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arxiv: 2602.11207 · v2 · pith:4CWILDHOnew · submitted 2026-02-10 · 🌀 gr-qc

Cylindrically Symmetric Black Holes Sourced by Dekel-Zhao Dark Matter

G. Alencar , V. H. U. Borralho , M. S. Cunha , R. R. Landim This is my paper

Pith reviewed 2026-05-21 12:59 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black stringsdark matterDekel-Zhao profilenaked singularitiescylindrical symmetryenergy conditionsBTZ black holeHawking temperature
0
0 comments X

The pith

Dekel-Zhao dark matter density sources exact solutions for cylindrical black strings in 3+1 dimensions and black holes in 2+1 dimensions, with the inner slope parameter a controlling whether an event horizon forms or naked singularities and

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

The paper derives analytical solutions to Einstein's equations in which the Dekel-Zhao dark matter density profile serves as the matter source for a cylindrically symmetric metric. It demonstrates that the event horizon radius shrinks with increasing values of the inner slope parameter a, disappearing entirely past a critical threshold and leaving naked singularities. The dark matter distribution also generates curvature singularities in the Ricci and Kretschmann scalars that are absent from the corresponding vacuum solutions. Energy conditions are examined, showing that the null, weak, and strong conditions hold while the dominant condition is violated in the lower-dimensional case because of steep tangential pressure. Thermodynamic quantities such as Hawking temperature and free energy are modified by the presence of dark matter, yet the solutions remain locally and globally stable.

Core claim

Exact analytical solutions exist for a (3+1)-dimensional black string and a (2+1)-dimensional black hole sourced by the Dekel-Zhao dark matter density profile. The event horizon radius depends on the inner slope parameter a; beyond a critical value the horizon vanishes and naked singularities appear. The dark matter induces singularities in the Ricci and Kretschmann scalars, converting the constant-curvature BTZ spacetime into a singular geometry. The effective energy-momentum tensor satisfies the null, weak, and strong energy conditions but violates the dominant energy condition in the (2+1)-dimensional case owing to large tangential pressure gradients. Dark matter alters the Hawking温度 and

What carries the argument

The Dekel-Zhao dark matter density profile with tunable inner slope parameter a, inserted directly into the cylindrically symmetric metric ansatz to determine the metric functions and resulting causal structure.

If this is right

  • Beyond the critical inner slope the solutions describe naked singularities instead of black holes or black strings.
  • Curvature singularities appear in both the Ricci and Kretschmann scalars wherever the dark matter density is present.
  • The null, weak, and strong energy conditions remain satisfied while the dominant energy condition is violated in 2+1 dimensions.
  • Hawking temperature and free energy shift with the dark matter parameters while local and global thermodynamic stability are preserved.

Where Pith is reading between the lines

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

  • Galactic dark matter halos with measured inner slopes could be checked against the critical value to see whether cylindrical black-hole-like objects would possess horizons or naked singularities.
  • The same density profile might be inserted into other symmetry ansatze to test whether similar horizon disappearance occurs in spherical or planar geometries.
  • Numerical simulations of collapse with this density profile could reveal whether the naked singularities form dynamically or remain artifacts of the static ansatz.

Load-bearing premise

The Dekel-Zhao density profile functions as a valid matter source for the chosen cylindrically symmetric metric without extra consistency conditions imposed by the Einstein equations.

What would settle it

Compute the metric component g_tt for an inner slope a larger than the reported critical value and check whether it fails to change sign at any finite radius, or evaluate the Ricci scalar at r=0 and verify that it diverges when the density profile is non-zero.

Figures

Figures reproduced from arXiv: 2602.11207 by G. Alencar, M. S. Cunha, R. R. Landim, V. H. U. Borralho.

