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REVIEW 4 major objections 5 minor 182 references

Compact Objects in Einstein-scalar-Gauss-Bonnet Theory and beyond

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A minimal scalar-Gauss-Bonnet extension of general relativity produces scalarised black holes, traversable wormholes, and regular particle-like objects, and a disformal transformation of a known black hole yields a smooth wormhole beyond…

desk verdict A clear review of the EsGB solution space, but the 'always emerge' claim overreaches the evidence. read the letter →

arxiv 2412.20296 v1 pith:H3MFZREU submitted 2024-12-28 gr-qc

Panagiota Kanti This is my paper · ORCID
classification gr-qc
keywords Einstein-scalar-Gauss-Bonnettheoryscalarisedblackholestraversablewormholesparticle-likesolutionsno-hairtheoremsHorndeskidisformaltransformationsGauss-Bonnetcoupling
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

This review argues that one minimal modification of general relativity — adding a scalar field coupled to the Gauss-Bonnet term — changes the allowed menu of compact objects. Scalarised black holes with a regular horizon emerge for essentially any choice of the coupling function $f(\phi)$, provided the scalar field satisfies the horizon regularity constraint, and they evade both the older and newer scalar no-hair theorems. The same theory supports traversable wormholes whose throat is held open by the geometry–scalar coupling rather than by exotic matter, and regular particle-like solutions that are ultra-compact and produce light rings and echo signals. In the broader beyond-Horndeski theory, a disformal transformation of an analytic black hole yields a smooth traversable wormhole with no matter layer at the throat. If this picture is right, strong-gravity observations of shadows, light rings, and echoes could distinguish these objects from ordinary black holes.

What carries the argument

The machine at the centre is the coupling function $f(\phi)$ multiplying the Gauss-Bonnet invariant $R^2_{GB} = R^2 - 4R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$. The near-horizon regularity constraint $\phi'_h = \frac{r_h}{4\dot{f}_h}\left(-1 \pm \sqrt{1 - 96\dot{f}_h^2/r_h^4}\right)$ fixes the scalar field's first derivative at the horizon, and together with the bound $\dot{f}_h^2 < r_h^4/96$ it lets the numerical integration start for any chosen coupling. The no-hair evasion is carried by the sign flip of the radial component of the energy-momentum tensor near the horizon, coming from the Gauss-Bonnet coupling. For wormholes, the flaring-out condition $b - r b' > 0$ is met through this geometry-coupling contribution, which violates the null energy condition without introducing a ghost field, while in the beyond-Horndeski part the disformal transformation converts a known Horndeski black hole into a smooth wormhole.

What would settle it

A concrete test: pick a smooth coupling function outside the tested families (for instance $f(\phi)=\alpha e^{-\phi^2}$ with a large coupling) and integrate the same boundary-value problem from horizon to infinity; if no regular interpolating solution exists, the 'always emerges' claim fails. For the particle-like solutions, the claim would be falsified by finding any curvature invariant or physical observable that diverges at the origin despite the reported finiteness of the standard invariants.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Einstein-scalar-Gauss-Bonnet action with an arbitrary coupling function $f(\phi)$ is a solution-generating mechanism: for every tested form of $f(\phi)$, numerical integration with the appropriate horizon regularity constraint produces scalarised black holes, wormholes with one or two throats, and particle-like solutions, and the scalarised black-hole line forms the boundary of the wormhole territory. Near the horizon, the Gauss-Bonnet term reverses the sign of the radial energy-momentum component, which the newer no-hair theorem would forbid, and this is why the theorem is evaded independently of the form of $f(\phi)$. For the particle-like solutions, the scalar field diverges as $1/r$ at the origin, but all curvature invariants and energy-momentum components stay finite, so the singularity is described as Coulomb-type and harmless. Beyond this class, a disformal transformation $g_{\mu\nu} = \bar{g}_{\mu\nu} - D(\bar{X})\nabla_\mu\phi\nabla_\nu\phi$ applied to a known Horndeski black-hole solution produces a traversable wormhole in beyond-Horndeski theory, with both metric functions regular and symmetric at the throat and no cusp or added matter.

Load-bearing premise

The central claim that scalarised solutions emerge for any coupling function is inferred from numerical integration for only a few chosen forms of $f(\phi)$, and the particle-like claim assumes that a scalar field diverging as $1/r$ at the origin is harmless because the usual invariants remain finite.

