Ellis-Bronnikov wormholes with NUT charge

Keisuke Nakashi; Masato Nozawa; Tomohiro Harada

REVIEW

Ellis-Bronnikov wormholes with NUT charge

Masato Nozawa, Tomohiro Harada, Keisuke Nakashi

General Relativity and Quantum Cosmology · Submitted 5 Oct 2026

Pith's verdict: Genuinely new NUT-charged Ellis-Bronnikov family in the ALF sector, checkable and internally consistent; the AdS extension rests on an asserted substitution and an unpublished citation that a referee should push on.

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arXiv ID2610.05656 (version 2)

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CitationKeisuke Nakashi, Masato Nozawa, Tomohiro Harada. "Ellis-Bronnikov wormholes with NUT charge." arXiv:2610.05656 (2026). https://pith.science/paper/CKEW5HKD

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Claim paper

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Editor's note

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The ALF sector is solid and genuinely new; the AdS sector is where a referee should push. The family (2.20) is a clean Ehlers rotation of the Ellis-Bronnikov seed, and the key steps check out: substituting (2.17) into (2.13) reproduces f and psi in (2.18)-(2.19), and the throat equation (3.23) plus the photon-surface equation (3.45) follow directly from dS/dr = 0 and dV_eff/dr = 0. The charge algebra (3.19), the vanishing-NUT limits, and the agreement of the Hamiltonian charge with the published holographic computation [67] all support the central claims. The authors are also honest about the pathologies - CTCs, Dirac-Misner string, energy-condition violation - and they do flag the distributional-singularity caveat in the body (b91-b92), even though the abstract's "free of curvature singularities" glosses it.

Soft spots, in proportion. The main one is the AdS generalization in Section IV B. The paper states that substituting (4.11)-(4.13) into the field equations with the engineered potential (4.7)-(4.8) yields Delta_sigma(r) = (4.13), but no derivation is shown, and the analogous hairy-Taub-NUT prescription is cited to the authors' own unpublished [59] (arXiv:2609.32139). The n -> 0 check reducing Delta_sigma to the static branches (4.6) is necessary but not sufficient. This doesn't collapse the paper - the ALF existence claim stands on its own - but the AdS wormhole, the vacua interpolation, and the mass formulas (4.26), (4.33) all hang on that single asserted substitution. A referee should ask for it to be verified or backed by a published source. The authors do openly acknowledge that the potential is engineered and beta thereby promoted to a theory parameter (b227-b228), so this is a verification gap, not a hidden defect.

Second, the wormhole interpretation depends on the orbit-space throat convention (3.22) following [26,30]. Without that convention, the spacetime remains an exact ALF stationary NUT solution, but "throat" is a three-dimensional-orbit-space construct rather than a minimal surface of the 4D metric. The authors state this plainly, and their horizonlessness and geodesic-completeness claims don't depend on it. Third, minor: the abstract should carry the body's distributional-singularity caveat.

Who this is for: exact-solutions and classical-gravity readers working on NUT charge, wormhole geometry, geodesic structure, and designer-gravity mass computations, plus people interested in ambi-Kahler Euclidean conformal classes. I'd send it to a serious referee with instructions to check the AdS substitution; the ALF sector is worth reporting as is.

Referee report

Significance

If the claims are correct, the ALF solution is the first exact Ellis-Bronnikov-type wormhole with NUT charge and a genuinely new member of the Taub-NUT family. The coset-based derivation is a strength: the transformation argument from the SL(2,R)/SO(2) structure of the reduced action is coherent, and the charges satisfy the independent check (3.19), the vanishing-NUT limits are consistent, and the Hamiltonian charge (4.26) is stated to agree with the published holographic computation [67]. The ALF sector is further supported by direct substitutions leading to (2.18)-(2.19) and by the independent throat and photon-surface equations. The AdS sector is more fragile: the existence of the solution hinges on the asserted substitution producing (4.13), which is not displayed or derived in the manuscript. The paper is well organized and, in the ALF sector, appears internally consistent.

