Spacetime-curvature induced uncertainty principle: linking the large-structure global effects to the local black hole physics

Ali \"Ovg\"un; Gaetano Lambiase; Nikko John Leo S. Lobos; Reggie C. Pantig

REVIEW 4 major objections 4 minor 3 cited by

Spacetime-curvature induced uncertainty principle: linking the large-structure global effects to the local black hole physics

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

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Pith's one-line read The paper derives a Schwarzschild-like black hole metric whose effective mass is rescaled by the cosmological constant and the Planck length, making the universe's large-scale curvature a participant in local black hole physics.

desk verdict The core effective-mass formula is asserted rather than derived, and internal errors in alpha, the Lyapunov exponent, and an undefined n make the paper unpublishable in its current form. read the letter →

arxiv 2412.00303 v2 pith:ZGR4ET2E submitted 2024-11-30 gr-qc

Reggie C. Pantig , Gaetano Lambiase , Ali Övgün , Nikko John Leo S. Lobos This is my paper · ORCID

classification gr-qc PACS 95.30.Sf04.70.-s97.60.Lf04.50.+h

keywords spacetimecurvatureuncertaintyprincipleblackholescosmologicalconstantcorpusculargravityholeshadowquasinormalmodesgravitationaldeflection

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 attempts to establish a concrete bridge between the largest scale in physics—the curvature of the universe—and the local quantum structure of a black hole. It does this by feeding the asymptotic generalized extended uncertainty principle (AGEUP), with the cosmological constant $\Lambda$ built into the position uncertainty, into the corpuscular picture of a black hole as a Bose-Einstein condensate of gravitons. The output is a Schwarzschild-like metric $f(r) = 1 - 2M_{\mathrm{eff}}/r$ whose effective mass $M_{\mathrm{eff}} = M - \frac{2\Lambda M^3}{3\pi^2} + \frac{\beta l_{\mathrm{Pl}}^2}{2M}$ contains a $\Lambda$-dependent term, so the cosmological constant would actively alter the black hole's mass rather than only bending spacetime at large distances. The paper then shows that this mass rescaling propagates into shadow radii, deflection angles, thermodynamics, and quasinormal ringing, and that current shadow and solar-system measurements can constrain the quantum modulation parameter $\beta$. This matters because it is a concrete proposal in which large-scale curvature and local quantum-gravity corrections leave imprints in the same observable quantity, the black hole mass.

What carries the argument

The load-bearing machinery is the dS/AdS form of the AGEUP relation, Eq. (7), together with the corpuscular scaling $N \sim R^2/l_{\mathrm{Pl}}^2$. The relation supplies a curvature-induced quadratic correction in position uncertainty, while the corpuscular scaling converts uncertainty data into a mass scale: $\sigma_x$ becomes the black hole radius and $\sigma_p$ becomes the graviton momentum, turning Eq. (7) into the effective mass $M_{\mathrm{eff}}$ of Eq. (10). Once $M_{\mathrm{eff}}$ is defined, the paper routes it through standard formalisms—photon-sphere shadow calculation, finite-distance weak deflection, strong-deflection integrals, holographic-screen temperature, and eikonal quasinormal modes—so that every observable is expressed as a Schwarzschild result plus $\Lambda$- and $\beta$-dependent shifts.

What would settle it

The decisive check is a first-principles computation of the graviton-condensate ground-state energy in Schwarzschild--de Sitter spacetime: if the $\Lambda$-dependent term in the effective mass is not $-2\Lambda M^3/(3\pi^2)$ with exactly that coefficient, then the metric of Eqs. (9)--(10) is refuted. Observationally, a shadow measurement of Sagittarius A* that pushes the allowed deviation $\delta/M$ below the existing range of $-0.364$ to $0.987$ would tighten the bound on $\beta$; if independent arguments force $\beta$ to be of order one, the model would be excluded.

