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Static regular black holes with a fixed scalar are ruled out in analytic Horndeski theories: only Schwarzschild survives, and it is singular at the center.

2026-07-10 10:59 UTC pith:KIKBHVTH

load-bearing objection Clean, carefully scoped no-go: analytic static Horndeski cannot support regular asymptotically flat BHs; sGB is the unique marginal nonanalytic escape and still singular at the center.

arxiv 2607.08228 v1 pith:KIKBHVTH submitted 2026-07-09 gr-qc hep-phhep-th

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

Antonio De Felice , Shinji Tsujikawa This is my paper
classification gr-qc hep-phhep-th PACS 04.50.Kd04.70.Bw04.20.Jb
keywords Horndeski theoriesregular black holesno-hair theoremscalar-Gauss-Bonnethorizon stabilitycurrent conservationanalytic branches
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks whether four-dimensional Horndeski gravity can support a static, spherically symmetric black hole that is regular at the center, stable at the horizon, and asymptotically flat, with a time-independent scalar. It finds that the near-horizon branch on which the scalar kinetic term X stays nonzero is generically blocked by divergent propagation speeds or ghost and gradient instabilities. On the remaining regular branch, where X vanishes at the horizon, analyticity of the theory functions at the three X=0 endpoints (infinity, horizon, and center) reduces the leading scalar equation to a finite set of Taylor coefficients. For nondegenerate shift-symmetric theories this yields a nonperturbative current no-hair theorem: the scalar must be constant and the metric must be Schwarzschild, which is centrally singular for any nonzero mass. For non-shift-symmetric positive-power couplings the same exclusion holds on the small-coupling branch continuously connected to Schwarzschild. Marginal nonanalytic completions are classified; covariant regularity forces them into the scalar-Gauss-Bonnet chain, whose known hairy black holes still have singular centers. Thus, within the static analytic sector, regular black holes are excluded by nonexistence rather than by a local center instability.

Core claim

On the regular horizon branch X(rs)=0 of static spherically symmetric asymptotically flat Horndeski solutions with a time-independent scalar, analyticity of the Gi functions at the X=0 endpoints reduces the scalar equation to finite Taylor jets. For nondegenerate shift-symmetric theories this produces a nonperturbative current no-hair theorem: the radial current vanishes in both exterior and interior patches, the scalar is constant throughout, and the metric is Schwarzschild, which is centrally singular for nonzero ADM mass. Non-shift-symmetric positive-power couplings are excluded on the perturbative branch connected to Schwarzschild. The unique covariantly regular marginal nonanalytic comp

What carries the argument

The finite-jet reduction of the leading scalar equation at each X=0 endpoint, together with the local factorization of the radial current Jr = h φ' [Aq + O(h φ')] and the resulting conserved-current (or perturbative) no-hair argument that forces φ' = 0 throughout each connected regular patch.

Load-bearing premise

The theory functions are analytic (Taylor-expandable in non-negative integer powers of X) at the three X=0 endpoints; without that the finite-jet reduction and current uniqueness argument do not close.

What would settle it

An explicit static spherically symmetric asymptotically flat solution of a nondegenerate analytic shift-symmetric Horndeski theory that has X(rs)=0, a regular center with finite curvature invariants, and a non-constant scalar profile.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies static, spherically symmetric, asymptotically flat black holes with a time-independent scalar in four-dimensional Horndeski theory, requiring both stable horizons and regular centers. It shows that the near-horizon branch with X(rs) ≠ 0 is generically obstructed by divergent propagation speeds or ghost/gradient instabilities unless special simultaneous degeneracies hold (Sec. II). On the regular branch X(rs) = 0, analyticity of the Gi at the X = 0 endpoints (infinity, horizon, center) reduces the leading scalar equation to finite Taylor jets. For nondegenerate shift-symmetric theories this yields a nonperturbative current no-hair theorem: the scalar is constant and the metric is Schwarzschild, hence centrally singular for nonzero ADM mass (Sec. III, Eqs. (24)–(29)). For non-shift-symmetric positive-power couplings the exclusion holds on the perturbative branch continuously connected to Schwarzschild. Marginal nonanalytic departures are classified; covariant regularity uniquely selects the scalar–Gauss–Bonnet chain (Appendix A), whose hairy solutions evade the analytic current step but remain centrally singular.

