Charged Fuzzy Dark Matter Black Holes

Bander Almutairi; Kazuharu Bamba; S. Khan; Z. Yousaf

REVIEW 4 major objections 5 minor 76 references

Charged Fuzzy Dark Matter Black Holes

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper introduces a charged, spherically symmetric family of regular black holes and self-gravitational droplets built from the Einasto dark matter density profile.

desk verdict The printed charged metric is internally inconsistent—it does not follow from the paper's own mass function—so the central claim fails as written. read the letter →

arxiv 2411.13871 v1 pith:AQS27SV4 submitted 2024-11-21 gr-qc astro-ph.COhep-th

Z. Yousaf , Bander Almutairi , S. Khan , Kazuharu Bamba This is my paper · ORCID

classification gr-qcastro-ph.COhep-th

keywords fuzzydarkmatterEinastodensityprofilechargedblackholesregularself-gravitationaldropletsnon-localequationofstateReissner-Nordstromanisotropicfluid

topics Dark Matter

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

This paper claims that the Einasto density profile, a standard fit for cold dark matter halos, can act as the matter source for a new class of charged, spherically symmetric solutions to the Einstein-Maxwell equations. With an anisotropic fluid satisfying $p_r = -\rho$, the solutions are regular black holes or self-gravitational droplets: the central region behaves like a de Sitter core, and the outer region approaches Reissner-Nordstrom. Replacing the equation of state with a non-local one produces a charged droplet with negative radial pressure beyond a finite radius. A sympathetic reader would care because these models propose that supermassive black holes at galactic centers could themselves be composed of dark matter, offering a concrete bridge between black hole physics and fuzzy dark matter.

What carries the argument

The key machinery is the Einasto density profile $\rho(r) = (\rho_0 + e_0) \exp[-(r/h)^{1/\beta}]$ together with the anisotropic charged fluid and the equation of state $p_r = -\rho$. The mass function is expressed through the lower incomplete gamma function, $m(r) = M \gamma(3\beta, (r/h)^{1/\beta})/\Gamma(3\beta)$ plus a charge integral, and the metric function $g_{00}$ is built from this mass function plus the charge term $q^2/r^2$. This structure is what makes the spacetime regular at the center (de Sitter core) and asymptotically Reissner-Nordstrom. The charge $q(r)$ remains an arbitrary function throughout; it enters only through the Reissner-Nordstrom terms.

What would settle it

Choose an explicit charge profile, such as $q(r) = Q r^3/(r^3 + b^3)$, insert it into the Einstein-Maxwell equations with the Einasto density and $p_r = -\rho$, and check whether a metric of the form $g_{00} = 1/g_{rr}$ solves the field equations; if no such profile yields the claimed two-horizon or droplet behavior, the central claim fails.

Watch

Extended reading notes

Core claim

The central discovery is a family of static, spherically symmetric charged black hole and droplet solutions whose energy density is the Einasto profile $\rho(r) = (\rho_0 + e_0) \exp[-(r/h)^{1/\beta}]$. For the de Sitter-like equation of state $p_r = -\rho$, the metric function $g_{00}(r) = 1 - 2m(r)/r + q^2/r^2$, with $m(r)$ determined by the lower incomplete gamma function, admits two event horizons when the rescaled mass $\omega = M/h$ exceeds a critical value $\omega_0$, one degenerate horizon at $\omega = \omega_0$, and no horizon for $0 < \omega < \omega_0$. The solution interpolates between a de Sitter core near $r = 0$ and the Reissner-Nordstrom metric as $r/h \to \infty$. For a non-local equation of state, the paper obtains a horizonless charged droplet whose radial pressure becomes negative beyond a finite radius.

Load-bearing premise

The derivation leaves the electric charge distribution $q(r)$ unspecified and never verifies that the metric ansatz $g_{00} = 1/g_{rr}$ is compatible with the charged stress-energy tensor, so the existence of a concrete charged solution is assumed rather than demonstrated.

Editorial extensions

If this is right

If the construction holds, supermassive black holes at galactic centers can be modeled as fuzzy dark matter halos themselves, using the Einasto index and scale length fixed by the halo fit.
The horizonless droplet solutions ($\omega < \omega_0$) would appear as black hole analogues: they produce the same effective potential for orbiting stars while lacking an event horizon.
The Hawking temperature formula reduces to the standard result at large radius, so the charged Einasto solutions inherit familiar black hole thermodynamics in the asymptotic regime.
For suitable parameters, the effective potential for massive particles matches the Schwarzschild potential near its minimum, which the paper uses to argue consistency with S-star orbits around Sagittarius A*.
The non-local equation of state yields a charged droplet with negative radial pressure outside a finite radius, giving a new class of bound dark matter configurations.

