arxiv: 2604.08953 · v1 · submitted 2026-04-10 · 🌀 gr-qc · quant-ph

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Finite Hilbert space and maximum mass of Schwarzschild black holes from a Generalized Uncertainty Principle

S. Jalalzadeh , H. Moradpour

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Pith reviewed 2026-05-10 17:47 UTC · model grok-4.3

classification 🌀 gr-qc quant-ph

keywords generalized uncertainty principleSchwarzschild black holesminimal lengthmaximal momentumdiscrete mass spectrumbounded entropyregulated Hawking temperaturequantum gravity constraints

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The pith

Applying a generalized uncertainty principle with minimal length and maximal momentum to Schwarzschild black hole phase space produces a finite discrete mass spectrum and a strict upper mass bound.

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

The paper imposes a GUP that incorporates both a smallest possible length and a largest possible momentum directly onto the reduced phase space variables of the Schwarzschild geometry. This produces only discrete allowed values for the black hole mass instead of any continuous range, together with an absolute maximum mass. The same modification bounds the entropy from above and fully regulates the Hawking temperature so that it remains finite. The authors also build a deformed lapse function that keeps the usual ADM mass and horizon radius unchanged yet reproduces the corrected temperature from surface gravity. Observations of the heaviest known supermassive black holes then fix an upper limit on the GUP deformation parameter β of order 10^{-98}.

Core claim

Implementing a generalized uncertainty principle (GUP) with both minimal length and maximal momentum directly on the reduced phase space of the Schwarzschild black hole (BH) leads to a finite and discrete mass spectrum, a strict upper bound on the BH mass, a bounded entropy, and a fully regulated Hawking temperature. We further construct a GUP-deformed lapse function that preserves the ADM mass and horizon radius while exactly reproducing the GUP temperature through the surface gravity. Using the most massive observed supermassive BHs, we derive the constraint on the GUP parameter, β≲10^{-98}, showing that present astrophysical data already impose robust bounds on minimal length quantum grav

What carries the argument

The GUP commutator with minimal length and maximal momentum imposed on the conjugate pair of horizon radius and ADM mass in the reduced phase space of the Schwarzschild geometry.

If this is right

The black hole Hilbert space is finite-dimensional rather than infinite.
Entropy remains bounded and does not grow without limit with horizon area.
Hawking temperature stays finite at all scales instead of diverging.
Astrophysical observations of the heaviest black holes already constrain the GUP parameter β to values ≲10^{-98}.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The discretization could limit the number of accessible microstates and thereby affect information retention during evaporation.
The same phase-space GUP construction might be applied to rotating or charged black holes to obtain analogous mass ceilings.
Tighter future bounds on β from gravitational-wave or shadow observations would test whether minimal-length effects remain undetectable at astrophysical energies.

Load-bearing premise

The specific GUP form can be imposed directly on the reduced phase space of the classical Schwarzschild geometry while preserving the ADM mass and horizon radius without inconsistencies with general relativity.

What would settle it

Discovery of a supermassive black hole whose mass exceeds the upper limit implied by the largest β still allowed by the observed population, or a direct measurement of Hawking temperature that fails to match the GUP-regulated surface-gravity expression.

Figures

Figures reproduced from arXiv: 2604.08953 by H. Moradpour, S. Jalalzadeh.

**Figure 2.** Figure 2: Specific heat CGUP(M) for a representative small GUP parameter β = 0.03. The heat capacity diverges at Mc = mP/ p 6β, marking the transition between an unstable Schwarzschild-like branch (C < 0) and a stable largemass branch (C > 0). The divergence approaches the maximal mass Mmax < mP/ p 2β from below, so the specific heat remains finite at the discrete endpoint of the spectrum. so the BH Hilbert space d… view at source ↗

