Extended uncertainty principle inspired black hole in a G\"odel Universe

Reggie C. Pantig

REVIEW 3 major objections 4 minor 1 cited by

Extended uncertainty principle inspired black hole in a G\"odel Universe

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A rotating universe, via a modified uncertainty principle, enlarges black hole horizons, shadows, and deflection angles.

desk verdict A heuristic EUP-to-mass bridge is asserted rather than derived, and the paper's headline prediction that rotation enlarges black hole observables is an artifact of a non-uniform expansion. read the letter →

arxiv 2506.00295 v1 pith:4TCUU5TB submitted 2025-05-30 gr-qc

Reggie C. Pantig This is my paper

classification gr-qc PACS 04.70.-s95.30.Sf97.60.Lf04.50.+h

keywords GödelspacetimeextendeduncertaintyprinciplecorpuscularblackholeshadowphotonspheredeflectionanglePPNconstraints

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

The paper proposes that large-scale rotation of a Gödel-type universe can leave a small quantum imprint on a local black hole. It takes a curvature-modified extended uncertainty principle derived in the Gödel spacetime and reinterprets its momentum bound as an effective energy, producing a Schwarzschild-like lapse function whose mass carries a rotation-dependent correction. In the weak-rotation limit $a \to \infty$, every gravitational observable -- horizon radius, photon-sphere radius, shadow radius, and weak-field deflection angle -- comes out slightly larger than its Schwarzschild value, with leading corrections of order $M^3/a^2$ and position-dependent terms of order $r_0^2/a^4$. Applying the formulas to Event Horizon Telescope shadow measurements of Sgr A* and M87* and to solar-system PPN light bending yields lower bounds $a/M \sim 10^5$ and $a/M_\odot \sim 5\times 10^4$. If correct, the model shows that global rotation, though observationally suppressed, is a coherent semiclassical source of positive corrections to black hole physics.

What carries the argument

The load-bearing mechanism is the Gödel-modified extended uncertainty relation $$\sigma_x \sigma_p \geq \pi\hbar\left[1 - \frac{\$sigma_x^{2}$}{3\$pi^{2}$ $a^{2}$}\left(1 - \frac{1}{C} - \frac{1}{$2C^{2}$}\right)\right]^{1/2}, \quad C = 1 - \left(\frac{r_0}{2a}\right)^2,$$ taken from the asymptotic generalized extended uncertainty principle. The paper's bridge is corpuscular: it identifies $\sigma_p$ with the energy of a condensate of $N$ gravitons confined in a width $\sigma_x$, sets $\sigma_x = 2M$ (the Schwarzschild horizon), and fixes $N$ by requiring $N\pi\hbar/(2M) = M$. This converts the local uncertainty correction into a global rescaling of the black hole mass, $M_{\mathrm{G\ddot{o}del}} = M[1 - 4M^2/(3\pi^2 a^2)(1 - 1/C - 1/(2C^2))]^{1/2}$, which is then placed into the standard lapse function $A(r) = 1 - 2M_{\mathrm{G\ddot{o}del}}/r$. All four computed observables are derived from this single modified lapse function.

What would settle it

Measure the shadow of a supermassive black hole with an uncertainty below the current $\sim M$ scale, or measure solar light deflection with PPN accuracy better than $\Delta = 3\times 10^{-4}$: the model predicts that any deviation from Schwarzschild is positive, so a negative deviation (smaller shadow or smaller deflection) at that precision would falsify it. Alternatively, compute the ground-state energy of the scalar Laplacian in the Gödel background directly from the perturbative eigenvalue problem and compare its implied mass renormalization with Eq. (7); agreement would test the single step on which every observable here depends.

