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Bounded compactness from G(E)UP

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

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Pith's one-line read From a generalized uncertainty principle, the paper derives that any self-gravitating object's compactness is capped at C ≲ 1/α, so black holes require α ≲ 2.

desk verdict The compactness bound is a valid formal consequence of the GUP, but the input Δp ~ M for a star is unjustified, so the astrophysical claim doesn't follow. read the letter →

arxiv 2507.14703 v2 pith:IM2ZTWRE submitted 2025-07-19 gr-qc

Roberto Casadio , Ilim Irfan Çimdiker , Octavian Micu , Jonas Mureika This is my paper
classification gr-qc MSC 83C5781S0783C45 PACS 04.60.-m04.70.-s03.65.-w
keywords generaliseduncertaintyprincipleGUPGEUPcompactnessblackholeminimumlengthquantumgravityphenomenologyshadow
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 aims to show that a minimal length of Planckian order, encoded in generalized uncertainty principles, generically caps the compactness of any self-gravitating object. Starting from the GUP relation $\Delta x\Delta p \ge \hbar(1+\alpha\Delta p^2/m_p^2)$ and a spherically symmetric object with isotropic uncertainties, it derives $C \lesssim M^2/(3m_p^2+\alpha M^2)$, which for $M\gg m_p$ becomes $C\lesssim 1/\alpha$. Since black holes are characterised by $C>1/2$, their existence forces $\alpha\lesssim 2$; the GEUP instead yields a two-sided band around the same limit. If correct, this links the Planck-scale parameter $\alpha$ to astrophysical observables such as the shadows of M87 and Sgr A*.

What carries the argument

The engine is the extremal Generalised Uncertainty Principle, $\Delta x\Delta p = \hbar(1+\alpha\Delta p^2/m_p^2)$, combined with the assignment that each Cartesian component of the momentum uncertainty satisfies $\Delta p_i = M/\sqrt{3}$. Solving the inequality for $\Delta x$ and substituting into $C = G_N M/R = \ell_p M/(m_p R)$ converts the quantum uncertainty into an algebraic cap on compactness; adding the GEUP's $\beta\Delta x^2$ term turns that single inequality into a two-sided band.

What would settle it

Estimate the rest-frame momentum dispersion $\Delta p_i$ of a realistic star from its internal velocity distribution and check whether $\Delta p_i \simeq M/\sqrt{3}$; alternatively, observe a horizonless compact object with measured compactness exceeding the right-hand side of Eq. (3.5) under an independently constrained $\alpha$.

Watch

Extended reading notes

Core claim

For a self-gravitating object described by a wavefunction in its centre-of-mass frame with isotropic uncertainties $\Delta x_i^2=R^2/3$ and $\Delta p_i^2=M^2/3$, the GUP implies $C \lesssim M^2/(3m_p^2+\alpha M^2)$, i.e. $C\lesssim 1/\alpha$ for $M\gg m_p$. A parameter of order one therefore caps the compactness of large astrophysical objects at order one, and black holes, which have $C>1/2$, can form only when $\alpha\lesssim 2$. The GEUP replaces the single cap by a band, with the large-mass limits $(1-\sqrt{1-4\alpha\beta})/(2\alpha) \lesssim C \lesssim (1+\sqrt{1-4\alpha\beta})/(2\alpha)$, and for $\beta\to 0$ the band collapses to the GUP bound. Using the strongest experimental limits on $\alpha$ and $\beta$, the paper concludes that the GUP correction is the relevant one for astrophysical compactness.

Load-bearing premise

The derivation hinges on the assumption in Eq. (2.5) that each Cartesian component of the object's momentum uncertainty equals $M/\sqrt{3}$; if a real astrophysical body has a much smaller momentum spread in its rest frame, the bound $C\lesssim 1/\alpha$ does not follow.

