Toroidal black holes in four dimensions

Alejandro Jim\'enez Cano; Carlos Barcel\'o; Gerardo Garc\'ia-Moreno

REVIEW 2 major objections 5 minor 23 references

Toroidal black holes in four dimensions

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims to have constructed the first four-dimensional, asymptotically flat black hole with a toroidal event horizon and no singularities outside the horizon, by matching a Minkowski exterior to a locally vacuum Rindler tube…

desk verdict A real and novel construction, but the event-horizon claim needs a global causal proof before it earns the title 'toroidal black hole'. read the letter →

arxiv 2504.16790 v2 pith:N4M26I5O submitted 2025-04-23 gr-qc hep-th

Carlos Barceló , Gerardo García-Moreno , Alejandro Jiménez Cano This is my paper · ORCID

classification gr-qchep-th

keywords toroidalblackholeeventhorizontopologyenergyconditionsthinshellsDarmois-IsraeljunctionWeylcoordinatesRindlermetricasymptoticallyflatspacetime

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

The paper sets out to build the first explicit four-dimensional black hole spacetime whose event horizon has the topology of a torus and whose exterior region is completely free of singularities. The construction works by gluing an exterior flat region (Minkowski space with a solid torus removed) to an interior flat 'Rindler tube'—a locally vacuum geometry with toroidal spatial sections—through an intermediate region filled with a non-trivial anisotropic fluid and bounded by thin shells. The authors prove that such a matching cannot be done with a single thin shell, provide an explicit two-shell interpolation, and show that the internal shell can be eliminated while the external one remains unavoidable for their family of geometries. They also identify where the energy conditions fail, proving that the weak and null energy conditions are violated near the outer shell, consistent with the classical theorem that non-spherical horizon topologies require such violations. If correct, this resolves a long-standing gap by demonstrating that non-spherical black holes in four dimensions can exist in asymptotically flat space without naked singularities.

What carries the argument

The load-bearing object is the matched three-layer metric: an exterior flat region with a torus removed, an interpolating region with line element $ds^2=-H(\ell)d\tau^2+d\ell^2+F(\ell,\beta)d\alpha^2+b^2d\beta^2$, and an interior Rindler tube. The argument is carried by the standard junction conditions, which fix the thin-shell energy-momentum tensors on the two matching surfaces; the key explicit result is Lemma 1 and Theorem 2, which show that the scalar $Z_\vartheta$ obtained by contracting the Einstein tensor with the vector field $H(\ell)^{-1/2}\partial_\tau + \vartheta b^{-1}\partial_\beta$ is asymptotically negative near $(\ell,\beta)=(L_+,\pi)$, implying weak and null energy condition violations there. The impossibility of a single-shell matching is proven by the geometric fact that a flat torus cannot be embedded $C^2$-isometrically in Euclidean 3-space, so the induced metrics from the two sides would differ.

What would settle it

Compute the causal past of future null infinity for the matched spacetime, for example by numerical null-geodesic integration from the outer shell inward: if the $\ell=0$ surface is not its boundary, or if null rays from the $\ell<0$ region reach the asymptotic flat exterior, then the constructed spacetime is not a black hole but a flat tube with an acceleration horizon, and the central claim collapses.

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Extended reading notes

Core claim

The central claim is that the three-region matched spacetime—Minkowski exterior minus a torus, the interpolating region with metric $ds^2 = -H(\ell)d\tau^2 + d\ell^2 + F(\ell,\beta)d\alpha^2 + b^2d\beta^2$, and the Rindler tube—provides the first explicit example of a four-dimensional toroidal black hole free of curvature singularities in the external region. The interior tube is locally vacuum and its $\ell=0$ boundary is identified as the event horizon; the matching surfaces $\Sigma_-$ and $\Sigma_+$ are the only places, together with the interpolating region, where matter (including thin-shell distributional matter) is present. The authors further claim that the external thin shell cannot be removed within the class of geometries they consider, while the internal shell can be suppressed by a suitable choice of interpolating functions, and that energy-condition violations are generically concentrated near the outer shell at the point $\beta=\pi$ (the inner equator of the torus).

Load-bearing premise

The construction treats the $\ell=0$ surface of the interior Rindler tube as the event horizon of the full matched spacetime, yet it never proves that this surface is the boundary of the causal past of future null infinity for the glued manifold, where the analogous flat-embedding surface is merely an acceleration horizon.