Editorial extensions

If this is right

  • If scalarised black holes exist for arbitrary $f(\phi)$, then every EsGB theory of this form carries a one-parameter family of hairy black holes that reduce to Schwarzschild at large mass, have smaller horizon areas than their GR analogues, and possess a lower mass bound set by the regularity condition.
  • Traversable wormholes can be built in EsGB theory without invoking a ghost scalar or other exotic matter, so the usual exotic-matter obstruction to wormhole physics is bypassed by the scalar–Gauss-Bonnet coupling.
  • The particle-like solutions are ultra-compact, bubble-shaped objects with negative energy density at the centre and a fast-falling shell profile, and they generically produce light rings and echo trains in scalar wave signals.
  • In beyond-Horndeski theory, disformally transformed black holes yield wormholes whose light rings all lie at radii smaller than $3M$, the Schwarzschild photon-sphere radius.
  • The solution space is connected: the scalarised black-hole line bounds the wormhole region, and families of particle-like solutions with different node numbers occupy further parts of the domain of existence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the review only gestures at is stability: the entropy comparison already used for black holes could be turned into a systematic criterion for which scalarised family is the thermodynamically preferred end state of collapse.
  • If the universality claim holds, the observational burden shifts: echo signals and light rings below the Schwarzschild photon-sphere radius become generic signatures of EsGB compact objects, and current gravitational-wave ringdown data could in principle constrain the coupling constant—this is not claimed in the paper.
  • The harmlessness of the Coulomb-type scalar divergence is supported by the finiteness of the standard invariants; a stricter test would be to check completeness of geodesics and higher-order curvature invariants, which the paper does not report.
  • The disformal recipe suggests that every Horndeski black hole with an appropriate scalar profile can be dressed into a beyond-Horndeski wormhole, potentially making wormhole solutions as numerous as known Horndeski black holes.
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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

4 major / 5 minor

Summary. This manuscript is a review article on compact objects in Einstein-scalar-Gauss-Bonnet (EsGB) theory and in Horndeski/beyond-Horndeski theories. After a brief discussion of black holes, wormholes, and particle-like solutions in GR and Einstein-scalar theory, the paper summarizes a series of results, largely from the author and collaborators: scalarised black holes for various coupling functions, traversable wormholes obtained by a cut-and-paste construction near the throat, scalarised particle-like solutions with a 'Coulomb-type' scalar-field singularity, and wormholes obtained by disformal transformation of the Lu-Pang black hole in beyond-Horndeski theory. The stated central message is that scalarised black holes 'always emerge' in EsGB theory independently of the form of the coupling function, that the theory supports traversable wormholes without exotic matter, and that regular particle-like solutions arise with observable signatures such as photon rings and echoes.

Significance. If its claims are accepted, the review would be a useful consolidated reference for a substantial body of work on scalarised compact objects in higher-order scalar-tensor theories. The manuscript is clearly written, covers a wide literature (183 references), and presents a helpful taxonomy of black-hole, wormhole, and particle-like solutions, including domains of existence and phenomenological features such as light rings and echoes. The main value lies in its survey character, and several individual results (e.g., the analytic Horndeski black hole, the disformal wormhole construction, and the explicit form of the near-horizon regularity condition) are presented in enough detail to be informative. However, the review's universal and 'without exotic matter' claims go beyond what the presented arguments establish, and at least one regularity statement is in tension with the displayed expansions. These issues are local but affect the advertised scope of the paper.