Figure 1
Figure 1. Figure 1: FIG. 1: DZ dark matter density profile for some values of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Lateral pressure [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Left) Metric function [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: NEC for the parameters [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Continuous family of curves confirming DEC, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Ricci scalar for several values of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Hawking temperature for some values of [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Heat capacity for some values of [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Free energy for some values of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Effective potential for massless particles for some values of [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Metric function [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Tangential pressure [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: (Left) Dominant Energy Conditions (DEC) [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Ricci scalar for some values of [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Hawking temperature for BTZ black hole for some values of [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Heat capacity for some values of [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Free energy for some values of [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Effective potential for photons for BTZ black hole for some values of [PITH_FULL_IMAGE:figures/full_fig_p025_20.png] view at source ↗
read the original abstract

In this work, we obtain analytical solutions for a $(3+1)$-dimensional black string and a $(2+1)$-dimensional black hole, both sourced by the Dekel-Zhao dark matter (DM) density profile. Our results indicate that the event horizon radius is sensitive to the inner slope parameter $a$; specifically, beyond a critical threshold, the horizon vanishes, leading to the formation of naked singularities. We find that the DM environment induces curvature singularities in the Ricci and Kretschmann scalars, which are absent in the vacuum BTZ case. Furthermore, an analysis of the effective energy-momentum tensor shows that while the null, weak, and strong energy conditions are strictly satisfied, the dominant energy condition is violated in the lower-dimensional scenario due to the high tangential pressure gradient. We also observe that DM modifies the Hawking temperature and free energy without compromising local or global stability. Notably, the DM distribution transforms the originally constant-curvature BTZ spacetime into a singular one, suggesting that a inherent stiffness of the DM profile is a determinant factor in the causal structure of these solutions.

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 manuscript claims to obtain analytical solutions for a (3+1)-dimensional black string and a (2+1)-dimensional black hole sourced by the Dekel-Zhao dark matter density profile. It reports that the event horizon radius is sensitive to the inner slope parameter a, with naked singularities appearing beyond a critical value of a. The DM distribution is stated to induce curvature singularities in the Ricci and Kretschmann scalars (absent in the vacuum BTZ case), while the effective energy-momentum tensor satisfies the null, weak, and strong energy conditions but violates the dominant energy condition in the lower-dimensional scenario due to tangential pressure gradients. Thermodynamic quantities including Hawking temperature and free energy are modified by the DM without compromising local or global stability.

Significance. If the solutions are shown to be fully consistent, the work would provide explicit, closed-form examples of cylindrically symmetric black holes embedded in a realistic DM halo profile, allowing direct study of how the inner slope a controls the causal structure and how DM introduces curvature singularities into constant-curvature spacetimes such as BTZ. The parameter-free character of the density profile and the reported stability analysis would constitute concrete, falsifiable predictions for the effect of DM on lower-dimensional black-hole thermodynamics.

major comments (2)
  1. [Einstein equations and matter source (main derivation)] The derivation begins by inserting the Dekel-Zhao density ρ(r) directly into the Einstein equations for the static cylindrically symmetric metric ansatz. For this ansatz the Einstein tensor possesses multiple independent components; specifying only the energy density fixes the mass function but leaves the radial, tangential, and (in 3+1) z-directed pressures undetermined. The manuscript reports an effective energy-momentum tensor and checks energy conditions, yet does not demonstrate that the chosen pressures satisfy the remaining Einstein equations or that ∇_μ T^μν = 0 holds identically without additional tuning of a or the scale radius. This consistency check is load-bearing for the claim of analytical solutions.
  2. [Analytical solutions and curvature analysis] The abstract and results state that beyond a critical value of the inner slope parameter a the event horizon vanishes, producing naked singularities, and that the DM profile transforms the constant-curvature BTZ spacetime into a singular one. Explicit metric functions g_tt(r), g_rr(r) (or their 2+1 analogues) and the integration steps from the Einstein equations should be displayed so that the location of the horizon and the appearance of curvature singularities in the Ricci and Kretschmann scalars can be verified directly.
minor comments (2)
  1. [Energy-momentum tensor] The notation for the effective energy-momentum tensor components (energy density, radial and tangential pressures) should be introduced once and used consistently; currently the distinction between the 3+1 and 2+1 cases is not always clear from the text.
  2. [Results] A brief comparison table of the critical value of a for horizon disappearance in the 3+1 versus 2+1 cases would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions regarding our work on cylindrically symmetric black holes sourced by the Dekel-Zhao dark matter profile. We provide point-by-point responses to the major comments and will update the manuscript to include additional details on the derivations and explicit expressions as appropriate.

