REVIEW
Nonreciprocal black holes with primary hair in beyond Horndeski gravity: Exact solutions and stability
General Relativity and Quantum Cosmology · Submitted 8 Oct 2026 · 101 references
Pith's verdict: A systematic analytic construction for non-reciprocal hairy black holes in beyond Horndeski gravity; clear limitations but a genuine advance.
T0 · AI review
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Referee report
The paper
The paper claims
A new systematic method constructs exact, static, asymptotically flat black-hole solutions with primary scalar hair in beyond Horndeski gravity, without imposing the restrictive reciprocal condition g_tt g_rr = -1.
The reading
What carries the argument
The central machinery is the systematic reduction of the beyond-Horndeski field equations under the ansatz (2.1) and the scalar profile ϕ = qt + ψ(r). This yields equations (2.14)–(2.16) that naturally separate into three branches. Branch III is solved by assuming the functional relation G2/G4 = 2/$λ^{2}$ (Eq. 2.25), which fixes the invariant Ξ ≡ $Z^{2}$ X in terms of the radial coordinate. The auxiliary functions Z(X) and F(Ξ) then determine the model functions and, via Eq. (2.28), the metric function A(r). The unique purely disformal transformation (3.1) maps any non-reciprocal solution to a reciprocal (g_tt g_rr = -1) solution in a different DHOST theory, a fact used extensively to simplify the perturbation analysis. Axial perturbations are reduced to the odd-parity problem of GR on an effective metric (6.13), while monopole perturbations are studied via a Schrödinger equation in scalar-adapted coordinates.
Extended reading notes
Core claim
The paper establishes a systematic procedure for obtaining exact, non-reciprocal black-hole solutions in shift- and Z2-symmetric beyond Horndeski gravity, thereby generalizing earlier constructions that relied on the restrictive condition g_tt g_rr = -1. The solution space splits into two qualitatively distinct branches: one (Branch II) where the Noether current vanishes and only khronometric hair exists, and another (Branch III) where a nonvanishing current supports genuine scalar hair. For Branch III, under the simplifying ansatz G2/G4 = 2/$λ^{2}$, the method yields explicit families of asymptotically flat black holes parametrized by a primary scalar charge q. A regularity condition relating q to the mass M removes the central singularity, producing regular black holes (with two horizons) or solitons. The paper further shows that every non-reciprocal solution is uniquely disformally related to a reciprocal solution in a different DHOST theory, and that the axial and monopole perturbations of these solutions are governed by effective Schrödinger operators whose non-negativity implies linear stability for the modes considered.
Load-bearing premise
The entire Branch III construction relies on the ansatz G2/G4 = 2/$λ^{2}$ (Eq. 2.25), which imposes a fixed ratio between two of the theory’s free functions and is not forced by the field equations; if this relation is not satisfied, the analytic solution for the scalar profile and the derived solution families do not apply.
Editorial extensions
If this is right
- The method provides a systematic template for constructing additional exact black-hole families in beyond Horndeski theories by choosing different pairs of the auxiliary functions Z(X) and F(Ξ).
- If the regularity condition holds, the resulting regular black holes and solitons offer explicit, non-singular geometries that can serve as testbeds for strong-field gravity and possible observational signatures.
- The linear stability evidence for axial and monopole perturbations under the chosen self-adjoint domains suggests that at least some of these families could be viable ground-state black-hole configurations in modified gravity.
- The disformal mapping to a reciprocal theory implies that every non-reciprocal solution has a twin description; matter minimally coupled to the original metric becomes disformally coupled in the reciprocal frame, potentially leading to distinct phenomenology.
- The method naturally extends to slowly rotating configurations via a small-parameter expansion, as has been done for the reciprocal case, opening the way to testing these hairy solutions with gravitational-wave or shadow observations.
Reading between the lines
- The key simplifying assumption G2/G4 = 2/λ^2, though presented as an ansatz, may be the most restrictive part of the Branch‑III construction. Testing whether fully non‑reciprocal solutions exist without this relation would clarify whether the method captures the true generality of the theory or only a special subclass.
- The paper’s monopole stability analysis flags a potential strong‑coupling problem for scalar fluctuations in the asymptotically Minkowski region. If this strong coupling sets in at finite radius, the linear Schrödinger description may break down, and quasinormal‑mode calculations or time evolutions would require a nonlinear completion.
