Force-free Electromagnetic Field Research Articles

Generalized split monopole magnetospheres: The effects of current sheets

We show that energy should be dissipated or extracted in the current sheet (CS) of a split magnetosphere deviating from the Michel split monopole, with the CS heating up or cooling down. But the electromagnetic energy remains unchanged everywhere. Based on the de-centered monopole solution generated by symmetry in flat spacetime, we construct two generalized split monopole configurations, in which the field lines intersect with the CS at arbitrary angles. One configuration resembles the outer geometry of the so-called ``new pulsar magnetosphere model", for which up to $47\%$ of the spin down energy is transferred to the Joule heating process in the CS. In the other configuration, we observe that negative energy is dissipated in the CS, which is usually observed in magnetospheres on rotating black holes. This means that energy is extracted simultaneously from the central star and the CS to power the output Poynting flux at infinity. We interpret the extraction of energy from the CS as that thermal energy of charged particles in the CS is transferred to the ordered kinetic energy of these particles drifting in the force-free (FF) electromagnetic fields. Hence, the CS is like an ``air conditioner" in the sky, which can heat up or cool down, depending on the configurations.

The non-null and force-free electromagnetic field

In this paper, we present a covariant formalism that connects solutions to force-free electrodynamics in the non-null case and foliations of spacetime. In doing so, we are also able to derive an expression of the general non-null current density vector. Just as in the null case, solutions in the non-null case can give rise to a dual solution, however, as is shown below, this can happen only when the solution describes a vacuum field. All theorems are illustrated with previously known solutions.

The essence of the Blandford–Znajek process

From a spacetime perspective, the dynamics of magnetic field lines of force-free electromagnetic fields can be rewritten into a quite similar form for the dynamics of strings, i.e., dynamics of "field sheets". Using this formalism, we explicitly show that the field sheets of stationary and axisymmetric force-free electromagnetic fields have identical intrinsic properties to the world sheets of rigidly rotating Nambu-Goto strings. Thus, we conclude that the Blandford-Znajek process is kinematically identical to an energy-extraction mechanism by the Nambu-Goto string with an effective magnetic tension.

Metric Independence of Vacuum and Force-Free Electromagnetic Fields.

Electromagnetic fields which solve the vacuum Maxwell equations in one spacetime are well known to also be solutions in all spacetimes with conformally related metrics. This provides a sense in which electromagnetism alone cannot be used to measure certain aspects of geometry. We show that there is actually much more which cannot be so measured; relatively little of a spacetime's geometry is in fact imprinted in any particular electromagnetic field. This is demonstrated by finding a much larger class of metric transformations-involving five free functions-which preserve Maxwell solutions both in vacuum, without local currents, and also for the force-free electrodynamics associated with a tenuous plasma. One consequence of this is that many of the exact force-free fields which have previously been found around Schwarzschild and Kerr black holes are also solutions in appropriately identified flat backgrounds. As a more direct application, we use our metric transformations to write down a large class of electromagnetic waves which remain unchanged by a large class of gravitational waves propagating "in the same direction."

Force-Free Electromagnetic Fields within Spinor Framework

The article deals with spinor representation of the force-free electrodynamics. The equations of the force-free electromagnetic field describe the physics of pulsars and black holes whose magnetospheres are filled with magnetically dominated relativistic plasma. The paper is a brief pedagogical introduction to the mathematics of the subject, based on 2-spinor calculi. The objective is to present the nonlinear theory of force-free fields in a compact and elegant form that the spinor framework provides. First, the algebraic classification of the Maxwell tensor is presented. Then, the reduced system of differential equations is obtained for two types of electromagnetic field and the basic features of the solutions are described. The null force-free field is connected with the shear-free geodesic null congruence in a space-time and is derived from a linear equation for a complex function. The magnetic force-free field is associated with the time-like 2-surface that represents the world-sheet of magnetic field line. The simplified system includes 4 linear differential equations for a real function. The article is educational in nature and there are no new solutions of force-free equations obtained.

The role of the ergosphere in the Blandford-Znajek process

The Blandford-Znajek process, one of the most promising model for powering the relativistic jets from black holes, was initially introduced as a mechanism in which the magnetic fields extract energy from a rotating black hole. We study the evolution of force-free electromagnetic fields on regular spacetimes with an ergosphere, which are generated by rapidly rotating stars. Our conclusive results confirm previous works, claiming that the Blandford-Znajek mechanism is not directly related to the horizon of the black hole. We also show that the radiated energy depends exponentially on the compactness of the star.