Major objections (2)

[Section IV B, Eq. (4.13)] The existence of the asymptotically locally AdS wormhole rests entirely on the sentence "Substituting the above configurations into field equations derived from the Lagrangian (4.3) with a potential (4.7) and (4.8), we find Δσ(r) = (4.13)." No substitution computation is shown, the equations of motion for (4.3) are not displayed, and the nearby reference [59] is an unpublished arXiv preprint describing a related hairy Taub-NUT construction rather than providing a published verification of this exact model. Every subsequent AdS claim—positivity of Δσ, interpolation between AdS vacua, the asymptotic expansions (4.14)-(4.19), and the Hamiltonian charges (4.26) and (4.33)—uses (4.13) as input, so an error there would invalidate the whole AdS construction. The authors' acknowledgement that the potential is engineered so that the ansatz works correctly identifies the restriction of the method, but it does not supply the missing check; the vanishing-NUT limit reducing (4.13) to (4.6) is necessary but not sufficient. I request that the substitution be shown explicitly, for instance in an appendix or in an accompanying computer algebra file, or that a published reference containing this exact verification be cited.
[Section III C, Eqs. (3.22)-(3.26)] The wormhole throat is defined as the minimum of the orbit-space areal radius S(r) = [(r^2 + m0^2)/f(r)]^(1/2), rather than as a minimal surface or a minimum areal radius of round two-spheres in the full four-dimensional spacetime. The paper correctly explains that the NUT fibration prevents globally defined round S^2 orbits, but the abstract and Section V call the spacetime a traversable wormhole without carrying this qualification. Since equations (3.23)-(3.26) and the throat-versus-photon-surface comparison (3.46) are statements about the reduced orbit space, the authors should state explicitly, at the point where the throat is defined, that the traversable-wormhole characterization is made with respect to the orbit-space convention of [26,30]. This is not an inconsistency with the definitions actually used, but the terminology should be pinned down to avoid overclaiming the geometric status of the throat.

Minor objections (4)

[Abstract and Section III D] The phrases "free of curvature singularities" in the abstract and conclusion should be qualified, since Section III D establishes only the absence of scalar-curvature singularities and leaves the Dirac-Misner string as a distributional singularity; the authors' own distinction in Section III A already suggests the wording "free of scalar-curvature singularities".
[Figure 1 and Section II C] The caption of Figure 1 gives m0 = 1 and beta = 0.6 but not the values of m and n for the three curves; since (2.17) determines them from the rotation angle, please state the resulting (m,n) pairs explicitly or add them to the caption.
[Section III B, Eqs. (3.7)-(3.8)] The notation phi^(infinity) is used for the scalar charge in (3.7)-(3.8) while an asymptotic field value appears in (2.9); please disambiguate the charge from the asymptotic scalar value, for example by writing phi_charge and phi_infinity.
[References] Several entries, including [5], [25], [42], [59], [72], and [81], are listed only by arXiv identifiers without a clear publication status; please provide journal references or state explicitly that they are preprints to appear.

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The paper

The paper claims

This paper constructs the first exact NUT-charged Ellis-Bronnikov wormhole, a horizonless, geodesically complete spacetime that nevertheless carries the Dirac-Misner string and closed timelike curves of Taub-NUT.

The reading

The paper constructs an exact solution of the Einstein-phantom-scalar system that merges two previously separate geometries: the traversable Ellis-Bronnikov wormhole and the gravitomagnetic NUT charge of Taub-NUT spacetime. The claim is that this is the first exact NUT-charged Ellis-Bronnikov wormhole, obtained not by solving the nonlinear field equations head-on but by applying an Ehlers-type rotation to the static wormhole seed. The result is a stationary, asymptotically locally flat (ALF) spacetime at both ends, free of curvature singularities, and geodesically complete, because the metric function $f(r)$ stays strictly positive and no Killing horizons form. It nevertheless inherits the Dirac-Misner string and closed timelike curves everywhere, the characteristic causal pathologies of Taub-NUT, and its throat is defined in the orbit space of the stationary Killing field. The same construction is extended to an asymptotically locally AdS wormhole whose phantom scalar interpolates between distinct AdS vacua, with finite Hamiltonian charges obtained by covariant phase space methods.