read point-by-point responses

  1. Referee: The derivation begins by inserting the Dekel-Zhao density ρ(r) directly into the Einstein equations for the static cylindrically symmetric metric ansatz. For this ansatz the Einstein tensor possesses multiple independent components; specifying only the energy density fixes the mass function but leaves the radial, tangential, and (in 3+1) z-directed pressures undetermined. The manuscript reports an effective energy-momentum tensor and checks energy conditions, yet does not demonstrate that the chosen pressures satisfy the remaining Einstein equations or that ∇_μ T^μν = 0 holds identically without additional tuning of a or the scale radius. This consistency check is load-bearing for the claim of analytical solutions.

    Authors: We appreciate the referee pointing out the need for explicit consistency verification. In our approach, the Dekel-Zhao density profile is used to determine the metric function through the relevant component of the Einstein equations, which defines the mass function. The other components of the Einstein tensor then yield the pressure terms of the effective energy-momentum tensor. By construction, all Einstein equations are satisfied. The covariant conservation of the energy-momentum tensor follows from the Bianchi identities without any additional parameter tuning. We will include a brief explanation of this in the revised manuscript to make the consistency explicit. revision: yes

  2. Referee: The abstract and results state that beyond a critical value of the inner slope parameter a the event horizon vanishes, producing naked singularities, and that the DM profile transforms the constant-curvature BTZ spacetime into a singular one. Explicit metric functions g_tt(r), g_rr(r) (or their 2+1 analogues) and the integration steps from the Einstein equations should be displayed so that the location of the horizon and the appearance of curvature singularities in the Ricci and Kretschmann scalars can be verified directly.

    Authors: We agree that providing the explicit metric functions and integration steps will facilitate direct verification by readers. In the revised manuscript, we will present the integrated forms of the metric components for both the (3+1)-dimensional black string and the (2+1)-dimensional black hole cases. This will include the expressions for g_tt(r) and g_rr(r), the steps used to integrate the Einstein equations with the Dekel-Zhao profile, the condition for the critical value of a leading to horizon disappearance, and the explicit forms of the Ricci and Kretschmann scalars demonstrating the DM-induced singularities. revision: yes

Circularity Check

0 steps flagged

No circularity: solutions derived directly from Einstein equations with external Dekel-Zhao density profile

full rationale

The paper starts from the Einstein field equations in a static cylindrically symmetric metric ansatz and inserts the externally given Dekel-Zhao density profile ρ(r) as the matter source. The metric functions are obtained by direct integration of the resulting differential equations, yielding analytical expressions for the black string and BTZ-like solutions. Curvature scalars, horizons, Hawking temperature, and energy conditions are then computed from these derived quantities. No step reduces a claimed result to a fitted parameter renamed as a prediction, nor relies on self-citation for a uniqueness theorem or ansatz. The inner slope a is treated as a free input varied across cases, and the effective T_μν is constructed from the Einstein tensor components, keeping the derivation self-contained against the input density profile.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The solutions rest on the Einstein field equations in 3+1 and 2+1 dimensions, the assumption that the Dekel-Zhao density profile can be used as a static matter source, and the cylindrical symmetry ansatz. No new particles or forces are introduced.

free parameters (1)
  • inner slope parameter a
    Controls the central density cusp of the Dekel-Zhao profile and is varied to find the critical value at which the horizon disappears.
axioms (2)
  • standard math Einstein equations hold with the given energy-momentum tensor derived from the Dekel-Zhao density.
    Invoked throughout the derivation of the metric functions.
  • domain assumption Cylindrical symmetry is preserved by the matter distribution.
    Used to reduce the metric to the black-string and BTZ-like forms.

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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.

  1. On regular black strings spacetimes in nonlinear electrodynamics

    gr-qc 2026-03 unverdicted novelty 6.0

    No regular purely electric black strings exist in NED recovering the Maxwell limit, but regular cylindrical Bardeen and Hayward analogues are constructed with finite curvature.

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