A Neutron Star Hidden Inside a Black Hole

Chen Tan; Yong-Qiang Wang

REVIEW 3 major objections 4 minor 56 references

A Neutron Star Hidden Inside a Black Hole

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A dark matter halo can enclose a neutron star in an event horizon while the star remains a regular, non-singular configuration inside.

desk verdict The TOV numerics are solid but the black-hole interpretation collapses: with their metric ansatz, the negative g_rr^{-1} shell is a signature change, not an event horizon. read the letter →

arxiv 2608.06224 v1 pith:3Y7S7FZJ submitted 2026-08-06 gr-qc astro-ph.HEhep-th

Chen Tan , Yong-Qiang Wang This is my paper

classification gr-qcastro-ph.HEhep-th PACS 04.40.Dg04.70.-s95.35.+d

keywords darkmatteradmixedneutronstarsEinastodensityprofilemodifiedTolman–Oppenheimer–VolkoffequationsregularblackholeseventhorizonformationstarofstateBSk19SLy4

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 tries to establish that a dark-matter halo described by the Einasto profile, with radial pressure equal to minus its energy density, can wrap an ordinary neutron star in an event horizon without destroying it. Solving the two-fluid Tolman–Oppenheimer–Volkoff equations for the baryonic star plus the halo, the authors find a parameter window in which the metric component $g_{rr}^{-1}$ dips below zero in a shell outside the stellar surface and then returns to positive values: an inner and outer horizon surrounding a regular neutron star. The configuration appears for both the BSk19 and SLy4 nuclear equations of state, and the paper argues it is numerically robust. If correct, it gives a concrete model of structure persisting inside a black hole, which is a new perspective on what hides behind a horizon.

What carries the argument

The argument runs on the modified TOV system (17)–(18), built from two-fluid stress-energy conservation and the metric function identification $e^{-2\beta}=g_{rr}^{-1}=1-2G m_{\rm all}/(c^2 r)$, where $m_{\rm all}$ sums the baryonic and dark-matter enclosed masses. The dark matter is the Einasto profile $\rho_d(r)=\rho_0 e^{-(r/h)^{1/n}}$ with the anisotropic equation of state $p_r^{(d)}=-\rho_d c^2$, which is smooth at the center and supplies negative radial pressure; the transverse pressure is fixed by conservation. The sign of $g_{rr}^{-1}$ is the horizon criterion: positive through the star, negative in a shell outside it, then positive again, defining the inner and outer horizons. This machinery converts the question of whether a neutron star can be hidden inside a black hole into a numerical scan over three halo parameters.

What would settle it

At the radius $r_h$ where $1-2G m_{\rm all}/(c^2 r_h)=0$, compute the metric component $g_{tt}(r_h)$ and the combination $m_{\rm all}(r_h)+4\pi r_h^3 p_{\rm all}(r_h)/c^2$. If $g_{tt}$ does not vanish there, or that combination is nonzero, the Einstein equations are singular at $r_h$, so the sign change of $g_{rr}^{-1}$ is not a smooth event horizon and the neutron-star-in-a-black-hole interpretation fails.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the coexistence of a neutron star with an Einasto dark-matter halo whose radial pressure satisfies $p_r^{(d)}=-\rho_d c^2$ produces, for a range of halo parameters, a static, spherically symmetric spacetime in which $g_{rr}^{-1}=1-2G m_{\rm all}/(c^2 r)$ changes sign outside the stellar surface while the neutron star solution continues to exist as a regular interior configuration. The region of negative $g_{rr}^{-1}$ is bounded by an inner and outer horizon, and the star is fully contained within it; the paper calls this a 'neutron star in a black hole.' It occurs for two different equations of state and is not tied to a specific EOS. The paper also maps the three solution regimes—ordinary dark-matter admixed neutron star, neutron star in a black hole, and no static solution—in slices of the Einasto parameter space and reports that at horizon-forming densities the dark matter mass is comparable to the neutron star mass.

Load-bearing premise

The whole result rests on the assumption that where the metric component $g_{rr}^{-1}$ crosses zero the spacetime remains smooth, so the crossing is a real horizon rather than a coordinate or signature failure; that smoothness is asserted, not verified in the paper.

