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Static regular black holes in Horndeski theories: analytic no-go and nonanalytic obstructions

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Pith's one-line read Static regular black holes with a time-independent scalar are ruled out in analytic Horndeski theories and in their unique marginal nonanalytic completion.

desk verdict Clean static-scalar no-go in Horndeski: analytic X(rs)=0 forces Schwarzschild (hence singular centers), and covariant regularity uniquely pins the only marginal nonanalytic escape to sGB, which still fails regular centers. read the letter →

arxiv 2607.08228 v2 pith:KIKBHVTH submitted 2026-07-09 gr-qc hep-phhep-th

Antonio De Felice , Shinji Tsujikawa This is my paper · ORCID
classification gr-qchep-phhep-th PACS 04.50.Kd04.70.Bw04.20.Jb
keywords Horndeskitheoriesregularblackholesno-hairtheoremscalar-Gauss-Bonnethorizonstabilitycurrentconservationanalyticity
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 asks whether four-dimensional Horndeski theories can support static, spherically symmetric, asymptotically flat black holes that are regular at the center and stable at the horizon when the scalar is time-independent. Near a non-extremal horizon the branch with nonzero scalar kinetic term X is generically obstructed by divergent speeds or ghost/gradient instabilities. On the remaining regular branch X vanishes at the horizon; analyticity at the three X=0 endpoints (infinity, horizon, center) reduces the leading scalar equation to a finite Taylor jet. For nondegenerate shift-symmetric theories a conserved current then forces the scalar to be constant, so the metric is Schwarzschild and any nonzero mass produces a central singularity. Positive-power non-shift-symmetric couplings are likewise excluded on the perturbative branch connected to Schwarzschild. The only covariantly regular marginal nonanalytic completion is the scalar-Gauss-Bonnet chain; its known hairy black holes still terminate at a curvature singularity rather than a regular center. The result therefore excludes a large class of candidate regular black holes and isolates precisely which assumptions must be relaxed for any future construction.

What carries the argument

Local current factorization Jr = h ϕ′ [Aq + O(h ϕ′)] at each X=0 endpoint, combined with analytic finite-jet reduction of the scalar equation and (for shift-symmetric theories) conservation of the radial current charge; uniqueness of regular ODEs then extends ϕ′=0 throughout each connected patch.

What would settle it

An explicit static, spherically symmetric, asymptotically flat solution of a nondegenerate analytic Horndeski theory (or of the sGB completion) that has X(rs)=0, a regular center with finite curvature invariants, and a stable non-extremal horizon would directly falsify the claim.

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Extended reading notes

Core claim

On the regular branch X(rs)=0, nondegenerate shift-symmetric analytic Horndeski theories obey a nonperturbative current no-hair theorem: the scalar is constant and the metric is Schwarzschild, hence centrally singular for nonzero ADM mass. Non-shift-symmetric positive-power couplings are excluded on the perturbative branch continuously connected to Schwarzschild. Covariant regularity singles out the scalar-Gauss-Bonnet chain as the unique marginal nonanalytic completion; hairy black holes in that completion remain centrally singular.

Load-bearing premise

The vacuum scalar kinetic coefficient is nonzero and the leading current factor never vanishes or becomes singular away from the endpoints, so a constant-scalar solution extends uniquely through the whole exterior and interior.

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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

0 major / 4 minor

Summary. The paper studies static, spherically symmetric, asymptotically flat black holes with a time-independent scalar in four-dimensional Horndeski theories, requiring both horizon stability and a regular center. On the near-horizon branch with finite nonzero X(rs), leading odd- and even-parity stability conditions generically produce divergent radial speeds or a negative no-ghost product FKB2 unless three simultaneous algebraic degeneracies hold (Eq. 11); those special cases are set aside. On the regular branch X(rs)=0, analyticity at the X=0 endpoints (infinity, horizon, center) reduces the leading scalar equation to finite Taylor jets. For nondegenerate shift-symmetric theories the conserved radial current vanishes at both the regular horizon and the regular center, the local factorization Jr=hϕ′[Aq+O(hϕ′)] with Aq finite forces ϕ′=0 near the endpoints, and regular-ODE uniqueness extends this through each connected patch, yielding the Schwarzschild metric and a central singularity for nonzero ADM mass. For non-shift-symmetric positive-power couplings the same exclusion holds on the small-coupling branch continuously connected to Schwarzschild. Marginal nonanalytic terms are classified by covariant regularity; the unique completion is the scalar–Gauss–Bonnet chain (Appendix A), whose known hairy solutions remain centrally singular (Appendix B).