Editorial extensions

If this is right

  • If the GUP bound holds, no self-gravitating object of mass $M\gg m_p$ can exceed a compactness of order $1/\alpha$, so singular zero-size configurations would be excluded for $\alpha\sim 1$.
  • Because black holes have $C>1/2$, horizon formation is possible only for $\alpha\lesssim 2$; larger values would make collapse end in horizonless compact objects.
  • The observed shadows of M87 and Sgr A* show that $C\sim 1/2$ is realised in Nature, so values of $\alpha$ up to about 2 are compatible with observation.
  • The current experimental bound $\alpha\lesssim 10^7$ leaves the compactness cap free to be of order unity, and string-theory-motivated values $\alpha\sim 1$ make the cap $C\lesssim 1$ physically relevant.
  • For the GEUP, the S2-orbit bound $\beta\lesssim 10^{-90}$ makes the EUP correction negligible, so the GUP bound remains the operative limit on astrophysical compactness.

Reading between the lines

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

  • The input relation $\Delta p_i = M/\sqrt{3}$ is assumed, not derived; for a composite astrophysical body the centre-of-mass momentum uncertainty need not scale with total mass, and a much smaller $\Delta p$ would weaken the cap.
  • The same ratio-symmetric argument could be run on angular variables to bound the dimensionless spin of a compact object from a GUP-like uncertainty relation, which the paper does not do.
  • A concrete testable extension would be to estimate the rest-frame momentum spread of a neutron star or white dwarf from its internal temperature and density; if $\Delta p/M\ll 1/\sqrt{3}$, the bound's reach for macroscopic bodies is much weaker than the $M\gg m_p$ formula suggests.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper derives upper (and for the GEUP, lower and upper) bounds on the compactness C = G_N M/R of self-gravitating objects from the GUP and GEUP. Starting from assumed position and momentum uncertainties in the centre-of-mass frame, Δx_i = R/√3 and Δp_i = M/√3 (Eqs. (2.4)–(2.5)), the authors obtain C ≲ M^2/(3 m_p^2 + α M^2) ≃ 1/α for M ≫ m_p (Eq. (3.5)), and the two-sided GEUP band (4.7). They then use the existence of black holes (C > 1/2) to infer constraints such as α ≲ 2, and combine these with experimental bounds on α and β. The central claim is that quantum gravity generically caps the compactness of large astrophysical objects at order unity for α of order one.

Significance. If the conclusion were correct, the paper would provide a simple, model-independent-looking link between GUP/GEUP parameters and the maximum compactness of astrophysical bodies, with falsifiable consequences such as α ≲ 2 from the existence of black holes and constraints on β from the same requirement. The algebraic steps from Eqs. (3.1)–(3.5) and (4.1)–(4.7) are clean and correctly executed, and the use of existing experimental bounds on α and β is transparent. However, the entire edifice rests on the unproven and, for many-body systems, physically implausible identification in Eq. (2.5) that the per-coordinate momentum uncertainty equals M/√3. For a localized star or black hole, the centre-of-mass momentum spread is set by the size of the system and is vastly smaller than M, so the derived bound does not apply to the systems the paper targets. The result therefore does not substantiate the abstract's claim about 'large astrophysical objects.'

