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Edited by: Vladislav Izmodenov, Space Research Institute, Russia

Reviewed by: Mikhail V. Medvedev, University of Kansas, USA; Sergey Moiseenko, Space Research institute, Russia

*Correspondence: Rudolf A. Treumann, ISSI Bern, Hallerstrasse 6, CH-3012 Bern, Switzerland e-mail:

This article was submitted to Space Physics, a section of the journal Frontiers in Physics.

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Magnetic fields in the universe are in general weak, of the order of μGauss only. However, in compact objects they assume extraordinarily large values. These are produced by gravitational collapse of massive magnetized objects. Clearly, fields in the massive progenitor are energetically limited by the available energy which can be fed into the generation of currents and magnetic fields. However, when collapsing down to small scales magnetic fields become superstrong exceeding any limits which can be reached in the laboratory. A brief review and discussion is given on the absolute limitation to the magnetic field strengths which can be obtained during such collapses.

The large-scale dynamics of the universe is governed by the general cosmic expansion and the gravitational field of the massive objects. Magnetic fields are believed not to play any major role in the former [

Magnetic fields are bound to electric current flow and thus, in contrast to electric fields whose sources are elementary charges and charge differences, be generated by processes which cause electric currents. Currents imply non-ambipolar transport of charges. The question of how strong magnetic fields can become is thus reduced to the question of how strong any currents can become. In classical electrodynamics this implies from Ampère's law for stationary magnetic fields that

if restricting to charge transport alone and assuming non-magnetic media of (for simplicity singly charged) ion and electron densities and bulk velocities _{i,e}_{i,e} respectively. Otherwise one would add a magnetization term

Assuming, without restriction, quasineutrality _{e}_{i}_{e}, a condition strictly holding in the ion frame of reference. Since velocities are limited by the velocity of light

suggesting that the magnetic field grows with _{cc} is in units of electrons per cm^{−3}, and _{km} is the length scale across a current filament in units of km. In the crust of a neutron star, for instance, we have _{km} ~ 1. If roughly all electrons in the crust would participate in current flow, we had _{cc} × ~ 10^{30}. Hence, the magnetic field strength could go up to ^{28} Gauss, a huge number compared with the maximum ^{15} − 10^{16} Gauss observed in magnetars.

This crude estimate needs to be commented on in order to avoid misunderstanding. Magnetic fields are believed to be generated preferentially by dynamo actions. Such actions are presumably not at work in white dwarfs, neutron stars, magnetars or any other compact objects. The fields are produced in their differentially rotating progenitors. Take the Sun as an example with dynamo action in the convection zone of thickness ^{☉} ~ 2 × 10^{5} km and average density ^{☉}_{cc} ~ 8 × 10^{23}. Using the total width of the convection zone grossly overestimates the current filament width. An absolute upper limit would be ^{☉}_{km} ≲ 2 × 10^{4}. Clearly velocities are also much less than ^{21} T. The comparably strong fields in neutron stars are subsequently produced in the rapid collapse of the magnetized heavy progenitor star not having had time within the time of collapsing to dissipate the magnetic energy which becomes compressed into the tiny neutron star volume. The compression factor being of the order of ~ 10^{12} yielding limit fields of ^{35} Gauss. The classical electrodynamic estimate clearly fails in providing an upper limit on magnetic field strength that would match the observational evidence.

Other no less severe discrepancies are obtained from putting the neutron star magnetic field energy equal to the total available rotational energy both in the progenitor or in the neutron star assuming equipartition of rotational and magnetic energy—clearly a barely justified assumption in both cases. Magnetic energy cannot become larger than the originally available dynamical energy of its cause of which it is just a fraction. It is presumably principally questionable whether magnetic fields could ever have been produced by any classical mechanism substantially stronger than observed in neutron stars (except for a brief ~10 s long post-collapse dynamo-amplification phase at the best yielding another factor of ~10–100 [

