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A rotating black hole in a Hernquist dark matter halo: horizon geometry, thermodynamics, and quantum emission

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A rotating black hole embedded in a Hernquist dark matter halo has a larger event horizon, a lower Hawking temperature, and a longer evaporation timescale than the same black hole without the halo.

desk verdict A competent rotating black hole with a Hernquist-like term, but the 'Hernquist halo' identification is asserted, not shown — the authors need to compute the stress-energy tensor or reframe the metric as phenomenological. read the letter →

arxiv 2606.30962 v2 pith:3FA4CAC7 submitted 2026-06-29 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 · ORCID
classification gr-qchep-th MSC 83C5783C4783C15 PACS 04.70.Dy04.70.-s95.35.+d
keywords rotatingblackholeHernquistdarkmatterhaloNewman-JanisalgorithmHawkingtemperaturethermodynamicsquantumtunnelingevaporationergosphere
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 constructs the rotating counterpart of a Schwarzschild black hole surrounded by a Hernquist dark matter distribution and works out the consequences for horizon geometry, thermodynamics, and quantum emission. The central claim is that the Hernquist halo displaces the outer event horizon to larger radii, suppresses the Hawking temperature, increases the Bekenstein-Hawking entropy, and, in the weak-halo and slow-rotation regime, reduces the Hawking luminosity and prolongs evaporation. If correct, the results quantify how ambient dark matter alters the basic black-hole observables of temperature, entropy, and lifetime.

What carries the argument

The central object is the rotating metric of Eq. (18) with the radial function Δ(r) of Eq. (19), whose zeros define the horizons. The construction relies on the noncomplexification Newman-Janis prescription, which replaces the static radial functions by real functions F and H; choosing H=Σ≡r^2+a^2 cos^2θ eliminates the G_{rθ} Einstein-tensor component, yielding the explicit Boyer-Lindquist form. The horizon equation becomes a cubic, solved perturbatively in the small-ρ regime, and the Hawking temperature, entropy, heat capacity, and tunnelling rate are all expressed through Δ'(r_h), r_h, and a.

What would settle it

Compute the full Einstein tensor of the metric in Eq. (18) and check whether the effective stress-energy tensor reproduces the Hernquist density in the slow-rotation limit and satisfies the weak or null energy conditions; alternatively, compute the Hawking temperature through an independent Euclidean path-integral or canonical-ensemble method and compare it with the value obtained from Δ'(r_h)/(4π(r_h^2+a^2)).

Watch

Extended reading notes

Core claim

Starting from a static metric with f(r)=1-2M/r-4πρ_s r_s^3/(r+r_s), the paper uses the noncomplexification formulation of the Newman-Janis algorithm to produce an axisymmetric geometry with Boyer-Lindquist form and horizon function Δ(r)=r^2+a^2-2Mr-4πρ_s r_s^3 r^2/(r+r_s). For the rest of the paper the halo scale is specialized to r_s=2M, giving Δ(r)=r^2+a^2-2Mr-32πρ M^3 r^2/(r+2M). The authors show that the positive halo parameter ρ shifts the outer horizon outward, enlarges the horizon area and entropy, lowers the Hawking temperature, introduces a Davies-type divergence in the heat capacity, and strengthens frame dragging. In the weak-halo, slow-rotation regime, the leading ρ and a^2 corre

Load-bearing premise

The load-bearing premise is that the rotating metric obtained by the noncomplexification procedure is actually a solution of Einstein's equations with a matter content that represents a rotating Hernquist dark-matter halo; the paper asserts that H=Σ eliminates G_{rθ} but does not verify the full field equations or the energy conditions for the rotated spacetime.

Editorial extensions

If this is right

  • If the halo parameter is positive, the outer event horizon moves to larger radii, so black holes embedded in dark matter halos have larger horizons than isolated black holes of the same mass and spin.
  • The Hernquist halo lowers the Hawking temperature, which weakens the thermal emission spectrum and shifts the extremal (zero-temperature) boundary toward smaller black-hole masses.
  • The entropy increases because the horizon area grows, meaning the halo adds thermodynamic degrees of freedom to the black hole.
  • In the weak-halo and slow-rotation regime, the Hawking luminosity decreases and the evaporation time increases, so dark matter halos delay black-hole evaporation rather than accelerating it.
  • The heat capacity acquires a divergence that separates stable from unstable branches, indicating a Davies-type critical point whose location depends on the halo density.

