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arxiv: 2605.22799 · v1 · pith:DF3T7FACnew · submitted 2026-05-21 · 🌀 gr-qc

Charged multi-sheet wormhole solutions

Yusuke Makita , Keisuke Izumi , Daisuke Yoshida This is my paper

Pith reviewed 2026-05-22 04:24 UTC · model grok-4.3

classification 🌀 gr-qc
keywords wormholescharged solutionsEinstein-Maxwellphantom scalarmulti-sheet geometriesHarrison transformationasymptotically flat regionsthroat regularity
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The pith

Regular charged wormholes with an even number of asymptotically flat regions exist beyond the usual bound on combined electric and magnetic charges relative to mass.

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

The paper constructs families of charged wormhole solutions in four-dimensional gravity coupled to Maxwell fields and a massless phantom scalar. These solutions connect an even number of asymptotically flat regions and are generated by applying the Harrison transformation to seed solutions. The throat radius is then fixed by requiring regularity, which extends the allowed parameter space to cases where the sum of the squares of the electric and magnetic charges exceeds the square of the mass. A new spheroidal coordinate system is introduced to describe one full asymptotic region together with its neighboring sheets in a compact way.

Core claim

In the Einstein-Maxwell-massless phantom scalar system the Harrison transformation produces charged wormhole metrics characterized by mass M, electric charge Q_e, magnetic charge Q_m, scalar charge P and sheet number 2n. Although the transformation is initially defined only for Q_e² + Q_m² < M², imposing regularity at the throat determines a positive radius that yields valid solutions for a wider range of charges, including values outside that inequality. The resulting geometries remain asymptotically flat on each of the 2n sheets.

What carries the argument

The Harrison transformation applied within the Einstein-Maxwell-massless phantom scalar system, with the throat radius fixed by the regularity condition at the minimal surface.

If this is right

  • The solutions remain regular and asymptotically flat on each of the 2n sheets for a broader set of charge and mass values.
  • The throat radius is uniquely determined once M, Q_e, Q_m, P and n are chosen.
  • The spheroidal coordinates cover one complete asymptotic region plus its adjacent sheets in a single chart.
  • The number of sheets can be increased while preserving the same charge and mass parameters.

Where Pith is reading between the lines

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

  • The extended parameter range may allow wormhole models with larger effective charges while still satisfying asymptotic flatness on multiple sheets.
  • The coordinate system could simplify calculations of geodesics or light propagation that cross between sheets.
  • Similar constructions might be attempted in other scalar-tensor theories to test how the regularity condition behaves.

Load-bearing premise

The Harrison transformation produces physically acceptable solutions in this theory and the regularity condition at the throat yields a positive radius even when the combined charges exceed the mass squared.

What would settle it

Explicit substitution of the constructed metric, scalar field and Maxwell potentials into the Einstein-Maxwell-phantom scalar field equations for a choice of parameters with Q_e² + Q_m² > M² and the regularity-fixed throat radius.

Figures

Figures reproduced from arXiv: 2605.22799 by Daisuke Yoshida, Keisuke Izumi, Yusuke Makita.

Figure 1
Figure 1. Figure 1: The contour curves of (r, θ) coordinates introduced in the individual ρz planes. The left panel describes the (+) sheet, while the right panel shows the (−) sheet. The thick curves denote the contours of r while the dotted denote those of θ. These coordinates approach to the usual spherical coordinates in the limit r → +∞. In our multi-sheet wormhole case, the contours of r coincide with those of the solit… view at source ↗
Figure 2
Figure 2. Figure 2: An example of covering by single spheroidal coordinates for [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
read the original abstract

We construct charged wormhole solutions with an even number of asymptotically flat regions in the four-dimensional Einstein-Maxwell-massless phantom scalar system via the Harrison transformation. The solutions are characterized by five parameters: the mass $M$, the electric charge $Q_\mathrm{e}$, the magnetic charge $Q_\mathrm{m}$, the scalar charge $P$ and the number of sheets $2n$. The regularity condition then determines the throat radius. Although the Harrison transformation directly generates the solutions only in the parameter region $Q_{\mathrm{e}}^2 + Q_{\mathrm{m}}^2 < M^2$, we show that regular solutions exist in a wider parameter region beyond this bound. In addition, we introduce a spheroidal coordinate system that covers one complete asymptotically flat region and its adjacent ones, and allows the solution to be expressed in a simple form.

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

2 major / 2 minor

Summary. The manuscript constructs charged wormhole solutions with an even number of asymptotically flat regions in the four-dimensional Einstein-Maxwell-massless phantom scalar system via the Harrison transformation applied to uncharged phantom wormhole seeds. The solutions are characterized by five parameters (M, Q_e, Q_m, P, and 2n), with the throat radius fixed by a regularity condition. The authors show that regular solutions exist in a wider parameter region beyond Q_e² + Q_m² < M² by analytic continuation of parameters, and they introduce a spheroidal coordinate system to cover multiple sheets without coordinate singularities while preserving asymptotic flatness in each region.