Reading between the lines

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

Because $q(r)$ is left arbitrary, the paper establishes a family of solutions rather than a concrete model; choosing a physical charge profile (e.g., proportional to the Einasto density) would be needed to compute observable signatures such as the shadow or photon ring.
The non-local droplet solutions could be tested for radial stability; the paper does not address stability, but the sign change in the radial pressure suggests a possible instability boundary.
The construction suggests that any smooth, asymptotically vanishing density profile with a finite central value and a de Sitter-like core could generate regular charged black holes, with Einasto being a convenient two-parameter family.
If such charged dark matter droplets exist, they would be distinguishable from standard black holes by the absence of a horizon and by modifications to the photon sphere; high-resolution observations of Sgr A* could in principle probe this distinction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This one should be rejected, and not just because the charge function is left unspecified. Substitute the mass function (30) into the line element (33): the charge integral enters as -(2/r)∫ q q'/x dx. Equations (34)-(35) print it as +∫ q q'/x dx with no prefactor. So the displayed line element does not solve the Einstein-Maxwell equations the paper sets up. This is a load-bearing algebraic error, not a cosmetic typo: the horizon condition, Hawking temperature, effective potential, and rescaled g00 all inherit the wrong metric.

To give credit where it is due: the charged extension is a legitimate continuation of Batic et al.'s Einasto FDM black holes. The authors cite the earlier work and check that the model reduces to it when q=0. The incomplete-gamma machinery is assembled cleanly, and the uncharged part reproduces known regular black-hole/droplet behavior. The citation pattern is unremarkable and mostly appropriate.

The soft spots are numerous. q(r) is never specified, so no concrete charged solution is constructed; the figures are effectively the q=0 limit. Equation (38) for the Hawking temperature is dimensionally inconsistent, Eq. (39) is a self-referential expression for the horizon radius, and Eq. (44) does not follow from Eq. (43). Section IV gives the Sagittarius A* mass as 4.1×10^-6 M⊙, off by twelve orders of magnitude. The central de Sitter-like behavior is an input of the Einasto profile, not a prediction, and the mass-matching condition (49) is imposed rather than derived. Any one of these might be fixable; the metric inconsistency is not.

The paper is for readers who want to see how the uncharged Einasto construction works and where a careless charged generalization goes wrong. Nobody should build on the charged results as printed. My recommendation: desk-reject or reject without further review. If the authors specify a charge profile and correct the sign/prefactor in the metric, the result might become a modest contribution to the regular-black-hole literature, but the current central object is not a solution to the equations it claims to solve.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a new class of spherically symmetric, self-gravitating charged black-hole and droplet solutions for fuzzy dark matter halos modeled by the Einasto density profile. It couples an anisotropic fluid with equation of state p_r = -rho to an electromagnetic field, adopts the metric ansatz g00 = 1/g_rr, defines a mass function in Eq. (30), and derives charged line elements, horizon conditions, Hawking temperatures, effective potentials, and a nonlocal-EoS variant. The abstract claims that the central density mimics a de Sitter core and that the outer region approaches Reissner-Nordstrom, with applications to Sagittarius A*.

Significance. If the construction were correct, it would extend the uncharged Einasto-based fuzzy dark matter black holes of Batic et al. to include electric charge and would connect the resulting regular spacetimes to supermassive galactic nuclei. The paper contains explicit formulas and several figures. However, the central charged solution is not actually a solution of the stated Einstein-Maxwell system: the printed line element does not follow from the mass function, the Maxwell equation is integrated incorrectly, the charge profile q(r) is never specified, and no charged example is plotted. The concrete numerical results are effectively the q=0 limit of earlier work. The claimed new charged class is therefore not established.

major comments (4)

