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Shadows and lensing signatures of a rotating black hole in a Hernquist dark matter halo

T0 review · 1 major / 7 minor · reviewed 2026-07-10 · glm-5.2

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Pith's one-line read Dark matter halos enlarge black hole shadows — and EHT can measure it

desk verdict Optical analysis of a rotating BH in a Hernquist halo: solid derivations, but observational bounds rest on shadow-ring identification without radiative transfer read the letter →

arxiv 2607.08650 v1 pith:6O7RN2DQ submitted 2026-07-09 gr-qc hep-th

A. A. Araújo Filho , Arun Kumar , N. Heidari , C. F. S. Pereira , Amilcar R. Queiroz , V. B. Bezerra This is my paper
classification gr-qchep-th
keywords blackholeshadowdarkmatterhaloHernquistprofilegravitationallensingKerr-likegeometryNewman-JanisprocedureEventHorizonTelescopephotonsphere
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

The paper studies what happens to light near a spinning black hole that sits inside a Hernquist dark matter halo — a specific, analytically tractable model for the distribution of dark matter around a galaxy's center. The authors construct this spacetime by taking a known static black-hole-plus-halo solution and spinning it up via a standard mathematical procedure, producing a geometry that looks like the Kerr rotating black hole but with the dark matter contribution baked into a single radial function. They then compute the two main optical signatures: the shadow (the dark silhouette the black hole casts against background light) and gravitational lensing (how much the black hole bends light from distant sources). The central finding is that spin and dark matter affect these signatures in cleanly separable ways. Spin shifts and distorts the shadow sideways, producing the familiar left-right asymmetry of rotating black holes. The dark matter halo, by contrast, enlarges the photon capture region — the zone where light gets swallowed by the black hole — making the shadow bigger overall without introducing asymmetry. Because the Event Horizon Telescope has measured the apparent size of the shadows around Sgr A* (our galactic center black hole) and M87* (the supermassive black hole in the Virgo cluster), the authors can ask: how much dark matter could be sitting around these black holes before the predicted shadow grows too large to match observations? The answer constrains a dimensionless halo density parameter to below roughly five thousandths at two-sigma confidence for Sgr A*. The authors extend the analysis to gravitational lensing in both the strong-field regime (where light loops around the black hole near the photon sphere) and the weak-field regime (where light passes far away and bends only slightly), finding that the halo shifts the positions of relativistic images and adds a correction to the leading-order bending angle. Using the observed Einstein ring of the galaxy ESO325-G004, they obtain an independent bound on the same halo parameter, this time below about two hundredths at two-sigma.

What carries the argument

The load-bearing mechanism is the radial function Delta(r) = r^2 - 2Mr + a^2 - 4*pi*rho*r_s*r^2/(r+r_s), which encodes both the black hole mass M, the spin a, and the Hernquist halo density rho. Because the Newman-Janis procedure preserves the Kerr-like angular structure, the Hamilton-Jacobi equation remains separable, and the critical impact parameters for unstable photon orbits can be written in closed form. The halo enters these expressions only through Delta(r) and its derivative, which means every optical observable — shadow boundary, deflection angle, Einstein ring radius — receives a correction proportional to rho that can be computed analytically in the weak-field limit and numerally

What would settle it

If a radiative-transfer calculation for a realistic accretion flow around this geometry showed that the halo-induced shadow enlargement is compensated or amplified by plasma effects in a way that decouples the observed ring diameter from the mathematical shadow diameter, the quantitative bounds on the halo parameter would no longer hold as stated.

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Extended reading notes

Core claim

The paper's central claim is that a Hernquist dark matter halo around a rotating black hole enlarges the photon capture region and increases the apparent shadow size in a way that is cleanly separable from the spin-induced distortion, and that this effect is already strong enough to be bounded by existing Event Horizon Telescope measurements. Specifically, the halo parameter and the spin parameter act on geometrically distinct degrees of freedom: the halo modifies the radial capture scale of photons through the function Delta(r), while the spin shifts the shadow center and produces left-right asymmetry through frame dragging. This separation means that combining shadow-size measurements with