major comments (4)
  1. [Section 5, first paragraph; Section 3.1] The Conclusions state that scalarised black-hole solutions 'always emerge, independently of the form of the coupling function, provided that appropriate boundary conditions are imposed.' This universal quantifier is not supported by the evidence in Section 3.1. Equation (14) is a local near-horizon regularity condition: it fixes phi'_h in terms of f'(phi_h) and r_h, but it does not guarantee that outward integration reaches the asymptotically flat regime (18)-(20) rather than encountering a singular point or failing to relax to phi_infinity. The paper reports numerical integration for a finite family of coupling functions (exponential, power-law, inverse-power-law, logarithmic, etc.), which is an inductive basis, not a proof. Moreover, the statement is literally false for f(phi)=const (e.g., f=0), for which f'(phi_h)=0 makes Eq. (14) degenerate and no nontrivial hair is expected; even for nonconstant f, Eq. (14) requires choosing phi_h with f'(phi_h)!=0 and satisfying the discriminant bound (15). The claim should be weakened to the classes of coupling functions for which complete numerical solutions have actually been constructed, and the f'(phi_h)!=0 requirement should be stated explicitly.
  2. [Abstract and Section 3.2, Eqs. (36)-(42)] The abstract and conclusions claim that EsGB theory supports traversable wormholes 'without the need for exotic matter.' In Section 3.2 the wormhole solutions are made regular by a cut-and-paste construction: the positive-l region is glued to a mirror image at l=0, and the cusps are 'justified' by introducing a thin shell described by Eq. (42). No analysis of the energy conditions of this shell is presented; the parenthetical claim that the perfect fluid is 'non-exotic' is asserted without derivation. In the Morris-Thorne framework, the flaring-out condition (36) combined with Eq. (38) gives rho+p_r<0 at the throat for a GR wormhole, and for EsGB the analogous statement must be checked with the full effective energy-momentum tensor including the scalar-GB coupling. As written, the review does not establish that the required shell matter satisfies any standard energy condition. The later disformal wormhole of Section 4 explicitly violates the NEC (Eq. (71)), with the text arguing that the violation arises from non-minimal couplings rather than exotic matter; this distinction between 'matter' and 'effective' energy conditions should be defined and applied consistently to the EsGB wormholes as well.
  3. [Section 3.3, Eqs. (44)-(48)] The paper advertises 'regular scalarised particle-like solutions,' but the displayed near-origin expansion (47) shows the scalar field behaving as phi ~ -c0/r + phi0 + ..., i.e., divergent at r=0. The text states that all gravitational scalar invariants and the components of T_mu_nu are finite 'despite the singularity in phi,' citing Refs. [166,167]. This is a nontrivial regularity claim that is not demonstrated in the review; a survey should at least specify which quantities were checked and in which of the cited papers the calculation appears. Without this, the word 'regular' in the abstract is misleading, because it refers only to the metric and derived invariants, not to the scalar field itself. The 'Coulomb-type' analogy is heuristic and should be labeled as such.
  4. [Section 4, Eqs. (64)-(70) and Fig. 12] The disformal wormhole construction is presented as giving a spacetime with r^2 = l^2 + r0^2, so the radial coordinate r is an even function of l. In that coordinate system the scalar field, which is a function of r through the seed solution (56), should be symmetric under l -> -l. The text, however, states that the profile of the scalar field 'is in fact asymmetric under the change l -> -l' (discussion following Eq. (70) and Fig. 12). This is an internal inconsistency in the presentation. If the two sides of the wormhole are obtained as two copies of the same r>=r0 solution, the scalar field must be even; if the asymmetry is intentional, the construction differs from what is described and the coordinate transformation needs to be clarified.
minor comments (5)
  1. [Equation (42)] The text introduces the shell action with constants (lambda1, lambda0), but immediately says '(lambda1, lambda2) are constants'. Please correct the label of the second constant.
  2. [Equation (20)] The 1/r^4 term in the scalar-field expansion is written in a way that is easy to misread: '12M^3D - 24M^2 \dot f - M D^3 / 6r^4' should probably be '(12M^3D - 24M^2 \dot f - M D^3)/(6r^4)' or an equivalent parenthesized form. Also, the dot on f is defined nowhere; please state that \dot f = df/d\phi.
  3. [Section 3.2, Fig. 6 caption] The caption refers to 'solutions for the scalar field for a family of dilatonic wormholes' but the plot axes are x and y and the curves appear to be trajectories; if the upper plot is indeed the scalar field and the lower plot the trajectories, the caption should be split or clarified.
  4. [Section 3.3, Eq. (49)] The geodesic Lagrangian is written as -epsilon, but for timelike particles the convention is usually 2L = -1 (or +1 depending on signature); please state the convention explicitly so that Eqs. (50) and the effective potential have a consistent sign.
  5. [References] Some references appear only as arXiv numbers without journal details (e.g., Refs. [68,69,116,117,121,122,123,124,125,126,127,128,129,146,147]); for a review, adding the final publication data would increase usability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the review reports prior numerical constructions rather than deriving its conclusions from its own inputs.

