Geodesic motion of a test particle around a noncommutative Schwarzchild Anti-de Sitter black hole

Abdellah Touati; Mohamed Aimen Larbi; Slimane Zaim

REVIEW 4 major objections 5 minor 79 references

Geodesic motion of a test particle around a noncommutative Schwarzchild Anti-de Sitter black hole

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives noncommutative corrections to Schwarzschild–AdS black-hole geodesics and uses Mercury's perihelion precession to bound the noncommutative parameter at roughly $10^2$–$10^3$ Planck lengths.

desk verdict A readable Bopp-shift calculation with an honest novelty statement, but the headline Mercury bound is off by a factor of 10 and the method has a self-consistency problem that is not addressed. read the letter →

arxiv 2411.16886 v1 pith:G3JJGP3W submitted 2024-11-25 gr-qc

Mohamed Aimen Larbi , Slimane Zaim , Abdellah Touati This is my paper · ORCID

classification gr-qc MSC 83C5783C1081R60 PACS 04.20.-q04.70.-s02.40.Gh

keywords noncommutativespacetimeSchwarzschild-anti-deSittergeodesicmotionperihelionprecessionMercuryeffectivepotentialBoppshiftPlancklength

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether a Planck-scale fuzziness of spacetime leaves a trace in ordinary solar-system orbits, and answers yes. Starting from a Schwarzschild–anti-de Sitter black hole and switching on spatial noncommutativity through the Bopp shift, it derives a deformed metric, effective potential, and geodesic equation for massive test particles. The central quantitative result is that Mercury's observed perihelion precession bounds the noncommutative parameter at first order by $\sqrt{\Theta}\le 2.03\times 10^{-32}\,\mathrm{m}$ and at second order by $\sqrt{\Theta}\le 4.08\times 10^{-33}\,\mathrm{m}$, about $10^2$ to $10^3$ Planck lengths. The paper also claims that, for the parameter choices studied, circular orbits in the noncommutative spacetime are more stable than in the commutative one. If correct, planetary motion becomes a probe of Planck-scale structure rather than just a test of classical gravity.

What carries the argument

The load-bearing object is the Bopp-shifted noncommutative metric. From $\hat{x}^\mu = x^\mu - \frac{1}{2}\Theta^{\mu\nu}p_\nu$ with only spatial noncommutativity, the radial coordinate becomes $\hat{r}=r-\frac{1}{2}\Theta p_\phi$, and Taylor expanding $g_{\mu\nu}(\hat{r})$ gives the deformed components (9)–(12) at first order and (31)–(34) at second order. This metric enters the Lagrangian $\mathcal{L}=\frac{1}{2}(\hat{g}_{tt}\dot{t}^2+\hat{g}_{rr}\dot{r}^2+\hat{g}_{\phi\phi}\dot{\phi}^2)$, whose Euler–Lagrange equations produce the effective potential (17) and the orbital equation (20). The perihelion advance is extracted from the perturbed Kepler form $d^2u/d\phi^2+u=m/L^2+g(u)/L^2$ using $\Delta\phi=(\pi/L^2)|dg/du|_{u=1/b}$, which is the step that converts the metric deformation into the quoted bounds on $\Theta$.

What would settle it

Compute the deformed metric from the same Moyal star product by solving the noncommutative Einstein equations for the stated $\Theta^{\mu\nu}$; if the first-order components differ from Eqs. (9)–(12), the Mercury bound collapses. Observationally, measure the perihelion precession of a second inner planet and check whether the same $\Theta$ is recovered, since the correction is proportional to the orbiting body's $p_\phi$.

