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arxiv: 2509.13335 · v2 · submitted 2025-09-10 · 🌀 gr-qc · hep-th

Regular black holes with gravitational self-energy as dark matter

Kimet Jusufi , Douglas Singleton This is my paper

Pith reviewed 2026-05-18 17:26 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords regular black holesgravitational self-energydark matterAyon-Beato-Garcia metricPlanck massnon-singular spacetimesHawking temperature
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The pith

Incorporating non-local gravitational self-energy modifies the ADM mass to yield regular neutral black holes including stable Planck-mass extremal objects that could act as dark matter.

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

The paper establishes that non-local gravitational self-energy, derived from the Newtonian gravitational potential and energy density, can be promoted to a coordinate-independent term and inserted into the spacetime metric. This changes the total ADM mass by adding a finite regularized gravitational mass term. The outcome is a regular Ayon-Beato-Garcia-type geometry without electric charge. As a direct result, extremal particle-black-hole configurations appear at the Planck mass. These objects are thermodynamically stable, have vanishing Hawking temperature, and are proposed as possible dark matter candidates.

Core claim

The total ADM mass is modified by a finite regularized gravitational mass term from the non-local self-interaction. This produces a regular Ayon-Beato-Garcia-type metric without electric charge. The construction yields extremal particle-black-hole objects of Planck mass that are thermodynamically stable with vanishing Hawking temperature and could be viable dark matter candidates.

What carries the argument

Non-local gravitational self-energy term derived from the Newtonian potential and energy density, promoted to a coordinate-independent quantity and inserted into the metric.

If this is right

  • The resulting spacetime geometry is regular and non-singular at the center.
  • Extremal particle-black-hole objects exist at the Planck mass scale.
  • These objects exhibit vanishing Hawking temperature.
  • The configurations are thermodynamically stable.
  • Such objects could serve as viable dark matter candidates.

Where Pith is reading between the lines

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

  • This construction suggests dark matter could arise from gravitational self-energy effects without new fundamental particles.
  • The regularization method might be applied to other singular solutions in general relativity to test consistency.
  • Astrophysical searches for compact objects near the Planck mass could provide indirect tests of the model.

Load-bearing premise

The non-local gravitational self-interaction obtained from the Newtonian gravitational potential and energy density can be directly promoted to a coordinate-independent term that is inserted into the spacetime metric while preserving the Einstein equations outside the smeared region.

What would settle it

A calculation showing that the modified metric still contains a curvature singularity at the origin, or that the Hawking temperature does not vanish for the extremal Planck-mass objects, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.13335 by Douglas Singleton, Kimet Jusufi.

Figure 1
Figure 1. Figure 1: FIG. 1: The plot shows the extremal configuration for the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The plot shows Hawking temperature for the extremal [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The plot shows the heat capacity for the extremal [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The plot shows constraints on the fraction of dark [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

We incorporate the effect of non-local gravitational self-energy to obtain a neutral, non-singular spacetime geometry. This is achieved by using a non-local gravitational theory inspired by T-duality, where particle mass is not point-like but smeared over a region. This non-local gravitational self-interaction is derived from the Newtonian gravitational potential and energy density, allowing us to define a coordinate-independent quantity. Thus, we incorporate the non-local gravitational field into the spacetime metric. We demonstrate that the total ADM mass is modified by a finite, regularized gravitational mass term, leading to a regular solution of the Ayon-Beato-Garcia type metric but without electric charge. We show the existence of extremal configurations known as \emph{particle-black hole} objects of order of the Planck mass, which are thermodynamically stable, have a vanishing Hawking temperature and could be a viable dark matter candidate.

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

3 major / 2 minor

Summary. The manuscript proposes incorporating a non-local gravitational self-energy term, derived from the Newtonian gravitational potential and energy density in a T-duality-inspired non-local theory, directly into the spacetime metric. This modifies the total ADM mass by a finite regularized gravitational mass correction, producing a neutral regular geometry of Ayon-Beato-Garcia type without electric charge. The paper identifies extremal particle-black-hole configurations of Planck mass that are claimed to be thermodynamically stable with vanishing Hawking temperature and viable as dark matter candidates.

Significance. If the construction can be shown to yield a consistent solution of the Einstein equations with a physically acceptable effective source, the approach would provide a novel route to singularity resolution via gravitational self-interaction and could motivate further study of Planck-mass extremal objects as dark-matter candidates. The work draws on established ideas in non-local gravity and regular black-hole metrics, but its impact hinges on resolving the consistency issues noted below.

major comments (3)
  1. [Abstract and metric construction] Abstract and metric-construction paragraph: the non-local self-energy is stated to be derived from the Newtonian potential and energy density and then promoted to a coordinate-independent term inserted into the metric, yet no explicit derivation steps, coordinate transformation, or verification that the resulting geometry satisfies the Einstein equations (with or without an effective stress-energy tensor) are provided. This leaves the central claim that the construction preserves the Einstein equations outside the smeared region unverified.
  2. [Abstract and extremal-configuration section] Abstract and extremal-configuration section: the regularization parameter (smearing scale l) is chosen so that the finite gravitational-mass correction produces a regular geometry and an extremal Planck-mass state; this renders the claimed 'prediction' of thermodynamically stable extremal objects a fitting outcome rather than an independent result of the dynamics.
  3. [Metric and Einstein-equation discussion] Metric and Einstein-equation discussion: because the self-energy term originates in the weak-field Newtonian limit, it is unclear whether the inserted term remains consistent with the nonlinear Einstein tensor for the strong-curvature Planck-mass objects; no explicit computation of the Einstein tensor or check of energy conditions near the origin is reported.
minor comments (2)
  1. [Notation and parameters] Define the smearing scale l explicitly in terms of the Planck length and state its numerical value or range used for the extremal configurations.
  2. [Metric comparison] Add a direct comparison table or paragraph contrasting the obtained neutral metric with the original charged Ayon-Beato-Garcia solution, highlighting differences in the effective source.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the changes planned for the revised version.

read point-by-point responses

  1. Referee: Abstract and metric-construction paragraph: the non-local self-energy is stated to be derived from the Newtonian potential and energy density and then promoted to a coordinate-independent term inserted into the metric, yet no explicit derivation steps, coordinate transformation, or verification that the resulting geometry satisfies the Einstein equations (with or without an effective stress-energy tensor) are provided. This leaves the central claim that the construction preserves the Einstein equations outside the smeared region unverified.

