Uncertainty Principles and Non-local Black Holes

Giuseppe Meluccio; Jonas R. Mureika; Salvatore Capozziello

REVIEW 3 major objections 4 minor 1 cited by

Uncertainty Principles and Non-local Black Holes

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper argues that the Generalized and Extended Uncertainty Principles are effective descriptions of non-local gravity, and that matching their Newtonian potentials to infinite-derivative gravity at the characteristic scales fixes…

desk verdict A plausible heuristic spoiled by an unmotivated single-point match: the claimed universal black hole laws are an artifact of the chosen radius. read the letter →

arxiv 2506.21048 v2 pith:RQP56IJQ submitted 2025-06-26 gr-qc hep-th

Salvatore Capozziello , Giuseppe Meluccio , Jonas R. Mureika This is my paper

classification gr-qchep-th

keywords GeneralizedUncertaintyPrincipleExtendedNon-localgravityInfiniteDerivativeBlackholethermodynamicsEventhorizonHawkingtemperature

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 tries to establish that the Generalized Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP) are not separate assumptions but effective descriptions of the non-local character of gravity, specifically of Infinite Derivative Gravity (IDG). The authors match the GUP and EUP Newtonian potentials to the IDG potentials at the respective characteristic length scales, which fixes the previously free parameters beta and alpha. The fixed parameters then imply universal deviations from general relativity: event horizons are about half (UV) or a tenth (IR) of the Schwarzschild radius, and evaporation temperatures are about twice (UV) or ten times (IR) the Hawking temperature. If true, this means black holes always evaporate hotter and are more compact than GR predicts, and the equivalence principle is violated at the quantum level.

What carries the argument

The key object is the matching condition: the Newtonian potential of the GUP-modified metric, $\Phi_{\mathrm{GUP}}(r) = -(G m/r)(1 + \beta/(16\pi G m^2))$, is set equal at $r = L_{\mathrm{UV}}$ to the non-local IDG potential $\Phi_{\mathrm{NL}}(r) = -(G m/r)\,\mathrm{erf}(r/(2 L_{\mathrm{UV}}))$, and similarly the EUP potential is matched to the IR non-local hypergeometric potential at $r = L_{\mathrm{IR}}$. This single-point equality converts the free parameters $\beta$ and $\alpha$ into definite functions of the mass $m$ and the non-locality scales, which then feed directly into the horizon and temperature formulas.

What would settle it

A direct test would be to compute the exact strong-field solution of the IDG theory and compare its event horizon to the predicted $R_{\mathrm{EH}} \approx 0.5\,R_S$ (UV) or $\approx 0.1\,R_S$ (IR); if the true horizon deviates from these values, the effective-description claim fails. Observationally, measuring the GUP parameter $\beta$ (e.g., through gravitational-wave or black-hole shadow data) and finding it positive, mass-independent, or otherwise different from $-m^2/M_P^2$ would also falsify the matching.

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Extended reading notes

Core claim

The central claim is that the dimensionless parameters $\beta$ of the GUP and $\alpha$ of the EUP are fixed by the non-locality scale of gravity. By setting $\Phi_{\mathrm{GUP}}(L_{\mathrm{UV}}) = \Phi_{\mathrm{NL}}(L_{\mathrm{UV}})$, the paper derives $\beta = -16\pi G m^2 [1 - \mathrm{erf}(1/2)] \approx -m^2/M_P^2$, which is negative and mass-dependent. Inserting this into the GUP metric gives an event horizon $R_{\mathrm{EH}} = 2\,\mathrm{erf}(1/2)\,G m \approx 0.5\,R_S$ and an evaporation temperature $T_e \approx 2\,T_H$. The analogous IR matching $\Phi_{\mathrm{EUP}}(L_{\mathrm{IR}}) = \Phi_{\mathrm{NL}}(L_{\mathrm{IR}})$ yields $\alpha \approx -137 (M_P^2/m^2)(L_*^2/L_P^2)$, giving $R_{\mathrm{EH}} \approx 0.1\,R_S$ and $T_e \approx 10\,T_H$. The paper reads these as universal laws for black hole physics beyond GR and as evidence that GUP and EUP are effective manifestations of non-local gravity.

Load-bearing premise

The argument assumes that equating the two potentials at exactly the non-local length scale $L_{\mathrm{UV}}$ (and $L_{\mathrm{IR}}$) captures the equivalence between the effective and fundamental descriptions; if the matching point were different, the derived values of $\beta$ and $\alpha$, and hence the universal horizon and temperature laws, would change.

