Investigating Optical and Ring-Down Gravitational Wave Properties of a Rotating Black Hole in a Dehnen Galactic Dark Matter Halo

Dhruba Jyoti Gogoi; Kalyan Bhuyan; Mrinnoy M. Gohain; Prabwal Phukon

REVIEW 2 major objections 5 minor 78 references

Investigating Optical and Ring-Down Gravitational Wave Properties of a Rotating Black Hole in a Dehnen Galactic Dark Matter Halo

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A dense galactic dark halo expands a black hole's horizon and shadow and shifts its ringdown.

desk verdict The shadow and ringdown calculations are competently done for the authors' metric, but that metric is not a Dehnen halo spacetime, so the paper's central claim doesn't hold. read the letter →

arxiv 2508.18053 v1 pith:S4KHJ4FQ submitted 2025-08-25 astro-ph.CO gr-qchep-th

Mrinnoy M. Gohain , Dhruba Jyoti Gogoi , Kalyan Bhuyan , Prabwal Phukon This is my paper · ORCID

classification astro-ph.COgr-qchep-th

keywords blackholeshadowdarkmatterhaloDehnendensityprofilequasinormalmodesrotatingWKBapproximationringdownergoregion

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 aims to show that a rotating black hole is observably different when it sits inside a Dehnen-type galactic dark matter halo than when it is isolated: the halo's central density and core radius push the event horizon and ergoregion outward, enlarge and reshape the black hole shadow, lower the energy emission rate, and change the oscillation frequency and damping time of gravitational-wave ringdown. If that claim is right, shadow images and ringdown signals are not purely probes of the black hole itself; they carry information about the dark matter environment, which future observations could extract. The paper constructs an axisymmetric rotating metric by applying a complex-coordinate rotation to a static seed solution derived earlier for the same halo profile, then computes null geodesics, shadow distortion, emission rates, and scalar-field quasinormal modes with a WKB treatment. The halo-dependent quantities are the central density $\rho_s$ and the halo radius $r_s$, with the black hole mass and spin playing the standard rotating-geometry roles.

What carries the argument

The load-bearing object is the rotating analogue of the halo-modified metric, encoded in the function $\Delta(r)=a^2-2Mr-\frac{4\pi r_s^3 r^2(r_s+2r)\rho_s}{3(r_s+r)^2}+r^2$, obtained by applying a complex-coordinate rotation to the static seed $f(r)$. This single function fixes the horizons as the roots of $\Delta=0$, the ergosurface through $g_{tt}=0$, the photon-region impact parameters $\xi$ and $\eta$ used to draw the shadow, the energy-emission cross-section, and the effective potential of the scalar perturbation equation whose peak the WKB scheme expands around. The halo enters only through the two parameters $\rho_s$ and $r_s$, and every reported observable is a functional of $\Delta$ and its derivatives.

What would settle it

Reconstruct the effective matter density of the seed metric from $f(r)$ through the Einstein tensor of the static line element and compare it with Eq. (2): a mismatch near the centre would mean the shadow and quasinormal-mode results describe a different halo. Alternatively, evolve scalar perturbations in the time domain with the same $\Delta(r)$ and locate the turning point of $|\omega_I|$ as a function of $\rho_s$ and $r_s$; the leading-order WKB prediction fails if the time-domain damping is monotonic.

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Extended reading notes

Core claim

The central claim is that the composite spacetime with lapse function $f(r)=1-\frac{2M}{r}-\frac{4\pi r_s^3(r_s+2r)\rho_s}{3(r_s+r)^2}$, once rotated to an axisymmetric metric, has geometry and observables that respond to the halo: for fixed spin, increasing $\rho_s$ or $r_s$ moves the event horizon and the ergosurface outward, and at high $\rho_s$ the inner and outer horizons approach each other, so the hole can approach an extremal or over-extremal configuration. The shadow in the celestial plane grows with $\rho_s$ and $r_s$, and its distortion increases with spin while decreasing slightly with the halo parameters. The energy emission rate falls as $\rho_s$, $r_s$, or the spin grow, implying longer-lived black holes in denser halos. For scalar-field perturbations, the WKB quasinormal frequency $\omega_R$ decreases monotonically with $\rho_s$ and $r_s$, while the damping rate $|\omega_I|$ is non-monotonic: it first rises and then falls, so dense or extended halos can either shorten or lengthen the ringdown depending on their parameters.