Editorial extensions

If this is right

A direct corollary is that a toroidal black hole horizon in four dimensions does not force a naked singularity outside, provided the exterior contains matter that violates the null and weak energy conditions.
The internal thin shell can be removed by imposing $H'(L_-)=2/L_-$ and $\partial_\ell F(L_-,\beta)=0$, making the construction rely on a single unavoidable external shell.
In any geometry of the form $ds^2=-H(\ell)d\tau^2+d\ell^2+F(\ell,\beta)d\alpha^2+b^2d\beta^2$ satisfying the hypotheses, weak energy condition violations occur in an open set arbitrarily close to the outer shell at $\beta=\pi$; if additionally $H'(L_+)=H''(L_+)=0$, the null energy condition is violated there.
The polynomial ansatz (4.37) provides an explicit, fully regular example of the construction, with the matter content consisting of a type-I anisotropic fluid plus the external shell.
The matching requires $b=m$ (equating the tube radius to the Rindler transverse period parameter), which is a necessary condition for the internal shell to have zero induced curvature mismatch.

Reading between the lines

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

Extension: if the $\ell=0$ surface is confirmed to be a true event horizon of the glued spacetime (the causal boundary of the past of future null infinity), the construction extends the no-hair paradigm beyond spherical topology in a direction the authors do not develop dynamically.
Extension: the same interpolation strategy could be adapted to replace the flat exterior with an asymptotically flat Schwarzschild or Curzon exterior, which the authors mention only as future work; a testable consequence would be a one-parameter family of toroidal black holes with tunable mass and torus parameters.
Extension: the persistence of null-energy-condition-violating matter near the inner equator $\beta=\pi$ suggests that any regular torus-to-flat transition must have a local negative-energy region there, a feature that numerical searches for such spacetimes could look for.
Extension: linear stability of the shells and fluid is not analyzed here; a natural follow-up would be a perturbation analysis of the matched construction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The construction is real and the mathematics checks out, but the paper's central claim—that this is a toroidal black hole—rests on a global causal identification that they never actually prove. The three-region matching is a legitimate piece of exact-solutions work: flat exterior with a torus removed, an interpolating anisotropic-fluid region, and an inner Rindler tube with toroidal sections. The Darmois-Israel junction conditions in Sec. 3 are worked carefully, the boundary conditions on the polynomial ansatz (4.37) hold, and the WEC/NEC violation asymptotics in Theorem 2 are supported by the expansions. The proof that a single flat-to-curved torus shell is impossible (Sec. 3.1) is clean and worth stealing for lectures.

The novelty is real. Peters's earlier attempt had naked singularities in the exterior; the Λ≠0 solutions are not locally vacuum; the five-dimensional phantom-brane examples are not 4D. This is the first explicit four-dimensional locally-vacuum toroidal horizon with a regular asymptotically-flat exterior, as far as the cited literature shows. The authors are appropriately careful about the energy-condition violations, and they correctly interpret them as required by Hawking's theorem.

The soft spot is precisely the event-horizon identification. In Sec. 2.1 they equate infinite-redshift surfaces in Weyl coordinates with event horizons. For the matched spacetime they never compute the causal boundary, never construct the Penrose diagram, and never specify what lies beyond ℓ=0. In the flat embedding, the Rindler ℓ=0 surface is an acceleration horizon, not an event horizon. To claim 'first explicit example of a toroidal black hole,' they need to show that ℓ=0 is ∂J⁻(I⁺) of the completed spacetime, or at least explicitly define the interior extension. Without that, the construction is an exotic static axisymmetric spacetime with a Rindler tube and a boundary—not yet a black hole.

That said, the gap is fixable, and the paper is not sloppy. It is a genuinely useful construction. I'd send it to a referee with a request that the authors either provide a global causal analysis or soften the abstract to 'locally vacuum toroidal horizon with regular exterior.' The xAct notebook being only 'available upon request' is a minor reproducibility ding; a public notebook would be better.

Verdict: this deserves peer review, not desk rejection. The referee should focus on the global horizon question and the completeness of the spacetime across ℓ=0.

Referee Report

2 major / 5 minor

Summary. The paper constructs static, axisymmetric four-dimensional spacetimes intended to describe toroidal black holes. The spacetime is built by matching three regions: an exterior flat Minkowski region with a torus removed (Omega_1), an intermediate region with an anisotropic fluid interpolating between a curved torus and a flat torus (Omega_2), and an interior Rindler tube with toroidal sections (Omega_3). The Darmois-Israel junction conditions are used to compute distributional energy-momentum tensors on the matching surfaces, and the energy conditions are analyzed, with particular attention to null and weak energy condition violations near the external shell. The authors claim that this is the first explicit example of a four-dimensional toroidal black hole free of singularities in the external region.