Significance. If the construction holds, this provides explicit examples of regular charged multi-sheet wormholes in a phantom scalar theory, extending the parameter space for such solutions beyond the standard bound. The use of the Harrison transformation ensures the field equations are satisfied by construction, and the spheroidal coordinates offer a practical description of the geometry. This could be useful for studies of wormhole traversability, energy conditions, and multi-asymptotic spacetimes.

major comments (2)
  1. [§3] §3 (solution generation via Harrison transformation): the extension of solutions beyond Q_e² + Q_m² < M² by analytic continuation of parameters is central to the claim of a wider region; please provide the explicit verification that the continued metric functions, scalar profile, and electromagnetic potentials continue to satisfy the Einstein-Maxwell equations and maintain regularity at the throat.
  2. [§4] §4 (regularity condition): the statement that the regularity condition determines a positive throat radius for parameters outside the bound is load-bearing; the manuscript should show the explicit algebraic step or equation that guarantees the radius remains positive without introducing singularities or violating asymptotic flatness.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'the regularity condition then determines the throat radius' is stated without identifying the condition; adding one sentence summarizing the condition would aid readability.
  2. [Introduction] Introduction: consider citing prior works on phantom scalar wormholes and the Harrison transformation in Einstein-Maxwell theory to better highlight the novelty of the multi-sheet charged extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below. We will revise the manuscript to incorporate additional explicit verifications as suggested.

read point-by-point responses

  1. Referee: §3 (solution generation via Harrison transformation): the extension of solutions beyond Q_e² + Q_m² < M² by analytic continuation of parameters is central to the claim of a wider region; please provide the explicit verification that the continued metric functions, scalar profile, and electromagnetic potentials continue to satisfy the Einstein-Maxwell equations and maintain regularity at the throat.

    Authors: The solutions are generated by applying the Harrison transformation to the uncharged phantom wormhole seeds, which ensures that the field equations are satisfied by construction within the original parameter domain. Because the field equations are analytic functions of the parameters, the analytic continuation to the region beyond Q_e² + Q_m² < M² preserves the validity of the equations. In the revised manuscript, we will provide an explicit check by substituting the continued forms of the metric, scalar field, and electromagnetic potentials into the Einstein-Maxwell equations and confirming that they are identically satisfied. We will also verify that the regularity at the throat, defined by the vanishing of the appropriate metric component derivative, is maintained without introducing coordinate or curvature singularities. revision: yes

  2. Referee: §4 (regularity condition): the statement that the regularity condition determines a positive throat radius for parameters outside the bound is load-bearing; the manuscript should show the explicit algebraic step or equation that guarantees the radius remains positive without introducing singularities or violating asymptotic flatness.

    Authors: We concur that an explicit algebraic derivation is necessary for clarity. The regularity condition is derived from setting the first derivative of the radial metric function to zero at the throat location and ensuring the second derivative is positive to guarantee a minimum. This leads to a quadratic or higher-order equation for the throat radius r_0 in terms of M, Q_e, Q_m, P, and n. We will include the step-by-step algebraic manipulation in the revised §4, demonstrating that the physical root for r_0 is positive in the extended parameter space, and that the asymptotic behavior in the spheroidal coordinates remains flat in each sheet without additional singularities. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs charged multi-sheet wormhole solutions by applying the Harrison transformation to an uncharged phantom wormhole seed in the Einstein-Maxwell-massless phantom scalar system. The regularity condition at the throat is imposed to fix the radius, and analytic continuation of parameters is used to extend solutions beyond the Q_e² + Q_m² < M² bound while preserving the field equations and asymptotic flatness. The spheroidal coordinate system is introduced to cover multiple sheets without singularities. No load-bearing step reduces by construction to its inputs, no self-citations are invoked for uniqueness or ansatzes, and the central claims follow from direct substitution into the transformed equations rather than tautological fitting or renaming. The derivation remains self-contained with independent content from the transformation and matching procedure.

Axiom & Free-Parameter Ledger

5 free parameters · 2 axioms · 0 invented entities

The construction rests on the Einstein field equations for the given matter content and on the applicability of the Harrison transformation; no new entities are postulated.

free parameters (5)
  • Mass M
    Input parameter that sets the scale of the solution.
  • Electric charge Q_e
    Input parameter characterizing the electromagnetic field.
  • Magnetic charge Q_m
    Input parameter characterizing the electromagnetic field.
  • Scalar charge P
    Input parameter for the phantom scalar field.
  • Number of sheets 2n
    Discrete parameter controlling the multi-sheet topology.
axioms (2)
  • standard math Einstein field equations hold for the Einstein-Maxwell-massless phantom scalar system
    The entire construction is performed within this gravitational theory.
  • domain assumption The Harrison transformation maps solutions of the system to new solutions
    Invoked to generate the charged multi-sheet geometries from seed solutions.

pith-pipeline@v0.9.0 · 5667 in / 1487 out tokens · 66355 ms · 2026-05-22T04:24:04.948787+00:00 · methodology

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