[Sec. III, Eqs. (30)-(35)] The charged line element is not the one generated by the stated mass function. With m(r) = M gamma(3beta,(r/h)^(1/beta))/Gamma(3beta) + integral_0^r q(x)q'(x)/x dx, Eq. (33) gives g00 = 1 + q^2/r^2 - 2M gamma/(r Gamma) - (2/r) integral_0^r q q'/x dx, whereas Eq. (35), and hence Eq. (34), contains + integral_0^r q q'/x dx with no 2/r prefactor and the opposite sign. The printed metric therefore does not satisfy the Einstein-Maxwell equations used in the derivation, and for nonzero charge it is typically not asymptotically flat because the integral tends to a constant. Every later quantity built on g00 - the horizon condition (39), the Hawking temperature (38), the effective potential (44), and the rescaled metric (51) - inherits this error.
[Sec. III, Eqs. (21)-(23)] The Maxwell equations are not integrated correctly. From Eq. (21) with V^0 = 1/sqrt(g00), the radial equation is (r^2 phi')' = 4 pi sigma_e r^2 sqrt(g00), so phi' = (4 pi/r^2) integral_0^r sigma_e x^2 sqrt(g00) dx. Instead, Eq. (22) sets phi' = q(r)/(r^2 sqrt(g00)) with q(r) = 4 pi integral_0^r sigma_e x^2 dx. These two expressions agree only in the trivial case g00 = 1 or under an additional relation that is neither stated nor verified. Moreover, q(r) is never specified anywhere in the manuscript, and no figure or example with nonzero charge is shown; the claimed 'charged' family is therefore not concretely defined.
[Sec. IV, Eq. (38)] The Hawking temperature formula is dimensionally inconsistent and not derived from the stated metric. In units G=c=1, q^2/r^2 is dimensionless, so q has dimension length; the first two terms in Eq. (38) then have dimension inverse length, while the final term q/r_H dq/dr_H is dimensionless. The notation dq/dr_H is also undefined. No derivation from the horizon condition g00(r_H)=0 for the charged metric is given, and because the metric itself is incorrect (see the first major comment), Eq. (38) is not a usable result.
[Sec. IV, Eqs. (43)-(48)] The effective potential is not derived consistently from the metric. Starting from Eq. (43) with the mass function (30), the final expression (44) should contain the charge integral I(r) = integral_0^r q q'/x dx both in the term coming from g00 and in the term -xi m/r; these contributions are absent. Additionally, Eq. (44) writes the incomplete gamma function argument as (r/h)^2 instead of (r/h)^(1/beta), and Eq. (48) repeats the missing exponent and charge terms. The claimed agreement of the effective potential with the Schwarzschild case in Figs. 3-7 is therefore not supported by the displayed formulas.

minor comments (5)

[Sec. IV, first paragraph] The text states 'MBH = 4.1 x 10^{-6} M_sun'; this should be 4.1 x 10^6 M_sun, since the quoted Schwarzschild radius 17.4 R_sun = 3.92 x 10^{-7} pc corresponds to the latter value.
[Sec. III, Eq. (31)] The sentence 'We require that T^0_0 = T^1_1 = -rho(r)' is inconsistent with the stress-energy components in Eq. (25) unless q=0; the subsequent equation implies p_r = -rho, which gives T^0_0 = T^1_1 = rho + q^2/(8 pi r^4). The wording should be corrected.
[Sec. IV, text near Fig. 2] The sentence 'There is only one degenerate horizon for omega_0 = 2.28378 with omega = omega_0 = 0.95206' assigns two different values to the same quantity; please clarify which value is the critical rescaled mass.
[Sec. V, Eqs. (52)-(55)] The nonlocal equation of state still involves the unspecified charge function q(r), and the plots in Figs. 8-10 do not demonstrate a charged nonlocal model; they appear to be the q=0 limit.
[Abstract and Sec. VI] The abstract and conclusion describe the de Sitter-like central core as an outcome, but it is an input: the Einasto density profile has a finite central value by construction and the metric is built from that density, so the regularity statement is not an independent prediction.

Circularity Check

1 steps flagged · score 6.0 of 10

The Schwarzschild-potential match in Sec. IV is imposed by the mass-matching inequality, not predicted; the charged metric also has a separate algebraic-sign defect that is a correctness issue rather than circularity.

fitted input called prediction [Sec. IV, Eqs. (40)-(41) and Eq. (49); also Figs. 3-7]
"Firstly, by imposing the condition that the total mass M appearing in the geometry (34) coincides with MBH. Nextly, when measured at the rmin of the eﬀective potential associated with a massive particle, the mass m oﬀers an accurate estimation for MBH. Alternatively, we require that 1 − m(rmin)/MBH ≤ 10−2. ... ∆γ ≡ 1/Γ(3β) γ(3β,(rmin/h)1/β) − 0.99 ≥ 0 ... the aforementioned criterion not only guarantees the equivalence between our diﬀused self-gravitating structure and the Schwarzschild eﬀective potential at their minimum."

The model's effective potential is constructed from m(r) = M γ(3β,·)/Γ(3β) (Eqs. 29-30, with the charge integral set aside). Equations (40) and (49) do not test the model against an external datum; they impose that m(rmin)/MBH lie within 1% of unity. Under that constraint, U_eff at rmin coincides with the Schwarzschild U_S by substitution, and the text explicitly says the criterion 'guarantees' the equivalence at the minimum. The subsequent figures then display this imposed coincidence as if it were a successful prediction of galactic dynamics. This is the standard 'fitted input called prediction' pattern: the inequality selects the parameters, and the agreement is the output rather than an independent check.