Significance. If the construction is a genuine black hole spacetime, it would be a valuable explicit model of a toroidal horizon in four dimensions, complementing the local Geroch-Hartle classification and demonstrating how Hawking's topology theorem is satisfied through energy-condition violations in the exterior. The Darmois-Israel computations appear internally consistent, the boundary conditions of the explicit ansatz (4.37) check out, and the energy-condition analysis in Sec. 4 is rigorous and supported by asymptotic expansions. However, the central black-hole interpretation rests on an unproven global causal premise, which is the main weakness of the manuscript.

major comments (2)

[Sec. 2.1; Sec. 3.2; Sec. 3.3] The central claim that the matched spacetime is a toroidal black hole is not supported by a global causal analysis. Section 2.1 equates event horizons with infinite-redshift surfaces, and Sections 3.2-3.3 treat the ℓ=0 boundary of the Rindler tube as the horizon without proving that this surface is ∂J⁻(I⁺) of the complete spacetime. In the flat-space embedding of Eqs. (2.9)-(2.10), the analogous surface is an acceleration horizon whose null generators reach future null infinity, so local Rindler character alone does not identify a black hole horizon. The authors should either extend the spacetime across ℓ=0 and prove that ℓ=0 is indeed the boundary of the causal past of the exterior's future null infinity, or explicitly adopt a local definition of 'horizon' and adjust the abstract's 'black hole' claim accordingly.
[Sec. 3.2; Sec. 3.3] The spacetime as defined comprises only ℓ>0, leaving the interior beyond ℓ=0 unspecified. Consequently, the manifold is geodesically incomplete at ℓ=0, and the 'horizon' is a boundary of the manifold rather than a regular null surface contained in the spacetime. To be a complete black hole solution, the authors need to specify a maximal extension across ℓ=0 and analyze the causal structure of the extended spacetime. Without this, the construction is at best a regular exterior with a would-be horizon, not a complete black hole spacetime.

minor comments (5)

[Eq. (3.37)] The notation '∂ℓF(L+,β +)' appears to contain a stray plus sign; it should read ∂ℓF(L+,β) for the limit ℓ↗L+.
[Fig. 8 and Fig. 9 captions] The sentence 'For the plots, we took L− = 1 L+ = 10' is missing a comma; it should read 'L− = 1, L+ = 10'.
[Proof of Lemma 1] The phrase 'the suppressed terms are order 1 in β−π and/or order 1 in ℓ−L+' is imprecise; it should state that the suppressed terms are O(1) as β→π and ℓ→L+.
[Acknowledgments] The paper states that the xAct notebook is 'available upon request'; making it publicly available (e.g., as ancillary files) would strengthen reproducibility.
[Sec. 2.1] The statement that 'the event horizon of a static configuration corresponds to a surface of infinite redshift' is a coordinate-dependent local characterization; given the central claim of the paper, the authors should emphasize that this is not by itself a proof of a global event horizon.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the metric defines the stress tensor, energy-condition violations are computed, and the cited author-overlapping result is context rather than input.

full rationale

The paper's construction is self-contained and reverse-engineered: it posits three regions (flat exterior minus torus, interpolating region, Rindler tube) and defines the stress-energy tensor via Einstein equations, so the matter distribution is the output of the metric choice, not an input fitted to a target. The Darmois-Israel junction conditions are then solved, or shown to require shells, from the metric data. The energy-condition violations are proved by asymptotic expansions using the stated boundary conditions, not assumed. The only author-overlapping citation, Ref. [2], is used for context about distorted toroidal horizons and for the flat-coordinate transformation (2.9)-(2.10), both of which are independently exhibited in the paper; it does not force the toroidal horizon construction. The asserted event-horizon interpretation of l=0 is a global causal claim that the paper does not fully demonstrate, but that is a correctness or support gap, not a circular reduction of a prediction to an input.

Assumptions & free parameters 3 free parameters · 6 assumptions · 2 invented entities

The central construction rests on standard mathematical theorems plus one domain assumption: that the Rindler horizon in Ω3 is a genuine black hole event horizon of the completed spacetime. The remaining inputs are the hand-chosen geometric parameters L+, L-, b and the ad hoc polynomial interpolation profiles. No free parameters are fitted to data, but the matter content is entirely reverse-engineered from the metric.

free parameters (3)

L+
Major radius of the external torus Σ+; chosen by hand, with L+>b required for a non-degenerate torus. Examples take L+=10.
L-
Radius of the inner flat-torus matching surface Σ-; chosen by hand with 0<L-<L+. Examples take L-=1.
b
Minor radius of the external torus; after Eq. (3.51), b=m, so it sets the transverse scale of the Ω3 torus. Examples take b=1,2,3.