read the original abstract

We show that implementing a generalized uncertainty principle (GUP) with both minimal length and maximal momentum directly on the reduced phase space of the Schwarzschild black hole (BH) leads to a finite and discrete mass spectrum, a strict upper bound on the BH mass, a bounded entropy, and a fully regulated Hawking temperature. We further construct a GUP-deformed lapse function that preserves the ADM mass and horizon radius while exactly reproducing the GUP temperature through the surface gravity. Using the most massive observed supermassive BHs, we derive the constraint on the GUP parameter, $\beta\lesssim 10^{-98}$, showing that present astrophysical data already impose robust bounds on minimal length quantum gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper imposes a GUP with minimal length and maximal momentum on the Schwarzschild reduced phase space to produce a discrete bounded mass spectrum, then uses observed supermassive black holes to set a tight limit on the deformation parameter.

read the letter

The core move here is taking the GUP directly on the phase-space coordinates of the classical Schwarzschild solution and extracting a finite mass spectrum plus an explicit upper mass bound. They also build a deformed lapse function that keeps the same ADM mass and horizon location while recovering the corrected temperature from surface gravity. That construction is the clearest new piece relative to earlier GUP black-hole papers, which usually stop at thermodynamic corrections without the discrete spectrum or the phase-space reduction step shown here. The resulting bound β ≲ 10^{-98} follows from matching the maximum mass to the heaviest known supermassive black holes, which is a straightforward phenomenological step once the spectrum is in hand. The work is internally consistent on its own terms and cites the relevant GUP literature without obvious omissions. The main soft spot is the assumption that the deformed metric remains a vacuum solution to the Einstein equations outside the horizon. The lapse is altered by hand to preserve ADM mass and horizon radius, but no check is given that the Einstein tensor still vanishes or that the stress-energy remains zero. If that deformation introduces a non-zero source term, the geometric identification with a Schwarzschild black hole becomes only formal. The data comparison also lacks detail on mass selection, error propagation, or whether the bound is dominated by a single object. These are fixable but need explicit treatment. The paper is aimed at researchers working on quantum-gravity phenomenology and modified black-hole thermodynamics. Anyone already using GUP in gravitational settings will see the targeted extension and can judge the consistency claim for themselves. It is coherent enough and grounded enough in existing techniques to merit a serious referee rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that applying a generalized uncertainty principle (GUP) incorporating both minimal length and maximal momentum directly to the reduced phase space of the Schwarzschild black hole produces a finite discrete mass spectrum, a strict upper bound on black hole mass, bounded entropy, and a fully regulated Hawking temperature. It further constructs a GUP-deformed lapse function that preserves the ADM mass (asymptotic coefficient) and horizon radius (lapse zero) while reproducing the GUP-corrected temperature exactly via surface gravity. Using masses of the most massive observed supermassive black holes, the GUP parameter is constrained to β ≲ 10^{-98}.

Significance. If the deformed-metric construction is internally consistent, the result would be significant for quantum-gravity phenomenology: it supplies a concrete mechanism for a finite Hilbert space in black-hole physics, derives an astrophysical upper bound on mass, and converts existing supermassive-black-hole observations into a quantitative limit on the GUP deformation parameter. The data-driven bound is a clear strength, though it renders the phenomenological claim observational rather than a pure prediction.

major comments (2)

[§3] §3 (imposition of GUP on reduced phase space): the central claim that the GUP can be imposed on the classical Schwarzschild reduced phase-space variables while preserving both the ADM mass and the horizon radius is load-bearing; the construction must be shown to leave the Einstein tensor identically zero outside the horizon, otherwise the deformed geometry is not a vacuum solution and the geometric identification of the black hole is only by fiat.
[Lapse-function construction] Lapse-function construction (following the mass-spectrum derivation): the deformed lapse is asserted to keep the same ADM mass and horizon radius while exactly matching the GUP temperature via surface gravity; without an explicit check that the resulting metric satisfies the vacuum Einstein equations, the preservation is only kinematic and does not guarantee consistency with general relativity.

minor comments (2)

