REVIEW
Perturbations and stability of black holes with static scalar hair in general GLPV theories
General Relativity and Quantum Cosmology · Submitted 1 Sep 2026 · 118 references
Pith's verdict: Thorough perturbation analysis for GLPV black holes; the quadratic-sector no-go results are robust, but the quartic-quintic extension rests on a conditional UV scaling criterion that the paper itself flags.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
Every static hairy GLPV black hole with nonzero horizon kinetic term is locally unstable or degenerate.
The reading
What carries the argument
The key machinery is the complete odd- and even-parity quadratic action in uniform-curvature gauge, the exact factorization of the radial characteristic polynomial (identifying tensor branch $c^{2}$_r_grav = G/F and scalar branch $c^{2}$_r_sc = F Delta_G/($f^{2}$ $h^{2}$ G Delta_K)), and the angular determinant identity that isolates Q_Omega as the single combination controlling the sign of the angular principal symbol. The compatibility condition (4.80) aligns covariant degeneracy directions and is shown to be necessary for finite scale-independent eikonal phase speeds in the interior; a local disformal map transfers the Horndeski near-horizon sign to the GLPV frame.
Extended reading notes
Core claim
The central discovery is that static hairy black holes with finite nonzero horizon kinetic term X_s ≠ 0 are generically locally unstable in GLPV theories, not just in Horndeski. The argument uses an exact factorization of the angular determinant for quadratic GLPV theories and a disformal map to Horndeski for the general class, showing that the sign of the angular stability invariant is inherited and negative. For X_s=0, the construction yields analytic power-law beyond-Horndeski deformations of scalar–Gauss–Bonnet black holes that preserve all local stability conditions at sufficiently small coupling.
Load-bearing premise
The angular-eikonal step assumes the formal L → ∞ limit must yield finite, nonzero, scale-independent phase speeds, which may not be required if the theory is regarded as an effective field theory with a finite cutoff in angular momentum.
Editorial extensions
If this is right
- Model building for static hairy black holes beyond scalar–Gauss–Bonnet must target the X_s=0 branch; the compatibility condition (4.80) becomes a necessary near-horizon regularity constraint.
- Weakly coupled scalar–Gauss–Bonnet black holes admit a one-parameter family of analytic beyond-Horndeski deformations (indexed by the power n of X in F_4) that preserve all local stability conditions.
- The complete quadratic actions provide the foundation for quasinormal-mode computations on stable branches, including beyond-Horndeski corrections to the ringdown spectrum.
- The near-horizon angular Laplacian instability predicted for X_s ≠ 0 branches should be testable via numerical evolution of the even-parity perturbation equations.
Reading between the lines
- The requirement of finite nonzero eikonal phase speeds for both angular branches in the interior forces the compatibility condition; if one accepts that the effective field theory need not respect this ultraviolet scaling, the near-horizon obstruction for the regular branch may be avoided, weakening the no-go to a conditional statement.
- The disformal map used in Sec. V C implies that stability of the GLPV frame is equivalent to stability of the Horndeski frame for the compatible class, so any stable branch in GLPV necessarily maps to a stable Horndeski black hole with X_s=0, restricting the search space to Horndeski.
- A direct test of the near-horizon angular Laplacian instability would be to compute the two-point function of metric perturbations on the exact quadratic GLPV solution; the imaginary part should contain a mode with growing angular harmonic at large ℓ close to the horizon.
- The construction of stable deformations suggests that scalar–Gauss–Bonnet may be the unique viable source of static scalar hair in scalar-tensor theories with second-order field equations; GLPV extensions only add perturbatively safe corrections on this branch.
Original abstract
We derive the background equations and complete odd- and even-parity quadratic actions for static, spherically symmetric black holes with radial scalar hair in general quartic-quintic Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, including Horndeski. On regular, nondegenerate branches, the formulation applies across Killing horizons and yields local no-ghost conditions and radial and angular eikonal characteristics. Tensor modes in both parity sectors share the same squared radial speed. When the radial coordinate is timelike, generic antisymmetric mixing induces nonstandard large-multipole scaling for both even-parity angular branches. Requiring both branches to have finite, nonzero eikonal phase speeds selects the Horndeski-related compatibility condition between the quartic and quintic beyond-Horndeski functions. This condition aligns the covariant degeneracy directions and permits a local disformal map to Horndeski when regular and invertible. For a finite, nonzero scalar kinetic term at the horizon, every nontrivial branch of the exact quadratic GLPV black hole is locally unstable or has a degenerate tensor cone near a simple outer horizon. Apart from identified exceptions, this obstruction extends to general shift- and reflection-symmetric quadratic GLPV theories and the regular Horndeski-related branch of the quartic-quintic class. Finally, for scalar-Gauss-Bonnet black holes with a vanishing horizon scalar kinetic term, we construct analytic power-law quartic beyond-Horndeski deformations whose associated quintic function is fixed by the same condition. At sufficiently small coupling, all local no-ghost and high-frequency gradient-stability conditions hold throughout the exterior. For a linear Gauss-Bonnet coupling, we estimate an interior scale below which the background and stability expansions lose perturbative control; this does not imply a physical instability.