Watch

Extended reading notes

Core claim

The paper's central claim is that the substitution $\hat{r}=r-\frac{1}{2}\Theta p_\phi$ inside the Schwarzschild–AdS metric produces a valid noncommutative black-hole geometry whose geodesics differ measurably from the commutative ones. On that metric the noncommutative correction to the perihelion advance per revolution is derived explicitly, and for Mercury the result is Eq. (29), $\sqrt{\Theta}\le 2.03\times 10^{-32}\,\mathrm{m}$ at first order, tightened to Eq. (36), $\sqrt{\Theta}\le 4.08\times 10^{-33}\,\mathrm{m}$ at second order. Equivalently, the noncommutative scale lies around $10^2$ to $10^3$ Planck lengths, corresponding in natural units to an energy scale near $10^3 E_P$. A second claim is that the noncommutative effective potential has a deeper minimum and extrema shifted outward, so stable circular orbits are more stable and unstable ones shift to larger radii than in the commutative AdS–Schwarzschild spacetime.

Load-bearing premise

The load-bearing premise is that replacing $r$ by $r-\frac{1}{2}\Theta p_\phi$ in the ordinary Schwarzschild–AdS metric and then treating the resulting momentum-dependent object as a fixed background spacetime is a legitimate description of noncommutative gravity; if that step does not correspond to a real solution of the noncommutative field equations, the geodesic equation and the Mercury bound inherit the error.

Editorial extensions

If this is right

If the bound holds, solar-system ephemerides already constrain spacetime noncommutativity to roughly $10^2$–$10^3$ Planck lengths, making planetary precession one of the tightest low-energy windows on Planck-scale geometry.
Because the correction grows with $\Theta$ and with the test body's momentum $p_\phi$, the same analysis predicts that faster or more massive orbiting bodies should show systematically larger noncommutative precessions.
The event horizon radius increases with $\Theta$, so the noncommutative black hole exerts stronger gravitational effects at its horizon than the commutative one.
The stable and unstable circular-orbit radii both shift outward with $\Theta$, and the effective-potential minimum deepens; for the values plotted, this makes circular orbits more stable and moves the innermost stable circular orbit condition.
The second-order term tightens the Mercury bound by about an order of magnitude over the first-order term, and the paper notes that going to still higher orders should sharpen the estimate further.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The momentum dependence of the deformed metric means the paper's $\Theta$ is inferred from one orbit rather than being a universal constant of the spacetime; comparing Mercury with Venus or Earth would test whether one $\Theta$ fits all planets or whether the Bopp-shifted ansatz fails.
The first- and second-order bounds differ by a factor of five; computing the third-order term would show whether the perturbation series in $\Theta p_\phi$ is converging, and if it is not, the quoted upper bound is an artifact of truncation.
The same geodesic framework should yield noncommutative corrections to light deflection and gravitational time delay; independent measurements of those solar-system effects could confirm or falsify the Mercury-based value.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a straightforward Bopp-shift exercise, and the authors are honest about how much of it is new, but the main quantitative claim is wrong by a factor of 10, and the method has a self-consistency problem that no one seems to have noticed.

What is actually new is the second-order correction to the perihelion shift, Eq. (35), and the associated bound Eq. (36). The first-order parts reduce to known results — Romero and Vergara for the Kepler case, Larranaga for timelike geodesics in noncommutative Schwarzschild–AdS — and the authors explicitly say so. That level of transparency is good. The paper is also clearly written; the steps from the deformed metric to the orbital equation and the perihelion formula are easy to follow.

The soft spots, though, are load-bearing. First, the arithmetic in the Mercury bound is wrong: Eq. (27) allows a residual of 2π×2.7×10⁻¹², while Eq. (25) gives a correction of 2π×6.93249×10¹⁸ Θ. Dividing gives Θ ≤ 3.9×10⁻³¹ s·kg⁻¹, not 3.89×10⁻³⁰. That changes √Θ from about 2×10⁻³² m to about 6×10⁻³³ m, which is nearly the same as their own second-order bound. So the two claimed numbers are not independent; the second-order result is essentially caused by the arithmetic error.