    Authors: We agree that additional explicit steps would strengthen the presentation. The non-local self-energy correction is constructed by integrating the Newtonian gravitational potential against the smeared energy density arising from the T-duality-inspired non-local model; the resulting finite term is expressed in a coordinate-independent manner through the spherically symmetric radial coordinate. In the revision we will insert a dedicated paragraph (or short subsection) that spells out these derivation steps and explicitly verifies that the Einstein tensor vanishes outside the smearing region, confirming that the geometry satisfies the vacuum Einstein equations with the corrected ADM mass. revision: yes

  2. Referee: Abstract and extremal-configuration section: the regularization parameter (smearing scale l) is chosen so that the finite gravitational-mass correction produces a regular geometry and an extremal Planck-mass state; this renders the claimed 'prediction' of thermodynamically stable extremal objects a fitting outcome rather than an independent result of the dynamics.

    Authors: The smearing scale l is not a free fitting parameter but is fixed by the underlying T-duality non-local framework to be of order the Planck length, the scale at which point-like sources are regularized. With this theoretically motivated value the extremality condition then yields a Planck-mass configuration as a derived consequence. We will revise the abstract and the extremal-configuration section to make this motivation and the dynamical origin of the Planck-mass result clearer. revision: partial

  3. Referee: Metric and Einstein-equation discussion: because the self-energy term originates in the weak-field Newtonian limit, it is unclear whether the inserted term remains consistent with the nonlinear Einstein tensor for the strong-curvature Planck-mass objects; no explicit computation of the Einstein tensor or check of energy conditions near the origin is reported.

    Authors: We acknowledge the importance of this consistency check. Although the correction is motivated by the Newtonian limit, the non-local smearing is intended to provide a regularization that remains valid in the strong-field regime. In the revised manuscript we will add an explicit computation of the Einstein tensor for the proposed metric together with a verification of the energy conditions near the origin, thereby confirming that the effective source is physically acceptable. revision: yes

Circularity Check

1 steps flagged

Regularized Newtonian self-energy added to ADM mass by construction forces extremal Planck-mass states

specific steps
  1. fitted input called prediction [Abstract; metric construction paragraph]
    "We incorporate the effect of non-local gravitational self-energy... This non-local gravitational self-interaction is derived from the Newtonian gravitational potential and energy density, allowing us to define a coordinate-independent quantity. Thus, we incorporate the non-local gravitational field into the spacetime metric. We demonstrate that the total ADM mass is modified by a finite, regularized gravitational mass term, leading to a regular solution of the Ayon-Beato-Garcia type metric but without electric charge. We show the existence of extremal configurations known as particle-blackhole"

    The regularization parameter is introduced to tame the Newtonian self-energy divergence and is then tuned so that the modified ADM mass produces a regular metric. The subsequent 'prediction' of extremal Planck-mass objects with vanishing temperature is therefore fixed by the same parameter choice that defines the input correction, rendering the dark-matter candidacy a direct consequence of the regularization rather than an independent output of the Einstein equations.

full rationale

The derivation begins by extracting a non-local self-energy from the Newtonian potential and energy density, then regularizes it with a parameter chosen to produce a finite correction. This term is promoted to a coordinate-independent quantity and inserted into the metric to modify the total ADM mass, directly yielding an Ayon-Beato-Garcia-type regular geometry without charge. The extremal particle-black-hole configurations of Planck mass, vanishing Hawking temperature, and thermodynamic stability then follow from the same regularization scale that was selected to enforce regularity. Once the mass correction is fixed by this choice, the subsequent checks of the Einstein equations outside the smeared region and the thermodynamic properties are independent, but the central claim that such objects arise as viable dark-matter candidates reduces to a fitting procedure rather than an independent first-principles result.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The construction rests on one free parameter (the smearing scale that regularizes the Newtonian self-energy), the assumption that Newtonian self-energy can be promoted to a relativistic metric correction, and the postulate of a new non-local gravitational interaction inspired by T-duality.

free parameters (1)
  • smearing scale l
    The length scale over which the mass is smeared; chosen so that the gravitational self-energy remains finite and produces the regular geometry.
axioms (2)
  • domain assumption Newtonian gravitational potential and energy density can be regularized to yield a coordinate-independent non-local gravitational self-energy that is inserted into the spacetime metric.
    Invoked when the authors state that the non-local gravitational field is incorporated into the metric.
  • domain assumption The resulting geometry satisfies the Einstein equations outside the smeared region.
    Implicit in the claim that a regular solution of Ayon-Beato-Garcia type is obtained.
invented entities (1)
  • non-local gravitational self-energy term no independent evidence
    purpose: To smear the mass and remove the central singularity without introducing electric charge.
    Postulated as a new interaction derived from T-duality; no independent falsifiable prediction outside the metric construction is given.

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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.

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    gr-qc 2025-12 unverdicted novelty 6.0

    Non-local gravitational self-energy induces spontaneous wave-function collapse with a model-independent collapse time inversely proportional to system mass.

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