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arxiv: 2606.06517 · v1 · pith:VDRYFSTTnew · submitted 2026-06-02 · 🌀 gr-qc

Odd-parity perturbations of trace-quadratic f(R,T) black holes with anisotropic matter: admissible branches, axial ringdown, and a coupled-PINN benchmark

Mushtaq Ahmad , M. Farasat Shamir , Adnan Malik , Ahdab K. Althukair This is my paper

Pith reviewed 2026-06-28 09:01 UTC · model grok-4.3

classification 🌀 gr-qc
keywords f(R,T) gravityblack hole perturbationsquasinormal modesaxial perturbationsanisotropic fluidodd-parity modestrace-quadratic gravity
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The pith

The odd sector around admissible trace-quadratic f(R,T) black holes reduces exactly to Einstein gravity plus a frozen anisotropic fluid, producing a 22% shift in axial quasinormal modes with no resolved alpha dependence.

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

This paper maps the parameter space of constant closure parameters (w_r, w_t) that permit regular horizons, asymptotic flatness, and hyperbolic evolution for odd-parity perturbations in trace-quadratic f(R,T) gravity. Only the negative-w_r branch satisfies these background conditions; the positive-w_r family commonly used in the literature fails the regularity test and is kept solely for numerical comparison. On admissible backgrounds the unreduced axial system collapses to a single gauge-invariant master equation identical to that of Einstein gravity coupled to a static effective anisotropic fluid. For the anchored closure (w_r, w_t) = (-0.2, 0.15) an exact Chebyshev solution shows the mass-normalized fundamental ell=2 mode frequency lies about 22% away from the Schwarzschild value, while direct dependence on the trace-coupling strength alpha remains statistically unresolved over 0 less than or equal to alpha/M squared less than or equal to 0.3. The dominant observable signature therefore traces to the existence of the matter-supported branch rather than to variation of the modified-gravity parameter.

Core claim

On static admissible backgrounds the odd sector is exactly equivalent to Einstein gravity coupled to a frozen effective anisotropic fluid, so the physical axial spectrum is governed by a single gauge-invariant master equation. For the anchored branch (w_r,w_t)=(-0.2,0.15) the mass-normalized spectrum differs from Schwarzschild by about 22%, whereas no statistically resolved direct alpha-dependence appears within the conservative spectral envelope over 0 less than or equal to alpha/M squared less than or equal to 0.3. Within this closure the main observable imprint in axial ringdown comes from the existence of the matter-supported branch itself, not from direct variation of the trace coupling

What carries the argument

the single gauge-invariant master equation that governs the axial sector once the odd-parity system is reduced on admissible backgrounds

Load-bearing premise

The constant closure parameters allow a regular horizon, asymptotic flatness, and hyperbolic evolution only for negative w_r.

What would settle it

Direct numerical integration of the background equations for any positive w_r value that produces a singular horizon or violates asymptotic flatness would confirm the claimed restriction to the negative-w_r family.

Figures

Figures reproduced from arXiv: 2606.06517 by Adnan Malik, Ahdab K. Althukair, M. Farasat Shamir, Mushtaq Ahmad.