Quantum mechanics provides a way of obtaining a first limit on the magnetic field from solution of Schrödinger's equation, originally found by Landau [_{0} is quantized with flux element Φ_{0} = 2πħ/e, _{0} is the number of elementary fluxes carried by the field, and ^{2}, putting ν = 1 defines a smallest magnetic length

This length, which is the gyroradius of an electron in the lowest lying Landau energy level, can be interpreted as the radius of a magnetic field line in the magnetic field

from which, for a given shortest “critical” length _{B}_{c}_{c}_{c}_{c}_{0} = 2πħ/^{q}_{ns}^{9} T = 3 × 10^{13} Gauss. It is of considerable interest that approximately this field strength was indeed inferred from observation of the fundamental (ν = 1) electron cyclotron harmonic X-ray line detected from the HerX1 pulsar [

Use of the Compton wavelength relates the limiting field strength in neutron stars to quantum electrodynamics. It raises the question for a more precise theoretical determination of the quantum electrodynamic limiting field strength accounting for relativistic effects. It also raises the question whether reference to other fundamental length scales may provide other principal limits on magnetic fields if only such fields can be generated by some means, i.e., if electric currents of sufficient strengths could flow under different conditions as for instance in quantum chromodynamics.

Very formally, except as for inclusion of relativistic effects, Equation (4) provides a model equation for a limiting field in dependence on any given fundamental length scale _{c}_{c}

_{c}_{Pl}_{Pl}^{q}^{9} T, the critical field of magnetized neutron stars (pulsars) in agreement with observation of the strongest cyclotron lines. Horizontal lines indicate the relation between other length scales and critical magnetic fields under the assumption of validity of the Aharonov-Bohm scaling. Space magnetic fields correspond to scales of ~ 1 mm. Strongest detected magnetar fields correspond to the first order relativistic correction on the lowest Landau level energy _{LLL}^{qed}^{28} T deep in the (shaded) relativistic domain which have not been observed. It is interesting that this limit coincides approximately with the measured [^{45} T, according to simple Aharonov-Bohm scaling. The dashed black curve indicates a possible deviation of the Aharonov-Bohm scaling near the quantum electrodynamic limit.

The Compton limit to magnetic fields was known from straight energy considerations [cf. e.g., _{ns}

with ^{q}^{q}^{q}

From here it follows that the lowest Landau level energy doubles only at magnetic fields of the order of ^{28} T (~ 10^{32} Gauss), way above any neutron star or magnetar surface magnetic fields. Relativistic self-energy corrections causing magnetic field decay will thus come into play only at these energies which may be the ultimate limit on magnetic field strengths.

It is notable that this limit approximately coincides with the [

Unless magnetic monopoles ever existed and survived in the universe, magnetic fields must have been produced at any times via generation of electric currents. Fields generated in the early universe have subsequently been diluted to today's low large scale values as discussed elsewhere [_{e}

The application of the Aharonov-Bohm scaling in Figure _{e}^{−22} m, the current upper limit on the electron radius [

where _{c}_{0}, and _{0} ≳ _{e}

What concerns the generation of magnetic fields before collapse by the generally accepted dynamo or battery effects, magnetic field strengths are strictly limited by the available dynamical energies, which are far below any quantum electrodynamic limit. One may argue that, as long as the scale of the electron radius is not reached during collapse, the quantum electrodynamic scaling provides a reasonable absolute limitation on any possible magnetic field strength. Neutron stars and magnetars have scales excessively larger than the electron scale. Heavier objects by decreasing their scale could possess substantially stronger fields, but the permitted range is narrowed by the condition that such objects readily become black holes when collapsing which, by the famous no-hair theorem, do not host any magnetic fields. It is not known what would happen to the field by crossing the horizon for no information about the field would be left to the external observer. The no-hair theorem suggests that the field is simply sucked in into the hole and disappears together with the collapsing mass. Ordinary reasoning assuming maintenance of the frozen-in state then suggests that the field inside the horizon should further increase in the presumably continuing gravitational collapse.

The available strong fields which come closer to the quantum electrodynamic limits are found in neutron stars and magnetars. So far no strange star magnetic fields have been positively detected. It has even been shown [_{ns}^{q}

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.