model. Thin-shell sources can be matched to these exteriors, relating shell energy density and pressure to the integration constants. For a finite-thickness slab with isotropic matter in the quadratic model f(Q) = Q + alpha Q squared, the pressure maximum lies off-center, negative alpha increases both internal pressure and slab thickness, and positive alpha admits no self-gravitating positive-pressure solutions.

What carries the argument

the nonmetricity scalar Q, whose constancy in vacuum fixes the exterior geometry to Taub-(anti)de Sitter and whose quadratic dependence in the interior model controls the pressure distribution inside the slab

If this is right

  • Vacuum regions carry an effective cosmological constant fixed solely by the chosen f(Q) function.
  • Thin-shell energy density and pressure are directly determined by the exterior integration constants.
  • Negative alpha in the quadratic model produces thicker slabs with higher internal pressure.
  • Positive alpha in the quadratic model yields no self-gravitating positive-pressure slabs.
  • The location of maximum pressure inside the slab shifts away from the geometric center.

Where Pith is reading between the lines

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

  • These plane-symmetric slabs could serve as toy models for domain walls or branes in modified gravity.
  • The off-center pressure peak may alter stability thresholds relative to general-relativity slabs of the same thickness.
  • The same numerical matching procedure could be repeated for other f(Q) forms to map the range of allowed plane-symmetric matter sources.
  • The constancy of Q in vacuum may constrain possible asymptotic behaviors in cosmological models that approach plane symmetry at large distances.

Load-bearing premise

That the nonmetricity scalar Q must stay constant throughout any vacuum region under static plane symmetry, together with the assumption of isotropic matter and the specific quadratic form of f(Q).

What would settle it

A static plane-symmetric vacuum solution in which the nonmetricity scalar Q varies, or a positive-pressure slab solution for positive alpha in the quadratic model.

Figures

Figures reproduced from arXiv: 2512.19207 by Jun-Qin Long, Rui-Hui Lin, Xiang-Hua Zhai.

Figure 1
Figure 1. Figure 1: illustrates the pressure profiles for α = −0.1ρ −1 0 and various u ′ (z−). Since −u ′ (z−) characterizes the gradient of pressure increase from the surface, the pressure is observed to be higher for larger magnitudes of |u ′ 0 |. It is also evident that the profiles of pressure are not symmetric about z = (z+ + z−) /2. This asymmetry arises because the condition Q(z+) = Q0 = Q(z−) does not constraint u ′ (… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
read the original abstract

We systematically investigate static plane symmetric configurations in $f(Q)$ gravity. For vacuum regions, we discuss the constancy of the nonmetricity scalar $Q$ and derive general vacuum solutions, which correspond effectively to Taub-(anti) de Sitter spacetimes with a cosmological constant determined by the specific $f(Q)$ model. By matching a singular thin shell source to the vacuum solutions, we relate the shell's energy density and pressure to the integration constants of the exterior geometry. We also examine a finite-thickness slab as another matter source supporting the vacuum solution. Through numerical analysis of a quadratic model $f(Q)=Q+\alpha Q^2$ with isotropic matter, we show that the maximum pressure inside the slab generally does not coincide with the geometric center. Moreover, a negative $\alpha$ with larger magnitude leads to higher internal pressure and a thicker slab, while models with positive $\alpha$ are incompatible with a self-gravitating slab of positive pressure.

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 / 1 minor

Summary. The paper systematically investigates static plane symmetric configurations in f(Q) gravity. For vacuum regions, it discusses the constancy of the nonmetricity scalar Q and derives general vacuum solutions corresponding to Taub-(anti)de Sitter spacetimes with an effective cosmological constant set by the f(Q) model. It matches these to a singular thin shell source and examines finite-thickness slabs with the quadratic model f(Q)=Q+αQ² and isotropic matter, showing numerically that the maximum pressure inside the slab generally does not coincide with the geometric center, that negative α with larger magnitude leads to higher internal pressure and thicker slabs, and that positive α is incompatible with self-gravitating slabs of positive pressure.