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Extended reading notes

Core claim

The paper's central claim is that the AGEUP uncertainty relation in de Sitter and anti-de Sitter backgrounds, $\sigma_p \sigma_x \geq \pi\hbar\left(1 - \frac{\Lambda}{6\pi^2}\sigma_x^2 + \beta l_{\mathrm{Pl}}^2 \sigma_p^2\right)$, can be combined with the corpuscular relation $N \sim R^2/l_{\mathrm{Pl}}^2$ for a graviton-condensate black hole to yield a Schwarzschild-like metric whose mass is rescaled to $M_{\mathrm{eff}} = M - \frac{2\Lambda M^3}{3\pi^2} + \frac{\beta l_{\mathrm{Pl}}^2}{2M}$. In this construction the uncertainty in position is identified with the horizon radius and the uncertainty in momentum with the graviton momentum, so the curvature term in the uncertainty relation becomes a correction to the black hole mass. From this rescaled mass the paper derives a critical mass $M_{\mathrm{crit}} = \frac{\pi}{2}\sqrt{6/\Lambda} \approx 3.14\times 10^{26}$ m at which the $\Lambda$ correction cancels the classical mass; a value $\alpha = 1/(3\pi)$ for the EUP modulation factor if the large fundamental length is the cosmological horizon; and constraints on $\beta$ from shadow radii, photon rings, and parametrized post-Newtonian light bending that range from about $10^{72}$ to $10^{96}$, far above laboratory bounds. The conceptual punch is that the cosmological constant is no longer a passive background but participates in the black hole's local gravitational field.

Load-bearing premise

The load-bearing premise is the uncalculated step from Eq. (7), an uncertainty relation in position and momentum, to Eq. (10), a mass formula: one must assume the position uncertainty is the horizon radius and the momentum uncertainty is the graviton momentum, with the exact coefficients presented, yet the paper asserts this corpuscular translation rather than deriving it.

Editorial extensions

If this is right

If the central claim is right, the cosmological constant enters the black hole's local quantities: horizon radius, temperature, entropy, and quasinormal-mode frequencies all acquire shifts proportional to $\Lambda M^2$, so the universe's large-scale curvature is not decoupled from the horizon.
There is a critical mass $M_{\mathrm{crit}} \approx 3.14\times10^{26}$ m at which the $\Lambda$ correction cancels the classical mass; for smaller masses the quantum term $\beta l_{\mathrm{Pl}}^2/(2M)$ can dominate, but the requirement $M_{\mathrm{eff}}>0$ forces $\Lambda>0$ and $\beta>0$ in that regime.
For astrophysical black holes the predicted shadow and deflection shifts are tiny, so the model is consistent with current measurements; the existing shadow and solar-system bounds translate into very large values of $\beta$ (approximately $10^{72}$ to $10^{96}$), so a positive detection would mean an unexpectedly strong quantum-gravity coupling.
In the strong-deflection limit the deflection coefficients take their Schwarzschild values ($\bar{a}=1$), with $\Lambda$ and $\beta$ entering only through the regular integral and the critical impact parameter, making strong lensing a comparatively weak probe of these corrections.
The derived value $\alpha = 1/(3\pi)$ ties the EUP modulation factor to the cosmological horizon; if independently measured, this relation becomes a test of the identification of the large fundamental length with $\sqrt{3/\Lambda}$.

Reading between the lines

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

Editorial inference: the exact coefficients in $M_{\mathrm{eff}}$ are not forced by the uncertainty relation alone; a first-principles derivation of Eq. (10) from the corpuscular Hamiltonian could change the coefficients while preserving the overall structure, so the metric should be regarded as a phenomenological template until that derivation exists.
Editorial inference: because the $\Lambda$ term grows as $M^3$, the largest supermassive black holes should show the largest fractional mass rescaling; comparing shadow or ringdown observations across a wide mass range would separate the $\Lambda$ effect from the $\beta$ effect.
Editorial inference: applying the same AGEUP-plus-corpuscular replacement to rotating or charged black holes would introduce spin-charge degeneracies in the shadow; multi-frequency imaging would be needed to isolate $\beta$.
Editorial inference: the framework suggests a testable relation between $\alpha$, $\Lambda$, and the cosmological horizon scale; an independent EUP experiment at cosmological distances could verify or break the relation $\alpha = 1/(3\pi)$.