Editorial extensions

If this is right

If true, the free parameters of GUP and EUP are no longer free: they are tied to the non-locality scale of gravity and to the black hole mass.
Black holes in this picture are always smaller and hotter than their GR counterparts, which changes evaporation rates, lifetimes, and the possibility of remnants.
The negative, mass-squared dependence of beta and alpha implies a violation of the equivalence principle at the quantum level, to be tested in other scenarios.
The predicted ratios (0.5 and 2 in the UV; 0.1 and 10 in the IR) can be compared with future observations of black hole shadows or gravitational-wave ringdowns.

Reading between the lines

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

The matching could also be performed at other radii, such as the horizon itself; because only a single-point equality is used, the claimed universal laws are sensitive to that choice, which the paper does not discuss.
If the identification is right, the same reasoning might extend to higher-order GUP corrections or to non-local theories with different smearing profiles, yielding a family of predictions that could discriminate among proposals.
The predicted factor-two UV temperature enhancement might be testable in the ringdown of small astrophysical black holes if $L_{\mathrm{UV}}$ is not far below the Planck scale, while the IR factor-ten enhancement could affect supermassive black hole observations.

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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

The paper has a clean core idea: treat GUP and EUP as effective descriptions of non-local gravity and use the equivalence to fix the free parameters beta and alpha. That is worth taking seriously, and the authors are honest that the link is an argument rather than a derivation. What is genuinely new is the application to general IDG potentials, both UV and IR, and the explicit constraints on beta and alpha that come out of the matching. The algebra in Sec. II is straightforward and the paper is well written.

The soft spot is the load-bearing one. The single-point match at r = L_UV is not motivated from the IDG action or from any physical principle; it is simply asserted as the scale where non-local effects emerge. The stress-test note is correct that substituting the solved beta back into the GUP potential gives Phi_GUP = -erf(1/2) Gm/r for all r, while the IDG potential becomes -Gm/r for r >> L_UV. At the horizon, where r ~ R_S >> L_UV for any astrophysical mass, the two potentials differ by about 48%. So the metric used to compute the horizon is not an effective description of IDG in the very regime where the predictions are made. This is not a minor technicality: any other choice of matching radius, say r = L_UV/2 or 2L_UV, changes the prefactors from 0.5 to 0.28 or 0.84 respectively. The same structural problem affects the IR case. The universal laws are therefore an artifact of the matching radius, not a robust consequence of non-locality.

There are two additional issues, one moderate and one minor. The moderate one is the use of weak-field Newtonian potentials to predict strong-field quantities like horizon radius and temperature. That extrapolation is not justified anywhere in the paper. The minor one is a likely typo in Eq. (22): the coefficient of the second hypergeometric term should be 2B/√π, not 2√π B, so the displayed result does not follow from Eq. (21) as written. Also, the "predictions" in Eqs. (13)-(14) and (22)-(23) are direct rearrangements of the matching equations themselves, so they are not independent checks, and no observational signature is identified.

All that said, the paper deserves a serious referee. It connects two active phenomenology programs, it is clearly argued, and the flaws are fixable in principle—derive the matching scale, show the potentials agree over a finite range, or find an external observable. The reader's conditional verdict is fair; I would not cite the universal laws as they stand, but I would want to see the revised version. Send it to peer review, with a referee who will push on the matching condition.

Referee Report

3 major / 4 minor

Summary. The paper argues that the Generalized Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP) are effective descriptions of non-local gravity, specifically Infinite Derivative Gravity (IDG). In Sec. II, the authors match the weak-field GUP Newtonian potential to the IDG potential of a smeared point source at the characteristic scale r = L_UV, solve for the GUP parameter beta, and obtain universal black-hole predictions R_EH ~ 0.5 R_S and T_e ~ 2 T_H. In Sec. III, the analogous matching at r = L_IR fixes the EUP parameter alpha and yields R_EH ~ 0.1 R_S and T_e ~ 10 T_H. The paper concludes that non-local gravity determines the GUP/EUP parameters and that these modified black holes always have smaller horizons and higher temperatures than their GR counterparts, with mass-dependent beta and alpha implying a possible quantum violation of the equivalence principle.