Significance. If the result holds, the paper substantially closes the static-scalar Horndeski window for nonsingular black holes by combining horizon stability with center regularity, going beyond earlier no-hair and stability analyses that allowed singular centers. The nonperturbative current argument for shift-symmetric analytic theories (finite-jet reduction, current factorization, and ODE uniqueness) and the derivation that covariant regularity fixes the sGB chain uniquely (Appendix A) are clear technical strengths. Degenerate Xs ≠ 0 and nonanalytic branches are scoped carefully rather than ignored, and the assumptions (analyticity, static scalar, nondegeneracy, asymptotic flatness) are stated explicitly. The work is of direct interest to the modified-gravity and regular-black-hole communities and cleanly identifies which broader constructions (time-dependent scalars, DHOST, etc.) remain open.

minor comments (6)
  1. Sec. III, after Eq. (25): the nondegeneracy condition that the leading current coefficient A_q remains nonzero (so that Jr = 0 forces φ′ = 0 near each endpoint) is used throughout the uniqueness argument but is stated less explicitly than η ≠ 0 at infinity. A short parallel sentence would make the theorem statement cleaner.
  2. Sec. IV, paragraph on sGB: the claim that hairy sGB black holes remain centrally singular rests on the near-origin behavior of known solutions. A brief local expansion (or a more specific citation to the absence of a regular Taylor series with finite curvature) would put this on the same footing as the analytic no-go and help readers who are not specialists in sGB numerics.
  3. Sec. II, Eq. (11) and surrounding text: the distinction between local algebraic degeneracies at a single point (φs, Xs) and an identity in X is important and well made; a single sentence reminding the reader that the fractional-power example (12) is nonanalytic at the vacuum X = 0 would further reduce the chance of misreading the scope.
  4. Appendix A: the integration-by-parts identity (A10) and the independence of the structures R, P2, [φ], and the scalar term are clear; it would help to note explicitly that the same cancellation conditions are required for the quadratic action (or that they follow from the action-level cancellation), so that the regularity criterion is not limited to the background equations.
  5. Introduction and Conclusions: the comparison with NED regular black holes and with infinite-tower higher-curvature constructions is useful; a one-sentence pointer that those spherical reductions are only effective two-dimensional Horndeski (already stated) could be repeated in the conclusions for readers who skip the introduction.
  6. Notation: the radial current Jr and the combination Y ≡ r² √(f/h) Jr are introduced cleanly; ensuring that the same symbol for the current is used consistently between the main text and Appendix A (A2) would avoid a minor reader pause.

Circularity Check

0 steps flagged

No significant circularity: the analytic current no-hair and sGB uniqueness follow from explicit assumptions and local cancellations, not from self-defined or fitted inputs.

full rationale

The load-bearing chain for the strongest claim (nonperturbative current no-hair on the nondegenerate shift-symmetric analytic X(rs)=0 branch) is self-contained: analyticity at the three X=0 endpoints reduces the scalar equation to finite Taylor jets (Sec. III); the current factorizes as Jr=hϕ′[Aq+O(hϕ′)] (Eq. 25); shift symmetry sets Pϕ=0 so Y is constant; horizon regularity of Jr^{2}/h and center regularity of Y force the constant to vanish; nondegeneracy plus ODE uniqueness then force ϕ′=0 throughout each patch, yielding Schwarzschild and the Kretschmann singularity for M≠0. The marginal nonanalytic completion is likewise derived ab initio in Appendix A by power-counting the unique lower singular terms that can cancel independent nonregular structures (R, P2, [ϕ], scalar) generated by the quintic logarithm, without presupposing the sGB form; identification with sGB occurs only after the coefficients are fixed. Self-citations ([31,32] etc.) supply prior expressions for propagation speeds and stability conditions that are restated with explicit near-horizon expansions and serve as independent inputs for branch classification, not as definitions of the target no-hair or uniqueness results. No fitted parameters, self-definitional loops, or smuggled ansätze appear.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely inside the standard four-dimensional Horndeski action. No free parameters are fitted. The load-bearing extra assumptions are analyticity at X=0, nondegeneracy of the vacuum kinetic term, the static spherically symmetric ansatz with time-independent scalar, and the leading near-horizon stability conditions imported from earlier work. No new entities are postulated.

axioms (4)
  • domain assumption All Gi(ϕ,X) are analytic (non-negative integer powers of X) at the three X=0 endpoints (infinity, horizon, center).
    Imposed at the start of Sec. III; enables reduction of the leading scalar equation to finite Taylor jets.
  • domain assumption The vacuum kinetic coefficient η = G2,X(ϕ0,0) is nonzero (nondegenerate scalar mode).
    Stated after Eq. (25); excludes strong-coupling vacua from the current theorem.
  • domain assumption Background is static, spherically symmetric, asymptotically flat, with time-independent scalar and a nonextremal outer horizon.
    Ansatz (2) and horizon conditions in Sec. II; time-dependent and beyond-Horndeski cases are explicitly excluded.
  • domain assumption Leading near-horizon stability conditions (F>0, G>0, κr, κ, etc.) from prior even/odd-parity analyses hold.
    Used in Sec. II to obstruct the Xs≠0 branch; imported from Refs. [31,36–38].

pith-pipeline@v1.1.0-grok45 · 18067 in / 2301 out tokens · 30023 ms · 2026-07-10T10:59:15.434564+00:00 · methodology

0 comments
read the original 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.

discussion (0)

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