REVIEW 4 major objections 5 minor 10 references
Geodetic Motion Around Rotating Blackhole in Nonlocal Gravity
T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper claims that in the RR nonlocal gravity model, the orbital frequencies of a test particle around a rotating black hole are Kerr values plus calculable nonlocal shifts.
desk verdict A short research announcement built on an unvalidated rotating metric; the frequency-shift idea is plausible but the paper as written is not independently checkable. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the RR action $S=\frac{1}{2\kappa^2}\int d^4x\sqrt{-g}\,[R+\frac{\mu^2}{3}R\,\Box^{-2}R]$, whose nonlocal term generates the perturbation $b_{\alpha\beta}$ in the rotating metric $g_{\alpha\beta}=g^{\rm Kerr}_{\alpha\beta}+b_{\alpha\beta}$. The argument runs through canonical perturbation theory: for bound Kerr geodesics the unperturbed motion separates in action-angle variables, and the first-order frequency shift is $\delta\omega_i = \frac{1}{m}\frac{\partial\langle H_1\rangle}{\partial \hat J_i}$, with $H_1=-(m^2/2)b_{\alpha\beta}u^\alpha u^\beta$ averaged over one unperturbed orbit. The right-hand panel of Fig. 1 is the numerical evaluation of these three shifts from the explicit $b_{\alpha\beta}$ extracted from metric (10).
What would settle it
Substitute metric (10) into the vacuum RR field equations (2) and check whether the residual tensor vanishes at order $\mu^2$; if it does not, the plotted frequency shifts are not predictions of the RR model. Independently, integrate the geodesic equation numerically in metric (10) and compare the resulting fundamental frequencies with the canonical-perturbation values shown in Fig. 1.
Extended reading notes
Core claim
The central claim is that in the RR model, the spacetime around a rotating black hole is Kerr plus a small nonlocal perturbation, metric (10), and that geodesic orbital frequencies are the Kerr frequencies plus calculable shifts $\delta\omega_i$ obtained from the averaged perturbing Hamiltonian. The magnitude of the shift is controlled by the nonlocal mass scale $\mu$, fixed by cosmology to $\mu \simeq 0.283 H_0$. The paper reports the resulting shifts in the observable frequencies as functions of the orbital semilatus rectum and finds them negative, of order $10^{-9}$ in the plotted range. This is the ingredient needed to build waveform templates in which nonlocal gravity leaves an imprint on EMRI signals.
Load-bearing premise
The entire frequency-shift calculation rests on the unverified assumption that metric (10), obtained by rotating the spherical RR solution with a standard solution-generating trick, is an actual solution of the RR field equations; if that assumption fails, the frequency shifts are not predictions of the model.
Editorial extensions
If this is right
- EMRI waveform templates built on pure Kerr will carry a systematic nonlocal correction; the paper's shifts give the leading piece of that correction.
- The plotted shifts are negative and as large as several parts in $10^9$ in the displayed range, so over the many cycles of an EMRI the accumulated phase difference can become observable.
- The weak-field rotating metric (8) predicts nonlocal modifications to frame-dragging and to the effective potentials $\Phi$ and $\Psi$, which can be checked in weaker-field observations.
- The same Hamilton-Jacobi and averaging machinery remains valid, so analytic templates for nonlocal-gravity EMRIs do not require abandoning Kerr-based methods.
Reading between the lines
- The absent field-equation check leaves a concrete computational task: substitute metric (10) into the vacuum field equations (2); until that is done, Fig. 1 should be read as a prediction of the metric ansatz rather than of the RR action itself.
- The same canonical-perturbation pipeline applies to any nearly-Kerr metric, so it offers a cheap way to compare alternative modified-gravity black-hole models before full waveform generation is attempted.
- Because the per-orbit shift is small but the number of orbits in an EMRI is large, the dephasing acts as a natural amplifier, making a Hubble-scale parameter potentially testable by future space-based gravitational-wave detectors.
- The weak-field rotating metric (8) provides an independent, nearer-term test of the same model in regimes where the effective potentials can be measured directly.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies geodesic motion around a rotating black hole in the RR model of nonlocal gravity. Section II derives a weak-field rotating metric (Eq. 8) with potentials Φ and Ψ given in Eq. (9). Section III presents a Kerr-like metric (Eq. 10) obtained by applying the Demianski-Janis-Newman algorithm in the authors' Ref. [8], and claims to compute the shift in orbital frequencies using canonical perturbation theory, with numerical results shown in Fig. 1. The text explicitly states that the detailed calculation of the frequency shifts and the solution of the geodesic equations will be presented in future work.