Reading between the lines

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

  • If the construction is physically valid, the same noncomplexification procedure could be applied to other halo density profiles; the qualitative pattern of a larger horizon and cooler temperature may be generic for positive-density halos, but the sign of the luminosity correction may depend on the profile and rotation rate.
  • The paper's claim of luminosity suppression is explicitly restricted to the weak-halo, slow-rotation regime; the correction C_ρ changes sign for sufficiently rapid rotation, so one should not extrapolate the suppression to near-extremal spins.
  • The rotating metric's physical interpretation as a 'rotating Hernquist halo' remains unverified: the paper does not compute the full stress-energy tensor or check energy conditions, so a direct check of the field equations would settle whether the thermodynamic results describe a genuine physical spacetime or an effective one.
  • If greybody factors were computed, the spectral emission rates in Eqs. (123)-(125) would allow concrete predictions for observable Hawking-like signatures from black holes in galactic centers, providing a testable link between dark matter density and black-hole radiation.
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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

3 major / 5 minor

Summary. The manuscript constructs a stationary, axisymmetric metric by applying the Azreg-Aïnou noncomplexification algorithm to a static, spherically symmetric black hole surrounded by a Hernquist dark matter halo (Eqs. (1)–(19)). It then analyzes the horizon and ergoregion structure, ZAMO frame dragging, surface gravity, Hawking temperature, Bekenstein–Hawking entropy, heat capacity, Hamilton–Jacobi tunneling rates, occupation numbers, and Stefan–Boltzmann estimates of luminosity and evaporation time, with perturbative expansions in the halo density parameter ρ and in slow rotation. The Kerr and Schwarzschild limits are recovered in the appropriate limits, and the authors explicitly acknowledge several caveats (nonuniformity near extremality, need for greybody factors, and the r_s=2M specialization used in the quantitative sections).

Significance. If the source identification were established, this would be a useful addition to the growing literature on black holes embedded in dark-matter halos. The horizon enlargement, temperature suppression, entropy enhancement, Davies-type critical point, and delayed evaporation are concrete, falsifiable predictions. The paper's careful limiting checks and its explicit statements about the validity regimes of the perturbative expansions are commendable. However, the central physical claim that Eqs. (18)–(19) describe a rotating Hernquist-halo black hole is not verified: the energy-momentum tensor is never computed, and the quantitative parts of the paper fix r_s=2M despite the abstract's mention of independent r_s and ρ.

major comments (3)
  1. [Section II, after Eq. (13); Eqs. (18)–(19)] The central physical claim is that Eq. (18) is a rotating Hernquist-halo black hole, but the matter source is never verified. The paper only states that H=Σ eliminates G_{rθ}; no component of G_{μν} or T_{μν} is computed. The Azreg-Aïnou algorithm does not guarantee that the generated metric solves Einstein's equations with an energy-momentum tensor corresponding to a Hernquist density profile; for seeds with g_tt=-1/g_rr the rotating source is generally anisotropic and θ-dependent. The authors should compute T_{μν}=G_{μν}/(8π), check the a→0 limit against the static seed's EMT, and verify the energy conditions in the parameter ranges used in Figs. 2–14. Without this, the 'dark matter halo' interpretation and all subsequent thermodynamic and emission results rest on an unverified effective geometry.
  2. [Section III A, Eq. (20); Section VI C] The abstract states the analysis is for independent halo parameters ρ and r_s, but all quantitative results set r_s=2M. The r_s-dependence is never studied; the manuscript itself acknowledges this limitation in Section VI C. Since r_s is the halo scale that characterizes the Hernquist profile, fixing r_s=2M ties the halo to the black hole mass and reduces the claimed two-parameter family to a one-parameter family. The authors should either perform the full r_s analysis or revise the abstract and conclusions to describe the single-parameter specialization.
  3. [Section IV C, Eq. (69)] The thermal stability analysis uses C_V = T(∂S/∂T)|_{a,ρ} and interprets its divergence as a Davies-type critical point. However, for a non-vacuum spacetime with matter sources, the first law is not established, and it is not clear that M is the relevant thermodynamic potential or that fixing a and ρ defines a canonical ensemble. Since the local stability claim is a central result, the authors should derive the applicable first law for the black-hole-plus-halo system (including matter contributions) or explicitly state the assumptions under which Eq. (69) is the correct heat capacity.
minor comments (5)
  1. [After Eq. (18)] The text refers to the 'rotating extension of the bumblebee black hole'; this should be 'Hernquist-halo black hole'.
  2. [Eq. (56)] The symbol ρ is used both for the Hernquist density parameter and, in Eq. (56), as a new azimuthal coordinate; this is confusing. Use a different symbol for the coordinate.
  3. [Captions of Figs. 6 and 7] 'paramters' and 'botom' should be 'parameters' and 'bottom'.
  4. [Eq. (109)] In the Stefan–Boltzmann formula, g⋆ appears as a multiplicative constant; clarify whether this is the effective number of species or a sum of spin degeneracies, and define ε_em.
  5. [Abstract and Section V] The occupation number in Eq. (98) and the spectral rates in Eqs. (127)–(128) are explicitly blackbody estimates; this is acknowledged in the text, but the abstract's phrase 'quantum emission' could be misread as a full greybody computation. A brief clarification would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the horizon, thermodynamic, and tunneling results follow algebraically from the externally imported static seed and the Azreg-Aïnou algorithm; the unverified matter content and the fixed r_s choice are correctness gaps, not circularity.