What carries the argument

The load-bearing machinery is the coset-space reduction of the stationary Einstein-phantom-scalar system together with the Ehlers transformation. The orbit-space reduction (2.5)–(2.8) recasts the field equations as a nonlinear $\sigma$ model on $\mathbb{R}\times SL(2,\mathbb{R})/SO(2)$ whose target metric (2.8) is invariant under a global SO(2) rotation (2.12). Applying this rotation to the static Ellis-Bronnikov seed (2.14), with the angle set by $\cos(2a)=m/m_0$ and $\sin(2a)=n/m_0$, produces the NUT-charged wormhole algebraically, thereby avoiding direct solution of the nonlinear system.

Extended reading notes

Core claim

The central discovery is the explicit metric (2.20) with $f(r)$ given by (2.18), claimed to be an exact solution of (2.2) describing a traversable wormhole carrying a NUT charge. The solution is generated by applying the Ehlers SO(2) rotation (2.13) of the coset-space $\sigma$ model to the static Ellis-Bronnikov wormhole seed (2.14), with the rotation angle fixed by $m_0=\sqrt{m^2+n^2}$. Because $f(r)>0$ on all of $\mathbb{R}$, the spacetime has no Killing horizons, is asymptotically locally flat at both ends, is of Petrov type D, and is geodesically complete. It inherits the Dirac-Misner string and ubiquitous closed timelike curves from the Taub-NUT geometry, and its throat is defined as the minimum of the orbit-space areal radius $S(r)$ in (3.22). The paper further builds an asymptotically locally AdS NUT-charged wormhole using the scalar potential (4.7) with an infinite sequence of AdS vacua, proves that the Hamiltonian charges (4.26) and (4.33) are finite, and shows that the Euclidean continuation's conformal class admits an ambi-Kähler structure.

Load-bearing premise

The wormhole characterization rests on defining the throat as the minimum of the orbit-space areal radius $S(r)$ in (3.22), following the convention of [26, 30]; if the throat must instead be a genuine minimal surface of the full 4D spacetime, the traversable-wormhole reading loses its geometric footing and the solution remains an exact ALF stationary NUT solution.

Editorial extensions

If this is right

The solution overlaps traversable wormhole physics with gravitomagnetic charge for the first time in an exact setting, giving a concrete arena for lensing and photon-surface studies in a wormhole that carries a Dirac-Misner string.
Because $f(r)>0$ everywhere, the spacetime avoids the horizon-induced infinite winding of geodesics found in the ordinary Taub-NUT spacetime and is geodesically complete.
For parameters with $m<0$ and $0<\beta<\beta_c$ with $\beta_c=\frac{1}{2\pi}\ln\frac{m_0-m}{m_0+m}$, the photon surface lies inside the throat relative to the $r\to+\infty$ end but outside it relative to the other end, a concrete arrangement for observational tests.
The asymptotically locally AdS generalization interpolates between distinct AdS vacua at its two ends, giving exact designer-gravity wormhole data on which scalar boundary conditions the Hamiltonian charge depends.
The Euclidean continuation's conformal class admits an ambi-Kähler structure, connecting the solution to gravitational instanton studies.
The conserved quantities satisfy $M_+^2+N_+^2=m_0^2\beta^2$ and $M_-^2+N_-^2=\bar m_0^2\beta^2\kappa$, tying the masses and NUT charges of the two ends to the wormhole parameters.
For null geodesics with $\mu=C=0$, the principal null congruences (3.48) are shear-free, placing the wormhole in the same algebraically special class as ordinary Taub-NUT.
The wormhole is horizonless and geodesically complete, so it provides a clean laboratory for studying CTCs and the Dirac-Misner string without the complications of Killing horizons.

