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arxiv: 2605.27600 · v2 · pith:Z3YCQH42new · submitted 2026-05-26 · 🌀 gr-qc · hep-ph· hep-th

Stationary generalizations for the vacuum ring wormhole

Mikhail S. Volkov This is my paper

Pith reviewed 2026-06-29 15:20 UTC · model grok-4.3

classification 🌀 gr-qc hep-phhep-th
keywords ring wormholestationary wormholesErnst equationsnumerical solutionsangular momentum relationsextremal Kerrscalarization
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The pith

Stationary ring wormholes satisfy M ~ J² for slow rotation and J = M² for fast rotation while approaching extremal Kerr when the ring shrinks to zero.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs stationary generalizations of the vacuum ring wormhole that remain invariant under reflections across the throat. These generalizations are obtained by solving the vacuum Ernst equations numerically under the corresponding symmetry conditions. The resulting solutions display a non-relativistic mass-angular momentum relation M ~ J² at slow rotation that transitions to the Regge relation J = M² at fast rotation, where centrifugal force stretches the ring without bound. When the static ring size is reduced to zero simultaneously, both M and J stay bounded as the throat velocity approaches the speed of light and the geometry approaches the extremal Kerr solution. The ring singularity can be eliminated by a scalarization procedure that introduces a phantom scalar field.

Core claim

The non-perturbative numerical solutions exhibit the non-relativistic relation M ~ J² for slow rotation, which transforms into the Regge relation J = M² in the fast-rotation regime; if the ring size is sent to zero simultaneously then M and J remain bounded as the throat linear velocity approaches unity and the geometry approaches the extremal Kerr solution.

What carries the argument

Numerical solutions to the vacuum Ernst equations subject to reflection symmetry across the wormhole throat.

If this is right

  • Slow rotation produces the non-relativistic scaling M ~ J² between mass and angular momentum.
  • Fast rotation produces the Regge relation J = M² while the ring stretches without bound due to centrifugal force.
  • Sending the static ring size to zero together with increasing rotation keeps M and J bounded as throat velocity approaches the speed of light.
  • The geometry approaches the extremal Kerr solution in that simultaneous limit.
  • The ring singularity can be removed by promoting the solutions to regular configurations with a phantom scalar field.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The imposed reflection symmetry distinguishes these wormholes from the Kerr family while still allowing them to approach Kerr in a controlled limit.
  • The bounded M and J in the shrinking-ring limit suggest these geometries could serve as singularity-free proxies for studying extremal Kerr properties.
  • Scalarization may open a route to fully regular, traversable wormholes whose asymptotic behavior matches that of extremal black holes.

Load-bearing premise

The numerical solutions satisfy the vacuum Einstein equations everywhere outside the ring singularity and respect the imposed reflection symmetry across the throat for all values of the rotation parameter.

What would settle it

A computation demonstrating that the numerical solutions violate the vacuum Einstein equations or the reflection symmetry condition for some value of the rotation parameter would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.27600 by Mikhail S. Volkov.

Figure 1
Figure 1. Figure 1: FIG. 1. Particles entering the ring are not seen coming from the other side [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The Kerr metric describes a wormhole that is not symmetric under swapping the asymptotic [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Profiles of [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Left: the ADM mass [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Parameters of spinning solutions. [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The value of [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plots of [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The ratios [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The isometric embeddings of the wormhole throat for the vacuum (left) and scalar-dressed [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The isometric embeddings of the equatorial sections of wormholes with [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The isometric realization of slices of the [PITH_FULL_IMAGE:figures/full_fig_p042_11.png] view at source ↗
read the original abstract

The ring wormhole is the zero-mass limit of the Kerr metric. Its geometry is locally flat, but the topology is nontrivial, with a throat connecting two asymptotic regions and a distributional curvature singularity on the ring encircling the throat. We construct stationary generalizations of this static wormhole that are different from Kerr and invariant under reflections across the wormhole throat. The problem reduces to solving the vacuum Ernst equations subject to the corresponding symmetry conditions. The slowly rotating perturbative solutions were constructed previously, while we now present a detailed analysis of non-perturbative solutions obtained within a numerical framework. For slow rotation, they exhibit the non-relativistic relation $M\sim J^2$ between the mass and angular momentum, which transforms into the Regge relation $J=M^2$ in the fast-rotation regime, when $J\to\infty$ and the ring is stretched without bound by the centrifugal force. However, if the ring size in the static limit is sent to zero at the same time, then $M$ and $J$ remain bounded as the throat linear velocity approaches unity. The wormhole geometry then approaches the extremal Kerr solution, thus ``mimicking'' it. The wormholes carry a curvature singularity at the ring, but this can be removed by via simple ``scalarization'' procedure that promotes the vacuum solutions to regular wormholes with a phantom scalar field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper constructs stationary, reflection-symmetric generalizations of the vacuum ring wormhole (zero-mass limit of Kerr) by numerically solving the vacuum Ernst equations. It reports that the resulting non-perturbative solutions obey M ∼ J² for slow rotation, transition to the Regge relation J = M² for fast rotation (with the ring stretched by centrifugal effects), and, when the static ring radius is simultaneously sent to zero, yield bounded M and J with the throat velocity approaching unity and the geometry approaching extremal Kerr. The ring singularity can be removed by a scalarization procedure introducing a phantom scalar field.