Watch

Extended reading notes

Core claim

The central claim is that the extended uncertainty principle of a rotating Gödel spacetime, grafted onto the corpuscular black hole picture, yields the effective metric function $$A(r) = 1 - \frac{2M}{r}\left[1 - \frac{$4M^{2}$}{3\$pi^{2}$ $a^{2}$}\left(1 - \frac{1}{C} - \frac{1}{$2C^{2}$}\right)\right]^{1/2}, \qquad C = 1 - \left(\frac{r_0}{2a}\right)^2,$$ where $a$ is the Gödel rotation parameter and $r_0$ the observer's distance from the rotation axis. Because $1 - 1/C - 1/(2C^2)$ is negative near $C\approx 1$, the corrected mass is slightly larger than $M$, and expanding in $a \to \infty$ gives $r_h \sim 2M + 2M^3/(3\pi^2 a^2) + \cdots$, $r_{\mathrm{ph}} \sim 3M + M^3/(\pi^2 a^2) + \cdots$, $R_{\mathrm{sh}} \sim 3\sqrt{3}M + \sqrt{3}M^3/(\pi^2 a^2) + \cdots$, and photon deflection $\hat{\Theta} \sim 4M/b + 4M^3/(3\pi^2 b a^2) + \cdots$. Every correction is positive, so global rotation makes black holes appear slightly larger and as stronger lenses than Schwarzschild, and the $r_0^2/a^4$ terms grow with the observer's distance from the rotation axis. The paper then argues that EHT shadow bounds and PPN solar bending bounds independently require $a/M \sim 10^5$ and $a/M_\odot \sim 5\times 10^4$, values large enough to keep the asymptotic expansion valid while leaving the rotation effect real but unobservable at current sensitivity.

Load-bearing premise

The load-bearing premise is that a local momentum-uncertainty correction computed in a geodesic ball of the Gödel spacetime can be equated with a global rescaling of the black hole mass through the corpuscular identification $N\pi\hbar/(2M) = M$, with no derivation that this local quantum effect should renormalize the Schwarzschild mass or that the result should be embedded as a static, spherically symmetric metric.

Editorial extensions

If this is right

The Schwarzschild limit is recovered smoothly as $a \to \infty$, so the model extends, rather than replaces, classical black hole physics.
The horizon, photon sphere, shadow radius, and deflection angle all acquire positive corrections of order $M^3/a^2$ plus position-dependent $r_0^2/a^4$ terms, so a rotating universe makes black holes look systematically larger and act as stronger lenses than their Schwarzschild counterparts.
EHT bounds on Sgr A* and M87* imply $a/M \gtrsim 1.3 \times 10^5$ and $a/M \gtrsim 1.55 \times 10^5$, respectively, placing a Gödel-type rotation five orders of magnitude above the local mass scale.
Solar-system PPN light bending gives $a/M_\odot \gtrsim 5.1 \times 10^4$, a four-orders-of-magnitude separation consistent with the asymptotic series used in the derivation.
The $r_0$-dependent corrections mean the imprint of global rotation grows with the observer's distance from the rotation axis, so the effect is not purely local.

Reading between the lines

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

The same corpuscular bridge could be applied to other rotating backgrounds, such as Kerr or Gödel-type spacetimes with a cosmological constant, predicting rotation-dependent corrections to the shadow asymmetry or the innermost stable circular orbit; the author does not perform this extension.
The positivity of all corrections is a distinctive signature: many alternative gravity theories shrink the shadow, so a future VLBI measurement finding a positive deviation of the predicted $M^3/a^2$ form would support the EUP-corpuscular picture, while a null result at the relevant precision would constrain $a$ from above.
The paper fixes the graviton number implicitly through $N\pi\hbar/(2M) = M$; a field-theoretic derivation of the condensate ground state in Gödel spacetime would turn the heuristic mass renormalization into a testable prediction.
Because the effective mass depends on the observer's position $r_0$, the same black hole would be assigned slightly different masses by observers at different radii, a position-dependent semiclassical mass that could in principle be probed through gravitational-wave ringdown observations from different geometric configurations.

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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 genuinely new thing here is the combination: the Gödel-space AGEUP of Dabrowski-Wagner is grafted onto Mureika's corpuscular mass formula, and the resulting horizon, photon sphere, shadow, and deflection corrections for a Schwarzschild-like lapse are worked out in closed form. That is not in the cited literature, and the algebra from Eq. (9) onward is internally consistent. The series expansions in a^-2 and r0^2/a^4 are executed carefully, and the leading-order checks against Schwarzschild all pass. Credit where due: this is a clean exercise in substituting one effective mass into a known metric and reading off standard observables.

