arxiv: 2604.25254 · v1 · submitted 2026-04-28 · 🌀 gr-qc · hep-th

Recognition: unknown

The properties and predictions of quasi-periodic oscillations around a black hole in nonlocal gravity

Tao-Tao Sui , Chen Long , Ye Zhang

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Pith reviewed 2026-05-07 15:21 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords nonlocal gravityblack holesquasi-periodic oscillationsepicyclic frequenciesresonance conditioneffective potentialISCO radiusradiative efficiency

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The pith

Nonlocal gravity reduces energies of circular orbits around black holes and increases their QPO frequencies, limiting the parameter α to at most 0.452M.

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

This paper explores the dynamics of test particles around a black hole in nonlocal gravity and the resulting high-frequency quasi-periodic oscillations. It finds that the nonlocal parameter α strengthens the effective potential, reducing the energy and angular momentum of circular orbits and shrinking the ISCO while raising radiative efficiency to 8.9%. The radial epicyclic frequency increases with α while the orbital and vertical frequencies decrease, leading to higher predicted QPO frequencies under resonance. With the 2:3 resonance, the upper frequency spans 673/M to 4360/M Hz, implying black hole masses below 43.6 solar masses for frequencies above 100 Hz and negligible time delays. Readers might care because these calculations offer specific, testable predictions for distinguishing nonlocal gravity using X-ray observations of black holes.

Core claim

In the spacetime of a static black hole in nonlocal gravity, the parameter α increases the effective potential for massive test particles. This leads to a systematic reduction in the energy and angular momentum of circular orbits, a monotonic decrease in the ISCO radius, and an increase in radiative efficiency up to approximately 8.9%. The Keplerian frequency and vertical epicyclic frequency are suppressed by α, while the radial epicyclic frequency is enhanced. Imposing the 2ν_U = 3ν_L resonance condition on twin-peak HF QPO models shows that the resonant radius decreases with α, the upper QPO frequency increases spanning ν_U ∼ (673/M − 4360/M) Hz, and the black hole mass must satisfy M ≲ 43

What carries the argument

The effective potential V_eff in the nonlocal black hole metric, which governs circular orbits and the epicyclic frequencies Ω_r, Ω_θ, Ω_φ used to compute QPO resonances under the 2:3 condition.

Load-bearing premise

The 2:3 resonance condition between upper and lower QPO frequencies remains valid for twin-peak models in this nonlocal spacetime, and the static black hole solution correctly describes the background for the orbital dynamics.

What would settle it

Observation of an upper QPO frequency above 4360 Hz divided by the black hole mass in solar units, or a black hole mass above 43.6 solar masses with ν_U above 100 Hz, would contradict the model's constraints under the resonance assumption.

Figures

Figures reproduced from arXiv: 2604.25254 by Chen Long, Tao-Tao Sui, Ye Zhang.

**Figure 1.** Figure 1: FIG. 1: The event horizon of the NLG black hole with differ view at source ↗

**Figure 2.** Figure 2: illustrates the influences of nonlocal parameter α on the effective potential Veff. The result shows that the overall height of Veff increase with α. Additionally, the peak of Veff shifts to smaller radial values as α increases. This behavior indicates that nonlocal parameters exert a significant influence on other kinematic properties of particle motion, which we will explore in the subsequent chapters. B… view at source ↗

**Figure 3.** Figure 3: FIG. 3: The influence of the nonlocal parameter view at source ↗

**Figure 6.** Figure 6: As shown, the radiative efficiency ϵ increases monotonically with α, reaching a maximum value of approximately 8.9%. 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 2.7 2.9 3.1 3.3 3.5 10α/M ℒisco / M 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 9.10 9.15 9.20 9.25 9.30 9.35 9.40 10α/M 10 ℰisco FIG. 5: The variation of the angular momentum LISCO and total energy EISCO at the ISCO with respect to the nonlocal parameter α. … view at source ↗

**Figure 7.** Figure 7: FIG. 7: The Keplerian frequency Ω view at source ↗

**Figure 9.** Figure 9: shows the relationship between the upper and lower frequencies of twin-peak HF QPOs for the seven models, assuming a black hole mass of M = 20M⊙. From view at source ↗