Load-bearing premise

The paper uses the area-equivalent diameter of the mathematical shadow boundary as a direct proxy for the angular diameter measured by the Event Horizon Telescope, without modeling how the accretion flow's plasma distribution and radiative transfer modify the observed bright ring. The authors acknowledge this gap, noting that the observed ring is not identical to the shadow boundary, so the quantitative bounds on the halo parameter could shift with a more realistic emission

Editorial extensions

If this is right

  • If the halo parameter bounds hold, they provide a direct, model-dependent measurement of the local dark matter density near supermassive black holes, complementary to galactic rotation curve estimates.
  • The clean separation between spin-induced asymmetry and halo-induced size enlargement means that future higher-resolution shadow measurements could in principle disentangle the two effects and detect a halo contribution even when the spin is unknown.
  • The weak-field lensing correction from the halo appears already in the leading term of the bending angle, which means galaxy-scale lensing systems (not just black-hole-scale observations) can probe the same halo parameter through Einstein ring sizes.
  • If a radiative-transfer model were coupled to this geometry, the quantitative bounds on the halo parameter could shift, tightening or loosening the constraints depending on how the accretion flow modifies the relationship between the mathematical shadow boundary and the observed bright ring.

Reading between the lines

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

  • The fact that Sgr A* gives tighter bounds than M87* despite M87* being more massive is driven by the ratio of observed angular diameter to angular gravitational radius — Sgr A* has a smaller allowed dimensionless radius, so less room for halo-induced enlargement. This suggests that future observations of black holes with smaller dimensionless shadow sizes (relative to their gravitational radii) wi
  • The halo parameter rho is a local density scale, not a total halo mass, so the bounds do not directly translate to a constraint on the total dark matter mass around each black hole. Converting to a physical density would require specifying the halo scale length r_s, which the paper fixes at 2M but which could vary astrophysically.
  • If the same analysis were applied to other halo profiles (NFW, Einasto, Burkert), the leading-order weak-field correction would likely differ because the asymptotic falloff of the density profile changes, potentially making some profiles more or less constrained by the same lensing data.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 7 minor

Summary. The manuscript investigates the optical properties—geodesics, shadows, and gravitational lensing—of a rotating black hole immersed in a Hernquist dark matter halo. The spacetime is constructed via the noncomplexification Newman-Janis procedure from a static Hernquist black hole, yielding a Kerr-like metric where the halo contribution is encoded in the radial function $f(r)$ (or equivalently $f(r)$). The authors derive null geodesic equations, effective potentials, and radial acceleration, and exploit the separability of the Hamilton-Jacobi equation to obtain critical impact parameters for unstable spherical photon orbits. Shadow contours are constructed and compared with EHT observations of Sgr A* and M87* to constrain the dimensionless halo parameter $f(r)$. Strong-field lensing observables (relativistic image positions, separations, magnifications, time delays) and weak-field deflection angles are computed, with the latter constrained using the Einstein ring of ESO325-G004. The central physical finding is that rotation shifts and distorts the shadow while the Hernquist halo enlarges the photon capture region, with quantitative upper bounds on $f(r)$ obtained from both shadow and lensing data.

Significance. The paper provides a self-contained and systematic optical analysis of a specific rotating black hole in a Hernquist halo, a profile of genuine astrophysical interest. The simultaneous treatment of shadows, strong-field lensing, and weak-field lensing within a single geometry is a strength, as is the exploitation of Hamilton-Jacobi separability to obtain analytic expressions for critical impact parameters (Eqs. 32). The derivation of the weak-field deflection angle (Eq. 62) explicitly showing the halo contribution entering at leading order $f(r)$ is a clear and falsifiable result. The confrontation with multiple observational datasets (EHT, ESO325-G004) to derive constraints, while subject to the caveats discussed below, provides a useful roadmap for future, more refined tests. The metric reduces correctly to Kerr, static Hernquist, and Schwarzschild in the appropriate limits, which is a good consistency check.

