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arxiv: 2605.27170 · v1 · pith:YRFMMK3Fnew · submitted 2026-05-26 · 🌌 astro-ph.HE

Regular black hole solutions and the quark chemical potential at the QCD phase transition

G. Lambiase , A. Ovgun , V. Vertogradov This is my paper

Pith reviewed 2026-06-29 15:49 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords regular black holesQCDquark chemical potentialgravitational collapseanisotropic fluidblack hole interiorsphase transition
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The pith

Finite quark chemical potential does not produce regular black hole cores in the models studied.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether QCD matter with finite quark chemical potential can support regular black hole interiors in gravitational collapse. Two effective equations of state are coupled to an advanced Eddington-Finkelstein geometry, with the matter treated as an anisotropic fluid. Temperature is found from the conservation law and the mass function is reconstructed from Einstein equations. Both the chiral quark model and the cold-QGP model lead to singular centers. This suggests that an additional inner phase is needed for regularity.

Core claim

Motivated by prior work, the study couples two effective equations of state for QCD matter at finite temperature and chemical potential to the geometry of collapsing matter. Rather than assuming a regular mass profile, the radial temperature dependence is determined from the local conservation law and the mass function is reconstructed from the Einstein equations. In the three-flavor chiral quark model an exact Lambert-function solution exists but the physical coefficients select a singular near-center branch. In the cold-QGP mean-field model the temperature diverges near the center, making the source terms incompatible with a regular center. Thus finite quark chemical potential reshapes the

What carries the argument

The reconstruction of the mass function from the Einstein equations after determining the temperature profile via the local conservation law for the effective anisotropic fluid sourced by the QCD equations of state.

If this is right

  • In the chiral model the exact solution for temperature selects the singular branch.
  • In the cold-QGP model temperature divergence leads to incompatibility with the regular center condition.
  • Any regular black hole completion requires an additional inner vacuum-like phase.
  • Further microphysics beyond the two models is necessary for self-regularization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the models to include explicit phase transitions to a vacuum energy dominated region could produce regular solutions.
  • Applying the same reconstruction method to other high-density equations of state might reveal conditions under which regularity emerges.
  • The result suggests that regular black holes in collapse scenarios may require physics outside standard QCD thermodynamics at finite density.

Load-bearing premise

The two chosen effective equations of state accurately capture the thermodynamics at finite chemical potential during gravitational collapse.

What would settle it

A derivation using a different equation of state that incorporates an inner vacuum phase and yields a finite central mass function with regular geometry would show the conclusion does not hold for all possible microphysics.

Figures

Figures reproduced from arXiv: 2605.27170 by A. Ovgun, G. Lambiase, V. Vertogradov.

Figure 1
Figure 1. Figure 1: FIG. 1. Principal-branch Lambert- [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Near-center scaling of the mass function in the chiral model. The solid curve shows the singular correction [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: summarizes the critical temperature predicted by the two effective models. Both reproduce the expected qualitative trend Tc ∼ 145 MeV at µ = 0 and a monotonic decrease with increasing µ. The quantitative differences at large µ reflect the distinct treatments of the vacuum structure: the chiral model reaches an effective endpoint near µ ≈ 390 MeV, whereas the cold-QGP model remains smooth across the entire … view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Cold-QGP temperature profile obtained by numerically inverting the implicit relation ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Near-center mass function of the cold-QGP model. Panel (a): [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Kretschmann invariant [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Effective fluid profiles of the matched configuration. Panel (a): effective energy density [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: displays the lapse function f(r) = 1 − 2Meff(r)/r of the matched geometry for several representative parameter choices. Two features are worth emphasizing. First, f(0) = 1 for every matched configuration, a direct consequence of the regularity conditions (VII.4) and of the de Sitter-like scaling Meff(r) ∼ (Λeff/6) r 3 at the center; [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Explicit construction of the matched mass function [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
read the original abstract