full rationale

This is a review article that surveys previously published numerical constructions of scalarised black holes, wormholes and particle-like solutions in Einstein-scalar-Gauss-Bonnet theory. The central claims are presented as results obtained by integrating the field equations with near-horizon and asymptotic boundary conditions, not as quantities that are fitted and then relabeled as predictions. The regularity constraint (14) is derived by demanding finiteness of phi'' at the horizon, and the subsequent sign computation (22) is an algebraic consequence of that constraint, not a circular restatement. The photon rings and echoes are computed from the effective potentials of the constructed solutions rather than being inputs of the construction. The wormholes are built by explicit ansatz choices or by disformal transformations designed to produce a throat, and calling the resulting objects traversable wormholes is applying a definition, not a disguised prediction. The main weakness is the wording that scalarised solutions 'always emerge, independently of the form of the coupling function'; the evidence is numerical integration for a finite family of coupling functions, so this is an inductive extrapolation and an overstatement, but it is not a circular derivation. Similarly, the extensive self-citations are normal for a review of the author's own body of work; they refer to external numerical studies and do not constitute a self-citation chain that replaces independent evidence. No equation is shown to be equivalent to an input by construction, and no fitted parameter is renamed as a prediction. The circularity score is therefore 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The review itself does not introduce new entities, but it relies on free parameters (coupling constant, horizon data, disformal scale) and domain assumptions about the validity of numerical integration and thin-shell constructions. The axioms listed are the load-bearing premises behind the claimed existence and regularity of solutions. Since the paper is a review, these are carried over from the cited works rather than derived anew.

free parameters (4)
  • alpha (coupling constant) = Varies by plot, e.g., 0.0009 < alpha < 0.919 for f = alpha/phi
    Coupling strength of the scalar field to the Gauss-Bonnet term; chosen by hand in numerical integrations for different coupling function forms.
  • phi_h (scalar field at horizon) = e.g., phi_h = 3 in Fig. 2
    Input boundary condition for the numerical shooting method; selected to satisfy the regularity constraint (14).
  • r_h (horizon radius) = Set to 1 in Fig. 2
    Horizon radius is chosen as a boundary condition; it determines the mass scale of the solution.
  • lambda (disformal scale) = Specified in Eq. (67)
    Free scale parameter in the disformal transformation that controls the location of the wormhole throat r0 via Eq. (68).
assumptions (4)
  • standard math The Gauss-Bonnet term yields field equations with at most second-order derivatives, avoiding Ostrogradski instabilities.
    Invoked in Section 3 to justify the viability of the theory as a physical scalar-tensor theory.
  • domain assumption The regularity constraint (14) for phi'_h ensures finite phi'' at the horizon.
    Used to derive the near-horizon asymptotic solution and to prove evasion of the no-hair theorem.
  • domain assumption The numerical integration of the field equations (12) yields solutions that smoothly interpolate between the near-horizon and asymptotic expansions.
    The review assumes the existence and uniqueness of such numerical solutions for any tested coupling function, without presenting code or convergence analysis.
  • ad hoc to paper The cut-and-paste construction at the wormhole throat can be supported by a thin shell described by action (42).
    Introduced to excise a singularity behind the throat; the stability of this shell is not analyzed in the review.

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Cite this review

Pith. "Pith review of Compact Objects in Einstein-scalar-Gauss-Bonnet Theory and beyond." pith.science (2026). https://pith.science/paper/H3MFZREU

@misc{pith2026241220296,
  author       = {Pith},
  title        = {Pith review of: Compact Objects in Einstein-scalar-Gauss-Bonnet Theory and beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H3MFZREU}},
  note         = {Machine review of arXiv:2412.20296}
}
read the original abstract

In the context of General Relativity, black holes are not allowed to possess scalar hair, wormholes are not traversable and particle-like solutions are irregular. Therefore, in order to derive novel and physically interesting solutions that describe compact objects one needs to address generalised gravitational theories. One popular class of such theories is the Einstein-scalar-Gauss-Bonnet (EsGB) theory with a general coupling function between the scalar field of the theory and the quadratic Gauss-Bonnet term. Starting from black holes, we present a variety of spherically-symmetric solutions for several different forms of the coupling function and discuss their main features. We then proceed to wormhole solutions and demonstrate that the EsGB theory naturally supports traversable wormholes without the need for exotic matter. Regular scalarised particle-like solutions also emerge in the context of the same theory which also possess interesting observable features such as photon rings and echoes. Moving beyond this class of theories, we then address the more extended scalar-tensor Horndeski theory, briefly mention the types of black-hole solutions that arise, and demonstrate that an appropriately constructed disformal transformation of a black-hole solution, such as the Lu-Pang solution, results into a traversable wormhole in the context of the beyond-Horndeski theory.