Editorial extensions

If this is right

For a window of Einasto halo parameters, the neutron star solution persists while $g_{rr}^{-1}$ dips below zero in a shell outside the star, so the star can sit wholly inside a horizon.
The configuration appears with both the BSk19 and SLy4 equations of state, so within this model it does not hinge on a specific nuclear equation of state.
At the horizon-forming densities, dark-matter mass and neutron-star mass are comparable, so the horizon is a joint effect of both components.
The parameter space has three regimes—ordinary dark-matter admixed neutron star, neutron star in a black hole, and no static solution—and increasing the halo mass on a static branch favors horizon formation.
The results are stable under changes in ODE solver tolerance, step size, and solver type, so the negative $g_{rr}^{-1}$ region is not a numerical artifact according to the paper's checks.

Reading between the lines

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

If the horizon interpretation survives a check of $g_{tt}$ and the regularity condition, the configuration gives a concrete two-horizon geometry whose interior contains baryonic matter; one could search for differences in quasinormal-mode spectra or tidal deformability relative to a vacuum Schwarzschild black hole.
The required halo densities far exceed current dark-matter predictions, so the paper's phenomenon is a proof-of-principle; a natural extension is to find dark-matter equations of state that produce the same negative-pressure effect at lower densities.
Stability under radial perturbations is unexamined; if these configurations are unstable, they would quickly collapse or migrate to another branch, which is a testable dynamical prediction.

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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

The headline is: the TOV numerics are fine, but the central 'neutron star in a black hole' claim is not supported by the paper's own metric ansatz. The authors find a shell where 1 - 2Gm_all/c^2r is negative and call it an event horizon, but with ds^2 = -e^{2α}c^2 dt^2 + e^{2β} dr^2 + ..., g_tt is always negative for real α. A horizon would require g_tt to vanish at a null surface. Instead, the negative g_rr^{-1} region means g_rr would be negative, which is impossible for real β, so the ansatz breaks down there. The paper never checks g_tt, never verifies the regularity condition at the zeros, and never constructs a global extension. The interior of the shell, if the metric were continued, would have signature (-,-,+,+), not a Lorentzian black hole interior.

What the paper does well: it applies the Konoplya–Zhidenko dark matter model (Einasto profile with p_r = -ρ c^2) to two-fluid TOV for the first time, and the numerical work is careful. The phase diagrams and M–R relations for BSk19 and SLy4 are computed with a nice robustness check against solver tolerance and step size. The authors also honestly state that the required halo densities are far above anything astrophysically plausible. That part is credible.

The soft spot is the interpretation. The sign change in 1 - 2m/r is a necessary but not sufficient condition for a horizon. In the standard Schwarzschild solution, g_tt and g_rr^{-1} vanish together. Here they don't. The authors look only at g_rr^{-1} and ignore g_tt. That is a load-bearing gap. The stress-test note is accurate: outside the star, c^2 m_all + 4π r^3 p_all is not zero at the zeros of 1-2m/r, so the α equation would diverge there. No regular horizon exists with their assumptions.

My recommendation: the paper should not be accepted as is, but it deserves a serious referee. The underlying framework is solid and the error is instructive. I would send it to review with a note that the central claim needs to either be fixed by demonstrating a genuine horizon (which I doubt is possible in this ansatz) or reframed as a signature-change configuration. A desk reject would be defensible, but I'd rather give the authors the chance to correct it.

Referee Report

3 major / 4 minor

Summary. The paper studies static, spherically symmetric configurations of a neutron star described by the BSk19 and SLy4 equations of state, embedded in an Einasto dark matter halo with anisotropic pressures and the radial equation of state p_r^(d) = -ρ_d c^2. Solving the modified two-fluid TOV equations, the authors find parameter regions in which the formal metric function g_rr^{-1} = 1 - 2G m_all/(c^2 r) becomes negative in a shell outside the stellar surface. They interpret this sign change as the formation of an event horizon containing a regular neutron star, and they map the corresponding phase diagrams and mass-radius relations for both equations of state.