Relativistic Poynting Jets from Accretion Disks

A model is derived for relativistic Poynting jets from the inner region of a disk around a rotating black hole that is initially threaded by a dipole-like magnetic field. The model is derived from the special relativistic equation for a force-free electromagnetic field. The "head" of the Poynting jet is found to propagate outward with a velocity that may be relativistic. The Lorentz factor of the head is Γ = [B/(8π)2ρextc2]1/6 if this quantity is much larger than unity. For conditions pertinent to an active galactic nuclei, Γ ≈ 8(10/)1/3(B0/103 G)1/3(next/1 cm-3)-1/6, where B0 is the magnetic field strength close to the black hole, ρext = next is the mass density of the ambient medium into which the jet propagates, = r0/rg > 1, where rg is the gravitational radius of the black hole, and r0 is the radius of the O-point of the initial dipole field. This model offers an explanation for the observed Lorentz factors of ~10 of parsec-scale radio jets measured with very long baseline interferometry.

A new approach to static force-free electromagnetic fields

In the early 1950s, Lundquist showed the significance of electromagnetic fields with a vanishing Lorentz force density to magnetohydrostatic fluids. Today, these fields are known as force-free electromagnetic fields. Later that decade, Lüst and Schlüter demonstrated that cosmic magnetic fields are force-free and Chandrasekhar and Kendall then constructed a large class of force-free fields whose electric charge density field is also vanishing. In this article, we constructed force-free fields without assuming that the electric charge density vanishes, and in some cases we established a connection between force-free fields and nonlinear Schrödinger equations.

The force-free magnetosphere around an oblique rotator

The force-free magnetosphere around an oblique rotator

9.15. Theoretical method for treating force-free black hole magnetosphere in non-stationary and non-axisymmetric state

The force-free approximation is a very useful method in studies of the global electromagnetic structure of the magnetospheres around the relativistic astronomical bodies. However, its application has almost been restricted to the stationary and axisymmetric configuration. In this work, we remove this restriction and construct a general method for treating the force-free electromagnetic field.

Theory of force-free electromagnetic fields. I. General theory

A general method to deal with the (relativistic) force-free electromagnetic field is developed. We formulate the theory without assuming symmetry of the electromagnetic field configuration. Thus we can apply it to any object where the force-free approximation is justified, e.g., the pulsar magnetosphere, the black-hole magnetosphere, and the magnetosphere around the accretion disk. We describe the force-free electromagnetic field by a classical field theory. The basic variables are the Euler potentials extended to the relativistic degenerate electromagnetic field. The basic equation is given by a two-component nonlinear equation for two Euler potentials. The theory has a close connection with geometry. It is based on the concept of the flux surface. The flux surface is a geometric entity corresponding to the world sheet of the magnetic field line. We give both the covariant and the 3+1 expression of the basic equation. By the latter form, the causal development of the force-free electromagnetic field is discussed. It is shown that the theory describes the causal development of the force-free electromagnetic field self-consistently as far as $F\ensuremath{\cdot}F>0$. The basic equation contains arbitrariness. It does not determine the solution uniquely. Although this arbitrariness originates from the gauge freedom of the electromagnetic field, it differs from the arbitrariness in the ordinary gauge field theories. Namely, the dynamics of the Euler potentials itself does not contain arbitrariness. It appears from nonuniqueness in correspondence between the Euler potentials and the electromagnetic field. Further, we discuss the breakdown of the force-free approximation.

Theory of force-free electromagnetic fields. II. Configuration with symmetry

We develop a method for applying the general theory of the force-free electromagnetic field given in the preceding paper to configurations with some symmetry. The electromagnetic field configuration invariant under the action generated by one Killing vector and the configuration invariant under the action generated by two Killing vectors are studied. The Euler potentials have specific forms when the electromagnetic field has symmetry. General forms of the Euler potentials in these two cases are determined by the symmetry assumed. As an example of the configuration with one invariant direction, the time-dependent axisymmetric configuration is studied. The example of the configuration with two invariant directions is the stationary and axisymmetric configuration. The relation between the present theory and the traditional way to treat the stationary and axisymmetric configuration is clarified. Lastly, using Noether's identities, we clarify the relation between the geometrical properties of the conserved fluxes and the symmetry of the configuration.