Significance. If the metric (10) were a verified solution of the RR field equations and the frequency shifts were correctly computed, the paper would provide an interesting first step toward strong-field tests of nonlocal gravity using EMRIs. The topic is timely, and the use of canonical perturbation theory on a Kerr background is a standard and promising approach. However, as it stands, the central result is not derived in this manuscript: the rotating metric is imported from a preprint by the same authors without verification, and the actual frequency-shift calculation is deferred to a future publication. No code, machine-checked algebra, or reproducible numerical parameters are provided for Fig. 1. The paper is better described as a research announcement than a complete derivation.
major comments (4)
- [Section III, Eq. (10)] The metric (10) is stated to be "obtained by applying Demianski-Janis-Newman algorithm ... in [8]", but the manuscript does not verify that this metric satisfies the RR field equations (2). This is load-bearing because all subsequent frequency-shift computations are built from the perturbation bαβ extracted from (10). Without such a check, the plotted shifts are not necessarily predictions of RR nonlocal gravity. The Newman-Janis algorithm is not a general solution-generating technique in modified gravity, so the paper should either demonstrate that (10) solves Eq. (2) or explicitly label it as a conjecture pending verification.
- [Section III, paragraph after Eq. (10)] The text states that "The detailed discussion on calculation of shift in orbital frequencies and solution of geodesic equations ... will be done in our future work." This directly contradicts the abstract and conclusion, which claim that the shift has been calculated. The manuscript omits the explicit form of H1, the averaging procedure ⟨H1⟩, the action-angle variables used, and the numerical parameters (p, e, θmin, a, µ) behind Fig. 1. Consequently, Fig. 1 is not reproducible and the central claim of the paper is not substantiated within the manuscript itself.
- [Section II, Eq. (9)] In the limit µ→0, the potentials in Eq. (9) reduce to Φ(r) ≈ −GM/(3r) and Ψ(r) ≈ −5GM/(3r). With Eq. (8), the weak-field metric then does not reduce to the standard weak-field Schwarzschild/Newtonian limit (g_tt ≈ −(1 − 2GM/r), g_rr ≈ 1 + 2GM/r in isotropic coordinates). This is a concrete internal inconsistency in a solution that is presented as a derivation from the field equations. It suggests an error in the metric input and must be resolved before the weak-field result can be trusted.
- [Section II, Eq. (6)] Equation (6) uses the symbol M in the nonlocal terms (e.g., 2M²□^{-1}) where the model's mass scale is µ, introduced after Eq. (1). If M here denotes the central mass, the equation is dimensionally or physically ambiguous because the nonlocal correction should be governed by µ. If it is a typo for µ², please correct it throughout the equation.
minor comments (5)
- [Title and affiliations] The title and affiliations contain typographical errors: "Grav ity" on the arXiv title line and "Isra el" in the first affiliation should be corrected.
- [References] Reference [7] spells the author's name as "Demiaski"; the standard spelling is "Demianski" as used in the text.
- [Figure 1] Figure 1 does not state the values of the orbital parameters (p, e, θmin), the mass ratio a/M, or the nonlocal scale µ. Without these, the numerical curves cannot be reproduced or compared with other work. The caption should define the units of p and the convention for Ω_i.
- [Section III, Eq. (10)] The dφ² term in Eq. (10) is written in a dense form with nested parentheses; adding a clarifying line or an explicit comparison with the standard Kerr metric would improve readability.
- [Concluding remarks] The Conclusion repeats the claim that the frequency shifts were calculated, but no equation for δω_i appears anywhere in the paper; please number and include the final result or explicitly reference a companion paper.
Circularity Check
The frequency-shift calculation is built on a same-author DJN black-hole metric that is never checked against the RR field equations; the plotted shifts are the response to that imported perturbation.