assumptions (6)

standard math Geroch-Hartle characterization: any static, axisymmetric, locally vacuum toroidal horizon can be written in Weyl form (2.1) with U,V solving (2.2)-(2.4).
Used in Sec. 2.1 and Sec. 3 to set the inner geometry; cited to Refs. [5,6].
standard math Darmois-Israel junction conditions (3.11) determine the distributional stress tensor on the matching hypersurfaces and are valid for the C^1 matchings used.
Used throughout Sec. 3 to compute shell sources at Σ+ and Σ-.
standard math Hawking's topology theorem (Ref. [4], Prop. 9.3.2): toroidal horizons in four dimensions require violations of energy conditions in the exterior.
Used in Sec. 2 and Sec. 4 to motivate and locate energy-condition violations; also frames the claimed novelty.
ad hoc to paper The interpolating ansatz (3.5), specialized to (3.7)-(3.8), is flexible enough to represent the transition from a flat torus to a curved torus.
This is a modeling choice; the paper does not prove all possible interpolations reduce to this form, and for this family the external shell is unavoidable.
domain assumption The ℓ=0 Rindler surface is a black hole horizon of the complete spacetime, not merely a local Killing horizon.
Assumed in calling Ω3 a locally vacuum toroidal black hole; not proved by a global causal analysis. This is the flagged weakest assumption.
standard math A C^2 flat torus cannot be embedded isometrically in R^3, used in Sec. 3.1 to rule out a single-shell construction.
Proof by contact with a sphere; the C^1 case is cited to Nash and Kuiper.

invented entities (2)

Exotic anisotropic fluid in Ω2
purpose: To source the interpolating geometry between the flat torus and the curved torus via T_ab=G_ab.
No equation of state or microphysical model is provided; the stress tensor is defined by the chosen metric, so there is no independent observational handle.
Thin shells on Σ+ and, in the two-shell model, on Σ-
purpose: To localize the distributional energy-momentum required by the discontinuity in extrinsic curvature.
The shell stress tensors (3.43)-(3.45) and (3.58)-(3.60) are computed from the junction conditions; they are idealized surface layers with no independent evidence.

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Cite this review

Pith. "Pith review of Toroidal black holes in four dimensions." pith.science (2026). https://pith.science/paper/N4M26I5O

@misc{pith2026250416790,
  author       = {Pith},
  title        = {Pith review of: Toroidal black holes in four dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N4M26I5O}},
  note         = {Machine review of arXiv:2504.16790}
}

read the original abstract

From a purely geometric (kinematic) perspective, black holes in four dimensional spacetimes can have event horizons with arbitrary topologies. It is only when energy conditions are imposed that the horizon's topology is constrained to be that of a sphere. Despite this, exploring exotic horizon topologies remains theoretically intriguing since it allows to unveil structural aspects of General Relativity and gain intuition on energy condition violations. In the axisymmetric case, besides the well-known spherical topology, only a toroidal topology is consistent with the symmetry. Complete solutions, describing the entire exterior region of such toroidal black holes without singularities, have not been reported yet. To the best of our knowledge, the construction we present here is the first explicit example of a toroidal black hole solution in four spacetime dimensions that is free of singularities in the external region.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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P. T. Chrusciel, J. Lopes Costa and M. Heusler,Stationary Black Holes: Uniqueness and Beyond, Living Rev. Rel.15 (2012) 7, [1205.6112]

arXiv 2012

[2] [2]

No-hair and almost-no-hair results for static axisymmetric black holes and ultracompact objects in astrophysical environments

C. Barceló, R. Carballo-Rubio, L. J. Garay and G. García-Moreno,No-hair and almost-no-hair results for static axisymmetric black holes and ultracompact objects in astrophysical environments, Class. Quant. Grav.42 (2025) 075020, [2410.08128]

work page Pith review arXiv 2025

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arXiv 2015

[4] [4]

S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2, 2023, 10.1017/9781009253161

doi:10.1017/9781009253161 2023

[5] [5]

R. P. Geroch and J. B. Hartle,Distorted black holes, J. Math. Phys.23 (1982) 680

work page 1982

[6] [6]

B. C. Xanthopoulos,Local Toroidal Black Holes That are Static and Axisymmetric, Proceedings of the Royal Society of London Series A388 (July, 1983) 117–131

work page 1983

[7] [7]

K. S. Thorne,A Toroidal Solution of the Vacuum Einstein Field Equations, J. Math. Phys. 16 (1975) 1860–1865

work page 1975

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P. C. Peters,Toroidal black holes?, Journal of Mathematical Physics20 (07, 2008) 1481–1485

work page 2008