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arxiv: 2603.08971 · v2 · pith:CJF3HPZRnew · submitted 2026-03-09 · ✦ hep-th · hep-ph

New Construction of Black Hole Solution in Non-Commutative Geometry and Their Thermodynamic Properties

Abdellah Touati This is my paper

Pith reviewed 2026-05-25 06:51 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords non-commutative geometryblack hole solutionsSeiberg-Witten mapHawking radiationphase transitiondark matterinformation paradox
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The pith

Non-commutativity applied via Seiberg-Witten map to potentials before solving Einstein equations yields black holes whose evaporation ends at a stable 2.73 Planck-mass remnant instead of diverging temperature.

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

The paper builds Schwarzschild and Reissner-Nordström-like black hole metrics in non-commutative geometry by first applying the Seiberg-Witten map to the interaction potentials and then solving the gravitational field equations. This ordering produces a dynamical non-commutative correction that preserves gauge covariance and leads to new geometric and thermodynamic features absent in the commutative case. The central result is that the Hawking temperature remains finite at the end of evaporation, a second-order phase transition appears, and a minimum-mass remnant forms. The remnant is presented as a purely gravitational cold dark matter candidate and a resolution to the information-loss problem because correlations between emitted particles are weakened and particle production is suppressed.

Core claim

By applying the Seiberg-Witten map directly to interaction potentials before solving Einstein's equations, both NC Schwarzschild and charged Reissner-Nordström-like black hole solutions are obtained in which the charged sector exhibits a novel branch dependence between attractive and repulsive electric interactions. Non-commutativity eliminates the temperature divergence at the final evaporation stage, inducing a second-order phase transition or a Hawking-Page-like transition in the presence of pressure. Linear response analysis shows high sensitivity to the NC parameter for small black holes, while quantum tunneling calculations demonstrate that the NC deformation suppresses particle-number

What carries the argument

The Seiberg-Witten map applied directly to interaction potentials before solving Einstein's equations, which introduces a dynamical spacetime non-commutativity effect while preserving gauge covariance.

If this is right

  • The Hawking temperature remains finite throughout evaporation and never diverges.
  • Evaporation ends with a stable remnant of fixed mass approximately 2.73 times the Planck mass.
  • A second-order phase transition occurs during the final stage of evaporation.
  • Particle emission is suppressed and correlations between successive emissions are weakened.
  • The charged solutions display an additional branch structure separating attractive and repulsive regimes.

Where Pith is reading between the lines

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

  • If produced in the early universe, the remnants could contribute to the observed dark-matter density without requiring new particle species.
  • The same mapping procedure might be applied to other singular solutions in general relativity to generate regular cores.
  • Gravitational-wave signals from small primordial black holes could carry signatures of the mass cutoff at 2.73 Planck masses.

Load-bearing premise

Applying the Seiberg-Witten map to interaction potentials before solving Einstein's equations produces consistent black-hole metrics that satisfy the modified non-commutative field equations.

What would settle it

A direct verification that the derived metric fails to solve the non-commutative Einstein equations obtained from the Seiberg-Witten mapped action would disprove the construction.

Figures

Figures reproduced from arXiv: 2603.08971 by Abdellah Touati.

Figure 1
Figure 1. Figure 1: FIG. 1. Behavior of the event horizons [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Behavior of the curvature function [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Behavior of the curvature function [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Behavior of the mass parameter [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Behavior of the Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Behavior of the heat capacity [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Behavior of the free energy [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Behavior of the Gibbs free energy [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Behavior of the NC susceptibility [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: shows the tunneling rate Γ across the NC event horizon. The left panel displays Γ versus particle frequency ω and for different Θ˜ ; the center panel shows Γ versus black hole mass m (for fixed ω), and the right panel shows Γ versus m for various ω. For the parameter ranges shown the NC correction reduces the tunneling rate, acting effectively as a barrier against particle escape. The qualitative behavior… view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Variation of the particle-number density [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Variation of the statistical correlation function [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
read the original abstract

In this work, we present a new construction of black hole solutions in non-commutative gauge theory by applying the Seiberg-Witten map directly to interaction potentials before solving Einstein's equations. This approach provides a dynamical effect of spacetime non-commutativity that preserves gauge covariance. We obtain both NC Schwarzschild and charged Reissner-Nordstrom-like black hole solutions, showing that the charged sector exhibits a novel branch dependence between attractive and repulsive electric interactions absent in the commutative limit. We analyze the geometrical properties, energy conditions, and thermodynamic properties of these spacetimes. Our results reveal that non-commutativity eliminates the temperature divergence at the final evaporation stage, inducing a second-order phase transition, or a Hawking-Page-like phase transition in the presence of pressure. Additionally, linear response analysis indicates high sensitivity to the NC parameter for small black holes. Finally, quantum tunneling investigations for both thermal and non-thermal radiation demonstrate that the NC deformation suppresses the particle-number density and weakens correlations between successive emissions, acting as a barrier to particle escape and supports the formation of a cold finite remnant. From a cosmological standpoint, since these stable remnant possess a fixed Planck-scale mass ($M^{\text{min}}\simeq2.73 M_{P}$), they provide a dynamically generated, purely gravitational cold dark matter candidate that aligns with dark universe phenomenology while simultaneously resolving the black hole information loss paradox.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims a new construction of non-commutative black hole solutions by applying the Seiberg-Witten map directly to interaction potentials before solving Einstein's equations, yielding NC Schwarzschild and RN-like metrics that preserve gauge covariance. It analyzes their geometry, energy conditions, and thermodynamics, asserting that non-commutativity removes the Hawking temperature divergence at the end of evaporation, induces a second-order phase transition (or Hawking-Page-like with pressure), produces a stable remnant at M^min ≃ 2.73 M_P, and that this remnant serves as a cold dark matter candidate while resolving the information loss paradox. Linear response and tunneling analyses are also presented.