Significance. If the central interpretation were correct, the paper would present a striking and computable model of a regular black hole with baryonic matter inside the horizon, going beyond the usual vacuum or matter-free regular black hole constructions. The numerical work has clear strengths: the TOV integrations are standard, the mass function formula is correctly derived, the two-EOS comparison supports the structural part of the study, and the numerical robustness checks (ODE tolerance, step size, solver) are well documented. However, the central claim is not established: the paper never verifies that g_tt vanishes at the would-be horizon, never checks the horizon regularity condition, and the region where the formal expression 1 - 2G m_all/(c^2 r) is negative has a negative-definite (t,r) block, which is a signature change rather than a Lorentzian black-hole interior. The authors also honestly acknowledge that the required dark matter densities exceed astrophysical expectations by orders of magnitude, which further limits the physical significance of the configuration as presented.

major comments (3)

[§III.B, Eq. (15)] The criterion that a sign change of g_rr^{-1} marks an event horizon is not justified. With the metric ansatz (9), g_tt = -e^{2α} c^2 < 0 for real α and g_rr = e^{2β} > 0 for real β, so the quantity e^{-2β} = 1 - 2G m_all/(c^2 r) cannot be negative without making β imaginary. A region where 1 - 2G m_all/(c^2 r) < 0 therefore has a negative-definite (t,r) block and is a Euclidean-signature region, not the interior of a black hole. The paper never shows that g_tt = 0 at the same radii, and Figs. 5-6 plot -g_tt as positive across the shell, which contradicts the horizon interpretation.
[§II.B, Eq. (18)] Even if one attempted to interpret the zeros of 1 - 2G m_all/(c^2 r) as Killing horizons, the regularity condition is missing. At a genuine horizon the numerator c^2 m_all + 4π r^3 p_all must also vanish at the same radius, otherwise α' diverges there. Outside the neutron star, where p_all = -ρ_d c^2, this requires ρ_d(r_±) = c^2/(8π G r_±^2). The paper does not check this condition, and for the representative parameters of §III.B and Fig. 6 the required value is an order of magnitude larger than the actual Einasto density at the would-be crossing. Thus the solution does not satisfy the necessary condition for a regular Killing horizon.
[§III.B, Figs. 5-8] The paper's central claim requires a global causal analysis that is absent. No null geodesics are computed, no trapped surfaces are identified, no extension across the would-be horizons is constructed, and the asymptotic structure of the spacetime outside r_+ is not analyzed. The term 'event horizon' is therefore used without the supporting global definition, and the plotted shell where the formal expression changes sign cannot, on the evidence provided, be identified as a black-hole horizon.

minor comments (4)

[§II.B, Eq. (13)-(15)] The quantity m(r) = 4π∫ρ x^2 dx is called a mass function, but it is a coordinate-density integral rather than the Misner-Sharp or ADM mass. In particular, M_all = m_all(R) in §III.C excludes the dark matter outside the neutron star radius, while the total gravitational mass relevant at infinity would be m_all(∞). This distinction should be stated when presenting the mass-radius relations.
[§III.B, Figs. 5-6] The text repeatedly says 'g_rr^{-1} changes sign', but with the ansatz (9) the actual metric component satisfies g_rr^{-1} = e^{-2β} > 0 for real β. The authors should consistently write that the formal expression 1 - 2G m_all/(c^2 r) changes sign, and should state explicitly that this implies a breakdown of the real-β ansatz rather than a horizon.
[§III.B, Fig. 1] The schematic in Fig. 1 labels r_- and r_+ as 'horizons', but the text never defines these surfaces as apparent horizons, trapping horizons, or event horizons. A brief definition in the text would help avoid ambiguity.
[§IV, Conclusion] The concluding caveat that the required dark matter densities are far above current predictions is appreciated, but the abstract's statement that the configuration 'appears in both equations of state' may overstate the physical relevance; the two-EOS comparison shows model-independence only within the same highly idealized dark matter model.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the modified TOV integration is self-contained and no fitted quantity is relabeled as a prediction; the horizon interpretation is a correctness concern, not a circularity.

full rationale

The paper's derivation chain is not circular. The metric functions are obtained by integrating the modified TOV equations (17)-(18) with the baryonic BSk19/SLy4 equations of state and the external Einasto dark-matter source (Eqs. 6-8 from Ref. [1]); the plotted quantity e^{-2β}=1-2Gm_all/(c^2r) (Eq. 15) is then evaluated from the resulting mass functions. No parameter is fitted to a target horizon, no data subset is used to force the sign change, and no uniqueness theorem is imported. The self-citations [55] and [56] are used only as stylistic or programmatic analogies ('filled hard candy'-like structure; possible generalization of the EOS) and are not load-bearing for the central claim. The assertion that min(g_rr^{-1})<0 'signals the formation of an event horizon' is physically questionable because with ansatz (9) g_tt=-e^{2α}c^2 cannot change sign for real α and the paper does not check the regularity condition or a global extension, but that is an incorrect or unsupported physical interpretation, not a case of the output reducing to the input by definition.

