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Novel electrically charged wormhole, black hole and black bounce exact solutions in hybrid metric-Palatini gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Electrically charged zero-potential HMPG solutions include traversable wormholes, black holes with double horizons, black bounces, and infinite-horizon spacetimes.

desk verdict The advertised 'expanding black universe' is not derived—every solution is static—but the Jordan-frame catalog of charged HMPG wormholes, bounces, and double-horizon black holes is careful, honest, and worth refereeing. read the letter →

arxiv 2412.10324 v2 pith:2XCLBKYX submitted 2024-12-13 gr-qc astro-ph.HEhep-th

Gabriel I. Róis , José Tarciso S. S. Junior , Francisco S. N. Lobo , Manuel E. Rodrigues This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th
keywords hybridmetric-Palatinigravityelectricallychargedsolutionswormholesblackholesbouncesscalar-tensortheoryconformalframesexact
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 paper aims to show that hybrid metric-Palatini gravity, in the special case of zero scalar potential and a pure electric field, contains a wide family of exact static spherically symmetric spacetimes in the physical Jordan frame. Starting from a known Einstein-frame electrovacuum solution with a linear scalar field, the authors transform back to the Jordan frame and classify the resulting geometries by the scalar-field parameters. The catalogue includes traversable wormholes, black holes with degenerate double horizons, black bounces with a throat behind a horizon, and a configuration with infinitely many double horizons; canonical scalar fields mostly give naked singularities, sometimes hidden behind a throat or an anti-throat, while phantom scalars produce the regular horizon structures. The paper also labels some of its horizon-interior geometries as 'black universe' solutions, in which the region beyond the horizon is said to lead to an expanding cosmological solution rather than a singularity.

What carries the argument

The load-bearing object is the conformal bridge between the Einstein-frame seed metric (15), the general static spherically symmetric electrovacuum solution with scalar field $\bar\phi=\bar C u+\bar\phi_0$ written in terms of the functions $s^2(k,u)$ and $s^2(h,u+u_1)$, and the Jordan-frame metrics obtained by multiplying by $\cosh^2(\bar\phi/\sqrt6)$ for a canonical scalar ($n=+1$) or $\cos^2(\bar\phi/\sqrt6)$ for a phantom scalar ($n=-1$). The constants are tied by $k^2\,\mathrm{sign}\,k = n\,3C^2 + h^2\,\mathrm{sign}\,h$. The mechanism does the classification work: the zeros of the cosine conformal factor, and their ordering relative to the zeros of the seed trigonometric functions, decide where the Jordan frame develops a singularity, a horizon, a throat, an anti-throat, a second spatial infinity, or a regular centre; the Kretschmann scalar and the embedding construction certify which candidate points are genuine structures.

What would settle it

Directly integrate the Jordan-frame field equations for action (11) with $V(\phi)=0$ and a pure Maxwell source, without assuming the Einstein-frame seed, for a static spherically symmetric metric; if any solution is found that is not a conformal transform of Eqs. (15)-(17), the classification is incomplete. For the 'black universe' reading, a more specific check is to compute the Jordan-frame scale factor and matter fluxes beyond the extremal horizon of a double-horizon solution and see whether the interior is homogeneous, isotropic, and expanding rather than merely regular.

Watch

Extended reading notes

Core claim

The central discovery, stated on the paper's own terms, is that the zero-potential HMPG action (11) with a Maxwell field admits Jordan-frame exact solutions whose qualitative global structure is controlled by the sign of the scalar kinetic term and the relative size of the constants $h$ and $k$ in the seed metric (15). For a canonical scalar the conformal factor $\cosh^2(Cu+\psi_0)$ never vanishes, and the resulting spacetimes are naked central singularities, in some cases preceded by a throat or a throat-plus-anti-throat pair because the radius function develops extrema. For a phantom scalar the conformal factor $\cos^2(Cu+\psi_0)$ has zeros; depending on whether these zeros occur before, after, or together with the zeros of the trigonometric functions $s^2(h,u+u_1)$ and $s^2(k,u)$, the geometry becomes a two-way traversable wormhole, a black bounce, a black hole with one degenerate horizon and a singularity, an asymmetric wormhole with a second flat infinity, a regular repulsive surface followed by a singularity, or a solution with infinitely many double horizons. The paper supports each classification with Kretschmann-scalar regularity checks, Penrose diagrams, embedding diagrams, and the summary Table I.