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ansatz smuggled in via citation
[Section III, Eq. (10) (paragraph following Eq. (10))]
"The metric for the spacetime around rotating blackhole in RR model was obtained by applying Demiaski-Janis-Newman algorithm[6, 7] on the spherically symmetric static solution[1] of the RR model in [8] as (in the form of gαβ = gKerrαβ + bαβ )"
This sentence is the only derivation given for the non-Kerr part bαβ that enters the calculation. The paper does not substitute (10) into the RR field equations (2), fix a gauge, or otherwise verify that the DJN-transformed metric is a solution; it cites the authors' own Ref. [8] for the metric. The frequency shift is then defined as mδωi = ∂⟨H1⟩/∂Ĵi with H1 = −(m2/2)bαβ(dxα/dτ)(dxβ/dτ), i.e. the shift is the first-order response to exactly the imported bαβ. Therefore the Fig. 1 curves are not independent predictions of the RR action but a re-expression of the same-author ansatz in orbital-frequency variables. The weak-field potentials (9) also fail to reduce to Schwarzschild as μ→0, which underscores that the metric input has not been validated.
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self citation load bearing
[Section II, Eq. (9)]
"where J is angular momentum, defined as v = r×J M r2 and Φ and Ψ are given by[5]"
The weak-field rotating metric (8) is stated as the solution of the linearized field equations, but the potentials Φ and Ψ are not derived in this paper; they are taken from Ref. [5], which is the same author group. Since these potentials are used in the conclusion's claim to have 'calculate[d] the axially symmetric stationary metric around the rotating object in RR model,' the central supporting input again rests on a self-citation rather than on a derivation shown here. This is secondary to the Section III metric, but it is part of the same load-bearing self-citation pattern.
full rationale
The paper's genuinely new computation is the canonical-perturbation frequency shift δωi, and that computation is formally self-contained once the perturbing metric bαβ is given. However, the only source for bαβ is the authors' own Ref. [8], where the rotating black-hole metric was produced by the Demiański-Janis-Newman algorithm. The DJN algorithm is a known solution-generating trick for vacuum GR; for a nonlocal modified-gravity action there is no guarantee that the transformed metric satisfies Eq. (2), and the present paper provides no check (no substitution, no gauge fixing, no explicit bαβ). The numerical shifts in Fig. 1 are therefore forced by an unvalidated same-author input: they are linear functionals of the imported bαβ. This is a partial circularity: the 'prediction' is a re-parametrization of the assumed metric perturbation, not a test of the RR field equations. The cosmological parameter μ=0.283H0 is a fitted input from Ref. [1], but it is explicitly stated as such and is not disguised as a prediction; this does not add circularity. The Kerr-frequency part of Fig. 1 is an independent numerical calculation, which prevents a score of 8-10. Because the central claim reduces to a self-citation/ansatz chain, the appropriate score is 6.
Assumptions & free parameters
free parameters (2)
- μ (RR mass scale) =
0.283 H0
- Orbital elements (e, θ_min) for Fig. 1 =
not stated
assumptions (4)
- domain assumption DJN transformation of the static RR solution yields a valid rotating solution of Eq. (2)
- domain assumption Linearized rotating metric (8) with Φ,Ψ from Eq. (9) correctly solves the RR field equations at first order
- domain assumption First-order canonical perturbation theory with averaged H1 gives the frequency shifts
- domain assumption μ remains the same at black hole scales as in cosmology and μr is small
Cite this review
Pith. "Pith review of Geodetic Motion Around Rotating Blackhole in Nonlocal Gravity." pith.science (2026). https://pith.science/paper/6DMEJYWK
@misc{pith2026190808188,
author = {Pith},
title = {Pith review of: Geodetic Motion Around Rotating Blackhole in Nonlocal Gravity},
year = {2026},
howpublished = {\url{https://pith.science/paper/6DMEJYWK}},
note = {Machine review of arXiv:1908.08188}
}
read the original abstract
Recently the non-local gravity theory has come out to be a good candidate for an effective field theory of quantum gravity and also it can provide rich phenomenology to understand late-time accelerating expansion of the universe. For any valid theory of gravity, it has to surmount solar system tests as well as strong field tests. Having motivations to prepare the framework for the strong field test of the modified gravity using Extreme Mass Ratio Inspirals(EMRIs), here we try to obtain the metric for Kerr-like blackhole for a non-local gravity model known as RR model and calculate the shift in orbital frequencies of a test particle moving around the blackhole. We also derive the metric for a rotating object in the weak gravity regime for the same model.
Figures
Reference graph
Works this paper leans on
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Reviewed August 14, 2026 · model on record in the stance chip above.
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