Significance. If the numerical solutions are confirmed to satisfy the Ernst equations to controlled accuracy, the work supplies explicit non-perturbative examples of rotating wormhole geometries that are distinct from Kerr yet can mimic its extremal properties in a controlled limit. This would be a concrete illustration of how nontrivial topology can produce stationary vacuum solutions whose asymptotic charges track those of black holes, complementing the earlier perturbative analysis.

major comments (1)
  1. [Numerical framework and boundary conditions] Numerical framework and boundary conditions: the central claims (M ∼ J² to J = M² transition, bounded M,J in the simultaneous zero-ring-size limit, and approach to extremal Kerr) rest on the assertion that the numerical solutions satisfy the vacuum Ernst equations everywhere outside the ring and respect the imposed reflection symmetry for all rotation parameters. However, the manuscript provides no quantitative validation—maximum residual norms, grid-refinement convergence studies, or direct comparison of the slow-rotation numerics against the known perturbative expansion. Without these, discretization artifacts cannot be ruled out as the source of the reported relations.
minor comments (1)
  1. The scalarization procedure that removes the ring singularity is mentioned only briefly; a short explicit description of the phantom scalar ansatz and the resulting regular metric would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for recognizing the potential interest of the non-perturbative rotating ring wormhole solutions. We agree that quantitative numerical validation is necessary to support the central claims and will incorporate the requested checks in the revised manuscript.

read point-by-point responses

  1. Referee: Numerical framework and boundary conditions: the central claims (M ∼ J² to J = M² transition, bounded M,J in the simultaneous zero-ring-size limit, and approach to extremal Kerr) rest on the assertion that the numerical solutions satisfy the vacuum Ernst equations everywhere outside the ring and respect the imposed reflection symmetry for all rotation parameters. However, the manuscript provides no quantitative validation—maximum residual norms, grid-refinement convergence studies, or direct comparison of the slow-rotation numerics against the known perturbative expansion. Without these, discretization artifacts cannot be ruled out as the source of the reported relations.

    Authors: We acknowledge that the present manuscript does not contain explicit quantitative validation metrics. In the revision we will add: (i) the maximum residual norm of the Ernst equations evaluated on the computational domain for representative solutions across the rotation range; (ii) results of grid-refinement studies demonstrating that the extracted asymptotic charges M and J converge under successive doubling of the grid resolution; and (iii) a direct side-by-side comparison, in the slow-rotation regime, between the numerical values of M and J and the analytic perturbative expressions obtained in our earlier work. These additions will confirm that the reported M ∼ J² and J = M² relations are not discretization artifacts. The boundary conditions are imposed to enforce exact reflection symmetry across the throat (by construction of the computational domain) together with the required asymptotic flatness and regularity at the axis, consistent with the vacuum Ernst formulation. revision: yes

Circularity Check

0 steps flagged

Numerical integration of Ernst equations yields independent M-J relations

full rationale

The derivation reduces the problem to solving the vacuum Ernst equations under reflection symmetry across the throat, then obtains non-perturbative numerical solutions whose M ~ J^2 (slow) and J = M^2 (fast) behaviors, plus the controlled limit to extremal Kerr, are direct outputs of that integration. No step renames a fitted parameter as a prediction, imports a uniqueness theorem from the same authors, or reduces the central claims to self-citation by construction. The prior perturbative work is cited only for context and is not load-bearing for the non-perturbative results. The derivation chain is therefore self-contained against the Einstein equations.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard vacuum Einstein equations in stationary axisymmetric form (Ernst equations) plus the imposed reflection symmetry; the ring size is a free parameter that is varied to zero in one limit.

free parameters (1)
  • static ring size
    Parameter controlling the throat radius in the non-rotating limit; sent to zero to obtain the extremal-Kerr mimicking regime.
axioms (2)
  • standard math Vacuum Einstein equations reduce to the Ernst system for stationary axisymmetric metrics
    Standard reduction used throughout the paper.
  • domain assumption Reflection symmetry across the wormhole throat
    Imposed to select the desired family of solutions.

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