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

The central claim rests on the exotic dark matter model (Einasto profile with p_r=-rho c^2) taken from Ref [1], on the two-fluid TOV framework, and on an unverified identification of the g_rr^{-1} sign change with an event horizon. The three Einasto parameters and the central baryonic density are free inputs scanned to produce the phenomenon.

free parameters (4)

Einasto central dark matter density rho0 = 0.1 to 0.4 times 10^19 kg/m^3 (scanned)
Controls the halo amplitude; the horizon appears only for sufficiently large rho0, so the central result depends on choosing values far above astrophysical dark matter densities.
Einasto scale radius h = 1 to 8 km (scanned)
Sets the radial extent of the halo; the location and existence of the negative g_rr^{-1} shell depend on h.
Einasto index n = 1/3 to 1.5 (scanned)
Sets the steepness of the density falloff; larger n broadens the region where g_rr^{-1} is negative.
Central baryonic density rho_c = 2.5 times 10^17 to rho_max (scanned)
Standard TOV boundary condition; scanned along the mass-radius curves in Section III.C.

assumptions (5)

domain assumption Dark matter is described by the Einasto profile with radial equation of state p_r^(d) = -rho_d c^2 (Eq. 7) and transverse pressure from conservation (Eq. 8).
Adopted from Ref [1]; no physical mechanism or observational support is given, and it is essential for the negative radial pressure that shapes the solutions.
domain assumption Baryonic and dark matter interact only gravitationally and are separately conserved.
Standard in two-fluid TOV models; supported by dark matter-baryon cross-section constraints cited in Sec. II.A.
ad hoc to paper The static, spherically symmetric metric ansatz (9) remains valid across the would-be horizon.
The ansatz with real alpha implies g_tt is always negative; a negative g_rr^{-1} then gives a non-Lorentzian metric, so the ansatz cannot cover the claimed black-hole shell without an unstated coordinate change.
ad hoc to paper A zero or sign change of g_rr^{-1} indicates an event horizon.
The paper uses this criterion in Sec. III.B but does not verify that g_tt vanishes at the same surface or that curvature invariants are finite. For the assumed ansatz, the sign change is a signature change rather than a proven horizon.
domain assumption The baryonic equations of state BSk19 [52] and SLy4 [53,54] are valid up to the causal limit rho_max.
Standard inputs from the literature; they affect the quantitative mass-radius relations but not the qualitative phenomenon.

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Cite this review

Pith. "Pith review of A Neutron Star Hidden Inside a Black Hole." pith.science (2026). https://pith.science/paper/3Y7S7FZJ

@misc{pith2026260806224,
  author       = {Pith},
  title        = {Pith review of: A Neutron Star Hidden Inside a Black Hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3Y7S7FZJ}},
  note         = {Machine review of arXiv:2608.06224}
}

abstract

We investigate dark matter admixed neutron stars in which the neutron star coexists with an anisotropic dark matter halo described by the Einasto density profile, a model recently shown to produce regular, singularity-free black hole solutions~[Phys. Rev. D \textbf{113}, 043011 (2026)]. Solving the modified Tolman--Oppenheimer--Volkoff equations with two different equations of state (BSk19 and SLy4), we find that the dark matter halo significantly alters the neutron star structure. Furthermore, for a specific range of halo parameters, $g_{rr}^{-1}$ changes sign outside the stellar surface, forming an event horizon with the neutron star persisting as a regular configuration inside it--- ``neutron stars in black holes''. This configuration appears in both equations of state and does not depend on a specific choice of the equation of state. The discovery of this configuration provides a new perspective and a concrete computable instance for the study of what lies inside a black hole.

Figures

Figures reproduced from arXiv: 2608.06224 by the authors.

**Figure 2.** FIG. 2: Radial pressure profiles (left) and metric functions [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: Radial pressure profiles (left) and metric functions (right) for different values of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4: Radial pressure profiles (left) and metric functions (right) for different values of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 7.** Figure 7: These diagrams, computed with the BSk19 equa [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 5.** Figure 5: FIG. 5: Approach to horizon formation as the central dark matter density [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6: Radial profiles of the normalized matter [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7: Phase diagrams of the three solution regimes in the Einasto parameter space, computed with the BSk19 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8: Radial profiles of the normalized matter [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9: Mass-radius relations for [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: FIG. 10: Mass–radius relations for [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11: Global minimum of [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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