Load-bearing premise

The argument assumes that the lapse function in Eq. (4) is the actual spacetime of a static black hole embedded in a Dehnen $(1,4,0)$ halo; the paper cites earlier work for this metric rather than deriving it from the Dehnen density profile here, and the effective density implied by $f(r)$ differs from the Dehnen profile near the centre, so all subsequent results inherit that identification.

Editorial extensions

If this is right

At fixed spin, denser or more extended halos enlarge the event horizon and ergoregion; at high $\rho_s$ the inner and outer horizons converge, suggesting that extremal-like black holes or naked singularities could arise in dense dark matter environments.
The black hole shadow is not determined by mass and spin alone: larger $\rho_s$ or $r_s$ increases the shadow radius, and the halo parameters also feed into the distortion parameter, so shadow measurements could in principle constrain the halo.
The energy emission rate decreases when $\rho_s$, $r_s$, or the spin increases, which lengthens the evaporation time of black holes inside dense halos.
Ringdown analysis is environment-sensitive: $\omega_R$ drops monotonically with $\rho_s$ and $r_s$, while $|\omega_I|$ is non-monotonic, so gravitational-wave spectroscopy could identify which side of the turning point a candidate halo lies on.
If real, these effects imply that very-long-baseline shadow images and future gravitational-wave ringdown measurements could serve as indirect dark matter probes in galactic centers.

Reading between the lines

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

The paper takes the static seed metric from its earlier work rather than deriving it here from the Dehnen density profile; a direct comparison of the effective density reconstructed from $f(r)$ with the quoted cored profile would show whether the printed shadow and quasinormal-mode curves belong to the Dehnen model or to a different effective density.
Because the WKB angular eigenvalue is taken in the eikonal limit, the non-monotonic damping rate is a leading-order result; a time-domain evolution of the perturbation equation, or a higher-order WKB calculation, would test whether the turning point in $|\omega_I|$ survives.
The dimensionless parameter ranges plotted, with $M=1$ and $\rho_s$, $r_s$ of order unity, are not calibrated to astrophysical units; converting them to solar masses and kiloparsecs would be the first step toward deciding whether real galactic halos fall in the interesting part of parameter space.
The same rotation-plus-shadow construction could be applied to the other Dehnen variants with $\gamma>0$, which would show whether the observable trends persist for cuspy halos and would widen the comparison with dwarf-galaxy measurements.

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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

Here's my read. The paper does what it says: it takes the authors' earlier static 'Schwarzschild–Dehnen' metric, spins it up with the Azreg-Aïnou version of Newman–Janis, and then works through the standard menu — horizons, ergoregion, shadow, energy emission, WKB quasinormal modes. The NJA step and the shadow/geodesic part are competently executed; if someone hands you that seed metric, the rest follows in a routine way. The figures are clean and the paper is readable.

The problem is the seed. Eq. (4) is not a Schwarzschild black hole in a Dehnen (1,4,0) halo. The Dehnen density in Eq. (2) is cored at the center and falls as r^-4 outside. But the Einstein equations for the static metric (3)-(4) give an effective density rho_eff = r_s^4 ρ_s (r_s + 3r) / (6 r^2 (r_s + r)^3), which diverges as r^-2 at small r and is a factor of two off at large r. The central region is exactly what controls horizons, photon sphere, and ringdown frequencies. So every observable claim — shadow size, QNM dependence, the abstract's headline about dense/extended halos — is computed for a cusped matter distribution that is not the profile the paper names. The paper imports Eq. (4) from the authors' own prior work [35] without re-derivation, and the derivation evidently produced the wrong mass–density relation.

The QNM section has problems too. The effective potential in Eq. (49) is written down with a 1/(2 a ω) term and a complicated ml dependence, but it is not shown to come from the Teukolsky equation in this spacetime. And the ω_R formula (51) does not obviously reduce to Kerr when ρ_s→0. That's a separate weakness, but secondary to the missing physical input.

For readers, this is only useful as an example of the pipeline; it does not tell you anything about Dehnen halos. My recommendation: desk reject in current form. A revision that either starts from a proper Dehnen solution or drops the Dehnen claim and studies the cusped metric as a model in its own right could be worth another look.

Referee Report

2 major / 5 minor

Summary. The paper constructs a rotating, Kerr-like black hole spacetime in the presence of a Dehnen (1,4,0) galactic dark matter halo by applying a modified Newman–Janis algorithm to a static seed metric imported from the authors' earlier work. It then computes horizons, ergoregion geometry, black hole shadows, distortion, energy emission rates, and scalar-field quasinormal mode (QNM) frequencies using a WKB approach. The central advertised result is that the dark matter halo parameters—central density and halo radius—leave observable imprints on the shadow and on gravitational-wave ringdown signals.