Significance. If the stated assumptions hold, the result supplies a clean, nonperturbative no-go for static regular black holes in a large and well-studied sector of Horndeski theory. The horizon obstruction is derived from explicit near-horizon expansions of the known odd/even speeds and no-ghost products; the analytic current argument combines standard conservation with local factorization and ODE uniqueness under nondegeneracy; and Appendix A derives the sGB chain by cancelling independent nonregular structures without presupposing the sGB form. These steps close a gap left by earlier no-hair and stability analyses that either allowed central singularities or treated only the small-coupling branch. The paper also correctly flags the excluded degenerate and nonanalytic cases (simultaneous horizon degeneracies, 4DEGB-type branches, strongly coupled Lovelock towers), so the scope is transparent. The work therefore provides a sharp benchmark for future constructions that relax staticity, analyticity, or the Horndeski restriction.

minor comments (4)
  1. In Sec. II the three simultaneous degeneracies (11) are correctly set aside, but a short clarifying sentence that an all-order cancellation would constitute a highly degenerate structural identity (rather than a local tuning) would help readers who might otherwise wonder whether higher-order terms could systematically cancel.
  2. Around Eqs. (25)–(26) the nondegeneracy assumption that the bracket Aq remains finite and nonzero away from the endpoints is stated, yet a one-line reminder that vanishing of Aq would itself signal a degeneracy of the kinetic sector would make the boundary of the theorem even more explicit.
  3. Appendix A, after Eq. (A11): the kinematic independence of the structures R, P2, [ϕ] and the pure scalar term is used to force each coefficient to vanish separately; a brief parenthetical that this holds for generic field configurations (as already implied) would forestall any pedantic objection.
  4. A few typographical inconsistencies appear (e.g., “ANAL YTIC”, “SCHW ARZSCHILD”, “NONANAL YTIC” in section headings; occasional missing spaces after commas in the arXiv metadata block). These are purely cosmetic.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: analytic no-go and sGB uniqueness are derived from stated assumptions, not forced by definition or load-bearing self-citation.

full rationale

This is a pure theoretical no-go paper with no fitted parameters and no empirical predictions that could reduce to inputs by construction. The central chain is self-contained: (i) Sec. II shows the Xs eq 0 horizon branch is generically obstructed by divergent speeds or FKB2 < 0 unless the three simultaneous degeneracies (11) hold (treated as special and outside the main argument); (ii) analyticity at the X = 0 endpoints reduces the leading scalar equation to finite Taylor jets (21)–(22); (iii) for nondegenerate shift-symmetric theories, current conservation plus Jr o 0 at both the regular horizon and regular center forces C = 0, the local factorization (25) with Aq finite and nonzero (secured by eq0 vacuum kinetic coefficient eq0) gives eq0 near the endpoints, and regular-ODE uniqueness extends eq0 through each connected patch, yielding Schwarzschild and Kretschmann eq0 for M eq 0; (iv) the non-shift-symmetric positive-power exclusion is correctly limited to the perturbative branch continuously connected to Schwarzschild; (v) covariant regularity uniquely fixes the marginal nonanalytic completion to the sGB chain (A13), derived in Appendix A without assuming the sGB form in advance; (vi) Appendix B then shows any nondegenerate analytic completion of that chain remains locally Minkowski at a regular center. Self-citations to prior stability/no-hair papers ([31, 32] etc.) supply background tools (current form, speed formulas) but are not load-bearing for the new regular-center and nonanalytic-uniqueness steps, which are derived here from the stated nondegeneracy assumptions. The reader’s weakest assumption (nondegeneracy of the X = 0 branch and non-vanishing of the current bracket) is the natural boundary of the theorem rather than a hidden circular gap. Score 1 only for ordinary self-citation of background tools; no reduction of the claimed result to its inputs by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The result is a conditional no-go inside classical Horndeski under static spherical symmetry. Load-bearing inputs are domain assumptions (static scalar, analytic Gi at X=0, nondegeneracy of kinetic sector, asymptotic flatness, nonextremal horizon) plus standard differential-equation uniqueness; no free parameters or new entities are introduced.