Load-bearing premise

The whole classification rests on the assumption that the imported Einstein-frame seed family is the complete solution set for the zero-potential, electrically charged, spherically symmetric case; if another seed solution exists that is not a conformal transform of Eqs. (15)-(17), the Table I catalogue misses it.

Editorial extensions

If this is right

  • HMPG with zero scalar potential and electric charge is not restricted to GR-like black holes: exact, globally regular two-way traversable wormholes with two asymptotically flat ends appear in the phantom branch.
  • Black holes in this theory can have a single degenerate double horizon instead of the usual inner/outer pair, with the metric remaining regular at the horizon.
  • Black bounce geometries, throats located behind an extremal horizon, exist within the same parameter family, giving regular black holes whose singularity is replaced by a throat.
  • In the phantom branch, when the zeros of the conformal factor align with those of the seed metric, the spacetime can contain an infinite sequence of double horizons separating static regions.
  • The regular-centre subcase 3C of class [3-] produces an Anti-de Sitter-like core that avoids a central singularity entirely.

Reading between the lines

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

  • The 'black universe' label in the abstract and conclusion implies a genuinely expanding cosmological interior beyond the horizon, but the body of the paper establishes regular interiors, throats, and horizons; it does not explicitly derive a Friedmann-like scale factor or isotropic expansion for those regions, so that cosmological reading should be treated as an open expectation rather than a demo
  • Because the paper notes that a magnetic charge would give analogous results, the same catalogue should extend to dyonic and purely magnetic sources without new machinery.
  • The parameter regions that produce double horizons and black bounces could be used to compute gravitational-wave ringdown and lensing signatures that distinguish these spacetimes from Schwarzschild and Reissner-Nordström; the paper does not perform those calculations.
  • The zero-potential assumption removes the scalar potential from the field equations, but the conformal zeros that create the horizons depend on the scalar charge $\bar C$ and $\psi_0$; adding a potential will generically shift those zeros, so the stability and survival of the catalogue under small potentials is an immediate open question.
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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

3 major / 4 minor

Summary. The paper constructs exact static, spherically symmetric, electrically charged solutions in hybrid metric-Palatini gravity with zero scalar potential by conformally transforming the known Einstein-frame electrovacuum family of Refs. [85-87]. It presents the Jordan-frame line elements for the canonical (n=+1) and phantom (n=-1) scalar branches, analyzes their asymptotic behavior, Kretschmann regularity, horizon and throat structures, and provides Penrose and embedding diagrams. The claimed outcomes are traversable wormholes, black holes with double horizons, black bounces, and 'black universe' models.

Significance. If the overstatement about expanding interiors is removed, the paper is a useful exact-solution catalog for HMPG. The conformal-frame method is standard and not circular: the solutions are obtained by transforming a previously known seed family, not by fitting a target metric. The explicit line elements, the systematic class-by-class treatment of throats and horizons, and the comparison with the Simpson-Visser black bounce are concrete strengths. The main limitation is that the most prominently advertised novelty, the 'expanding cosmological' black universe, is not actually derived; the paper's own analysis shows every solution remains in the static sector with unchanged signature across double horizons. Additionally, the completeness of the classification is inherited from an unverified seed-family claim.