Reading between the lines

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

The same Ehlers-type rotation likely generates NUT-charged versions of other static phantom-scalar wormholes, such as the accelerating C-metric wormholes, extending the family beyond the Ellis-Bronnikov seed.
Because the NUT fibration prevents globally defined round two-spheres, the throat is defined only in the orbit space; a direct minimal-surface computation on the full 4D metric (2.20) would test whether the orbit-space throat is a genuine minimal surface of the spacetime.
The radial instability known for the ordinary Ellis-Bronnikov wormhole should be re-examined with a nonzero NUT parameter: the absence of Killing horizons and the presence of a Dirac-Misner string may alter the unstable mode spectrum qualitatively.
The decrease of the equatorial critical impact parameter with increasing $n$ suggested by figure 1 offers a testable prediction for wormhole-mimicker searches, connecting this solution to ongoing black-hole-shadow observations.

Original abstract

We construct an exact NUT-charged generalization of the Ellis-Bronnikov wormhole in the Einstein-phantom-scalar system. The solution is stationary, asymptotically locally flat at both ends and free of curvature singularities. It inherits characteristic features of the Taub-NUT geometry, including the Dirac-Misner string and closed timelike curves, while the absence of Killing horizons eliminates the horizon-induced infinite winding of geodesics and renders the spacetime geodesically complete. By allowing the phantom field to have a nontrivial scalar potential with an infinite sequence of AdS vacua, we also obtain an exact asymptotically locally AdS wormhole with a NUT charge, which interpolates between different vacua at its two asymptotic ends. Using the covariant phase space formalism, we show that the Hamiltonian charges associated with the two asymptotic regions are finite under appropriate scalar boundary conditions. By performing a double Wick rotation to Euclidean signature, we demonstrate that the conformal class of the resulting metric admits an ambi-K\"ahler structure.

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.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-06: 1 request, 5,103 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.34 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We construct an exact NUT-charged generalization of the Ellis-Bronnikov wormhole in the Einstein-phantom-scalar system. The solution is stationary, asymptotically locally flat at both ends and free of curvature singularities. It inherits characteristic features of the Taub-NUT geometry, including the Dirac-Misner string and closed timelike curves, while the absence of Killing horizons eliminates the horizon-induced infinite winding of geodesics and renders the spacetime geodesically complete. By allowing the phantom field to have a nontrivial scalar potential with an infinite sequence of AdS vacua, we also obtain an exact asymptotically locally AdS wormhole with a NUT charge, which interpolates between different vacua at its two asymptotic ends. Using the covariant phase space formalism, we show that the Hamiltonian charges associated with the two asymptotic regions are finite under appropriate scalar boundary conditions. By performing a double Wick rotation to Euclidean signature, we demonstrate that the conformal class of the resulting metric admits an ambi-K\"ahler structure.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We construct an exact NUT-charged generalization of the Ellis-Bronnikov wormhole in the Einstein-phantom-scalar system.	contribution 1.00	0.97	0.51	asserted 1.00
s1	The solution is stationary, asymptotically locally flat at both ends and free of curvature singularities.	result 0.97	0.86	0.15	asserted 1.00
s2	It inherits characteristic features of the Taub-NUT geometry, including the Dirac-Misner string and closed timelike curves, while the absence of Killing horizons eliminates the horizon-induced infinite winding of geodesics and renders the spacetime geodesically complete.	result 0.80	0.79	0.22	asserted 1.00
s3	By allowing the phantom field to have a nontrivial scalar potential with an infinite sequence of AdS vacua, we also obtain an exact asymptotically locally AdS wormhole with a NUT charge, which interpolates between different vacua at its two asymptotic ends.	contribution 0.60	0.96	0.21	asserted 1.00
s4	Using the covariant phase space formalism, we show that the Hamiltonian charges associated with the two asymptotic regions are finite under appropriate scalar boundary conditions.	result 0.74	0.97	0.11	asserted 1.00
s5	By performing a double Wick rotation to Euclidean signature, we demonstrate that the conformal class of the resulting metric admits an ambi-K\"ahler structure.	result 0.61	0.95	0.08	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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references and citing sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Typical behavior of effective potential for null geodesics. The plot is shown for

Figures reproduced from arXiv:2610.05656 by the authors.

References

Discussion

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