Figures

Figures reproduced from arXiv: 2412.20296 by the authors.

Figure 1
Figure 1. The profile of the scalar field for a variety of choices for the coupling function f(ϕ) (left plot), and the two metric functions (in absolute value, right plot) [55][56]. where we have used the constraint Eq. (14) for the regularity of the horizon. It is clear that the above expression is always positive-definite, in contrast to the requirement of the novel no-hair theorem. The presence of the GB term near the hori… view at source ↗
Figure 2
Figure 2. The scalar charge D (left plot), and the ratios Ah/ASch and Sh/SSch (right plot, lower and upper curve respectively) in terms of the mass M, for f(ϕ) = a /ϕ [55]. function of the black-hole mass, a result that renders the scalar hair secondary. Also, a common characteristic in all cases is that, as the mass of the black hole increases, the scalar charge decreases and eventually vanishes as our black-hole solution ma… view at source ↗
Figure 3
Figure 3. The solution for the scalar field ϕ (left plot), and for the metric components |gtt| and grr (right plot) in terms of the radial coordinate r, for f(ϕ) = aϕ2 [56]. [59][60], however, it may be shown numerically that it holds in the perturbative limit of small GB coupling constant a, for all forms of the coupling function. We notice that the dominant term in the expression of ϕ at large distances has a logarithmic fo… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: The profile of the scalar field in terms of the radial coordinate for different (polynomial) choices of the scalar potential (left plot); the lines of existence of the black-hole solutions for the same choices of V (ϕ) (right plot) [61]. As the degree of the polynomial…
Figure 5
Figure 5. Figure 5: The embedding diagram of a typical traversable wormhole. and is known as the flaring-out condition of the wormhole. Therefore, in order to have a throat, the shape function b(r) should satisfy b − rb′ > 0, which fully justifies its name. In fact, the flaring-out condit…
Figure 6
Figure 6. Figure 6: The solutions for the scalar field for a family of dilatonic wormholes (upper plot), and particles trajectories in the background of a dilatonic wormhole (lower plot) [152][153]. the EsGB theory with alternative forms for f(ϕ), as in the case of black holes? In the con…
Figure 7
Figure 7. Figure 7: A wormhole solution with an equator and two throats (left plot), and the domain of existence of wormhole solutions in the case where f(ϕ) = aϕ2 (right plot) [154]. The metric functions and scalar field assume regular forms both near the throat and the far asymptotic re…
Figure 8
Figure 8. Figure 8: The solutions for the metric and scalar field (left plot) and the corresponding energy-density (right plot) in the case of the quadratic coupling function [166][167]. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 F = αφ2 , φ∞=0 Veff=e(f0 -f1 )/2/r^ Veff R ^ c ∧α, d 4.0, 0.6 3.4, 0.6…
Figure 9
Figure 9. Figure 9: The effective potential for null particles (left plot) and for test scalar particles (right plot), for a quadratic coupling function [166][167]. origin are given below: ρ(0) = − 3 32α , p(0) = 2 32α . (48) We may therefore conclude that the singularity of the scalar fi…
Figure 10
Figure 10. Figure 10: The two metric functions after the disformal transformation [179]. in the expression of W(X¯) at a distance r0 > rH, where rH is the event horizon of the “seed solution”. The presence of this root would result into the vanishing of g rr at this point but not of gtt, w…
Figure 11
Figure 11. Figure 11: The two metric functions after the change of variable r → l [179]. throat (all curves in [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: The Kretchmann scalar K (left plot), and the profile of the scalar field (right plot) [179]. One may wonder whether the emergence of these wormhole solutions must again be supported by the violation of the energy conditions. If we focus again [PITH_FULL_IMAGE:figures…
Figure 13
Figure 13. Figure 13: The violation of the NEC (left plot) and the effective potential of photons propagating in the wormhole background (right plot) [179]. on the Null Energy Condition, namely Tµνn µn ν ≥ 0, we arrive at the result 8πG(T r r − T t t ) = G r r − G t t = − f ′ (r0) r0 . (71…
Figure 14
Figure 14. Figure 14: The different types of solutions emerging in the context of the Einstein-scalar￾GB theory [166][167]. result that a series of echoes should characterize the wave signal at infinity, a characteristic observable associated to our particle-like solutions. The different t…

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Reviewed August 10, 2026 · model on record in the stance chip above.