REVIEW
A Geometric No-Go Theorem for Regular Black Holes in Scalar-Tensor Theories
General Relativity and Quantum Cosmology · Submitted 4 Nov 2026 · 82 references
Pith's verdict: A clean geometric no-go for two-horizon regular black holes in scalar-tensor theories; the central identity holds up for the covariant families, with a narrow but real gap in the SCG branch that needs closing.
T0 · AI review
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Referee report
The paper
The paper claims
The paper proves a geometric no-go theorem: in a broad class of scalar-tensor theories, no static, spherically symmetric regular black hole supported by a single radial scalar field can be stable, because the two-horizon geometry forces…
The reading
What carries the argument
The load-bearing object is the ratio $q(\tau) = a(\tau)/b(\tau)$ of the longitudinal scale to the sphere radius inside the trapped region of a two-horizon regular black hole, together with the turning-point identity $(\ddot{q}/q)|_{\dot{q}=0} = c_{\rm odd}^2/b^2$, which is Eq. (10) of the paper. The geometry part is elementary: inside the T-region the metric takes homogeneous Kantowski–Sachs form with $q>0$, and $q\to 0$ at both bounding horizons, so compactness forces an interior local maximum with $\dot{q}=0$ and $\ddot{q}\le 0$. The dynamics part is the claim that, across a broad class of scalar-tensor theories (defined as TPST precisely by this property), the background equations evaluated at such a turning point identify $\ddot{q}/q$ with the squared speed $c_{\rm odd}^2 = F_T/K_T$ of odd-parity tensor perturbations, where $K_T$ and $F_T$ are the kinetic and gradient coefficients of the physical odd-tensor mode. A nondegenerate Lorentzian tensor cone requires $c_{\rm odd}^2>0$, which through the identity demands $\ddot{q}>0$—a local minimum—contradicting the geometric maximum. The proof machinery includes the volume–shape parametrization, the background shape equation, and the off-shell radial shape identity for quadratic HOST theories, which is derived without any degeneracy relation and without dividing by $J'$ or $\phi'$.
Extended reading notes
Core claim
The paper's central claim is the turning-point no-go theorem: in any TPST theory, no static, spherically symmetric regular black hole supported solely by a real radial scalar field can possess both a bounded two-horizon trapped region with $q\to 0$ at both horizons and a nondegenerate Lorentzian odd-parity tensor cone throughout the interior. The argument runs through a single identity, $(\ddot{q}/q)|_{\dot{q}=0} = c_{\rm odd}^2/b^2 = F_T/(K_T b^2)$, where $q=a/b$ is the ratio of the longitudinal scale factor to the sphere radius inside the Kantowski–Sachs trapped region and $c_{\rm odd}^2$ is the squared speed of odd-parity tensor perturbations. Geometry alone forces $q$ to vanish at the inner and outer horizons and hence to peak somewhere between them, giving $\ddot{q}\le 0$; the identity then forces $c_{\rm odd}^2\le 0$ there, meaning the odd-tensor sector must suffer a gradient instability, an odd ghost, or a degenerate cone. The paper verifies the identity for spatially covariant gravity, k-essence, full Horndeski theory including the quintic interaction, general quadratic HOST theories with no degeneracy assumption (covering quadratic GLPV, DHOST classes Ia/IIa, and U-DHOST), and theories obtained from these by admissible disformal, higher-derivative, Weyl-conformal, and mimetic transformations, and it derives an extremal corollary $c^2_{r,T}|_e = -L^2/(2r_e^2) < 0$ at a double-zero horizon, checked against explicit class-Ia DHOST Hayward and arctangent solutions.
Load-bearing premise
The load-bearing premise is that the regular black hole is supported solely by a single real radial scalar field $\phi=\phi(r)$, so that inside the trapped region the scalar depends only on proper time and its spatial pressure is isotropic; the authors themselves list time-dependent scalar profiles, multiple scalars, and additional anisotropic matter as escape routes, but if the interior stress is not isotropic the turning-point identity need not hold.
Editorial extensions
If this is right
- All theory families in the paper's scope—spatially covariant gravity, k-essence, full Horndeski, quadratic HOST with its GLPV/DHOST/U-DHOST subclasses, and admissible disformal, higher-derivative, and mimetic extensions—are excluded from hosting stable two-horizon regular black holes supported by a single radial scalar, with no case-by-case stability analysis required.
- The even-sector scalar–metric mixing that stabilizes beyond-Horndeski bounces and wormholes cannot rescue regular black holes, because the turning-point identity is saturated by the odd-parity tensor coefficients alone.
- At a double-zero extremal horizon the radial odd-tensor speed is fixed to $c^2_{r,T}|_e = -L^2/(2r_e^2) < 0$, and the explicit class-Ia DHOST Hayward and arctangent realizations confirm the sign and the near-extremal instability window $1 < \mu/\mu_e < 2(2/5)^{2/3}$.
- Constructing a viable regular black hole in these theories requires abandoning at least one premise: a single static radial scalar, a two-horizon trapped region with $q\to 0$ at both horizons, or TPST-class dynamics.