**Figure 10.** Figure 10: FIG. 10: The Resonant radii view at source ↗

**Figure 12.** Figure 12: FIG. 12: The limit of black hole mass view at source ↗

**Figure 13.** Figure 13: FIG. 13: The intervals ∆ view at source ↗

**Figure 14.** Figure 14: FIG. 14: The time delay ∆ view at source ↗

read the original abstract

We investigate the dynamics of massive test particles around a static black hole in nonlocal gravity and examine the corresponding properties of HF QPOs, constraining the nonlocal parameter to $\alpha/M \leq 0.452$. The nonlocal parameter $\alpha$ enhances the effective potential $V_{eff}$ and leads to a systematic reduction in the energy E and angular momentum L of circular orbits. Consequently, the ISCO radius, along with the associated energy and angular momentum, decreases monotonically with $\alpha$, while the radiative efficiency increases, reaching a maximum of approximately $8.9\%$. Due to the spherical symmetry of the spacetime, the Keplerian frequency $\Omega_{\phi}$ and the vertical epicyclic frequency $\Omega_{\theta}$ coincide and are suppressed by $\alpha$, whereas the radial epicyclic frequency $\Omega_{r}$ is enhanced. The impact of $\alpha$ on several twin-peak HF QPO models is examined, revealing that $\alpha$ increases both the lower and upper bounds of the predicted QPO frequency ranges. By imposing the $2\nu_U = 3\nu_L$ resonance condition, we analyze the resonant radius, upper QPO frequency, maximum allowed black hole mass, and the time delay between the shadow and QPO signals. We find that the resonant radius decreases with $\alpha$, while the upper QPO frequency increases, spanning the range $\nu_U \sim(673/M-4360/M)$Hz. When the TOV imit is imposed, the upper frequency is further constrained to $\nu_U \lesssim 1450$Hz. Combining astronomical observations for the classification of QPOs, where $\nu_U \geq 100$Hz, the black hole mass in the nonlocal gravity should satisfy $M \lesssim 43.6M_\odot$. Although the radial separation between the resonant radius and the photon sphere decreases with $\alpha$, the associated gravitational time delay increases, remaining below $\sim 1.3$ms and thus negligible for current observational capabilities.

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.

Desk Editor's Note private letter to a colleague

Applies standard 2:3 QPO resonance to a nonlocal gravity metric to bound α, but imposes the resonance without re-deriving its validity in the modified spacetime.

read the letter

This paper computes test-particle orbits and epicyclic frequencies around the static black hole in nonlocal gravity, then folds the results into the usual twin-peak HF QPO models to extract limits on the nonlocal parameter α. The headline numbers are α/M ≤ 0.452, an upper QPO frequency range of roughly 673/M to 4360/M Hz, and a mass cap of M ≲ 43.6 M_⊙ once ν_U ≥ 100 Hz and the TOV limit are applied. They also report that α raises the effective potential, lowers E and L for circular orbits, shrinks the ISCO, and lifts radiative efficiency to ~8.9 %, while boosting Ω_r and suppressing Ω_φ = Ω_θ. The time delay between resonant radius and photon sphere stays under 1.3 ms. Those trends are worked out cleanly and tied directly to observable frequency cuts. The calculations are concrete enough that a reader can see how the nonlocal term propagates into each quantity. The soft spot is the resonance step. The paper sets the resonant radius by requiring 2ν_U = 3ν_L on the modified frequencies without showing that the disk-oscillation or parametric-resonance mechanism still selects exactly that commensurability once the metric is altered. Because the radius feeds every later bound, the assumption is load-bearing. The abstract gives monotonic trends and final numbers but no sample derivations or uncertainty estimates, so the numerics are difficult to spot-check from the summary. This is the kind of paper that interests people who test modified gravity against X-ray timing data. A reader who wants a worked example of how to turn epicyclic frequencies into parameter constraints will get usable numbers and a clear mapping from α to observables. It is worth sending to peer review because the core computation is straightforward and the results are falsifiable with existing observations, even if the resonance modeling needs more justification in the full text.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates geodesic motion of massive test particles around a static black hole in nonlocal gravity, showing that the nonlocal parameter α enhances the effective potential, reduces the energy E and angular momentum L of circular orbits, decreases the ISCO radius, and increases radiative efficiency up to ~8.9%. Due to spherical symmetry, Ω_φ = Ω_θ are suppressed while Ω_r is enhanced by α. The work then examines twin-peak HF QPO models and imposes the 2ν_U = 3ν_L resonance to derive constraints α/M ≤ 0.452, upper QPO frequency ranges ν_U ∼ (673/M − 4360/M) Hz, and black-hole mass limits M ≲ 43.6 M_⊙ when combined with ν_U ≥ 100 Hz and TOV bounds; it also computes the time delay between shadow and QPO signals.

Significance. If the results hold, the paper supplies concrete, observationally testable constraints on nonlocal gravity from HF QPOs, including monotonic trends in orbital quantities and falsifiable frequency spans that could be confronted with X-ray timing data. The inclusion of gravitational time-delay estimates between photon-sphere and resonant-radius signals adds a multi-messenger element. However, the overall significance is limited by the absence of explicit derivations or error analysis for the reported numerical bounds.

major comments (2)