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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

Colleague,

This one is a reject, and for the reasons the reader gives. The central formula, the effective mass in Eq. (10), is simply asserted. The text says it is obtained 'by using the corpuscular framework' and then writes it down, but nothing in Section III shows how the AGEUP uncertainty relation (7) becomes M_eff = M - 2ΛM^3/(3π^2) + β l_Pl^2/(2M). You have to guess the identifications of σ_x and σ_p with horizon radius and graviton momentum, the graviton counting N∼R^2/l_Pl^2, and a saturation convention. Without those choices, the exact coefficients are unfalsifiable. Since everything else—shadow, deflection, QNMs, lensing—inherits this M_eff, the gap is load-bearing.

What the paper does passably: it gives a clear review of AGEUP, and the observation that Mureika's EUP black hole length scale can be identified with the cosmological horizon is a reasonable short step. The shadow and weak-deflection calculations are standard, and the authors are honest that the resulting bounds on β are very weak and that the model might be observationally excluded. That honesty is worth some credit.

The soft spots are not just the derivation gap. The α value is internally inconsistent: with L* = √(3/Λ), Eq. (13) gives α = 1/(2π²), not the printed 1/(3π) or the ≈0.034. The Lyapunov exponent in Eq. (42) has the wrong dimension (1/√length instead of 1/length). Eq. (20) has an undefined n. And the β constraints from EHT and VLBI are fits to observed deviations, not predictions; with positive β selected by hand. The bounds, β ~ 10^72–10^95, are so weak that the paper's own conclusion questions whether the model is already excluded.

Who gets value from this? Possibly someone cataloguing EUP/GUP black hole metrics, but they would have to redo the central derivation themselves. As is, I would not send it to a referee. The authors should supply the missing derivation and fix the internal errors; then the paper could be worth another look.

Recommendation: desk reject.

Referee Report

4 major / 4 minor

Summary. The manuscript combines the Asymptotic Generalized Extended Uncertainty Principle (AGEUP) with the corpuscular graviton-condensate picture to propose a static, spherically symmetric metric with effective mass M_eff = M - (2ΛM^3)/(3π^2) + β l_Pl^2/(2M). It then applies this metric to black hole shadows, weak and strong deflection angles, Unruh-Hawking thermodynamics, eikonal quasinormal modes, and EHT/VLBI constraints on the quantum parameter β. The central claim is that the cosmological constant enters local black hole mass through the uncertainty relation, rather than only through a global dS/AdS background term.

Significance. If Eq. (10) were derived rather than asserted, the idea that large-scale curvature can enter the black hole mass through an uncertainty relation would be a genuinely interesting phenomenological contribution, and the paper covers a standard set of observational probes in a transparent way. The authors also cite the relevant corpuscular and AGEUP literature and present their numerical bounds explicitly. However, the load-bearing formula is currently introduced without derivation, and several quantitative results in Sections III, IV, and VI are internally inconsistent. These issues prevent the reader from assessing the validity of the claimed predictions, so the significance of the work cannot be established from the manuscript in its present form.

major comments (4)

[III, Eq. (10)] The step from Eq. (7) to Eq. (10) is not shown. The text states 'we determine the M_eff by using the corpuscular framework' and cites Refs. [65-69], but it does not specify how the uncertainty variables σ_x and σ_p are identified with the horizon radius and the graviton momentum, how the corpuscular counting N ~ R^2/l_Pl^2 is imposed, or what saturation convention is used. Without these identifications the coefficient 2/(3π^2) in the Λ-term and the coefficient 1/2 in the β-term cannot be checked. Since Eqs. (15), (18), (32), (45), and (60)-(64) all inherit M_eff, this missing derivation is the load-bearing pillar of the paper.
[III, Eq. (13) and following text] The derived value of α is inconsistent with the preceding formula. If L_* = sqrt(3/Λ), then Eq. (13), L_*^2 = 6π^2 α/Λ, gives 6π^2 α = 3 and hence α = 1/(2π^2) ≈ 0.0507. The paper instead states α = (3π)^(-1) ≈ 0.034. This arithmetic error affects the claimed derivation of the EUP modulation parameter from the cosmological horizon scale.
[VI, Eqs. (41)-(42)] The Lyapunov exponent in Eq. (42) is incorrect. Evaluating Eq. (41) for f(r) = 1 - 2M_eff/r at r_0 = 3M_eff gives λ = 1/(3√3 M_eff), not 1/(3√(2M_eff)). The printed expression has the wrong dimension, and the imaginary part of the eikonal QNM frequency in Eq. (44) should share the same 1/(3√3 M_eff) factor as the real part. This changes the expansion in Eq. (46) and the subsequent interpretation of damping rates.
[IV, Eq. (17) and the numerical bounds] The substitution of the EHT deviations is inconsistent. If δ in Eq. (17) is the shadow-length deviation, as implied by the quoted range -0.364 ≤ δ/M ≤ 0.987, then for Sgr A* the first term gives β ≈ 5.9 × 10^88, not the reported 9.24 × 10^78. If δ is instead taken as δ/M, the first term of Eq. (17) has the wrong dimension. In addition, these 'constraints' attribute the entire observed Schwarzschild deviation to the β-term; they are fits to the model, not predictions, and the choice of only the positive sign in the VLBI comparison in Eq. (20) further weakens the exclusion claim in Section VIII.