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
We derive the background equations and complete odd- and even-parity quadratic actions for static, spherically symmetric black holes with radial scalar hair in general quartic-quintic Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, including Horndeski. On regular, nondegenerate branches, the formulation applies across Killing horizons and yields local no-ghost conditions and radial and angular eikonal characteristics. Tensor modes in both parity sectors share the same squared radial speed. When the radial coordinate is timelike, generic antisymmetric mixing induces nonstandard large-multipole scaling for both even-parity angular branches. Requiring both branches to have finite, nonzero eikonal phase speeds selects the Horndeski-related compatibility condition between the quartic and quintic beyond-Horndeski functions. This condition aligns the covariant degeneracy directions and permits a local disformal map to Horndeski when regular and invertible. For a finite, nonzero scalar kinetic term at the horizon, every nontrivial branch of the exact quadratic GLPV black hole is locally unstable or has a degenerate tensor cone near a simple outer horizon. Apart from identified exceptions, this obstruction extends to general shift- and reflection-symmetric quadratic GLPV theories and the regular Horndeski-related branch of the quartic-quintic class. Finally, for scalar-Gauss-Bonnet black holes with a vanishing horizon scalar kinetic term, we construct analytic power-law quartic beyond-Horndeski deformations whose associated quintic function is fixed by the same condition. At sufficiently small coupling, all local no-ghost and high-frequency gradient-stability conditions hold throughout the exterior. For a linear Gauss-Bonnet coupling, we estimate an interior scale below which the background and stability expansions lose perturbative control; this does not imply a physical instability.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We derive the background equations and complete odd- and even-parity quadratic actions for static, spherically symmetric black holes with radial scalar hair in general quartic-quintic Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, including Horndeski. | contribution 0.99 | 0.96 | 0.07 | asserted 1.00 |
| s1 | On regular, nondegenerate branches, the formulation applies across Killing horizons and yields local no-ghost conditions and radial and angular eikonal characteristics. | result 0.63 | 0.89 | 0.06 | asserted 1.00 |
| s2 | Tensor modes in both parity sectors share the same squared radial speed. | result 1.00 | 0.88 | 0.05 | asserted 1.00 |
| s3 | When the radial coordinate is timelike, generic antisymmetric mixing induces nonstandard large-multipole scaling for both even-parity angular branches. | result 1.00 | 0.87 | 0.04 | asserted 0.99 |
| s4 | Requiring both branches to have finite, nonzero eikonal phase speeds selects the Horndeski-related compatibility condition between the quartic and quintic beyond-Horndeski functions. | result 0.96 | 0.86 | 0.10 | asserted 0.99 |
| s5 | This condition aligns the covariant degeneracy directions and permits a local disformal map to Horndeski when regular and invertible. | interpretation 0.83 | 0.75 | 0.06 | asserted 0.86 |
| s6 | For a finite, nonzero scalar kinetic term at the horizon, every nontrivial branch of the exact quadratic GLPV black hole is locally unstable or has a degenerate tensor cone near a simple outer horizon. | result 1.00 | 0.89 | 0.67 | asserted 1.00 |
| s7 | Apart from identified exceptions, this obstruction extends to general shift- and reflection-symmetric quadratic GLPV theories and the regular Horndeski-related branch of the quartic-quintic class. | result 0.96 | 0.84 | 0.29 | asserted 0.99 |
| s8 | Finally, for scalar-Gauss-Bonnet black holes with a vanishing horizon scalar kinetic term, we construct analytic power-law quartic beyond-Horndeski deformations whose associated quintic function is fixed by the same condition. | contribution 0.93 | 0.95 | 0.05 | asserted 1.00 |
| s9 | At sufficiently small coupling, all local no-ghost and high-frequency gradient-stability conditions hold throughout the exterior. | result 1.00 | 0.91 | 0.08 | asserted 1.00 |
| s10 | For a linear Gauss-Bonnet coupling, we estimate an interior scale below which the background and stability expansions lose perturbative control; this does not imply a physical instability. | limitation 0.64 | 0.93 | 0.04 | asserted 0.72 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.