The soft spot is not the algebra; it is the physics at the front end. Equations (6)-(8) convert a local momentum-uncertainty bound for a geodesic ball into a global rescaling of the Schwarzschild mass, with N silently fixed by N pi hbar/2M = M. That is asserted, not derived, and no argument is given that a position-dependent uncertainty correction should be representable as a static, spherically symmetric lapse A(r) = 1 - 2M_Godel/r. If that bridge fails, every formula downstream fails with it. The reader flagged this correctly, and I agree it is load-bearing.

The stress-test note goes further, and here I think it lands. The exact correction factor in Eq. (7) is M sqrt(1 - (4M^2/3 pi^2 a^2) f(C)). For fixed r0 and a to infinity, f -> -1/2 and the correction is positive, which is the regime of Eqs. (12)-(18). But f changes sign at r0/a ~ 2.337. For the parameter values actually used in the EHT and PPN constraints, r0/a ~ 10^5 or larger, so C is large and negative and the exact correction is negative: the horizon, shadow, and deflection are slightly smaller than Schwarzschild, not larger. The positive r0^2/a^4 terms come from expanding in a^-1 at fixed r0 without tracking the size of r0/a; at the quoted points the omitted terms dwarf the kept ones. So the bounds in Eqs. (19) and (21), which require delta > 0, are artifacts of a misapplied expansion. The paper even appears to notice the tension, since the text at the end of Section VI says the large-a values "resolve the earlier tension" when in fact the earlier tension was a symptom of the expansion failing exactly where it was used.

This is a paper that would deserve referee time if the corpuscular bridge were argued rather than assumed, because the computation itself is workmanlike and the combination is new. But as it stands, the central claim is not established even on its own terms: the exact result contradicts the sign of the headline effect in the constrained regime. My recommendation is to engage with it as a borderline desk reject: the model-building idea is worth a referee's hour, but the paper needs a serious derivation of Eqs. (6)-(8) and a uniform expansion before it can support observational bounds. I would not cite it in its current form.

Referee Report

3 major / 4 minor

Summary. The paper constructs a semiclassical black hole spacetime by combining an asymptotic generalized extended uncertainty principle (AGEUP) relation derived in Gödel spacetime with a corpuscular black-hole energy argument. The resulting lapse function A(r)=1-2M_G/r, with M_G given by Eq. (7), is used to compute the event horizon, photon sphere, shadow radius, and weak-field deflection angle. Series expansions in a→∞ are claimed to show that Gödel rotation increases all these observables relative to Schwarzschild, and EHT and PPN data are used to derive lower bounds a/M~10^5 and a/M_sun~5×10^4. The paper concludes that Gödel-type rotation is observationally suppressed but theoretically coherent.

Significance. If the central construction were sound, the paper would offer a novel way to connect curvature-modified uncertainty relations in a rotating spacetime to black-hole observables, with concrete, falsifiable predictions for shadow size and light deflection. The author also deserves credit for carrying out the geodesic/observable calculations consistently from the assumed metric and for attempting to compare with EHT and PPN data rather than leaving the model untested. However, the central bridge from a local uncertainty relation to a global Schwarzschild-like mass is asserted, and the sign and validity of the asymptotic expansion are not controlled in the regime used for the observational constraints. These issues are load-bearing, so the significance of the claimed result is not established.

major comments (3)

[Section II, Eqs. (6)-(8)] The passage from the AGEUP uncertainty relation (5) to the effective mass (7) and then to the lapse function (8) is asserted rather than derived. Setting σx=2M and E_eff=Nπℏ/σx fixes N implicitly, but there is no argument that a local uncertainty bound for a geodesic ball should rescale the total gravitational mass of a black hole, nor that the resulting object is described by a static, spherically symmetric Schwarzschild-like metric. The text itself says 'we assume this effective mass acts as a static, spherically symmetric source,' but no field equations or embedding construction are given. Since every subsequent observable is computed from A(r)=1-2M_G/r, this unsupported identification is the foundation of the paper; if it fails, the horizon, shadow, and deflection formulas all fail with it.
[Section III, Eq. (12), and Section VI, Eqs. (19) and (21)] The asymptotic expansion in a^{-1} is not uniform in r0/a, and the paper's central sign claim fails in exactly the regime used for the EHT and PPN constraints. The exact correction in Eq. (7) depends on C=1-(r0/2a)^2 through f(C)=1-1/C-1/(2C^2). For fixed r0 and a→∞, C→1 and f→-1/2, which gives M_G>M and positive corrections. But f changes sign at C=(1-√3)/2, i.e., r0/a≈2.337; for larger r0/a, f>0 and M_G<M. At the quoted constraints r0/a is of order 10^3 to 10^5, so the exact horizon, photon sphere, shadow, and deflection are slightly smaller than Schwarzschild, not larger. The terms proportional to r0^2/a^4 in Eqs. (12), (14), (15), and (18) come from Taylor expanding at fixed r0 and omitting higher powers of r0^2/a^2, which are enormous at the quoted parameters; the truncation is therefore invalid. The EHT/PPN bounds in Eqs. (19) and (21), which require δ>0, are artifacts of this misapplied expansion and do not constrain a in the claimed way.
[Section IV, Eq. (15), and Section VI] There is an internal inconsistency between the claimed sign of the Gödel correction and the text of Section VI. The paper's expansions predict that global rotation 'consistently increases all observables,' yet the text states that 'the Gödel correction ... always acts to reduce the effective radius' when discussing the constraint. This is not merely a wording issue: the constraint formula (19) is derived by assuming the observed shadow deviation δ is positive and then taking the square root of δ; if the exact correction is negative in the observational regime, the constraint is inverted and the inferred lower bounds on a are not valid.