Figure 1
Figure 1. Figure 1: Analytic admissibility wedge for the constant-(wr, wt) closure. The shaded region satisfies the horizon-regularity condition −1/3 ≤ wr < 0 and the finite-mass condition wt > −wr/2. The star marks the anchored admissible branch used for physical inference, while the cross marks the positive-wr comparison family retained only for numerical benchmarking [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Coupled PINN architecture for the compactified two-field axial eigenproblem. The shared encoder extracts common radial features, the two decoder branches generate field-specific latent outputs, and the boundary transform enforces Eq. (47) to produce the regular fields g0 and g1. The physical perturbations are reconstructed as ha = Fga. 3.2. Coupled PINN architecture and exact horizon regularity The PINN le… view at source ↗
Figure 3
Figure 3. Figure 3: Production scan on the anchored admissible negative-wr branch, (wr, wt) = (−0.2, 0.15) with ρs = 0.01835620783535. Left: ℜ(Ω). Right: −ℑ(Ω). Points show the finest-grid frequencies from the exact axial master equation, dashed horizontal lines mark the α = 0 baselines, and error bars show the conservative spectral envelope from inter-grid spread. The scan is effectively flat: the pointwise central values va… view at source ↗
Figure 4
Figure 4. Figure 4: Background quantities along the anchored admissible negative-wr branch. Left: M/rH. Right: γH. Dashed horizontal lines mark the α = 0 baselines. Both remain essentially constant across the full scan, which explains the negligible direct α-dependence of the ringdown spectrum. 4.3. Mass-normalized spectra and ringdown diagnostics Mass-normalized frequencies are the quantities most directly tied to fixed-mass… view at source ↗
Figure 5
Figure 5. Figure 5: Mass-normalized axial QNM frequencies on the anchored admissible negative-wr branch. Left: ℜ(Mω). Right: −ℑ(Mω). Dashed horizontal lines mark the α = 0 baselines, and error bars show the same conservative spectral envelope used in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mass-normalized axial QNM frequencies for the positive-wr comparison branch. Left: ℜ(Mω). Right: −ℑ(Mω). Dashed horizontal lines mark the α = 0 values, while the dotted vertical line and shaded band indicate the same near-boundary regime shown in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Frequencies on the positive-wr comparison branch for wr = 0.2 and wt = 0.15. Left: ℜ(Ω). Right: −ℑ(Ω). Dashed horizontal lines show the Schwarzschild values. The dotted vertical line marks the interpolated hyperbolicity zero, αcrit/M2 ≈ 0.336, and the shaded band indicates the near-boundary regime where the linear problem becomes marginal or inadmissible. Circles denote externally validated points (α/M2 ≤ … view at source ↗
Figure 8
Figure 8. Figure 8: Admissibility diagnostics for the positive-wr comparison branch. Left: the minimum characteristic-speed diagnostic Hmin. Right: the final total loss on a logarithmic scale. The dashed horizontal lines mark Hmin = 0 and Ltot = 0.1, respectively. The dotted vertical line marks the interpolated hyperbolicity zero at α/M2 ≈ 0.336, and the shaded band denotes the near-boundary regime where the extracted frequen… view at source ↗
Figure 9
Figure 9. Figure 9: Independent outward integration on an admissible background, (wr, wt, α) = (−0.2, 0.15, 0) with rH = 1 and ρs = 0.01835620783535. Left: the matter profile ρ0(r) together with the reference power law r −3.5 predicted by Eq. (19). Right: the metric function N(r) and its asymptotic approach to 1 − 2M/r. The fitted exponent nfit = 3.497 agrees with the analytic prediction n∞ = 3.5. C. WEAK-COUPLING FREQUENCY S… view at source ↗
Figure 10
Figure 10. Figure 10: Percentage frequency shifts in the anisotropy scan on the positive-wr comparison branch at fixed α/M2 = 0.1 as a func￾tion of wt for three values of wr. The left panel shows 100 ∆ℜ(Ω)/ℜ(ΩSchw), and the right panel shows 100 ∆[−ℑ(Ω)]/[−ℑ(ΩSchw)]. The steeper dependence on wt quantifies the stronger sensitivity to tangential pressure; the branch is shown here only for numerical comparison [PITH_FULL_IMAGE:… view at source ↗
read the original abstract

We study odd-parity gravitational perturbations of static black holes in trace-quadratic $f(R,T)=R+\alpha T^2$ gravity supported by an anisotropic effective fluid with constant closure parameters $(w_r,w_t)$. From the unreduced axial system and its principal symbol, we identify the sector of parameter space that supports a regular horizon, asymptotic flatness, and hyperbolic odd-sector evolution. Within this closure the admissible branch lies at negative $w_r$, while the commonly used positive-$w_r$ family fails the background regularity test and is kept only as a numerical comparison branch. On static admissible backgrounds the odd sector is exactly equivalent to Einstein gravity coupled to a frozen effective anisotropic fluid, so the physical axial spectrum is governed by a single gauge-invariant master equation. For the anchored branch $(w_r,w_t)=(-0.2,0.15)$ we compute the fundamental axial $\ell=2$ quasinormal mode with an exact Chebyshev solve. The mass-normalized spectrum differs from Schwarzschild by about $22\%$, whereas no statistically resolved direct $\alpha$-dependence appears within the conservative spectral envelope over $0\le \alpha/M^2\le 0.3$. We also construct a coupled physics-informed neural network for the unreduced two-field eigenproblem and use it to benchmark the inadmissible comparison branch. A closure-level audit of the anchored family shows positive diagnostic combinations associated with the null, weak, and dominant energy conditions, denominator safety in the modified balance law, and an effective exterior mass fraction of about $20\%$, while indicating that the constant-$(w_r,w_t)$ model should be read as an effective anisotropic stress rather than as a microphysical fluid. Within this closure, the main observable imprint in axial ringdown comes from the existence of the matter-supported branch itself, not from direct variation of the trace coupling.