Linear perturbations in force-free black hole magnetospheres -- I. General theory

A Lagrangian perturbation theory for the treatment of small disturbances of forcefree electromagnetic fields is developed in a covariant way. The geometrical nature of degenerate electromagnetic fields permits us to develop a Lagrangian treatment. A Lagrangian displacement similar to the one used in the perturbation theory of the ideal fluid is introduced. In the Lagrangian displacement, the basic equation is given as a set of two-component second-order partial differential equations. The action that yields our basic equation is also obtained. Using this action, conserved quantities and the energy-momentum tensor of disturbances are studied. Further, it is shown that there exists intrinsic arbitrariness in our theory. It corresponds to the so-called trivial displacement and arises from the gauge freedom of the electromagnetic fields. We apply the theory to the stationary and axisymmetric force-free black hole magnetosphere. The boundary condition at the event horizon is considered. It is shown that the boundary condition for pure-mode solutions takes the same form as that of the case of electromagnetic waves and gravitational waves scattered by the black hole in vacuum. Lastly, in a special limiting example, we discuss the consistency of our theory with the result from the black hole physics.

Comment on “On the possibility of force-free electromagnetic fields”

A Letter by Zaghloul claims to refute a proof due to Parker concerning the impossibility of configuring current-carrying conductors so as to experience no net magnetic stresses. We here point out that Parker's result stands, but that one must pay proper attention to the boundary conditions implicit in Parker's result.

Force-free electromagnetic pulses in a laboratory plasma.

A short, intense current pulse (I=150 A, \ensuremath{\Delta}t\ensuremath{\simeq}0.2 \ensuremath{\mu}s, N\ensuremath{\simeq}2\ifmmode\times\else\texttimes\fi{}${10}^{14}$ electrons) is drawn from an electrode immersed in a magnetized afterglow plasma (n\ensuremath{\simeq}2\ifmmode\times\else\texttimes\fi{}${10}^{11}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$, ${\mathit{kT}}_{\mathit{e}}$\ensuremath{\simeq}1.3 eV, ${\mathit{B}}_{0}$=10 G). The induced magnetic field B(r,t) assumes the shape of a helical double vortex which propagates along ${\mathbf{B}}_{0}$ through the uniform plasma as a whistler mode. The observations support a prediction of force-free (J\ifmmode\times\else\texttimes\fi{}B+neE=0) electromagnetic fields and solitary waves. Energy and helicity are approximately conserved.

Force-free electromagnetic waves

The time-dependent Force-Free Electromagnetic Field (FFEMF) is studied. In contrast to the case of Force-Free Magnetic Field (FFMF), it is shown that the FFEMF can occur in the form of waves. The FFEMF wave equation is solved in the case of one spatial dimension. Besides a periodical linear FFEMF wave solutions, the existence of solitary wave solutions is demonstrated. The possible application of FFEMF solutions to solar flares is discussed.

Cylindrical oscillations of force-free electromagnetic fields

The force-free electromagnetic field represents a natural generalization of the well-known force-free magnetic field model and allows the magnetic field to maintain electric charge separation. The basic equation for the cylindrical oscillations of the force-free electromagnetic field is obtained and solved for a linear case. The spectrum of possible resonances in a magnetized atmosphere is discussed.

Rotating charged dust as a source for cylindrically symmetric electrovacs

Exact solutions for stationary rotating charged dust with cylindrical symmetry, for which one of the spacelike Killing vectors is hypersurface orthogonal, are considered. The general solution for the case of rigid rotation and constant charge to mass ratio is given. It is proved that all rigidly rotating solutions can be joined smoothly to the exterior electrovac solutions involving third Painlevé transcendents. Also the solutions for differentially rotating charged dust, which have a force-free electromagnetic field can be matched to a particular class of the electrovac metrics.

Differentially rotating charged dust with a force-free electromagnetic field. II

For pt.I see ibid., vol.1, p.199 (1984). It is proved that a stationary axisymmetric (non-cylindrically symmetric) differentially rotating charged dust distribution has a non-vanishing Lorentz force or a non-constant ratio of the charge density to the mass density.

Differentially rotating charged dust with a force-free electromagnetic field

The problem of a cylindrically symmetric configuration of charged rotating dust is studied under the additional assumption that the Lorentz force on a typical particle of the fluid vanishes. Exact solutions with differential rotation are presented both in Newtonian physics and in general relativity. In the Newtonian case one particular solution exists which is regular, but which has a vanishing matter density on the axis. For the relativistic case the general solutions of the problem can be given in terms of hypergeometric functions; particular families of solutions are discussed for which the relevant physical quantities are regular on the axis.