major comments (3)
  1. [Abstract and Sec. VI] The abstract and conclusion claim 'black universe' models in which spacetime beyond the horizon leads to an expanding cosmological solution, but no such time-dependent or interior-region solution is derived. All metrics in the paper are static and spherically symmetric (line element (13)); the horizons identified are double zeros of g00, and the text itself states in Sec. IV that the metric signature remains (+---) beyond such horizons. Hence t remains timelike and r remains spacelike across the horizon, so there is no region with a timelike radius or a scale factor. The only regular-centre solution, class [3-] subcase 3C in Sec. V F, has g00 ~ c (u - u_max)^2 and is Anti-de Sitter-like, not an expanding cosmology. I recommend removing the phrase 'expanding cosmological solution' from the abstract and conclusion, or replacing it with a precise statement about regular static interiors and black bounces.
  2. [Sec. III, Eq. (15)] The paper adopts Eqs. (15)-(17) as 'a general solution' from Refs. [85-87] without derivation or field-equation verification. Since the classification in Table I is presented as exhaustive within the listed classes, the central completeness claim rests on the seed family being the full static spherically symmetric electrovacuum solution of action (12). The authors should either state this inheritance explicitly and qualify the completeness claim, or include a verification that Eqs. (15)-(20) satisfy the field equations derived from action (12).
  3. [Sec. V, Kretschmann analysis] The regularity and singularity classification is made on the basis of K1 expressions (e.g., Eqs. (38), (42), (49)-(51), (56)-(57), (62)-(63), and (68)), with the repeated assertion that 'the remaining terms of K' follow the same behaviour. Because K is a sum of four independent components (Eq. (34)), the unsupplied K2-K4 terms are needed to rule out additional divergences at the same points. I request that the authors include these terms, or provide a supplementary computational file, so that the singularity statements can be checked directly.
minor comments (4)
  1. [Sec. IV] The paper defines a black bounce as 'a particular case of a black universe' but never gives a precise definition of 'black universe'; align this terminology with the abstract and conclusion to avoid the impression that an expanding interior is being asserted.
  2. [Eq. (33)] The mass formula is written as 'm = lim eβγ′/β′', which is ambiguous; it should be typeset as e^{β} γ'/β' with the exponential made explicit.
  3. [Eq. (21)] The condition s2(h,u1) = 1/q^2 imposes both asymptotic flatness and real-valuedness restrictions (for h < 0 it requires |q| ≥ |h|); state this restriction explicitly where u1 is introduced.
  4. [Figs. 2-4, 9-10] The z(x) versus r(x) panels would benefit from explicit axis labels and units; several figures currently rely on the caption alone to identify the plotted functions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Jordan-frame solutions follow by conformal transformation of an imported Einstein-frame solution; the 'black universe' wording is an overclaim, not a circular reduction.

full rationale

The paper's derivation chain is: HMPG action (7) is conformally transformed to the Einstein-frame action (12); the static spherically symmetric electrovacuum solution (15)-(17) is imported from refs. [85-87]; the inverse conformal transformation (24)-(27) yields the Jordan-frame line elements (36)-(67); and each class is then analyzed for horizons, throats, and singularities. Every step is an explicit mathematical operation, and no parameter is fitted to a target outcome. The imported Einstein-frame solution is not a self-citation of the present authors: refs. [85-87] are by Bronnikov and collaborators, with no author overlap with the present paper, and the paper does not invoke a uniqueness theorem to exclude alternatives. It adopts a known solution family and systematically classifies the sign combinations (22)-(23). The 'black universe' phrasing in the abstract and conclusion, which mentions 'an expanding cosmological structure' beyond the horizon, is not supported by the static solutions derived in the body; the body itself uses 'black universe' only as a generic term for regular black-bounce geometries, citing Simpson-Visser [90]. That is an overclaim or inconsistency, but it is not a circular reduction of a derived result to an input. No fitted quantity is renamed as a prediction, and no load-bearing self-citation chain forces the stated conclusions. Therefore no circular step is exhibited, and the appropriate finding is no significant circularity.