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 / 3 minor

Summary. The paper studies odd-parity perturbations of static black holes in trace-quadratic f(R,T)=R+αT² gravity supported by an anisotropic fluid with constant closure parameters (w_r,w_t). It identifies the admissible branch (negative w_r) that permits regular horizons, asymptotic flatness, and hyperbolic evolution; shows that on such backgrounds the odd sector reduces exactly to Einstein gravity plus a frozen effective anisotropic fluid, yielding a single gauge-invariant master equation with no direct α dependence; computes the fundamental ℓ=2 axial QNM for the anchored branch (w_r,w_t)=(-0.2,0.15) via exact Chebyshev spectral method, finding a ~22% mass-normalized shift relative to Schwarzschild; benchmarks the inadmissible positive-w_r branch with a coupled PINN; and audits the anchored family for energy conditions and effective mass fraction.

Significance. If the claimed exact reduction holds, the result cleanly separates background effects from direct modified-gravity corrections in axial ringdown, showing that the dominant imprint arises from the existence of the matter-supported branch itself. The explicit energy-condition audit, denominator-safety check, and reproducible Chebyshev/PINN numerics strengthen the contribution; the absence of resolved α-dependence within 0≤α/M²≤0.3 is a falsifiable statement that can be tested against future data.

minor comments (3)
  1. [Abstract / Numerical results] The abstract states an 'exact Chebyshev solve' for the master equation; the main text should specify the polynomial degree, truncation error bound, and convergence test used to obtain the quoted 22% shift (e.g., in the section reporting the spectrum).
  2. [Reduction to master equation] The principal-symbol analysis establishing decoupling of the T² term is central; a brief appendix or inline derivation of the symbol for the unreduced axial system would make the 'exact equivalence' claim fully self-contained.
  3. [Background and closure] Notation for the effective fluid parameters (w_r,w_t) and the mass-normalized frequency should be introduced once with a clear table or equation reference to avoid ambiguity when comparing branches.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment, clear summary of our results, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; central reduction is derived from unreduced equations

full rationale

The paper states that the odd sector equivalence to Einstein gravity plus frozen anisotropic fluid follows from explicit analysis of the unreduced axial system and its principal symbol on admissible backgrounds. The master equation and spectrum are then obtained by standard methods (Chebyshev solve) on backgrounds selected for regularity and energy conditions. No self-citations, fitted parameters renamed as predictions, or self-definitional steps are quoted or required for the load-bearing claims. The 22% shift is presented as a background property, consistent with the stated scope. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The model rests on the choice of constant closure parameters (w_r,w_t) and the assumption that the odd sector reduces exactly to Einstein gravity plus a frozen fluid; no new particles or forces are introduced.

free parameters (2)
  • (w_r,w_t) = (-0.2,0.15)
    Constant closure parameters chosen to satisfy horizon regularity and asymptotic flatness; the anchored values (-0.2,0.15) are selected by hand.
  • α/M² = 0–0.3
    Trace-coupling strength scanned over the interval 0 to 0.3; no statistically resolved dependence is reported.
axioms (1)
  • domain assumption The odd sector on admissible backgrounds is exactly equivalent to Einstein gravity coupled to a frozen effective anisotropic fluid.
    Invoked to reduce the two-field system to a single master equation.

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discussion (0)

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