minor comments (4)

[VII, Eqs. (60) and (62)] The regular-integral expression in Eq. (60) contains -6M^2π, while the coefficient k in Eq. (62) contains -6M^2π^2; one of these is a typographical error and the two equations should be made consistent.
[IV, Eq. (20)] The quantity n appearing in the second term of Eq. (20) is not defined, which makes the PPN expression difficult to reproduce.
[II and V] There are several typos and encoding artifacts, including 'Schrdinger' for Schrödinger and 'Godel' for Gödel, and Eq. (14) would be clearer as β_crit = 4ΛM^4/(3π^2 l_Pl^2) rather than β_crit/M^4 = ... .
[VIII] The conclusion refers to a 'weak bound β < 10^120' and suggests the model may be observationally excluded, but the paper actually derives enormous positive β values by fitting deviations; the language should distinguish an upper bound on β from a fitted value under the model.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central M_eff formula is an underived phenomenological input rather than a self-referential prediction, and the EHT/VLBI beta bounds are honest fits; self-citations are not load-bearing.

full rationale

The paper's central step, Eq. (10), is introduced with the sentence 'Note that we determine the Meff, by using the corpuscular framework' and no algebraic derivation is shown. I checked whether Eq. (10) reduces to Eq. (7) by construction. Reproducing the coefficients would require choosing sigma_x = 2M and sigma_p = 1/(sqrt(2) M), but the paper never states these identifications or derives them from N ~ R^2/l_Pl^2; alternative identifications change the M-dependence. So this is a derivation gap, which is a correctness risk, not a circular equivalence. The shadow, deflection, temperature, QNM, and strong-lensing results all follow consistently from f(r) = 1 - 2M_eff/r and are standard consequences once the metric is adopted. The beta constraints in Eqs. (17) and (20) are obtained by solving the model's expressions against EHT/VLBI data and are explicitly labeled as constraints or bounds, not parameter-free predictions, so the fitted-input-as-prediction pattern does not apply. The alpha value alpha = (3 pi)^-1 is a conditional consequence of choosing L* = sqrt(3/Lambda), not a hidden restatement of the input. Self-citations [46], [86], and [88] are used for calculational methodology and a comparative remark; none is load-bearing for the AGEUP-to-metric step, which rests on external references [44, 57, 65-69]. Hence no circular step meeting the quoted-reduction standard can be exhibited.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new particles or forces are introduced. The free parameters beta and alpha are carried over from the uncertainty principle and fitted or assumed; the central metric inherits all of them.

free parameters (2)

beta (AGEUP/GUP modulation parameter) = 9.24e78 (Sgr A* shadow), 1.25e82 (M87* shadow), 1.02e72 (VLBI), 9.96e95 (photon ring)
Dimensionless parameter modulating the GUP term in Eq (7); constrained by fitting observed shadow deviations and deflection angles rather than predicted independently.
alpha (EUP modulation parameter) = 1/(3pi^2) approximately 0.034 as claimed, inconsistent with Eq (13) which gives 1/(2pi^2)
Free in Eq (1)/(13); the paper sets it by assuming the large fundamental length is the cosmological horizon L*=sqrt(3/Lambda).