Significance. If the construction were physically sound, the paper would provide a useful survey of how a cored galactic dark matter halo affects strong-field black hole observables, and the explicit formulas for the rotating metric, shadow boundary, and QNM spectra would be convenient for follow-up work. The paper is clearly organized and the parameter scans in Figs. 1–7 are systematic. However, the physical interpretation rests entirely on the claim that the static seed metric in Eq. (4) describes a Schwarzschild black hole embedded in the Dehnen (1,4,0) halo of Eq. (2). That claim is not demonstrated, and the metric itself implies a different matter distribution, so the subsequent shadow and ringdown results are computed for an ad hoc cusped spacetime rather than for the Dehnen halo named in the title and abstract.

major comments (2)

[II, Eq. (4)] The seed metric f(r) in Eq. (4) is imported from Ref. [35] but it is not shown to be a solution sourced by the Dehnen (1,4,0) density profile in Eq. (2), and the metric itself contradicts that identification. For the static metric ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 dΩ^2, the Misner–Sharp mass is m(r)=r(1-f(r))/2, which gives the Einstein-frame effective density ρ_eff = m'(r)/(4πr^2) = r_s^4 ρ_s (r_s+3r)/(6 r^2 (r_s+r)^3). This behaves as ρ_eff ∼ ρ_s r_s^2/(6r^2) near r=0, so it diverges, whereas the claimed Dehnen (1,4,0) density is cored with central value ρ_s. At large r it behaves as ρ_s r_s^4/(2r^4), which is a factor 1/2 off from Eq. (2). Since the central region controls the horizon, photon sphere, and ringdown frequencies, all results in Sections III–VI are computed for a different, cusped matter distribution rather than for the Dehnen halo named in the title and abstract. This is a load-bearing assumption, and it must be established before any physical or observational interpretation can be made.
[VI, Eqs. (43)–(52)] The QNM calculation assumes that the scalar wave equation in this Newman–Janis generated rotating spacetime separates in the Teukolsky form (43)–(46), but the paper does not demonstrate that the spacetime is of Petrov type D or that the radial and angular parts decouple for nonzero halo parameters. The effective radial potential in Eq. (49) is introduced without a derivation from the separated field equation, and the WKB result in Eqs. (51)–(52) is not validated against the Kerr limit ρ_s=0 or against known Kerr QNM frequencies. Because the abstract's ringdown claim depends on this analysis, the QNM part must be redone from the explicit scalar wave equation for the rotating metric.

minor comments (5)

[Eq. (2)] The printed form of the Dehnen (1,4,0) density appears as ρ_D = ρ_s (r/r_s + 1)^4, which grows with radius and is unphysical; the intended expression is ρ_s/(1 + r/r_s)^4, with the denominator lost in typesetting.
[III, after Eq. (16)] The phrase 'the NED BH assumed in our work' is inconsistent with the rest of the paper: the seed metric is not a nonlinear electrodynamics black hole, and the variables ζ and Q in the following sentence are never defined.
[Fig. 3 and surrounding text] The text describing Fig. 3 is inconsistent with the panels: the left panel is described as varying ρ_s at fixed a, but the panel legend shows varying a, while the middle panel is said to vary a although the caption fixes a=0.99.
[Figs. 4–7] The halo parameters ρ_s and r_s are varied as dimensionless numbers alongside M=1, but no conversion to physical units is given; this makes the claimed observational relevance of the plots difficult to assess.
[III, Eqs. (8) and (12)] The imaginary unit is rendered as '˙ι' throughout the null tetrad expressions; this should be typeset as the standard i.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation of the shadow, horizon, or QNM results; the only self-referential element is the seed metric Eq. (4), which is imported from the authors' own prior work and whose claimed link to the Dehnen profile is not demonstrated here.