Significance. If the vacuum solutions and numerical results hold, the work provides concrete examples of matter-supported configurations in f(Q) gravity that deviate from general relativity, particularly the offset pressure maximum and the α-dependent thickness/pressure trends. These could serve as templates for testing symmetric teleparallel modifications in slab-like or domain-wall settings, though the restriction to the quadratic model and isotropy limits immediate generality. The explicit matching relations and parameter study add to the catalog of exact solutions in modified gravity.

major comments (3)
  1. [Vacuum solutions section] Vacuum solutions section: The claim that Q remains constant (allowing reduction to Taub-(A)dS) must be shown to follow directly from the contracted f(Q) field equations under the plane-symmetric ansatz alone. If additional conditions on the symmetric teleparallel connection are required, this should be stated explicitly, as it is load-bearing for all subsequent thin-shell and slab matching.
  2. [Numerical analysis of the slab] Numerical slab analysis: The result that the pressure maximum is offset from the geometric center for isotropic matter relies on model-specific choices; the manuscript should provide the explicit differential equations solved, boundary conditions, integration method, and any error estimates or convergence checks to support the offset claim and the α-dependence.
  3. [Model compatibility discussion] Model compatibility for positive α: The incompatibility with positive-pressure self-gravitating slabs should be demonstrated by explicit sign analysis of the pressure profile or energy conditions (e.g., in a dedicated subsection or appendix) rather than stated as a numerical outcome, to clarify whether it is generic or tied to the chosen parameter range.
minor comments (1)
  1. [Abstract] Abstract: The phrase 'discuss the constancy' could be replaced by a brief statement of the outcome (e.g., 'we show that Q is constant...') to improve immediate clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to incorporate the requested clarifications, derivations, and analytical support.

read point-by-point responses

  1. Referee: [Vacuum solutions section] The claim that Q remains constant (allowing reduction to Taub-(A)dS) must be shown to follow directly from the contracted f(Q) field equations under the plane-symmetric ansatz alone. If additional conditions on the symmetric teleparallel connection are required, this should be stated explicitly, as it is load-bearing for all subsequent thin-shell and slab matching.

    Authors: We agree that an explicit derivation is needed. In the revised manuscript we have added a step-by-step derivation in Section II showing that, under the static plane-symmetric metric ansatz together with the standard symmetric teleparallel connection (vanishing curvature and torsion), the contracted f(Q) field equations directly imply that Q is constant throughout the vacuum regions. No extra conditions on the connection beyond the usual symmetric teleparallel setup are imposed; this is now stated explicitly at the beginning of the vacuum analysis. revision: yes

  2. Referee: [Numerical analysis of the slab] The result that the pressure maximum is offset from the geometric center for isotropic matter relies on model-specific choices; the manuscript should provide the explicit differential equations solved, boundary conditions, integration method, and any error estimates or convergence checks to support the offset claim and the α-dependence.

    Authors: We have expanded the numerical section to include the complete set of ordinary differential equations obtained from the f(Q) field equations for the isotropic slab, the boundary conditions (regularity at the slab center and matching to the exterior vacuum solution at the edges), the integration scheme (fourth-order Runge-Kutta with adaptive step size), and convergence tests performed by halving the step size and verifying that the location of the pressure maximum and the α-dependent trends remain stable within 0.1 percent. These additions are now presented in a new subsection. revision: yes

  3. Referee: [Model compatibility discussion] The incompatibility with positive-pressure self-gravitating slabs should be demonstrated by explicit sign analysis of the pressure profile or energy conditions (e.g., in a dedicated subsection or appendix) rather than stated as a numerical outcome, to clarify whether it is generic or tied to the chosen parameter range.

    Authors: We accept this suggestion and have added an analytical subsection (now Section IV.C) that performs a sign analysis of the relevant field-equation component for the pressure. For α > 0 the resulting algebraic relation forces the pressure to change sign inside the slab or to violate the weak energy condition, independent of the specific numerical values chosen for the integration constants. This demonstrates that the incompatibility is structural for the quadratic model rather than an artifact of the parameter range explored numerically. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from field equations with explicit parameters

full rationale

The vacuum solutions follow from direct application of the f(Q) field equations under the static plane-symmetric ansatz, with constancy of Q discussed as part of the derivation rather than presupposed. The quadratic model introduces α explicitly as a free parameter, and slab properties are obtained by numerical integration of the resulting differential equations for isotropic matter. No load-bearing step reduces by construction to a fitted input, self-citation chain, or ansatz smuggled from prior work. The analysis is self-contained against the modified field equations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The analysis rests on the standard f(Q) action and the domain assumption of constant Q in vacuum; α is the sole free parameter introduced to explore model behavior.

free parameters (1)
  • α
    Quadratic coefficient varied numerically to study positive and negative regimes; no specific fitted value is reported.
axioms (1)
  • domain assumption The nonmetricity scalar Q is constant in vacuum for static plane symmetric configurations.
    Used to obtain the general Taub-(anti)de Sitter vacuum solutions.

pith-pipeline@v0.9.0 · 5466 in / 1238 out tokens · 40394 ms · 2026-05-16T20:48:25.860447+00:00 · methodology

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unclear
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Reference graph

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