Significance. If the central identification were sound, the paper would provide a concrete bridge between two active phenomenological programs and would yield testable predictions distinguishing non-local black holes from GR. The manuscript is clearly written, the linearized IDG potentials are useful explicit inputs, and the algebra in Sec. II is easy to follow. However, as detailed below, the load-bearing matching condition is arbitrary, the strong-field predictions are extracted from weak-field expressions in a regime where the matched potentials disagree, and one displayed result in Sec. III contains a coefficient error. These issues undermine the main claims as stated, so the present version does not establish the advertised universal laws.

major comments (3)

[II, Eq. (11)] The single-point matching at r = L_UV is not physically derived and completely determines the claimed universal prefactors. Substituting Eq. (12) into Eq. (6) gives Phi_GUP(r) = -erf(1/2) Gm/r for every r, while the IDG potential (10) tends to -Gm/r for r >> L_UV. The horizon in Eq. (13) lies at r ~ R_S, and for any astrophysical or black-hole mass R_S >> L_UV under the usual identification L_UV ~ L_P, so the two potentials differ by the constant factor 1 - erf(1/2) ~ 0.48 precisely in the regime where the horizon and temperature are computed. Matching at r = c L_UV would give beta = -16 pi G m^2 [1 - erf(c/2)] and hence R_EH/R_S = erf(c/2), e.g. 0.28 for c = 1/2 and 0.84 for c = 2; the value 0.5 is therefore an artifact of choosing c = 1. Equations (13) and (14) are direct rearrangements of the matching equation, so they do not provide an independent test of the identification.
[II and III, Eqs. (10), (19)] The paper uses weak-field Newtonian potentials to derive strong-field horizon and temperature predictions. The potentials (10) and (19) are solutions of the linearized field equations and are valid only for r >> R_S, where the curvature is weak, but Eqs. (13)-(14) and (22)-(23) evaluate quantities at r ~ R_S. For Schwarzschild-like masses, R_S >> L_UV, so at the horizon the IDG potential is essentially Newtonian while the matched GUP potential remains -erf(1/2) Gm/r. The manuscript provides no argument that a non-local theory that reproduces the Newtonian potential in the weak field must produce the same horizon and temperature as the GUP-inspired metric. A strong-field comparison is required before the claimed equivalence is established.
[III, Eq. (22)] The displayed result in Eq. (22) does not follow from Eq. (21). Substituting Eq. (21) into Eq. (18) gives R_EH = [0F2(;1/2,1;1/4) - (2/sqrt(pi)) 0F2(;3/2,3/2;1/4)] Gm, with coefficient 2/sqrt(pi), not 2 sqrt(pi). This changes the numerical coefficient and therefore also affects the claimed value 0.1 R_S and the temperature factor 10 in Eq. (23).

minor comments (4)

[Title page] The author affiliation contains the typo 'Cintxhia' in the first address; it should read 'Cinthia' as in the second address.
[III, Eq. (19)] The generalized hypergeometric notation 0F2 with the semicolon placements is not defined; a brief definition or reference would make the expression accessible to readers.
[IV] The statement that beta and alpha are 'predicted to be either directly or inversely proportional to the square of the mass' is presented as an equivalence-principle violation, but because beta and alpha are solved from the matching conditions rather than measured independently, this conclusion should be explicitly framed as a consequence of the assumed identification.
[III, Eq. (21)] The approximate numerical value alpha ~ -137 (M_P^2/m^2)(L_*^2/L_P^2) would be easier to check if the hypergeometric values at argument 1/4 were quoted explicitly.

Circularity Check

2 steps flagged · score 6.0 of 10

The 'universal' horizon and temperature laws are algebraic rearrangements of the single-point matching equations used to fix beta and alpha.

fitted input called prediction [Section II, Eqs. (11)-(14)]
"We argue that this length scale is the one at which non-local gravitational effects emerge, i.e. L_UV. To understand this claim and test its predictions, one can then study the equation Φ_GUP(L_UV) = Φ_NL(L_UV). (11) Solving for β yields β = −16πGm^2[1−erf(1/2)] ≈ −24.1Gm^2 ≈ −m^2/M_P^2. (12) ... Substituting this expression for β into that for the event horizon (7) then yields R_EH = 2 erf(1/2)Gm ≈ 0.5 R_S. (13) ... T_e ≈ 2T_H, (14)"

The free parameter β is solved from the single equality Φ_GUP(L_UV)=Φ_NL(L_UV). But Eq. (7) defines R_EH = [1+β/(16πGm^2)]R_S, so inserting the matched β gives R_EH = 2 erf(1/2)Gm by pure algebra. The claimed universal factor erf(1/2), and the evaporation temperature T_e≈2T_H, are therefore rearrangements of the same matching equation. No independent regime or datum is used; choosing any other matching radius r=cL_UV would instead give R_EH/R_S=erf(c/2).
fitted input called prediction [Section III, Eqs. (20)-(23)]
"The useful equation to be studied is then Φ_EUP(L_IR) = Φ_NL(L_IR). (20) Solving for α leads to α = ... (21) ... By substituting this expression for α into that for the event horizon (18), one obtains R_EH = ... ≈ 0.1 R_S. (22) ... T_e ≈ 10T_H. (23)"