full rationale

The paper's derivation chain is not circular in any of the enumerated senses. The static Hernquist black hole seed f(r) in Eq. (2) is imported from an external source, Ref. [66], and the rotating metric, Eqs. (18)-(19), is obtained by applying the Azreg-Aïnou noncomplexification algorithm with H = Σ and F = (r^2 f + a^2 cos^2 θ)/Σ. From the resulting Δ(r), the horizon displacements, stationary limits, frame dragging, surface gravity, Hawking temperature, entropy, heat capacity, tunneling rate, occupation number, and Stefan-Boltzmann luminosity are all obtained by explicit algebra or standard semiclassical identifications. No parameter is fitted to the quantities that are later called predictions; ρ and a are free external parameters, and the ρ→0 and a→0 limits are independently checked against Kerr and Schwarzschild. The self-citations that appear are background references to tunneling methods and thermodynamics and never carry the load of the central argument. The real weaknesses are non-circular: the rotating metric's Einstein tensor and stress-energy tensor are never computed, so the physical identification of Eq. (18) with a rotating Hernquist halo is not verified, and all quantitative results fix r_s = 2M despite the abstract mentioning independent r_s; the paper itself acknowledges this limitation in Sec. VI C. The stray phrase 'rotating extension of the bumblebee black hole' before Eq. (18) is an apparent copy-paste artifact and does not affect the mathematical derivation.

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

The central claim rests on the validity of the imported static seed and on the unverified assumption that the Azreg-Aïnou rotation procedure produces a spacetime sourced by a rotating Hernquist halo. The r_s=2M specialization is an ad hoc reduction of the parameter space advertised in the abstract.

free parameters (3)
  • ρ (Hernquist density parameter)
    Treated as an independent input and as a small perturbative parameter; figures use values up to 0.5. Not fitted to data.
  • r_s (halo scale radius) = 2M (set by hand)
    The halo scale is specialized to r_s=2M in Section III, contrary to the abstract's claim of an independent r_s; this restricts the rotating solution.
  • ε_em (emissivity) = 1 (ideal blackbody limit)
    Phenomenological frequency-averaged emissivity introduced in Eq. (109) for the Stefan-Boltzmann luminosity; not derived.
assumptions (4)
  • domain assumption The static Hernquist-halo black hole seed (Eqs. 1–2) from Ref. [66] is an exact solution of the Einstein equations with the Hernquist density profile.
    The paper imports this seed without re-deriving it; all rotating results reduce to it when a→0.
  • domain assumption The Azreg-Aïnou noncomplexification prescription generates a valid rotating solution with the same matter content for any static seed.
    Used in Section II to construct Eq. (18); the paper checks G_{rθ}=0 via H=Σ but does not verify the full field equations or the stress-energy interpretation.
  • ad hoc to paper Specializing the rotating metric to r_s=2M does not lose the essential physics of a Hernquist halo.
    Section III Eq. (20) fixes r_s=2M; this is a modeling choice not justified astrophysically and it limits the abstract's claimed two-parameter independence.
  • domain assumption The Stefan-Boltzmann law with horizon area as effective emitting area approximates the Hawking luminosity.
    Eq. (109) and Section VI use this to estimate lifetimes; the text acknowledges greybody factors are required for exact fluxes.

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Pith. "Pith review of A rotating black hole in a Hernquist dark matter halo: horizon geometry, thermodynamics, and quantum emission." pith.science (2026). https://pith.science/paper/3FA4CAC7

@misc{pith2026260630962,
  author       = {Pith},
  title        = {Pith review of: A rotating black hole in a Hernquist dark matter halo: horizon geometry, thermodynamics, and quantum emission},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FA4CAC7}},
  note         = {Machine review of arXiv:2606.30962}
}
abstract

We investigate the geometrical, thermodynamic, and quantum emission properties of a rotating black hole immersed in a Hernquist dark matter halo. Starting from a static black hole spacetime surrounded by a Hernquist distribution, we construct its rotating counterpart through the noncomplexification formulation of the Newman-Janis algorithm and analyze the modifications induced by the independent halo parameters $\rho$ and $r_s$ and the rotation parameter $a$. The horizon structure is determined from the roots of the radial function $\Delta(r)$, while the stationary limit surfaces and the corresponding ergoregions are obtained from the condition $g_{tt}=0$. We show that the Hernquist contribution displaces the outer event horizon toward larger radii and modifies the size of the ergoregion, whereas rotation controls the oblateness of the horizon and the strength of frame dragging. We further derive the surface gravity, Hawking temperature, Bekenstein-Hawking entropy, and heat capacity. The quantum tunneling rate is obtained from the Hamilton-Jacobi method, leading to the corresponding occupation number and a thermal estimate of the particle creation density. Finally, we estimate the Hawking luminosity and evaporation timescales within a Stefan-Boltzmann approximation. All standard Kerr and Schwarzschild results are recovered in the appropriate limiting cases.