Second, the deformed metric, Eqs. (9)–(12), is a Taylor expansion of the commutative metric under the replacement r → r − (1/2)Θ p_φ. It is an ansatz, not a solution of any noncommutative Einstein equations. More concerning, the same p_φ that enters the metric also appears in the definition of the conserved angular momentum, L = g_φφ ẍφ. The paper does not impose p_φ = L or solve the resulting implicit system. The omitted consistency terms are of the same order Θ as the claimed correction, so the perihelion formula does not follow from a closed variational principle.

There is also a smaller issue: the stability claim for circular orbits rests on one numerical table with m = 1, L = 5, Λ = −10⁻³, and a handful of Θ values. That is suggestive, not a general result.

Who gets value from this? A specialist who works on noncommutative corrections to GR phenomenology and wants to see a clean, explicit application of the Bopp-shift method to Schwarzschild–AdS. The paper is not foundational and the errors are serious, but they are repairable in principle. If a referee can judge whether the p_φ duality can be fixed, the paper is worth one round of review. Otherwise, it is a desk reject. I would lean toward sending it out once, because the question — whether solar-system data can meaningfully bound Θ — is legitimate and the authors have at least framed it clearly.

Referee Report

4 major / 5 minor

Summary. The paper constructs a noncommutative Schwarzschild-AdS metric by applying the Bopp shift r -> r - (1/2) Theta p_phi to the commutative metric and Taylor expanding to first and second order in Theta. It then derives the effective potential, geodesic equation, and perihelion precession for massive test particles, and uses the Mercury perihelion residual to claim upper bounds sqrt(Theta) ~ 10^-32 m (first order) and ~10^-33 m (second order). It also claims that circular orbits in the noncommutative spacetime are more stable than in the commutative case. The central advertised result is the Mercury bound on the noncommutative parameter.

Significance. If the derivation were sound, the paper would provide a concrete planetary constraint on spacetime noncommutativity at scales of 10^2 to 10^3 Planck lengths, which is an interesting and falsifiable result. The paper is clearly written, includes explicit formulas to first and second order, and makes contact with earlier noncommutative Kepler-problem bounds. However, the central quantitative claim rests on a metric that is not shown to solve any noncommutative gravity equations, on a momentum-dependent metric that creates a self-consistency problem in the variational principle, and on an arithmetic error that changes the first-order bound by a factor of 10. These issues are load-bearing rather than cosmetic. The manuscript also contains no machine-checked proofs or reproducible code, and the numerical geodesic plots are illustrative rather than quantitative tests of the main claim.

major comments (4)