α is solved from the matching condition at r=L_IR, and Eq. (18) defines R_EH = [1+4αG^2m^2/L_*^2]R_S. Substituting the matched α reduces directly to the hypergeometric combination in Eq. (22), so the 0.1 R_S prefactor and T_e≈10T_H are consequences of the assumed matching radius rather than independent predictions. The displayed Eq. (22) also appears to contain an arithmetic typo—the coefficient of the second 0F2 should be 2/√π, not 2√π—but in any case the numerical 'law' is fixed by Eq. (21).

full rationale

Both the UV and IR derivations rest on a single-point matching ansatz: the GUP/EUP potential is assumed to equal the IDG potential at the characteristic non-locality scale, Eqs. (11) and (20). The parameters β and α are solved from those equations, and the 'universal' horizon and temperature results (13)-(14) and (22)-(23) are algebraic consequences of the same equations. Thus the central quantitative claims reduce, by construction, to the chosen matching radius. The paper gives no independent argument that r=L_UV (or r=L_IR) is the correct identification, nor a benchmark outside the fitted values; and for astrophysical black holes the matching point lies far from the horizon r~R_S where the laws are applied. This is the fitted-input-called-prediction pattern, not a self-citation chain: the IDG potentials (10) and (19) are taken from external references [73] and [78], and the self-references [4] and [22] are background, not load-bearing. There is also an apparent algebraic inconsistency in Eq. (22) (coefficient 2√π should be 2/√π), but that is a correctness issue rather than circularity. Overall, the paper's headline predictions are forced by the matching ansatz, so the circularity score is 6.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on the matching assumption (one point at L_UV and L_IR) and on treating the weak-field Newtonian potentials as valid for horizon-scale predictions. The parameters beta and alpha are not measured or derived from first principles; they are solved from the matching equations. The paper introduces no new entities, so the invented_entities list is empty.

free parameters (3)

beta (GUP parameter) = beta ≈ -m^2/M_P^2
Solved from Eq. (11), the equality of the GUP and non-local Newtonian potentials at r = L_UV. Mass-dependent, so not a fixed constant of nature.
alpha (EUP parameter) = alpha ≈ -0.2 L*^2/(G^2 m^2) ≈ -137 (M_P^2/m^2)(L*^2/L_P^2)
Solved from Eq. (20), matching the EUP and IR non-local potentials at r = L_IR. Mass- and scale-dependent.
Matching point r = L_UV and r = L_IR = unspecified
The identification of the effective-description scale with the IDG characteristic scale is assumed; the universal prefactors depend on this choice.

assumptions (5)

domain assumption GUP (4) and EUP (15) are the correct modifications of Heisenberg uncertainty.
Taken over from the quantum-gravity phenomenology literature (Refs. [27,39,51,54]).
standard math The weak-field IDG potential (10) is the correct non-local gravity potential.
From Ref. [73]; presumably derived from the IDG action. The paper cites it without re-derivation.
ad hoc to paper A single-point equality of potentials at r = L_UV (or L_IR) is sufficient to fix the effective GUP/EUP parameters.
This is the key assumption (Eqs. 11 and 20). No justification for why a single point captures the equivalence, nor why that point is L_UV or L_IR.
ad hoc to paper Weak-field Newtonian potentials can be used to predict strong-field horizon and temperature behaviour.
Phi_GUP and Phi_NL are weak-field limits; substituting into the horizon formula (7) or (18) extrapolates them to r ~ R_EH where the weak-field approximation breaks down.
standard math Hawking temperature formula T = 1/(4*pi*R_EH) applies to the modified metrics.
Standard relation for spherically symmetric black holes; assumed without derivation.

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Pith. "Pith review of Uncertainty Principles and Non-local Black Holes." pith.science (2026). https://pith.science/paper/RQP56IJQ

@misc{pith2026250621048,
  author       = {Pith},
  title        = {Pith review of: Uncertainty Principles and Non-local Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQP56IJQ}},
  note         = {Machine review of arXiv:2506.21048}
}

read the original abstract

We discuss the Generalized Uncertainty Principle and the Extended Uncertainty Principle in the context of black hole solutions coming from non-local theories of gravity, focusing, specifically, on Infinite Derivative Gravity. We argue that these modifications of the Heisenberg Uncertainty Principle are effective descriptions arising from the non-local features of gravitational interaction. By comparing the predictions of both the modified uncertainty principles and non-local gravity, we find theoretical constraints on otherwise free parameters as well as universal laws for black hole physics beyond General Relativity.

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Forward citations

Cited by 1 Pith paper

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.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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