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

The central derivation rests on the imported Einstein-frame solution family, the conformal transformations, and a set of physicality cuts (m>0 and allowed psi0 ranges). There are no fitted parameters in the sense of observational data; the listed constants are integration constants of the exact solution family. No new particles, forces, or dimensions are introduced.

free parameters (4)
  • q (electric charge)
    Integration constant in the Maxwell field (16); sets the charge scale and determines u1 via Eq. (21).
  • C (scalar field slope)
    Linear coefficient in phi_bar(u) = C u + psi0; with h and sign n fixes k through Eq. (28); must satisfy m>0 constraints.
  • h
    Integration constant in the seed metric through s2(h, u+u1); controls trigonometric versus hyperbolic behavior and the class.
  • psi0 (scalar field offset)
    Nonzero offset of the scalar field at u=0; changes conformal factors (26)-(27) and asymptotics; central to the paper's claimed novelty and restricted by physicality cuts.
assumptions (4)
  • domain assumption Eqs. (15)-(17) with relation (20) is the general static spherically symmetric electrovacuum solution of the Einstein-frame action (12).
    Imported from [85-87] in Sec. III; not re-derived in this paper.
  • domain assumption The conformal transformations (26)-(27), with the Maxwell term unchanged as in Eq. (12), correctly map the HMPG Jordan frame to the Einstein frame.
    Taken from [80,85]; used to construct every Jordan-frame line element in Secs. IV-V.
  • ad hoc to paper Physical solutions are selected by imposing m>0 and by excluding some psi0 values (e.g., psi0 = pi/2 + c pi) as non-physical.
    This post-selection shapes the classification in Table I; without it, additional singularity and horizon combinations would appear.
  • domain assumption The coordinate transformations (30)-(32) and analytic continuation through zeros of the Einstein-frame metric are valid.
    Used to define the x coordinate, extend solutions beyond endpoints (case 3 in class [2-]), and support the Penrose diagrams; relies on [92,94].

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Cite this review

Pith. "Pith review of Novel electrically charged wormhole, black hole and black bounce exact solutions in hybrid metric-Palatini gravity." pith.science (2026). https://pith.science/paper/2XCLBKYX

@misc{pith2026241210324,
  author       = {Pith},
  title        = {Pith review of: Novel electrically charged wormhole, black hole and black bounce exact solutions in hybrid metric-Palatini gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XCLBKYX}},
  note         = {Machine review of arXiv:2412.10324}
}
read the original abstract

This paper presents a systematic exploration of exact solutions for electrically charged wormholes, black holes, and black bounces within the hybrid metric-Palatini gravity (HMPG) framework. HMPG combines features of the metric and Palatini formulations of modified gravity, offering a powerful approach to address challenges in General Relativity, particularly those related to cosmic acceleration and dark matter. We examine configurations characterized by a zero scalar potential under spherical symmetry, and present solutions in both the Jordan and Einstein conformal frames. A diverse set of solutions emerges, including traversable wormholes, black holes with double horizons, and ``black universe'' models in which spacetime beyond the horizon leads to an expanding cosmological solution rather than a singularity. Each configuration is categorized according to the properties of the scalar field, with an in-depth analysis of the horizon and throat structures, asymptotic behaviour, and singularities. These findings underscore the versatility of HMPG in capturing complex gravitational phenomena and broadens the scope of the theory, offering a robust framework for modelling gravitational phenomena across a range of astrophysical contexts. Future work will benefit from extending these solutions to include scalar potentials, addressing both early-universe inflation and late-time acceleration, and applying observational data, such as gravitational lensing and gravitational wave measurements.

Figures

Figures reproduced from arXiv: 2412.10324 by the authors.

Figure 1
Figure 1. FIG. 1: Left plot: Penrose diagram of a naked, light-like singularity ( [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Left plot: depicts the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Left plot [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Left plot [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Left plot: Penrose diagram of a naked light-like singularity ( [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Representation of the three cases of class [2 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Penrose diagram of a two-way traversable [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Left plot: Penrose diagram of space-time composed of an double event horizon (diagonal lines at [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Left plot [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Embedding diagrams developed from the Penrose diagrams in Fig. [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Left plot: Penrose diagram of a space-time composed of infinite double horizons (only three are drawn) [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Left plot: Penrose diagram of a black hole solution with a central, time-like singularity. In our universe are [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Embedding diagrams developed from the Penrose diagrams in Fig. [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Penrose diagram of a space-time composed of [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Left plot [PITH_FULL_IMAGE:figures/full_fig_p035_16.png]

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Forward citations

Cited by 1 Pith paper

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Reference graph

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