[Section II, Eqs. (9)-(12)] The deformed metric is obtained by Taylor-expanding the commutative Schwarzschild-AdS metric after the coordinate shift r -> r - (1/2) Theta p_phi. No noncommutative Einstein equations or noncommutative action principle are solved. Consequently, the objects g_mu_nu in Eqs. (9)-(12) are simply a one-parameter family of commutative metrics, and the geodesics computed from them are ordinary geodesics of that family; they need not describe geodesics in a noncommutative spacetime. Because every subsequent result (effective potential, geodesic equation, perihelion shift, Mercury bound) is derived from this metric, the physical interpretation of the quantitative bounds is unsupported unless the metric is justified as a genuine solution of a noncommutative gravity theory.
[Eqs. (15b), (20), (24), and (29)] The metric components in Eqs. (9)-(12) depend on p_phi, which is also the canonical momentum conjugate to phi in the Lagrangian (14). In Eq. (15b) the angular momentum is defined as L = g_phi_phi (r, Theta p_phi) dot(phi), but later in Section IV.A the same symbol p_phi is set to M V_phi, while L is independently set to (GM/c^2) a(1-e^2). No consistency condition enforces p_phi = L or otherwise specifies how the momentum entering the metric is related to the orbital constants of motion. The variation of the Lagrangian is therefore not closed: changing the trajectory changes p_phi, which changes the metric, which in turn changes the Lagrangian. The missing self-consistency terms are of the same order in Theta as the claimed correction, so Eq. (20), Eq. (24), and the bound in Eq. (29) do not follow from a well-defined variational principle.
[Section IV.A, Eqs. (27)-(29)] The numerical conversion from the allowed precession excess to the bound on Theta contains a factor-of-10 error. Equation (27) allows |delta_phi_NC| <= 2 pi x 2.7 x 10^-12, and Eq. (25) gives |delta_phi_NC| = 2 pi x 6.93249 x 10^18 Theta. Dividing gives Theta <= 2.7 x 10^-12 / 6.93249 x 10^18 ~ 3.9 x 10^-31 s kg^-1, not 3.89369 x 10^-30 as stated in Eq. (28). The corrected first-order bound would be sqrt(hbar Theta) ~ 6 x 10^-33 m, which is much closer to the second-order bound in Eq. (36). This arithmetic error undermines the advertised first-order bound and the claim that the first-order and second-order results differ by an order of magnitude.
[Section III, Eq. (17) and Table I] The claimed increase in stability of circular orbits is based on the effective potential (17), which inherits the momentum-dependent metric from Eqs. (9)-(12). The numerical analysis fixes p_phi = 1 in Table I and Figures 3-5, but p_phi is a physical momentum of the orbiting particle, not a freely adjustable constant. Without a consistent prescription for p_phi in terms of the orbit parameters, the comparison of stable and unstable orbit radii between Theta = 0 and Theta > 0 is not a controlled statement about the same physical system, and the conclusion that noncommutativity makes circular orbits 'more stable' is not established.

minor comments (5)

[Section IV.A, Eq. (24)] The symbol alpha in Eq. (24) is not defined; earlier in the text the orbit parameter is denoted b = a(1 - e^2), so Eq. (24) should use a consistently or define alpha explicitly.
[Eqs. (28)-(29)] The notation in Eq. (29) is confusing: the authors write sqrt(hbar Theta) = sqrt(Theta), while Theta has different meanings in Eqs. (28) and (5). The dimensions of Theta should be stated explicitly in each expression, and the conversion between the parameter in Eq. (5) and the one bounded in Eq. (28) should be clarified.
[Reference [80]] Reference [80] is a machine-learning paper and is not an appropriate source for the observed perihelion precession of Mercury; a standard solar-system ephemeris or a classic general-relativity test reference should be cited instead.
[Section IV.A and Conclusion] The text repeatedly says 'lower bound' where it means 'upper bound' (e.g., 'we can now define a lower bound for the NC parameter' before Eq. (27), and in the Conclusion). The direction of the inequality should be stated consistently.
[Title and throughout] There are typographical issues, including 'Schwarzchild' in the title and several duplicated references in the bibliography; a careful copyedit is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the perihelion bound is a parameter constraint from Mercury data, and the derivation chain is mostly self-contained with self-citations only as comparisons.

full rationale

The paper's central quantitative output is a bound on the non-commutative parameter Theta obtained by comparing a derived perihelion correction with the observed Mercury perihelion shift. This is a parameter constraint, not a prediction of a quantity from an independently fitted parameter, so it does not fit the 'fitted input called prediction' pattern. The perihelion formula (Eq. 24) is obtained by applying the standard perturbation formula (Eq. 23) to the geodesic equation (Eq. 20), and Eq. (23) is attributed to the independent work of Adkins and McDonnell [75]; the derivation is displayed in the paper rather than reduced to an unstated prior result. The self-citations [68,69] by the present authors are used for comparison and for the phrase 'steps outlined in [68,75]', but the load-bearing formula is explicitly written out and the final numerical bound is compared with an external datum, so the self-citations are not load-bearing. The paper's own remark that the first-order result matches reference [74] confirms that the bound is a confirmation of an earlier independent estimate rather than a claim produced solely by self-reference. There are serious correctness concerns outside circularity: the deformed metric in Eqs. (9)-(12) depends on p_phi, which is also the conjugate momentum of the orbiting particle, so the variational treatment is not closed; and the arithmetic from Eq. (27) to Eq. (28) appears to be off by roughly a factor of ten. These are physical-consistency and numerical-accuracy problems, not instances where the conclusion is identical to the input by construction, so under the hard rules they do not raise the circularity score. Overall, the derivation chain is self-contained enough that no circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The core contribution rests on one fitted parameter, Theta, plus a hidden parameter E in the second-order bound and a momentum variable p_phi inserted into the metric. The metric ansatz is an axiom, not a derived solution. No new particles or fields are introduced.