- The disformal equivalence to a reciprocal solution is a double‑edged tool: it simplifies perturbations, but the reciprocal theory has a different matter coupling. Connecting these solutions to astrophysical observations (e.g., shadows, gravitational waves) must therefore be done in the original frame, where matter is minimally coupled, or must account for the disformal coupling in the reciprocal f
Original abstract
We develop a novel systematic method for constructing exact, static, and asymptotically flat black-hole solutions in shift-symmetric and $Z_2$-symmetric beyond Horndeski theories. In contrast to previous studies, our approach does not rely on the restrictive reciprocal condition $g_{tt}g_{rr}=-1$, allowing for more general geometries characterized by two independent metric functions. The resulting solution space contains two qualitatively distinct types of hair originating from different branches of the theory. One branch is characterized by khronometric hair, while the other by genuine scalar hair. In both, the scalar profile defines a nontrivial preferred foliation, but the shift-symmetry Noether current and the associated propagating degree of freedom are trivial for the former and nonvanishing for the latter. Focusing mainly on the second branch, we derive several distinct families of exact black-hole solutions. In general, these solutions are singular and exhibit primary scalar hair associated with an independent scalar charge. Nevertheless, we demonstrate that they can be regularized to yield regular black-hole or solitonic configurations, for which the scalar hair becomes secondary. We further investigate the stability properties of these solutions under linear perturbations by analyzing the axial sector and the monopole $(\ell=0)$ mode of the even-parity sector. For representative solutions with scalar hair, the perturbative analysis provides evidence for linear stability under the modes considered.
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 develop a novel systematic method for constructing exact, static, and asymptotically flat black-hole solutions in shift-symmetric and $Z_2$-symmetric beyond Horndeski theories. In contrast to previous studies, our approach does not rely on the restrictive reciprocal condition $g_{tt}g_{rr}=-1$, allowing for more general geometries characterized by two independent metric functions. The resulting solution space contains two qualitatively distinct types of hair originating from different branches of the theory. One branch is characterized by khronometric hair, while the other by genuine scalar hair. In both, the scalar profile defines a nontrivial preferred foliation, but the shift-symmetry Noether current and the associated propagating degree of freedom are trivial for the former and nonvanishing for the latter. Focusing mainly on the second branch, we derive several distinct families of exact black-hole solutions. In general, these solutions are singular and exhibit primary scalar hair associated with an independent scalar charge. Nevertheless, we demonstrate that they can be regularized to yield regular black-hole or solitonic configurations, for which the scalar hair becomes secondary. We further investigate the stability properties of these solutions under linear perturbations by analyzing the axial sector and the monopole $(\ell=0)$ mode of the even-parity sector. For representative solutions with scalar hair, the perturbative analysis provides evidence for linear stability under the modes considered.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We develop a novel systematic method for constructing exact, static, and asymptotically flat black-hole solutions in shift-symmetric and $Z_2$-symmetric beyond Horndeski theories. | contribution 0.92 | 0.97 | 0.11 | asserted 1.00 |
| s1 | In contrast to previous studies, our approach does not rely on the restrictive reciprocal condition $g_{tt}g_{rr}=-1$, allowing for more general geometries characterized by two independent metric functions. | contribution 0.64 | 0.75 | 0.09 | asserted 1.00 |
| s2 | The resulting solution space contains two qualitatively distinct types of hair originating from different branches of the theory. | result 0.98 | 0.92 | 0.14 | asserted 1.00 |
| s3 | One branch is characterized by khronometric hair, while the other by genuine scalar hair. | result 0.97 | 0.86 | 0.10 | asserted 1.00 |
| s4 | In both, the scalar profile defines a nontrivial preferred foliation, but the shift-symmetry Noether current and the associated propagating degree of freedom are trivial for the former and nonvanishing for the latter. | result 0.84 | 0.85 | 0.09 | asserted 1.00 |
| s5 | Focusing mainly on the second branch, we derive several distinct families of exact black-hole solutions. | contribution 0.90 | 0.96 | 0.07 | asserted 0.99 |
| s6 | In general, these solutions are singular and exhibit primary scalar hair associated with an independent scalar charge. | result 0.98 | 0.91 | 0.15 | asserted 0.96 |
| s7 | Nevertheless, we demonstrate that they can be regularized to yield regular black-hole or solitonic configurations, for which the scalar hair becomes secondary. | result 0.89 | 0.95 | 0.18 | asserted 1.00 |
| s8 | We further investigate the stability properties of these solutions under linear perturbations by analyzing the axial sector and the monopole $(\ell=0)$ mode of the even-parity sector. | method 0.79 | 0.96 | 0.05 | asserted 0.96 |
| s9 | For representative solutions with scalar hair, the perturbative analysis provides evidence for linear stability under the modes considered. | result 1.00 | 0.96 | 0.22 | hedged possibility 0.91 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:2610.11712 by the authors.
References
Discussion
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