Significance. If the metrics are confirmed to solve the non-commutative Einstein equations, the work would offer a gauge-covariant dynamical incorporation of spacetime non-commutativity into black hole solutions and a concrete mechanism for stable Planck-scale remnants. This could impact discussions of black hole evaporation endpoints, information preservation, and gravitational dark matter candidates, provided the minimum mass is shown to be a robust prediction rather than parameter-dependent.

major comments (2)
  1. [Construction section / abstract] The central construction (described in the abstract and presumably §2–3) applies the Seiberg-Witten map to interaction potentials prior to solving Einstein's equations and claims this produces valid NC black hole metrics. However, no explicit check is provided that the resulting NC Schwarzschild and RN-like metrics satisfy the modified Einstein field equations obtained from the mapped action to the relevant order in the non-commutativity parameter. Without this verification, the subsequent thermodynamic claims (no T divergence, phase transition, remnant mass) lack a demonstrated dynamical foundation.
  2. [Thermodynamics and remnant discussion] The quoted minimum mass M^min ≃ 2.73 M_P is presented as a fixed, dynamically generated value, yet it depends on the non-commutativity parameter whose value is not independently fixed by any first-principles constraint or matching condition. This makes the dark-matter candidacy and information-loss resolution claims sensitive to an arbitrary scale rather than a genuine prediction of the framework.
minor comments (2)
  1. Clarify the precise order in θ to which the Seiberg-Witten map is applied and whether higher-order terms are neglected consistently in the metric and thermodynamic calculations.
  2. Ensure consistent notation for the non-commutativity parameter across equations and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we respond point-by-point to the major concerns, indicating where revisions will be made to address the dynamical verification and parameter dependence.

read point-by-point responses

  1. Referee: [Construction section / abstract] The central construction (described in the abstract and presumably §2–3) applies the Seiberg-Witten map to interaction potentials prior to solving Einstein's equations and claims this produces valid NC black hole metrics. However, no explicit check is provided that the resulting NC Schwarzschild and RN-like metrics satisfy the modified Einstein field equations obtained from the mapped action to the relevant order in the non-commutativity parameter. Without this verification, the subsequent thermodynamic claims (no T divergence, phase transition, remnant mass) lack a demonstrated dynamical foundation.

    Authors: We agree that an explicit verification of the field equations would strengthen the presentation. The construction solves the Einstein equations with the Seiberg-Witten-mapped potentials by design, but we will add an appendix in the revised manuscript that explicitly substitutes the obtained metrics back into the modified Einstein tensor (to O(θ)) and confirms consistency with the mapped stress-energy source. revision: yes

  2. Referee: [Thermodynamics and remnant discussion] The quoted minimum mass M^min ≃ 2.73 M_P is presented as a fixed, dynamically generated value, yet it depends on the non-commutativity parameter whose value is not independently fixed by any first-principles constraint or matching condition. This makes the dark-matter candidacy and information-loss resolution claims sensitive to an arbitrary scale rather than a genuine prediction of the framework.

    Authors: The numerical value is obtained by solving T(M,θ)=0 for the specific normalization θ=1 in Planck units used throughout the paper; the existence of a remnant where T→0 is independent of that choice. We will revise the text to present the general expression M^min(θ), discuss the scaling, and note that a first-principles fixing of θ remains an open question, while the qualitative features (finite remnant, suppressed tunneling) hold for any θ in the relevant regime. revision: partial

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper introduces a new construction by applying the Seiberg-Witten map to interaction potentials prior to solving Einstein equations, obtains explicit NC Schwarzschild and RN-like metrics, and derives thermodynamic quantities including the remnant mass from those metrics. No step reduces by definition or construction to its inputs; the minimum mass value emerges from the explicit solution of the modified equations rather than being presupposed or fitted to the target result. The derivation chain is self-contained with independent content at each stage.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The construction rests on the Seiberg-Witten map applied to potentials and the resulting metrics satisfying Einstein equations; the NC parameter functions as a free scale whose specific value produces the quoted 2.73 M_P remnant mass; the remnant is then interpreted as dark matter without additional evidence.

free parameters (2)
  • non-commutativity parameter
    Scale introduced to deform the geometry; its value determines the minimum mass 2.73 M_P
  • M^min
    Numerical value 2.73 M_P presented as fixed outcome of the NC deformation
axioms (2)
  • domain assumption Seiberg-Witten map applied directly to interaction potentials yields a gauge-covariant dynamical effect of non-commutativity
    Core methodological premise stated in the abstract
  • domain assumption Resulting metrics satisfy the non-commutative version of Einstein's equations
    Invoked after the map is applied to obtain the black-hole solutions
invented entities (1)
  • cold finite remnant as dark matter candidate no independent evidence
    purpose: Resolves information loss and supplies gravitational dark matter
    Derived from thermodynamics but identification as dark matter is an interpretive step without independent falsifiable handle

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Reference graph

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