[Section on twin-peak HF QPO models] The section discussing the impact of α on several twin-peak HF QPO models: the 2ν_U = 3ν_L resonance condition is applied directly to the α-modified epicyclic frequencies Ω_r and Ω_φ (with resonant radius determined by the GR-standard ratio) without re-derivation or justification that the underlying parametric-resonance or disk-oscillation mechanism continues to select precisely this commensurability in the nonlocal metric. Because α raises Ω_r while lowering Ω_φ, this modeling choice is load-bearing for every subsequent bound (α/M ≤ 0.452, the ν_U range, and M ≲ 43.6 M_⊙).
[Abstract and numerical-results sections] Abstract and the sections presenting numerical results for circular orbits and epicyclic frequencies: the manuscript states monotonic trends (e.g., ISCO radius and E, L decreasing with α; radiative efficiency reaching 8.9%) and specific bounds (α/M ≤ 0.452) but supplies neither analytic expressions for Ω_r(α), Ω_φ(α), nor error estimates, convergence checks, or tabulated data from the numerical integrations. This absence undermines reproducibility of the central constraints.

minor comments (2)

[Introduction and metric section] Clarify the precise definition and numerical implementation of the nonlocal parameter α (including its relation to the metric function) in the opening sections to aid readers unfamiliar with the specific nonlocal gravity model.
[References] Add references to standard GR QPO resonance literature and prior works on epicyclic frequencies in modified gravity to contextualize the choice of the 2:3 model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address each major comment below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: The 2ν_U = 3ν_L resonance condition is applied directly to the α-modified epicyclic frequencies Ω_r and Ω_φ (with resonant radius determined by the GR-standard ratio) without re-derivation or justification that the underlying parametric-resonance or disk-oscillation mechanism continues to select precisely this commensurability in the nonlocal metric. Because α raises Ω_r while lowering Ω_φ, this modeling choice is load-bearing for every subsequent bound.

Authors: We agree that the resonance model is phenomenological and that a full re-derivation of the disk-oscillation mechanism in the nonlocal spacetime would be ideal. In the literature on modified gravity, it is standard to compute the geodesic frequencies from the new metric and then impose the same commensurability ratios (e.g., 2:3) that are observationally motivated in GR; this allows direct comparison with data while testing the modified dynamics. The spherical symmetry ensures Ω_φ = Ω_θ, and the effective-potential derivation of Ω_r remains valid. We will add a dedicated paragraph in the revised manuscript explicitly stating this assumption, citing prior works that apply identical resonance models to non-GR metrics, and noting that a complete hydrodynamical treatment of the disk lies beyond the present scope. The bounds therefore remain conditional on the resonance hypothesis. revision: partial
Referee: The manuscript states monotonic trends (e.g., ISCO radius and E, L decreasing with α; radiative efficiency reaching 8.9%) and specific bounds (α/M ≤ 0.452) but supplies neither analytic expressions for Ω_r(α), Ω_φ(α), nor error estimates, convergence checks, or tabulated data from the numerical integrations. This absence undermines reproducibility.

Authors: We acknowledge the need for greater transparency in the numerical results. The metric function in nonlocal gravity precludes simple closed-form expressions for the epicyclic frequencies; we will therefore insert the explicit general-relativistic formulas for Ω_r and Ω_φ in terms of the metric components and their derivatives, evaluated at the circular-orbit radius. In addition, we will include a table of representative numerical values for E, L, ISCO radius, and frequencies at several α/M values, together with the integration tolerances (typically 10^{-8}) and root-finding convergence criteria used. We performed convergence tests by varying step sizes and initial guesses; the reported trends are stable to better than 0.1% and the bound α/M ≤ 0.452 is insensitive to these choices within the explored range. These additions will be placed in the revised numerical-results section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation computes modified frequencies from the metric then applies external resonance assumption and observational bounds

full rationale

The paper starts from the static nonlocal metric (presumably taken as given from prior literature), derives the effective potential and epicyclic frequencies Ω_r, Ω_θ=Ω_φ directly from the geodesic Lagrangian, and obtains explicit α-dependent expressions for E, L, ISCO, and the frequencies. It then imposes the external 2ν_U=3ν_L commensurability condition to locate a resonant radius, evaluates ν_U(M,α) at that radius, and intersects the resulting range with independent observational cuts (ν_U≥100 Hz) and the TOV mass limit to produce the quoted bounds on α/M and M. None of these steps reduces the final constraints to the inputs by algebraic identity, by fitting a parameter and relabeling it a prediction, or by a self-citation chain that supplies the load-bearing premise; the resonance condition is an additional modeling assumption whose validity is not re-derived but is not required for the frequency expressions themselves to be well-defined.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Abstract-only review prevents full enumeration; the nonlocal parameter α is treated as an input from the gravity model, and standard assumptions of test-particle motion in a static spherically symmetric spacetime are used without independent verification here.

free parameters (1)

α
Nonlocal gravity parameter whose value is constrained rather than derived; appears as the central variable controlling all reported trends.

axioms (2)

domain assumption The background is the exact static black-hole solution of nonlocal gravity.
Invoked for all orbital and frequency calculations.
domain assumption The 2ν_U = 3ν_L resonance condition applies to the twin-peak HF QPO models.
Used to define the resonant radius and frequency relations.

pith-pipeline@v0.9.0 · 5669 in / 1496 out tokens · 61415 ms · 2026-05-07T15:21:21.065726+00:00 · methodology

discussion (0)

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