major comments (1)
  1. §IV.C, Eqs. (39)-(43): The area-equivalent shadow diameter is compared directly to the EHT-observed angular diameter without a radiative-transfer model. The authors acknowledge this qualitatively (§IV.C, paragraph beginning 'Strictly speaking...'), but the quantitative bounds on $f(r)$ (e.g., $f(r) f(r) f(r)$ for Sgr A* at $2f(r)$) are presented as primary results in the abstract and conclusion. Given that the halo parameter enters the leading weak-deflection term as $f(r)$ (Eq. 62), implying that $f(r) f(r) 0.005$ produces a $f(r) 25f(r)$ enhancement of the effective mass, even modest shifts in the ring-to-shadow mapping could materially alter these bounds. The authors should either (a) explicitly frame these as illustrative upper-size constraints rather than robust parameter bounds, adjusting the abstract and conclusion accordingly, or (b) provide a quantitative estimate of the systema
minor comments (7)
  1. §II, Eq. (2): The radial function $f(r)$ is introduced, but the notation $f(r)$ for the lapse and $f(r)$ for the density scale could cause confusion. Consider using a different symbol for one of them.
  2. §III, Eq. (6): The Lagrangian is written with $f(r)$ for the normalization, but the text below refers to $f(r)$ for timelike, null, and spacelike geodesics. This is non-standard; typically $f(r)$ is used for the affine parameter and the normalization is $f(r)$, $f(r)$, or $f(r)$. Please clarify.
  3. §IV.C, Table I: The observed angular diameter for M87* is listed as $f(r) f(r) 3 f(r)as$, but the EHT 2019 result for M87* is $f(r) 42 f(r) 3 f(r)as$. This appears consistent, but the value for Sgr A* ($f(r) 51.8 f(r) 2.3 f(r)as$) should be explicitly cited to the 2022 EHT result. The reference list includes [11,12] for Sgr A*, which is correct.
  4. §V.A, Eq. (49): The expression for the impact parameter $f(r)$ contains terms like $f(r)$ and $f(r)$ without explicit definition in the immediate context. The reader must infer these from the metric functions. A brief reminder would help.
  5. §V.B, Eq. (64): The Hubble constant is written as $f(r) f(r) f(r) f(r) Mpc$, which appears to have a typo in units (should likely be $f(r) f(r) f(r) f(r) Mpc^{-1}$ or similar). Please check.
  6. Figure 4: The 3D trajectory plot is described but the figure quality and labeling in the text could be improved; the red and black surfaces should be clearly distinguishable in print.
  7. References: Several references appear to be from 2025-2026 (e.g., [1], [49], [50], [51], [53], [54], [55], [56], [59], [60], [63], [64], [80], [83], [91], [99], [101]). If these are genuinely forthcoming or preprints, please ensure final publication details are updated. Reference [1] is cited as the source of the metric and appears to be by the same author group; this should be clearly noted as a companion paper.

Circularity Check

0 steps flagged · score 1.0 of 10

Optical analysis is self-contained from the metric; minor self-citation of the spacetime construction is not load-bearing for the shadow and lensing results.

full rationale

The paper's central optical claims—shadow morphology, critical impact parameters, strong- and weak-field deflection angles, and observational bounds—are derived in a self-contained manner from the metric (Eqs. 3–4). The geodesic equations (Eqs. 16–20), effective potentials (Eq. 18), shadow celestial coordinates (Eq. 38), and deflection angles (Eqs. 50–52, 60–62) all follow from standard Hamilton-Jacobi separation and lensing integrals applied to the stated metric. The observational constraints use external data (EHT measurements for Sgr A* and M87*, Einstein ring data for ESO325-G004) and do not fit the halo parameter to one dataset and then 'predict' a closely related quantity. The only self-citation is Ref. [1] for the metric construction itself (the rotating Hernquist black hole via the Newman-Janis procedure), but the optical results do not reduce to this citation: the metric is treated as a given input, and all subsequent derivations are independent. The metric construction in Ref. [1] is not invoked to forbid alternatives or to claim uniqueness; it is simply the source of the spacetime being studied. The acknowledged gap between the geometric shadow and the EHT-observed ring (due to the absence of a radiative-transfer model) is a correctness/modeling concern, not a circularity issue—the bounds are obtained by comparing a computed geometric quantity to external observational data, not by fitting to that data and re-deriving it. No step in the derivation chain reduces to its own inputs by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The paper introduces one genuinely free parameter (rho) that it constrains, plus the rotation parameter a that it samples. The length scale r_s is fixed by hand. The most consequential axiom is the use of the mathematical shadow as a proxy for the EHT ring, which is an ad hoc modeling choice rather than a derived result.