full rationale

The paper's observable predictions (horizon and ergoregion structure, shadow radius and distortion, energy emission, and QNM frequencies) are computed from the seed metric f(r) in Eq. (4) through standard, self-contained manipulations: the modified Newman–Janis algorithm for rotation, Hamilton–Jacobi null geodesics for shadows, and WKB/Teukolsky methods for QNMs. No parameter is fitted to data, and no predicted observable is fed back into the model, so the derivation chain from the assumed metric to the stated results is not circular. The genuine concern is different: the metric f(r) in Eq. (4) is claimed in Sec. II to follow from the Dehnen (1,4,0) density in Eq. (2) by citing the authors' own Ref. [35], and the paper does not re-derive that identification. Moreover, the Einstein equations applied to the static metric (3)–(4) imply an effective density ρ_eff = ρ_s r_s^4 (r_s + 3r) / [6 r^2 (r_s + r)^3], which diverges as ρ_s r_s^2/(6 r^2) near the origin instead of tending to the finite central value ρ_s required by Eq. (2), and it differs by a factor of about 1/2 in the outer falloff. Thus the computed shadow and QNM properties are properties of the imported metric, not of the cored Dehnen halo named in the title. This is a correctness/validity problem and a load-bearing self-citation, but it is not a circular derivation: the calculations do not assume the conclusions they claim to establish. Score 2 reflects the mild self-referential input rather than a closed logical loop.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central results depend on the halo parameters ρs and rs, the spin a, and mass M as manually chosen inputs. No independent derivation of the seed metric is given, and the effective density does not reproduce the Dehnen profile, so the model should be regarded as an ad hoc parametrized deviation from Schwarzschild/Kerr rather than a solution sourced by Dehnen dark matter.

free parameters (3)

ρs (central halo density) = chosen by hand; plots use 0, 0.2, 0.3, 0.4, 0.5, 0.6
Controls the strength of the halo term in the metric; not fitted to data.
rs (halo radius) = chosen by hand; plots use 0.2, 0.3, 0.4, 0.5, 1.0
Controls the spatial extent of the halo term; not fitted to data.
a (spin parameter) = chosen by hand; plots use 0.2 to 0.99
Standard black hole spin; varied to show spin dependence.

assumptions (4)

ad hoc to paper The metric f(r) in Eq. (4) is the correct spacetime for a Schwarzschild black hole in a Dehnen (1,4,0) halo.
Taken from the authors' previous work; no derivation is given in this paper, and the implied density does not match the Dehnen profile.
domain assumption The modified Newman-Janis algorithm (Azreg-Ainou) produces a valid rotating solution for this seed metric.
The method is standard, but its validity for this particular seed is assumed, not verified by solving the field equations.
domain assumption The WKB approximation is accurate for the chosen quantum numbers l=2, n=0.
WKB is applied at low l where its accuracy is doubtful; standard higher-order WKB is not used.
domain assumption The Teukolsky equation and the approximate angular eigenvalue in Eq. (46) apply to scalar perturbations of the rotating metric.
The paper assumes the separable Teukolsky form and uses an eikonal-limit eigenvalue expression at l=2.

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Cite this review

Pith. "Pith review of Investigating Optical and Ring-Down Gravitational Wave Properties of a Rotating Black Hole in a Dehnen Galactic Dark Matter Halo." pith.science (2026). https://pith.science/paper/S4KHJ4FQ

@misc{pith2026250818053,
  author       = {Pith},
  title        = {Pith review of: Investigating Optical and Ring-Down Gravitational Wave Properties of a Rotating Black Hole in a Dehnen Galactic Dark Matter Halo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4KHJ4FQ}},
  note         = {Machine review of arXiv:2508.18053}
}

abstract

We present a comprehensive study of the optical and dynamical properties of a rotating black hole immersed in a Dehnen-type $(1,4,0)$ galactic dark matter halo, modeled by a double power-law density profile commonly used to describe realistic galactic cores. By extending our previous Schwarzschild-Dehnen solution using a modified Newman-Janis algorithm, we construct a Kerr-like axisymmetric spacetime that smoothly incorporates both black hole rotation and the influence of the surrounding dark matter halo. We systematically investigate the effects of the halo parameters-the central density and halo radius-on horizon structure, the shape and extent of the ergoregion, and the null geodesics associated with black hole shadows. Our results show that the presence of a dense or extended halo expands the event horizon and ergoregion, and significantly alters the size and distortion of the black hole shadow. Furthermore, by applying the WKB approximation to scalar field perturbations, we compute the quasinormal mode (QNM) spectra and demonstrate that the frequencies and damping times of ringdown signals are highly sensitive to the halo profile. These results open promising avenues for probing the dark matter environment of astrophysical black holes through black hole imaging and gravitational wave observations.

Figures

Figures reproduced from arXiv: 2508.18053 by the authors.

**Figure 2.** FIG. 2. Two-dimensional slices of the ergoregion in the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. Shadows of the BH for different values of the model parameters. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Distortion of the BH for different values of the model parameters. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5. Emission rate of the BH for different values of the model parameters. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6. Variation of QNMs with respect to model parameter [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7. Variation of QNMs with respect to model parameter [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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