Figures

Figures reproduced from arXiv: 2606.30962 by the authors.

Figure 1
Figure 1. Radial behavior of ∆(r) for different values of the model parameters. In the left panel, we fix a = 0.1 and M = 1 and vary the Hernquist parameter ρ, whereas, in the right panel, we set ρ = 0.1 and M = 1 and vary the rotation parameter a. Increasing ρ shifts ∆(r) downward, while increasing a shifts it upward. its first-order correction becomes singular in this limit. The extremal configuration must instead be obtain… view at source ↗
Figure 2
Figure 2. Behavior of the outer event horizon radius rh for different values of the Hernquist parameter ρ in the left panel and of the rotation parameter a in the right panel. The remaining parameters are fixed at the values indicated in the respective panels [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Three-dimensional representation of the outer event horizon radius rh in the (a, ρ) parameter space for a fixed value of the black hole mass. Since Σ(r, θ) is nonvanishing outside the curvature singularity, the stationary limit surfaces follow from ∆(r) − a 2 sin2 θ = 0, or, equivalently, r 2 f(r) + a 2 cos2 θ = 0. After multiplying by (r + 2M), the corresponding cubic equation becomes E(r, ρ, θ) ≡ [PITH_FULL_IMAGE… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Two-dimensional representations of the outer event horizon for selected values of the Hernquist and rotation parameters. The parameter choices are indicated in the respective panels. The resulting first-order correction is r ± e,1 (θ) = 16πM3 [PITH_FULL_IMAGE:figures/…
Figure 5
Figure 5. Figure 5: Ergosphere profiles for different configurations of the system. In the left panel, we fix ρ = 0.001 and M = 1 and vary the rotation parameter a, whereas, in the right panel, we set a = 0.5 and M = 1 and consider different values of ρ. In [PITH_FULL_IMAGE:figures/full_…
Figure 6
Figure 6. Figure 6: Three-dimensional representations of the event horizon and ergosphere for ρ = 0.001 and different values of the rotation parameter a: a = 0.6 (top left), a = 0.7 (top right), a = 0.8 (bottom left), and a = 0.9 (bottom right). For the rotating black hole surrounded by t…
Figure 7
Figure 7. Figure 7: Three–dimensional representations of the event horizon and ergosphere for a = 0.9 and different values of the dark matter parameter ρ: ρ = 10−4 (top left), ρ = 5 × 10−4 (top right), ρ = 10 × 10−4 (bottom left), and ρ = 20 × 10−4 (bottom right). The Kerr result is recov…
Figure 8
Figure 8. Figure 8: The angular velocity ω(r, ρ, a) as a function of the radial coordinate r for different values of ρ (left panel) and the rotation parameter a (right panel). The influence of the Hernquist parameter can also be established directly from Eq. (49). For positive a, M, and r…
Figure 9
Figure 9. Figure 9: Hawking temperature T(ρ, a, M) as a function of the black hole mass M for different values of the Hernquist dark matter parameter ρ, with the rotation parameter fixed at a = 0.1. The curves were obtained from the complete temperature and horizon equations. Increasing ρ…
Figure 10
Figure 10. Figure 10: Entropy S(ρ, a, M) as a function of the black hole mass M for different values of the Hernquist dark matter parameter ρ, with the rotation parameter fixed at a = 0.9. The entropy increases with both M and ρ, reflecting the enlargement of the event horizon area induced…
Figure 11
Figure 11. Figure 11: Heat capacity CV (ρ, a, M) as a function of the black hole mass M for different values of the Hernquist dark matter parameter ρ, with the rotation parameter fixed at a = 0.4. The divergence separates locally stable configurations with CV > 0 from unstable configuratio…
Figure 12
Figure 12. Figure 12: Particle occupation number ⟨NωJ ⟩ in the low–frequency regime for small values of J. The top panel shows the effect of varying the Hernquist parameter ρ at fixed a, whereas the bottom panel shows the effect of varying the rotation parameter a at fixed ρ. In both panel…
Figure 13
Figure 13. Figure 13: Particle creation density dn/dω as a function of the frequency ω. The top panel shows the effect of varying the Hernquist parameter ρ at fixed a, whereas the bottom panel shows the effect of varying the rotation parameter a at fixed ρ. Therefore, the tunnelling method…
Figure 14
Figure 14. Figure 14: Particle number density n as a function of the Hernquist parameter ρ for different values of the rotation parameter a. T can produce a more pronounced modification in the number of particles created. In [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]

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