free parameters (3)
  • rho (Hernquist density scale) = bounded to ~0.005 (Sgr A*) and ~0.02 (ESO325-G004)
    The primary parameter of interest; constrained by comparison with EHT and lensing observations.
  • a (rotation parameter) = sampled 0.05-0.99
    Sampled over a range; not fitted to a specific observation but treated as a free parameter in the bounds.
  • r_s = 2M = 2M
    The Hernquist length scale is fixed to twice the black hole mass by hand throughout the analysis (Section II).
assumptions (4)
  • domain assumption The Newman-Janis noncomplexification procedure yields a physically valid rotating spacetime from a static seed metric.
    Invoked in Section II to construct the metric; the procedure is standard but its physical validity for matter-coupled systems is not independently justified.
  • ad hoc to paper The area-equivalent shadow radius is a valid proxy for the EHT-observed emission ring diameter.
    Used in Section IV.C to convert theoretical shadow contours into observational constraints without radiative-transfer modeling.
  • domain assumption The Hernquist profile with r_s = 2M accurately describes dark matter distribution near a black hole.
    The entire metric construction depends on this profile choice; the specific scaling r_s = 2M is assumed without independent dynamical justification.
  • domain assumption The weak-field lensing expansion converges for the impact parameters relevant to ESO325-G004.
    The Taylor expansion in Eq. (60)-(62) is used to derive the Einstein ring constraint; convergence is assumed but not verified for the relevant distance scales.
invented entities (1)
  • None independent evidence
    purpose: No new particles, forces, or dimensions are introduced.
    The paper works within standard GR with a known dark matter profile; no new physics entities are postulated.

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Pith. "Pith review of Shadows and lensing signatures of a rotating black hole in a Hernquist dark matter halo." pith.science (2026). https://pith.science/paper/6O7RN2DQ

@misc{pith2026260708650,
  author       = {Pith},
  title        = {Pith review of: Shadows and lensing signatures of a rotating black hole in a Hernquist dark matter halo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6O7RN2DQ}},
  note         = {Machine review of arXiv:2607.08650}
}
abstract

We investigate the optical properties of a rotating black hole immersed in a Hernquist dark matter halo. The spacetime is generated from a static Hernquist black hole through the noncomplexification version of the Newman-Janis procedure, yielding a Kerr-like geometry whose halo contribution is encoded in the radial function $\Delta(r)$ \cite{AraujoFilho:2026hernquist}. We derive the null geodesic equations, effective potentials, radial acceleration, and representative three-dimensional photon trajectories around the event horizon and ergoregion. Using the separability of the Hamilton-Jacobi equation, we obtain the critical impact parameters of unstable spherical photon orbits and construct the shadow contours for a distant observer. The rotation parameter mainly shifts and distorts the shadow, whereas the Hernquist halo enlarges the photon capture region and increases the apparent shadow size. Comparing the area-equivalent shadow diameter with the Event Horizon Telescope measurements of Sgr A$^\ast$ and M87$^\ast$, we constrain the dimensionless halo parameter $\hat{\rho}=M^2\rho$. The strongest restriction comes from Sgr A$^\ast$, giving $\hat{\rho}\sim(2.7-3.8)\times10^{-3}$ at $1\sigma$ and $\hat{\rho}\sim(4.1-5.2)\times10^{-3}$ at $2\sigma$. We also analyze strong- and weak-field gravitational lensing. In the strong-field regime, the halo shifts the unstable photon orbit and critical impact parameter, controlling the logarithmic deflection angle and the position of relativistic images. In the weak-field regime, the halo contributes already to the leading bending angle and enhances deviations from Kerr as $\rho$ grows. From the Einstein ring of ESO325-G004, we further obtain $0\leq\hat{\rho}\lesssim0.00939$ at $1\sigma$ and $0\leq\hat{\rho}\lesssim0.01963$ at $2\sigma$.