assumptions (4)

domain assumption AGEUP uncertainty relation Eq (7): sigma_p sigma_x >= pi hbar (1 - Lambda/(6pi^2) sigma_x^2 + beta l_Pl^2 sigma_p^2)
Taken from Ref [57] as the starting point; not derived here. It presupposes that curvature modifies the uncertainty principle in this exact algebraic form.
domain assumption Corpuscular framework: black holes are Bose-Einstein condensates of N similar R^2/l_Pl^2 weakly interacting gravitons
Invoked via Refs [65-69,44] to justify replacing the mass in the metric by M_eff; the N-R relation is assumed.
domain assumption The dS/AdS curvature enters only through Lambda in Eq (7), and the resulting metric is Schwarzschild-like with f=1-2M_eff/r
The paper assumes spacetime remains static and spherically symmetric with the same lapse structure after quantum and cosmological corrections; Section III.
ad hoc to paper Mapping from uncertainty variables to black hole mass: the unshown step from Eq (7) to Eq (10)
No derivation is presented; without it the central metric is asserted. This is the weakest premise.

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Cite this review

Pith. "Pith review of Spacetime-curvature induced uncertainty principle: linking the large-structure global effects to the local black hole physics." pith.science (2026). https://pith.science/paper/ZGR4ET2E

@misc{pith2026241200303,
  author       = {Pith},
  title        = {Pith review of: Spacetime-curvature induced uncertainty principle: linking the large-structure global effects to the local black hole physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGR4ET2E}},
  note         = {Machine review of arXiv:2412.00303}
}

abstract

This paper links the advanced formulation of the Generalized Uncertainty Principle, termed the Asymptotic Generalized Extended Uncertainty Principle (AGEUP), to the corpuscular framework to derive the AGEUP-inspired black hole metric. The former incorporates spacetime curvature effects to explore black hole dynamics under quantum gravitational corrections, while the latter is a view that black holes are Bose-Einstein condensates of weakly interacting gravitons. In a particular case, the phenomenological union between the AGEUP with cosmological constant $\Lambda$ to the corpuscular framework enabled a black hole metric that has a scaled mass, which depends on $\Lambda$ and the Planck length $l_{\rm Pl}$. Interesting implications occur, such as the maximum limit for mass $M$ where $\Lambda$ ceases to influence the black hole. Another is the derived value of the modulation factor of the EUP term, $\alpha$, if the large-scale fundamental length is defined solely as the cosmological horizon. Additional analyses were done through the shadow and deflection angle phenomena, deriving constraints on the quantum gravity modulation parameter $\beta$. Constraints from the Event Horizon Telescope (EHT) and Very Long Baseline Interferometry (VLBI) are discussed as avenues for verifying AGEUP-related deviations in black hole shadow radius and deflection angles, offering potential observational evidence of quantum gravitational effects at astrophysical scales. The findings suggest that AGEUP could be instrumental in providing hints on the quantum gravity nature of black holes, particularly in high-energy astrophysical contexts. By linking local black hole physics with large-scale curvature effects, AGEUP paves the way for further research at the intersection of quantum gravity and cosmology, with implications for observational astrophysics and the fundamental structure of spacetime.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Bounded compactness from G(E)UP
gr-qc 2025-07 conditional novelty 6.0 of 10

The generalized uncertainty principle bounds the compactness of any object much heavier than the Planck mass by about 1/α, and the existence of black holes forces the GUP parameter to satisfy α ≲ 2.
Combining the Generalized and Extended Uncertainty Principles
gr-qc 2026-02 reject novelty 5.0 of 10

An alternative combined uncertainty relation (EGUP) is introduced, and the GEUP is shown to gain a third branch that the authors label—without a derived metric—a new type of black hole.
Extended uncertainty principle inspired black hole in a G\"odel Universe
gr-qc 2025-05 reject novelty 4.0 of 10

A Godel-rotation-modified uncertainty principle is used to define a corrected black hole mass, producing enlarged horizon, shadow, and deflection with lower bounds a/M ~ 10^5 from EHT and PPN data.

Reference graph

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