- Earlier exclusions obtained case by case in Horndeski, GLPV, and DHOST settings are unified into a single geometric mechanism, so future searches for stable regular black holes can be screened by the sign of $\ddot{q}$ at the turning point alone.
Reading between the lines
- Because the interior maximum of $q$ follows from compactness alone, the obstruction should persist for any matter content whose spatial stress inside the trapped region is isotropic; the natural next test is whether time-dependent or multi-scalar hair (the authors' escape route i) can restore $c_{\rm odd}^2>0$ without reintroducing inner-horizon instabilities.
- Identity (10) offers a cheap diagnostic for anyone constructing candidate solutions: evaluating $\ddot{q}/q$ at the interior stationary point and comparing it with $c_{\rm odd}^2/b^2$ screens a candidate before any full perturbation calculation, and a mismatch would signal a non-TPST structure or an error.
- The near-extremal corollary turns the no-go into a potentially observable signature: a near-extremal regular black hole in the covered classes would carry a split-signature odd-tensor principal metric just outside the horizon, which could in principle be probed by gravitational-wave ringdown or horizon-scale observations.
- A direct stress test of the theorem is to relax only the isotropy assumption: adding a single anisotropic matter field, such as nonlinear electrodynamics, to the same two-horizon geometry should break identity (10), and the framework predicts that stability then hinges on how strongly the added field modifies the shape equation—quantifying that threshold is a concrete extension of escape route (ii
Original abstract
Perturbative instabilities generically puzzles singularity-free spacetimes, including non-singular bounce, traversible wormholes and regular black holes (RBHs). Can the scalar-tensor (ST) theories that stabilize the first two also stabilize RBHs? We show that the answer is no for static, spherically symmetric RBHs supported by a radial scalar field within a broad class of ST theories. In any RBHs with two-horizon structure, the ratio of the longitudinal scale to the sphere radius must attain an interior maximum. The proper-time derivative of that ratio is associated to the squared propagation speed of odd-parity tensor perturbations by the background field equations in a broad class of scalar-tensor theories. A Lorentzian odd-tensor cone then requires a local minimum instead of a maximum. This conflicts indicate a geometric no-go theorem, covering a broad class of ST theories including spatially covariant gravity, Horndeski theory, quadratic GLPV, DHOST, and U-DHOST theories, and their admissible disformal extensions. Viable RBHs therefore require relaxing at least one geometric or dynamical assumption.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Perturbative instabilities generically puzzles singularity-free spacetimes, including non-singular bounce, traversible wormholes and regular black holes (RBHs). Can the scalar-tensor (ST) theories that stabilize the first two also stabilize RBHs? We show that the answer is no for static, spherically symmetric RBHs supported by a radial scalar field within a broad class of ST theories. In any RBHs with two-horizon structure, the ratio of the longitudinal scale to the sphere radius must attain an interior maximum. The proper-time derivative of that ratio is associated to the squared propagation speed of odd-parity tensor perturbations by the background field equations in a broad class of scalar-tensor theories. A Lorentzian odd-tensor cone then requires a local minimum instead of a maximum. This conflicts indicate a geometric no-go theorem, covering a broad class of ST theories including spatially covariant gravity, Horndeski theory, quadratic GLPV, DHOST, and U-DHOST theories, and their admissible disformal extensions. Viable RBHs therefore require relaxing at least one geometric or dynamical assumption.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Perturbative instabilities generically puzzles singularity-free spacetimes, including non-singular bounce, traversible wormholes and regular black holes (RBHs). | background 1.00 | 0.06 | 0.02 | asserted 0.94 |
| s1 | Can the scalar-tensor (ST) theories that stabilize the first two also stabilize RBHs? | gap 0.86 | 0.05 | 0.02 | unclear 0.84 |
| s2 | We show that the answer is no for static, spherically symmetric RBHs supported by a radial scalar field within a broad class of ST theories. | result 0.93 | 0.96 | 0.86 | asserted 1.00 |
| s3 | In any RBHs with two-horizon structure, the ratio of the longitudinal scale to the sphere radius must attain an interior maximum. | result 1.00 | 0.78 | 0.11 | asserted 1.00 |
| s4 | The proper-time derivative of that ratio is associated to the squared propagation speed of odd-parity tensor perturbations by the background field equations in a broad class of scalar-tensor theories. | result 0.88 | 0.71 | 0.05 | asserted 1.00 |
| s5 | A Lorentzian odd-tensor cone then requires a local minimum instead of a maximum. | result 0.69 | 0.62 | 0.10 | asserted 1.00 |
| s6 | This conflicts indicate a geometric no-go theorem, covering a broad class of ST theories including spatially covariant gravity, Horndeski theory, quadratic GLPV, DHOST, and U-DHOST theories, and their admissible disformal extensions. | interpretation 0.53 | 0.77 | 0.53 | asserted 0.82 |
| s7 | Viable RBHs therefore require relaxing at least one geometric or dynamical assumption. | interpretation 0.87 | 0.53 | 0.43 | asserted 0.93 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:2611.06460 by the authors.
References
Discussion
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