major comments (3)
  1. [§2, Eq. (2.5)] The assignment Δp_x^2 = Δp_y^2 = Δp_z^2 = M^2/3 is the sole input that converts the GUP inequality (3.1) into the compactness bound (3.5), but it is neither derived nor defensible for the systems under consideration. In the centre-of-mass frame the operators \p_i in Eq. (2.3) are total-momentum components with zero expectation value; their variances are determined by the centre-of-mass wavefunction, not by the rest mass. For a localized object of size R, a coherent state satisfies Δp_i ≳ ℏ/(2R), and for R ~ G_N M this gives Δp_i ≲ m_p^2/M, which is many orders of magnitude smaller than M. With this realistic Δp_i, the GUP inequality (3.1) yields a trivial bound (essentially C ≲ (2/3) M Δp/m_p^2 plus small corrections) that is automatically satisfied and does not restrict compactness at order 1/α. In addition, the 'dispersion relation' M^2 ≃ p_x^2 + p_y^2 + p_z^2 used before Eq. (2.5) is not the dispersion relation of a massive body at rest; at rest p^2 = 0. The paper's own invocation of the soccer-ball problem in §2 makes this point, but Eq. (2.5) reintroduces exactly the unphysical scale that soccer-ball arguments are meant to suppress. Consequently Eq. (3.5) and the α ≲ 2 conclusion in §5 do not follow for stars or black holes.
  2. [§5, Eqs. (3.5) and (4.7)] The observational constraints on α and β derived from the existence of black holes inherit the defect of Eq. (2.5). The statements that α ≲ 2 (from C ~ 1/2) and that β is restricted to 0 ≲ β ≲ (2−α)/4 for 1 ≲ α ≲ 2 are obtained by combining the compactness bounds (3.5) and (4.7) with black-hole formation. Since the underlying compactness bound is not valid for a realistic localized many-body state, these parameter-space restrictions are unsupported. The experimental bounds quoted from Refs. [53] and [36] are independent of this issue, but the paper's main new constraint—that black-hole existence forces α ≲ 2—disappears once Eq. (2.5) is dropped.
  3. [§2, Eq. (2.4)] A further gap is the identification of the position uncertainty Δx_i = R/√3 with the gravitational-radius input R in the compactness definition (2.6). For a many-body object, Δx_i is the spread of the centre-of-mass wavefunction, which need not equal the stellar radius or the horizon radius. No argument is given that the width of the quantum state corresponds to the classical radius R used in C = G_N M/R. This is relevant because even a 'smeared' centre of mass does not change the gravitational radius of the collective mass distribution unless additional assumptions about the internal state are supplied.
minor comments (4)
  1. [§5, first paragraph] There is a typo: 'one the strongest constraint' should read 'one of the strongest constraints.'
  2. [§5, paragraph on GEUP parameter β] The word 'paramater' appears in 'the GEUP paramater β'; it should be 'parameter.'
  3. [References] Several reference entries contain formatting inconsistencies, e.g., 'Phys,Lett.' in [11] and 'Mod. Phys. Lett.' in [12]; these should be normalized to the journal's style.
  4. [§3, paragraph after Eq. (3.5)] The phrase 'The unity of α is further reinforced' is awkward; consider rephrasing to 'The order-one value of α is further supported.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the compactness bound is a transparent algebraic consequence of the stated GUP and the explicit isotropic-momentum assumption (2.5); the questionable status of that assumption is a validity concern, not a circular reduction.

full rationale

The paper's derivation is explicit: assume a center-of-mass wavefunction with vanishing means (2.1), spherical symmetry (2.4), and isotropic momentum dispersion Δp_i²=M²/3 (2.5). Substituting Δx=R/√3 and Δp=M/√3 into the GUP inequality (3.1) gives Eq. (3.4), and then Eq. (3.5), C≲M²/(3m_p²+αM²)≃1/α. This is a conditional algebraic derivation, not a fit or a renamed input. The parameters α and β are not fitted to the target compactness; they are model parameters subsequently confronted with external bounds (oscillator experiments, S2 precession, EHT shadows), so the resulting constraints are not statistically forced by construction. The cited self-references, including [41] for the center-of-mass/soccer-ball discussion and [6,7,56] for context, are not load-bearing uniqueness claims and do not substitute for the derivation. The main weakness is physical rather than circular: Eq. (2.5) treats a many-body object's per-coordinate total-momentum uncertainty as M/√3 via the dispersion relation M²≃p_x²+p_y²+p_z², an assumption that is not derived and would fail for localized nonrelativistic many-body states. But the paper states this assumption openly, and the compactness bound follows from it by algebra. Therefore no specific circular reduction can be exhibited.