full rationale

The strongest genuinely circular step is in Sec. IV: the mass-matching condition (40)/(49) forces m(rmin)/MBH ≥ 0.99, and because U_eff is built from exactly that m(r), the advertised equivalence of the model potential with the Schwarzschild effective potential at its minimum is guaranteed by construction. That is a partial circularity and is scored accordingly. I do not separately count the 'de Sitter core' language as a self-definitional step, because the finite central density is explicitly part of the input Einasto density (Eq. 5) and the paper phrases the de Sitter behavior as a deduction rather than a falsifiable prediction. No load-bearing self-citation chain was found: the main comparison works [31,32] are by Batic, Abuhejleh, Nowakowski and Batic, Faraji, Nowakowski, not by the present authors; the authors' own references [26-28] appear only as contextual motivation. Separately, I note a serious non-circular defect: substituting the mass function (30) into the line element (33) gives a -(2/r)∫qq'/x dx term, whereas Eqs. (34)-(35) print a +∫qq'/x dx term, and q(r) is never specified. This algebraic inconsistency undermines the charged-metric claims, but it is a correctness/completeness problem, not a circularity, so it does not raise the circularity score beyond the value already set by the fitted-input step.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or forces. Its central claim rests on the assumed Einasto density profile, the de Sitter EoS, the metric ansatz, and an arbitrary charge function. The free parameters (beta, h, omega) are taken from prior literature or chosen by hand, and the charge function is left completely unspecified, so the 'charged' models are not concrete.

free parameters (4)

Einasto index beta = 0.7072 in figures; varied in 0.1-13 range
Taken from Einasto [51,52] and DM halo simulations; not derived in this paper.
Scale radius h = 2.121e-9 kpc (hE) or 3.92e-9 kpc (H=10)
Chosen to match DM halo scales and the Schwarzschild radius of Sgr A*; effectively fitted.
Rescaled mass omega = M/h = varied; critical values omega0 = 0.95206 and 0.5206
Parameter controlling existence and number of horizons; chosen by hand.
Charge function q(r) = unspecified
The central quantity for the claimed 'charged' models, but never given a functional form. All equations keep it arbitrary.

assumptions (5)

domain assumption Einasto density profile as the dark matter halo density
Section II, Eq. (5). The entire construction uses this profile as the matter source.
domain assumption De Sitter-like equation of state p_r = -rho
Section III, Eq. (31). Imposed to obtain a regular core, not derived from any microphysical model.
ad hoc to paper Metric ansatz g00 = 1/g_rr
Eq. (10). This Schwarzschild-like ansatz forces G^t_t = G^r_r, which imposes a consistency condition on the stress tensor; the paper does not verify this for nonzero charge.
domain assumption Non-local equation of state from Hernandez and Nunez
Section V, Eq. (52). Imported from [71] without justification for fuzzy dark matter.
standard math Maxwell equations with spherical charge distribution
Section III, Eqs. (17)-(21). Standard, but the charge density sigma_e(r) is never specified.

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Pith. "Pith review of Charged Fuzzy Dark Matter Black Holes." pith.science (2026). https://pith.science/paper/AQS27SV4

@misc{pith2026241113871,
  author       = {Pith},
  title        = {Pith review of: Charged Fuzzy Dark Matter Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AQS27SV4}},
  note         = {Machine review of arXiv:2411.13871}
}

abstract

We investigate the impact of fuzzy dark matter (FDM) on supermassive black holes (SMBHs) characterized by a spherical charge distribution. This work introduces a new class of spherically symmetric, self-gravitational relativistic charged models for FDM haloes, using the Einasto density model. This study enables the dark matter (DM) to appear as the matter ingredient, which constructs the black hole and extends the non-commutative mini black hole stellar solutions. By considering the charged anisotropic energy-momentum tensor with an equation of state (EoS) $p_{r}=-\rho$, we explore various black hole solutions for different values of the Einasto index and mass parameter. Our approach suggests that the central density of the resulting black hole model mimics the usual de Sitter core. Furthermore, we discuss the possibility of constructing a charged self-gravitational droplet by replacing the above-mentioned EoS with a non-local one. However, under these circumstances, the radial pressure is observed to be negative. Ultimately, we consider various possibilities of constructing DM black holes, featuring intermediate masses that could evolve into galaxies. Consequently, some of these theoretical models have the potential to replace the usual black hole solutions of the galactic core. Simultaneously, these models are physically beneficial for being comprised of the fundamental matter component of the cosmos. Due to the outcomes of this paper, we would be able to study the connection between BH and DM by formulating stable stellar structures featuring fuzzy mass distributions derived from the Einasto distribution of DM halos.

Figures

Figures reproduced from arXiv: 2411.13871 by the authors.

**Figure 2.** FIG. 2: Behavior of the relativistic metric potential [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3: Pictorial representation of the effective potentials [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4: Diagrammatic scheme of the effective potentials [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5: Behavior of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: FIG. 6: Behavior of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7: Behavior of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8: Diagrammatic scheme of the rescaled form of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 10.** Figure 10: FIG. 10: Pictorial representation of [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 9.** Figure 9: FIG. 9: Pictorial representation of the rescaled pressure [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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