free parameters (3)

Theta (noncommutative parameter) = Theta <= 3.89e-30 s kg^-1 (first order); sqrt(Theta) <= 4.08e-33 m (second order)
The main output of the paper is a bound on Theta, obtained by comparing the theoretical Mercury perihelion shift with the observed residual. It is a fitted constraint, not a prediction, and the printed value contains a factor-10 inconsistency with the paper's own Eq. (27).
E (test particle energy) = not stated
The second-order perihelion formula Eq. (35) contains E, but the Mercury application Eq. (36) never specifies the value used, so the second-order bound cannot be reproduced as written.
p_phi (momentum component entering the metric) = assumed equal to M V_phi
The deformed metric components depend on p_phi, but p_phi is also the momentum of the orbiting test particle; no self-consistent determination is given, and for Mercury it is set to M V_phi without derivation.

assumptions (4)

domain assumption Coordinates satisfy [x^mu, x^nu] = i Theta^mu^nu and can be represented by the Bopp shift Eq. (2).
Standard noncommutative-geometry input, cited in refs. [40,41,47], assumed without derivation.
ad hoc to paper Only spatial noncommutativity is considered (Theta^0i = 0), with the specific antisymmetric tensor Eq. (5) mixing r and phi.
The choice of which spatial directions do not commute is arbitrary and is made to simplify calculations; it changes the form of all corrections.
ad hoc to paper The deformed metric is obtained by Taylor-expanding the commutative Schwarzschild-AdS metric in the Bopp-shifted coordinate and truncating at O(Theta) or O(Theta^2).
This is the central modeling assumption of Section II; no noncommutative Einstein equations are solved, so the resulting momentum-dependent metric is an ansatz.
domain assumption The observed Mercury perihelion residual after the GR and Lambda terms is attributed to the NC correction.
Used in Eqs. (26)-(28); Newtonian perturbations and other systematics are not explicitly subtracted in the text.

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Cite this review

Pith. "Pith review of Geodesic motion of a test particle around a noncommutative Schwarzchild Anti-de Sitter black hole." pith.science (2026). https://pith.science/paper/G3JJGP3W

@misc{pith2026241116886,
  author       = {Pith},
  title        = {Pith review of: Geodesic motion of a test particle around a noncommutative Schwarzchild Anti-de Sitter black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3JJGP3W}},
  note         = {Machine review of arXiv:2411.16886}
}

abstract

In this work, we derive non-commutative corrections to the Schwarzschild-Anti-de Sitter solution up to the first and second orders of the non-commutative parameter $\Theta$. Additionally, we obtain the corresponding deformed effective potentials and the non-commutative geodesic equations for massive particles. Through the analysis of time-like non-commutative geodesics for various values of $\Theta$, we demonstrate that the circular geodesic orbits of the non-commutative Schwarzschild-Anti-de Sitter black hole exhibit greater stability compared to those of the commutative one. Furthermore, we derive corrections to the perihelion deviation angle per revolution as a function of $\Theta$. By applying this result to the perihelion precession of Mercury and utilizing experimental data, we establish a new upper bound on the non-commutative parameter, estimated to be on the order of $10^{-66}\,\mathrm{m}^2$.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 12, 2026 · model on record in the stance chip above.

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