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

No new particles, forces, or dimensions are introduced. The only free numbers are the GUP and GEUP parameters α and β, imported from the quantum gravity literature and constrained, not fitted, here. The most consequential input is the ad hoc per-coordinate uncertainty identification Δp = M/√3, which appears as an axiom because the paper does not derive it.

free parameters (2)
  • GUP parameter α = Not fitted; constrained to α ≲ 2 by black hole existence (Sec. 5)
    The central bound C ≲ 1/α is a function of α; the paper does not determine it from first principles but constrains it using the requirement that black holes with C > 1/2 exist.
  • EUP/GEUP parameter β = Not fitted; constrained to β ≲ 10^-90 by S2 precession and β ≲ (2-α)/4 by black hole existence (Sec. 5)
    Enters the GEUP compactness band Eq. (4.7); its value is imported from prior observational constraints.
assumptions (4)
  • domain assumption The GUP/GEUP inequalities (1.1), (1.3) are valid and characterize quantum-mechanical measurements.
    The entire paper derives consequences of these proposed relations; their validity is assumed from the quantum gravity literature.
  • ad hoc to paper Per-coordinate position and momentum uncertainties of a self-gravitating object satisfy Δx_i = R/√3 and Δp_i = M/√3 (Eqs. 2.4-2.5).
    This is the key modeling step that turns the single-particle GUP into a statement about a star; it is stated as a definition without physical derivation.
  • domain assumption Spherical symmetry, isotropy, and the center-of-mass frame restriction remove the soccer ball problem (Sec. 2).
    Invoked to justify applying the GUP to a composite object, citing ref [41].
  • domain assumption Black holes exist and require compactness C > 1/2.
    Used to turn the upper bound into the parameter constraints α ≲ 2 and β ≲ (2-α)/4; based on standard Schwarzschild geometry.

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Pith. "Pith review of Bounded compactness from G(E)UP." pith.science (2026). https://pith.science/paper/IM2ZTWRE

@misc{pith2026250714703,
  author       = {Pith},
  title        = {Pith review of: Bounded compactness from G(E)UP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IM2ZTWRE}},
  note         = {Machine review of arXiv:2507.14703}
}
read the original abstract

We analyse how different Generalised Uncertainty Principles could place bounds on the compactness of self-gravitating systems. By considering existing experimental bounds on the relevant parameters, we conclude that the compactness of large astrophysical objects is bounded above by the inverse of the GUP parameter, which would naturally be of order one. Conversely, the existence of black holes imposes stronger bounds on those parameters.

Figures

Figures reproduced from arXiv: 2507.14703 by the authors.

Figure 1
Figure 1. ∆p = ∆p− from Eq. (3.2), with ∆x in units of ℓp and ∆p in units of mp. The above Eq. (3.2) for α > 0 yields complex momenta for ∆x smaller than a minimum length determined by r 1 − 4 α ℓ 2 p ∆x 2 = 0 ⇒ ∆x = 2 √ α ℓp ≡ ℓGUP , (3.3) which is also called the remnant limit in the thermodynamic community [21, 44]. This limit for the physical solution ∆p = ∆p− from Eq. (3.2) is shown in [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 2
Figure 2. ∆p = ∆p− from Eq. (4.2), with ∆x in units of ℓp and ∆p in units of mp. which, in the large M limit, reduces to 1 − √ 1 − 4 α β 2 α ≲ C ≲ 1 + √ 1 − 4 α β 2 α , (4.7) where we again assumed β > 0 and 0 ≲ 4 α β < 1. Note that the GUP bound (3.5) is recovered for β → 0, whereas the limit α → 0 for finite β > 0 yields a lower bound C ≳ β. In [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Compactness as function of the mass M (in units of mp). Shaded region represents values allowed by the inequality (4.6) with α = 1 and β = 0.1. 0 2 4 6 8 10 -0.5 0.0 0.5 1.0 1.5 M C Upper Bound Lower Bound Allowed Region [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Compactness as function of the mass M (in units of mp). Shaded region represents values allowed by the inequality (4.6) with α = 1 and β = −0.1. still in qualitative agreement with independent analyses, like those stemming from string theory [46–49]. 3 Moreover, using …

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Cited by 1 Pith paper

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

  1. 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.

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