assumptions (6)
  • domain assumption Background is static, spherically symmetric, asymptotically flat, with time-independent scalar ϕ=ϕ(r) and nonextremal outer horizon f(rs)=h(rs)=0.
    Stated in Sec. II, Eq. (2) and surrounding text; excludes time-dependent and non-spherical branches by construction.
  • domain assumption Gi(ϕ,X) are analytic (Taylor expandable in non-negative integer powers of X and ϕ−ϕq) at the X=0 endpoints infinity, horizon, and center.
    Sec. III, Eq. (16); fractional/negative/log powers are deferred to Sec. IV.
  • domain assumption Nondegenerate vacuum kinetic coefficient η=G2,X(ϕ0,0)≠0 (and Z*≠0 at a regular center after diagonalizing scalar-metric mixing).
    Sec. III after Eq. (25); Appendix B Eq. (B12). Strong coupling or vanishing principal part is excluded.
  • standard math On nondegenerate analytic branches, local ϕ′=0 at endpoints extends uniquely through each connected regular patch by ODE uniqueness unless the current bracket vanishes or becomes singular.
    Invoked in Sec. III after Eq. (25) to promote local vanishing of ϕ′ to global constancy.
  • ad hoc to paper Special simultaneous horizon degeneracies G5,X=κr=κ=0 and exceptional nonanalytic Xs≠0 branches (e.g. 4DEGB) are set aside and not claimed to be covered by the no-go.
    Sec. II, Eqs. (11)–(12) and discussion of 4DEGB; scope limitation rather than a derived impossibility.
  • domain assumption Horndeski action (1) is the most general second-order scalar-tensor theory in four dimensions used as the dynamical framework.
    Opening of Sec. II; standard field content for the paper’s class.

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Pith. "Pith review of Static regular black holes in Horndeski theories: analytic no-go and nonanalytic obstructions." pith.science (2026). https://pith.science/paper/KIKBHVTH

@misc{pith2026260708228,
  author       = {Pith},
  title        = {Pith review of: Static regular black holes in Horndeski theories: analytic no-go and nonanalytic obstructions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KIKBHVTH}},
  note         = {Machine review of arXiv:2607.08228}
}
abstract

Regular black holes in Horndeski theories must have stable horizons and regular centers. We study static, spherically symmetric, asymptotically flat configurations with a time-independent scalar. The horizon branch on which the scalar kinetic term $X$ remains nonzero is generically obstructed by divergent propagation speeds or ghost/gradient instabilities, aside from special degeneracies. On the regular branch, where $X$ vanishes at the horizon, analyticity at the relevant $X=0$ endpoints reduces the leading scalar equation to finite sets of Taylor coefficients. For nondegenerate shift-symmetric theories this gives a nonperturbative current no-hair theorem: the scalar is constant and the metric is Schwarzschild, hence centrally singular for nonzero ADM mass. For non-shift-symmetric positive-power couplings, the corresponding exclusion applies to the perturbative branch continuously connected to Schwarzschild. We also classify marginal nonanalytic departures: covariant regularity fixes the scalar-Gauss-Bonnet chain as the unique marginal nonanalytic completion. Hairy black holes in this completion evade the analytic current step but remain centrally singular.

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