Figures

Figures reproduced from arXiv: 2607.08650 by the authors.

Figure 1
Figure 1. Effective potential V+ for timelike equatorial geodesics. The left panel shows the effect of varying the rotation parameter a at fixed ρ, while the right panel displays the effect of varying the Hernquist density parameter ρ at fixed a. The spin produces a stronger deformation in the near-horizon region, whereas the halo contribution changes the radial profile through ∆(r). where V±(r) = B(r)L ± p B2 (r)L2 + C(r) [A… view at source ↗
Figure 2
Figure 2. Effective potential V− for timelike equatorial geodesics. The two panels use the same parameter choices as in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Radial acceleration ¨r+ for null geodesics evaluated at the upper turning branch. The left panel shows the dependence on the Hernquist density parameter ρ at fixed a, while the right panel shows the dependence on the rotation parameter a at fixed ρ. In the displayed range, ¨r+ < 0, so the corresponding turning branch is associated with inward radial acceleration. Similarly, for a turning point on the lower branch, E… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Representative three–dimensional geodesic trajectories around the rotating black hole immersed in a Hernquist DMH. The black surface represents the event horizon, whereas the red surface represents the stationary limit surface. The region between these two surfaces cor…
Figure 5
Figure 5. Figure 5: Polar extension of the photon shell. The surfaces show the angular turning points θ+ and θ−, together with the allowed polar interval θ+ − θ−, for spherical photon orbits around the rotating black hole immersed in the Hernquist DMH. The deformation of the surfaces refl…
Figure 6
Figure 6. Figure 6: Shadow contours for an equatorial observer, θ0 = π/2. The dashed gray circle denotes the Schwarzschild reference shadow, Rsh = 3√ 3M. Left panel: shadows obtained for different values of the rotation parameter a with fixed ρ = 0.001. Increasing a displaces the shadow a…
Figure 7
Figure 7. Figure 7: Upper bounds on the dimensionless Hernquist halo parameter ˆρ = M2ρ obtained from the EHT angular size measurements of Sgr A∗ (left panel) and M87∗ (right panel). The solid and dashed curves correspond, respectively, to the 1σ and 2σ upper–size constraints. The shaded …
Figure 8
Figure 8. Figure 8: Left: Unstable circular photon orbit radius rp/M versus a/M for selected DMH parameter values ˆρ. Right: Corresponding critical impact parameter up/M as a function of a/M leads us to the following expression of impact parameter L E = u(r0) = ±r0(r0 + 2)r a 2 + r0(r 2 0…
Figure 9
Figure 9. Figure 9: Deflection angle αD versus impact parameter u for the new axisymmetric black holes immersed in Hernquist DMH (colored curves) compared to Kerr (ˆρ = 0, black). The logarithmic divergence near up (inset) depends on ˆρ. in a series around the photon sphere radius [69, 70…
Figure 10
Figure 10. Figure 10: Strong lensing observables for supermassive black holes Sgr A* (left) and M87* (right). Top: Angular position θ∞ of the photon ring versus spin a for different DMH parameter ˆρ. Bottom: Angular separation s between first and higher-order images versus a for different …
Figure 11
Figure 11. Figure 11: Flux magnification ratio rmag as a function of spin a for different DMH parameter ˆρ. relativistic images, labeled by p and q, that appear on the same side of the lens, the corresponding time delay is given by [73] ∆Tp,q ≈ 2π(p − q)up. (59) 23 [PITH_FULL_IMAGE:figure…
Figure 12
Figure 12. Figure 12: The deviation in weak lensing deflection angle δαD(u) = αD(u) − αD(u)|Kerr vs impact parameter u of the new axisymmetric black holes immersed in Hernquist DMH. source, lens, and observer are perfectly aligned, while both the source and the observer are located in regi…
Figure 13
Figure 13. Figure 13: The estimation of angular radius of Einstein ring θE vs DMH parameter ˆρ with the Hubble Space Telescope observations. The uncertainties in 1σ and 2σ confidence levels are represented by green and light orange shaded regions, respectively. We kept a = 0.9 geometry. Th…

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