minor comments (4)

[Throughout] There are repeated typographical errors, most notably 'photon spherehere' appearing in the abstract, Section IV, and the conclusion; these should be corrected.
[Section II, Eqs. (2) and (5)] The symbol C is used both for the Cartan-type curvature invariant in Eq. (2) and for the Gödel parameter C=1-(r0/2a)^2 in Eq. (5); this overloaded notation invites confusion and should be changed.
[Section VI, Eqs. (19) and (21)] The numerical bounds for Sgr A* and M87* are quoted but the values of r0 used in the constraints are not specified; since the correction depends explicitly on r0, the bounds are not reproducible without stating the assumed observer or fluctuation distance.
[Section III, after Eq. (12)] The text says that for r0≫a the r0^2/a^4 term dominates and 'we may observe a horizon radius that is considerably larger than 2M.' This directly contradicts the exact expression (11), which approaches M_G<M for r0≫a; the statement is a symptom of using an expansion outside its domain of validity.

Circularity Check

1 steps flagged · score 3.0 of 10

Constraint loop in Sec. VI: the EHT/PPN bounds on a are generated by the same shadow/deflection expansion that they are then used to certify.

fitted input called prediction [Section VI, paragraph after Eq. (19); echoed in Section VII]
"With these constraints, which now fall squarely within the regime of the expansion a → ∞, it confirms that the series approximation used to derive horizon, photon spherehere, and shadow corrections remains fully valid."

Equation (19) is obtained by equating the observed shadow deviation δ to the positive r0^2/a^4 term of the shadow expansion in Eq. (15) and inverting for a. The resulting fitted value of a is then used to 'confirm' the validity of that same expansion. This is circular: the confirmation input is manufactured by the expansion being confirmed. The claim also ignores that the expansion is not uniform in r0/a: at the quoted Sgr A* values r0/a ~ 2e5, the exact correction in Eq. (7) has the opposite sign, so the constraint and the validity assertion are artifacts of the truncated series rather than independent evidence.

full rationale

The paper's central construction imports the Gödel-AGEUP uncertainty relation (Eq. 5) from the external Ref. [10] and the corpuscular energy-mass identification from Ref. [11]; neither is a self-citation, and Refs. [23, 24] by the same author are not load-bearing for the key formulas. The mass rescaling in Eqs. (6)-(8) — setting σx = 2M and Eeff = M times the AGEUP correction — is a model assumption rather than a circular derivation: given that assumption, the horizon, photon sphere, shadow, and deflection follow from standard Schwarzschild formulas evaluated at the rescaled mass. The genuine circularity is confined to Section VI, where a parameter a is fitted to the observed shadow deviation using the r0^2/a^4 term of the same expansion that the fitted value is then invoked to validate. A separate, serious correctness issue is the non-uniform expansion in r0/a: for the constrained EHT/PPN regime the exact Eq. (7) predicts smaller observables, whereas the truncated expansions in Eqs. (12)-(18) predict larger observables. That inconsistency is a mathematical error rather than circularity, but it makes the circular validation loop harmful rather than benign. Overall, the paper does not fit its observables to tune the free parameters after the fact — a is constrained after the formulas are fixed — so the circularity is moderate and localized, not pervasive.