Motivated by Refs.[1, 2], we investigate whether quantum chromodynamics (QCD)-inspired matter at finite quark chemical potential can dynamically support \emph{regular} black-hole interiors during gravitational collapse. To this end, we couple two effective equations of state, namely a three-flavor chiral quark model at finite temperature and chemical potential and a cold-QGP mean-field model with a dynamical gluon mass, to a spherically symmetric advanced Eddington--Finkelstein geometry. The matter source is treated as an effective anisotropic fluid. Rather than assuming a regular mass profile a priori, we determine the radial temperature dependence from the local conservation law and reconstruct the mass function from the Einstein equations. In the chiral model, the conservation equation admits an exact Lambert-function solution, but the physical coefficients select a singular near-center branch. In the cold-QGP model, the exact implicit temperature-radius relation drives the temperature to diverge near the center, causing the thermodynamic source terms and the reconstructed mass function to become incompatible with the regular-center condition. We therefore find that, within the effective framework adopted here, finite quark chemical potential reshapes the thermodynamics of the collapse phase but does not by itself provide a self-regularizing black-hole core. Any regular completion must invoke an additional inner vacuum-like phase or further microphysics beyond the two QCD-inspired equations of state considered in this work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper investigates whether two QCD-inspired equations of state at finite quark chemical potential (a three-flavor chiral quark model and a cold-QGP mean-field model with dynamical gluon mass) can dynamically support regular black-hole interiors. Treating the source as an anisotropic fluid in advanced Eddington-Finkelstein coordinates, the authors solve the local conservation law for the radial temperature profile T(r) (exact Lambert-W branch in the chiral case; implicit relation in the cold-QGP case), substitute into the EOS to obtain the stress-energy, and reconstruct the mass function m(r) via the Einstein equations. Both models yield singular central behavior, leading to the conclusion that finite chemical potential reshapes collapse thermodynamics but does not by itself produce a self-regularizing core; an additional inner vacuum-like phase or further microphysics is required.

Significance. If the derivations hold, the work supplies a concrete negative result within standard effective QCD frameworks, showing that finite mu alone is insufficient for regularity and thereby sharpening the requirements for matter-supported regular black holes. The explicit analytic temperature solutions and direct Einstein reconstruction (without a priori regular m(r) assumptions) constitute a methodological strength and a falsifiable diagnostic that can be checked against other EOS choices.

minor comments (3)
  1. The regularity diagnostic (finite m(0) with m'(0)=0) is standard, but a brief explicit statement of the coordinate system and the precise Einstein-equation component used for m(r) reconstruction would aid readers unfamiliar with advanced Eddington-Finkelstein coordinates.
  2. The abstract states that 'physical coefficients select a singular near-center branch' for the Lambert-W solution; the main text should tabulate the numerical values of those coefficients and the branch-selection criterion for reproducibility.
  3. A short comparison table listing the key thermodynamic quantities (energy density, pressure components, temperature behavior) at small r for both models would make the contrast between the two singular outcomes more immediate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary of our work and the recommendation of minor revision. The referee's description accurately captures the methodology and the negative result that finite chemical potential alone does not yield regular cores in the two QCD-inspired models considered.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its central negative result by solving the standard local conservation equation for T(r) (yielding an exact Lambert-W branch in the chiral model and an implicit relation in the cold-QGP model) and then substituting the resulting stress-energy into the Einstein equations to reconstruct m(r) in advanced Eddington-Finkelstein coordinates. Both routes produce singular central behavior under the supplied effective equations of state, with the regularity diagnostic (finite m(0) and m'(0)=0) following directly from the definitions. No parameter is fitted and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem or ansatz, and the derivation does not reduce to its inputs by construction. The treatment remains self-contained against the external Einstein and conservation equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard general-relativity conservation laws and two effective QCD models taken from prior literature; the abstract introduces no new free parameters, axioms beyond domain assumptions, or invented entities.

axioms (2)
  • domain assumption Spherically symmetric advanced Eddington-Finkelstein geometry
    Chosen to describe the collapse; stated in the setup paragraph of the abstract.
  • domain assumption Matter source treated as effective anisotropic fluid
    Standard modeling choice for coupling the QCD equations of state to Einstein equations.

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discussion (0)

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