[Abstract] The abstract states the bound β ≲ 10^{-98} but does not specify the precise selection criteria or error treatment for the supermassive black holes used; this detail should be added for reproducibility.
[Introduction] Notation for the GUP parameter β and the precise form of the minimal-length/maximal-momentum GUP should be introduced once in the introduction and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [§3] §3 (imposition of GUP on reduced phase space): the central claim that the GUP can be imposed on the classical Schwarzschild reduced phase-space variables while preserving both the ADM mass and the horizon radius is load-bearing; the construction must be shown to leave the Einstein tensor identically zero outside the horizon, otherwise the deformed geometry is not a vacuum solution and the geometric identification of the black hole is only by fiat.

Authors: We agree that an explicit demonstration that the Einstein tensor vanishes outside the horizon would be required for the deformed metric to qualify as a vacuum solution. Our construction applies the GUP directly to the reduced phase-space variables to obtain the discrete mass spectrum and then deforms the lapse function phenomenologically so that the ADM mass and horizon radius are preserved by construction while the surface gravity reproduces the GUP-corrected temperature. This is an effective approach motivated by thermodynamic consistency rather than a modified Einstein equation. In the revised manuscript we will add an explicit statement clarifying that the deformed geometry is not claimed to satisfy the vacuum Einstein equations and that the black-hole identification rests on the preserved asymptotic and horizon properties. revision: partial
Referee: [Lapse-function construction] Lapse-function construction (following the mass-spectrum derivation): the deformed lapse is asserted to keep the same ADM mass and horizon radius while exactly matching the GUP temperature via surface gravity; without an explicit check that the resulting metric satisfies the vacuum Einstein equations, the preservation is only kinematic and does not guarantee consistency with general relativity.

Authors: The referee is correct that the matching of ADM mass, horizon radius, and temperature is kinematic. We have not verified that the resulting metric satisfies the vacuum Einstein equations. The lapse is chosen solely to enforce the boundary conditions and to yield the GUP temperature from the surface-gravity formula. We will revise the text to state clearly that the construction provides a thermodynamically consistent effective description within the GUP framework but does not constitute a solution of general relativity. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation applies GUP to phase space independently

full rationale

The central derivation imposes the GUP (with minimal length and maximal momentum) on the reduced phase-space variables of the Schwarzschild solution to obtain a discrete mass spectrum, upper mass bound, bounded entropy, and regulated temperature. This follows directly from the GUP commutation relations without reducing to fitted parameters or prior results by construction. The deformed lapse function is separately constructed to preserve ADM mass and horizon radius while matching surface gravity to the GUP temperature, but this is an explicit extension rather than a tautological redefinition of the spectrum. The β ≲ 10^{-98} bound is obtained by direct comparison to observed supermassive black-hole masses and functions as a phenomenological constraint, not a self-referential prediction. No load-bearing self-citations, uniqueness theorems, or ansatzes reduce the claims to their inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard Schwarzschild solution plus a specific two-parameter GUP; the only free parameter introduced is β, whose value is constrained rather than freely fitted.

free parameters (1)

β (GUP deformation parameter)
Introduced in the GUP commutator and constrained to ≲10^{-98} by matching the largest observed black-hole masses; no other numerical constants are fitted.

axioms (2)

domain assumption The Schwarzschild geometry remains valid outside the horizon and its ADM mass and horizon radius are preserved under the GUP deformation.
Invoked when constructing the GUP-deformed lapse function that reproduces the new temperature via surface gravity.
domain assumption The GUP with both minimal length and maximal momentum can be consistently imposed on the reduced phase space of a classical black-hole solution.
Central modeling choice stated in the abstract; no derivation from a more fundamental quantum-gravity framework is provided.

pith-pipeline@v0.9.0 · 5415 in / 1520 out tokens · 38298 ms · 2026-05-10T17:47:49.792281+00:00 · methodology

discussion (0)

Reference graph

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