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

No new particles, fields, or forces are introduced. The effective mass M_God is a derived quantity, not an independent entity. The free parameters are a, r0, and the implicit graviton number N.

free parameters (3)

a (Godel rotation parameter) = a/M ~ 1.30e5 (Sgr A*), 1.55e5 (M87*); a/M_sun ~ 5.1e4
Enters the metric through Eq. (9); determined from EHT and PPN constraints rather than measured independently.
r0 (radial location or observer distance) = r0/M_sun = 1.017e8 for the Sun; not specified for Sgr A* or M87*
Sets the scale of the r0^2/a^4 corrections and is used as the observer distance in Eq. (15); its values in the EHT constraints are not stated.
N (number of gravitons in the corpuscular condensate) = unspecified
Appears in Eq. (6); the paper implicitly assumes N pi hbar / (2M) = M, but this relation is never written or justified.

assumptions (5)

domain assumption The AGEUP for the Godel metric (Eq. 5) is a valid uncertainty relation and can be extended to horizon scales.
Imported from Ref. [10]; no independent derivation or check is provided in this paper.
ad hoc to paper A local uncertainty correction can be reinterpreted as a global rescaling of black hole mass via the corpuscular relation E_eff = N pi hbar / sigma_x with sigma_x = 2M.
Eqs. (6)-(7) perform this identification; it is the bridge between the uncertainty principle and the metric.
ad hoc to paper The resulting effective mass behaves as a static spherically symmetric Schwarzschild source, despite its rotating Godel origin.
Eqs. (8)-(10) assume this; no field equations or embedding are given.
domain assumption Observed EHT shadow deviations and PPN residuals are attributable to the Godel correction, and only positive deviations are allowed.
Section VI uses delta greater than 0 to derive lower bounds and discards negative deviations without independent justification.
standard math The Gauss-Bonnet deflection formula of Ref. [31] applies to this effective metric.
Used in Section V to derive the deflection angle; it is an established method but its applicability to the assumed metric is not discussed.

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Cite this review

Pith. "Pith review of Extended uncertainty principle inspired black hole in a G\"odel Universe." pith.science (2026). https://pith.science/paper/4TCUU5TB

@misc{pith2026250600295,
  author       = {Pith},
  title        = {Pith review of: Extended uncertainty principle inspired black hole in a G\"odel Universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TCUU5TB}},
  note         = {Machine review of arXiv:2506.00295}
}

abstract

We explore analytically the implications of a curvature-modified extended uncertainty principle (EUP) derived in a rotating G\"odel spacetime and apply it to the construction of a semiclassical black hole model. Adapting techniques from corpuscular black hole frameworks, we reinterpret the G\"odel-type uncertainty relation as an effective energy bound, leading to a modified lapse function with explicit dependence on the global rotation parameter $a$ and the radial coordinate $ r_0 $. Analytic expressions are derived for key gravitational features, including the event horizon, photon spherehere, shadow radius, and deflection angle, with curvature corrections scaling as $ a^{-2} $ and $ r_0^2 / a^4 $. Series expansion in the limit $ a \to \infty $ shows that global rotation consistently increases all observables relative to the Schwarzschild case. Applying these results to astrophysical data, we use Event Horizon Telescope (EHT) measurements of Sgr A* and M87* to infer lower bounds of $ a/M \sim 10^5 $, while solar system light-bending observations in the parametrized post-Newtonian (PPN) framework yield $ a / M_\odot \sim 5 \times 10^4 $. These large but finite values validate the asymptotic expansion and confirm that G\"odel-type rotation remains observationally suppressed, yet theoretically coherent. Our results demonstrate that global rotation, when treated semiclassically via curvature-modified uncertainty, introduces detectable signatures in principle, though well below current observational sensitivity. The framework offers a consistent path toward exploring the quantum-gravitational interplay between global geometry and local black hole structure.

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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

Uncertainty Principles and Non-local Black Holes
gr-qc 2025-06 conditional novelty 5.0 of 10

By matching GUP and EUP potentials to non-local gravity at a single point, the authors derive mass-dependent GUP/EUP parameters and universal horizon and temperature scalings for black holes.

Reference graph

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