Melvin's magnetic universe — the static, whole-cylinder-symmetric solution of the Einstein-Maxwell equations describing an infinite bundle of parallel magnetic flux lines in gravito-magnetostatic equilibrium — was proven absolutely stable by Thorne (1965) against arbitrarily large radial perturbations confined to a finite region about the symmetry axis. The proof rests on the fact that the Melvin configuration absolutely minimizes the localizable cylindrical energy (C-energy) at fixed magnetic flux, and nonlinear numerical integrations confirmed damped, turbulent oscillation, with the excess energy radiated away as gravitational and electromagnetic waves before the fields settle back into the static configuration. Several pieces of the dynamical picture were left open and remain unsettled: (i) the finite-amplitude numerics imposed the restriction A2=A˙2=0A_2=\dot A_2=0, excluding the azimuthal magnetic field BφB_\varphi and the longitudinal electric field EζE_\zeta, and no later computation has removed it; (ii) stability against nonradial perturbations is established only at linearized order (Safko 1968) and against unconfined (global) perturbations not at all, with a non-perturbative black-hole pair-creation caveat (Gibbons-Herdeiro 2001); (iii) the toroidal magnetic geon, the finite-major-radius counterpart of which the Melvin universe is the infinite-major-radius limit, is conjectured to be major-radius unstable but this has never been demonstrated in full general relativity; (iv) the underlying principle of flux resistance to gravitational collapse — nonzero electric or magnetic flux through every two-surface in a collapsing region halts the collapse — remains an unproven conjecture with only qualitative support; and (v) the emitted radiation has never been characterized quantitatively (energy partition, decay law). The problem is to complete this stability characterization.

Background

Melvin's magnetic universe (Melvin 1964; Melvin 1965) is a static solution of the Einstein-Maxwell equations of general relativity exhibiting whole-cylinder symmetry: invariance under rotation about and translation along an axis of symmetry, and under reflections in planes containing that axis or perpendicular to it. Its only matter content is a magnetic field whose lines are everywhere parallel to the symmetry axis (a purely longitudinal magnetic field, accompanied by a purely azimuthal electric field only when perturbed), strongest on the axis and decaying outward so that the total magnetic flux per unit length is finite. Physically it is a bundle of magnetic flux lines held in gravito-magnetostatic equilibrium: the gravitational attraction between the field lines (field energy gravitates) is balanced by their Maxwell-Faraday repulsion and by magnetic tension. Wheeler had shown (as Thorne recounts in the 1965 stability paper, citing Wheeler's Les Houches lectures) that in the Newtonian limit the configuration maximizes the energy per unit length at fixed magnetic flux — the basis of the then-current Newtonian expectation that the configuration should be unstable, an expectation the relativistic analysis overturned. The Melvin universe is also the infinite-major-radius limit of a toroidal magnetic geon — a geon being Wheeler's term for a self-gravitating configuration of electromagnetic (and gravitational) field energy held together by its own attraction, and a toroidal magnetic geon being a finite donut-shaped bundle of magnetic flux. To analyze such cylindrical systems Thorne (Phys. Rev. 138, B251 (1965)) introduced C-energy, a localizable energy per unit length that exists for whole-cylinder-symmetric spacetimes, and proved that at fixed magnetic flux the Melvin universe absolutely minimizes the C-energy inside any radius. Thorne (Phys. Rev. 139, B244 (1965)) then proved the absolute stability of the Melvin universe against arbitrarily large radial perturbations confined to a finite region about the symmetry axis: no such perturbation can cause gravitational collapse to a singularity or electromagnetic explosion to infinite dispersion; instead the fields undergo damped, turbulent oscillation, radiating away all perturbation energy as gravitational and electromagnetic waves, and settle back into the static configuration. On the basis of this result Thorne proposed the principle of flux resistance to gravitational collapse, stated verbatim in the abstract of his C-energy paper, Phys. Rev. 138, B251 (1965): "In any configuration of electromagnetic fields collapsing toward a singularity, each electric and magnetic-field line is either entirely ejected from the collapsing region or entirely swallowed by it as collapse proceeds; there can be no flux threading a collapsed region." Equivalently: in a configuration of electromagnetic fields collapsing gravitationally toward a singularity, a nonzero total electric or magnetic flux across the two-surfaces threading the collapsing region will stop the collapse. Later literature approached these questions from several directions: linearized stability analysis of nonradial perturbations (Safko 1968); non-perturbative decay of the Melvin background via black-hole pair creation mediated by the Ernst instanton, a Euclidean solution describing the nucleation of a pair of oppositely magnetically charged black holes (Gibbons-Herdeiro 2001); the Raychaudhuri equation — the evolution equation for the expansion of a congruence of worldlines, central to the singularity theorems — whose magnetic-tension (magneto-curvature) terms can counteract gravitational focusing (Tsagas 2006; Tsagas-Mavrogiannis 2021; Bini-Mashhoon 2022); exact gravitational and electromagnetic waves propagating on the fixed Melvin background (Garfinkle-Melvin 1992; Ortaggio 2004); and the role of rotation in halting cylindrical collapse (Apostolatos-Thorne 1992). None of these closes the questions left open by the 1965 proof.

Problem Statement

The common objective is to complete the dynamical stability characterization of Melvin's magnetic universe beyond the class for which absolute stability was proven (radial perturbations, confined to a finite region about the symmetry axis, within the unperturbed field class), and to settle the broader questions that the 1965 proof motivated. The problem is to determine: (1) the fully general finite-amplitude radial evolution, removing from the nonlinear analysis the restriction A2=A˙2=0A_2=\dot A_2=0 on the initial data — where A2A_2 is the axial component of the electromagnetic vector potential (the component along the symmetry axis, which vanishes in the unperturbed Melvin configuration), so that the restriction excludes the azimuthal magnetic field BφB_\varphi and the longitudinal electric field EζE_\zeta — thereby confirming or refuting Thorne's expectation of no qualitative change; (2) stability against nonradial perturbations beyond linearized order (Safko 1968 covers only small nonradial perturbations) and against perturbations not confined to a finite region about the axis, with the non-perturbative pair-creation channel delineated as its own sector; (3) the stability of the toroidal magnetic geon — the finite-major-radius counterpart whose infinite-major-radius limit is the Melvin universe — in particular whether it is stable against minor-radius collapse and unstable against major-radius collapse, as Thorne conjectured; (4) the status of the principle of flux resistance to gravitational collapse: a theorem, a counterexample, or a precise delineation of its domain of validity; and (5) a quantitative characterization of the radiation emitted in the relaxation — total radiated energy, its partition between gravitational and electromagnetic channels, and the decay law of the turbulent oscillation. Scope caveat, recorded honestly: deliverables (1)-(2) concern the Melvin background itself, (3) its finite-torus counterpart, and (4)-(5) generalizations motivated by it. They form one coherent program — completing the stability picture that the 1965 proof began — rather than five independent problems, though the literature treats them separately. A solution determines, for (1)-(3), whether the stated stability or instability claim holds, with the proof or disproof at the rigor of the original radial proof; for (4), the status of the principle of flux resistance to gravitational collapse as a theorem, a counterexample, or a delineated domain of validity; and for (5), the emitted radiation as a functional of the initial data for an explicitly specified class of perturbations.

References

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